Critical Phenomena

Home , Critical phenomena, Noble gas (data page)

UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

CRITICAL PHENOMENA

Proceedings of a Conference Held in Washington, D.C., April 1965

Edited by

M. S. Green and J. V. Sengers

Institute for Basic Standards National Bureau of Standards Washington, D. C. 20234

Sponsored by

National Bureau of Standards National Science Foundation Office of Naval Research

National Bureau of Standards Miscellaneous Publication 273

Issued December 1, 1966

For sale bv the Suuerintendent of Documents. I S. Government Printing Office Washington. D.C.. 20402 • Price S2.50 Library of Congress Catalog (lard Number: 66-60040 Conference on PHENOMENA IN THE NEIGHBORHOOD OF CRITICAL POINTS

Held at

The National Bureau of Standards

Washington, D.C.

April 5 to 8, 1965

Organ izing Com m ittee:

M. S. GREEN. Chairman National Bureau of Standards

G. B. Benedek Massachusetts Institute of Technology

M. E. Fisher University of London King s College

E. W. Montroll Institute for Defense Analysis and University of Maryland

C. J. Pings California Institute of Technology

iii

FOREWORD

One of the important responsibilities of the National Bureau of Standards is to help the Nation meet its needs for accurately measured, critically evaluated, and well-understood properties of materials. To aid in fulfilling this responsibility, the Bureau joined with the National Science Foundation and the Office of Naval Research in sponsoring a Conference on Phenomena in the Neighborhood of Critical Points, which was held at NBS in Washington. D.C.. April 5 to 8. 1965. The Conference was attended by approximately 200 scientists from this country and abroad, representing many laboratories and disciplines. Most of the papers presented at the Conference, as well as the subsequent discussions, are included in this volume. Primary responsibility for their technical content must rest, of course, with the individual authors and their organizations.

A. V. Astin, Director.

PREFACE

In that system for denning, determining, communicating, and recording numerical descriptors of physical systems which has recently been called the National Measurement System, the Na- tional Bureau of Standards has had from its earliest beginnings a concern with the part having to do with properties of substances. Indeed, accurately determined values of key data on properties of substances are an essential means for the dissemination of measurement capability. The experimental determination, the critical evaluation and the theoretical understanding of well-defined properties of well-characterized substances is one of the major modes through which the scientific enterprise contributes to the technological enterprise and to the well-being of the nation. To facilitate and to contribute to this process and to bring it into close coupling with the national standards is one of the primary responsibilities of \the Bureau and its component Institute for Basic Standards. The program of the Conference on Phenomena in the Neighborhood of Critical Points, whose proceedings are recorded in the present volume, serves excellently to illustrate some of the ways in which this process takes place. An important part of the program of the conference was concerned with measurement techniques. New techniques were introduced. Advantages and limitations of traditional techniques were discussed. In many instances it became apparent that new techniques were necessary not only to provide more accurate data in new experimental ranges but to answer fundamental theoretical questions. The Institute for Basic Standards has a deep and perennial interest in techniques for measuring properties of substances. The technological enterprise needs accurate data on the substances and materials from which the designs of our engineers are constructed, which constitute the ambience which they must take into account,! or which enter into the various stages of the productive process. In meeting this need the task of the scientific enterprise does not end with the complete measurement of a property. Competent scientists must critically review and evaluate various measurements which inevitably conflict and contradict each other. Many of the papers at the conference were just such critical reviews and evaluation of experimental data. The literature was searched for relevant data. Factors affecting the accuracy and validity of the experimental approach were investigated. Patterns, meanings, and conclusions were drawn insofar as the data warranted. These reviews represent an essential step in the process by which primary experimental data become the reliable and easily accessible reference data which the technological enterprise requires. Anyone who attempts even a rough estimate the dimensions of the need for data will realize that not even all the laboratories and data centers of the whole world are sufficient to provide an accurately measured and critically evaluated datum for every required property in every required condition. The scientific enterprise has, of course, another resource available to satisfy the data requirements of technology, the web of well-established theoretical relationships which relate properties in various substances and conditions among themselves and to the fundamental constants of nature. One of the purposes which has motivated the present conference was the expec- tation or hope that a common thread of explanation would be found among such diverse phenomena as liquid-vapor phase transitions, magnetism, superconductivity, order-disorder transitions. If these expectations are realized, a few theoretical concepts will provide knowledge of properties which is significant for a large number of very disparate fields of technology.

The task of providing adequate data on properties of substances is, of course, one which must be carried out by the scientists of the nation and indeed the world. The Bureau and Institute contribute to this task through services they provide to the scientific community as well as by the scientific work of its own laboratories. As I remarked in my address of welcome to the conference, the Bureau and Institute were happy and honored to serve as a place of congregation for the distinguished group of speakers and participants for the purpose of discussing a subject so germane to our function. We were happy also for the stimulus to our technical program which the conference provided through the activities of our staff as participants, speakers, and organizers. More than these, however, we were happy to have this important group of scientists in our midst so that we might learn from them about opportunities and needs to improve our services and they in turn might perhaps learn from us about those of our services and activities which might be of help to them.

R. D. HUNTOON, Director Institute for Basic Standards. INTRODUCTION

As their temperature is raised, the properties of a liquid and of the vapor with which it is in equilibrium become more and more similar until, at a particular temperature, the difference disappears. The state of the fluid at this temperature is the critical point. At room temperature iron has the power to acquire a magnetic moment which remains even after the magnetic field is removed. As its temperature is raised this remanent magnetism diminishes until, at a temperature of some several hundred of degrees, it suddenly disappears. This state of iron is called the Curie 4 point. Below the temperature of its critical point He may exist in an ordinary liquid form which is qualitatively the same as liquids existing at room temperature. At a temperature several degrees below its critical point the ordinary liquid suddenly acquires new properties, especially those relating to flow, which are so extraordinary that the state has been called superfluid. The state at which this transition occurs is called the Appoint. All these phenomena have in common that at a definite transition point a substance gains or loses all at once what another age would have called a virtue. In this they differ from first order phase transitions, melting, evaporation, sublimation, in which the physical change is not sudden, but takes place by a piecemeal transformation of small portions of the substance from one state to another. They differ also from first order phase transitions in that they seem to be the focus or source of anomalous behavior in a wide region of the surrounding thermodynamic phase plane. Critical points of liquid vapor transitions, consolute points in binary liquid mixtures, Curie and Neel points in magnetism, second order phase transition points in solids, A-points in liquid helium together with the anomalous phenomena which occur in their neighborhood were the subject of the conference whose proceedings are reported in this volume.

That critical points in this general sense have much in common with each other is attested by the fact that they have been treated from the beginning by very similar models and theories. These models and theories are based on the idea that phase transitions and critical points are brought about through the mutual cooperative interaction of many particles. For this reason these phenomena are often called cooperative phenomena. The essential similarity of these early theories, the Van der Waals theory of the liquid-vapor critical point, Curie-Weiss theory of ferromagnetism, the Bragg-Williams and Bethe theory of order-disorder phenomena is emphasized in the introductory historical lecture of Uhlenbeck. They give, as Uhlenbeck pointed out, an excellent qualitative picture of critical phenomena, but fail more and more seriously in their quantitative predictions as the critical point is approached. Their ultimate inadequacy is demonstrated by their disagreement with Onsager's exact solution of the two-dimensional Ising model of ferromagnetism. Several recent scientific developments, both experimental and theoretical, contributed to the feeling on the part of a number of scientists, and in particular, on the part of those who formed themselves into an ad hoc committee to organize it, that April 1965 was an appropriate moment for a conference on critical phenomena. Firstly, there was the theoretical prediction of nonclassical critical behavior for three dimensional lattice models of gases and magnets based on the numerical generation and summation of a large number of terms of exact low and high temperature series expansions. A second recent development, in this case experimental, was the discovery that the specific heat at constant volume of a simple fluid near the critical point is apparently singular and very similar in its behavior to the A-singularity in the specific heat of liquid helium. A third development was the confirmation through measurement of the internal magnetic field by nuclear magnetic resonance that the shape of the magnetization versus temperature curve of a ferro- or antiferromagnet is similar if not identical to the shape of the coexistence curve of the liquid-vapor phase transition and similar to the shape predicted by the series summation method. A fourth development was the critique of the Ornstein-Zernike' theory of critical opalescence and the experimental confirmation of deviations from this theory.

A common feature of these developments and of others not listed is that they point up a relationship between phenomena which are normally investigated by different institutional group- ings of scientists using different experimental techniques who perhaps normally do not meet each other at scientific conferences. It was the purpose of the conference to present these recent developments, to provide an opportunity to assess their significance both individually and as a total picture, to bring together for mutual enlightenment scientists working in different specialities and disciplines, to provide a forum for theoretical speculations. The program of the conference consisted primarily of papers reviewing the experimental situation for various important critical phenomena. In addition to these, each session contained a selection of shorter papers presenting new data, a novel experimental method, or new theoretical ideas. These proceedings contain the text of the papers substantially as presented except for

the paper of Passell which is represented only as an abstract. Also included is a "'dictionary" of corresponding parameters in various critical systems together with a tabulation of results on critical exponents prepared by Fisher. An attempt was made by the session chairmen to elicit as much discussion as time permitted

and this is included almost verbatim in these proceedings. Certain discussion remarks were in effect short papers. In some of these cases a manuscript was available and was included in the

text. In other bases it was possible to make references to a recent or shortly to appear publication.

It will be noted that the bulk of the papers at the conference were experimental. This does not represent a bias towards empiricism on the part of the conference organizers but rather their feeling that the most appropriate way to further an eventual theoretical understanding of critical phenomena was by a complete and critical presentation of the experimental situation. However in addition to the theoretical papers of Domb, Fisher, Buckingham, and Marshall which were pre sented in the main program, the last session of the conference was almost entirely devoted to short theoretical presentations. Not all of these are published here. This supplementary session als included the experimental paper of Lorentzen. Although not actually presented at the confer

ence, a review article by I. M. Firth, a participant, of certain techniques of low temperature calo

rimetry was felt to be relevant and is included in this volume.

The contribution of a conference such as this one is to be judged not only on the questions it

answers but also for the issues which it brings to clear statement. I close this introduction wit my own opinion of what some of these outstanding issues are.

It may perhaps be regarded as established that the thermodynamic properties of substances near critical points are singular functions in the mathematical sense. A primary issue then is what is the nature of these singularities? One of the ways of expressing these singularities is through critical exponents as defined in the paper by Fisher. The fact, coming out of the papers by Rowlinson, Domb, Benedek. and Fisher, that values of critical exponents predicted by the series summation method are in very good agreement with experiments both in fluids and in magnets is an important success of theory Nevertheless, several questions remain. The summation theory predicts different results for different physical models such as the Heisenberg and Ising models of ferromagnetism. Can we exhibit these differences experimentally? The paper of Wolf describes an experimental realization of an Ising ferromagnet. What is the significance of the quantum effects on critical exponents pointed out in the papers of Sherman and M. H. Edwards?

Another aspect of the nature of the critical singularity is the behavior of the specific heat. Is the logarithmic specific heat singularity, so convincingly demonstrated for the K-transition in liquid helium in the paper of Fairbank, a paradigm for the behavior ofspecific heat on all critical systems? The paper of Moldover and Little reports experiments on this question for the liquid vapor critical point of helium while Yamamoto reviews the situation for some solid-state second order transitions. Teaney discusses the specific heat of some ferromagnetic and antiferromagnetic transitions from this point of view, while Kierstead presents results on the singular behavior of the pressure coefficient near the \-point. A question which was not represented in the program but which was touched upon in the discussion is what is the relation of the superconducting transition point to the k-point and to the liquid vapor critical point? The paper of Chu and the earlier work of Mclntyre referred to in the discussion indicate deviations from the classical, Ornstein-Zernike theory of critical fluctuations within a few hundredths of a degree Kelvin from the critical consolute point of a liquid mixture. An obvious experimental problem is to confirm these deviations, to correlate the results of light and x-ray scattering and to make precise the nature of the critical singularity in these phenomena. The present status of results on elastic light and x-ray scattering was reviewed by Brumberger while Dietrich and Als-Nielsen presented their results obtained by elastic neutron scattering on /3-brass. A related question brought up in the discussion is why do deviations from the classical theory of critical opalescence occur only within hundredths of a degree of the critical point while deviations in thermodynamic quantities show nonclassical behavior much farther away? Phenomenological theory predicts that magnetic fluctuations become slower and slower near the Curie point, and therefore that critical magnetic scattering should be elastic, while the experiments of Passell and Jacrot show a definite and unexpected amount of inelastic scattering.

Although Marshall's paper suggests a possible and even likely solution, the question still remains why is critical magnetic neutron scattering inelastic? The review of transport phenomena by Sengers indicates that the thermal conductivity of simple fluids exhibits a large anomaly near the critical point while the viscosity exhibits no such anomaly or a very minor one. The opposite seems to be the case for binary liquid mixtures. The question arises, which transport coefficients are anomalous near critical points and why? Several new techniques or techniques newly applied to critical phenomena were presented at the conference. One of these was the inelastic scattering of laser light, which was represented in the papers of Alpert and of Ford and Benedek. Another was nuclear magnetic resonance whose application to fluids was given in the paper by Bloom and in a discussion remark of Trappeniers. The application of this method to antiferromagnetism was presented in the talk of Heller.

Not a new method but one which has not perhaps been fully exploited is the diagnosis, using a light probe, of the density gradients produced by gravity in a fluid near its critical point. The papers of Schmidt and L,orentzen present results obtained by this method. A somewhat related method was brought up by Webb in the discussion in which the meniscus, which becomes more and more diffuse as the critical point is approached, is probed by its reflectivity to light. Results on ultrasonic propagation in fluids near critical points were reviewed by Sette while the application of this method to a solid state second order transition is given in the paper of Garland and Renard and to helium near its critical point by Chase and Williamson. Perhaps the foregoing questions can be subsumed under two fundamental questions: the first question is are all the phenomena considered in the conference under the rubric of critical phenomena truly analogous? This question can be perhaps answered quickly and affirmatively when, say, the liquid-vapor transition is compared to the phase separation of a binary liquid mixture or even to order-disorder phenomena in solids. The analogies become less clear when these are compared to magnetic phenomena, superfluidity, and superconductivity. When available in any given case, the best experimental and theoretical evidence seems to show that the singular behavior of analogous properties in different systems is very similar. Of course, the series summation method indicates small but definite differences between different systems. Without prejudging the significance of these differences, the similarity of behavior already suggests a common explanation. A second question is then, what is the common reason for the singular behavior of analogous properties ?

Melville S. Green. ACKNOWLEDGEMENTS

The organizing committee wishes to express appreciation for the generous support of the National Science Foundation and the Office of Naval Research which enabled the conference to benefit from the participation of many scientists from western Europe, Japan, and Australia whose contribution otherwise would have been sorely missed. The committee also wishes to express its thanks to the many members of the NBS staff who contributed to making the conference arrangements pleasant and convenient. It wishes especially to thank Mr. Robert Cooke of the

Office of Technical Information and Publications and Mrs. May J. Lum and Mrs. Hattie N. Brown of the Heat Division. It also wishes to thank Mrs. Vivian R. Green for her help in organizing the social program of the conference. CONTENTS

Page FOREWORD V PREFACE VII INTRODUCTION , IX ACKNOWLEDGEMENTS XII

EQUILIBRIUM CRITICAL PHENOMENA IN FLUIDS

Chairman: A. M. J. F. Michels G. E. Uhlenbeck — The Classical Theories of the Critical Phenomena 3 R. H. Sherman — The Coexistence Curve of He 3 7

J. S. Rowlinson — Critical States of Simple Fluids and Fluid Mixtures: a Review of the Experimental Position 9

E. H. W. Schmidt — Optical Measurements of the Density Gradients Produced by Gravity in C0 2 , N 2 0, and CCIF3 Near the Critical Point 13 M. E. Fisher — Notes. Definitions, and Formulas for Critical Point Singularities 21 Discussion 25

CRITICAL PHENOMENA IN FERRO- AND ANTIFERROMAGNETS Chairman: H. B. Callen C. Domb — Critical Properties of Lattice Models 29 G. B. Benedek — Equilibrium Properties of Ferromagnets and Antiferromagnets in the Vicinity of the Critical Point 42 W. P. Wolf—Experimental Studies of Magnetic Ising Systems Near the Critical Point 49 D. T. Teaney — Specific Heats of Ferro- and Antiferromagnets in the Critical Region 50 P. Heller — Nuclear Resonance Studies of Magnetic Critical Fluctuations in MnF> 58 Discussion 64

LOGARITHMIC SINGULARITIES Chairman: C. N. Yang W. M. Fairbank — The Lambda Transition in Liquid Helium 71 M. R. Moldover and W. A. Little-The Specific Heat of He 3 and He 4 in the Neighborhood of Their Critical Points 79 M. H. Edwards — The Coexistence Curve of He 4 82 T. Yamamoto. O. Tanimoto, Y. Yasuda, and K. Okada — Anomalous Specific Heats Associated with Phase Transitions of the Second Kind 86 H. A. Kierstead — The Logarithmic Anomaly in the Pressure Coefficient of Helium Close to the Lambda Line 92

M. J. Buckingham — The Nature of the Cooperative Transition 95 Discussion 101

SCATTERING: ELASTIC Chairman: P. Debye P. Debye — Introductory Remarks 107 M. E. Fisher — Theory of Critical Fluctuations and Singularities 108 H. Brumberger— Scattering of Light and X Rays from Critically Opalescent Systems 116 B. Chu — Experiments on the Critical Opalescence of Binary Liquid Mixtures: Elastic Scattering 123 Discussion 129

SCATTERING: INELASTIC Chairman: K. S. Singwi W. Marshall — Critical Scattering of Neutrons by Ferromagnets 135 L. Passell — Critical Magnetic Scattering of Neutrons in Iron ^2 0. W. Dietrich and J. Als-Nielsen — Critical Neutron Scattering from Beta-Brass 144 N. C. Ford, Jr., and G. B. Benedek — The Spectrum of Light Inelastically Scattered by a Fluid Near Its Critical Point... 150 S. S. Alpert — Time-Dependent Concentration Fluctuations Near the Critical Temperature 157 Discussion IgO

xi CONTENTS— Continued

TRANSPORT AND RELAXATION PHENOMENA Chairman: R. W. Zwanzig

J. V. Sengers — Behavior of Viscosity and Thermal Conductivity of Fluids Near the Critical Point 165 M. Bloom — Nuclear Magnetic Resonance Measurements Near the Critical Point of Ethane 178 D. Sette — Ultrasonic Investigation of Fluid System in the Neighborhood of Critical Points 183 C. E. Chase and R. C. Williamson — Ultrasonic Investigation of Helium Near Its Critical Point 197 C. W. Garland and R. Renard — Ultrasonic Investigation of the Order-Disorder Transition in Ammonium Chloride 202 Discussion 209

SUPPLEMENTARY SESSION. H. L. Lorentzen and B. B. Hansen — Effect of Gravity on the Equilibrium Mass Distribution of Two-Component Liquid Systems. The Observation of Reversed Diffusion Flow 213 S. Katsura and B. Tsujiyama — Ferro- and Antiferromagnetism of Dilute Ising Model 219 S. F. Edwards — The Statistical Mechanics of a Single Polymer Chain 225

I. M. Firth — Some Comments on Techniques of Modern Low-Temperature Calorimetry 233 LIST OF PARTICIPANTS 239

xiv EQUILIBRIUM CRITICAL PHENOMENA IN FLUIDS

Chairman: A. M. J. F. Michels

The Classical Theories of the Critical Phenomena

G. E. Uhlenbeck

The Rockefeller University, New York, N.Y.

1. When the organizers of this conference on found already in Andrews' paper critical phenomena asked me to start the dis- plus the suggestion that this was a cussions with a review of the classical theories general phenomenon. in order to provide the proper historical perspective, 1873 Dissertation of J. D. van der Waals

I had hoped that I would find the time to do this in on the continuity of the gas and a responsible way. However I soon found out liquid state. You all know that the that is not easy! The literature is large and very miraculous van der Waals equation: few (if any!) good historical studies have been made about the development of thermodynamics and kT statistical mechanics in the second half of the last v — b century. I recommend it to the historian of science! There are many themes and questions which seem together with the Maxwell equal area to me very worth while to study as for instance the rule (added by Maxwell in 1874 in a penetration of the ideas of Gibbs and his methods long review of the work of van der (especially his graphical methods) into the Dutch Waals) explains the vapor-liquid school of thermodynamics of van der Waals, Bakhuis equilibrium and the existence of the Roozeboom. Kamerlingh Onnes. and others; the critical point: relation of the Strassbourg school of Weiss on fer-

, 1 a . „ 8 a romagnetism with the ideas of van der Waals: the 0 =-- vc = 3b; pc =--; kTc early controversies about the critical region versus b the critical point which is now again so much in the center of attention: and so on! Of course this confirmed the idea

I hope that you will take therefore the short that the critical point was a general historical remarks I will present with a grain of phenomena and that there were no salt, and that you will consider my talk more as a "permanent" gases. Made explicit the kind of keynote address! I will try to mention some by of the names and some of the notions which you will 1880 law of corresponding states (van der hear over and over again, and I will try to explain Waals) why I think the recent revival of the study of the critical phenomena is of such deep interest for the student of statistical mechanics. ^ = tt= F(a>, m w = - # = ^ Pc vc T,

2. Let me begin with some names and dates. 1869 Discovery of the critical tempera- with F a universal function. The ture of CO2 by Andrews. The gen- role which this law played in the

eral form of the set of isotherms for liquefaction 1 of gases and especially the liquid-vapor equilibrium can be of Helium (Kamerlingh Onnes, 1908)

209-493 O-66— was spectacular and I hope well a. At the critical point the specific heat makes u known. finite jump and then decreases again. For a van der The discovery of the magnetic transi- Waals gas at the critical density, one gets: tion point came much later. So far

as I know, it was only in: 0 for T > Tc 1890 Hopkinson found that iron above a Ac, 2STr -T certain temperature ceased to be A 1 25 Tc ferromagnetic. Confirmed and thoroughly investigated by Pierre Curie For a Weiss ferromagnet: in a classical paper of 1895, and therefore now always called the 0 for T>Tr - Curie temperature. Acp = 13 T(.-T 1907 P. Weiss introduced the notion of the 5 Tc inner field and explained the exist-

ence of the Curie temperature Tc For a Bragg-Williams binary alloy: and of the spontaneous magnetiza- 0 for T > Tr tion for T < Tc . The third critical Ac 3 8 Tc - T transition phenomena is that of the A 2 order-disorder transformation in bi- 5 Tc nary alloys or better substitutional solid solutions. Suspected in 1919 b. At the critical point the coexistence curve is by Tammann, proved in 1925 by the parabolic. For a van der Waals gas (with reduced u x-ray work of Johanson and Linde, quantities): 0)g — m = 4(1 — d) -. For a Weiss the theoretical explanation given in: ferromagnet: 1934 by Bragg and Williams, was clearly patterned after the Weiss theory.

One could still mention the discovery of the critical mixing tempera- [also with reduced quantities; W= constant of ture of two fluid phases for a binary = mixture. Found by Alexejew around Weiss inner field, M = magnetization; Tc 3k 1800. Explanation by van der Waals and a series of coworkers = Curie temperature. Putting H*=j-j, H. then

1890-1905 (Kuenen, Kohnstamm, van Laar, the ^magnetic equation of state becomes: I* — —L(H* + I*) where L(x) is the Langevin Korteweg) 1890-1905, by generaliz- u ing the van der Waals equation for Function]. mixtures and applying for the thermodynamic discussion the graphical c. At the critical point the compressibility or the

methods of Gibbs. susceptibility become infinite as \j\T— Tc \. For ai 3. Leaving for a moment this sort of schoolbook van der Waals gas: history let me emphasize the great similarity of the dir\ three classical theories of van der Waals, Weiss, = -6(#-l) r'»aJ„ and Bragg-Williams although the derivations are =i, r>rc often quite different. It is now clear and generally accepted that they are all based on the assumption 12(1 - &). coex, T Tr T< Tc

4 For a Bragg-Williams alloy, properties b and c can centered on the so-called rigorous theories. I be denned but tbey have less direct observational think one is now coming back from these theories meaning. because one has become dissillusioned about the 4. At this point one may well already raise the series expansions since no general point of view question, why the classical theories represent so emerges from them. well, at least qualitatively, the observed critical 5. The first and unfortunately till now phenomena. Of course, as long as little was known 1944 practically the only step forward beyond the about the intermolecular forces, one could think classical theories was made in the famous that the attractive forces had a very long range. paper by Onsager on the two-dimensional

In fact van der Waals never doubted that, and prob- Ising model. I will not say much about it, ably Weiss also had always the long range dipole- because you will hear much more about it dipole forces in mind. However the later later. It has dominated the subject for the development showed more and more clearly that last 20 years. Let me remind you only that the cooperative forces were not of a very long the Ising model (proposed by Ising in 1925) range, that each atom was influenced by at best a can be considered as a simplified version of few shells of neighboring atoms. Of course, it the ferromagnetic, and of the order-disorder took a long time before this was realized. It re- and even of the vapor-liquid problem (through quired the elucidation, of the nature of the van der the lattice gas). Each atom on a lattice site Waals attractive forces by London and of the Weiss can be in two states (spin up or down) and inner field by Heisenberg which came in the late the interaction is only between neighboring twenties as a byproduct of the quantum mechanical pairs and is ±J depending on whether the revolution. spins are parallel or antiparallel. It is there- Partially because of this fact, the classical the- fore a model with short range forces. The ories became more and more discredited in the most striking result of Onsager was that there thirties. This was also because it turned out that is a critical point and that the specific heat the classical theories although qualitatively cor- is logarithmically infinite there in contrast to rect could not be fixed up so as to give a precise the classical result. quantitative description of the phenomena. For 6. So far I have only talked about the criti- the van der Waals theory this became accepted cal point and its properties for the equation of already around 1900, and as a result Kamerlingh- state. A deeper question is the question Onnes started then to represent the experimental what happens with the correlation function data by series expansions in the density, so- the near a critical point. That the density fluc- called virial expansion, for which Ursell and Mayer tuations will get large near the critical point then gave the theoretical analysis in the thirties. was pointed out by Smoluchowski, but that in For the Weiss and the Bragg-Williams theory the addition the correlation of the density fluc- attempts for improvement lasted a bit longer, but tuation at different points will become of also here one soon began to turn more and more to very long range and that this is the cause of series expansions. The reason is clear. The the critical opalescence was first clearly terms in the expansions can be calculated exactly 1914 stated by Ornstein-Zernike in 1914. and then (at least in principle) and related to the inter- worked out in the Groningen dissertation of molecular forces. And since one began to learn 1916 Zernike of 1916. more and more about these forces, this was of Angular distribution of the scattered radi- course of the greatest interest. However, one lost ation is related to the Fourier transform of the general point of view which the classical the- the correlation function: ories provided, and perhaps because of this one also lost interest in these theories. They were 1 Kk - — g(k, v, T) dre n>{r: v.T) considered to give "convenient interpolation formula" or "semiphenomenological descriptions."

2tt . e Remarkable for instance is the fact that in the great , A= sm 2" international congress in 1938 to celebrate van der T Waals' 100th birthday, the van der Waals equation was mentioned only once, and all attention was Ornstein and Zernike give specifically for

5 > c» T Tc v = vc :

g(k; iv. T)=— j

with

cfrv-Vattr(r)

* 3

—77- versus k 2 (0-Z plot) should be a straight

line whose intercept with the ordinate axis — with B+ = B~ and C~-C+= 10k. must go to zero if T > Tc as (T— Tc ). There are many ways of justifying the O-Z This surprising result by the Russians, is also found theory. I think though that always one again for He" — He 1 transition and I believe also for has to assume that the forces are of long some ferromagnetic transitions. You will hear range, so that it can be considered a natural more about it tomorrow. generalization of the van der Waals theory. If there is such an universal, but nonclassical The corresponding theory for neutron behavior, then there must be an universal explana- scattering near the Curie point is so recent tion which means that it should be largely inde- that it does not belong in a historical survey! pendent of the nature of the forces. The only 7. Outlook. What are the problems, and what corner where this can come from is I think the fact can one hope from the present revival of interest? that the forces are not long range. The Onsager As always in statistical mechanics one must dis- solution gives I think a strong hint. It may well be tinguish the equilibrium from the nonequilibrium so that away from the critical points the classical properties. theories give a good enough description but that A. I think it is wonderful that so much new ex- they fail close to the critical point where the sub- perimental work is being done. For a long time stance remembers so to say Onsager. I think that experiment was too much dominated by the theory, to show something like this is the central theoretical and except in a few places (like in the van der Waals problem. One can call it the reconciliation of experimental work on laboratorium in Amsterdam) Onsager with van der Waals. critical point nearly the properties of matter near the With regard to the correlation function and the I glad that it is different now. What ceased. am Ornstein-Zernike theory I hope that the new ex- I that it will turn out can one hope to find? hope perimental work will show that there is a critical that again there is a deep analogy between the region, where already deviations from the classical various critical points, that there are similarities van der Waals-Ornstein-Zernike theories begin to a lot like in the classical theories. You will hear show up. I think there are indications (as the about the new experimental work, so I will only specific heat anomaly) that such a region exists. mention a few points which seem to me very sug- Of course there is also a singularity or a critical gestive. point.

1. Near the critical point the coexistence curve B. Finally a remark about nonequilibrium phe- seems much better represented by the empirical nomena. There are very surprising experimental 7 results found about various transport coefficients, 1/3 Guggenheim law: &>,,— coj = -(l — 0) . Also the which clearly are worth pursuing. Till now there magnetization curve near Tc seems to show such a is no theory, not even a van der Waals like theory. cubic behavior. The development of the proper transport theory with

2. For Argon and O2, and now also for He, it long range forces, van der Waals-like but probably impossi- seems that c r does not show a jump but becomes with one more constant, seems to me not logarithmically infinite. For T, ble and very worth while.

6 p

The Coexistence Curve of He 3 *

Robert H. Sherman

Los Alamos Scientific Laboratory, Los Alamos, N. Mex.

In this report we wish to present the results of some recent P-V-T measurements on He3 in the critical region. A significant departure was found in the shape of the coexistence curve from the behavior expected for classical systems such as Xe and CO2. The P-V-T surface of liquid He 3 was investi-

1 gated by Sherman and Edeskuty [1] between 1.0 and 3.3 °K and for pressures from the vapor pressure to solidification. The measurements reported by

3 2 these authors lacked detail in the region from 3.0 I0" I0" 10"' Tc-T, °K to 3.324 °K, the critical point, [2] and for pressures less than ~ 3 atm. In addition, there no were Figure 1. A plot — versus T along the co- of p x p% c — T 3 existence curve He . measurements reported on the fluid for T > Tc . for Recently 23 isochores of He 3 were measured with densities ranging from 0.015 to 0.066 g/cm 3 using the apparatus of reference with only slight modi- [1] deviates from the "classical" value of ~ V3, and fication. Along each isochore data were taken with (3^ 0.48 + 0.02 as T^ Tc . This slope appears to temperature decreasing from 4 °K, a final observa- 7'= *be constant up to Tc — 0.07°. tion being made along the liquid-vapor equilibrium

curve (vapor pressure). Temperatures were re- 3 Table 1. Densities of He along the coexistence curve. 4 duced to the 1958 He Scale of Temperatures [3]. Pressures were measured with a fused quartz bourdon tube gage having an accuracy of 0.05 mm T Pq Pi Hg. While the cell has a vertical dimension of ~4.6 cm, measurements were not made very close to the critical point and as a result, density gra- °K gm cm~ 3 gm cm~ 3 dients within the cell were not of any significance. 2.4 0.00801 0.07404 P-V-T data taken by the isochoric method have 2.5 .00915 .07288 the advantage of yielding P-T relations which are 2.6 .01044 .07159 almost linear. Thus each isochore was extrapolated 2.7 .01192 .07015 to the vapor pressure to provide a p-T point on [4] 2.8 .01371 .06852 the coexistence curve. Values of pi and t] were p 2.9 .01573 .06668 read from a smooth p-T graph and — p, (/ was 3.0 .01808 .06458 plotted as a function of T — T using logarithmic ( 3.05 .01951 .06338 coordinates as shown in figure 1. It is easily seen 3.1 .02113 .06200 that the critical point exponent /3 [5, 6] in the relationship 3.15 .02317 .06033 3.2 .02557 .05815 p,-p^ (Tc -T)i 3.25 .02867 .05498 3.3 .03373 .04970 3.324 .04178 .04178 Work performed under the auspices of the U.S. Atomic Energy Commission.

1 Figures in brackets indicate the literature references at the end of each paper.

7 3 Table 2. Properties of He at the critical point calculated using virial coefficients [8]. In table 1 values are given for the densities of He :i liquid and vapor along the coexistence curve from the present series of experiments and in table 2 some of the T = 3.324 ±0.0018 °K a c properties of He3 at the critical point are summarized.

a p c = 873.0 ±1,5 mm References

3 g/cm- [1] R. H. Sherman and F. J. Edeskuty, Annals of Physics 9, 522 p c = 0.0418 ±0.001 (1960). An error of +0.30 percent was made in the cell volume calibration. As a result all volumes and entropy a Reference 2. changes reported should be reduced in magnitude by 0.30 percent. Expansion and compressibility coefficients are unaffected.

[2] S. G. Sydoriak and R. H. Sherman. J. Res. NBS 68A (Phys. and Chem.) No. 6, 547 (1964). Kerr [7] has reported results of P-V-T measure- [3] F. G. Brickwedde. H. Van Dijk. M. Durieux. J. R. Clement ments for He'5 along the coexistence curve. Above and J. K. Logan, NBS Mono. 10. (U.S. Govt. Printing Office. 1960). increasing 3.0 °K, however, his results show de- [41 R. H. Sherman, S. G. Sydoriak. and T. R. Roberts. J. Res. viations from the present measurements probably NBS 68A (Phys. and Chem.) No. 6, 579 (1964). [5] For the critical point exponents the notation of Fisher [6] is due to the increasingly larger obnoxious volume used. corrections which must be made as the critical [6] M. E. Fisher, J. Math. Phys. 5, 944 (1964). [7] E. C. Kerr, Phys. Rev. 96, 551 (1954). is 2.9 °K, Kerr's gas point approached. Below [8] T. R. Roberts. R. H. Sherman. S. G. Sydoriak. J. Res. NBS densities are in excellent agreement with those 68A (Phys. and Chem.) No. 6, 567 (1964).

8 Critical States of Simple Fluids and Fluid Mixtures; a Review of the Experimental Position

J. S. Rowlinson

Imperial College, London, Great Britain

At a critical point the intensive properties of two An elementary and ultimately erroneous analysis coexisting phases become identical. This review of the behavior of a system of one component may is of our experimental knowledge of the a summary be based on the assumption that the Helmholtz rate of the approach to identity as T— Tc approaches free-energy is an analytic function of V and T at zero. Consider some property of the fluid X(T, V) and define an index x by and near the critical point [1]. Let the first non- vanishing derivative of A with respect to volume be of the order 2n. Thermodynamic stability requires ± X = \im[\nX(T,V)l\n±(T-Tc )]. (1) that n be integral, and van der Waals's equation

and all similar equations, require that n — 2. This assumption leads to the following values of the The limit is usually taken along the path V— V c , indices above + and, in general, \ an d X~ wih be different. The following indices + are discussed below: a = c*7 = 0 oq = — (2 — n)l(n— 1) (2)

Property Index Conditions

+ — . Cv a ct , limit along the critical isochore V= Vc af, limit in the one-phase fluid along the outside of the coexisting curve. cii, limit along the critical isochore in the two ^hase fluid. + (v-vx The subscript

r=Tc . (d*pldT% -e The derivative is the curvature of the vapor pressure line and so 6+ does not exist.

9 )8 = l/2(n-l) (3) above and below the critical, but no reliable value of ar could be obtained from them. + Fisher concludes that the singularity is a little y = yr = l (4) [9] sharper than logarithmic, both above and, probably, below. 8 = 2/i-l (5) Hence

a + -0.1-0.2 0 = 0. (6) a 2 ~ 0.0-0.1

Oil ? Thermodynamic stability requires that certain inequalities hold between the indices. Thus Rush- large fi. — Here there is substantial agreement of a brooke [2] showed that for a magnetic critical number of measurements. The most useful of these point (and Fisher [3] extended the proof to the fluid appear to be those of Michels, Blaisse, and Michels of one component) [11] (C0 2 ), those analyzed by Guggenheim [12]

(inert gases), Cook [13] (N 2 0), Weinberger and (7) Schneider [14] (Xe) as analyzed by Fisher [3], Lorentzen [15] (CO2), and Edwards and Woodbury Griffiths [4] has obtained the inequalities [16] (He) as analyzed by Miller [17]. The conclusions are, as follows. cx.7 + /3( 1 + 5)^2 Gl (8)

MBM (3 = 0.36 a; + G2. (9) f3 G, C =0.33 WS(F) =0.345 ±0.015 Widom [5] has shown that R becomes an equation if L =0.33 ±0.02 the conditions attached to a are that it represents EW(M) =0.41 the difference (CV)* — (C r)i across the orthobaric boundary. He has also made two conjectures Here, Lorentzen's value is that obtained with a [6, 7] for which there are no rigorous proofs; namely freshly filled cell and includes points to within 10" 3 °C of Tc . All figures except the last suggest + y = 2(1-/3) Wl (10) that is close to V3 but probably above it. y. — An analysis of the results of Michels, Blaisse, and Michels [11] (CO2) and of Habgood and Schnei- yr = 0(8-1) W2 (11) der [18] (Xe) is shown in figure 1.

It is seen that W2, if correct, implies that R and Gl y+ = 1.2 are the same inequality. y~ is uncertain but probably exceeds unity. The classical equations (2) to (6) satisfy R, Gl,

G2, and W2 for all values of n. cr. — Widom and Rice [19] have analyzed the re-

The best experimental values do not fit to (2) (6) sults of Michels, Blaisse, and Michels [11] (C0 2 ), with integral any value of n, and certainly not with Habgood and Schneider [18] (Xe) and Johnston, ra = 2. One concludes that A is nonanalytic. Keller, and Friedman [20] (H 2 ). The results are The most probable values of the indices will now be reviewed in turn. Xe, H 2 4.2 a.— The best measurements of Cv appear to be C0 2 = 4.0. those of Voronel' and his colleagues for argon and oxygen [8]. Little experimental detail is given but However more recent discussion [21, 22] of the the results are more consistent and extend to lower optical determinations of density gradients near

values of (T— Tc ) than any previous sets. However, the critical point suggest that measurements of they are restricted to the critical isochore. Fisher 8 from P-V-T may not be as reliable as the analysis [9] has analyzed them in detail, and his analysis need of Widom and Rice suggests. Because of the high not be repeated. The results of Michels and value of 8 the determination from P-V-T rests essen- Strijland [10] for carbon dioxide are at densities tially on results at densities well-removed from the

10 Hence,

Gl, the most useful inequality, probably requires

a-7 > 0 and suggests that both (3 and 8 are

near to their upper limits, that is /3 = 0.36 and 8 = 4.4.

G2 is trivially satisfied. Possibly [5] 6=a^.

R is too uncertain to be of practical value for fluids.

W\ is probably wrong.

W2 is probably correct but is subject to the same

uncertainty as R. It also requires /3 and 8 to be at their upper limits. A positive value

of 6 implies a "infinite asymmetry" in (Cv)2

as T approaches (Tc )~.

All these results can be extended to the critical

point of a two-component system. Here there is no real distinction between a gas-liquid, a gas-gas, Figure 1 and a liquid-liquid point, but all precise work has

been done on systems at which P is essentially zero and which are therefore called conventionally critical and hence possibly they are not a true guide liquid-liquid critical points. The behavior of a to the limiting behavior. two-component system can be obtained by analogy 6. —Careful measurements of vapor pressure in the from that of a one-component system by replacing critical region [11, 23] suggest that 9 may not be A(V, T) by G(x, T), where x is the mole fraction of zero but might have a small positive value, since one component and where the pressure of the sys- the fitting of a smooth curve through experimental tem is fixed. Hence the indices above now refer points usually gives a deviation graph that shows to the following properties a sharp rising upwards as Tc is reached. In summary, a Cp x0 ? once an equation [24] for (Gp )-2. The experimental results are now more frag- mentary, although this probably reflects lack of ft is the most certain of these, although 8 is certainly effort rather than inherent difficulty. Precise of high apparent precision. If and 8 are taken calorimetry and measurement of vapor pressures as known within the limits shown then the inequal- should be easier than at the critical pressure of a ities above lead to the following results one-component system since these critical points can be studied at negligible pressures.

Gl 0.06-0.40 a.— The best measurements of C,, are those of

R oq ^ (0.08 — 0. 16) ± uncertainty in yf Schmidt, Jura, and Hildebrand [25] (CCL4

W\ y+= 1.28-1.36 + C K FiiCF.i): Jura, Fraga, Maki, and Hildebrand

W2 y~ = 0.96 -1.22. [26] (/ — ChHik + n — CtFk;): Amirkhanow. Gurvich, and Matizen [27] (PhOH + rL>0). These suggest References either a weak infinity in C,, or possibly no infinity at all. [1] J. S. Rowlinson. Liquids and Liquid Mixtures, ch. 3 (Butter- worths. London. 1959). /3. — The recent and, precise results of Thompson [2] G. S. Rushbrooke.J. Chem. Phys. 39,842(1963). and Rice [28] make it unnecessary to discuss earlier [3] M. E. Fisher, J. Math. Phys. 5, 944 (1964). [4] R. B. Griffiths, Phys. Rev. Letters 14,623 (1965). work, including their own. They studied CCI4 [5] B. Widom. private communication. 10~ 4 + 7? — C7F16 to within °C of Tc and obtain [6] B. Widom, J. Chem. Phys. 37,2703(1962). [71 B. Widon, J. Chem. Phys. 41, 1633 (1964). [8] M. I. Bagatskii, A. V. Voronel. and B. G. Gusak. Soviet Phys. £ = 0.33 + 0.02. JETP 16, 517 (1963); A. V. Voronel' Y. R. Chashkin, V. A. Popov, and V. G. Simkin, ibid. 18, 568 (1964). [9] M. E. Fisher. Phys. Rev. 136, A1599 (1964). y, 8. — The results of Croll and Scott [29] for [10] A. Michels and J. Strijland. Physica 18, 613 (1952). + [11] A. Michels. B. Blaisse. and C. Michels, Proc. Roy. Soc. CH4 + CF4 suggest that is about unity. The y A160, 358, (1937). authors may have unpublished points on these [12] E. A. Guggenheim. J. Chem. Phys. 13, 253 (1945). [13] D. Cook. Trans. Faraday Soc. 49, 716 (1953). systems to justify some attempt to determine y [14] A. M. Weinberger and W. G. Schneider, Canad. J. Chem. and 8; no earlier work suffices. 30, 422 (1952). [15] H. L. Lorentzen. Paper read at conference on Statistical If, by analogy with the more extensive work on Mechanics at Aachen. 1964. + systems with one component, it turns out that y > 0 [16] M. M. Edwards and W. C. Woodbury, Phys. Rev. 129, 1911 (1963). then there are interesting consequences. Thus [17] D. Miller, private communication. 2 2 2 2 both {d H/dx and (d S/dx rmust be zero at the [18] H. W. Habgood and W. G. Schneider, Canad. J. Chem. 32, )p, T ) P , 98. 164(1954). critical point. That is curves of the heat of mixing [19] B. Widom and O. K. Rice. J. Chem. Phys. 23, 1250 (1955). as a function of composition must have a point of [20] H. L. Johnston. K. E. Keller, and A. S. Friedman. J. Amer. Chem Soc. 76 (1954) 1482. inflexion. This is not inconsistent with, for ex- [21] S. A. Ulybin and S. P. Malyshenko. Third Symposium on ample, the measurements of Copp and Everett Thermophysical Properties. Amer. Soc. Mech. Eng., Purdue. 1965. p. 68. [30] and of Matizen and Kushova [31] on the lower [22] S. Y. Larsen and J. M. H. Levelt Sengers. Third Symposium critical solution point of Et 3N + H 2 0. There on Thermophysical Properties. Amer. Soc. Mech. Eng.. Purdue. appear to be no other accurate measurements close 1965. p. 74. [23] A. Michels. J. M. H. Levelt and W. De Graaff. Physica 24, to Tc . 659 (1958): A. Michels. J. M. H. Levelt, and G. J. Wolkers. + ibid. 24, 769 (1958). A further consequence of y > 1 for which there [24] Reference 1, p. 168. is some experimental evidence is that coexistent [25] H. Schmidt. G. Jura and J. H. Hildebrand, J. Phys. Chem. curves calculated from excess Gibbs free-energies 63, 297 (1959). [26] G. Jura. D. Fraga. G. Maki. and J. H. Hildebrand. Proc. that have been extrapolated to Tc on the implicit Mat. Acad. Sci. 39, 19 (1953). + Kh. Amirkhanow, 1. G. Gurvich and E. M. Matizen. Dokl. assumption that y = 1 must always be too high [27] Akad. Nauk. SSSR 100, 735 (1955). at an upper critical solution temperature and too [28] D. R. Thompson and O. K. Rice. J. Amer. Chem. Soc. 86, 3547 (1964). low at a lower critical solution temperature. It is [29] I. M. Croll and R. L. Scott, J. Phys. Chem. 68, 3853 (1964): common experience that this is so [32, 33]. A see also N. Thorp and R. L. Scott, ibid. 60, 670. 1441 (1956). recent paper by Scatchard and Wilson [33] shows [30] J. L. Copp and D. H. Everett. Discuss. Faraday Soc. 15, a closed solubility loop calculated by such extrapo- 174 (1953). lations which shows simultaneously an upper critical [31] E. M. Matizen and N. V. Kushova. Russ. J. Phys. Chem. 34, 1056 (1960). point that is too high a and lower one that is too low. [32] G. E. Williamson and R. L. Scott. J. Phys. Chem. 65, However more direct evidence of the magnitude 275 (1961). [33] G. Scatchard and G. Wilson, J. Amer. Chem. Soc. 86, 133 of (y+ — 1) would be welcome. (1964).

12 Optical Measurements of the Density Gradients Produced by Gravity in

C0 2 , N 2 0, and CC1F 3 Near the Critical Point

E. H. W. Schmidt

Technische Hochschule, Miinohen, Germany

1. Introduction 2. The Pressure Chamber and Its

When Dr. Green asked me to read a paper dealing Accessories with my work on density gradients produced by gravity in fluids near their critical points, I hesitated For our experiments we use now the installation to follow this invitation because I already presented shown in figure 1. The fluid is in a strong cylin-

some of such experiments with CO > at the Second drical chamber of stainless steel with horizontal Symposium on Thermophysical Properties held in axis closed on both ends by parallel windows of Princeton University January 1962 [1]. However, optical glass with 28 mm free diameter. meanwhile together with my collaborator J. Straub, For equalizing the temperature of the pressure we repeated these measurements with greater pre- chamber it is placed in a big copper cylinder, which cision and extended them to N >0 and CC1F :! (Freon on its outer side is supplied with screw-shaped 13). groves through which water from a thermostatic

First I may remind you of the original purpose and apparatus flows bifilarly. In order to minimize the the start of these experiments. The old conven- fluctuations of the temperature produced by the tional method for measuring the critical data of a switching in and out of the heating coil of the ther- substance by observing the disappearance of the mostat with the help of a mercury contact ther- meniscus i.e., the interface between the liquid and mometer, three thermostats are used in series so the vapor phase is not very exact because it needs a that the first with its contact thermometer de- slow change of temperature or pressure and there- livers warm water for heating the next one, and this fore the fluid is not in a state of equilibrium. More- again for the third, as shown in figure 2. In this over it is difficult and depends upon the ability of way the temperature of the copper cylinder with the observer to state exactly when the meniscus has the pressure chamber in it could be held constant disappeared. This is the reason why the critical for many hours and even for several days within point of most substances is not very well known, and deviations of less than ±0.0015 °C. some observers even had the conception, that the The copper cylinder has on both ends conical critical point had to be extended to a critical region. holes for the optical observation. In these holes In order to avoid the difficulties of the conven- two more plates of thick optical glass are placed,

tional method I came to the idea of using the refrac- which together with two outer thinner windows and tive index for the determination of the critical electrically heated coils inside these windows state data, which index depends in a sjmple way prevent the cooling of the pressure chamber by upon the density of the substance irrespective of the surrounding air. its liquid or gaseous state. The temperature of the room, in which the whole

As long as a meniscus is visible, the densities experimental arrangement has its place is also and the refractive indices are different in the liquid regulated by way of contact thermometers together and in the vapor. However, when the critical state with electric heaters and the fluctuations of the is attained, refractive index and density should be room temperature do not exceed ±0.1 °C. Never- equal below and above the level where the meniscus theless, the copper cylinder is covered with a thick disappeared. layer of polystyrol as heat insulation.

13 Figure I. The pressure chamber used for the experiments: (a) Axial cross section (b) Cross section A-B (c) Cross section perpendicular to axis

Dimensions in mm; a. Copper cylinder, b. Pressure chamber, c. Strong glass windows, d. Sealing foils, e. Glass prism, f. Tube, g. Filling valve, h. Screw shaped channels for water heating, i. Protective heating of the valve, k. Heating tube, 1. Glass plate with protective heating coil behind it. m. Glass plates, n. Holes for thermocouples, o. Welded joints.

11 I ^

d C 0

FIGURE 2. Scheme of experimental arrangement: a. First thermostat, b. Water container, c. Second thermostat, d. Third thermostat, e. Screw pump

for mercury, f. Pressure balance, g. Manometer, h. Pressure chamber, i. Dehydrator for gas, k. Container for pressurized gas, I. Glass tube for testing the vacuum, m. Valve, n. Joint to vacuum pump. help of a screw pump more or less mercury can be pumped into the pressure chamber changing the

volume and the density of the fluid contained in it.

3. Optical Installation

The optical installation for measuring the refrac-

tive index and from it the density of the fluid is shown in figure 4. The pressure chamber a with the

glass prism in it is placed in the middle of the

disk c, around which two arms can be swung of which one supports the sodium lamp d. the con-

densor e, the vertical slit f, and the lens g; whilst

Figure 3. Picture of the pressure chamber {heat insulation the other arm bears the telescope h, the horizontal removed). slit i, and a small concave mirror k. The sodium lamp with its monochromatic light illuminates with A picture of the pressure chamber and the copper the help of the condensor the vertical slit which is cylinder before surrounding it with the heat insula- placed in the focus of the lens g. tion is shown in figure 3. All tubes leaving the In this way a parallel beam of light enters the pressure chamber are provided with protective pressure chamber and is refracted by the glass prism electrical heating in order to prevent any loss of and the prism-shaped fluid in the chamber. If the heat which would cause local cooling. The purpose telescope is turned in the right direction, a vertical of all these arrangements is to improve the con- picture of the slit is to be seen in it, and from the stancy of temperature in the pressure chamber and angle of the position of the telescope the refractive to avoid natural convection in the fluid, which dis- index of the fluid in the pressure chamber can be turbs the optical measurements. calculated. This angle can be determined very

The temperature of the fluid is measured with the exactly with the help of a small concave mirror help of manganin-constantan thermocouples placed placed above the middle of the pressure chamber in vertical holes drilled into the copper cylinder. and moving with the telescope, to which it is fixed.

The pressure of the fluid is measured with a piston A light pointer reflected by the mirror throws a manometer f shown in figure 2. This manometer mark on a circular scale with 2.05 m radius. In this is filled with oil which is separated from the fluid way the angle of the telescope can be determined in the pressure chamber by mercury. With the with precision of better than 0.01°.

c b a

FIGURE 4. Optical installation for measuring the refractive index.

a. Pressure chamber, b. Glass prism, c. Circular disk, d. Sodium lamp. e. Condensor. f. Vertical slit, g. Lens. h. Telescope,

i. Horizontal slit, movable up and down, k. Concave mirror. 1. Lighting of the observation mark, rn. Circular scale, n. Picture of the observation mark.

15 FIGURE 6 Photographs taken below and at the critical tempera-

ture of CO ..

These experiences gave rise to use a horizontal

slit before the objective of the telescope, which can I FIGURE 5 Samples of photographs taken with the telescope. be moved up and down, whereas the position of

If the density of the fluid in the pressure chamber the slit can be measured with the help of a cathe- does not depend on height, the telescope shows a tometer. In this way the density of the fluid can vertical picture of the slit. If there is a meniscus be measured as a function of height. in the pressure chamber separating liquid and vapor, we get two different pictures in the telescope, one 4. Some Results of the Point by Point produced by the light passing the liquid phase, the Measurements of the Density of CO2 other produced by the light passing the vapor phase, Near Its Critical Point as a Function the distance of these lines being proportional to of the difference in density of liquid and vapor. Height

One of the first photographs which I got in this As an example figure 7 shows the results of way with CO> at its critical pressure is shown in density figure 5. Here two horizontal slits are placed before measurements depending on height for CO2 at its critical pressure and at a temperature the telescope so that the upper slit gets light having passed the vapor phase, the lower light having passed the liquid phase. On the right-hand side we see what the telescope shows. The distance of the two lines decreases if we approach the critical temperature, but they do not coincide when the meniscus has completely disappeared. -x The last two pictures of figure 5 are taken without the horizontal slits. Here both lines are indistinct, 0 showing that in the fluid there must be a change of density with height. A better picture of the same b kind, also taken with two horizontal slits, is shown in figure 6; here even at the critical temperature of 31.01 °C the two lines have a substantial distance, \ as the third frame shows. The last frame was taken after stirring the fluid thoroughly with the help of 0.90 0.95 1.00 1.05 1.10 a magnetic stirrer. However, if the stirrer is REDUCED DENSITY pip stopped the two lines of frame three reappear after

FIGURE 7. Reduced density c of C0 2 at the critical pressure some hours and we learn that in the field of gravity p\p and § = 0.007 °C above the critical temperature. the stable state has gradients of density. a. Curve with measured points, b. Corrected curve.

16 = 0.007 °C above the critical temperature. We see how well the measured values lie on the curve a the scattering being less than the thickness of the line. All curves which we shall see later are determined by measurements with the same precision and thus it may be allowed to omit the measured points. The curve b is a slight correction of curve a due to the fact that in a field with a gradient of density the ray of light is slightly curved. In the figure the abscissa is the reduced density p/p c where p c is the density at the critical point. We see that for a change in height of only 10 mm (from + 5 until — 5) the change in density is about 10 percent.

Now already in 1892 Gouy [2] has emphasized that the critical pressure can only exist at a certain level of the fluid, because the pressure and thus the density decreases with height due to the weight of the 0.95 1.00 1.05 REDUCED DENSITY^ fluid. However, the measured gradients of densities are 30 to 50 times larger than those produced by with mean Figure 8. Density plp c against height for C0 2 a hydrostatic pressure if these are derived from a van the following temperatures relative density 0.985 p c and for -0.005 °C; b. 0.000 C: der Waals equation. # relative to the critical vilue: a. c. 0.003 °C: d. 0.012 °C; e. 0.035 °C;f. 0.095 °C. Figure 7 is measured at a temperature &= + 0.007 °C above the critical temperature and at a mean density very near the critical value. If we change the temperature at constant mean density of in the gaseous phase. The curve b is observed at a the fluid we get a set of curves as shown in figure 8, temperature very near the critical value, the curves 0.003 taken at a mean density of plp c = 0.985. The curve c, d, e, and / are taken at temperatures from a corresponding to a temperature # = — 0.005° below until 0.095 °C above the critical. Here the jump the critical shows a jump in density corresponding in density has changed to an inflection point with to a visible meniscus, but nevertheless there is still a tangent growing steeper the more we surpass the a substantial gradient of density in the liquid and critical temperature.

(A)

1 e d c bo ?

0.90 0.95 1.00 1.10 0.90

REDUCED DENSITY pip

FIGURE 9. Density plp c against height for CO>.

A. Same as figure 8. B. Mean relative density of 1.033 pc and the following temperatures # relative to the critical value: a. -0.008 °C; b.- 0.003 °C; c. 0.002 °C; d. 0.005 °C;

e. 0.015 °C;f. 0.035 °C; g. 0.065 °C; h. 0.125 °C; i. 0.232 °C. 17 The immediate neighborhood of the critical point In figure 10 all curves are taken at the criti- is difficult to observe because here the gradient cal pressure and at the same temperature # = of density has its highest value, and the light is — 0.006 °C below the critical value, only the mean deflected down. Therefore, in what follows we will densities differ from curve to curve and are denoted calculate the temperature difference # using as the by the small circle in each curve. starting point the highest temperature at which the Figure 11 shows the same very near the critical density versus height curve still shows a discon- temperature. Figure 12 for # = + 0.007 °C; figurj tinuity. 13 for # = + 0.03 °C; figure 14 for # = + 0.06 °cJ

If the mean density is increased by pumping more These figures show how difficult it is to determine mercury into the pressure chamber the level where the critical density without optical observation be- the meniscus disappears ascends as to be seen in cause the inflection point of the density curve (the figure 9, where the left-hand picture repeats figure level where the meniscus disappears) may have a 0.985, the right- higher or lower position. 8 with the mean density p/p<= and If it is high, more than I

. half of the hand picture corresponds to a mean density p/p ( chamber is filled with fluid of larger dens-

ity, if it is = 1.033. The different curves are isotherms from low, only a smaller part of volume has I — 0.008 up to 0.252 °C above critical temperature. the higher density.

US 100 105 REDUCED DENSITY/?//? REDUCED DENSITY pip

FIGURE 10. Density against height for C0 2 at a relative tem- FIGURE 12. The same as figure 10. but for a relative temperature perature i? = — 0.006 °C and at different mean densities in- $ = 0,007 °C. dicated by the small circle at each curve.

I.OS 1.10 1,15 UO B.60 0.85 0 50 0.9S 1J>0 |

REDUCED DENSITY pip REDUCED DENSITY p/p^

temperatures FIGURE 11. The same as figure 10 but for a relative temperature Figure 13. The same as figure 10, but for a relative § = 0.000 °C. that is at the critical temperature. $ = 0,030 °C.

18 3

0.91 100 1.01 110 REDUCED DENSITY pip

FIGURE 14. The same as figure 10, but for a relative temperature $=0,060 °C.

0.95 1,00 1.05 1.10 REDUCED DENSITY pip

5. Measurements With N 2 0, CC1F :! , Figure 15. Density against height for N2O at a mean relative density 0.998 p c and for temperatures ?? relative to the critical and H 2 0 temperature tc = 36.585 °C: a. -0.027 °C: b. -0.012 °C; c. -0.006 °C; d. 0.000 °C;

e. 0.012 °C: 0.021 °C. All measurements up to here are made with COi- f. In order to show that not only this substance under- = from 0.000 until goes gradients of density in the field of gravity, plp< 0.955 and temperatures # = tempera- experiments were also made with N^O and with + 0.056 °C, the right with p/pc 1.003 and tures # from -0.002 until +0.218 °C. The figure CC1F :! . Figure 15 shows isotherms of N2O for tempera- is very similar to figure 9. Figure 17 also relates to — with different tures # = — 0.027 until +0.021 °C at a mean density N 2 0 at a temperature & = 0.004 °C plpr — 0.998. In figure 16 two sets of isotherms are mean densities given by the small circles in the placed side by side, the left with a mean density curves.

REDUCED DENSITY/0//3,

FIGURE 16. Density against height for N2 0.

A. Mean density 0.995 pc and for the following relative temperatures: a. 0.000 °C, b. 0.004 °C, c. 0.015 °C; d. 0.020 °C; e. 0.044 °C; f. 0.056 °C. B. Mean density 1.005 pc and for the following relative temperatures: a. -0.002 X; b. 0.003 °C; c. 0.010 X; d. 0.020

°C; e. 0.032 °C;f. 0.038 °C; g. 0.063 °C; h. 0.119 °C: i. 0.167 °C; k. 0.218 °C. 19 209-493 O-66— 0.66 0.90 1.00

REDUCED DENSITY pip REDUCED DENSITY pip

17. with the mean: FIGURE Density against height for N 2 0 at a relative tempera- FIGURE 18. Density against height for CC1F : i

' ture i? = — 0.004 °C and at different mean densities the following temperatures relative to the given by the density 0.989 pc and small circle at each curve. critical temperature tc = 28.715 °C: a. -0.065 °C; b.- 0.025 °C; c. 0.000 °C; d. 0.021 °C; e. 0.040 °C; f. 0.093 °C; g. 0.183 °GM h. 0.265 °C.

Finally figure 18 is measured with CClF :i at a Finally we found that sapphire windows withstand mean density — 0.989. the different curves pip, and corrosion and these are used now. correspond to temperature differences with the critical value from # = -0.065 until 4- 0.265 °C. In the COj-molecule the two O-atoms are placed 6. Summary symmetrically on opposite sides of the C-atom. corresponding to the scheme 0=C=0. The N^O- The above experiments show undoubtedly that molecule is asymmetric and therefore polarized near the critical state of substances considerable following the scheme N=N=0. The CClF :i mol- gradients of density are produced by gravity. For ecule has the shape of a tetraeder with the C-atom the three substances encountered, the density at in the middle and the halogen atoms at the corners the critical pressure and near the critical tempera- of the tetraeder. As these three molecules, differ- ture changes by about 10 percent over a height of ing in shape and chemical constitution show the about 10 mm. These distributions of density are same behavior in the neighborhood of the critical stable and reproducible. After any changes of state, it is to be expected that this behavior is a state of the fluid, they are restored by diffusion and genera] quality of all substances. some kind of sedimentation within times of about 2 The critical data which we found are in good hr for heights of some centimeters. agreement with the results of other authors. Measurements with FFO are now being done.

This molecule has the shape of a triangle and its 7. References behavior near the critical state seems to be the same as for the other substances. However, the measurements are much more difficult due to the [1] Ernst H. W. Schmidt anc? K. Traube. Gradients of Density j of Fluids Near their Critical States in the Field of Gravity, pp. higher temperature ?,=374.15 °C and the higher | 193-205. Progress in International Research on Thermodynamic | pressure p, =221.3 bar. Windows of glass and and Transport Properties A.S.M.E. (Academic Press. New York and London 1962). even of quartz are impossible because these sub- [2] R. Gouy. Effets de la pesanteur sur les fluides au point stances are corroded and lose their critique, Compt. Rend. optical quality. Acad. Sci. Paris 115, 720-22 (1892). I

20 .

Notes, Definitions, and Formulas for Critical Point Singularities

M. E. Fisher

University of London, King's College, London, Great Britain*

1. Introduction then being either continuous or discontinuous as x passes through zero. For the convenience of participants a set of The definitions of the most important critical definitions and notes concerning the various criti- point exponents for gases and ferromagnets are cal point singularities and a table of formulas set out in the following table together with certain relating the lattice gas and Ising ferromagnet relations between them. Analogous formulas models were distributed at the Conference. These hold for binary alloys and binary fluid systems. are reproduced here with a few corrections and Note that the exponent tj enters directly into the extensions. scattering of waves at the critical point through the Most of the results quoted are described and formula discussed in detail in references 1. 2. and 3 where further references to the literature are given. A few more recent results are contained in references l//,(k) ~ k-~\ 4 to 8 and, of course, in the work reported at this Conference particularly in connection with critical where 1(h) is the intensity of scattering at wave scattering by Chu, Passel, Nathans, Dietrich, and vector k with A = (47r/A.) sin h 0 (for d=3>). Als-Nielsen.

The meanings of the symbols used in table 1 and below are 2. Critical Point Exponents Specific heats at constant volume, Cr, Cp , Ch- Cm pressure, field, and magnetization, For theoretical discussions and in the analysis respectively. of experimental data it is useful to have a precise T. T Temperature, critical temperature, definition of a critical point exponent. This may c Density, critical density; densities be given as follows: p, p,-; p/., pa of coexisting liquid and gaseous phases. if lim log f{x)l logx= k. jr-»0 + M, Mo Magnetization, spontaneous mag-

K — netization. we say fix) ~ x as x > 0 + Kt, Ks Isothermal and adiabatic com-

A pressibilities. Note that with this definition the statement f(x) ~ x Isothermal and adiabatic suscep- does not exclude the possibility of a logarithmically Xt, Xs K x tibilities. divergent factor, i.e., Ax log x\ ~ x . In particu- \ p. pc po- Pressure, at the critical point and at lar if A =0 the function might diverge as log fix) | | coexistence (vapor pressure), for example, or f(0) might be finite, the function f(x) respectively.

H Magnetic field.

gi(r) Pair correlation function (in a fluid). *Now at Cornell University, Ithaca. New York.

21 p T

TABLE 1. Critical point exponents

Exponent Fluid Magnet

- Below Tc At coexistence as T—> Tc At // = 0 as T^TC a ' a ' a' Cv~(Tc -T)- C„~(Tc -D- 3 In, — n, \ ~ IT — TV — IT — T\0 P VPL pc) \1 c I r MjT\ y' KT ~(Tc -T)-y' XT~(Tc -T)-y' a* Idtpv/d-Pl-iTc-T)-"*

-» At T= Tc As p pc As H^O

8 \P-Pc\~\p-Pc\ 6 \H\~\M\ 8 Note the exponents j8, y', and 8 might differ on the gas and liquid sides.

* 00 °° At p = pc as i At # = 0 as 2+ " ~ \g2 (r)-l\~ l/r"-

1 —» = Above Tc At p = c , as 7 rc + At // 0, as T-* Tc + a ^-(r-r,.)- 0 CH ~ (T- TcY a )-y y KT ~{T-Tcyy xr ~ (T- c V Inverse Range of Correlation K T) ~ (T-TcY

Rigorous a' + (2)8 + y')mi„^2 a' + 2p + y' ^2 inequalities «'+i8 max + (/3S) min ^2 a' + /3(l + 8)^2 a*^a'+/3max — Conjectured a' + 2 3 + y' = 2, 7' = £(8-1), y=(2 t/)i/ i relations

o? orous Dimensionality of the system. inequality recently derived by Griffiths [7] ij

Si Spin variable at site r. (see table 1). ffi Hamiltonian. For the standard three-dimensional (d=3) spher- a' 0 z Activity. ical model one may add the results: = a =

Chemical potential, at coexistence. (continuous), fB—i, y' not defined, 8 = 5, 77 = 0, p, p„ | y = 2, v = l. For a spherical model with long range ferromagnetic forces decaying as l/rlt+ "~ {0 In table 2 the exact, estimated and observed 1 < cr < 2) one has more generally : a' — a = 0, values of the critical exponents defined above are j /3=f as before, but (a) for cr < |e/: y' — 7=1,1 presented for various theoretical models and for 8 = 3, r) = 2 — cr, v undefined; while (b) for cr 3= \d\\ real fluids and ferromagnets [1-6]. The queries ! y' undefined, 8 = (d+ ar)/(d— cr), 7) — 2 — cr, l indicate significant uncertainties or total ignorance! y =

imental results for a', /3, and 8 satisfy the rig- 1 These results have been obtained by G. S. Joyce.

22 Table 2. Values of critical point exponents

Expo- "Classical" Ising Heisenberg Observation nent theory d=2 d=3 d = 3

Fluids Magnets

? a' 0 0 > 0? > 0 (log) > 0 ? (discon.) (log)

i - 5 9 P 2 Vs 0.312 /ie 0.33 to 0.36 0.33 0.303 to 0.308 ±0.015

3 y' 1 l /4 1.23 to 1.32 9 =*1.2 ? ?

a* 0 0 >0 5=0 ? (log) 8 3 15 5.20 ? 4.2 4.22 ±0.15 ±0.1 ±0.05

V 0 V4 0.059 - Vis 0.08 >0 ? >0 ? ±0.006 ± 0.04

a 0 0 2=0 -0 =3 0? >0 ? (discon.) (log) ^0.2

3 7 1 l /4 1V4 - 1V3 > 1.1 ? 1.35 1.32 to 1.38 ±0.02

V v% 1 0.644 0.69 > 0.55 ? >0.66 ±0.002 ±0.02

3. Relation Between a Lattice Gas mental results included in table 2 suggest that the fluid-magnet analogy might be deeper than the Ising and an Magnet simplicity of the Ising model would indicate. We consider a general (/-dimensional lattice of

The mathematical equivalence of the lattice ./V sites, each occupying a cell, of volume vo- The gas model of a fluid and the Ising model of a magnet, pair interactions are not necessarily limited to

[stressed originally by Yang and Lee, is well known nearest neighboring sites. Positive values of the

1 but it was felt that a more complete account of exchange integral J(r) correspond to ferromagnetic

I the relation between the two models than normally coupling but the formulas are valid for arbitrary given would be useful. Indeed some of the experi- coupling.

23 Table 3. Correspondence between a lattice gas and an Ising magnet

l//.«r) Lattice of N sites; J-dimensional volume v () per site. (/3=

= s Total Potential -yj(r iJ)s is -Hy i &s s j Energy of n atoms

Occupation ti = 1 or 0 5 , = 2Sf = ±l Spin ti= Hi- si) Variables Variables

= Potentials (0) £ (r u) Exchange Integral

Density Magnetization per Spin pjpc =\-M ^raax 1 Pmax = 1/W0 = 2pf

Chemical Potential p Magnetic Field, H 2 il - units chosen so that magnetic P/jL = \n(zA") A = (h l27rmk H T) = -2H moment per spin is unity. at condensation

p = p,r = ^(O)-AT ln,i>»/A")

Free Energy per Spin Pressure p(p, T) PVu F*-H-Vs(t>(0)= [l-M(H')]dH' F*(//, T)

Configurational cunfiu — C * Entropy per Spin Entropy Density

2 Susceptibility Isothermal Compressibility ^Vi)p Kr = Xi Isothermal

Configurational Specific Heats pvoCf«t* = CM =CH -Tc?Jxr Specific Heats

Note 1/xt = 0 in the "two-phase" region. p^qonfm = T JlL^ + + 2T Xr CH f-^n | (dM\ ( 1-M l-M

aH = Q for H=0, T > Tc Note Cp/Ci' — Kt/K.s ChICu = XtIxs

Thermal Pressure f„y. = v = S* (1 - t) (dpldT) y + M)anlxi Coefficient

Correlation - 2 - ( Vi)Qr) = Vi) [g.j{r) 1 = < (s ) = Functions p p ] W 0y-)/((sl) - Hi

= at = . p p„ T>,TC W(r) = TC

24 From'the correlation functions G(r) and T(r) one S. Y. Larsen: I would like to make a remark concerning the comment of Dr. Pings. Dr. Mountain, Dr. Zwanzig, and I have made may derive the critical scattering intensities through a theoretical studv of the behavior of the index of refraction [3]. the formulas We find no anomaly in the behavior of the real part of the index of refraction in the supercritical region when the critical point is approached. /(k) = x(k)/o(k)

O. K. Rice: I may mention that some years ago Rowden and I where /o(k) is the form factor for scattering from made vapor pressure measurements on the cyclohexane-aniline system [4]. We approached the critical point to within 0.1 or the noninteracting system and where 0.2 °C and it looked like y= 1.

J. S. Rowlinson: Did it?

ik k v e e' X(k)=l+p 2 l) *G(r)=l+ T(r). r ft 0 r * 0 O. K. Rice: We could not say definitely whether y was either greater or smaller then 1, but it did look very much like y= 1, as The fluctuation relation may then be written far as we could tell from the measurements.

J. S. Rowlinson: I am sorry that I omitted your results in my KtIK%=xW) = XtI7& review. I did not try to deduce a value for y from your results on the cyclohexane-aniline system. where the superscript zero refers to an ideal noninteracting system. M. E. Fisher: I would just like to emphasize one point, which Dr. Rowlinson made very well. There is an inconsistency with the rigorous inequalities if one takes the values of the exponents at their face value. Although one cannot be too sure that any of them are as good as they seem to be, the fact is that for 8 to 4. References be consistent with the relationship (Gl) a rather big change in its value is needed. It means that 8 should be near 5 rather than [1] M. E. Fisher. J. Math. Phys. 124 (1963). 4, 4.2, is not that right? [2] M. E. Fisher. J. Math. Phys. 5 , 944 (1964). [3] M. E. Fisher, Proc. Int. Conf. on Magnetism. Nottingham It on what lower limit one puts on 1964, 79 (The Physical Society, London, 1965). J. S. Rowlinson: depends aj. allows = 0.06, can get away with /3 = 0.36 and [4] J. S. Kouvel and M. E. Fisher. Phys. Rev. 136, A. 1626 (1964). If one one = [5] M. E. Fisher. Phys. Rev. 136 A. 1599 (1964). 8 4.4. [6] D. S. Gaunt. M. E. Fisher. M. F. Sykes. and J. W. Essam. Phys. Rev. Letters 13,713(1964). M. E. Fisher: That is correct. However, since you multiply R. B. Griffiths. [7] Phys. Rev. Letters. 14,623(1965). 8 with 1/3 in relation (Gl), the effect of a change in 8 in order [8] R. J. Burford. E. Fisher, and M. to be published. to satisfy the relation gets damped down. If you take the central values of 8 and 8. there really is quite a big discrepancy.

Discussion J. S. Rowlinson: I agree that the central values lead to a discrepancy, and if you prefer 8 = 5, I cannot say no.

R. L. Scott: Dr. Rowlinson made some reference to our work on mixtures of CF4 with methane. Recently Croll and I published E. Helfand: In a mixture there are many new thermodynamic variables some rather old work on mixtures of CF4 with ethane [1]. I did because one has one extra variable and consequently not actually consider the work good enough to apply this kind many more derivatives. One of the interesting derivatives of consideration, but at the request of Dr. Widom, I looked at may be dp/dp. This property depends on the liquid correlation functions, that is density correlation function, whereas the the results again. The data give the total pressure of CF4 with ethane at about 150.6 °K, which is about 0.5° above the critical second derivative of the free energy with respect to the concen- solution temperature. The curves are, in fact, nearly flat. If tration depends on the concentration correlation function. One one ignores the fact that the only relevant data are considerably might well expect these correlation functions to have the same removed from the critical composition (just the point that Dr. range and the same analytic behavior. Thus perhaps dpldp Rowlinson made and that really vitiates the whole result), then would reflect the same y behavior. the results are consistent with an exponent 8 = 4, but it would not be very difficult to make the data consistent with 8 = 5. J. S. Rowlinson: It would be very difficult to measure the dpldp in a dense fluid with the precision that is needed to get C. J. Pings: The one enigma in the data showed by Dr. Rowlinson these indices. for fi was in the results obtained by Edwards and Woodbury for 4 He . I believe those were deduced from refractive index W. W. Webb presented measurements. We have some unpublished data on the Lo- measurements of the reflectivity of the interface between two phases in rentz-Lorenz function for argon and methane which suggest a equilibrium just below the critical consolute temperature possible anomaly in the Lorentz-Lorenz function in the critical [5]. These measurements indicate that the effective equilibrium thickness of region [2]. Until this is resolved, we must be dubious about any the transition region increases roughly — 7 conclusions of this type deduced from refractive index as [Tr 7V"- as the critical temperature is approached, reaching ~ 1.000 at {T,. — ~ 10"3 measurements. A T)ITV .

S. I J. Rowlinson: mentioned the results of Edwards and Wood- S. Y. Larsen: Dr. J. M. H. Levelt-Sengers and I have looked at bury only in passing. I did not use them in any way in the anal- the disagreement between the optical data and the isotherms ysis. They are clearly out of line with the results on gases at derived from PVT measurements [6]. This disagreement was high temperatures. Whether this is a particular quantum effect first noticed by Dr. Schmidt and noted in the account of his or not, I do not know. optical work dven at the 1962 Symposium of the A.S.M.E. on

25 Thermophysical Properties and also has been taken up by Ulybin because one could still change the leading coefficient. I think and Malyshenko in a recent paper delivered at the Third Sym- that one would like to start with the beautiful new data of Dr. at posium on Thermophysical Properties [71. Schmidt and try again to look these questions. Our conclusion is that the optical work may be in agreement with the PVT data. The disagreement lies between the optical results and the analytical forms for the isotherms which have References been fitted to the PVT data. These include both the Taylor type expansions and the fit of Widom and Rice. We believe that one of the main reasons for the disagreement is that these I. Croll and R. L. Scott, Phys. 3853 (1964). expressions are really fitted to data somewhat outside the critical [1] M. J. Chem. 68, C. P. Abbiss, C. M. Knobler, R. K. Teague, and C. Pings. region. We find that the conventional way of effecting PVT [2] J. measurements provides an exceedingly insensitive method of J. Chem. Phys. 42, 4145 (1965). determining the pressure versus density relationship very close [3] S. Y. Larsen, R. D. Mountain, and R. Zwanzig, J. Chem. to the critical point and that optical methods are certainly to be Phys. 42, 2187 (1965). preferred. Accordingly, we believe that the isotherms are very [4] R. W. Rowden and O. K. Rice, J. Chem. Phys. 19, 1423 much flatter near the critical point, and the compressibilities (1951); Changements de Phases, Comptes rendus de la very much larger, than has been inferred from PVT data. 2e Reunion Annuelle tenue en commun avec la Commis- sion de Thermodynamique de 1' Union Internationale de Physique, Paris, 1952, p. 78. G. I if those remarks could be translated B. Benedek: wonder [5] G. H. Gilmer, W. Gilmore, J. Huang, and W. W. Webb, into the following terms: What do you think is the proper value Phys. Rev. Letters 14, 491 (1965). of 6, and what think is the proper value of do you y? [6] S. Y. Larsen and J. M. H. Levelt Sengers, Third Symposiun on Thermophysical Properties, A.S.M.E., Purdue, 1965, p. 74. S. Y. Larsen: At the present time we have not attempted to reeval- [7] S. A. Ulybin and S. P. Malyshenko, Third Symposium on uate the old data. The result that we have does not imply that Thermophysical Properties, A.S.M.E., Purdue, 1965, the S of Widom and Rice, for example, is not the correct one p. 68.

26 CRITICAL PHENOMENA IN FERRO- AND ANTIFERROMAGNETS

Chairman: H. B. Callen

Critical Properties of Lattice Models

C. Domb

University of London King's College, London, Great Britain

Introductory Remarks in the first few terms will not diminate asymptotically (and I shall later quote an example of such I shall assume that you all know what is meant behavior for the Heisenberg model), but the fol- by the Ising and Heisenberg models and are rea- lowing physical considerations enable one, never- sonably familiar with Onsager's exact solution theless, in many cases, to make confident predic- [l] 1 of the two-dimensional Ising model . My tions about the singularities of functions from a task is to discuss the three-dimensional model, finite number of terms of series expansions. and the effect of changing the spin and range of (a) The two-dimensional solution can act as a interaction, and for these problems exact methods guide since we expect the three-dimensional solu- have so far proved to be of no avail. Some people tion to be similar to it in essential features. have no enthusiasm for methods which are not (b) We have a few exact theorems which apply exact and consider that detailed investigation of to three-dimensional solutions. higher order approximation terms is not the task (c) Physical considerations of what is expected* of a theoretical physicist. I am reminded of a of the model can lead us to transformations of the conversation which I had with Yvon about his own series which reveal peculiarities of behavior. important contribution to this field (which I shall We must be sure at the outset that we have refer to later). He told me that he developed the enough terms of our series to get over small num- method, tried it in lower order, and found that ber effects. No general rule can be given since the it gave the same result as other approximations. number of terms needed varies from problem to

I asked if he had not been interested to go to higher problem. However, careful analysis of the data orders to see whether it gave any improvement. for each particular problem usually shows if the He replied: "my task as a theoretical physicist is number of terms derived is adequate. to develop methods, and see that they work and Before going into details of the behavior of the are correct: I then hand over to the engineers." Ising and Heisenberg models, I should make the

By these standards, what I have to say to you this connection with this morning's lectures on liquids afternoon, is engineering, but at least I will spare and gases. Fortunately, Professor Fisher has made you the technical details of the engineering methods my task very easy by circulating a sheet giving the and discuss only the results. Let us also remember detailed connection between the Ising model and that engineering methods led to the discovery of the lattice gas. I should like, first, to point out

Fourier series, and I hope to terminate my talk that two quite different lattice models are used in with some remarks about the possibility of an en- connection with fluids. The first, with which we gineering approach to an exact solution. are concerned here, is a model of liquid vapor

The methods I shall discuss depend on develop- equilibrium, and it corresponds to the Ising model

ing large numbers of terms of series expansions of a ferromagnet [2]. From the relations given by

at high and low temperatures which are used to Professor Fisher, I shall draw attention to three estimate critical behavior. Now any second-year which are of relevance to my talk. In critical be- mathematics student will tell you that nothing can havior C\ for the fluid corresponds to Ch for the be learned about the singularities of a function magnet in zero field at all temperatures: the density from a finite number of terms of a series expansion: of one component of the fluid corresponds to minus for example, even if the first 15 terms of an expan- the spontaneous magnetization: and the isothermal

sion are positive and well-behaved, there is nothing compressibility of the fluid corresponds to the iso- to ensure that an effect which is small and unnoticed thermal susceptibility of the magnet. It is impor-

29 for tant, also, to note that the chemical potential larity is on the positive real axis [5]. If they alter- magnetic the fluid is proportional to minus the nate regularly the dominant singularity is on the field. negative real axis. If they manifest more compli- The second model applies to solid-fluid equilib- cated behavior, the dominant singularities are rium and corresponds to the Ising model of an anti- elsewhere in the complex plane. Physical interest ferromagnet [3]. This model has been extensively centers largely on singularities on the real axis. used as an analog of the condensation of a fluid of For the Ising and Heisenberg models in zero field, further the real hard spheres [4], and will not be discussed we expect to find a positive singularity on in the present talk. axis corresponding to the Curie point, and possibly a singularity on the negative real axis cor- Short range forces; Series expansions responding to the Neel point. It is convenient to divide the series expansions to be considered into three groups. Group I May I start with a few elementary observations all positive and from table 1 we select about power series expansions. If all the terms have terms the coefficients in the high tempera- of a power series are positive, the dominant singu- as an example

Table 1. Coefficients a n in the susceptibility expansion for a number of lattices

Type of lattice Simple Face-centered n tjudurciiic/Til Onl'oi'lP cubic Triangular Simple cubic

0 i 1 1 1 1 4 6 6 12 2 12 30 30 132 3 36 138 150 1,404 4 100 606 726 14,652

5 276 2,586 3,510 151,116 6 740 10,818 16,710 1,546,332 7 1,972 44,574 79,494 15,734,460 8 5,172 181,542 375,174 139,425,580 9 13,492 732,678 1,769,686

10 34,876 2,935,218 8,306,862 11 89,764 11,687,202 38,975,286 12 229,628 46,296,210 13 585,508 14 1,486,308

15 3,763,460 16 9,497,380

30 ture series for the susceptibility of the Ising model Passing now to "even" lattices like the S.C., for triangular and F.C.C. lattices. These lattices B.C.C. and Diamond, we again find the same value are chosen because the absence of antiferromag- for y. However, now because of the symmetry netic ordering leads one to expect a single clear between ferromagnetism and antiferromagnetism, and dominant singularity. The general behavior we expect a corresponding singularity on the nega- of the terms suggests comparison with a binomial tive real axis which is much weaker and therefore expansion of the form masked by the positive singularity. We should therefore try to look for a function of the form

(1)

and we therefore investigate the ratio of successive where the second term represents the antiferromagnetic singularity. In order to focus attention terms a„/fl„-i as a function of — . For a binomial n on this second factor, Fisher and Sykes [7] took expansion (1) we should have w In Xo, subtracted — - In and obtained a 4 Wc well-behaved series with terms alternating in sign. a„lan -\ 1 + (2) After some ingenious manipulation they concluded H, that the second factor in (3) could be represented by

Figure 1 shows that the terms converge rapidly and i H In -^ smoothly; for the triangular lattice the estimates C +D[l + 11 + (4) of wc and the slope are very close to the correct values [6] 2- V3 and 3/4. For the F.C.C. lattice The form of the susceptibility at the Neel point is a good estimate can be obtained of w and one can c shown in figure 2; it is characterized by a maximum u 5 above the Neel temperature and a logarithmically conjecture that y — —. infinite slope at the Neel temperature.

The second example in this group is the high temperature expansions for the specific heat of the Ising model for the triangular and F.C.C.

lattices [8]. The terms are again all positive but

converge more slowly (eq. (5)).

qa„-i face-centred cubic

triangular

kT/q J l/n

FIGURE 2. Comparison of the antiferromagnetic susceptibility Figure 1. Ising model. [33] of (a) the simple cubic lattice (q = 6) and (b) the plane honey- Successive ratios in the susceptibility expansions of the triangular and face-centered cubic lattices comb lattice (q = 3). [7] as functions of l/n. tq = Coordination number of lattice = number of nearest neighbors ot each atom.) The critical point is indicated by a small circle.

31 — OC/

2 2 Table 2a. ChIR = 0.06257* (1 + 0.8169? + 0.6778; Triangular lattice + 0.5680i3 + 0.4818?4 + 0.4190r 5 6 7 + 0.3725r + 0.3364r +. . . .) (F.C.C.) n h

(5)

O ChIR = 0.2263f 2 (l + 1.0986* + 0.8298; 2 8 0.117270 3 4 a + 0.5525i + 0.402k + 0.3384? 5 9 .122611 6 7 1 A 1 OOA/1 o + 0.297k + 0.2614r +. . . .) (Triangular). 10 .122048 11"1 "1 .1130001 1 O AAA 112O .1047051 A/1 TAC It is clear from the analysis shown in table 2 that 13 .09858AAOCC7 not enough terms are yet available to draw definite 114A .09ooo8AA OOOO AAA A A/; conclusions; the slope for the three-dimensional 15 .088486 lattice seems to be sharper than a logarithm cor- 16 .084055AO A AC responding to the two-dimensional lattice, and a 17 .080042

figure of (l could be regarded as reason- 18 .076592 1 a YJ 19 .073056 ably consistent with the data. Table 2a shows an 20 .069993 extension of the triangular lattice (calculated by 21on .067174 D. L. Hunter) and it will be seen that for values of 22 .064570 n greater than 11 the series settles down to steady 23 .062158A/oi CO behavior; a linear extrapolation would yield a value 24O/I .059918ACAAl of close zero. steady behavior oc AC7O0O a to The same 25 .05 /832 seems to set in earlier (ra = 5) in the three dimen- 26OA .055886AC COOA sional case, and this increases confidence in the 2/(17 .054065AC/I AAC above estimate of critical behavior. 28 .052358 It is interesting to ask why the convergence to 29 .050755 the asymptotic value of the specific heat coefficients 30 .049246 is so much slower than for the susceptibility. I think that this can best be understood by comparison with an analogous problem in the lattice model of polymer configurations. If c„ represents

the number of simple chains of n links on a lattice and u» the number of simple closed polygons, the' n series Xc„:r" and %unx , both have the same radius Table 2 of convergence; but the cn reach their asymptotic value much more rapidly than the «„, as can be seen from figure 3. The most important configu- n h (triangular) h (F.C.C.) rations contributing to the susceptibility are "chain-like," and those contributing to specific! heat are "polygon-like." 1 1.0986 0.8169 Group II refers to series not consistent in sign; 2 0.5106 .6594 in which the singularity of physical interest is 3 -.0025 .5140 masked by a spurious singularity. Examples of 4 -.0887 .3930 this group are given in table 3 of the coefficients' 5 .2079 .3483 in the low temperature expansion for the spon-| 6 .2676 .3341 taneous magnetization. Only a real positive!

7 .1599 .3215 singularity is of physical significance here and it! can be assumed that this singularity has beenj

32 tion of the type

u = A

and we should expect it to be at least as useful in improving the convergence as the best such homographic transformation. Clearly, for a function of

the type iMu) it is best to approximate to -7- In Mu), clu and the zeros of the denominator then locate the

singularities uc , u.\, u > . . . . Our interest is focused

on B. and it is best to assume the value of uc from the high temperature susceptibility expansion and

approximate to fi by

[(« — u c ) *M")]- (10)

There are a number of devices which can be used FIGURE 3. Excluded volume problem. [33] Asymptotic limits of c„ and u„ for self-avoiding walks. to improve the estimate of /3. After careful exami-

nation it has been conjectured that B = — for three 16 dimensional lattices. located by the high temperature susceptibility For series of Group II we also find examples of series. The triangular and S.C. lattices give rise slow convergence, and these are illustrated in table to a regular alternation in sign, and the dominant 4, which gives the low temperature susceptibility singularity is therefore on the negative real axis: expansion for three-dimensional lattices [10]. The for the F.C.C. lattice a more complicated distri- Pade approximant technique has again been applied bution in the complex plane is indicated. We know assuming a function of the form (7), but even though the exact solution for the triangular lattice [6], of a substantial number of terms are available, it will the form be seen from the analysis in table 5 that the limits ~ — y of the index A'(u u,) ) are still between * y'(xo (1 - 3uyl*(l + u? 1 1.37. (6) 1.27 and a + 3 U yi«a-uf*' Similarly, the specific heat at low temperatures

[11] converges slowly and present data are consist- for the spon- ? We might therefore reasonably try ent with logarithmic behavior three-dimensional lat- s taneous magnetization of

c tices a function of the form Ch ~ BY n \TC — T), (11)

* V(u) = E(u- ucf{u -mfiu- u»f (7) but are not sufficiently convergent to draw reliable conclusions. of particular interest It is for this type of series that the Pade approxi- These last two quantities are mant. introduced into this field by G. A. Baker because of thermodynamic inequalities derived by Rushbrooke and Griffiths [12] which estimates [9], is of prime importance. The [N, M] approxi- satisfy (this aspect will be mant is defined by of the indices fail to discussed in more detail by Professor Fisher).

P(u) J.p r u' (degree M) The best hope of obtaining more reliable infor- flu) _ = Q(u)~2q,W (degree AO' (8) mation would seem to lie with the development of more terms for the diamond lattice [13] since

The [N, N] approximant is the most useful since they belong to Group I. Even with the powerful

it is invariant under any homographic transforma- help of the Pade approximant. it is still true to say

33 1 !

Table 3. Coefficients b n in expansion of {d(t. T)ldr}ofor large t

Type of lattice Face-

ft cciiicx cu Simple Triangular Simple cubic cubic quadratic

0 8 12 12 24 1 34 -2 -14 -26 2 152 78 90 0 3 714 -24 -192 0 4 3,472 548 792 48

5 17,318 -228 -2,148 252 6 88,048 4,050 7,716 -720 /inn 7 454,378 - 2,030 -23,262 438 8 2,373,048 30,960 79,512 192 9 1 9 51 5 634 — 1 7 670 — 95? 054 984

10 66 01 ft 242,402 Oti U,UZ.O 1,008 1L 1 — 1 ^9 ^90 — 9 7^3 ^90 — 1 9 994 12 1 Q39 000 0 90^ 800 19,536 13 — 1 312 844 — 30 371 124 - 3,062 14 15,612,150 8,280

15 -11,297,052 — 26,694 16 - 153,536 17 507,948

that series of Group I provide the most reliable estimate of Curie temperature and a susceptibility information. (Of course, the Pade approximants 1 index of 4/3, can be used to good effect on these series as well.)

This program is being currently pursued. 4/3 [Xo~C'(7V-D- ] (12)! Group III consists of series in which there is unexpected asymptotic behavior of the coefficients (and this index is given more clearly by the F.C.C. which does not manifest itself for many terms. lattice [15]). Table 6 shows the coefficients in the high tempera- However, physical considerations lead us to ture susceptibility series for the Heisenberg model | think that there is a corresponding antiferromag- of the B.C.C. lattice [14]. If they are analyzed ! netic singularity which is no longer symmetric in the manner of Group I, they lead to a good j with the ferromagnetic singularity. The position of this Neel point was located very strikingly by Rushbrooke and Wood recently [16], who used the 1 Kor subsequenl work (spin Vj) see Baker. Gilbert. Eve. and Rushbrooke. Phvs Letters 20. 146 (1%6I. staggered susceptibility, (i.e., susceptibility in a

34 4

Table 4

F.C.C. B.C.C. S.C. n

q=\2 q = S q = 6

1 0 0 0 2 0 0 12 3 0 16 — 14 4 0 -18 135 5 24 0 -276

6 -26 252 1 520 7 0 -576 -4 056 o rirv 8 U 1 i (to 9 72 3 264 -54 392 10 378 - 12 468 213 522

11 -1 080 20 568 - 700 362 12 665 26 662 2 601 674 13 384DO/1 — 215 568 — 8 836 812 14 1 968 528 576 31 925 046 15 2 016 - 164 616 -110 323 056

16 - 25 698 - 3 014 889 17 39 552 10 894 920 118o — 3 872 19 20 880 20 - 65 727

21 -379 856 22 1 277 646

field which alternates in sign from one lattice point to the Ising model, and if so. we can anticipate an to the next). They found that the Neel temperature asymptotic form for the coefficients of the type is 10 percent higher than the corresponding ferromagnetic Curie temperature. If we take off the 4 (9 7Q1 )" |(2.537)" + (-D" • (13) ferromagnetic singularity (table 6) as for the Ising o n model, we find clear evidence of this dominant factor although insufficient terms are available to analyze the detailed critical behavior. Fisher The terms would then ultimately alternate in sign has advanced general theoretical arguments [17] but the first negative coefficient would not occur which lead us to expect a similar critical behavior until after about 60 terms.

35 209-493 0-66— Table 5. Pade approximants for y'

N F.C.C. B.C.C.

7=1 /=0 /=-l j=l 7 = 0

6 1.207 1.254 1.241 1.246 1.264 7 1.259 1.251 1.254 1.280 1.402 1.374

1 C% A A 1 C% A T 8 1.287 1.244 1.247 1.375 1.405 1.285

9 1.249 1.262 1.236 1.295 1.273 1.268

10 1.240 1.360 1.332 1.269 1.272 1.276

n 1.332 1.353 1.350 1.277 1.274 1.273 12 1.350 1.353 1.270 1.274 13 1.273 1.273

Table 6

Heisenberg B.C.C. lattice Magnetic model susceptibility n d log xo (4/3) X Delta dK a-KiKc )

1 4.00000 4 3.383236 + 0.616764 2 12.00000 8 8.584714 - .584714 3 34.66666 24 21.783088 + 2.216912 4 95.83333 52.6 55.272995 -2.606328 5 262.7000\) 154.16 140.251202 + 13.915465 6 708.04166 350.11663 355.8773 -5.760670 7 1893.28968 943.01682 903.0126 + 40.004 8 5012.10863 2189.44354 2291.329 -101.885 9 13235.59898 6111.4476 5814.018 + 297.43

Kc taken as 0.3941, inverse 2.5374. Summary of Critical Behavior of at the Curie point. For the Ising model the index = h Nearest Neighbor Lattice Systems S {H M ) is about 15 in two dimensions and about 5.2 in three dimensions [20]. Data for the Heisen- We have already referred to the critical be- berg model are available but have not yet been havior of the magnetic susceptibility of ferro- put into a convenient form for detailed analysis. and antiferromagnets in the previous section. Critical properties of the correlations are being Change in spin value from V2 to general s does not discussed in detail by Professor Fisher. change this critical behavior either for the Ising or Heisenberg model [15]. Longer Range Forces Regarding the critical behavior of the spontaneous magnetization, accurate information is available The precision of our knowledge of the effect of only for the Ising model of spin V2 (see previous longer range forces has increased appreciably dur- section). A Green's function interpolation method ing the past few years. The idea had often been

used by Tahir-Kheli [18] has suggested an index of expressed intuitively that if the number of neighbors

1 V3 for /3; this differs only slightly from the Ising participating equally in the interaction tended to 5 value of /i6. infinity the mean field or Bragg-Williams approxima- Critical values of thermodynamic functions, tion would result exactly. This has now been

particularly entropy, provide insight into the mag- established rigorously [21] by taking an interaction nitude of the "tail" of the specific heat curve. For of the form eJe~ (R and allowing e to tend to zero; the

a model of spin V2 the total entropy change is result is valid even for a one-dimensional model. k\n2 per system. With the Ising interaction in two- A solution of Bethe type including a small residual

dimensions the tail is large and corresponds to specific heat above Tc results if the force is long more than 50 percent of the entropy; in three- range in one direction and short range in the other

dimensions the tail is very much reduced and [22]. It is significant to observe this physical inter- corresponds to about 15 percent. For the three- pretation of the Bethe result which had previously dimensional Heisenberg model, the tail is increased been regarded as a first approximation to the solu- by a factor of more than two and corresponds to tion for a nearest neighbor model, or the exact

about 35 percent of the entropy [19]. solution for an unphysical "pseudo-lattice" contain-

When we change from spin V2 to general 5, the ing no closed circuits [23].

total entropy changes from Aln2 per system to It is interesting to see how the mean field solution

£ln(2s+l). However, most of the change takes is approached as the range of force increases. For

place below the Curie temperature and even in this purpose it has been found convenient to intro-

going from 5 = 1/2 to s = °°, the increase in entropy duce an "equivalent neighbor (r shell)" model in

above the Curie temperature is less than 30 percent. which all interactions are equal up to the rth neigh- This can readily be understood since the high tem- bor shell, and zero outside. The series expansion

perature portion is derived from averages which methods described in the previous section can be

are not likely to be very sensitive to change in spin applied to this model, and Dalton and I have been value. However, the behavior at low temperatures able to take account of the second and third neigh-

is determined by the excitation spectrum which is bor shells for a number of two and three dimen-

extremely sensitive to the form of interaction. sional lattices [24]. We find that the critical We have referred to the critical behavior of the behavior (e.g., susceptibility index) is identical with

specific heat in the previous section. It is worth that of a nearest neighbor model. pointing out that this critical behavior is more However, we noticed empirically that for large q

dependent on the form of interaction than that of simple asymptotic formulae fit the Curie tempera- the magnetic susceptibility. Although only limited ture and thermodynamic functions. For example, numbers of terms are available, they indicate an appreciable sharpening on the high temperature side as s changes from V2 to infinity; similarly, there are marked differences between the Ising and and (14) Heisenberg interactions [15]. Attention has recently been focused on the be- = Llq - havior of the magnetization as a function of field —k~ 37 Here H and L are constants which depend on the bonds on the lattice. We realized that for a range dimension and the type of interaction. Although of interaction which included many lattice points, this is the kind of asymptotic formula given by these lattice constants could be regarded as lattice

closed form approximations, the numerical values sum approximations to hard sphere cluster in-

of H and L which we obtained were quite different tegrals, many of which have already been evalu- 1 from the closed form values. ated. The efficacy of this approximation can be judged from table 7 which gives typical estimates, We then tried to find a justification for this type these are of limiting form. Series expansions for an equiva- and compared with the correct valuesi based on exact enumerations. lent neighbor model can be expressed in terms of "lattice constants" which are the number of independent multiply connected graphs of various We can thus readily see why the constants kinds which can be formed from the interacting H, L differ for different dimensions since the cor

TABLE 7. Asymptotic formula from cluster integrals

Three Dimensions

3 2 p 5n = 0.05903(g + D - 0.8360 q + l + 3q + 1.25

p 5 r/(calc.) Q p 5 n(calc.) p 5 „(exact) p 5 rt(exact)

6 -0.46 0 12 25.7 36 0.71 18 158.3 174,186 .88 26 637.8 630,690 .97 42 3274.8 3,318 .99

4 3 2 p 5 = 0.02381

5(calc.) / t \ , ^ p q p 5 (calc.) p 5 (exact) p 5 (exact)

6 25.1 0 12 245.8 168 1.5 18 1382.2 1212 1.14 26 6954.0 6852, 6276 1.06 (mean) 42 55418.3 54 564 1.02

38 ) )

responding cluster integrals will then have different For example, taking the interaction between i-j approximation gives the correct ; values. The Bethe pairs as J/rjj in one dimension (assuming unit lat- limiting value as the dimension becomes large, and tice spacing), the series expansion of magnetic large in | not as the coordination number becomes suceptibility is

' a given dimension. Using the above approximation based on cluster kTXo. 1 + 3.290 £/ + 8.659 (pjf + 20.686 (#/ integrals, we find that the simple polygons dominate m

I for the early terms, and these can be calculated

I to any order using the methods of Montroll and 4 5 6 46.411 (/3J) + 99.699 (/3J ) + 207.338 ((3J) . (15) Mayer. The approximation must break down for

higher order terms since it does not give correct This leads us to a Curie temperature critical behavior for magnetic susceptibility, spe-

cific heat, etc. However, it can give valid approxi- j ^0.5ix£ (16) mations for the Curie point and critical values of J «j thermodynamic functions since higher order terms and a susceptibility index of about three. With an make only a small contribution to these. The interaction in two dimensions the correspond- I approach to the mean field limit (linear inverse ing susceptibility expansion is susceptibility, finite and discontinuous specific

heat, etc.) occurs in a singular manner, and this is

illustrated diagramatically for the specific heat 2 1 + 9.033622 (#/) + 76.94741 (/3J) rn- in figure 4.

In the light of the above discussion, it is con- 3 4 5 + 642.168 (/8J) + 5302.1 + 43508 (/3J + . structive to consider the effect of a 1/r'' force for (£/) (17) varying p and this is being currently investigated at King's College by one of my research students, and the susceptibility index is reduced from V4 for G. S. Joyce. It is much more difficult to derive a short range force to about 1 • 13. This change in series expansions for such a law of force since all neighbors must be taken into account. A sub- susceptibility index is demonstrated in figure 5. has also investigated the properties of the stantial simplification occurs for the Ising model Joyce spherical model in one dimension for long range of spin V2 if the method of Yvon (see next section) forces. Whilst one must be cautious in drawing is used to eliminate all but multiply connected con- detailed conclusions about critical behavior from figurations. Joyce finds that a sufficiently small value of p can change critical behavior, and produce critical effects, even in one dimension.

LONG RANGE FORCE FINITE CUTOFF

Figure 4. Diagrammatic illustration of the difference in spe- 5. Susceptibility two-dimensional Ising model with cific heat curve between a finite range force and a force extend- Figure of 3 interaction, ing to infinity sufficiently slowly. a long range 1/R law of force: {a) long range the residual entropy For a finite cutoff the specific heat is always infinite although {b) short range nearest and second neighbor interaction. [14] tends to zero as the range becomes large. (The final slope represents the susceptibility index.)

39 the spherical model, I think it can act as a general T guide to the influence of long range forces on

critical behavior. Joyce finds that there is a crit- 1-8- ical point in one dimension when 1 < p < 2. When 3 —3 1 < p < — the susceptibility index is 1 and when —

For the general spin s and for the Heisenberg model, the calculations are more complicated since

all connected diagrams must be taken into account.

However, if the spin is allowed to become infinite in the Heisenberg model (classical vector model), the conditions of applicability of the second Mayer theorem are satisfied, and only multiply connected clusters enter. The expansions can then be pushed

much further, and this simplification is being currently exploited.

1 0 1-5 20 T* 2-5

Figure 6. Plot of y(T*) against T*(=T/TC ) for the triangular Expansions; Approach to the | Diagram lattice. [27] For each successive approximation passes through a maximum and then falls y j Exact Solution to unity. The locus of the maxima extrapolates to a value 1.75 for T*=l. Thefl limiting curve is derived from series expansions. Early work using closed form approximations provided no reliable information on detailed behavior near the critical point. One of the major defects of this work was the inability to develop a series of successive approximations which would which should tend to 7/4 for a two dimensional and solution. Even when tend in the limit to the exact 5/4 for a three dimensional model as T—» Tc . Figure the methods introduced could in principle produce 6 shows the results for a two dimensional model. such a series of approximations, the practical cal- For each stage of the closed form approximation culations were prohibitive. y(T) rises to a maximum and then falls to the mear

The cluster integral development of Mayer can field value of 1. For temperatures nearer to T, be adapted to the Ising model [25], and can be made than this maximum, the approximation cannot be to furnish a steady series of approximations based regarded as reliable. The locus of maxima tends on all multiply connected graphs which can be to 7/4, and we see that the approach of the closed

constructed from bonds of the lattice. The Ath form approximations to the exact solution is stage of the approximation could correspond for nonuniform. example, to including all graphs of A lines where k We may follow an alternative approach classi- is allowed to increase steadily. It was Yvon [26] fying graphs into groups of increasing complexity who showed how to take all graphs into account, and summing each group to infinity. Let us and not merely regular graphs in which vertices look at the high temperature partition function in and bonds are equivalent. His method involved zero field which can be written in the simple

using the Mayer theory of multicomponent mixtures. form [28] Hiley and I [27] investigated the critical behavior aN of such a series of approximations to see how they 1— In Z.v = /V In 2 + In (cosh K) + ^ pHXf>M approached the exact solution as estimated from series expansions. To focus attention on the sus- (19 ceptibility index, for example, we calculated K = JlkT w = tanhK

40 where the p„x are the "lattice constants" repre- arises for a single self-avoiding walk; for more senting numbers of multiply connected graphs and complex graphs the mutual interference of two the fnxiw) can be determined quite simply. Con- or more self-avoiding walks must be considered. fining attention first to simple polygons which Even if this difficult empirical investigation is com- provide the dominant contribution, we obtain the pletely satisfactory the theory could only be con- function S/J„ln(l + w"). Sykes [29] has suggested sidered respectable if the asymptotic estimates

that for three dimensional lattices are established rigorously. But this at least is a far less formidable problem than the Ising problem itself, and Professor Edwards has taken an p„=^(A~ 1.75) (20) important step recently towards its solution [32].

where /jl is the self-avoiding walk limit. Hence, if Helpful discussion with colleagues and research we ignore all configurations other than simple students mentioned in the text is gratefully ac- polygons, we will obtain a specific heat singu- knowledged. The remarks at the end of the final )~ l/4 larity of the form (T—Tc , and an estimate of section were stimulated by a series of lectures given the Curie point which differs from the correct value at King's College recently by Dr. R. Brout. by about 2V2 percent (the close connection between

I [i and the Ising critical value was pointed out sev- References eral years ago by Fisher and Sykes [30]).

To make further progress we must introduce [1] L. Onsager, Phys. Rev. 65, 117 (1944). C. N. Yang and T. D. Lee. Phys. Rev. 87, 404. 410 (1952). a suitable topological classification of multiply [2] [3] C. Domb, Phil. Mag. 42, 1316 (1951). connected graphs, and here Sykes [31] has sug- [4] C. Domb. II Nuovo Cimento Supplement to Vol. 9, 9 (1958); M. Burley, Proc. Phys. Soc. 77, 262 (1960); D. M. gested using the cyclomatic number c (number of D. Gaunt and M. E. Fisher, J. Chem. Phys. 43,2840(1965). bonds — number of vertices + 1); the types of graph [5] P. Dienes, The Taylor Series Oxford 1931. ch. 14. C. Domb, Advan. in Physics 9, 149, 245 (1960). corresponding to c= 1, 2. 3. and 4 are shown in [6] 7] M. F. Sykes and M. E. Fisher, Physica 28, 919, 939 (1962). figure 7. Even here the number of types increases [8] C. Domb and M. F. Sykes, Phys. Rev. 108, 1415 (1957). A. Phys. Rev. 124, 768 (1961). quite rapidly, but can be substantially reduced by [9] G. Baker, [10] J. W. Essam and M. E. Fisher, J. Chem. Phys. 38, 802 ( 1963). ignoring in the first instance multiple connections [11] D. S. Gaunt and J. W. Essam, Proc. International Conference on Magnetism, Nottingham 1964, p. 88. I between two vertices (thus p. 0, 8, and Q can also [12] G. S. Rushbrooke, J. Chem. Phys. 39, 842 (1963); R. B. be considered basically as one type). Griffiths, Phys. Rev. Letters 14, 623 (1965).

[13] J. W. Essam and M. F. Sykes, Physica 29, 378 (1963). 1 One must then consider the asymptotic numbers [14] C. Domb and D. W. Wood, Proc. Phys. Soc. 86, 1 (1965).

1 of each type of graph to understand how the as- C. Domb, N. W. Dalton, G. S. Joyce and D. W. Wood, Proc. International Conference on Magnetism, Notting- ymptotic value of the coefficients in the high tem- ham 1964, p. 85. 1 perature series is constituted. The constant /jl [15] C. Domb and M. F. Sykes, Phys. Rev. 128, 168 (1962); J. Gammel, W. Marshall and L. Morgan, Proc. Roy. Soc. A275, 257 (1963). [16] G. S. Rushbrooke and P. J. Wood, Mol. Phys. 6, 409 (1963). [17] M. E. Fisher, Phil. Mag. 7, 1731 (1962). [18] R. A. Tahir Kheli, Phys. Rev. 132, 689 (1963). [19] C. Domb and A. R. Miedema, Progress in Low Temperature Physics, IV (1963), p. 296. [20] D. S. Gaunt, M. E. Fisher, M. F. Sykes and J. W. Essam, Phys. Rev. Letters 13, 713 (1964). [21] G. A. Baker, Phys. Rev. 126, 2071 (1962); M. Kac and E. Helfand, J. Math. Phys. 4, 1078 (1963). [22] G. A. Baker, Phys. Rev. 130, 1406 (1963). [23] M. Kurata, R. Kikuchi and T. Watari. J. Chem. Phys. 21, 434 (1953); see also ref. 6, p. 284. [24] N. W. Dalton, Thesis, London (1965); see also ref. 14. [25] G. S. Rushbrooke and I. Scoins, Proc. Roy. Soc. 230, 74 (1955). [26] J. Yvon, Cah. Phys. No. 28 (1945), Nos. 31, 32 (1948). L G H l J K [27] C. Domb and B. J. Hiley, Proc. Roy. Soc. 268, 506 (1962). [28] Reference 6, p. 322, eq (59). [29] B. J. Hilev and M. F. Sykes, J. Chem. Phys. 34, 1531 (1961). [30] M. E. Fisher and M. F. Sykes, Phys. Rev. 114, 45 (1959). [31] M. F. Sykes, private communication. To be submitted N O P Q M to J. Math. Phys. [32] S. F. Edwards, This Conference. Figure 7. [33] C. Domb. J. Royal Stat. Soc. London B26, 373 (1964).

41 Equilibrium Properties of Ferromagnets and Antiferromagnets in the Vicinity of the Critical Point*

G. B. Benedek

Massachusetts Institute of Technology, Cambridge, Mass.

Within the past few years there has been a con- FERROMAGNET: SPONTANEOUS MAGNETIZATION FLUID: COEXISTENCE CURVE siderable advance both theoretically and experi- P Mill mentally in our knowledge of the behavior of M(O) , magnetic systems in the vicinity of the critical H = 0 temperature. The present paper consists of a presentation of some of the new experimental information and its connection with the new theoretical predictions. (a) Some of the equilibrium properties of magnetic systems which are important in the description of FIGURE 1. (a) A schematic plot of the temperature dependence a ferromagnet in the critical region are the following: of the magnetization of a ferromagnet. (b) The fluid analog, the liquid and gas densities along the coexistence curve. 1. The spontaneous magnetization versus temperature in zero external field,

2. the magnetic susceptibility (dA//d//)w=0 = x. 3. the magnetization versus field at the critical

temperature Tc , *DU (l (1) 4. the specific heat at constant volume C , and v \M(0)) H=0 V T 5. the thermal expansion coefficient. In the discussion which follows, we shall present — D' 1 (2) the experimental data and theoretical predictions 2Pc now available on each of the properties listed above. In view of the fact that each of these properties has an analog in the liquid-gas system, as has been The first experimental measurements of D and (3 were reported in 1962 by P. Heller and G. Benedek' emphasized by Fisher [1] and his coworkers [2], we shall make the connection between each mag- [3] through a study of the nuclear resonance of' 19 the F nucleus in the antiferromagnet . netic property and its fluid analog. MnF 2 1 1. The spontaneous magnetization. In figure In table we show the temperature range AT/T(

1 we present a schematic drawing of the tempera- over which (1) was found to fit the data, and the ture dependence of the spontaneous magnetization corresponding values of D and /3. Equation (l)lj 10~ 5 M(T) of a ferromagnet normalized to its value at holds in MnF> over a temperature range 7 X

=S (AT/T ) < 0.1. The closest point measured! r=0, (M(0)), in zero external field (H = 0). Along C critical point. This side this we present the fluid analog, namely the was about 3 mdeg from the \

1 temperature dependence of the liquid-gas density data shows that 0 = 0.333 ± 0.003. This result is field theories which along the coexistence curve. In the critical region in disagreement with molecular

= . of the the magnetization and orthobaric density data can give j8 V2 in the limit T—> Tc In view that of be fit to equations of the form: marked disparity between this result and the classical theories of magnetism, Heller and

Benedek [4] later carried out a careful investigation This research was supported bv the Advanced Research Projects Agency under Contract SD 90. of the insulating ferromagnet EuS. This crystal

42 Table 1. Experimental and theoretical values for the parameters describing the temperature dependence of the spontaneous magnetization of a ferromagnet or antiferromagnet

D (f)\ 1 c /

<-> 5 MnF 2 0.10 7 x 10" 1.200 ±0.004 0.333 ±0.003

EuS 0.08 ^lx 10" 2 1.145 ±0.02 0.33 ±0.015

2 CrBr3 0.04-^ 0.7 x 10~ 1.32 ±0.07 0.365 ±0.015

Molecular field 1.48 (S = 5/2) 0.500 theory 1.44 (S = 7/2)

Ising model 1.49 0.3125

is cubic, the magnetic europium spins are localized only a small three-dimensional admixture. Never- on the lattice sites and each magnetic ion has zero theless, this system behaves somewhat like the orbital angular momentum and S = 7/2. This ideal three-dimensional Heisenberg ferromagnet

crystal represents the closest known physical EuS in that /3 = 0.365. The critical region for CrBr 3 approximation to the ideal "Heisenberg'' ferromag- sets in a A777V ~ 5 X 10"- at which point M(T)/M(0)

i net. The results of this investigation are sum- ~ 0.41.

marized in table 1. The one-third power law was In table 1 we also present the theoretical predic- found to apply also to this system. tions of the Ising [5] and molecular field models.

It is worth noting that the 1/3 power law starts No theoretical prediction has yet appeared for the

to fit the data on EuS and MnF 2 when M/Tc ~ 9 Heisenberg model. It should also be mentioned

percent. The magnetization at this point is about that H. Callen and E. Callen [6] have calculated 1/2 the saturation value. Thus the 1/3 power law M(T) theoretically using a sophisticated cluster applies to the magnetization data over a rather approximation, and report that M(T) follows the 1/3 2: wide range in M, namely 1/2 > (M(T)/M(0)) 0. power law provided that T is not too near Tc . In

In row 3 of table 1 we also present new measure- the limit T—> Tc , of course, their cluster theory

ments of D, f3 and AT/TC made very recently by goes over to the /3= 1/2 value. S. D. Senturia at MIT on the ferromagnetic insu-

lator CrBr.3. This material is particularly interest- As an example of the analogous behavior of a ing from the theoretical point of view because the fluid we point out the work of Weinberger and

ferromagnetic chromium spins are arranged in Schneider [7, 1]. They found that eq (2) fits the 10~ 5 planes, and each plane is separated from the next data on xenon for AT/TC as small as 3 X , and plane by a plane of nonmagnetic Br ions. As a that /3' = 0.345 ±0.015 and D' — 1.61. The van

result, the coupling between the magnetic spins der Waals equation gives /3 = 1/2. In fact, Landau's

is very anisotropic. The exchange constant for theory [8] of the critical region shows that /3 = 1/2 coupling in the plane is 16 times stronger than the follows from the very general assumption that the constant for coupling between the planes. Thus, free energy can be expanded as a power series in this ferromagnet is almost two-dimensional, with density and temperature around the critical point.

43 Thus the breakdown of this law reflects a rather ceptibility of the ferromagnet is defined as unusual mathematical and physical character of the critical point. x = (dM(T,H)ldH)H=o. (3)

From an historical standpoint it should be noted that the departure of from the classical value for As one approaches Tc from the high temperature magnetic systems was discovered at about the or paramagnetic side, x diverges like same time that important theoretical advances were being made. In 1961 and 1962 Domb and x = CI(\-TITc )y. (4) Sykes [9, 10] were able to show from a study of their high temperature expansions that the sus- ceptability of a ferromagnet in the Ising and Heisen- As one approaches Tc from the ferromagnetic side, berg models did not diverge as slowly as was X can be fit to an equation of the same form as (4) expected by the molecular field theories. Also, but one must leave the possibility that the coefficient and exponent have different values Baker [11, 12] introduced at this time the method D y of Pade approximants as an accurate and reliable for T < Tc . Thus, on this side of Tc we write means of extrapolating the high temperature expansions into the critical region. This extrapola- x = C'ia-(TITc ))y\ (5) tion showed [13] that the susceptibility as calculated on the Heisenberg and Ising Hamiltonians diverged In the fluid the analog is the isothermal compres rapidly than that predicted by the molecular more sibility (3t- field theories. These developments and the observation, which was stressed particularly by M. E.

Fisher, of the close similarity in the behavior of (6) systems as disparate as the ferromagnet and the fluid greatly contributed to the present expansion This can also be written in terms of the chemical of work in this field. potential p, using the thermodynamic identity that 2. The susceptibility. In figure 2 we show the along an isotherm temperature dependence of the isothermal sus-

ceptibility of a ferromagnet as T~* Tc . Along side

it is shown the temperature dependence of the (7): analogous property of the fluid, the isothermal bulk modulus along the critical isochore. The sus- Th us

(8)

FERROMAGNET: SUSCEPTIBILTY FLUID: COMPRESSIBILTY As T—>TC from above, the compressibility at p — pc diverges as follows:

(3r, Pc = dl{l-(TITc)f. (9) (T+ ->TC)

Below Tc the i ^levant compressibility is that evaluated right at the coexistence line for either the liquid or the gas

= or d'Kl- (T/TcW. ( 10) fa. p e P,, (T Tc) (a) (b)

FIGURE 2. (a) The temperature dependence of the isothermal We see that in the liquid case, the pressure, or the susceptibility of a ferromagnet as . T-*TC (b) The tempera- chemical potential plays the same role as does the ture dependence of the isothermal compressibility of a fluid evaluated at p = c for T > T and at = or = magnetic field in the ferromagnetic case. p c p p, p pQ forT < Tc . 44 The first comparison between the new theoretical by S. Arajs [17], when analyzed in terms of eq (4) predictions [9, 10. 13. 14] and experiments was also gave a nonclassical value for y. This observa- provided by the work of Jacrot [15] who determined tion was later confirmed by Arajs [18] who has X through a study of the scattering of neutrons measured y from his studies on Fe, Co, and Ni. above. 7 from iron as T—> T, from His measure- His important results on y and AT ,,,/ 7V are given in ments indicated that y=1.3. Stimulated by the table 2. Kouvel and Fisher also examined the departure of y from the classical value y = I, Noakes divergence in the susceptibility of nickel by going and Arrott [16] reexamined their older measure- back to the very early careful work of P. Weiss ments of the susceptibility of iron for T> 7V, and and R. Forrer [20]. This gave y=? 1.35 ±0.02. they also carried out new measurements of this Graham [21] has studied the susceptibility of Gd quantity. They found y— 1.37 ±0.04. This result above the Curie point and concludes 7 = 1.3 but is tabulated in table 2 where we give y for each does not give error limits for this estimate. Fur- ferromagnet which has been studied up till now. thermore, his data indicates that y depends on We also give in column 2 the maximum range over whether the magnetic field is applied along the which eq (4) actually fits the experimental data. c-axis or perpendicular to it.

The range is 0 < A777V < &Tu/Tc , and &T.u/Tc rep- In table 2 we also list in the last three rows the resents the outer limit of the critical region insofar values of y as obtained theoretically using the as the susceptibility is concerned. In their paper molecular field, Ising and Heisenberg models along Noakes and Arrott pointed out that earlier data with literature references to these calculation.

Table 2. Experimental and theoretical values for the parameters describing the temperature dependence of the susceptibility of a

ferromagnet as T—*TC from above

Ferromagnet (ArM/rc) y References

Fe 3 x 10" 2 1.3 15, 13 1.37 ±0.04 16 1.33 ±0.04 18

2 Co 1 x 10- 1.21 ±0.04 18

Ni 2 x 10- 2 1.35 ±0.02 19 1.29 ±0.03 18

Gd 4 x 10-2 1.3 21

Theory

Molecular field 1.00

Ising 1.25 9, 11

Heisenberg 1.33 ±0.01 13, 14

45 It is worth stressing that we do not know of data ing derivative of H (or pS) relative to M (or p). In on y' the parameter which describes the divergence magnetic systems 6 has been found for two materials, nickel and gadolinium. of x as T^> Tc from the ferromagnetic side of the The information on Curie point. nickel was obtained from Weiss and Forrer's 1926

In the case of a fluid, the classical theories give measurements [20] by Kouvel and Fisher [19]. They r=1.0. Because of the experimental difficulties, found 8 = 4.22 ±0.04. Graham [21] reports that data on the isothermal compressibility appears to his studies of gadolinium give 8 = 4, without plac- be scanty and insufficiently precise in the critical ing error limits on this figure. The molecular field region. The measurements of Habgood and theory predicts 8 = 3 while the best present anal- Schneider [22] indicate only that [1] T > 1.1. The ysis of the Ising model in three dimensions strongly

Ising theory for the liquid predicts r=1.25. indicates that the value 8 = 5.2 is the correct the- 3. The critical isotherm. The next equilibrium oretical value for this Hamiltonian. There are as property of a ferromagnet which is of particular yet- no calculations of 8 using the Heisenberg Ham- interest in the critical region is the dependence of iltonian. These results are collected in table 3.

M on H at T— Tc . The divergence of the sus- In the fluid there is considerably more data [23] ceptibility as T^-Tc shows that M is an extremely on the shape of the critical isotherm. The data strong function of H at Tc . In figure 3 we show a on Xe, C02, and H2 indicates [2, 24] that for the 8' close schematic plot of M versus H at Tc . Along side fluid = 4.2 ±0.1. Once again we see the it we present the fluid analog, namely the depend- similarity in behavior between the ferromagnet — pressure for i.e., the and the fluid. Theoretically the van der Waals ence of (p pc ) on the T=TC , shape of the critical p—p isotherm. The data on theory, which is the analog of the molecular field both these systems may be fit to equations of the theory of ferromagnetism, predicts a cubic equa- following form: tion for the p — p critical isotherm, i.e., 8' = 3. The Ising model as analyzed by Gaunt, Fisher, Sykes,

1 S H) = ' and Essam gives 8' = 5.2. We tabulate these M(TC , AH (11) [2] results in table 4. and 4. The specific heat. Of all the physical quan-

tities that vary rapidly near the critical point, the I

l 6 ' i = to delicately. (p-pc )= A'{p-pc ) A'\tJL-^yi6' (12) specific heat appears diverge most r=7v Onsager's theoretical treatment [25] of the two- dimensional Ising system indicates that the specific where p. is the chemical potential. The index 8

. a or 8' is a measure of the verticality of the critical heat diverges logarithmically as T~^TC Such in isotherms as sketched schematically in figure 3, as slow divergence requires very high precision these indices are measures of the lowest nonvanish- the measurement of the specific heat, and accurate temperature control very near to the critical point. A very careful study of the specific heat of a magnetic system has recently been carried out by Skalyo FERROMAGNET FLUID and Friedberg [26] on the antiferromagnet CoCl • 6H2O. These workers find that as one approaches M(TC ) <>P-Pc)

Tc from above (+) or from below (—) that the specific COEXISTENCE heat appears to diverge like CURVE cP(T) L y a In (13) R /h=o Tc (p-p f c ) This behavior begins when (T—TC)ITC is of the order of a few percent. However, in the immediate 10~ 3 5 vicinity of Tc {(T-TC)ITC ~-5X and + 1 X lO"" ) this does not continue. Presumably (a) (b) divergence this is due to the presence of crystal strains and the FIGURE 3. (a) The dependence the of spontaneous magnetiza- resulting coexistence of the antiferromagnetic and tion M on the magnetic field H at the Curie point (T = TC ). (b) The shape of the critical isotherm of a fluid. paramagnetic phases.

46 T

Table 3. Experimental and theoretical values of parameters describing the shape of the critical isotherm for a ferromagnet

Ferromagnet 8 Reference

* NiTVT 4.22 ±0.04 19

Gd 4 21

Theory

Molecular field 3.0

Ising 5.2 ±0.15 2

Table 4. Experimental and theoretical values ofparameters describing the shape of the critical isotherm for a fluid

Fluid 8 References

C0 2 , H 2 , Xe 4.2±0.1 2, 23, 24

Theory

van der Waals 3.0 Ising 5.2±0.15 2

In addition, Yamomoto, et al. [27] have recently the range over which the logarithmic dependence - 3 10"2 surveyed the literature for specific heat measure- applies is 5 X lO" < (T- c)/Tc < 5 X . ments on magnetic systems. They fit the experi- Measurements examining the question of continu- mental data to a logarithmic divergence of the form ance of the logarithmic divergence closer to the of eq (13), and present a table listing the resulting critical point are needed. Along these same lines

values of a and (A+ — A_) for 28 ferromagnetic and it should be pointed out that what is measured ex-

antiferromagnetic elements and compounds. perimentally is Cp and what is calculated theoret-

It may be well to stress that while the data does ically is Cv . The difference is proportional to 2 show that Cp does get large very slowly as T^TC cx //3r where a is the thermal expansion and /3t

47 the isothermal compressibility. It is known that Hamiltonian for the metals is not known. Also, the thermal expansion increases markedly as it would be most desirable to measure several

T—> Tc , but accurate data is not now available giv- equilibrium properties on a single ferromagnet. ing the precise behavior of a{T) and fi(T). In On the theoretical side, the clear need is a cal- view of the slow divergence of Cp it is particularly culation of the magnetization, the critical isotherm, necessary to determine the importance of the the susceptibility below Tc and the specific heat using Cp — C v correction term before reaching conclu- the Heisenberg Hamiltonian. And finally, sions about the precise mathematical nature of the close analogy between the fluid and the ferro- the divergence of C v . magnet, combined with the clear breakdown of In the case of the fluid, while the van der Waals the conventional expansions of the free energy in a series theory predicts no divergence in C v as Tc , the power around Tc suggest that the be- experimental data has indicated [23] that Cv along havior of three-dimensional ferromagnets and fluids near the critical the critical isochore grows very large as T—* Tc . point may be a result of some very Recently the work of Voronel and his coworkers general property connected with the ordering and

[28, 29] has shown that the specific heat at constant is relatively independent of the details of the volume in argon and oxygen apparently diverges Hamiltonian of the system. logarithmically as T-+ T . The data of these work- c References ers has recently been very carefully examined and M. E. Fisher, Math. Phys. 944 (1964). discussed in the light of the best present theoretical [1] J. 5, [2] D. Gaunt, M. E. Fisher, M. Sykes, and J. W. Essam, Phys. information [30, 31]. Rev. Letters 13, 713 (1964). [3] P. Heller and G. B. Benedek, Phys. Rev. Letters 8, 428 Conclusions (1962). [4] P. Heller and G. B. Benedek, Phys. Rev. Letters 14, 71 of ex- Since 1961-1962 a considerable amount (1965). perimental data on the equilibrium properties of [5] J. W. Essam and M. E. Fisher, J. Chem. Phys. 38, 802 (1963). [6] H. Callen and E. Callen, Conference on Magnetism, J. Appl. magnetic systems has been obtained. The be- Phys. (1965), in press. havior of each of these properties in the critical [7] M. A. Weinberger and Schneider, Can. J. Chem. 30, 4222 (1952). region is in striking clear-cut disagreement and [8] L. D. Landau and Lifshitz, Statistical Physics (Addison- with the predictions of the "classical" molecular Wesley, 1958). [9] C. Domb and M. F. Sykes, J. Math. Phys. 2, 63 (1961). field theories of ferromagnetism. The develop- [10] C. Domb and M. F. Sykes, Phys. Rev. 128, 168 (1961). ment of the method of Pade approximants, in com- [11] G. A. Baker. Jr., Phys. Rev. 124, 168 (1961). G. A. Baker, Jr., Phys. Rev. 129, 99 (1962). bination with the high temperature expansions of [12] [13] J. Gammel, W. Marshall and L. Morgan, Proc. Roy. Soc. the partition functions have enabled calculations (London), A275 (1963). G. A. Baker, Jr., Phys. Rev. 136, A1376 (1964). of the equilibrium properties of the ferromagnet [14] [15] B. Jacrot, J. Konstantinovic, G. Parette, and D. Cribier, using the Ising Hamiltonian. The results of such Symposium on Inelastic Scattering of Neutrons in Solids and Liquids, Chalk River (1962). (See also reference 13.) calculations are closer to the data than the "classi- [16] J. E. Noakes and A. Arrott, J. Appl. Phys. 35, 931 (1964). cal" theories, however, by and large the experi- [17] S. Arajs and P. S. Miller, J. Appl. Phys. 31, 986 (1960). [18] S. Arajs, Conference on Magnetism and Magnetic Materials, mental results indicate that the Ising model is J. Appl. Phys. (1965), in press. inadequate to describe a real ferromagnet. The [19] J. Kouvel and M. E. Fisher, Phys. Rev. 136, A1626 (1964). Heisenberg Hamiltonian has been used success- [20] P. Weiss and R. Forrer, Ann. Phys. (Paris), 5, 153 (1926). [21] G. D. Graham, Jr., Conference on Magnetism and Magnetic fully to calculate the divergence of the suscep- Materials, J. Appl. Phys. (1964), in press. [22] H. Habgood and W. Schneider, Can. J. Chem. 98(1954). tibility and the results are in very good agreement 32, [23] J. S. Rowlinson, Liquids and Liquid Mixtures, ch. 3, pp. with the experiment. 93ff. (Butterworth's (London), 1959). B. Phys 1250 (1955). At this point in the development of the subject [24] Widom and O. K. Rice, J. Chem. 23, [25] L. Onsager, Phys. Rev. 65, 117 (1944). certain directions for the future work seem clear. [26] J. Skalyo, Jr. and S. A. Friedberg, Phys. Rev. Letters 13, 133 (1964). On the experimental side there is a need for much [27] T. Yamamoto, H. Matsuda, O. Tanimoto, and Y. Yasuda, more information particularly on the shape of the Department of Chemistry, Kyoto University, Kyoto, Japan. critical isotherm, the specific heat, the thermal (See Proceedings of Conference on Critical Phenomena, Washington, 1965.) expansion, and the susceptibility below Tc . Also, [28] M. Bagatskii, A. Voronel, and V. Gusak, Soviet Phys. we have to have more data on the localized spin JETP 43, 728 (1962). [29] A. Voronel, Y. Chashkin, V. Popov, and V. Sinkin, Soviet systems rather than the ferromagnetic metals. Phys. JETP 45, 828 (1963). In the insulators we may expect that the Heisen- [30] M. E. Fisher, Phys. Rev. 136, A1599 (1964). [31] C. N. Yang and C. P. Yang, Phys. Rev. Letters 13, 303 berg Hamiltonian will apply, while the appropriate (1964).

48 ,

Experimental Studies of Magnetic Ising Systems Near the Critical Point*

W. P. Wolf

Yale University, New Haven, Conn.

The Ising model has long been used as a con- Tc — 2.493 °K is critical (Neel) temperature in zero for field. values for strictly venient approximation the investigation of mag- The £c , B+, and B- are not netic order-disorder transistions, but until recently comparable with those given by any of the theoret- no real magnetic substances were known in which ical estimates since none of these have considered the assumptions of the model coincided with reality. the garnet structure explicitly, but it is interesting

The advent of many new rare earth compounds to note that the ratio B--.B+ is close to the value 3 has changed this position. There are now com- found for other three dimensional structures. pounds in which it is an excellent approximation The "reduced" susceptibility measurements to treat the magnetic ions as effective spins s' = 1/2, have also been used to compare Fisher's predicted whose magnetic moments /x = g-s' are constrained relation for the specific heat [11], C — 4 — (X^' to one direction by an extremely anisotropic g dl tensor [1]. A simple material of this kind is with the observed values [7]. (A is a slowly varying dysprosium aluminum garnet (DAG) which orders function of temperature whose value at Tc can be antiferromagnetically at about 2.5 °K. The prop- estimated from other measurements.) Excellent erties of DAG in the critical region have been agreement was found over a range of temperatures investigated by a number of workers over the past 0.5 TC

49 i

[3] M. Ball, M. J M. Leask, W. P. Wolf, if critical point theories could be extended to in- and A. F. G. Wyatt. J. Appl. Phys. 34, 1104 (1963). clude these effects. Magnetic dipole forces are [4] M. Ball, W. P. Wolf, and A. F. G. Wyatt, Physics Letters 10, 7 (1964). of course present also in all other materials, and as a [5] A. F. G. Wyatt, Thesis, Oxford University, 1963. matter of principle their effect should be considered [6] A. Herpin and P. Meriel, C. R. Acad. Sc. Paris 2 5 9, 2416 even for materials in which exchange interactions (1964). [7] W. P. Wolf, and A. F. G. Wyatt, Phys. Rev. Letters 13, 368 are dominant. (1964). [8] J. M. Hastings, L. M. Corliss, and C. G. Windsor, PhysJ Rev. 138, A176 (1965). [9] A. H. Cooke, K. A. Gehring, M. J. M. Leask, D. Smith, and H. M. Thornley. Phys. Rev. Letters 14, 685 (1965). [10] See for example M. E. Fisher, and M. Sykes, Physica 28, It is a pleasure to acknowledge a number of 939(1962). [11] M. E. Fisher, Phil. Mag. stimulating discussions with Professor Michael E. 7, 1731 (1962). [12] The temperature at which the transition changes from being Fisher. I should also like to thank Dr. A. F. G. first order has recently been determined with greater accuracy at (0.65±0.02)r Wyatt, Dr. B. Schneider, Dr. B. E. Keen and Mr. r by B. Schneider and W. P. Wolf (to be published). Above 0.657V it is apparently D. Landau for making available to me the results higher than second order. [13] B. E. Keen, D. Landau of their experiments prior to publication. and W. P. Wolf (to be published). [14] Substances already known to have Ising-like magnetic- References properties include several rare earth ethyl sulphates (A. H. Cooke, D. T. Edmonds, C. B. P. Finn, and W. P. Wolf, J. Phys. Soc. Japan 17, Suppl. B-l, 481 (1962)) [1] See for example W. P. Wolf, M. Ball, M. T. Hatchings, and a number of the anhydrous rare earth chlorides M. J. M. Leask, and A. F. G. Wyatt, J. Phys. Soc. Japan (J. C. Eisenstein, R. P. Hud son, and B. W. Mangum, Phys. 17, Suppl. B-l, 443 (1962). Rev. 137, A1886 (1965). However all these materials [2] M. Ball, M. T. Hutchings, M. J. M. Leask, and W. P. Wolf, order at such low temperatures that it has not yet been Proc. 8th Int. Conf. on Low Temperature Physics, p. 248 possible to investigate their critical properties in any] (Butterworth, London, 1963). detail.

Specific Heats of Ferro- and Antiferromagnets in the Critical Region

D. T. Teaney

IBM Watson Research Center, Yorktown Heights, N.Y.

1. Introduction heats will be discussed, and a number of other hazards will be described. The prospects of even more refined experiments are examined and found to be encouraging. In this paper we shall describe for the first time the results of high resolution specific heat measurements of two classical magnetic systems: a ferro- 2. Choice of Material and the Lattice magnetic insulator, EuO, and an antiferromagnetic Subtraction Problem insulator, MnF 2 . For completeness we shall also review the earlier results of Friedberg and Skalyo The heat capacity of a typical magnetic system [1] on CoCl 2 • 6H 20, a rather different kind of is shown in figure 1. The anomaly at the Curie antiferromagnet. A logarithmic form of the specific point of EuO [2] (69 °K) is clearly displayed on top I heat is found in all cases, the experimental results of the lattice specific heat, which is of about the! being the most clear in the case of MnF2 . The same magnitude. From the figure it seems that I problem of separating lattice and magnetic specific EuS [3] would be a better case to study, but sufficiently large single crystals are not yet available.

*W,,rk But the choice of supported in part by the USAFOSR of the Office of Aerospace Research material goes beyond the avail- under i:.>ntract AK 4916381-1379 and Contract AF 49(6381-1230. ability of crystals. 50 slope by a factor of two produces only faintly per- ceptible kinks in the highest |AT| data presented here. The apparent magnitude of the magnetic

specific heat, that is, the value of 5, is more directly influenced by the lattice subtraction, but in view of

the extensive work by the above authors it seems

unlikely that an error larger than 10 percent is

introduced in the MnF 2 results by incorrect lattice subtraction.

In the case of EuO our results are only tentative. The diamagnetic isomorphs CaO and BaO are extremely hygroscopic, and we have not been able to make specific heat measurements on reliably dry samples, nor can we find reference to these measure- TCK) ments in the literature. The curve fitting method of

Hofman et al., must then be relied upon, and it is Figure 1. The specific heat o/EuO and EuS. here we wish to interject a word of caution. In order to obtain the entropy from the measured specific heat the usual conversion

At least for a start, it is desirable to study critical c = c (l — AcpT) point phenomena in simple systems which are char- v p acterized by a clearly dominant strong interaction. is required, where the Nernst-Lindeman coefficient In the case of a magnetic system this means we want an exchange interaction that is large compared to dipolar crystal field or effects. From this point kpc-„ of view the lattice becomes an inevitable complica- tion briefly and we now discuss what can be done is assumed to be temperature independent. The about it. value of this coefficient strongly affects the estimate

Fortunately the problem has been investigated in of lattice background since this estimate is based detail for the case of MnF 2 by Stout and Catalano [4] only on high temperature data. (Clearly, accurate and by Hofman, Paskin, Tauer, and Weiss [5]. specific heat data is also essential.) The choice of

Stout and Catalano use a corresponding state meth- thetas is strongly influenced by the slope of the od to deduce the lattice heat capacity of MnF2 from entropy curve because at high temperatures the that of isomorphous ZnF 2 . Hofman et al., use a Debye functions are becoming independent of the method of matching two Debye entropy functions thetas. Since remarkably little expansivity or to the experimental entropy of MnF 2 at high tem- compressibility data exists for compounds it would peratures, having first subtracted an amount be hazardous to guess a value for ,4, and we believe R In (2S + 1). That is, they find that for two Debye recourse to experiment is necessary. We are temperatures, 0=220 °K and 0' = 575 °K. grateful to Professor R. Stevenson of McGill Univer- sity for providing us with compressibility data prior SMnFz(T) -R In 6 = SD(dlT) + 2SD(d'IT) to publication. We have obtained the expansivity

from our own x-ray diffraction study [6], The ex- for temperatures above ~ 2 T\. This procedure perimental value for A for EuO, accurate to about when applied to 10~ 6 ZnF 2 gives a two parameter Debye 20 percent, is found to be 7 X . For compari- curve that fits the 10~ 6 observed specific heat to within son, the value for MgO is 1 X . Tentative theta

1/2 percent in the temperature range 35 to 250 °K. values of 175 and 560 give fair fits to the entropy The results obtained by the two methods are in above 150 °K, and the full scale numerical fitting excellent agreement. program will be completed shortly. Again we It is important to stress that the apparent char- emphasize that the uncertainties in lattice subtrac- acter of the singularity depends only weakly on tion have very little effect on the apparent form of the slope of the lattice background: changing this the specific heat in the critical region. The magni-

209-493 0-66-5 51 50 1 1 1 1 tude of the magnetic specific heat is, of course, very 1 r sensitive to the lattice subtraction, and we refrain from quoting a number for this magnitude until a detailed numerical analysis has been completed.

3. Experimental Technique

We use the standard discontinous heating method in these experiments. The sample is surrounded by a radiation shield the temperature of which could be held constant to about 1 mdeg by referencing a platinum thermometer against a Mueller bridge. Sample and shield are inside a vacuum jacket surrounded by double Dewars of 66 67 68 69 70 71 72 liquid nitrogen. The temperature of the sample is TEMPERATURE - DEGREES KELVIN monitored by a thermister of the type used to sense the level of liquid nitrogen [7]. The thermister Figure 2. Comparison of specific heat data of two powder samples (histograms) with the single crystal (circles) o/EuO. comprises one arm of a constant amplitude 100 c/s 7 resistance bridge stable to 1 part in 10 . The 10" 6 overall sensitivity was limited to about 30 X °K results, we believe that this indicates a similarity by Johnson noise of the bridge detector. The in the rounding of the transition in two different thermister is calibrated in situ against the platinum powder samples and not the presence of some un- thermometer on the shield, equilibrium to within accountable systematic temperature gradient. 5 X 10~ 4 °K being accomplished by condensing liquid nitrogen into the vacuum chamber. The 4. Transition Rounding platinum thermometer is calibrated against the vapor pressure of the condensed pure nitrogen, and While it may not be surprising to find a smearing the less than ideal arrangement for this part of out of the critical point in a powder sample, we the calibration is the cause of the relatively large have two further remarks about transition rounding. systematic uncertainties quoted below. The func- (We feel that "transition truncation" is an expres- tion ZU(T) is used for interpolation [8]. Our ap- sion to be avoided.) The first is an inference as to paratus records drift rate and heating interval data the effect of impurities on single crystals. The directly on punched cards, and all the data analysis position of the transition in the EuO crystal is notice- including addendum corrections and lattice sub- ably on the low side of the distribution indicated by traction is done in a minute or two of computer time the powder histogram. This crystal (1.7 g) con- I [9]. tained filamentary inclusions of tantalum metal, The importance of single crystal calorimetry is amounting to as much as 2.0 percent by volume, indicated in figure 2. The powder results are shown which were observed by microscopic examination as the solid and dashed histograms, the width of of the polished surfaces. The rounding of the

the horizontal steps being equal to the heating in- transition which is suggested in figure 2 and is dis- I terval. The single crystal results are shown by the tressingly evident in figure 3 may well be correlated circles. The rounding of the powder results is with this impurity. A new method of growing crys- not a consequence of temperature gradients since tals has been found to avoid this problem, and hope- the calorimeter is fairly small 3 the monitor- (25 cm ), fully we will have more to say on this point in the I ing power is only 10/xW, the drift rates are on the future. order of 1 mdeg per hour, and the calorimeter re- remark on transition rounding is con- The second |

1 sponse time is about 3 min as compared with an cerned with our results on MnF 2 . These experi- overall discontinuous heating rate of 100 mdeg per ments were performed on two single crystals. One

hour. While a similar experiment with the same crystal (0.02 mole, closed circles in fig. 4) of Bell I calorimeter loaded with a powder sample, the anti- Telephone Laboratories origin, was obtained from

ferromagnet RbMnF 3 (r.v = 83 °K), produced similar Heller and Benedek and is the same one used by

52 |t-t ck) crystal had Ts = (67.23 ± 0.02) °K. (The error limits m | represent confidence in the absolute calibration of the thermometers in each of the two runs.) The

difference is well outside experimental error and also well outside combined uncertainties in locating Ts- The discovery of such a large difference be-

tween two high quality crystals is remarkable. The transition in both crystals was sharper than our experimental resolution of about 10 mdeg K. and we are therefore unable to comment on the rounding which Heller found to amount to about 15 mdeg. The difference in Neel temperatures between the

two crystals is not so large as the difference in Curie temperatures between the EuO crystal and the "best" powder, so that the suggestion made above

|l-(T/T about correlation between rounding and depression c )| of the critical point need not be invalidated.

FIGURE 3. Logarithmic plot of the magnetic specific heat of the EuO crystal. 5. Results for EuO and MnF.

|T-TH |(°K) We turn now to the results themselves. The EuO IP"* KT" 10^ K)' data is shown in figure 3. The rounding is much more pronounced on the low side of the transition

(cf. fig. 6). The Curie temperature was taken as the apparent vertical tangent as seen on a linear plot of the data on the high temperature side. This tangent was apparent to an accuracy of 10 or 20 mdeg,

so that the linearity of the high temperature data is

not the result of choosing a fortuitous Tc within the range of the rounding. The slope of the logarithmic

divergence for T > T is given in table 1. recall ( We

that EuO is a rock salt structure and that the domi-

nant exchange interaction is between nearest ++ neighbor Eu ions [11]. The appropriate coordina-

tion number is therefore 12. The coefficient A+

for this crystal of EuO is seen to be noticeably larger

than that for MnF 2 for which the coordination num-

ber is 8. Such a large difference is not expected theoretically; the somewhat disappointing character of this data may be, however, license for not taking FIGURE 4. Logarithmic plot of the magnetic specific heat of MnF> near the Neel temperature. it too seriously. Open circles. IBM crystal: closed circles. Heller and Benedek crystal. The closed The experimental results for MnF2 are shown by squares are the specific heat density of argon. The dashed curve is for a = 0.1. the open and closed circles in figure 4. A loga- them in their classic NMR experiments [10]. The rithmic form of divergence is clearly indicated over ~ 4 _1 other crystal (0.06 mole, open circles) was *'on the about three decades from 10 to 10 T\. The shelf at our laboratory and is of lost commercial striking resemblance to the configurational specific pedigree. The character of the singularity is seen heat density of argon in the same region is shown by to be identical in both crystals, but we found a sig- the closed squares which are taken from Fisher's nificant difference in Neel temperatures. The Hel- analysis [12] of the results of Voronel' and coworkers ler and Benedek crystal had 7\ = (67.33 ±0.01) °K [13]. Our data covers almost an additional decade in agreement with their result, while the IBM in | A 77 7* | and has a rather better signal-to-noise

53 ~ !

Tabi.E 1. Logarithmic parameters for magnetic and confifiarational specific heats

The specific heat is in units of R and the logarithm is to the base 10.

1 ? 1 attir*** crtic^ FuO MnFo1YJ.11X °£ IA11rerangun 1 j Cl l llv I ii, d o CoClo^HoO

A fi — 1.15 1.15 1.1 (U.2/)

A + 1. 0.6 (0.5) 0.5 0.27 A-IA+ 2.0 (2.3) 2.3 (1.0) B- 0.82 1.16 0.1 0.56 B+ .14 (-0.3) -.3 -.015

ratio the than argon results. The parameters that 6. Results for Co02*6H 2 0 result from fitting the data with a logarithmic divergence of the form C ± /R = A ± \\ogu)\AT/T\\ -\- B ± are We turn now from our work on three dimensional summarized in table 1. Comparison is made with Heisenberg-like systems to the work of Friedberg the argon experimental results and with the theo- and coworkers on CoCl2 -6H 2 0, which has a transi- retical estimates for a lattice gas. In reference 12 tion temperature of 2.29 °K. Since detailed ac- the results of various approximations are listed count of this work has been published elsewhere along with detailed to their origins. references We [1, 14] we mention here only the results of specific have tabulated here only rounded values. It is heat measurements, which are shown in figures 5j clear that while the present experiments reveal the and 6 and in table 1. The principal point estab- character of the singularity, the signal-to-noise ratio lished is that this salt resembles a two-dimensional is not good enough to warrant a discussion of the Ising antiferromagnet. Now that we have results theoretically rather small effects of lattice structure. on a three-dimensional antiferromagnet this re- i It is important to consider the possibility of a semblance becomes all the more impressive. stronger than logarithmic i singularity, particularly CoCl 2 -6H 2 0 forms a layer lattice with the spins on the high temperature side. For purposes of com- at Co + + arranged in planar tetragonal nets. The; a parison a divergence of the form C+ = A \hT/TN \ structure and exchange interactions in CoCl^F^O: with a = 0.1 is plotted as the dashed curve in figure have been described by Shinoda et al., [15]. The 4. Deviation of the experimental points from the Co + + form a tetragonal net, the plane of which logarithmic straight line, i.e., that for which a = 0, also contains the CI". The exchange interaction in is seen to be not inconsistent with a small positive the plane is .//A: = 2.03 °K while that between a. Notice that shifting TN down by 5 mdeg would planes, which are separated by waters, is found to improve the linear fit for T TN . The assign- results by careful theoretical analysis [16, 17] of ment of some definite value to a would place unjusti- the high temperature heat capacity tail as measured fied weight on the noisiest experimental points, earlier [18]. (They have also analyzed their ownl

those closest to . TN We feel that these data can results for CoCl2 -2H 2 0 (TN = 17.20 °K) and find it only be said to suggest that 0=£a:<0.1. Subject to be a chain structure; the resolution of their ex- to the physical reservations expressed in sec. 7, perimental results is unfortunately such that they a much better value of a is not inaccessible to ex- can only be said to indicate a logarithmic singu- periment. Although complicated behavior and/or larity.) In addition to the sheet-like structure of rounding of the transition might be expected in the CoCl2 -6H 2 0, there are two other reasons to suggest next decade closer to TN , improved precision in the its similarity to the Ising model. First, the effec- ++ present range would allow us to set improved limits tive spin of the Co is S = l/2 at these tempera- on a. tures and the moment is highly anisotropic. Sec-

54 f i in two-dimensions [16, 19]. The magnitude of the

/ I

/ 4.4- I logarithmic coefficient is comparable with the values

/ 4.3- I calculated for the two-dimensional Ising model, / 42- I and can even be gotten quantitatively by a suitable 4.1- / choice of anisotropy. While the data fall on a 4.0- / 5 3.9- / straight line above Ts for about 1.5 decades, not O / 3.8 a great deal of confidence may be derived from the

data below TN . However, the specific heat results y 3.6 /. are strongly ^ 3.5 supported by their comparison with < \ 34 '":.-\ magnetic susceptibility measurements [14]. Good ~ 3.3 confirmation of the theoretical relation, [20] C ° 3.2- \ — A 8(x\\T)/dT, is found for experimental points in 3.1 \

; 3.0 a range of ±0.1 TN . A good case is thus made for Z9- believing that outside of the complicated, rounded 2.8- region, the actual 27- specimens measured approximate 2.6- rather well the behavior of an ideal crystal. 2.5- 2.4- _L 7. Lattice Stability 2.2600 2.2700 2.2800 2.2900 T (°K) We have not mentioned in the discussion above

Figure 5. The specific heat of CoCl2-6H 2 0 after Skalyo and the problem of lattice stability at the critical point, Friedberg. since there was no evidence of its being relevant to

the experiments reported. Clearly, if the magnetic

|T-TJ CK) transition is accompanied by a crystallographic

10 10" transition or even a lattice parameter anomaly the 4.5 rigid lattice approximation no longer applies and

4.0 the interpretation of experimental results becomes TN = 2.2890°K complicated. This problem has been considered 3.5 in three dimensions by Bean and Rodbell [21] using

2 3.0 the molecular field approximation, and more re- o 2 V t

!3 2o that in the presence of a strain dependent exchange < FRIEDBERG 8 SKALYO interaction the magnetic transition may become first- 1.5 £ H;!:, v t,:V+H order unless the lattice is "clamped." Extensive T>TN 1.0 experiments [23] on MnAs have demonstrated these effects, and Wangsness [24] pointed out the evi- 0.5 dence for structure change in MnO on the basis of 0.0 specific heat data. In the case of MnF2, the c-axis 10 10 10 10 undergoes a pronounced anomaly which has been

|I-(T/TN )| carefully studied by Gibbon [25]. It has been pointed out that this anomaly could lead to a Figure 6. Logarithmic plot of the specific heat of CoCl2-6H 20. [26] Note that the horizontal scale is the same for all three logarithmic plots. first-order transition, and that, furthermore, the existence of this sort of incipient first-order transi- ond, the earlier c p data [18] reveal a spin entropy tion could complicate the interpretation of a more gain of very nearly (1/2) R log (2S+ 1) above the rapid than logarithmic divergence, particularly for Neel temperature. temperatures very close to 7V The critical point specific heat measurements, As an example of what can happen we would like made on large aggregate specimens of single crys- to present some results of a perovskite antiferromag- tals, are difficult to interpret due to the fairly severe . variation of the net, KMnF :i The temperature rounding. T\ is chosen in such a way as to produce lattice parameters is shown in figure 7. and the a symmetric logarithmic singularity as is expected specific heat [27] in the same range is shown in

55 ;

+1.0

KMnF, DRIFT METHOD AT TN

5 dT,

100 200 300 TEMPERATURE, °K

AT»0.27°K- FIGURE 7. Lattice parameters o/KMnF.i after 0. Beckman and K. Knox, Phys. Rev. 121, 376 (1961). FlGURE 9. Qualitative measurement of the specific heat of

powder sample . of KMnF :) near T N

i —1 1 1 —

KMn F 150 3 about 0.05 °K [29]. Furthermore, no hysteresis was observed in neutron scattering experiments per- 186 "K formed under conditions of even higher resolution

- 88.0 °K ^^"^ Perhaps the fact that is TN [30]. there no anomaly in cell volume accounts for the apparent stability of / '^—TETRAGONAL CUBIC — the magnetic transition. It is tempting to identify 50 the lower specific heat peak with the Neel temperature, but we lack the absolute accuracy necessary to make such an identification positively.

i i i 50 100 200 250 For EuO. the hardness of this material, especially TCK) in comparison with the fluorides, makes questions of stability less important. No structure change at Figure 8. The specific heat of KMnF .,. low temperatures was found using standard powder

x-ray techniques, [6] and Dillon [31] observes cubic anisotropy in ferromagnetic resonance. Since the figure 8. The instability of this crystal is presum- rock salt structure is stable set of ably due to a consequence of the low Goldschmit for the whole europium chalcogenides we would hardly expect any tolerance factor (t = 0.94). The situation may be structure change; the hardness argument should described as follows: the pseudocubic unit cell is - therefore made up of Mn ++ on the corners and F on the be adequate assurance that first-order effects will be small. edges. The size of the cell is dominated by the + Mn-F-Mn bonding, and the K fits rather loosely in its position at the body center. This "looseness" 8. Conclusion has been invoked to explain the anomalously low thermal conductivity of KMnF. { [28]. A qualitative We have shown that an antiferromagnet makes a investigation of the specific heat of a powder sample good gas — perhaps even better than argon. A near the Neel temperature is shown in figure 9. logarithmic divergence of the specific heat is clearly It seems remarkable that the character of the nu- found on both sides of the Neel temperature of cleation on warming and cooling would be so clearly MnF 2 . while the possibility of a more rapid diver- i displayed in a powder sample. The much larger gence on the high temperature side cannot be ex- | lattice parameter hysteresis was observed in the eluded. The specific heat of the ferromagnetic x-ray study of a tiny single crystal. In contrast to insulator EuO. though rounded from below, shows the x-ray and specific heat results, however, a care- logarithmic divergence on the high side of the Curie ful study of the antiferromagnetic resonance near temperature. We have reviewed the work of Fried- T\ produced no evidence of hysteresis larger than berg and collaborators on CoCl.;-6H;>0 which makes 56 the a convincing case for the resemblance of mag- 9. References netic ordering of this crystal to the two-dimensional Ising model of antiferromagnetism. The experi- [1] J. Skalyo, Jr., and S. A. Friedberg, Phys. Rev. Letters 13, 133 (1964). results reported are thus in substantial mental [2] C. F. Guerci, V. L. Moruzzi and D. T. Teaney, Bull. Am. agreement with existing theory. Phys. Soc. II. 9, 225 (1964). V. L. Moruzzi and D. T. Teaney, Solid State Comm. 1, 127 what is [3] It is worthwhile to comment briefly on (1963) . heat yet to be done experimentally on the specific [4] J. W. Stout and E. Catalano. J. Chem. Phys. 23, 2013 (1955). magnetic systems in the critical region. Clearly of [5] J. A. Hofman, A. Paskin, K. J. Tauer and R. J. Weiss, J. better results should be obtained on EuO because of Chem. Phys. Solids 1, 45 (1956). [6] D. T. Teaney, V. L. Moruzzi and J. E. Weidenborner, to be this material; these results should the ideal nature of published. also be extended to include finite applied fields. A [7] We are indebted to Dr. H. B. Sachse of the Keystone Carbon Company for samples of these thermisters and advice on should be studied. Nickel ferromagnetic metal their characteristics. already would be a good case because of the amount [8] J. M. Los and J. A. Morrison, Canadian J. Phys. 29, 142 (1951). about it, but because the difficulties of spe- known [9] The computer program, expressly written to cover a wide cific heat measurements above about 20 °K increase range of laboratory circumstances, is available to interested experimenters. least the third power of temperature, a more with at [10] P. Heller and G. Benedek, Phys. Rev. Letters 8, 428 (1962). rare earth likely candidate may be found among the [11] B. A. Calhoun and J. Overmeyer, J. Appl. Phys. 35, 989

(1964) . metals. As for antiferromagnets, MnF 2 should be [12] M. E. Fisher, Phys. Rev. 136, A1599 (1964). put on measured more accurately and better limits [13] M. I. Bagatskii, A. V. Voronel, and B. G. Guzak. Zh Eksperim

i Theor. Fiz. 728 English translation, for 43 , (1962); Soviet a. Another ideal antiferromagnet is available Phys. JETP 16, 517 (1963). structure remains . perovskite study. RbMnF :! This [14] J. Skalyo, A. F. Cohen and S. A. Friedberg. Proceedings of the Ninth International Conference on Low Temperature simple cubic at low temperatures so that it is un- Physics, Columbus, Ohio, 1964. Good likely that lattice stability will be a problem. [15] T. Shinoda, H. Chihara and S. Seki, J. Phys. Soc. Japan 19, 1637 (1964). results for this system will give some insight into [16] L. Onsager, Phys. Rev. 65, 117 (1944). well as con- the problem of coordination number as [17] A. J. Wakefield, Proc. Cambridge Phil. Soc. 47, 419, 799 (1951). tribute to the general validity of the experimental [18] W. K. Robinson and S. A. Friedberg, Phys. Rev. 117, 402 conclusions. (1960). [19] See, for example: M. E. Fisher, J. Math. Phys. 4, 278 (1963) [20] M. E. Fisher, Phil. Mag. 7, 1731 (1962). [21] C. P. Bean and D. S. Rodbell. Phys. Rev. 126, 104 (1962). We are grateful for the stimulating and enlighten- [22] C. W. Garland and R. Renard. to be published. [23] R. W. De Blois and D. S. Rodbell. Phys. Rev. 130, 1347 ing discussion with P. Heller, G. Benedek, M. (1963) . Fisher, T. Yamamoto, and R. Nathans. The expert [24] R. K. Wangsness, Science 116, 537 (1952). [25] D. F. Gibbons, Phys. Rev. 115, 1194 (1959). preparation of material M. Shafer and C. Guerci by [26] P. Heller, G. Benedek, and T. Yamamoto, private com- have made it possible to study EuO. The en- munication. [27] D. T. Teaney and V. L. Moruzzi, Bull. Am. Phys. Soc. II, 9, the collaboration couragement of J. S. Smart and 225 (1964). of V. L. Moruzzi are greatly appreciated. The [28] Y. Suemune and H. Ikawa, J. Phys. Soc. Japan 19, 1686

(1964) . technical assistance of S. Leddy has made it J. [29] D. T. Teaney, Thesis, University of California, 1960, (un- possible to concentrate on these experiments, and published). [30] R. Nathans, private communication. the programming talents of Miss E. Bergonzi have [31] J. F. Dillon, Jr., and C. E. Olsen, Phys. Rev. 135, A434 been indispensible. (1964).

57 * Nuclear Resonance Studies of Magnetic Critical Fluctuations inMnF 2

P. Heller

Brandeis University, Waltham, Mass.

1 1 1 Atomic nuclei with magnetic moments can serve 1 1 to probe the local fields within magnetic materials. o PRESENT WORK

By studying the nuclear resonance frequency and fi SHULMAN and JACCARINO PHYS- REV 10.S (1219) 1957 line width in the critical temperature region near the - SHULMAN and JACCARINO PHYS, REV [07 (1196) 1957 it to magnetic transition, is possible obtain precise H 0 ALONG C AXIS and detailed information [1, 2] on the establishment of magnetic order. Here we shall be concerned with the effect of the critical fluctuations on the *0.S NMR line width: associated with the approach to the critical temperature Tc the nuclear resonance lines in ferro- and antiferromagnets exhibit a remarkably rapid broadening. A very detailed study of this effect has been made for the case of the antifer-

1 1 l 1 l 50 100 150 200 romagnet IVMs. (Tc = 67.336 °K, also denoted by TEMPERATURE IN OEGREES KELVIN Tn). These experiments shed light on the behavior 19 Figure 1. Temperature dependence o/F nuclear resonance of the electron spin correlation functions in the line width in MnF2 from 4 °K to 300 °K. critical region. In this account we show that the Applied field along c-axis. experimental data implies a "slowing down" of the spin fluctuations in the critical region. A simple 3000 physical model for the time dependence of the correlation functions will then be presented. This model yields a theory which is in satisfactory accord with the observed broadening of the F 19 resonance in

MnF-2 as T approaches Tc from above. The line width anomaly at the critical point 19 for the F resonance in MnF2 is shown in figure 1. shall largely ' We be the situa- ' I 'I concerned with ' 3 0 L^J ' ' OO 10 0.030 0.100 0.300 1.00 3.00 10.0 tion in the paramagnetic state . T -T - IN T > Tc The de- NE EL DEGREES tailed behavior of the line width in the region COMPARISON OF THEORY WITH EXPERIMENT, T>TN Tc + 0.006 °K < T< °K is shown by the 7V+10 12 1 x = 2.7 x 10 sec." experimental points in figure 2. (The drawn lines 19 Figure 2. Experimental F line widths in MnF-> just above Tc . correspond to the 12 -1 theory to be outlined below.) Drawn curves represent theoretical results with \ = 2.7 X 10 sec . The line -width is anisotropic with respect to the direction of the externally applied d-c field H>. For Ho along c (the antiferromagnetic axis), the

Just above Tc , the line widths level off at the values line broadens by about a factor of ten as Tc is approached. In the region of severe broadening, 1100 kc/sec the ratio of the line width for H 0 along the c-axis 8vc(Tc)= to the line width for H 0 along the a-axis is very nearly 2 to 1, 8i/,X7'c)=550kc/sec. (2)

dpJ8v n = 2. (1)

One observation, made at 200 mdeg above Tc , "This work was supported by the Advanced Research Projects Agency under Con- that line monotonically tract SD-90 at the showed the width increases Massachusetts Institute of Technology. Also reported in part at the International Conference on Magnetism, Nottingham, England, 1964. as Ho is turned from the a to the c direction. At 58 77 °K, where the line is relatively narrow, the line ence of the spin correlation functions becomes

width is essentially isotropic. "slow" near Tc . Let

We would like to understand these results. One

f possible mechanism for the line width broadening < {&S ( )8S (0)} > G^t)- M M 2 ' will now be stated and then shown to be incompat- <[SSM (0)] > ible with the experimental facts. Suppose that as

Tc was neared, the electron spins in the sample /x = where 1 , 2, 3 denotes a Cartesian component, became permanently ordered to a degree which and define correlation times rM by writing varied from one region to another. This would result in a variation AH in the time average local field seen by different nuclei. We assume the direc- tm = G»{t)dt. tion of antiferromagnetic spin alinement within each Jo region to be along the c-axis, the direction of anti-

ferromagnetic alinement below Tc . The direction Then (3) becomes of AH will then also be along c. The nuclear resonance would then be inhomogeneously broadened.

This broadening would be severe if the applied field 1 (A = 2 8p tz < (8S > hand, if Z ) H ( i were colinear with AH. On the other Ho were perpendicular to AH, the observed broadening would be of the second order in |A//|. A

simple calculation shows that if this were the 2 + Tr < (8SX ) >+\r < (8Syf > (4) mechanism responsible for the line width, the aniso- \ y tropy factor 8vcl8v a would be very much greater than

2, in contradiction with the experimental results. The experimental fine width was found to be iso- The line width is therefore to be attributed to tropic above Tc + 10 °K. We then surmise that in fluctuations in the local hyperfine fields seen by this temperature range the spin fluctuations are the nuclei; it can then be related to the electron isotropic and tx = Ti, = tz = t. If the correlation spin correlation functions in a way that may be times were constant as T—> Tc we would have expressed quantitatively as follows. If the interaction between a nucleus and an electron spin has

the form ^f—Al-S, where A is the hyperfine cou- < {8S z f > pling constant, and if z is the direction of the average local field at the nucleus and furthermore

if the correlation time for the spin fluctuations is 2 2 + ^<(§5J.) > + ^<(SS, > short compared with the nuclear Larmor frequency, y )

then [3, 4]

But for T>TC , = 0, and the following iden- 1 tity holds 8v- (4 dt < {8Sz(t)8Sz{0)} > 7T V3 W Jo 5 7 2 2 2 = < (8SX ) > + < (8S y ) > + < (8SZ ) >=S(S + 1) 2 2 {8S (t)8S > + {8Sy(t)dS > x x(0)} l< y(0)} (3) From this it may be seen that the temperature dependence of the time independent correlations Here 8S(t) — S(t) — < S > denotes the departure of 2 < (SS,x) > cannot be responsible for the tenfold the spin from its thermal average value and {AB} = increase in line width observed as T^> Tc . The f(AB + BA). correlation times evidently increase in the critical We now show that in order to explain the line region. width anomaly within the framework of the for- The observed fine width anisotropy is now inter- mula (3), it must be assumed that the time depend- preted as follows. Denoting the crystal axes by

59 c, a, and a', the line widths for the average local spin deviations decay exponentially with time, i.e., field along c and a are given respectively by < {ds^wdstfl)} > = < |ss^(0)| 2 > e-ry™. T < (8S 2 > C C ) (7)

~~2 2 + | T« < (8S„) > +| Ta< < (SS„>) > Using eq (7) and (6) in eq (3) it follows that~~

~~2 1 (AV 1_ < |gS,-c(0)| > and 8vc = ta/3 w n rc(Ao 2 ra < (8S n ) > 2 2 1 < |5S/,-„(0)| > 1 < loSA-q'(0)| > (8) 2 ra(k) 2 ra,(k)~~

~~2 2 r (8S < (8S <) > + | c < C ) > +| TV a~~

when the average local field is along the c-axis. Therefore, the factor of 2 anisotropy actually ob- If the average local field is along the a-axis one interchanges c a. served just above Tc implies that at those and temperatures The tenfold increase in the fine width that is to as from can 2 observed occur T—> Tc above TC <(8SC ) > >Ta<(dS n f>. (5) 1 —>• now be understood as follows. As 7 Tc , 2 the quantities < |8Sa c(0)| > become very large theory of the electron spin correlation functions A while the corresponding decay rates rc(&) become in the critical region was given by Moriya [5]. very small for wave vectors k near the antiferro- expressed The physical content of the theory may be magnetic made ko- This corresponds to the physi- in the following terms. Just above Tc there exists cal picture of increased correlation range and a short range antiferromagnetic correlation between cluster lifetime near Tc . the spins. Roughly speaking, the spin arrangement To make a quantitative theory of these effects at any instant of time consists of clusters of anti- we thus need to know the temperature dependence ferromagnetically alined spins. The time average of both the quantities < 8Sa 2 > and T^k). In | m | value of the spin at any site is zero, however, and his theory, Moriya used the fluctuation-dissipation so these clusters are constantly forming, dissolving theorem to relate <|8Sa 2 > to the wavelength m | and reforming. The average size and lifetime of dependent d-c susceptibilities. The relation is

the clusters increases as Tc . This latter point, k T which corresponds to a "slowing" of the 2 B < |8SA- > = (9) M | spin fluctuations is an essential part of the theory. The direction of antiferromagnetic alinement within

the clusters is mainly along the c-axis; this is re- where ks is the Boltzman constant, [An the Bohr sponsible for the inequality (5) and the consequent magneton, and g—2. The susceptibilities Xm(^) fine width anisotropy. can be calculated on a molecular field model. The

results are given by Moriya [5]. On the basis of this picture it is clear that we should rewrite eq (3) in terms of the Fourier trans- To calculate the decay rates V^k), Moriya em- form spin deviations 8Sk(t). These are defined in ployed a technique developed by Mori and Kawa- terms of the ionic spin deviations SSj(i) as follows saki [6] for describing spin diffusion in ferromagnets. Here, however, we shall employ a very simple

physical model. According to this view [7], the (6) decay of a staggered spin configuration is inhibited by the staggered molecular field generated by that configuration, an effect which leads to a particu-

The theory is greatly if simplified we assume that larly slow decay near Tc . On this model the decay the correlation functions for the Fourier transformed rates are related to the wavelength dependent d-c

~~60 k susceptibilities as follows Thus the "staggered susceptibility" is effectively~~

~~(T) r /; \ \ Xcurie l — - Xcurie ^(K) A =r, (10) = (13) XM«, T) Xs i-/oy~~

~~(13) with (12) find where A is a temperature independent parameter, Comparing we and~~

\ XCurie g2 ^ 2 S(S+l) Xs Xcurie(^) : (11) SkBT The same argument may be applied to relate the relaxation of spin arrangements of arbitrary wave- is the Curie susceptibility. length to the corresponding susceptibilities. We The motivation behind eq (10) is as follows. thus obtain eq (10). In applying this relation to a Imagine that at some temperature T > T we arti- c calculation of the time dependence of the correla- ficially produced a "staggered magnetization" M s tion functions, we are assuming that the time decay by applying a staggered field Hs . If H$ is turned in the Ath Fourier component of the magnetization off, eventually relaxes to zero. Consider the Ms which occurs as a result of spontaneous fluctuations situation at the instant that Hs is turned off. Ac- is similar to that which characterizes the return to cording to the viewpoint of the Weiss theory, where equilibrium following an artificially produced non- the coupling between the spins is replaced by an equilibrium state. effective field, the spins would find themselves The line width may now be expressed in terms in a staggered "molecular field." We suppose this of the wavelength dependent susceptibilities and field to be strong enough to produce a staggered the parameter A. Using (9), (10), and (11) in (8) we magnetization at equilibrium. Evidently M* find

M* =f{T)Ms where/fT) is a fraction which becomes 2 1 1 A 1 kB T 1 equal to unity at Tc , that being the temperature at 8vc = 2 2 which the magnetization becomes self-sustaining. XirVzH \g fJig' S(S+1)N As a model to describe the decay of the staggered magnetization we now write the following differ- 2Mi(*)+xi(ft)] k ential equation

~~(14) dMs = -m*-M*) = s -\[\-ffl\M 2 dt -1 1 A ( B T 8va = X7TVSt 2 \g2 fJLp S(S+l)N rs' This leads to an exponential decay Ms =Ms (0)e where xP)+§xI.W~~

r,=A[i-/(7U (12) where x\\(k) an ^ XJ.W denote the components of the wavelength dependent susceptibilities respec- Now let us calculate the staggered magnetization tively parallel and perpendicular to the c-axis.

Ms produced at equilibrium by a staggered field Hs . The discussion so far has assumed that each In the absence of the molecular field we would have fluorine nucleus is coupled to a single manganese ms = HsXcurie where Xcurie is given by (11). In the spin. In actual fact, there are three spins to which presence of the molecular field the coupling is important [8]. We denote these by S, S', and S", their positions in the lattice by

Ms = HsXCune+ AT)MS r, r', and r" and the corresponding hyperfine coupling tensors by A, A' and A". We then replace the or coupling AI S in the preceding discussion by the coupling I-(A-S + A'-S' + A"-S"). The line width formulae (14) then become revised in a way which Ms HsXc, can be expressed compactly as follows. For each

~~61 wave-vector k construct the tensor Also~~

~~2 iAw ' "oik-r" = ikr ' A"e (15) g /4S(S+l) A(k) Ae + A'e + XAko+q)- ; 3knTc~~

~~For the present purposes, the principal axes can be taken to be along the c = [100], x = [110] 1 (19) and [ lTO] directions. Then the line widths can y= 8 2 2 2 8 + anis +ka () + c q ] be written~~

~~ti-z 1 kB T 1 ( 8 = 0.02 is ratio = where anis . roughly the of the dipolar 8vc 7= I 2 A 77V3\g ^/ S(S+1)N anisotropy energy to the exchange energy. Note~~

2 that not diverge at while \A Ak)\*+\A (k)\ X-l(^o) does Tc , xil(^o) does. 2 x yy 2 \A cc(k)\ xn(k) + X M Using (17), (18), and (19) in (16), the temperature dependence of the line widths is found to be

~~(16) asymptotically~~

~~1 (k T s B 14 8vc — S(S+1)J« AW 3 V S(S + A3 5| /r~~

~~2 2 3 \A xAk)\ +\A tw(k)\ 2 1 2 \A cc(k)\ x\\(k)+2 A±Xl(k) [S-^ + O.lSfS + ' 2 Sa^)- ]~~

Since the modes k = k 0 make the most significant 2 1 A(k) A(& ). 4 2 \A cc(ko)\ contribution near Tc , we can replace by 0 8^ = 7^ \/4 S(S+l) The numerical values of the tensor components can A3 V3 be obtained from the electron spin resonance 1 2 ++ [- 8-^ 0.26(8 S )- / ]- (20) experiments [8] on Mn in ZnF 2 and may be ex- + + anis pressed as follows

~~The quantity A, which we regard as an adjustable = Mc/s parameter, is now chosen to fit the line width data | |J cc(A:o)| 160 for 7V + 0.015 °K < T < 7V +1 °K. We find~~

~~2 2 s_y\A ko)\ +\A (k 0 )\ 12 1 xA yy A = 2.7X 10 sec" . =m Mc/s)2 (17) (21) h ] 2~~

The fit is shown in figure 2, the solid lines corre- where S — 5/2. sponding to eq (20). Some deviation in the fit is To obtain line width formulae which are asymp- apparent above 7V+1 °K. This is attributable to to totically correct as f-^ Tc , it will be sufficient the use of the approximate expression (18) and (19) express the wavelength dependent susceptibilities and the replacement of A(Zr) by A(&o) in (16). Both in a form valid near the antiferromagnetic mode X(k) approximations tend to underestimate the line = molecular k0 . Letting k k 0 + q where < h, the \q\ width at temperatures where contributions from field approximation yields the following results modes k not very close to ko become important.

~~above Tr : A more serious defect in the theory occurs in~~

the immediate vicinity of Tc where eq (20) predicts 2 1 -1 g nlS(S+l) ' 2 = divergent line widths proportional to S . This + <7) X||(&o MbTc 2 2 2 2 2 contrasts to the constant experimental values re- 8 + l[a (q + q ,) + c q ] ported in eq (2). The infinity in the theory arises the suscepti- (18) from divergence at Tc of the staggered bility XI|(^o, T). We now suggest that in reality

~~Here a and c refer to the unit cell dimensions, and this quantity does not become infinite at Tc , but is merely very large over some narrow range of tem-~~

~~T-Tr peratures centered on Tc . In other words we sug- 8 = gest that the critical point is not perfectly sharp,~~

~~62 there being a distribution of critical temperatures just calculated. Thus the divergent theoretical~~

~~of width 8TC . Such a situation might well arise formulae, when modified for the experimentally when on account of crystalline strains, different observed Curie point distribution, yield a good~~

parts of the sample have different Curie points. account of the line width broadening above Tc . We do not wish to assert that this is really the Let us see whether our model eq (10) for the source of the broadening, however. In any case, relaxation rates is still valid well above the critical point. a distribution of Curie points should lead to two For T>TC , X(k) = XCuTie and hence F{k) = \ other experimentally observable effects near Tc : for all k. Then the decay rate for the single-ion

(1) There should be a coexistence region in which correlation functions is also A.. This enables us 19 nuclear resonances corresponding to both the para- to compute A. from the F line width observed at magnetic and antiferromagnetic phases are simul- high temperatures [9], taneously observable.

(2) The NMR lines observed in the antiferro- 8v x =170 kc/s. magnetic state T < T, should be inhomogeneously broadened. The line width should be maximum The result is when the average local field is along c and mini-

12 1 mum when along a. Furthermore, the anisotropy k= 1.8 X 10 sec" . (24) factor 8vmaJ8vmin should be greater than the 2:1 ratio associated with fine broadening by fluctuating Comparing this with the result (21) deduced from fields. the critical fluctuations, we see that the model Both of these effects were observed and both eq (10) gives a satisfactory picture of the fine width were consistent with a Curie point distribution of behavior above Tc . width We now turn briefly to the situation below Tc , where the line width data is shown by the experi-

8T( = 17 millidegrees. (22) mental points in figure 3. In the antiferromagnetic state, the average local field We now show that such a distribution would at the nucleus HNUCL is the resultant of the applied field and the field result in a finite observed line width at the (aver- produced at the nucleus by the antiferromag- age) Curie point Tc , and we calculate this width. netically ordered ++ spins: At that temperature, the observed line correspond- Mn ing to the paramagnetic phase will be due to those 19 members of the distribution which have Curie nucl— H0 + HMn++ . points 7^ somewhat below Tc . The larger the difference — the Most of the observations were with Tc Tc , narrower and hence the stronger made H 0 and will be the contributed NMR line. Roughly speak- hence H NUCL along c. Some observations were ing, the observed line should then be due to those made with H 0 so oriented that H NUCL was along a. members of the distribution for which Tc — Tc

~~= 2 8TC . The observed line width should then be given by (20) by inserting the value MEASUREMENTS~~

~~= *|?£=1.3 4 S xlO- .~~

~~When this is done, using for A. the value given in - THEORY (21) we find INHOMOGENEOUS BROADENING FOR~~

~~H NU(. L I|C DUE TO ASSUMED NEEL oV c =1200 kc/s TEMPERATURE DISTRIBUTION~~

~~I I . , I I (23) I I 0.003 O.OI 0.03 0.10 0.30 1.0 3.0 8v„ = 600 kc/s~~

in agreement with the experimental values reported Figure 3. Experimental 19 F line widths below Tc . Curves drawn through experimental points represent smoothed data. Other in eq 2. The theoretical line width curves in drawn curves represent theoretically calculated line width for along a HNm . and inhomogeneous broadening with along c, calculated for a Curie point distri- HNm . figure 2 have been "leveled off" at the values (23) bution of widht 6T, = 17 mdes.

~~63 1 r— i 1 i 1 'I 1 1 the molecular field model to calculate the wavelength dependent susceptibilities, the theoretically~~

~~calculated line width is found to fall off much more~~

_ T< T H SMOOTHED DATA CORRECTED FOR FOR INHOMOGENOUS BROADENING rapidly with decreasing temperature than the experimentally observed width. This discrepancy is - illustrated in figure 3, where the theoretical line

~~- width for along a is compared with the data. HNLn . T>T \ M SMOOTHED DATA \ X. At present, therefore, we do not understand the~~

~~line width behavior below Tc.~~

~~200~~

~~1 i i i i i il i 1 i i i i ul i~~

~~0.003 0.01 003 0.1 0.3 10 3.0 100 author is deeply grateful to Professor I G. B. T-T„ I IN DEGREES KELVIN The Benedek for many important comments and sug-~~

4. Line widths along c above and below T . Figure for H Nlcl . c gestions, to Professor T. Moriya for making avail- Line widths for T< T, have been corrected for the inhomogeneous broadening due to I he Curie point distribution. able his theory prior to publication, and to Dr. Y. Obata, Dr. T. Izuyama, and Dr. K. Kawasaki for this condition was satisfied the lines When NMR many helpful theoretical discussions. were quite narrow. This is because the inhomogeneous field AH arising from the distribution References of Curie points is along c; only a second order broadening results with HNUCL perpendicular to AH. [1] P. Heller and G. B. Benedek, Phys. Rev. Letters 8,428(1962). [2] P. Heller and G. B. Benedek, Phys. Rev. Letters 14, 71 (1965). An upper bound on the second order inhomoge- [3] R. Kubo and K. Tomita, J. Phys. Soc. Japan 9, 888 (1954). T. Moriya, Prog. Theor. Phys. 23, 641 (1956). neous broadening for H NUCIj along a was calculated [4] 16, [5] T. Moriya, Prog. Theor. Phys. 28, 371 (1962). by postulating that all of the observed broadening [6| H. Mori and K. Kawasaki, Prog. Theor. Phys. 25, 723 (1962). for H NUCL along c was inhomogeneous. This upper [7] P. Heller, Thesis, Harvard University 1963 (unpublished). [8] A. M. Clogston, J. P. Gordon, V. Jaccarino, M. Peter, and bound was considerably smaller than the line L. R. Walker, Phys. Rev. 117, 1222 (1960). width 8v„ observed for H NUC l along a. We conclude [9] R. G. Shulman and V. Jaccarino, Phys. Rev. 108, 1219 (1956). that the 8v„ line widths shown in figure 3 do not reflect the inhomogeneous broadening. They are Discussion due presumably to fluctuations.

It was possible roughly to correct the data for G. S. Rushbrooke: May I ask Dr. Domb how summing over further from applying along c for the broadening. graphs besides ring graphs differs fundamentally HNLJC | inhomogeneous say the Percus-Yevick equation to the Ising model, which one This was done in two ways as follows: knows is not very good?

(1) The corrected value of 8vc was set equal to C. Domb: The approximation which I propose attempts to find twice 8v„ at those temperatures where the 8v„ an asymptotic representation of polygons and other graphs on data was available. the lattice. In terms of Mayer theory it corresponds to summing all cluster integrals which can be formed using all the bonds Assuming that (2) the 8vc line widths observed of these configurations on the lattice. Thus even in lowest order

it it still account just below Tc were entirely inhomogeneous, the when is concerned only with polygons takes into a much more complex set of integrals than those of simple ring width of the Curie point distribution to was found type. In this respect the lowest order approximation is the gen- be about 17 mdeg. Then the inhomogeneous part eralization to infinity of the ring approximation of Rushbrooke and Scoins. It seems to me that approximations like those of of the broadening was calculated as a of function Percus-Yevick are derived by summing Mayer ring integrals, temperature and "subtracted off' from the observed and make no serious attempt to estimate asymptotically the number of polygons on the lattice and the corresponding excluded line width. volume effect._ The estimated line width for H NlJCL along c, cor- M. E. Fisher: I would like to make a comment on the point that rected for the inhomogeneous broadening is shown Dr. Rushbrooke raised. It is in the spirit of the Van der Waals as a function of \TC — T\ in figure 4, together with equation to take into account the short range repulsion exactly. the actual The short range repulsion with excluded volume effects is also a 8vc line widths for T > Tc . Note that difficult problem, but the point is that it is possible to split the the corrected line is width continuous at Tc. problem this way. This approach using the data from lattice The theory described above, which gave a satis- models represents it, one hopes, in a better way. factory account of the behavior above Tc , can be C. N. Yang: Dr. Domb said he believed that for some three used to calculate the line widths for T < Tc . Using dimensional lattice models the specific heat singularities below

64 the critical temperature are logarithmic. What is the constant in As the averages are calculated by some kind of approximation front of the logarithm? the two methods do not lead to the same result. However it is possible to make a guess as to which of the two methods is better at low temperature and which one better at high temperature. C. Domb: Unfortunately, I have not got the figures at this mo- One therefore introduces a parameter and says that the total ment, but I can easily provide them at a later stage. Sz is a linear combination of (2) and (3), the ratio being (V2S) (S2/SK. This has been investigated for various values of the exponent x. At first we had carried out the method for B. R. Livesay presented measurements of the magnetization of [5] x= 1 but at the meeting in Kansas City in March 1965 Copeland thin nickel films near the Curie temperature [1]. and Gersch indicated that they had investigated the method for general x, and that for x = 3 the theory leads to excellent results C. Eisenstein: I would like to report some measurements on J. [6J. For the shape of the critical isotherm, Copeland and Gersch NdCla, which have been carried out by Hudson and Mangum reported 8 = 4.22, whereas various experiments that have been here at the National Bureau of Standards [2]. This substance reported here give 8 = 4.2 ±0.2. For the coefficient of the sus- has highly anomalous magnetic properties. It has two extremely 4 ceptibility above the Curie temperature they reported y — /a sharp spikes in its susceptibility at approximately 1.0 and 1.7 quite accurately. As my brother told you, all Green's function susceptibility the =1 °K. With regard to the dependence of the on theories give /8 /3, but only at some distance below the Curie temperature in the vicinity of the 1.7 °K peak, we found that temperature (much further below than Dr. Benedek found). y' the coefficients y and are approximately the same. How- In fact, one finds that the value of (i changes from V2 close to ever, they are not 1.3, but approximately 0.69. Also the sus- the Curie temperature to fi= lh some distance away from the ceptibility of CeCl3 was measured [3]. It was found that the critical temperature. Incidentally this is very reminiscent of susceptibility of this substance has a very sharp spike at 0.345 the data in He3 that Dr. Sherman presented, where they found °K. We think CeCl3 has a second susceptibility peak at a lower experimentally a narrow region where the coexistence curve temperature, but it has not been located yet. For CeCl3 the follows a V2-power law going over into a V3-power law beyond value of y on the high side of the transition is about 0.61. The this region. Finally, I remind you that the coefficient D in (1) measurements on NdCl3 extended from about 5 to 55 mdeg on is correctly predicted to within 2 percent. either side of the transition. For CeCl3 the range was 10 to Anonymous: How is the value 3 chosen for the exponent xt 100 mdeg.

M. E. Fisher: By comparison with experiment? E. Callen: There exists some theoretical analysis of the Heisen- berg Hamiltonian. If one looks for the asymptotic behavior, H. B. Callen: Copeland and Gersch find the magnetization to any cluster theory or Green's function theories give the value be double valued if one takes x in the range 1 < x < 3. The j8=V2 . That is in selection of x = 3, rather than x > 3. is on the ad-hoc basis of the results cited. M = D(\-TITr f (1)

the behavior of M asymptotically at T=TC is given by f3=V%. G. E. Uhlenbeck: I would like to make a comment with regard On the other hand, however, if one is content with some sort to the remarks of Dr. Callen. I am not in favor of saying that of a numerical procedure, it turns out that many results are al- the theories you mentioned have not led to interesting results.

ready given by simple models. In particular, if one carries out However, I do not think it is right to treat them on the same level the variational procedure as described by Dr. Livesay, in which as the classical theory. The classical theory corresponds to a one considers dMjdT as a function of /3 and takes the value of definite model namely long range forces, while the theories you j3 which gives the best fit to the magnetization curve, the Green's =1 mentioned contain what I would call uncontrolled approximations. function theories give /3 /3 for a broad temperature range. Also the cluster theory including nearest-neighbor and next- nearest-neighbor interaction leads to the V3 power [4]. Further- H. B. Callen: At the risk of a rebuttal which may call forth an- more, by taking the nearest and next-nearest exchange constant other rebuttal, I remark that the molecular field theory only from experimental data without using any adjustable parameter, became an exact model many years after it was phenomenolog- the cluster theory predicts the observed value of the coefficient ical. That is, the model for which it is exact was not discovered D in (1) within 2 percent. Although these theories are not cor- until long after the theory has been presented. I imagine that rect for any so called phenomenological model, one can find, if one is asymptotically at T=TC , numerically one obtains rather clever enough, some obscure model for which it is a rigorous good results from the present existing theory away from Tr . Perhaps only higher and higher sophistication will extend this solution. In the case of the molecular field solution one contrives

range of validity to temperatures closer and closer to T,-. a model in which every spin interacts equally strongly with every other spin. It seems to me to be a technicality to call that a physical model, and to make a distinction between that theory H. B. Callen: I would like to amplify that. There has been a and other phenomenological theories. great deal of discussion, it seems to me, about rigorous theories. Nevertheless, the molecular field theory, the cluster theory and L. Tisza: The internal field theories can be discussed from two i other phenomenological theories have historically turned out points of view and these are not in conflict if properly distin- to be very convenient. These theories to my mind have not guished. We may search for models or types of interactions 1 been considered sufficiently in the discussion so far. As my for which such a theory is more or less adequate, or else, we may brother just observed, they are in fact considerably more suc- consider the method as an approximation to the real state of I cessful than they have been given credit for. I would like to its point breakdown. I wish to review very rapidly some of the successes these theories have matters and observe of make the point of view. It is possible to for- had, choosing specifically a form of Green's function theory. a remark from second mulate the statistical thermodynamics of equilibrium in a fashion If one asks for the equation of motion of a spinwave, one has finite appears to with its infinite to take into account the interaction between two spinwaves. that a system be coupled en- 1 vironment. The system is specified by additive random vari- This interaction is described by the appearance of Sz in the ables the distribution of which contains the intensities of the Hamiltonian [5]. Now one can represent S in either of the , z environment as parameters. The theory is still valid for small < , following two forms [5]: systems, but it fails for critical points where an important + Jacobian vanishes [7]. This argument justifies the use of an S* = S-S-S (2) internal field as the intensity of the environment of a small or open system, independently whether the interaction is of short or long range. However, the theory fails near the critical point + + ! S Z = »/2(S S--S-S ) (3) and here more incisive methods have to take over.

65 O. K. Rice: I would like to say something about the accuracy C. Domb: Nevertheless, I think the susceptibility might possibly with which one can measure ft. As was mentioned by Dr. Row- be manageable. linson, Thompson, and I [8] measured the coexistence curve of the binary liquid system perfluoromethylcyclohexane-carbon- P. W . Kasteleyn: I have been asked to report some measure- tetrachloride. We had a temperature control to about 0.0001 ments which have been performed by Dr. W. J. Huiskamp, °C and we measured a curve which extended to about 0.02 °C Dr. A. R. Miedema and Mr. R. F. Wielinga in the Kamerlingh from the critical temperature. We obtained data which looked Onnes laboratory in Leiden. Measurements were carried out as good as the magnetic data under discussion. However, we on two groups of salts, one group representing the Ising system =1 tried to distinguish between the value /3 /3 and the value [9] and the other group representing the Heisenberg model [10]. 5 /3 = /i6 that Fisher suggested. For this purpose we plotted The first group consists of cobalt potassium sulfate and cobalt our data both ways. We figured we did not know the critical cesium chloride, which have very anisotropic g-factors and which temperature better than within 0.0001 °C, so that we had a small are neat Ising systems also in other respects. It was found, range in which the critical temperature could be adjusted. We e.g., that for the first salt y, the index for the susceptibility, is found in doing this that one cannot distinguish between these about 1.22, whereas the theoretical conjecture is y=1.25 for two exponents. Therefore, I wonder whether in the case of the Ising systems and y = 1.33 for Heisenberg systems. For T> Tr magnetic data the critical temperature is really definitely fixed. the specific heat seems to diverge with a very high value of a in Perhaps with a slight shift of the critical temperature, one might those cases. Namely, for the first salt a = 0.57 ±0.06 and for

~~get a fit of the data with exponents that are slightly different. In the second salt even a = 0.68 ±0.05 which is rather surprising.~~

~~other words, I wonder whether even in the best cases, the data In most other salts measured up to now this coefficient of diver-~~

are really good enough to fix the exponent /3 to be Va. gency was found to be much smaller. For T > Tr it seems that the experiments can be fitted very reasonably by a logarithmic P. Heller: In response to that last question, it should be re- divergence. That these systems are Ising systems can also be marked that there is no uncertainty about the value of /3. If seen from the difference in entropy S at infinite temperature plot you the cube of the magnetization as a function of the tem- and at the Curie temperature. This difference was found to be perature you get a straight line. If you plot the magnetization (S x —S r )IR = 0.14 and 0.125 for the two salts mentioned, whereas to a power slightly different from 3, you do not get a straight for a simple cubic spin V2 Ising system one finds 0.133 and for a line. In this procedure of finding the exponent that rectifies Heisenberg system 0.25. the observed data, one does not need to know the temperature In the second series of salts they took potassium chloride, of the Curie point. rubidium chloride, ammonium chloride, and ammonium bromide.

The structure is nearly body centered cubic. {S x —Sr )/R = 0.222, D. H. Douglass: I wish to point out, in addition to the magnetic whereas for the Ising body centered cubic model this should be and liquid-gas systems, that superfluid liquid helium is another 0.107 and for the Heisenberg model about 0.25. The value of y system for which the exponent V3 probably appears. The quan- turned out to be 1.36, which is close to the value 1.33 expected tity in liquid helium which corresponds, in the magnetic case, to for the Heisenberg system. It seems that the specific heat data the magnetization is (I get this from analogy with supercon- p'J- can be fitted accurately by a logarithmic type of divergence both ductivity). I have used the data of Dash and Taylor [Phys. Rev. below and above the Curie temperature. Only above the Curie 105, 7 (1957)] to determine if, near the A-point, the superfluid temperature the curve is much lower. The specific heat data density follows the relation pl^aiTy—Ty. I find that y = 0.30 the reduced temperature T/T fit very 5 plotted as a function of r ±0.03. which includes the value V3 (and /i6, too). closely the same curve for all four salts. This gave the authors reason to think that these salts are fairly ideal systems, because C. Domb: With regard to the paper of Dr. Wolf, I would like all deviations from ideality would be expected to be different to make two comments. for the various salts. As the data lie on the same curve, these Firstly, I think the first order phase transition may be sig- deviations are apparently small. The data can be represented nificant and is worthy of further investigation. The antiferro- by a logarithmic curve over a broad temperature range from magnetic transition is the analog of an order-disorder transition 0.1 to 0.001 °K from the Curie temperature. The authors also with unequal ratios of the constituents. I believe the experi- tried to fit the data to some power law, but it turns out that this mental data on the order-disorder transition of this kind in can only be done for a much smaller temperature range, namely, Cu 3 Au are also thought to represent a first order transition. between 0.01 and 0.001 °K from the Curie temperature. So We might well expect something different to happen when the it seems that the logarithmic shape is really the best one. symmetry of equal ratios of constituents is removed in an order- transition, for anti- disorder and when the field is nonzero an S. A. Friedberg presented experimental data for the specific ferromagnet. I would think this is a result worth following up. heat and susceptibility of the antiferromagnet CoCL>-6H:;C) near Secondly, I do not think one can really say any one shape of its Neel point [11] drawing attention to the fact that these data specimen is better than another for testing the theory. I really satisfy Fisher's relation C = Ad(\T)ldT. It was further noted think we just have to do more theory. In fact what has held that the coefficient A yields a value for the curvature of the us up with long range dipole forces is the terrifying nature of antiferro-paramagnetic phase boundary in the H-T plane at the calculations. However, if the dominating term is the short H~0 in good agreement with experiments. range interaction, and the dipole force, although it produces a I want to mention a quantity that one can measure shape dependent effect, is nevertheless a relatively small cor- E. Callen: that gives essentially the same information as the specific rection, I think the calculations may be manageable, and then and

contribution to the thermal expan- I, we should be able to get a better comparison. heat. This is the magnetic sion coefficient. This quantity is easier to obtain and in any | case would allow an independent measurement of the same W . P. Wolf: Let me make that point quite clear. In our anti- thing. ferromagnet the energy of three cells of nearest neighbors accounts for the major part of the energy. However, when you W. Marshall: Commenting on the paper of Dr. Heller, it is worth apply a magnetic field, the change in dipolar energy, the denoting that the neutron scattering results of Turberfield [12] magnetizing energy, that is, is comparable with the energy due on manganese fluoride showed again explicitly the fact that the to the applied magnetic field, and in a large field it can become correlations in the a and b direction are more erratic than in rather comparable with all the other energies. I didn't give any the c direction as we would expect. numbers for this but for DAG, the effects are in fact all about to' about the theory you de- comparable. I have only one comment make scribed which, of course, pulls out all the essential physics but C. Domb: Well, perhaps you get cancellation in the higher which is not actually strictly correct. This is because in the fluc- order terms. tuations which matter for you, there is a fluctuation near the superlattice vector and at the superlattice vector at high tempera- W. P. Wolf: Yes, when it is antiferromagnetic. However, not tures your response function does not follow an exponential when you magnetize the system and polarize the spins. decay. In fact you can show that its Fourier transform has a

~~66 6~~

Gaussian shape. Therefore, for example, the discrepancy References between the A you needed and the A which was observed at high temperature may be due to a temperature dependence. However, a more important correction is probably due to the [1] B. R. Livesay and E. J. Scheibner, J. Appl. Phys. 36, 3240 fact that the shape changes from Lorentzian to Gaussian as you (1965). go to high temperatures. [2] J. C. Eisenstein, R. P. Hudson, and B. W. Mangum, Phys. Rev. 37, A1886 (1965). [3] J. C. Eisenstein, R. P. Hudson, and B. W. Mangum, Appl. P. Martin: I am a little surprised that your theory agrees so Phys. Letters 5, 231 (1964). well with experiment. We have been hearing that the molecu- [4] E. Callen and H. Callen, J. Appl. Phys. 36, 1140 (1965). lar field theory is not adequate near T, where it gives incorrect [5] H. B. Callen, Phys. Rev. 130, 890 (1963). values for the temperature dependence of the susceptibility. [6] J. A. Copeland and H. A. Gersch, Bull. Am. Phys. Soc. 10, Since the theory you have presented is basically a time depend- 304 (1965). ent version of the same molecular field theory and the quantity [7] L. Tisza and P. M. Quay, Ann. Phys. (N.Y.) 25,48 (1963). you are calculating so closely related to the susceptibility, why [8] D. R. Thompson and O. K. Rice, J. Amer. Chem. Soc. 86, does your theory work so well? 3547 (1964). [9] A. R. Miedema, R. F. Wielinga, and W. J. Huiskamp, Phys. Letters 17, 87 (1965).

P. Heller: I agree that the molecular field theory should not (10] A. R. Miedema, R. F. Wielinga, and W. J. Huiskamp, Phys- be very accurate. In principle we could use the line width data ica 31, 1234 (1965). to determine the actual temperature dependence of the sus- [11] J. Skalyo and S. A. Friedberg, Phys. Rev. Letters 13, 133

ceptibilities in the critical region. I think that both the data (1964). J. Skalyo, A. F. Cohen and S. A. Friedberg, Proc. and the theory are too crude for this purpose. The fact that 9th Int. Conf. Low Temperature Physics, Columbus, theory and experiment agree as well as they do may be fortuitous. Ohio, 1964. However,- I feel that the qualitative nature of the line width [12] A. Okasaki, R. W. H. Stevenson and K. C. Turberfield, anomaly has been correctly described by the theory. Proc. Intern. Conf. on Magnetism, Nottingham, 1964.

~~67 209-493 0-66—~~

~~LOGARITHMIC SINGULARITIES~~

~~Chairman: C. N. Yang~~

~~The Lambda Transition in Liquid Helium~~

~~W. M. Fairbank~~

~~Stanford University, Stanford, Calif, and C. F. Kellers~~

~~Wells College, Aurora, N.Y.~~

This talk is based on work done at Duke Univer- Onsager, and Feynmann have suggested that liquid sity several years ago by C. F. Kellers, M. J. Buck- helium as a superfluid is required to be irrotational. ingham and myself [1]. A major part of the work If the helium rotates, it rotates according to the constituted the Ph.D. thesis of Dr. Kellers [lb]. Onsager-Feynmann theory, with quantized vortex

~~Much of what I will say has been previously reported motion [2]. Helium rotating slowly enough is pre-~~

[1]. Still, when Dr. Green asked me to give this dicted to rotate in one single quantized state of talk I consented because I believed a review of the rotation with each helium atom having angular lambda transition in liquid helium is particularly momentum fi regardless of the size of the bucket in pertinent to the present conference on critical which it is contained [2, 3]. The theoretical sugges- phenomena. Not only were the Ehrenfest relation is then the following: If a bucket of helium no tionships for a second order transition enunciated matter how large is cooled through the lambda first for the lambda transition in liquid helium, but transition into the superfluid state while rotating liquid helium also served, through the experiments slowly enough, each superfluid helium atom will which I will describe, to first call attention to the attain the same angular momentum. For the logarithmic character of the lambda transition which slowest stage the superfluid will stop rotating and is so much a part of the discussion of this conference. the bucket, if suspended freely, will rotate faster

The lambda transition in liquid helium is, of [4]. This experiment has never been performed. course, of particular interest because of the unique Some physicists, including a graduate student momentum ordering that takes place below the George Hess at Stanford, are trying to perform this lambda point which makes it perhaps similar but experiment [5]. This then is the type of order tran- certainly different from all other transitions. In sition which is predicted for liquid helium, long- addition, however, of all the cooperative transitions, range order in momentum space of a very special it is the one which offers the best chance for experi- kind. mental observations very close to the transition. Blatt, Butler, and Schafroth have taken another Because liquid helium does not have any crystal approach to the problem [6, 7, 8, 9]. They questioned whether a real liquid, as contrasted with structure, because it is a pure liquid changing at an ideal gas exhibit the lambda point only in its momentum ordering, it Bose can arbitrarily long-range order or correlation in is apparently possible with sufficiently good thermal momentum space. They that equilibrium to make measurements as close as suggest there may be a finite correlation length 10~ 5 experimental techniques permit to the lambda point of approximately cm over which helium atoms without observing any of the broadening character- are correlated in momentum space, but beyond istics of other transitions. which there is no correlation at all. In particular, Ever since London suggested that liquid helium they developed a theory of a Bose liquid in which represents in its superfluid phase a long-range order helium atoms are correlated over a distance of -5 in momentum space, physicists have been puzzled about 10 cm, the correlated atoms representing by exactly what this means. London, Landau, the superfluid. By calculating the amount of this correlated phase p, they calculate the ratio of uncorrected to correlated phase (normal to superfluid,

~~Pnlps) as a function of temperature, the correlation Supported in pari by the National Science Foundation and the Army Research Office Durham. length being an adjustable parameter. With a cor-~~

71 relation length of 10~ 5 cm, they were able to accom- 10 r plish something which to my knowledge had not been accomplished before. They were able to derive n lp.i as a function of temperature in agree- p 8 - ment with experiment up through the lambda point and with the same theory, to derive the specific heat at the lambda point in quite good agreement with experiment both as to shape, magnitude, and temperature of the transition. Their theoretical curves are shown along with the then existing experimental data in figures 1 and 2. One characteristic new feature of this specific heat transition is that it is a quasi-transition. Instead of being a second-order transition with a discontinuity in the specific heat at the lambda point, the specific heat is rounded at the lambda point for a temperature interval of ap- 3 proximately 10" deg which is a direct result of a

finite 1 correlation length. This rounding could not i i I 0 I i i i 1.95 2.0 2.1 2.2 2.3 have been observed in the then existing data. It also follows from their theory that curl v = 0 is not s FIGURE 1. The specific heat of liquid helium versus temperature an equilibrium property of superfluid helium II. in the neighborhood of the k-point. The curve is theoretical, the experimental points are taken from Keesom: Helium Feynmann [10] took issue with the premise that (Amsterdam, 1942), Table 4.20. The theoretical curve is a continuous and differenti- able function of temperature everywhere, represented by the same theoretical expres- the correlation length in liquid helium could be sion throughout the temperature region covered in ref. [8]. cutoff at some finite distance, and suggested that the lambda point in liquid helium is not only sharp at one-thousandth of a degree from the lambda point but it is sharp at a hundred-thousandth of a degree. He believed that even though the correlation in liquid helium II gets weaker and weaker as the distance gets bigger, it still remains finite. Thus, no matter how large the bucket, liquid helium would still rotate in a single quantum state if the rotation were sufficiently slow. At the expense of bringing you a free quote which was only told to me by Feynmann and therefore is not quoted from the literature, I am going to quote an exchange between Feynmann and Blatt because

I think it illustrates the problem of long-range order very clearly for this conference. "Blatt, 'according to Feynmann', said to him, 'Feynmann, how is your intuition so good that you can say that there is a correlation between helium atoms separated by the size of the universe.' 'Feynmann said he replied', 'Blatt, how is your intuition so good that you can say

~~a correlation which decreases as 1/r2 and is down by something like 1070 at the edge of the universe,~~

~~doesn't exist'." That is just the this point, correla- 1 I i 01 i ' 1 ' sH- 195 2.0 2.1 2.2 2.3 tion falls off very rapidly. If the helium is to stop~~

rotating in a rotating bucket when the bucket is FIGURE 2. The normal fluid concentration pjp in liquid helium cooled below the lambda point, then the rotational versus temperature, in the neighborhood of the k-point. soun The curve is theoretical, the experimental points are based on the second 2 function o speed must decrease as 1/r as the size of the bucket data of Pellam. The theoretical curve is a continuous and ditferentiable temperature everywhere, represented by the same theoretical expression throughou becomes larger and larger. the temperature region covered in the ref. [9].

72 These conflicting theoretical predictions served as an incentive for us to perform a specific heat measurement to at least 10~ 5 deg of the lambda point. This experiment was performed by Buckingham, Kellers, and Fairbank, and has been reported in

~~reference 1. Ultimately Cs was measured to within 10 -6 deg of the lambda point T\, with equal precision~~

both above and below 7\. I would like to recall here just a few highlights of the experiment and comment on the possible significance of the data. Many previous measurements have been made on the lambda point, including an unpublished master's degree thesis from our laboratory. The

difficulty with all these earlier experiments was that reasonably large amounts of liquid helium was used in containers which were connected to the outside world. To obtain the needed high temperature resolu-

tion it is essential that the attainment of equilibrium be unaffected by the drastic change of thermal conductivity of liquid helium at the lambda point, or by the onset of the creeping film. Both of these requirements were met by permanently sealing the

helium (0.0587 g) in a cooper container (200 g), the inside of which was in the form of fins so placed that the helium was everywhere within 0.003 in. Figure 3. Schematic diagram of adiabatic chamber for specific [7c]/ (A stainless of the copper surface (fig. 3). With a heat input of heat measurements ) steel wire for closing heat switch; (B) brass cap on filling capillary; (C) wires connecting 10 erg/s temperature equilibrium of greater than to heater and resistors on sample; (D) cotton plug dyed with 6 "0" 10" deg could be obtained, even in the helium I carbon black as radiation trap; (E) indium ring; (F) filling capillary; (G) Kovar seal used as thermal short for wires to region. to for In order eliminate the need removing sample; {H) radiation shield and thermal short for heat switch; exchange gas, the sealed container was suspended (/) nylor cord; (J) three prongs of heat switch {copper); (K) indium coating and suspension for sample; (L) temperature- in a vacuum and a mechanical heat switch provided sensitive resistor; (M) heater; (N) sample cavity; (O) tempera- for contact with the bath when required. ture-sensitive resistor; (P) copper shield over resistor; (R) calorimeter wall. Measurements were made by means of a carbon Figure taken from ref. (lc]. resistor with a minimal detectable change represent-

ing 2.10~ 7 deg. It was possible to make measurements both while increasing and decreasing the 4 temperature. the first and last curve is about 5 X 10 . Thus if Figure 4 shows the data. These of course are not the projected slide of the first diagram is 10 ft,

~~new data just taken, but the reason I was asked to it would have to be expanded to 100 miles to obtain~~

~~present them is that perhaps they are very similar a 10 ft projection of the third figure. to data which are now being obtained on other transi- It can be seen that as the specific heat is displayed~~

~~on a more and more expanded scale, it maintains tions. The specific heat, C.s , is plotted as ordinant~~

7^ 7"^ the same geometric shape. There is certainly no — | versus . order to the nature of the 1 In show transition very near the lambda point, the data are indication of a rounded quasi-transition as predicted shown on successively expanded temperature scales. by the theory of Blatt, Butler, and Schafroth. To aid in a visualization of the very large amount of Figure 5 shows the data in a form with which you

expansion of each successive curve, a small ver- are all now familiar. C s has been plotted as ordi- tical line has been drawn just above the origin. nant against T—T\ in degrees Kelvin on a logarith- The width of the line indicating the fraction of the mic scale. It is seen in this kind of a plot that

curve which is shown expanded in the curve directly near the lambda point on each side there is a factor 4 to the right. The ratio of the expansion between of 10 in T—TK over which the data fall in two

~~73 -1.5 -1.0 -.5 0 .5 1.0 1.5 -4 -2 0 2 4 6 -20 -10 0 10 20 30~~

~~( - degrees (T-T ) millidegrees (T-T ) microdegrees T Tx ) x x~~

Figure 4. Specific heat of liquid helium versus T — T* in °K [1]. 0 represent data cif Kellers, Fairbank and Buckingham [1]. <8> represent, above 1.5 °K, data of Hill and Lounasmaa [20] and Lounasmaa and kojo [18] and. below 1.5 °K, data of Kramers. Wasscher, and Gorter |21[. Solid line near \-point represents eq (1). Width of small vertical line just above origin indicates portion of diagram shown expanded (in width) in the curve directly to the right. Figure from ref. [lc].

~~I 1 I l I l l -8 -4 0 4 8~~

T-Tx (MICRODEGREES) Figure 5. The specific heat liquid helium versus log\T— T x [1]. of | 3 0 represent data of Kellers, Fairbank and Buckingham 1 1 ]. # represent, above 1.5 K near \-point. data of Hill and Lounasmaa [20] and Lounasmaa and Kojo |18] and, below 1.5 °K, data Figure 6. Plot of Cs versus T — Tx very of Kramers, Wasscher, and Gorter [21]. Two parallel straight lines represent eq (1). Data taken from one single run randomly selected from five runs where there were, The broken line represents formula shown in the figure. Figure from ref. [lc], no disturbing influences. Figure from ref. [lb]. parallel straight lines which are branches of the Figure 6 shows previously unpublished data from expression a single run taken from C. F. Keller's Ph.D. thesis [lb]. About 20 runs were taken back and forth

C s = 4.55-3.00 log 10 |:T-7\|-5.20A (1) through the lambda point over ± 10 ;u,deg pen minute. About half of the runs were discarded = = at the where A 0 for T < TK , and A 1 for T > 7\. One because the rate of cooling or heating begin reason that this was very interesting and exciting ning or end of the run was different. In about half at the time, is that Onsager had published the exact of the remaining runs there was a big spike in the! solution of the two-dimensional Ising model [11]- data due to someone opening the door and so forth. The two-dimensional Ising model gives a logarithmic Of the remaining 5 runs, the one shown in figure t

~~transition but only for an exact solution. is typical and picked at random. The data is~~

~~74 ~ averaged over 1 /udeg so the lambda point is still which has been fitted to the data for T —T\ between~~

1 3 clearly observable. The three solid curves are 2 X 10" and 2 X 10" as shown. This is the funda- placed in the di'agram to show the effect of averag- mental form suggested by the Pade approximation ing over finite temperature intervals 10" 6 deg of the three-dimensional Ising model. It is seen 6 Kelvin and 2 X 10" deg Kelvin. The curve labeled that although it fits the data rather well for -3 infinity is the curve given in eq (1) and represents [T— 7\) > 2 X 10 °K, it departs very dramatically infinite resolution. As stated above these data were for temperatures closer to the lambda point. Un- averaged over 1 fideg interval and should corre- less data are obtained close enough to the lambda 10 6 point one could not differentiate this ex- spond to that curve labeled . between Thus we see from the experimental data that pression and the logarithmic singularity. the lambda point, instead of being a rounded quasi- Much of this conference will be taken up with the transition as suggested by the theory of Blatt, exact nature of other kinds of lambda transitions.

Butler, and Schafroth assuming a finite correlation However, I wish to compare quickly two results by length, is in fact, sharp to at least 2 orders of mag- Robinson and Friedberg [12] and Skalyo and Fried- nitude closer in temperature to the lambda point berg [13] on the lambda transition in hydrated nickel than predicted by their theory. The simple loga- and cobalt chloride with specific heat data on liquid rithmic behavior of the data shown in figure 4 which helium. extrapolates to a logarithmic singularity of the Figure 7 shows the specific heat data of nickel lambda point has stimulated more detailed experi- chloride to within 0.07 °K of the lambda point mental and theoretical investigation of the lambda from above, and within 0.2 °K from below the transi- point, as witnessed so completely by this confer- tion temperature. Figure 8 s hows the same data ence. Although the two-dimensional Ising model with Cp plotted against \og\T—Tk \. If one de- has been solved exactly by Onsager giving such a fines a reduced temperature by t = — \ T 7\|/7\ and' logarithmic singularity, it is obvious from this compares the curve for nickel chloride with the conference that an exact solution to the three- helium curve, one sees for the same value of t dimensional Ising model will not be easily found. both curves show nonparallel straight fines. In figure 5 the curve of the form Figure 9 shows the data for cobalt chloride taken A to smaller values of f. These data ~^ Tx\ C show two parallel ll5 ^ + T>,\ | f— straight lines and then a flattening out of the data.

75 was first made by Tisza [14]. The first experimental evidence concerning the logarithmic nature of the lambda transition was obtained by Atkins and Edwards [15]. They suggested that a logarithmic term could be used to derive the result of their measurements of the thermal expansion coefficient below the lambda point. Figure 10 shows the data on the expansion coefficient of Atkins and Edwards,

~~Chase and Maxwell, and Kerr and Taylor. It is seen that the data can be represented by two parallel straight lines from the plot of the expansion~~

~~coefficient versus log(r— T\). This curve is re-~~

produced from the paper by Kerr and Taylor [16]. Figure 11 shows the relative molar volume of helium Figure 8. C versus p |T-T | and C versus log|T-T for N p N | in the vicinity of the lambda point as given by Kerr NiCl2 -6H 2 0. Dala by Robinson and Friedberg taken from ref. [12]. and Taylor. It is seen that the lambda point

~~comes where the slope is infinite rather than at the~~

A comparison with helium indicate agreement point where the molar volume is minimum. as to the two parallel straight lines. The flattening Buckingham [lc] has derived rigorously the ther- out at values of t closer to the lambda point would modynamic consequences of lambda transitions be expected in either an experiment or theoretical characterized by the absence of a latent heat, but

model where long-range correlation is cutoff. These at which the specific heat at constant pressure two sets of data are then seen to be consistent with becomes infinite. Pippard [17] had previously the lambda transition for helium and indicates the considered such a transition and worked out ther- possibility that other lambda points have the same modynamic relationships based on the assumption form as the helium transition. that the entropy surface is cylindrical near the The suggestion that the specific heat of helium lambda point. The thermodynamic relationships might become infinite at the lambda transition worked out by Buckingham and Pippard can be

~~4.5~~

~~4.0~~

~~.,s.~,^>,".v. •>>: 3.5 UJ V o 3.0 "V, X~~

~~S X Q 2.5 \ X < 2.0 V X. o 1.5~~

~~•T SKALYO AND FRIEDBERG MSl x T>TJ »~~

~~0.5 ' ROBINSON AND FRIEDBERG [|2]~~

~~00 -3 L 10 10 10" l 10 ' I0 10' IT-TJ CK)~~

Fici re 9. The heat capacity, C,„ 0/C0CI2 6H.O as a function o/ln |T-T for T = 2.2890 °K. N | N Figure taken from ref. [13]. 76 FIGURE 10. The expansion coefficient, a s . near the \-point as a function o/log|T— Curve from Kerr and Taylor. Figure taken from ref. (16].

~~should give asymptotically a single straight fine. Two parallel straight lines are obtained by Kerr and Taylor using the data shown in figures 4 and 10.~~

~~This shows that the fundamental form of the data is~~

correct but the details need to be refined. It is difficult to obtain exactly the same experimental conditions in different experiments very close to the lambda point. It seems worthwhile to make simultaneous specific heat and expansion coefficient measurements to within 10" 6 °K of the lambda transition. The behavior of the velocity of sound in the vicinity of the lambda point can also be predicted using zjTS 2J6 2J7 tie f!i9 T5i 2 specific heat and other thermodynamic data [lc]. TEMPERATURE (°K) Here again there is approximate but not exact agree- FIGURE 11. The molar volume of He* in the vicinity of the k-point. ment. Since there will be a separate talk on this Data by Kerr and Taylor. Figure from ref. [16]. subject I will not present further data here. used to compare the behavior of various thermody- So in summary, the specific heat expansion coeffi- namic properties in the neighborhood of the lambda cient and velocity of sound all give evidence to the line [lc]. In reference lc and 16 the specific heat logarithmic nature of the lambda transition in helium and coefficient of expansion are compared using the although refined details of comparison still need to

Buckingham relationship. The slope of the lambda be worked out. I would like to end by again men- line (dSjdT)\ is taken from the data of Lounasmaa tioning that liquid helium presents the unique oppor- and Kojo [18] and Lounasmaa and Kaunisto [19]. tunity to obtain data close to the lambda point. The

~~The data can be plotted in a parametric plot which question naturally arises, is the transition in liquid~~

77 helium similar to lambda transition in other sub- out the entire container. Part of the atoms have stances, and does liquid helium present an idealized one special density and they are all together in the

~~experimental model for all lambda transitions? At container, and all the rest of the atoms have a dif-~~

~~first glance one might think that liquid helium ferent density. This sudden transition is to a state~~

doesn't. I would, however, like to point out a sense of order which depends only on the size of the j in which liquid helium does present such a model; container. a sense that has been emphasized by Buckingham in I want to close with this question. Is the lambda private discussions. transition in liquid helium, except for the fact that When one goes from the normal liquid to the one has a lambda line, an exact model for other superfluid in liquid helium, one gets a change from a transitions when long-range correlations are not normal system to something which is suddenly cutoff in other experiment systems or theoretical ordered throughout the whole container. In this models?

particular case, it is an ordering in the superfluid. The superfluid atom in one part of the container References knows what is happening to the superfluid atom in a container, long that different part of the and how [1] (a) W. M. Fairbank, M. J. Buckingham, and C. F. Kellers, 5th Int. Conf. Low Temp. Phys. (Madison, Wis., order is depends only on the size of the container. Proc. 1957), p. 50; (b) C. F. Kellers: Thesis (Duke University,} destroying this One can disturb the system without 1960): (c) M. J. Buckingham and W. M. Fairbank, Progress; order throughout the entire system provided the in Low Temperature Physics, Vol. 3, edited by C. J." Gorter (Amsterdam, 1961), ch. Ill; (d) W. M. Fairbank, disturbances, for example rotational speeds, become Estratto da Rendiconti della Scuola Internazionale djj smaller and smaller as the size of the system Fisica (E. Fermi) XXI Corso. [2] R. P. Feynmann, Progress in Low Temperature Physics,) becomes larger and larger. ch. 2. In the case of superconductors one has a quan- [3] F. London, Superfluids, Vol. II, (John Wiley & Sons, New? York, N.Y., 1954). tized flux over supposedly the distance of magnetic [4] H. London, Physical Society 1946 Cambridge Conference; the length of the wire in superconducting magnets Report (London, 1947) p. 48. [5] W. M. Fairbank and C. B. Hess, Angular Momentum of} that feet long. may be thousands of The length Helium II in a Rotating Cylinder (Proceedings of the IX of the ordering depends only on the size of the super- International Conference on Low Temperature Physics,!! York, 1965). P : 188 (Plenum Press. New conducting loop and especially in the case of super- [6] J. M. Blatt, S. T. Butler, and M. R. Schafroth, Phys. Rev. conductors, exists in the presence of large quantities 100, 481 (1955). [7] S. T. Butler and J. M. Blatt, Phys. Rev. 100, 495 (1955). of impurities. Now there is different kind of order [8] S. T. Butler, J. M. Blatt, and M. R. Schafroth, Nuovo Cimento parameter in superconductors and that is the length 4, 674 (1956). [9] J. M. Blatt, S. T. Butler, and M. R. Schafroth, Nuovojj of the correlation between electrons in a pair. This Cimento 4, 676 (1956). 4 is only 10~ cm, but the two electrons in a pair have [10] R. P. Feynmann, private communication.

1 [11] L. Onsager: Phys. Rev. 65, 117 (1944); G. G. Newell and equal and opposite momentum. momentum of The E. W. Montroll: Rev. Mod. Phys. 25, 353 (1953). 402: each pair is zero and therefore every pair throughout [12] W. K. Robinson and S. A. Friedberg, Phys. Rev. 117 , (1960). the whole superconductor has the same zero [13] J. Skalyo and S. A. Friedberg, Phys. Rev. Letters 13, 133 momentum. Thus there is a long-range order in (1964). See also the discussion remark of Dr. Friedbergjj on page 66. momentum throughout the entire superconductor [14] L. Tisza, Phase Transformations in Solids, edited by; -4 even though each pair is paired over 10 cm. Smolychowski, Mayer, and Weyl (New York, 1951). [15] K. R. Atkins and M. H. Edwards, Phys. Rev. 97, 1429, One doesn't have this momentum ordering in any (1955). other kinds of transition besides superconductors [16] E. C. Kerr and R. D. Taylor, Annals of Physics 26, 292-306! (1964). and liquid helium. But on the other hand, as Buck- [17] A. B. Pippard: The Elements of Classical Thermodynamics ingham is going to point out at the end of the morn- (Cambridge, 1957), ch. IX.

O. V. Lounasmaa and E. Kojo, Physica 36 , 3 (1959). ing, one has a transition from a state where ordering [18] [19] O. V. Lounasmaa and L. Kaunisto, Ann. Acad. Sci. Fennicae ( exists within a small cluster in a particular part of A6, No. 59 (1960). [20] R. W. Hill and O. V. Lounasmaa, Phil. Mag. 2, 145 (1957).! a container, to a system where one has suddenly, [21] H. C. Kramers, J. D. Wasscher, and C. J. Gorter, Physica for example in a liquid gas transition, order through- 18, 329 (1952).

78 The Specific Heat of He 3 and He 4 in the Neighborhood of Their Critical Points*

~~M. R. Moldover and W. A. Little~~

~~Stanford University, Stanford, Calif.~~

~~Introduction Experimental Procedure~~

Recently Bagatskif, Voronel', and Gusak [1] There are several practical advantages to using showed that the specific heat at constant volume helium rather than another noble gas for C r meas- of argon exhibited what appears to be a logarith- urements near the critical point. The low heat mic singularity at the critical temperature (7V) for capacity of metals at liquid helium temperature measurements taken at a density near the critical permits one to use a massive calorimeter of large density. This singular behavior is in sharp con- surface to volume ratio. Thus the path for heat trast to the predictions of the traditional view of transfer through the helium may be kept short. this phase transition by Landau and Lifshitz [2]. In addition, more nearly adiabatic conditions and However, the behavior is precisely that to be ex- high resolution thermometry are most easily at- pected for the so-called "lattice gas" model for tained at liquid helium temperatures. the liquid-gas transition. Lee and Yang [3] have Our calorimeter was built of two OFHC copper shown that the partition function of a classical parts. The helium was contained in the lower gas of particles moving on a discrete lattice with part in 50 slots. Each slot was 0.01 cm wide to a repulsive force preventing double occupancy of facilitate good thermal contact between the calorim- any site, and a nearest neighbor attraction can eter and the helium. The slots were made only be mapped precisely onto the partition function of 0.3 cm deep in an effort to minimize possible gravi- an Ising model of a spin system in an external mag- tational effects. [In contrast a thin stainless steel netic field. The specific heat for this Ising model shell 10 cm high and 4 cm in diameter containing in zero field exhibits a logarithmic singularity at a magnetic stirrer was used for the work on argon the Curie point. The specific heat for the corre- [1, 5]. This construction was necessary to obtain sponding lattice gas on the critical isochore exhibits a low ratio of heat capacity of the calorimeter to a logarithmic singularity at the critical point. its contents while maintaining constant volume at The measurements on argon then indicate that the high critical pressure of argon.] The lower part for a real gas the specific heat behaves in a similar of our calorimeter was wound with a constantan manner to that of a lattice gas. We have investi- heater and had a carbon resistor clamped and gated this point further by studying the specific cemented to it. The calorimeter was supported on 3 4 heat at constant volume (C r) of both He and He nylon threads in an evacuated chamber. Thermal at densities close to the critical density. We contact to the bath was made with a mechanical have done this for two main reasons. Firstly, to heat switch. A stainless steel filling capillary see whether the behavior observed for argon is 5 in. in length and 0.006 in. I.D. led from the calorim- also observed for helium, for which quantum effects eter to a needle valve. The dead volume was should be important, and secondly, to investigate about V2 percent of the total volume of the calorim- the detailed nature of the singularity in the pres- eter. The helium was admitted to the calorim- sure-density plane, not only on the critical density, eter via a Toeppler pump which was used to but also in its immediate neighborhood. Yang measure the volume of gas to an accuracy of about and Yang [4] have conjectured that the quantum 0.2 percent. effects would reduce the magnitude of the singular One of the precautions taken was the measure- contribution to the specific heat in helium. Our ment of the stray heat input to the calorimeter results confirm this view. before and after each data point. The approach to temperature equilibrium of the calorimeter was *Work supported in part by the National Science Foundation and the Ghee of Naval Research. observed after each heating interval. As Tc was

79 approached, the equilibrium time increased from a few seconds to several minutes and became one of the limiting factors in this measurement. A Vio W Ohmite carbon resistor of nominal resistance 560 fi was used as a secondary thermometer. Its resistance was measured with a 100 c/s bridge using a lock-in amplifier. Temperature changes of about 2 X 10 « °K at 5.2 °K and 1 X 10" fi °K at 3.3 °K could be observed. At the end of a run exchange gas was admitted to the vacuum space and the resistor was calibrated against the o I I I i

~~4 0 1 2 3 4 5 6 vapor pressure of the He bath using the T5S TEMPERATURE °K scale either from 4.2 to 2.9 °K or from 5.15 to 4.2 °K~~

3 4 3 4 depending on whether He (TV = 3.3 °K) or He FIGURE 1. C v o/He and He at their res/iective critical densities is plotted as a function of temperature. (TV = 5.2 °K) was being studied. A smooth resist- The solid curves are the present work. ance-temperature relation was fitted. Hence relative specific heat measurements near the critical points are not sensitive to errors either in the calibration or in the vapor pressure-tempera- phase region below C, exhibits logarithmic ture scale near Tr . Calibration was always done Tc , as the bath temperature was reduced. A hydro- behavior. static head correction was made. All residuals In the one phase region above Tc two alternatives 4 1 were less than 8 X 10~ °K from a fit to the Clement- have been suggested: First, C r 9 log {T— Tc ) has -4 Quinnell [6] formula and less than 3 X 10 °K from been suggested by an exact calculation on a two! dimensional Ising measurements a four constant formula. Each run, Tc was ten- model [9], by C, tatively defined as the temperature at which on argon [1] and oxygen [8], and by velocity of sound measurements on helium four [10]. Second, C, for a density near p c fell most abruptly. The C, ^ a for small positive a (such as final value of Tc was then chosen to give the best (T—TcY some fit to Vs) has been suggested by Fisher [11] on the basis of approximate calculations on a three dimensional = -alo + 6 Ising model and his analysis of the argon and | & (^) ,1) oxygen data. We do not believe the present data permit one to distinguish between these two over the 10" 3 10"' range °K<(T( ~T)< °K. The alternatives. adjustment was less than 3 X 10" 4 °K. This value In figure 3 similar plots are given for He3 for a of Tc was also for used analyzing data taken during 5 density of 0.98 p,.. Logarithmic behavior is again the same run at densities far from pc . On four observed for T< Tc . The data for argon, oxygen separate runs with 4 He , Tc was found to be 5.189 [8], and both isotopes of helium expressed in the ±0.001 °K. This is considerably below the value dimensionless form (1) are summarized in table 1. Tc = 5.1994 °K defined on the Tf, H temperature scale. We see that the coefficient a decreases from Ar to A single run with 3 He also yielded a value for T, 4 He to He 3 in agreement with the conjecture of well below the accepted value. Yang and Yang [4]. We note that the coefficient 4 0.62 for He is very close to that observed for the Results specific heat along the saturated vapor pressure curve near the A-point of He 4 (0.64) by Buckingham Figure 1 is a general view of the specific heats and Fairbank [12] and may have particular sigl at the critical density of He3 and He 4 at low tem- nificance in explaining the quantum transition. peratures [7]. We have indicated the temperature Yang and Yang [4] have pointed out that one may range covered by these measurements. In figure 2 write in the two phase region, we have plotted on both a linear and logarithmic scale measurements of C, of He 4 at a density within c'=- w * 0.5 percent of the critical density. In the two (S),^(f) r T

~~3 P = -98% 40 _ \- He c~~

~~T| T - ^ - -.37 log L )+I.7 ^ ( 30~~

~~20 t < tV-.~~

~~0 i 10 10 10 10 10~~

~~( T - T ) °K ( T - T ) °K~~

~~<-> 1.4~~

~~4 3 TABLE 1. Coefficients a and b of eq (1) for 0>, Ar, He , and He~~

~~60 .,...07,, Element a b: T~~

~~o 2 2.4 10 Ar 1.8 8 He 4 0.62 2.0 He3 .37 1.7~~

~~0 1 i i 10 10 10 where is the chemical potential. They suggest - p ( T T ) °K 2 2 2 that both ((Pfi/dT ^- and (d PldT ) v become infinite~~

~~2 2 Figure 4. 4 is plotted against |T at densities at = C v of He — c | of Tc for a real gas, whereas {d p,ldT ) v Q for the 1.07 pc and 0.96 pc . two dimensional Ising model. Figure 4 shows~~

~~4 data on He taken at the densities 0.96 pc and 1.07 p c . Tc for these plots was chosen from data does not approach zero as T approaches Tc . A~~

(not shown) taken during the same run at 0.99 p c . similar plor with the ordinate heat capacity/volume The two phase region (lower temperatures) is the rather than heat capacity/mole indicates that upper 2 2 branch of each curve. The difference be- (d p/dT ) v also does not approach zero as T ap-

~~tween the upper branches in this region is pro- proaches Tc . This is consistent with the suggestion 2 portional to (d^PldT ),. It is clear that (cPP/dP) . of t Yang and Yang.~~

81 In the one phase region the two curves on figure 4 [2] L. D. Landau and E. M. Lifshitz, Statistical-Physics (Addi- son-Wesley Publishing Co., Inc., Reading, Mass., 1958). differ drastically. This suggests particular atten- [3] T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952). tion to density measurements will be required to [4] C. N. Yang and C. P. Yang, Phys. Rev. Letters, 13, 303 (1964). determine behavior at the critical density. We [5] A. V. Voronel' and P. G. Strelkov, PTE 6, 111 (1960). also note the transition to the one phase region is [English Transl: Cryogenics (G. B.), 2, 91 (1961)]. [6] J. R. Clement and E. H. Quinnell, Rev. Sci. Instr. 23, 213 particularly well marked in the case of p=1.07 . p c (1952). 4 This suggests specific heat measurements may be [7] The dashed curve for He was calculated from data of ref. 12; R. W. Hill and O. V. Lounasmaa, Phil. Mag. 2, Ser. 8, a sensitive method of determining the coexistence 145 (1957); O. V. Lounasmaa and E. Kojo, Series A., VI. curve. Physica 36, 3 (1959); H. C. Kramers, J. D. Wasscher and C. J. Gorter, Physica 18, 329 (1952); M. H. Edwards, Canad. J. Phys. 36, 884 (1958); H. van Dijk, M. Durieux, J. R. Clement, and J. K. Logan, J. Res. NBS 64A (Phys. We thank Dr. Derek Griffiths, Dr. R. S. Safrata, and Chem.) No. 1, 1 (1960). The dashed curve for He3 Dr. C. F. Kellers, and Dr. M. H. Edwards for many was calculated from data of Henry Laquer, Stephen G. Sydoriak. and Thomas R. Roberts, Phys. Rev. 113, 417 useful discussions and Dr. Derek Griffiths and Don- (1959): R. H. Sherman, S. G. Sydoriak, and T. R. Roberts, ald Moldover for much help in analyzing data. We J. Res. NBS 68A (Phys. and Chem.) No. 6, 579 (1964); and T. R. Roberts and S. G. Sydoriak, Phys. Rev. 98, thank Professor C. N. Yang for his suggestions, 1672 (1955). interest and advice. Some of this data was pre- [8] A. V. Voronel', Yu R. Chashkin, V. A. Popov, and V. G. Sim-

kin, Zh. Eksperim. i Teor. Fiz. 45, 828 (1963) [English sented at the LT9 Conference in 1964. Transl: Soviet Phys.-JETP 18 , 568 (1964)]. [9] L. Onsager, Phys. Rev. 65, 117 (1944). [10] C. E. Chase, R. C. Williamson, and Laszlo Tisza, Phys. References Rev. Letters 13, 467 (1964). [11] Michael E. Fisher. Phys. Rev. 136, A1599 (1964). [12] M. J. Buckingham and W. M. Fairbank, The Nature of the [1] M. I. BagatskiT, A. V. Voronel', and V. G. Gusak, Zh. \-Transition, Progress in Low Temperature Physics,

~~Eksperim. i Teor. Fiz. 43, 728 (1962). [English Transl: III. P. 86 (North Holland Publishing Co., Amsterdam, I Soviet Phys.-JETP 16, 517 (1963)]. 1961).~~

~~The Coexistence Curve of He 4~~

M. H. Edwards*

~~Stanford University, Stanford, Calif.~~

~~During the years between 1956 and 1963 I meas- We analyzed our data on the coexistence curve 4 ured the density of saturated He liquid and vapor between 0.98 and 0.993 Tc by extending Landau~~

~~along the coexistence curve from 0.30 to 0.99 T( and Lifshitz's theory using a fit to a power series , jj~~

~~by refractive index measurements with a modified expansion in volume and temperature [3]. If, >~~

~~Jamin interferometer [1-3]. Between 0.95 and as is now believed, the critical point is a singular ,~~

0.993 Tc , 76 experimental points were taken. point, this whole expansion procedure is question- Advantages of this method of density measurement able, since the coexistence curve cannot in prin- near the critical point are ciple be represented asymptotically by a Taylor 1. High resolution of density to ~ 0.01 percent series in the density about such a nonanalytic is possible. point. A reanalysis of this same data will now| 2. There are no dead space corrections to the be presented, without the use of a Taylor series;* observed densities. expansion. of saturated liquid andl 3. The density of a horizontal "slice" only 1 mm The coexistence curve vapor densities and as a function of tempera-jj deep is measured and hydrostatic head effects pi p tf were always less than 0.05 percent in density. ture, is symmetrical, not about the critical density,! 4. If temperature inhomogeneities of ~ 10 4 °K but about a "rectilinear diameter." This fine of! appear in the optical cell the fringes disappear mean densities of vapor and liquid passes through and no results are obtained. Pc the critical density, and for He 4 would extrap-| olate linearly to 1.1 p c at T=0. The symmetry of On leave I'rnm K.>\al Militar) College, Kingston. Ontario. Canada. the coexistence curve is obscured on a plot of sat-

~~82 7~~

V\ V,, liquid vapor urated molar volume and of and ' • • since the mean volume curves sharply away from the temperature axis at lower temperatures. In examinations of the shape of coexistence curves, — it is customary to consider the quantity ) (p/ p fl or (pi — p

1 suggest that /3 = /3 for a certain range of temperatures. Accurate values of Tr are needed for meaningful tests of such a relationship. Furthermore, the slope of the "rectilinear diameter," or line of mean densities, differs from substance to substance,

that FIGURE 1. Temperature dependence 2 3 4 so the similarity of coexistence curves may be ofX and X for He . 3 X is linear above 0.8 7V. obscured by comparing experimental data in that manner. M. J. Buckingham has suggested that where a is a constant, and t = — T. the shape of the coexistence curve should be Tc The quantity 2 — 2 3 analyzed using the natural variable X l(l \n X) lies between X and X for the whole temperature range. The question of the asymptotic form of the co- = Pi- Pa Kl^P'-py X existence curve of He 4 will now be examined p, + P!l V,,+ V,' 2p (2) using the 76 experimental points listed in table III

of reference 3. All these points were taken within Advantages of this variable are 250 mdeg of 5.1994 °K (the critical temperature of (1) X ranges from 1 to 0 as T ranges from 0 to 4 4 He on the 1958 He scale of temperatures [5]). Tc for any substance, The temperature of each data point was obtained (2) there is equal symmetry using either density directly from the measured saturated vapor pres- or molar volume, and the effect of the slope of the sure. Although both V and V\ were not often "rectilinear diameter" is entirely removed, and g measured at precisely the same temperature, we (3) if we plot X", ^where n= against T, we may evaluate an X for each experimental point by ~j^J- need writing not know Tr or p c or Vc and in fact may determine Tc by such plots. 2Va 2V, 2 1 = Figure 1 shows how plots of X (or /3 = /2), and X -i. orX=l-- (4) s X (or /3= V3), appear for He 4 over the whole range of measurements [1, 2, 3] from 0.3 to 0.99 Tc . For each temperature at which either V or Vi was s g Clearly, X is nearly linear, (or fi = Vz), above about measured, the quantity (Vg + Vi), which varies rather

0.8 Tc (but not too near T , see later), in agreement c slowly and smoothly with temperature, was read with many other measurements for many fluids. by interpolation, or, within 50 mdeg of T by linear c , 2 Note that the classical X is not linear over any extrapolation. Thus 76 values of X are obtained extended range of temperature. within 250 mdeg of Tc . The added uncertainty M. J. Buckingham has shown [4] that the simplest in X produced by the uncertainty in the value of singular entropy surface which is consistent with Vy+Vi falls from 0.6 percent at £ = 35.9 mdeg to a logarithmic infinity in C r at the critical point, below 0.3 percent above £ = 50 mdeg. would imply a coexistence curve whose asymptotic Figure 2 shows a graphical test of the three form as T—> T is given by c functional forms for the coexistence curve of He 4 X2 for all points within 250 mdeg of 5.1994 °K. The at (3) 2 \-\nX straight line drawn on the plot of X is a least-

83 209-493 0-66— 4 for the 65 points within 200 mdeg of Tc . Clearb He COEXISTENCE CURVE over this temperature range the data are repre 2 sented best by the function X I(1 — In X). Note 3 particularly the marked curvature of X as T—> Tc Table 1 shows the slopes and intercepts, will

~~standard errors, computed by least squares fits o the data in the form~~

~~f[Xi) = ati + aATc (5~~

~~1 2 is 3 where j{X) X , X I(1 -In X), or X , and when — £ = 5.1994 r58 , in millidegrees, and kT= the shif in critical point implied by a linear extrapolatioi~~

~~of fiX) to zero. None of these asymptotic fit~~

~~extrapolates to the value rc = 5.1994 °K assumed ii~~

~~the r58 scale of temperatures. Yang and Yang [t~~

~~have suggested that the T5S scale may be in erro~~

~~near Tc , and that the actual critical temperatur should be lower. As a more sensitive test of the experimental ev~~

~~200 150 100 50 0 dence in favor of any of these functional form m deg K for the coexistence curve, we plot the residual~~

Figure 2. Tests of the asymptotic form of the coexistence curve 4 for He above 0.95 Tc . Ati = ti--f(Xi)-ATc (i I a squares-fitted straight line for the 39 points within 2 2 100 mdeg of T . The lines shown on the plots of — c for each of six fits with f(X) equal toX , X l(l \n), j X2 /{1— In X) and of X3 are least-squares-fitted fines 3 and X , over the ranges t < 100 mdeg and

~~Table 1. Linear fits of coexist ce curve data to three functional fc~~

~~Range of data in fit~~

~~t < 100 mdeg (39 points) t < 200 mdeg (65 points)~~

~~3 3 10 a ATC 10 a ATC f(X) (mdeg K)- 1 mdeg K (mdeg K)" 1 mdeg K~~

~~X2 1.005 ±0.004 + 4.2±0.3 0.882 ± 0.007 + 14.2±0.9~~

~~X2 \-\nX 0.527 ±0.002 -7.5 ±0.3 0.513 ±0.001 -5.9 ±0.3~~

~~X3 0.404 ±0.003 - 18.1 ±0.4 0.429 ±0.002 -20.8 ±0.4~~

84 t < 200 mdeg. Figures 3 and 4 are plots of these deviations from linearity, expressed in millidegrees. Figure 3 shows clearly that both X2 and X3 show systematic deviations fitted within 200 mdeg of T. Figure 4 shows that, using only the data points within 100 mdeg, both X2 and X2/(l-ln X) ap-

~~pear to give almost equally good fits. However, 2 X /(l — In X) gives much the best fit when all~~

~~points within 200 mdeg of Tc are included.~~

~~Conclusions~~

~~The main conclusions of this analysis of the form of the He4 coexistence curve are 3 1. X is not asymptotically linear in T above about~~

~~0.98 T despite its linearity from 0.8 to 0.98 Tc . c , 2. X2 can be shown to be asymptotically linear~~

~~only above 0.98 Tc , with the critical temperature I50mdeg 100 4 mdeg above the T^h value. Such a shift is in the~~

~~opposite sense to that expected [6], and this func-~~

tional form is inconsistent with the observed [7] linearity, expressed in IGURE 3. Deviations from millidegrees, logarithmic singularity in C r below Tc - for linear fits of the data for t < 200 mdegfor the three functions 2 2 3 3. X I(1 — In X) is the best asymptotic form, with = . f(X) X , XV(l-ln X) and X the critical point lowered 6 to 8 mdeg below the

~~+ 2 r 8 value. This shift is in the sense expected r 5 [6], and the functional form is consistent [4] with the~~

~~observed [7] logarithmic singularity in C r below Tc.~~

~~4 + 2~~

~~I am grateful to M. J. Buckingham and M. R. Moldover for many discussions. -2 1-inX 4 References + 2 —~~

~~[1] M. H. Edwards, Can. J. Phys. 34, 898 (1956): 36 , 884 (1958). [2] M. H. Edwards, Phys. Rev. 108, 1243 (1957). -2 [3] M. H. Edwards, and W. C. Woodbury, Phys. Rev. 129, 1911~~

(1963) . [4] M. J. Buckingham, These proceedings. 150 100 mdeg 50 [5] F. G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement, and J. K. Logan, J. Res. NBS 64A (Phys. and Chem.) = t (5.I994-T ) 58 No. 1, 1 (1960). [6] C. N. Yang and C. P. Yang, Phys. Rev. Letters 13, 303 IGURE 4. Deviations from linearity, expressed in millidegrees, (1964) . [7] M. Moldover and W. A. Little, Proc. Ninth International for linear fits of the data for t < 100 mdegfor the three functions Conference on Low Temperature Physics (Plenum Press, 2 f(x) = x-. x 1 - :i /( In x), and x . New York, 1965), to be published, and These proceedings.

~~85 Anomalous Specific Heats Associated With Phase Transitions of the Second Kind~~

~~T. Yamamoto, O. Tanimoto, Y. Yasuda, and K. Okada~~

~~Kyoto University, Kyoto, Japan~~

There are many reasons to expect a logarithmic is given by an average over those of the homoge- singularity in the anomalous specific heat accom- neous subsystems: panied with the phase transition of the second kind.

From the theoretical side one may refer to the C„{T)=\ Chomo,(T; T,)f\TK )dT,. (1) Jo classical work of Onsager [1] and the recent intensive investigations by Domb and Sykes, Fisher II. The distribution of the transition points of and others [2]. From the experimental side one 1 subsystems is Gaussian : may refer to at least three elaborate measurements of specific heat: the first is the experiment by

Fairbank, Kellers on the A.-transi- 2 Buckingham, and /(7\) = a -exp [-aCTx-To) ]. (2) 4 77 tion of liquid He [3], the second is that by Voronel' and his collaborators on the liquid-gas transition at the critical point [4], and the third is that by To is identified with the Curie or Neel point of the Skalyo and Friedberg on the antiferromagnetic real system.

transition of CoCl 2 6H,0 [5]. III. The specific heat of the homogeneous system We have analyzed the experimental data avail- consists of two parts: able at present on the specific heat of many magnetic substances. The purpose is to disclose the analytical nature of the singularity which the specific heat has at the magnetic transition point = CSym a function of \T—T\ (3) of the second kind. We would like to report the results of our analyses. Our main conclusion is f0 for T>n, as follows: In the case of ferromagnetic transition ^step in terbium and antiferromagnetic transition in U for T

~~! C£r) = A 1(t: D) + A • F(t) + const., (5) We start with the following assumptions:~~

I. Any real system is heterogeneous and composed of a large number of subsystems. Each subsystem is homogeneous and has its own transi- 'This obvious assumption has already been adopted in, for example, Belov's book tion temperature [5]. The observed specific heat "Magnetic Transitions," Consultant Bureau, N.Y., 1961.

~~86 where The process of analysis consists of the following~~

~~four steps. First the normal specific heat is sub- x 2' I(t;D) = —^=\ e~ In (\x-t \+D)dx. (6) tracted from the observed values if it can be done Vjt. without danger of introducing an appreciable~~

error. When the transition temperature is in the F{r) = —=\ e-^dx, region of liquid helium temperature, we omitted vVJ (7) 7 this step. In most cases we approximated the normal part by a linear equation of temperature, T^V^tt, t = T-T0 , and D= 8. (8) because we are interested in only a rather narrow interval of temperature. For our purpose we do /(t; D) is even with respect to t so that one has from not have to know the absolute values of the specific

(5) heat. This makes it considerably easier to find a linearized equation for the normal part with L} sufficient A-\t\)-C (\t\)} = F(-\T\)-F(\r\). accuracy even by inspection. The ^ ^l{CI p second step of the analysis is to plot Cv or its (9) anomalous part against log \T—Tq\. To is chosen so that, except in the closest neighborhood of To, It should be noted here that (9) is independent of as many observed points as possible form two paral- the assumption IV. lel straight lines, corresponding to the high- and The values of I(t; D) were machined — calculated low-temperature sides. Generally speaking, it is and tabulated, r runs from 0 to 1 with step of not always easy to carry through this procedure 0.1, from 1 to 2 with step of 0.2 and from 2 to 5 without any ambiguity. In several cases we have with step of 0.5. D runs from 0 to 0.2 with step succeeded in locating the transition point quite of 0.02 and from 0.2 to 0.7 with step of 0.05. Fig- definitely and in finding two parallel straight lines. ure 1 gives examples of the specific heat calculated Examples are shown in figures 2, 3, 4, and 5 [6, 7, 8]. from (5). When we could not get rid of some ambiguity, To was found such that the plot resembles the characteristic shape as much as possible in the sense / v -I(7;0)+2F(r)— -*/ that the observed points begin to form two parallel » . -1(7-; 0.3) + 2F(7") 7^ straight lines after — T^o | less than 1 becomes

~~(figs. '/ \ several hundredths of To 6 -and 7) [9, 10]. / / \\ \ / / Once To is fixed in this way, the values of the step ' / \ \ \\ \ / / ! A and the slope A of the fines are read from the~~

~~// * 1 ' \ plot at once.~~

~~' \ * In not a small number of cases examined, the -in(lrl +0.3)—z_w \ \ \~~

1 \ V semilogarithmic plot gave a bill-shaped figure in ^ "X', \ -I(r;0.3) \ \ v the closest neighborhood of the transition point. \\ ^ W v We can proceed one step further in these cases and determine the value of a by making use of (9).

~~v Namely we plot (f)/A against log \t\ \ AC/; and compare \ \ \ \ it with the plot F(— |t|) — F(]t|) versus log |t|. ^\ The difference in the abscissae of the two figures~~

~~is just equal to V2 log a. If the figures coincide with each other through horizontal translation, this provides an empirical justification for the~~

~~2 validity of our assumptions I, II, and III. Figures A/IAI= 2 8 and 9 show examples of the third step. The~~

~~-The anomalnus specific heat at the superconducting transition in Sn was measured -2 very accurately by Kellers and Fairbank. Their data can almost exactly be repro- "~~

(5) with 7\-=3.7230. ; ( duced by °K.^ = 0 and V2^r,J ' = 2.39 X \0~\ This fact can Figi re 1. Calculated curves for the specific heat: = be supposed as another supporl for the Gaussian distribution of the local transition C„ -I(r: D) + 2F(r). temperatures.

~~87 ( cal /deg-moie ) Cp (cal / deg-mole)~~

~~- T. = 175.3 K ! 2.00 T. =52,39 "K A = -1.34 A = -0 31 Kcal/deg-mole) & - 5 94 A = 0 730(cal/deg-mole)~~

*\ M Griffei, R E. Skochdopole, o >• * and F. H. Spedding T T 1.50 < " (1956) T

~~T > Tn \^ 1.00~~

~~• \ • ^~~

~~\o • - C = 0.1402 X(T - IB) (cal/deg-mote) 0.50 L X o T Tn N. > R. H. Busey and W F Giauque (1952) \°o° • \ oo, "Op 3 2 \ 3 2 °0 16 10 1 IT-lil/T. ip io V IT-Jil/T. \ 0ln~~

~~1 10 10 IT-Tnl('K)~~

~~Figure 3. C versus log\T — T plot dysprosium. Figure 6. C„ — versus log\T — T plot NiCh. C,, | p N | of N of~~

~~Cp(cal/ deg-mole ) Cp(cal/deg-mole)~~

~~FeCI, CoCI; T. = 23.55 "K T.= 24 69 "K A = -1.26 (cal /deg-mole) A: -P.4A7(cal/deg-mole) A = 2.8 ± 0.2 (cal /deg-mole) A= 1 08 (cal /deg-mole) (Cl subtracted )~~

~~= I cal /deg-mole) CL 0 1092X(T-K) C L =0.16A5X(T-K.A )(cal/deg~~

~~0. N Trapeznikowa and L.W. Schubnikow R. C. Chisholm and I W Stout ( 1962 ) ( 1935)~~

~~0" 1 IT-T»I/T„ 0* 1 1 o' 1 10 IT-T>I/T»~~

~~10' 1 0~~

~~Figure 4. C„-C, versus log |T-T plot o/FeCF. Figure 7. C„ — C versus log\T — T plot C0CI2 N | L N | of 1 0 CI -6H Co 2 20 ACp JoiTn = 382~~

~~0 5~~

~~— F(-r)-F(r)~~

~~1 6" 1 1 7~~

~~Figure 9. Determination of a in C0CI2 -6H:>0.~~

~~FlGl'RE 8. Determination of a in terbium.~~

CoCl>-6H 2 0. In the other cases we can't say anything definite without reservation. Table 1 value of 8 is adjusted in the last step to reproduce collects the numerical results of our analyses. ' the bill shape as truly as possible. Now we would like to point out the following We have examined about 30 phase transitions of features common to all the magnetic transitions

the second kind, mainly magnetic, and have suc- taken up in table 1. First, \A\/A is less than one ceeded to carry through the analysis up to the half. This seems to be one of the characteristics final step in about one fourth of these cases. of three-dimensionally coupled spin systems 3

~~Figures 10 [11], 11 [5]. 12 [13], 13 [14], and 14 [15] 'show examples of the final results. The agreement~~

:l Even when the spins substantially couple two-dimensionally, we observe that this between the observed and calculated values is ratio is less than one half. It becomes larjier than unity only when the spins couple satisfactory in at least two cases, terbium and essentially one-dimensionally.

~~i ll~~

~~cal Reference Substance To -A ^ X10 3 x 10 3 dee — mol deg— mol T0 No. 2aT0~~

~~(°K) Samarium 105.8 (Tc) 0.471 3.04 3.01 0.85~~

~~Terbium 228.3 (Tc) 1.97 5.20 2.06 .88~~

~~CoCl,-6H,0 2.2890 (TN) .538 1.14 1.85 .79~~

~~MnF, 67.05 (TN) .530 2.73 4.86 3.4~~

~~NiF, 73.33 (TN) .643 2.08 2.51 .71~~

~~Cr,03 306.5 (TN) 1.25 2.64 4.45 1.9~~

~~/3-UH3 172.0 (Tc ) .469 1.81 6.55 2.8~~

~~89 C C~~

~~Secondly, the half-width of the Gaussian distribu- dependent quantity," as Heller and Benedek put~~

it in their Phys. Rev. Letters [17]. To make the tion, divided by T0 , is always of the order of magni- 10" 3 discussion more specific, let us assume that, when tude of . Thirdly, the degree of pinching, -3 of 10 we approach the transition point from the high tem- 8/7o, also has the same order magnitude, , -1 the contracts and the value of and is almost always less than (V2q: To) . This perature side, sample fact has been preventing us from directly observing the exchange coupling increases. This means that, the pinched logarithmic anomaly. if we stay at a distance from the transition point, we As regards the origin of pinching in the logarith- overestimate the distance therefrom. When we mic anomaly in an ideally homogeneous system, we are in the low temperature side, we again over- would like to propose the following explanation. estimate it for the same reason. Since the anom-

~~As is well known, a transition of the second kind alous expansion is really large only in the closest is accompanied with an anomalous expansion. neighborhood of the transition point, the shift of~~

~~Now the transition temperature 7\ is a "temperature the estimated transition point from the real transi-~~

~~tion point is almost constant, so long as we stay outside this narrow region. We identify this con-~~

~~stant shift with our 8. Then the thermal expansion effect may be separated off by neglecting 8 in the Cp(cal/deg-mole) expression for the symmetric part of specific~~

~~14.0- Tb heat at constant pressure to get that of specific heat;~~

, Tt = 228.3 K 12.0- at constant volume. This interpretation leads us] A=-1 97cal/deg-mole A=5 20cal/deg-mols to the important conclusion that, if the system isf 10.0- 6/Tc=8.76X10* T!c\ able by a step function. 400 Before closing our report, we would like to add Y= 0 cal/ mole-deg1 2.00- a few words: The conclusion of the present analysis

~~Jennings-Stanton-Spedding(l957) is tentative and the possibility of some other kind 0.00- of singularity still remains in the cases other than~~

3 Id 10* 10"' IT-Tcl/l terbium and CoClo^rTO. To establish ouij ! 10'' 1 10 10 IT- conclusion more definitely, we have to make furthei investigations from both the experimental anc Figure 10. versus log\T — T plot terbium. Cp — L — E c | of -4 The solid and dotted lines correspond to S/7V=o.76 X 10 and zero, respectively. theoretical sides.

~~4.5 6H CoCl 2 2 0~~

~~4.0 h T„= 2.2890 °K A = -0.538(cal/deg-mole) 3.5 A = 1.1A(cal/deg-mole) 4 ° V. T < Tn 6/Ii=7 86 X 10 3.0 X.~~

~~?2.5 •o "v. V 3 2 0 T>T- 'N a x <-> 1.5 Skalyo Jr. and S. A. Friedberg ( 19W ) 1.0~~

~~0.5~~

~~1.0' 1.0"' 10 1.0 1 IT-TJ/T. 0.0 10" ; 10 1 0 1 0 10" 10 IT-ThICK)~~

~~Figure 11. C versus log\T — T plot o/CoCl2-6H 2 0. p N | The solid and dotted lines correspond R 8/7\=7.86x 10^ and zero, respectively.~~

~~90 C )~~

~~Cp(cal/deg-mole) Cp (cal/deg mole~~

~~NlF 5 00 2 Cr 0 T. =73.33 "K 2 3 = -0.614 cal/deg-mole \ T. =306 5 'K = 2.10 uU deg-mote 30 0 6/Tti.7.06 X10* " a = 2.64 {cal/deg-mole) \ 3 (Ci subtracted) \ 6/Tt.1.89X10"~~

~~29.0 T~~

~~280~~

~~T>T«~~

~~C L = 0 0840( T-20 )cal/deg- mole \. T>Tm 27.0 \~~

~~E Catalano and J. W. Stout (1955) 26.0~~

~~C. T. Anderson ( 1937 )~~

~~3 1,0'* 1,0" IT-T.I/T. 3 ? 2 10' 0' 0'' 1 d" ,o IT-T.I/T. 1 1 1 10 IT-T~~

~~0"' 1 IT- T.I (K)~~

~~12. C,. versus log\T — T plot o/NiF-> Figure C p — N | solid lines correspond to S/7V=7.08 x 10 ^. Figure 13. C versus log\7-T plot Cr,0 . The p N \ of 3~~

~~10~ : '. The dotted lines correspond to 6/TN =1.89x~~

~~Cp (cal/deg-mole)~~

~~P-UH 3 Tc = 172.0 'K~~

~~\ A = -0.469 (cal/deg-mole) 3.00 A = 1.81 (cal/deg-mole) 3 6/Tc=2 79X10"~~

~~(Ci subtracted)~~

~~1.00 T>Tc C = 0.0257 X( T-1 00) + 6 00 (cal/deg-mole) \ L~~

~~Flotow, Lohr, Abraham and Osborne ( 1959 )~~

~~6" 3 2 1 1 o" 10 /\ IT-Tcl/Tc 1 1~~

~~1 0 10 IT-Tcl('K)~~

Figure 14. versus log |T — T plot o//3 UH . C p — L c | — 3 10" 3 The dotted lines correspond to 8/7'r =2.79 x . References S. A. Friedberg, Proc. 9th Int. Conf. Low Temp. Phys., Columbus, Ohio 1964. (To be published.) L. [1] Onsager, Phys. Rev. 65, 117 (1944). [6] M . Griffel. R. E. Skochdopole. and F. H. Spedding, Phys.

[2] A. J. Wakefield, Proc. Cambridge Phil. Soc. 47 , 419, 799 Rev. 93, 657 (1954); J. Chem. Phys. 25, 75 (1956). (1951); C. Domb and M. F. Sykes, Phys. Rev. 128, 168 [7] O. N. Trapeznikowa and L. W. Schubnikow, Phys. Z. Soviet.

~~(1962) ; J. W. Essam and M. E. Fisher, J. Chem. Phys. 38, 7,66(1935).~~

I 802 (1963); J. Gammel, W. Marshall, and L. Morgan, Proc. [8] L. D. Roberts and R. B. Murray, Phys. Rev. 100, 650 (1955). Roy. Soc. (London) A2 75, 257 (1963). G. S. Rushbrooke. [9] R. H. Busey and W. F. Giauque, J. Am. Chem. Soc. 74, J. Chem. Phys. 39, 842 (1963): G. A. Baker, Jr., Phys. 4443 (1952).

~~Rev. 129, 99 (1963); M. E. Fisher, J. Math. Phys. 4, 278 [10] R. C. Chisholm and J. W. Stout, J. Chem. Phys. 36 , 972~~

(1963) . (1962). [3] W. M. Fairbank. M. J. Buckingham and C. F. Kellers, Proc. [11] L D. Jennings, R. M. Stanton, and F. H. Spedding. ibid. 5th Int. Conf. Low Temp. Phys., Madison, Wisconsin 27, 909 (1957).

~~(1957), p. 50. . [12] L D. Jennings, E. Hill and F. H. Spedding, ibid. 31, 1240 [4] M. I. Bagatskii, A. V. Voronel', and B. G. Gusak, Zh. (1959).~~

Eksperim. i Teor. Fiz. 43, 728 (1962) [Translation: Soviet [13] E. Catalano and J. W. Stout, ibid. 23, 1284 (1955). Phys. -JETP 16, 517 (1963)]; A. V. Voronel', Yu. R. [14] C. T. Anderson, J. Am. Chem. Soc. 59, 488 (1937). Chashkin, V. A. Popov, and V. G. Simkin, Zh. Eksperim. [15] H. E. Flotow, H. R. Lohr, B. M. Abraham, and D. W. Os- i Teor. Fiz. 45, 828 (1963) [Translation: Soviet Phys. borne, ibid. 81, 3529 (1959).

-JETP 18 , 568 (1964)]. [16] J W. Stout and H. E. Adams, ibid. 64, 1535 (1942). [5] J. Skalyo, Jr. and S. A. Friedberg, Phys. Rev. Letters 13, [17] P. Heller and G. B. Benedek, Phys. Rev. Letters 11, 428 133 (1964). See also J. Skalyo, Jr., A. F. Cohen and (1962).

~~91 |1~~

The Logarithmic Anomaly in the Pressure Coefficient of Helium Close to the Lambda Line*

~~H. A. Kierstead~~

~~Argonne National Laboratory, Argonne, 111. A~~

~~Introduction~~

Several investigators have reported logarithmic anomalies in the heat capacity [1], thermal expansion [2-6], and pressure coefficient [7-8] of liquid helium in the neighborhood of the lambda transition. Unfortunately, all of these experiments except those of Lounasmaa [7] and of Lounasmaa and

~~Kaunisto [8] were done on the vapor pressure curve, where the anomalous behavior is confined to a very narrow temperature interval. The experiments of~~

Lounasmaa and Kaunisto [8] at various pressures along the lambda curve show that the anomalous Figure 1. Schematic drawing of the apparatus. (A) needle valve; (B) vacuum case; (C) pumping and vapor pressure tubes (2) f(|| behavior is spread out over a larger temperature (E); (D) 30 percent Cu-Ni capillary tubing, 0.1 mm ID; (E) temperature and vapilk pressure compartment; (F) heater; (G) piezometer packed with fine copper wire fiJI interval at higher pressures. Consequently, we rapid equilibrium (volume 39.83 cm 3 at 1.76 °K, height 10 mm to keep hydrostatlr pressure differences small); (H) dashed lines enclose space immersed in liquid heliu bath: (I) thermometer; (K) differential oil manometer, 0.060 in. ID constat/ have measured the pressure coefficient, {S v={dP/dT) v , germanium

1 bore glass capillary (left arm 40 cm long, right arm 90 cm long); (L) oil reservoir aril of liquid helium along the isochore which crosses leveling device filled with Apiezon B oil; (M) to He*1 supply tank and purifier: (N) pump: (O) to atmosphere; (P) 25-cm dial test gauge calibrated against a dead weiglj 3 the lambda line at the melting curve. This inter- tester; (Q) ballast volume (1461 cm ) for balancing the right-hand side of K, held constant temperature by an ice bath; & needle valves. section is called the upper lambda point (Ty, Py).

Experimental Procedure Small changes in the sample density could be mad by raising the oil in the left limb of the manometer The apparatus used in these experiments is essen- forcing gas to condense into G. A change of 0.0' tially the as that same used by Lounasmaa [7]. mm in the oil level caused a change of 2 X 10 Therefore, it will be described only briefly here. It g/cm 3 in the density. is shown schematically in figure 1. Temperatures were measured with a germaniun Helium gas was purified in a trap (not shown) resistance thermometer in a potentiometer circui] immersed in liquid helium, and was condensed into using a double potentiometer. At the uppej the piezometer G through the low-temperature valve lambda point the thermometer resistance way A, which was kept closed during measurements. G 1483 fl, and its sensitivity was 636 microdegreesi I was isolated from the liquid helium bath by the per ohm. With 50 /jlA of measuring current, thfL vacuum case B. Its temperature was controlled resistance could be measured with a precision aL by the heater F and by pumping on liquid helium 0.002 fi. The measuring current heated the thei' v in E. mometer above the cell temperature; but this effecL Changes in the pressure on the sample were was reproducible, and the same measuring curren measured by the oil manometer K and read with a It , was used in calibrating the thermometer. wa : cathetometer to 0.01 mm, which corresponded to calibrated against the vapor pressure of helium in li about a millionth of an atmosphere. The absolute on the T5 g scale [9], using a tube separate from th< pressure was read to 0.01 atm on the Bourdon gage pumping tube. The change in the thermometer P, which was calibrated against a dead weight tester. resistance at the upper lambda point after warminjj to room, temperature and recooling corresponded * Based on work performed under the auspices of the U.S. Atomic Energy Commission. to a temperature change of less than 10~ 4 °K. AJ

92 temperatures were referred to the upper lambda point in order to correct for small changes in the thermometer or the measuring circuit. Points on the lambda curve were found by warni- ng the sample slowly at constant volume. Because

i)f the large change in thermal conductivity at the ambda transition, the heating curve showed a sharp oreak, which could be located to within a micro- degree. The sample was held at this temperature until equilibrium was established before reading he manometer. Then the density was changed slightly by raising the oil in the left limb of the nanometer, and the new lambda point was found. The break in the heating curve was especially sharp

FIGURE 2. (vith solid present in the cell, because the heat of Pressure coefficient ^v = (flP/ST)v of He I along the isochore V = V(T\). melting increased the apparent heat capacity. Equation of line: /3, =9.39 + 7.69 log,,, l7"-7\) atm/°K. Thus the upper lambda point could be determined directly, rather than as the intersection of two inde- In connection with these experiments we have pendently measured curves. measured the change of pressure and density with

~~The pressure coefficient (3 v = (dP/dT) r was deter- temperature along the lambda line. These experi-~~

Inined by keeping the oil level in the left limb of ments, which are reported in detail elsewhere [10]. manometer constant and measuring the height j he can be summarized by the equations |)f oil in the right limb at successively lower tempera- lures. After the lambda point was passed, as k = - 55.54 - 96(7\ -TV) atm/°K sludged by the thermal response, the temperature dT (2) Ivas raised slowly in order to determine the lambda i>i joint exactly.

~~(^;)a = -43.7-230(7\-7Y) mg/cm :i"°K (3) Results~~

~~A The pressure coefficient /3, = {dP/BT) v was meas-~~

~~:i ured along an isochore which crossed the lambda X = 0.7868 + 2. 8(7\-7V) mg/cm atm. (4)~~

Mine very close to TV. Results are shown in figure 2, l" blotted against logio (T — 7\). The equation of where T\ is the lambda temperature for the pressure ' he line is P and density p, and 7Y is the temperature of the upper lambda point, 1.7633 °K.

j8 r = 9.39+ 7.69 logtoOT-Tx) atm/°K, (1) Discussion ai vhere T\ is the lambda transition temperature for e ;he isochore. This graph contains points taken on The logarithmic dependence of /3 r on T— T\

I>lwo different cooling runs. The lambda point for is similar to that found by Lounasmaa [7] and by o|»ne run was 15 /xdeg below TV, and for the other it Lounasmaa and Kaunisto [8] at lower densities, c'lvas 5 /Ltdeg above. These small differences are not and is consistent with their observation that the

~~5 ' egarded as significant, but the observed lambda two constants in eq (1) both increase with increas- i| mint was used in each run in plotting T—T\. ing density. The measurements of Lounasmaa~~

~~>a ! >ince this isochore meets the melting curve at the at 13 atm pressure (7\ = 2.023 °K) are of particular~~

>' ambda point, the measurements in figure 2 refer to interest since he found that /3 r fitted an equation Te /. One point was measured in supercooled of the same form as (1) to within 2 X 10" 5 °K of the t( =le//, where /3 r was found to be —27.6 at T—T\ lambda point for both He I and He II, the constants

111 = 5 — 1.08 X 10" . is figure being different for the two cases. ; This point not plotted in In the same e ' T, but it is approximately the value to be expected paper Lounasmaa presents measurements of the A l V T-TK = 1.08X 10-\ compressibility kt= p~ {dpldP)T along an isotherm

93 passing through the same lambda point. He finds no evidence for anything but a slow linear 3 variation of kt with P — P\ to within 10~ atm from He I the lambda line (corresponding to 2 X 10 5 °K for 1.763 °K T— T\) and a small discontinuity at the lambda point.

~~Since an isochore cannot cross the lambda line with a slope greater in magnitude than that of the~~

~~lambda line, {dPldT) k is a lower bound for fi v (both~~

~~are negative); so eq (1) must break down for some small value of T—T\. On the other hand, there~~

~~is no such restriction on kt- In fact, Buckingham~~

~~and Fairbank [11] have shown that if Cp becomes~~

infinite at the lambda line, kt must also. The exper- -8 -6 -2 0 2 4 10 12 3 I0 (p-P ),atm iments seem to suggest that fi v becomes infinite and x Kt does not. This situation can be examined more FIGURE 3. Compressibility k = l {dpldP) of liquid helium. closely with the help of the equation t p T Open circles: Points measured by Lounasmaa [7]. Dashed lines: lines drawn b> Lounasmaa through his points. Solid lines: calculated from logarithmic relations foi

~~= {dPldT) -{ K )- x pr x P T (dPldT)K , (5)~~

which should apply to any point close enough to line the lambda that fi v and kt depend only on the We plan to measure Kt near the upper lambda point distance from the lambda line. Solving for kt in the near future. we have To investigate the nature of the lambda trans-

~~formation, it is very important to study how kt goes, (dpldT)k ' kt (6) to infinity and how fi r approaches (dP/dT)\ (if (dPldT) K -p e indeed they do) at temperatures so close to 7\ that~~

~~eq (1) breaks down. This is probably only pos This equation shows that kt will not vary much sible near the upper lambda point. In Lounas~~

until becomes nearly equal to . fi r {dPldT) k If fi r maa's experiment at 13 atm the logarithmic relation becomes equal to (dP/dT)\, as must happen if CP that, for = 10~ 6 will less than predicts T—T\ , fi v be becomes infinite, then kt becomes infinite. If a fourth of (dPldt)\. In the present experiment^ fir were to exceed (dP/dT)\ in magnitude, k would t :i fir was a fourth of (dP/dTh at T-Tk =lO- - and become negative and a first-order phase transfor- if eq (1) remains valid, fi v will be two-thirds ofj mation would ensue. Equation (6) has been used (dP)dT) k at T-TK = lCTl We are at present en- to calculate k t along the isotherm studied by gaged in extending the measurements to smaller Lounasmaa, [7] using his equation for r and fi temperature intervals. Lounasmaa and Kaunisto's [8] data for (dP/dT)\ and (dp/dT)\. These calculated values are plotted

as the solid line in the lower part of figure 3. The circles are Lounasmaa's measured points and the dashed lines are the ones he drew through his data. Although the agreement is not very good, The author expresses his indebtedness to Pro; it is apparent that he could not have expected to fessor O. V. Lounasmaa of the Institute of Tech! see an anomaly at 1 the pressure resolution he at- nology, Helsinki, who originally constructed tht tained. A similar calculation using the data re- apparatus and has contributed valuable advice; ported in this paper is shown in the upper part of to Professor Dillon Mapother of the University oj figure 3. The much larger change in k t results Illinois for fruitful discussions during the progress from the larger magnitude of fi v and the smaller of the work; and to S. T. Picraux and Z. Sungail; magnitude of (dPjdT)x at the higher pressure. for assistance in modifications to the apparatus.

~~94 References [5] E. Maxwell and C. E. Chase, Physica 24, 5139 (1958). [6] E. C. Kerr and R. D. Taylor, Ann. Phys. (N.Y.) 26 , 292~~

[1] W. M. Fairbank, M. J. Buckingham, and C. F. Kellers.. (1965). Proc. 5th Int. Conf. Low Temp. Phys., Madison, Wis- [7] O. V. Lounasmaa, Phys. Rev. 130, 847 (1963). consin (1957), p. 50: C. F. Kellers. Thesis, Duke Uni- [8] O. V. Lounasmaa and L. Kaunisto, Ann. Acad. Sci. Fenn. versity (1960). AVI. No. 59 (1960): Bull. Am. Phys. Soc. 5, 290 (1960).

~~[2] K. R. Atkins and M. H. Edwards, Phys. Rev. 97, 1429 [9] H. van Dijk. M. Durieux. .1. R. Clement, and J. K. Logan. (1955). National Bureau of Standards Monograph 10, 1960.~~

[3] M. H. Edwards, Can. J. Phys. 36, 884 (1958). [10] H. A. Kierstead. 138, A1594 (1965). [4] E. Maxwell. C. E. Chase, and W. E. Millet. Proc. 5th Int. [11] M. J. Buckingham and W. M. Fairbank, Progress in Low Conf. Low Temp. Phys., Madison, Wisconsin (1957), Temperature Physics (C. J. Gorter, ed.) Vol. Ill, ch. Ill p. 53. (North Holland Publishing Company, Amsterdam, 1961).

~~The Nature of the Cooperative Transition~~

~~M. J. Buckingham~~

~~University of Western Australia, Nedlands, Western Australia~~

solution, each cell is properly described as a 1. Introduction action piece of one or other of the phases, but in the ab- A thermodynamic singularity arises in statis- sence of the interaction these are mixed at random. tical mechanics only in the limit N^>°°, where N The qualitative effect of the interactions between

~~is the number of atoms in the system. By regard- cells is thus the segregation or phase separation of ing an infinite system as made up of an infinite the cells. The entropy associated with this sepa-~~

™8 ' number of large but finite parts, called cells, with ration is constant (depending on composition) in M atoms each, the singularity is seen to arise from the two phase region and is, of course, zero in the the interactions between the cells — which for M one phase region — except for a very small tempera- al ' large correspond to surface interactions, since the ture interval in the immediate neighborhood of

10 ' interatomic forces are short -ranged. For the same the transition temperature. * reason interactions between other than nearest At first sight the very small magnitude of the mix- m neighbor cells are zero. Thus any thermodynamic ing entropy — associated with but one degree of ,ia11 singularity may be regarded as an example of a freedom out of M, would suggest that it is entirely generalized near-neighbor lattice problem. negligible. However the extreme smallness of the 1/3 For very large M, one might expect the approxi- temperature interval (~ 1/M) over which it can mation of neglecting the surface interactions change means that, in some circumstances, it may

11 between cells to be a very good one since it cor- be far from negligible — indeed accounting for the " ft responds to the neglect of only a fraction 1/M 1/3 of asymptotic form of the thermodynamic singularity.

~~all interactions. Indeed for a homogeneous stable An example is the ordinary first order transition from thermodynamic system the approximation is ex- gas to liquid phase, as mentioned below. The case~~

tremely good. However it is not good near a singu- of the critical transition is much more subtle, but larity or in a two phase region, since the correct, by incorporating the assumption that the pair corre-

~~partition function for the system is then singular, lation function is just a function of r/ro where r~~

Pro< between-cells problem results in some power of the with finite range forces only, ostensibly possessing partition function for a finite system of M atoms, a "classical" van der Waals critical point would in

which is nonsingular. fact have to have a logarithmic singularity in its ice; It is easy to see the nature of the solutions in the specific heat. The discussion that follows is not two phase region. The correct solution describes intended as a "theory of the critical point" but as a 0\ a system with two phases separated in space by preliminary account of the viewpoint that is being some minimum-area boundary. In the zero-inter- developed.

~~95 )~~

~~2. Division Into Cells When Z\ is at a normal, homogeneous stable thermodynamic point, then the operation of changing~~

Consider a system with N atoms in a volume V, A. from 0 to 1 has a negligible effect for large M. described by a Hamiltonian with finite range forces However, near a point where say the specific heat at constant volume for the system, only, and with an exact partition function Z(N, ft). Cy— A=l Imagine a division of the system into Jf "cells" C\ remains finite but dependent on M, for the A = 0 each with M atoms (N = M.A-"') but with no restriction system, see figure 1. on the volume or energy of a cell (or of the mag- It is easy to show at a first-order transition for netization in the case of a magnetic system). Num- example that whereas C,,(T) has a 8-function singu-! larity, the corresponding (C ,(T))M reaches a value ber the cells with i = 1, . . . . Jf. t In any configuration contributing to the parti- of order M with a width — (all thermodynamic quan- tion function, separate the interactions linking atoms in different cells. Imagine, for convenience, tities being expressed per atom). Thus near

8T 1 v the latter all multiplied by a parameter A say. U , • U" 1 U enough (within — — say) to a singularity the , Thus 1 c M operation of putting A—>1, far from being negligible indeed produces the dominant effects and can H = H; (tf X {H ~ > + 2 int ) + X. >nt (Hint J 1 i=l i.j i.j be regarded as leading for example to the infinity in the specific heat. where the first term contains all interactions link- Considering now the problem of obtaining the 1 ing atoms all of which are in the same cell; inter- exact solution for the system from the solution actions between atoms in different cells contribute Z(M, ft) we observe the following: The solution for to the last term. (1) denotes mean value. Clearly A = 0 represents an ensemble of cells with extensive for M large enough, interaction between atoms parameters distributed according to the partitiom linking' cells other than near neighbors can be function Z(M, ft). Thus at a temperature a finite: finite of the forces, neglected because of the range amount above a critical temperature, say at 7V + t,i so the sum of interaction terms can be written as the density of cells is distributed about the meani a sum over near neighbors only. Note that the value (say the critical density p c) with a Gaussian! exact Hamiltonian is obtained with A=l. distribution of width corresponding to the fluctua-i

~~Let Z\(N, ft) be the partition function for the~~

~~A.. is system for some value of Then Z\{N , ft) the exact Z(N, ft) while for A = 0 we have a collection of noninteracting cells which can exchange energy, volume, etc. Thus~~

~~z„(/v, ft)={Z(M. p>y~~

This relation is exact, with Z(M, ft) the exact partition function for a system of M atoms. We use Z somewhat schematically for the appropriate partition function — the constant pressure one in the case of a gas liquid system for example. We will use the fact that, considering ft=l/kT as a com-

~~1 plex variable, Z(N, ft) is analytic in the right half-plane for /V finite, but not necessarily so in the limit N —> oo.~~

~~Even when there is a singularity in Z{N, ft) as~~

~~N—>, the corresponding Z 0(/V, ft)(k — 0) is analytic. T~~

~~FlGl'RE 1. Sketch of the qualitative temperature dependence o 'Z is the Laplace transform of the nonnegative degeneracy function~~

~~96 tions of the density for cells of size M.~~

~~T »T f~~

~~— 2 ~ X (compressibility). Pip) Pip) ((p p c ) )~~

On the other hand, at a finite temperature below the critical temperature, Tc~t, (in a 2-phase coexistent region) the possible density values will be distributed in two Gaussians separated by the FIGURE 2. Probability P(p) of finding a cell with density p,

density difference of the two phases in equilibrium, at temperatures above and below Tc . figure 2. — We can regard the A. > 1 problem as a generalized, near neighbor, lattice problem and we can Thus "switching on" the interaction between cells estimate the magnitude of the effective interaction does nothing to the distribution of the states of parameter. It is the total interaction energy due cells when T> Tc + t since the effective tempera- to interactions finking atoms across the boundary, ture of the lattice problem is then T^ff ~ 00 whereas say for the interaction between cells 1 and 2. This for T

"ii (piPi) these are mixed at random for A — 0 but separated 2 I 3 i.e., M € 0 or Em 1. 2 for \=1. The entropy change involved is thus ve In 2 —^j- (per atom) and all this is lost in a very small where e0 is the strength of the interaction (eo ~ kTc) ite is at the of cell 1. In the and pi the density surface temperature interval around the critical -. problem with A = 0, pi is typical of the density any temperature. anj where in cell 1 so p!~pi. We now approximate an E- by dropping the superscript on p\. The effec- ln 2 mt , AC ZV5 = —r— per atom. (2.1) ia- 1 ti ve interaction parameter of the lattice-interaction

~~[problem is the difference between the E int in oppo-~~

~~site configurations: i.e., 3. Long Range Density Fluctuations~~

~~— ). _1 EintiPu pi) + Em {p-2, p2) 2Emt(pi, p 2 This entropy contribution, M ln 2, arising for~~

Thus the effective interaction in the lattice problem T < Tc as sl result of the sorting of the previously is of order (A. = 0) randomly mixed "fluctuations of size M" might seem at first sight completely negligible. ApiAp 2 213 2 / 3 There are, after all, surface terms of order M e0 M times greater. However we note that the contribu- _1 tion remains constant, M ln 2, throughout the The equivalent lattice variable is the ratio of the temperature range T < Tc — t, and all its effect oc- extreme values of to E int kT curs in a small temperature interval, approaching zero as M— about the critical temperature. (It may be noted that if the density had a value other than the critical one, say corresponding to a fraction / of the atoms in one component and <£LZ££. (1—/) in the other, then the entropy would be ^.5±M*tz x tTr + T transition at temperatures and pressures below kT M- the critical values.) This mixing entropy term corresponds, of course, oo T < Tr—T. to just one degree of freedom per cell of size M,

~~97 L~~

say, cells of sufficiently large size will be statisti- but it describes the relative configuration of the large size density fluctuations — and these are what cally independent of each other, whereas those of ultimately become correlated near enough to the smaller size will not be. We can determine the transition temperature. Thus if Im is the linear number of statistically independent regions or dimension of a cell of size M, the number of Fourier elements by examining the fluctuations. components of the density with wavelength greater 5. Number of Statistically Independ- than is just the same number, namely 1/M of Im ent Elements all the degrees of freedom.

While it would seem quite possible for many We are concerned with the development of cor-j more than one degree of freedom per cell to be relations in a particular extensive property of our involved in the cooperative transition — and in- system — namely the one .that is becoming long- deed this is apparently the case for the two-dimen- range ordered and identifies the two components sional Ising problem, we offer a tentative proof in the two phase region. The fluctuations of this that, in fact, just this contribution is sufficient to quantity determine the number of statistically in- make impossible the "classical" critical point — dependent elements we require. with properties typified by van der Waals equation. A number n of things each with an extensive statistical with finite range forces That is, a system 2 property x, with mean square deviation dx , will, (in any number of dimensions) would not be self- if they are statistically independent, have a mean consistent if it possessed a "Classical" critical — 2 = 2 = square deviation ((X X) ) n8x , where X that a point. In particular, we show system whose n compressibility diverges as a simple pole as a func- 2 x\. In a gas-liquid system x\ is the number of] tion of temperature must have at least a logarith- atoms in a volume element bV i.e., p8V. This mic singularity in its specific heat. number is either 0 or 1, and has mean value pdV. The number n of statistically independent fluctuat- 4. Calculation of the Mixing Entropy ing units in a large volume with mean number N\

In order to obtain quantitative information con- of atoms is then cerning the interaction problem we shall invoke the 2 fact that the nature of the problem is independent _ p n of the size, M, of the cells (provided M is sufficiently ((p-m large); the size only determining the scale of the which reduces, as it should, to /V in the absence of temperature interval in which the transition oc- any correlation, i.e., at infinite temperature. The curs. For this purpose we now define a sequence mean number of atoms per independent element of cell divisions with ever increasing size of cell. is then In the cell division defined above in section 2, let us replace M by M0 and A. by A 0 , and introduce = a new parameter \i multiplying the interactions M=^=^p-N^kTpKT X(t), (5.1) n pz at the boundaries between cells of size M\ > M0 . Thus marks the interactions which would have \i say where Kt is the isothermal compressibility. Inl been marked by A. before, had we chosen Mi in- the case of a magnetic system, the corresponding! stead of M. For convenience let M\ = JfM&< V>1). number of statistically independent magnetization 1

~~Similarly let us also introduce A2, A . . . \„ . . ., 3 elements is the magnetic susceptibility in units of~~

~~dividing into cells of size M->, :i , ...... , M M n the magnetic moment per particle.~~

~~respectively, with M„=. VM,,-\ = . Y"Mu.~~

, — Now by setting Aa- = 1 k # n and k„ 0, we obtain 6. Recurrence Relation a division of the system into cells of size M„, which we call the nth order. The exact solution can now We have seen that the fluctuation of the natural! be regarded as that obtained in the nth order as extensive variable of the system determines thel — n —* °°, or as A.,, * 1. We now determine the tem- size M(t) of the statistically independent regions.! perature interval required for the transition in the Returning to the cell division of our system, welf various orders. At some very small temperature assume that the temperature interval over which thej

~~t = T/Tc—l, above the transition temperature, nth order undergoes its transition is the tempera-~~

98 using the result (5.1). derivative of /has been is. The ture, tn , such that M„ ~ M(t„) = x(tn)- That using written /'. the result (2.1) obtained in section 2, the nth order ^ solution would lead to a change of entropy For a "Classical" critical point xW but 2 ~ ^ f y = in inter- for other suggested singularities x where AS„ ln 2/M n , occurring a temperature > 1. val t„. y Let write We can now assert that for large enough cells the us functional form of this entropy change is given by X{t)- l = a'ty y*?l (7.2)

~~In 2 ASn(t) = f(tltn) (6.1) eq (7.1) becoming~~

where the function / is independent of n. This CJ$-CAfi) = atl-y%tltn), (7.3) assertion enables us to write a recurrence relation 1 where a is written for a In 2. The cases 1 and which we solve for the function /, and is the main y= > 1 are different first the case source of our argument — it depends on the fact y and we discuss that for large enough M (or small enough t) the y=l. interaction problem has ceased to depend on any detailed property of the interaction between atoms, 8. The "Classical" Critical Point, or of any finite number of atoms and has taken an y=l asymptotic form. We are effectively assuming the existence of an asymptotic "lattice problem" The equation for the singularity becomes, for the such that the description of the nth order in terms case y=l,

of the (n— l)th is the same as that of the (n+l)th = in terms of the nth. Cjt)-Cn(t) af'(tlt ll ) (8.1) The function f(x) is continuous monotonic in- 7 creasing and satisfies the boundary conditions It is easy to see that the function/ ^) cannot be

/(*)- 0, x> 1 and f(x)^> — l, x< — \. finite at x = 0. For let such finite value of the right The partition function corresponding to the side of (8.1) be Co. Then C„,(0) - C,(0) = 0 for choice k,, = 0, \a- = 1, k n, i.e., the nth order, is any m, n and yet C x (0) — C„(0) = C0 ^ 0. We find a valid approximation to the exact one for t > t that the asymptotic solution requires J'ix) n , below since regions of size M„ are then statistically inde- In x as x^ 0. pendent. Similarly, except for the constant mixing Taking the difference of (8.1) for n and n + r, we entropy, it is valid for t < — t„. The exact solution obtain A.ji=1, contains the additional contribution to the entropy given by (6.1). Cn+At)-Cn(t)=a{f'(tlt,) -/'(, I "'*/«„)}

~~where tn\tn-v\ — Jf- Setting x = tjt n we write this 7. Difference Equation for Specific equation as Heat~~

~~"'*)} Cn+r(xtn)-Cn(xtn) = a{f'(x)-f% I (8.2) Let C»(t) be the specific heat corresponding to~~

X„ = 0. C„{t) will have the partition function with where the right-hand side is independent of n and the shape of the exact specific heat except that the rr can be written ag(J x) say. Now {t) is finite , Cn singularity will be smoothed out over an interval for all t and by putting r—\ and, repeatedly, n + 1

~~~ t with C„(t) finite for all t. can now write, n , We for n, we find using 7V as the unit of temperature,~~

~~C ~ = ar g{yf , 0) = ar ln Jf n+M CM g() C x (t)-Cn (t) = ^±fltltn) M„ at where gfo is a constant (independent of r, and n).~~

~~ln2 f'(tltn) (7.1) - See for example: M. E. Fisher. J. Malli Pliys. 5,944(19641. The Iwn dimensiimal Xitn)tn lsin~~

~~99 209-493 O-66--8 o~~

~~Thus furthermore, the coefficient of the logarithm, a in~~

~~eq (7.2), is proportional to the inverse of the co- C„+i(0) - C„(0) = ago In JIT, (8.3) efficient of (T—Tc)~ l in the compressibility (5.1).~~

That is we expect so that, increasing the order n by one increases C„(0) by a constant independent of n, and thus C„(0) ~ k-iCv — a In fcp) + C0 r 1 increases linearly with n. Now = J t„-i = • = Jf~ nT say, so that

~~where a ~ In 2 pMT— T< )k t , (8.6) - In (tjT) n " - InJf A- being Boltzman's constant, kt the compressibility~~

~~and p c the critical density. Thus~~

~~9. Nonclassical Critical Point, > 1 Cn(0) = Co + nago In yT, y~~

~~becomes We do not give a detailed discussion of this case, since it is not hard to show that the solution of eq~~

~~(7.3) for > 1 is not singular. This suggests, in C„(0) = Co — a go In U„/t). y terms of the discussion in section 3 above, that a~~

~~Thus we see that the specific heat becomes infinite singularity with y > 1 involves more than just the like the logarithm of the temperature. We can one degree of freedom per cell which describes~~

easily find the asymptotic form of the function f'(x), the spacial arrangement of the large scale fluctuations. since by (8.2) and (8.3) we can write It is not a pleasing circumstance that the simplest -> possibility in the present theory appears inade- f'(x)-f'(.\x) g0 lnJT. (8.4) x->0 quate to account for the behavior of the two dimen- Now the equation sional Ising problem, but further discussion of these points must be left to another occasion. F(x)-F(bx) = C 10. Further Properties of the has the solution Transition

F(x) = —.— h-Fo We have discussed at some length the nature In o of the singularity to be expected at the critical point of a system which could be said to have a where F 0 is an arbitrary constant. Hence the "Classical" critical point in the "zeroth order" asymptotic solution of the eq (8.4) is approximation. We have seen that the specific heat diverges logarithmically with a characteristic /'(*) -> -g In 0 x+f0 . (8.5) t —» coefficient. The singularity is not necessarily symmetric however. Certainly the constant Co in

~~The eq (8.1) whose asymptotic solution we have (8.6) is not the same on the two sides of the transi-~~

found contains an arbitrary multiplicative constant tion temperature. In fact it differs by approxi- which should be determined from the normalization mately the amount of the discontinuity in the original condition given in section 6. We can see from "Classical" specific heat. We have obtained a

~~(8.5) together with this condition that g0 will be a first approximation in which the coefficient, a, is number of order unity. different above and below the transition, but there~~

Summarizing the results of this section we assert are indications that this is not the final asymptotic that a system whose compressibility diverges like form and that the latter may well be symmetric. a simple pole as T—» Tc cannot possess the "Classi- The preliminary results of this approximation lead cal" specific heat temperature dependence, but to a coexistence curve different from the classical

~~that it must have a logarithmic divergence and one, which is a quadratic relationship between the~~

100 density difference of the two phases and the tem- hoped that some will soon be available. In passing perature difference from the critical temperature. we note that the van der Waals equation would 9 This could also be described as a quadratic relation yield for the value of a in (8.6) the number U In 2, between the density and the energy difference from (for T > Tc) which is near values that have been the critical energy — the energy being taken on observed. the critical density line. This relation between

~~the extensive parameters is preserved in our result, but the relation between energy difference and temperature is no longer linear — in fact This work was carried out with the enthusiastic~~

E — Ec^ t In t. This results in a coexistence curve and able assistance of J. D. Gunton. Much of which can be represented: it was performed during the tenure of a National Science Foundation Senior Foreign Scientist Fel-

~~0C 1'2 lowship at Stanford University, which is gratefully Ap = p,-p 9 (~t In t) acknowledged. or, equivalently, Discussion~~

C. N. Yang: We are very grateful to Dr. Fairbank for presenting his most interesting results. One cannot emphasize too much — In Ap that one of the most important results of recent years is the verification that in some quantum mechanical systems there exists a new long range order which is macroscopic as is demonstrated by this experiment. In 1961 Dr. Fairbank was also involved in It is useful to express this in terms of the natural an experiment which demonstrated the existence of quantized variable x, in this case x — pi~ pjpi + py, and in the flux in a superconductor showing the macroscopic effects of long range order [1]. These together with the recent discovery case of a magnetic system, x = M/Mo where M is of the Josephson effect [2J, have proved once and for all that the spontaneous magnetization and Mo the satu- quantum mechanical systems at very low temperatures often ration magnetization. The natural variable ranges exhibit correlations which are different from crystalline correlations, but which also extend to infinite distances. Dr. Fairbank between 0 and 1 as t = 1 — T/Tc, ranges from 0 to 1. said that this experiment was brought about on a fifteen dollar In this variable the expression for the asymptotic bet from theoreticians. Let us hope that if we make a bet at the thirty dollar level, we will obtain more results. coexistence curve is

W . M. Fairbank: I would like to point out that Fred Kellers carried out the experiment on the X point I just described. x 2 oc i — In x M. E. Fisher: It is perhaps worth pointing out that, although you do have a X-line in liquid helium and not a X-line in the ferromagnet, in the antiferromagnetic transition you do have a very but this is not very suitable for comparison with close analog of the X-line. In fact, in a spin flop antiferromagnet, experiment because of the spurious singularity I think, you have something rather closely analogous to the off-diagonal long range order which Dr. Yang was talking about at x—l. An expression with the same asymptotic [3]. form, but with the correct limit at T=0 is

L. Tisza: The X-"point" in a ferromagnet is a point only in the x2 same sense as the melting "point" is. Both refer to atmospheric oc i pressure and a variation of this parameter yields 7\ (P), a line 1 — In x in the P-T plane. This is in contrast with the critical point in fluids.

This expression (in which the coefficient unity J. S. Rowlinson: Even at the liquid-vapor critical point of fluids multiplying In x is quite arbitrary) has been com- you do have a X-line if you have a two component system, where there is an extra degree of freedom. pared with his measurements of the coexistence

curve for liquid helium by Dr. M. Edwards who L. Tisza: This is correct. The prediction of Gibbs' phase theory has presented the comparison to this conference. is that the dimensionality of critical states in the thermodynamic space of intensities is 8 = c— 1 where c is the number of inde- It is interesting that he obtains an agreement pendent components. The difficulty in interpreting X-lines as rather better than with the usually accepted cubic critical states is that this would mean 8=1 for c=l, a contradiction to this rule. Ehrenfest [4] sought to remedy the situation relation. Concerning the experimental test of the by assimilating the X-lines to lines of ordinary phase equilibrium. the case 1.) His problem was then predicted relation (8.6) between the coefficient of (I confine this remark to c= to explain the absence of discontinuities in the first derivatives the logarithm and the compressibility, we have of the Gibbs function that are characteristic, say of melting. this transitions for which unfortunately not been able to obtain data with He did by postulating higher order nth derivatives of the Gibbs function exhibit discontinuities which to make a useful comparison but it is to be along a line in the space of intensities, while the lower derivatives

101 are continuous. Shortly thereafter Landau showed that such the critical temperature on the r58 scale. I would therefore transitions can be indeed expected to occur on the basis of the suggest that one should treat the plots made to fractions of a

thermodynamic theory of stability. The seemingly satisfactory millidegree from the critical temperature with reservation, if situation was upset by Onsager's discovery of the singularity this is indeed that uncertain. In view of the thermometry in- of the Ising model, particularly since experiment showed that volved and my inability to put a thermometer into the actual logarithmic singularities provide good descriptions also of real optical path, I do not know whether this difference is precisely

~~systems. These developments posed two problems for the 0.006 °K, but I think that there is a significant difference. It is~~

~~student of Gibbsian thermodynamics: (i) identify the flaw in conceivable that my whole curve is shifted somewhat. I should~~

Landau's argument; (ii) generalize Gibbs' phase rule to account have said that for the A^-plot this would require a shift of 0.020 °K, for X-lines. It can be indeed shown that [5] (i) at critical states which is certainly beyond the realms of possibility. The A^-plot the discriminant of the so-called stiffness matrix vanishes and leads to a critical temperature somewhat above the listed one this invalidates Landau's power series expansion; (ii) at critical depending on where you put the tangent. states there are two modifications that become identical to each other. In the classical case these modifications differ energeti- M. J. Buckingham: I would like to make two remarks. The cally (liquid and vapor). At \-points we have, say, the two polar- first is a plea for the use of the proper natural variable in studies ized states of the Ising model which are energetically equivalent of the coexistence curve of gas-liquid systems. This is the var- — and which transform into each other by a symmetry operation; in iable X = {pi p„)(pi + pg ) = (va — vdivy + vi), rather than the often this case the dimensionality of singular points becomes 8 = c — used (pi pg )l2pc . The variable X ranges from 0 to 1 as t=l is of and the above mentioned difficulty removed. The nature — 777V ranges from 0 to 1 and is indifferent to the choice of this symmetry is evident in all cases involving solids where the number of atoms or volume as the basic extensive variable. The mechanism underlying the X phenomenon is understood. A other variable and X are equal if the "rectilinear diameter" is remarkable exception is the X-line of liquid helium. I find not only straight but constant as in the case of the Ising cell myself incapable of refuting the argument that this system model. It has other advantages for plotting experimental re- exhibits some elusive intrinsic symmetry. This is interesting sults as mentioned by Dr. Edwards. also in view of the recently discovered singularity of C r at Concerning the particular form of the suggested coexistence fluid critical points which seems to point into a similar direction 2 function, namely X I(\ — In X), I should say that the theoretical (see the contribution of Dr. Chase on Wednesday afternoon). argument which I will mention in my talk later, merely indicates A precise formulation of this symmetry may well be at present 2 -1 the asymptotic form — X j\n X for X —* 0. The factor (—In A') the most challenging problem of the theory of critical points. in this particular expression is not useful because of the spurious infinity at X= 1. One is led therefore to consider the factors

P. W . Anderson: For a study of fluctuation phenomena in super- — ( 1 n In X)~* which have the required asymptotic form and which conductors, perhaps the best direction there is not to go to infor all n are zero at X = 0 and unity at X= 1 where the slope is n. creasingly pure systems but to increasingly impure systems, These properties are shared with the set of functions X"; indeed because the amount of the critical point fluctuations will increase their general shape is remarkably similar to tjjie corresponding when one shortens the correlation length. So one would hope to simple power function. bring out the critical point fluctuation phenomena by decreasing We can now emphasize the great difficulty of obtaining re- the path, thus by making an increasingly impure system, which, liably the correct asymptotic behavior of empirical results how- however, should be as homogeneous as possible. ever accurate, over a limited range of the variable. As Dr. 2 Edward's illustrations showed, the function X /{1 — In X) is over /. Rudnick reported measurements of the sound velocity near the his liinge of parameters, numerically much closer to the func- X-point in liquid :< 1 [6] helium. tion X than X , although the corresponding value for Fisher's 2 coefficient /3 would be V2 as for X' , rather than V3.

M. J. Buckingham: With regard to the paper of Dr. Rudnick, I would like to point out that if one would find a violation of the L. Tisza: Even if the Buckingham variable X is "natural" for Pippard relations, this means either that the system is not in the analysis of the coexistence curve, it is not a substitute for p thermodynamic equilibrium or that the singularity is not qualita- used with its conjugate intensity p. (see paper of Chase and Wil- tively the same all along the X-line. The latter property is used liamson). In fact, X is defined only on the coexistence curve in deriving the Pippard relations [7]. and has no meaning for the homogeneous fluid.

E. Helfand: I want to mention something which surprisingly the has not been mentioned yet. Perhaps, it may be important here. J. S. Rowlinson: Dr. Moldover gave us some information on There has been some thought recently about whether some of variation of C, with the density in the two phase region which he 2 these apparent second order phase transitions really should not said implied (

R. E. Barieau: Recently, I have analyzed the data of Hill and C. N. Yang: Could I ask Dr. Edwards what happens to the critical Lounasmaa [8] for helium in this way to obtain the quantities 2 2 2 2 temperature if you make a plot of the variable X I(\ -\nX) that (d p/dT )\ and (cPpldT )^. Now ordinarily below the critical 2 2 Buckingham suggested? point (d p,/dT )t is negative, so that the two contributions to C, 2 2 2 2 namely — NT(d p.ldT )v and VT{d pldT )\ are both positive down around the boiling point. Now when the critical point is ap- M. H. Edwards: I forgot to point out that on this basis the best proached, the data of Hill and Lounasmaa indicate that straight line goes through a temperature which is 0.006 °K below 2 2 — NT(d u.ldT )i- is first positive and then becomes negative.

102 M. R. Moldover: I believe the data of Hill and Lounasmaa were M. E. Fisher: Well, I do not want to defend the measurements taken at intervals of 0.050 °K or larger and were not taken par- of Weinberger and Schneider either, but they did a series of ticularly at the critical density. experiments to make a special study of the density effect. (The experiments on xenon, hydrogen. He 4 and He 3 have since been R. E. Barieau: That is correct. They made the measurements analyzed systematically with a suggestion as to the nature of at different densities and the measuring points were not chosen the quantum deviations [3].) to get these two quantities with the maximum accuracy. How- ever, it is possible to deduce some figures for these quantities M. H. Edwards: Yes. but only down to 1 cm. They had a cell and there are. two series of experiments between 4.2 and 4.5 °K that was approximately 1.2 cm by approximately 10 cm. normally in which the densities are far apart so that one can obtain these quantities with considerable accuracy. used in the vertical position. They rotated to the horizontal position to change the gradient from a height of 10 cm to a height

of 1 cm. but not further. I claim that with a smaller slice of M. R. Moldover: Yes, there is a conflict between your conclusion the fluid I got a different result. from the Hill and Lounasmaa data and mine. With respect to the data of Hill and Lounasmaa in the vicinity of the critical point it should be remarked that they used a large chamber in which G. B. Benedek: I wonder whether we could clear up the question complete thermal equilibrium is difficult to realize. not of the difference in flatness between your data and those of Schneider, but between your data and those of Moldover. M. H. Edwards: They also smoothed their data.

Dr. Moldover made the point that he had evidence that his den- M. S. Green: 1 would like to ask Dr. Edwards and Dr. Moldover sity curve was flatter than a dotted curve of yours. how much the density dependence on the height in the fluid would affect their experiments. I think Dr. Edwards had some control of this height. Is that correct? M. H. Edwards: Not entirely clarified. On the density versus temperature plot Woodbury and I had results that were extrapolated to go through a which to the M. H. Edwards: Yes. Of course, this effect is not important temperature corresponded further away from the critical point. At least at 5 °K the cor- T^h value of the critical temperature. That is the temperature to he is it with. evidence suggests rection due to the compressibility, which incidentally, I have which comparing My now that critical If I fit measured in this range, is less than 0.05 percent in any of the the temperature should be reduced. now individual density measurements. The liquid had only a helium the data not with the power series but with the logarithmic function, which calculation I have not yet completed, and extrapolate, liquid pressure between 1 mm and perhaps in the extreme case the curve becomes flatter. 4 mm above it. This correction was completely trivial even while the extrapolated compressibility is going to infinity. However, the real answer is that our experimental data do not yet overlap so that the disagreement is between an extrapolation

based on a misconception and a clear experiment. I would say M. R. Moldover: Most of the helium in my cell is in the slots that the weight of evidence is in favor of the direct measurements which are 3 mm deep. There will be a distribution of densities of Dr. Moldover with respect to this point. As he pointed out, in this helium column as you approach the critical point. Dr. he is measuring a very large quantity. Edwards measured the compressibility far from the critical point and also gave an equation of state, which he derived in the earlier days when one used a power series expansion. Using this I R. Renard: I would like to point out that Dr. Garland and I have calculated that in my cell the distribution of densities is less than developed a theory in which not only the spin contribution but 2 percent at the temperatures closest to the critical point. So also the lattice contribution to the specific heat of a crystal is on the plot I presented this gives a contribution to the curvature. taken into account. For an Ising model in two dimensions it can be shown that the system is mechanically unstable around temperature, leading to a hysteresis. This must also M. S. Green: In computing this correction one should probably the critical for a three dimensional system if it is correct that C, use the nonanalytic character of the density dependence as Dr. be true to infinity at the critical temperature for this model, where Larsen pointed out yesterday. It actually makes a significant goes replace by dT/dp along a line of instability difference. One cannot be satisfied to use an analytic expansion one has to dTJdp that C,, should go to infinity but that C, in computing this correction. points. It is predicted would not.

M. E. Fisher: I want to ask Dr. Edwards: what is the actual value of the parameter X in your experiments? In their experi- References ments on xenon, Weinberger and Schneider [9] approached the 10"" 5 critical . = point to within A7'/7'( 3 X , while you essentially [1] B. S. Deaver and W. M. Fairbank. Phys. Rev. Letters 7, 43 approached the critical point only to within 2 percent. So (1961). Weinberger and Schneider approached the critical point a few [2] B. D. Josephson, Phys. Rev. Letters 1, 251 (1962). decades closer, but even as close as that this parameter X was [3] See, M. E. Fisher. Phys. Rev. Letters 16, 11 (1966). 0.07. In other words, the curve is so flat that the density dif- [4] P. Ehrenfest. Leiden Comm. Suppl. 75b (1933). ference pt — fig is still very large. Dr. Rowlinson made the point, [5] L. Tisza, Ann. Physics (N.Y.) 13, 1 (1961). See also L. Tisza which I think is a relevant one, that you have to approach the "Phase Transformations in Solids." Smoluchowski. Mayer. critical point considerably closer. and Weyl. eds.. ch. 1 (John Wiley & Sons. Inc.. New York, N.Y., 1951). M. H. Edwards: Yes. this is a curious feature. I went to within [6] I. Rudnick and K. A. Shapiro. Phys. Rev. Letters 15, 386 2 percent from the critical temperature. My corresponding .Y (1965). values — and consequently my density differences pi pu , however, [7] M. J. Buckingham and W. M. Fairbank, Progress in Low are smaller. In other words, their xenon data look flatter than Temperature Physics, ed. by C. Gorter (North Holland my helium data. After studying the data of Weinberger and Publ. Comp. Amsterdam. 1961. Vol. Ill, p. 80). Schneider. I concluded that they did not adequately guard against [8] R. W. Hill and O. V. Lounasmaa. Phil. Trans. A252, 357 the gravitational gradients effe 'ts. They had a sample that was (1960).

~~1 cm deep and the compressibility correction is 15 times what [9] M. A. Weinberger and W. G. Schneider. Can. J. Chem. 30, it would be for helium in the corresponding case. 422(1952).~~

~~103~~

~~SCATTERING: ELASTIC~~

~~Chairman: P. Debye~~

~~Introductory Remarks~~

~~P. Debye~~

~~Cornell University, Ithaea, N.Y.~~

Since Dr. Montroll will not talk, I will take the relaxation time, if you want to call it that way de- opportunity to speak two minutes to give you my pending on the circumstances. This would be conception of what he would have said. I think he completely phenomenological. would have said that, if you want to look at what I was prepared to tell him that this is a linear radiation is going to do when it falls on a medium theory, which you generally call a Born approxima- with irregular fluctuations, you have to characterize tion. If you come in the neighborhood of the these fluctuations first in a phenomenological way. critical point, the correlations become bigger and

He would have started (don't believe what I say; bigger and bigger, so that probably your linear this is only my conception) by characterizing these theory will not work anymore. If you want to tell fluctuations by a correlation function. Then he something about the angular distribution of the would have insisted that this correlation function frequency distribution of the scattered radiation has to be a correlation function in space as well as derived from actual observations, you have to intro- in time. Then he would have said that from this duce the right theory. The linear theory may not correlation function you can calculate the intensity be correct at the critical point, because you get of the scattered radiation directly by a simple such big fluctuations. So I had hoped to induce Fourier transformation. Then he would have him to say something about the second approxima- said that you have to look at two things in the tion and about what we are going to see there at intensity distribution. First, how is the intensity least in a qualitative way. distributed over the angles? This geometrical feature of the intensity gives you an opportunity to Now, this is, of course, a phenomenological characterize a length. Then he would have said background. If you want to know anything about that you can also make a spectral investigation of your correlation function, you have to build that the scattered light, which would be characteristic upon a molecular theory. I hope that we will for time correlations. Then he would have said that hear something about this molecular background. this is characterized by a correlation time or by a What about it, Dr. Fisher? From you?

~~107 Theory of Critical Fluctuations and Singularities~~

~~M. E. Fisher~~

~~King's College, London, Great Britain 1~~

(1) In the previous lectures the behavior of the and macroscopic thermodynamic functions of systems in equilibrium near critical points has been dis- f/maS netic ^-^yWHr), (6) cussed. Notations for the various exponents r describing the critical singularities have been in- where is the pair potential, with Fourier trans- troduced and the connections between them have (f(r) form V T(r), (8)

~~r=t=0 2 G(r) = g(r)-\ = v n2 (r)-l (1) and in a magnetic system in zero field which are directly related to the fluctuations of the Fourier coefficients of density and spin deviation~~

~~r(r)=r^(r)=/gS(S + i) (2) through~~

2 2 2 both of which reduce to (s Sr) in a simple Ising (|8S <|8p k ! ) k | ) = X(k) 2 (9) spin system (s = ± 1) or a lattice gas at critical den- Vp /V<|8So| ) sity. In terms of these functions the compressibility and magnetic susceptibility of the systems are The thermodynamical formulas (3) to (6) may then given by be rewritten as

~~63 eal ' = - K T Z^ 1 Xt/Xt , (10) \ x(0)~~

~~1 + G(r)dr (3) and~~

~~deai « At/ a xr=xV 1+2 T(r) (4)~~

Furthermore the scattering intensity observed with Furthermore the internal energy can also be found light, x rays or neutrons at wave number k = (4>irlX) since X sin (6/2) is given (in first Born approximation) by

~~^coring— hp~~

~~where 7o(k) is the intensity which would be ob- ' Present address: Cornell University, Ithaca, New York. served from the ideal, noninteracting, system.~~

~~108 These results show that x(k) is the quantity of most fundamental and direct theoretical and experimental interest. In as much as a critical point is associated with an infinite divergence of~~

Kt and \t we see from (3) and (4) that the correlations must become of increasingly long range near a critical point. This, in turn, implies some singularity in the behavior of x(k) for small k as T

~~approaches Tc . Indeed a rapid increase of x(k) for small k is the mathematical expression of the observed critical opalescence.~~

~~(2) Let us review briefly the "classical" approxi-~~

2 mate theories of the behavior of the correlations FlGl RE 1. Plot of inverse scattering intensity versus k near the critical point to be expected on the basis of the Ornstein- near a critical point. The first theory was that due Zernike and equivalent theories. to Ornstein and Zernike [2] for a fluid; in many

ways it is still to be preferred since its theoretical assumptions are clear and easy to appreciate. along the critical isochore or in zero field. It fol- _1 Firstly, the direct correlation function C(r) is intro- lows that a plot of 7 (k) should intersect the origin duced via the relation for its Fourier transform. when T=TC (see fig. 1). Inverting the transform (15) for a (/-dimensional X(k)=l/[l-pC(k)]. (13) system yields:

~~°° (a) for fixed T > Tc and r—>~~

It is then argued, or assumed [3], that C(r) is of ~ e-^/rW- 1 *'/ 2 short range and remains so up to and at the critical C(r) , (18) point. If this is so one may expand its transform °o in a convergent Taylor series as (b) for T= Tc and r^>

~~2 2 = . • • , G(r) = (r)~ l/^" C(k) C(0)-rik + (14) Gc , (19)~~

~~2 since C(k) is analytic in k even at the critical point. where l/r° corresponds to logr foro?=2. In general~~

The "interaction length" r0 is constant (or slowly these results can be combined into a formula of the varying) and remains finite at the critical point. form Truncation yields the well-known result Kr G(r)**D(er li*-')[l + Q(Kr)'], (r-> «) (20) 2 xW-roW+K ), (15) where

where k is related to C(0) and turns out to be the ()(*)-»• 0 as *->0, (21) inverse range of correlation. From (15) a plot of the experimentally observed inverse scattering and intensity I~ l {k) should be linear in k 2 at all tem-

~~(d - 3) 2 peratures approaching Tc as shown in figure 1. 1 + (?(*) ~jC / as x-^°°. (22)~~

~~This Lorentzian line shape is a central prediction of the theory. If we assume a van der Waals equation of state we~~

~~As k approaches zero we see from (10) that have [1] from (16)~~

~~= 2 , lim 1/k and KT Xt~ x(k) x(0)~ , (16) y=l v=h, (23)~~

~~so that k must vanish as the critical point is ap- but more generally according to Ornstein and proached. We may assume, rather generally, that Zernike we would have only the relation~~

~~k(T)~(T-Tc as T-*TC Y , ( i 7) y=2v. (24)~~

109 We may cite briefly many later alternative treat- they are also valid for small r (specifically of order ments that are essentially equivalent as regards of the range of direct interaction) we may calculate their (but in > consequences which the nature of the the specific heat as T— Tc . If, as is often fashion- basic assumptions are frequently somewhat ob- able, we perform the calculations entirely in terms scure). For fluids there are the semithermody- of Fourier transforms it is quite easy to overlook namic or phenomenological theories based on the this implicit assumption since the "derivation" introduction of gradient terms into the free energy runs roughly as follows: For example for a magnet, due to Rocard, Klein and Tisza, Debye, Fixman, assuming (15) and using (11), we have and Hart. Yvon has argued that C(r) could be replaced by the Mayer function fir) i.e., by its leading J(k)dk ' (27) virial 2 2 term in a expansion. In a somewhat similar + k ) way the Percus-Yevick equation always predicts a short range C(r) if the potential T c ) range. Brout's self-consistent field approach yields, with p=HkT,

~~- - X(k) XH.c.(k)/ [1 )8p£'(k)xH.c.(k)] , (25) so that from (17) the exponent for the specific~~

~~heat [1] would be where H.C. denotes the corresponding pure hard core system and <£'(k) is the Fourier transform of a=l-(d-2)v. (29)~~

^WgH.c.M- This is again equivalent near Tc . (Some of these theories have been reviewed in For d— 3 and the usual approximation, v— i, we more detail recently [3].) For binary alloys one thus find may mention work by Landau, Krivoglaz, Zernike, yl*, and Cowley amongst others. For magnetic sys- C„~~1I(T-Tc (30) tems the phenomenological theory of Van Hove which is a surprizingly common prediction in the was modeled on the Ornstein-Zernike and van der literature! The analysis, however, is quite un- Waals theories and is quite analogous. The gen- justified as is rather easier to see if one works in eralized mean field treatments of Brout, and El- real space using (18) or (20) and (5) or (6) and ex- liott and Marshall [4], etc., yield results of the form plicitly inserts the short range character of the potential by cutting off the integral or sum at some X(k)~l/[l-£/(k)], (26)

~~r= ri [5] . In fact the specific heat is determined by~~

1 the variation of G(r) or T(r) with 7 —* which, for short range exchange potentials, are T as Tc for small, r (i.e., for small in essentially equivalent. The more recent Green's fixed by Q(x) x (20); see also below). Thus although in a general sort of functions treatments of Tyablikov and others, way the long range behavior of G(r) is a direct "indi- Mori and Kawasaki, and Kubo and Izuyama (for cator" of the transition it is not sufficient to deter- the itinerant electron model) lead again to virtually mine the specific heat singularity. the same results; in particular the direct correla- (4) We have listed the "classical" approximate tion function is always of short range. The spheri- theories which concur with the predictions (15) cal model of Berlin and Kac is also similar but to (20) of Ornstein and Zernike. Recently, how- with v—1 rather than v=\ as in most of the other ever, other answers have been proposed, in the theories, since y — 2 in that case. first place by M. S. Green [6] on the basis of the (3) I would now like to point out a conclusion hypernetted chain integral equation. Green argued drawn from many of these theories 2 concerning the that certain irreducible diagrams might be neglected nature of the specific heat singularity (CV or C«=o). at the critical point and thereby obtained an equa- Firstly note that one can hope to justify most of tion for Xc(k)- As observed by Stillinger and these approximate expressions for G(r) at fixed Frisch [7] his argument may be applied in d dimen- T> Tc only as r—> °o [3]. If, however, we assume sions and yields [3]

~~2 But not all! See the discussion comment by Marshall concerning an honorable exception [4]. Gc {r)^DI^\ (T-Tc) (31)~~

~~110 1~~

with will enable us to judge the approximate theories. Firstly, as to the question of the exponential decay r = of the correlation the critical ) 2-(d/3). (32) functions away from point (in the case envisaged of short range inter-

~~Taking the Fourier transform of (31) with 17 > 0 actions). With fair generality one may define the~~

leads to the important prediction, contrasting with inverse range of correlation through [3] that of the O-Z theory, that a plot of inverse scat- 2 tering intensity versus k at T= Tc should be curved K(D = -lim sup (1/r) log |G(r)|, (T>TC ). (37) — 00 upwards according to r *

The existence of this limit with k > 0 implies the ~ 1/Xc(k) ~ (33) /oU) asymptotic exponential decay of G(r). For the Ising model the matrix method provides expres- (see fig. 2). This point will be returned to. sions for the correlation functions from which (37) More recently another expression for x(k), is an immediate consequence in a system of finite namely cross-section (but infinite length). Onsager's [10] explicit solution for d=2 shows this remains true 1 X(k) ~ l/(A + k ^), (k -> 0), (34) for a system of infinite width above Tc . For a continuum classical gas one may construct a gen- has been presented by inspired Abe [8] who was eralized integral kernel for which one similarly by the recent paper of Patashinskii and Pokrovskii finds an exponential decay, the parameter k being on the lambda anomaly in helium. This latter [9] determined, as in the matrix case, by the ratio of work has attracted a great deal of attention since the two largest eigenvalues. it claimed to predict theoretically the logarithmic The convergence of the activity and virial ex- specific heat anomaly which is observed in the pansions for gases recently established rigorously beautiful experiment described by Professor Fair- by Penrose, Lebowitz, and Ruelle implies a similar bank this morning. Indeed the formula (34) is result for G(r) in the domain of convergence — and, "designed" to predict a logarithmic specific heat by continuation along the physical axis, up to some by just the same false argument we outlined above! singularity which would have a physical significance [In the analysis of Patashinskii and Pokrovskii (probably the critical or condensation point). We and of Abe it is assumed that A(T) varies as (T— T c) can thus expect rather generally that x(k) should to the power unity. However some other power 2 be analytic for small k above Tc (in disagreement would not alter the logarithm but only its ampli- with Abe's result but in agreement with the O-Z tude.] Even if (34) is correct, cannot, there- we theories). fore, believe in the "derivation" of the logarithmic The vanishing of k as T approaches Tc according specific heat. Actually, as we will see below, (34) to (17) is confirmed rigorously by Onsager's result, is not correct for the Ising model and its analog for helium, in terms of the one-body density matrix, exp(-Ka) = v(l + i;)/(l-v), » = tanh UlkT), (38) is equally to be doubted. From (34) we find

for the d=2 Ising model. This yields the non- Gc(r) « D/r'W (35) classical value v=\ for the exponent defined in (17). Further details of the decay of correlation above which is equivalent to (31) with Tc for the simple Ising model follow from an analysis

of the matrix method [3, 11] which confirms the T) = 2-hd. (36) O-Z result (18). For the more general integral Note that in significant contrast with all other the- kernels an appropriate form of perturbation theory 3 2 ories, G(r) will still have a long-range tail (1/r ' reveals a spectrum of eigenvalues from which a for d = 3) when T exceeds Tc according to (34) since similar conclusion will follow. The analyses of X(k) is still singular at k2 = 0. (Patashinskii and nonuniform fluids by Lebowitz and Percus in terms Pokrovskii were more circumspect on this point!) of the direct correlation function also serve to

~~(5) Now I will turn to some more rigorous results justify the asymptotic validity of the O-Z theory~~

~~on the behavior of the correlation functions which away from the critical point [3].~~

~~1 1 3 The most important rigorous result for our present long-range part of the pair potential it is worth~~

purpose, however, follows [3, 7] from the work of pausing to ask how Onsager's famous result that Onsager and Kaufman [12] on the pair correlation the specific heat of the d = 2 Ising model diverges as log fits into the picture. Accepting the functions of the d=2 Ising model at the critical \T—TC \ point. Their result shows that form (40) as the leading contribution to G(r) for all

~~/' and using (6) and (38) one can easily see that the logarithmic singularity in the specific heat Gc(r) « fl/ri/4, i.e., v= 1/4, (d=2). (39) means that~~

~~This is in clear contradiction with the predictions Q{x) ~ Bx log x, as x —» 0. (44) of Ornstein and Zernike (tj = 0) and all other the-~~

ories! [See eqs (32) and (36).] As explained, it is only the To summarize these considerations, we can still behavior for small x hope generally to describe the correlations near which is relevant. More generally if a + 2v>l (as the critical point in terms of two lengths, the is probably always the case) a specific heat "range of correlation" I/k(T) which diverges as exponent a > 0 would follow from (40) if

~~T^>TC and a "range of direct interaction" ro{T)~~

~~which remains finite. However, generalizing (20), (1 ~ a)lv Q(x) = x-bx +. . . asx^O. (45) we must expect for all small k that~~

~~Kr d ~ 2 G(r) = D(e- lr ^) [1 + Q{kt)\ (r -> oo) (7) In the absence of exact results for the three- , (40) dimensional Ising model (or for the Heisenberg~~

+r model) one must return to approximate methods of where D = d nlprl > and Q(x) satisfies (21) and, in calculation. However, by developing systematic place of (22), successive approximations, notably sufficiently long series expansions, one may hope, as in the -BrfOMV***, -» oo. 1 +Q{x) as x (41) case of the thermodynamic functions, to draw re-

liable conclusions about the critical behavior. I If will report on progress these lines in we neglect Q (which is valid near Tc) the Fourier along work transform of (40) is well approximated by undertaken in collaboration with Mr. R. J. Burford. Firstly, for the d = 3 Ising model one may develop

2 2 a series expansion for the range of correlation k in X(k)~DI(K + k y^\ (42) powers of i; = tanh (J/kT). In the standard Ising problem the partition function can be expanded which extends (33) to temperatures above T - (See c in terms of configurations of (closed) polygons on also fig. 2.) In view of (39) we might expect the the lattice (each bond being associated with a factor exponent 17 to have a relatively small but positive v) subject to the condition that an even number of value. Letting k approach zero and using (10) bonds meet at each vertex. To calculate e~ K" and (17) shows that the divergence of the sus- (where a is the lattice spacing) one needs in addition ceptibility and compressibility (along the isochore) to the polygons a single chain of bonds that stretches [1] is now given by right across the lattice (in the direction of interest). By enumerating these configurations one finds for y={2-ri)v (43) the propagation of correlation parallel to a principal axis of the simple cubic lattice

~~in place of the Ornstein-Zernike formula (24). As~~

~~observed some time ago [13] this result applied to -*a 2 6 e = v + 4v + 12ir3 4. 36t;4 + iQgv5 + 356|; the d = 2 Ising model using (38) and (39) yields~~

~~y = 7/4 which is just what is found by careful 7 8 9 + 1204t; + 4420i; -fl6124i; + .... (46) extrapolation of the series expansion [14, 15] as explained yesterday by Professor Domb.~~

(6) In view of my remarks on the impossibility 3 The specific heal divergence can, however, be found from the long range part of the fourth order correlation functions.. See remarks in the discussion following tins of determining the specific heat singularity from the talk.

112 Evaluation of this series with the aid of Pade ap- This series and the corresponding ones for the proximants enables accurate plots of k(T) to be plane square lattice and other three-dimensional drawn which reveal clear deviations from the pre- lattices are very regular and may be extrapolated

~~dictions of the simpler approximate theories [16]. by the ratio and Pade approximant techniques. As~~

~~The series is not, however, sufficiently regular to a check on the consistency of our analysis let us~~

~~yield a firm estimate of the exponent v although one first look at the two-dimensional series for p,-i. Using~~

may conclude that it lies in the range 0.60 to 0.7. the exactly known critical point and the result 7 Since y=1.25 for the simple cubic this is con- y = /4 the Pade approximants to (v — vr )(d/dv) log p-i

~~sistent with some small value of tj in (43) but evi- yield the successive estimates: dently cannot decide for or against the Ornstein-~~

Zernike prediction 17 = 0. 7) = 0.2495, 0.2532, 0.2485, 0.2511, 0.2507, (d = 2). A more fruitful approach has been to calculate the series expansions for the correlation function These are evidently approaching the exact limit G(r) at all values of r on the lattice. The relevant 7] = V4 rather rapidly which confirms the validity configurations consist again of polygons plus a chain of our approach. For the simple cubic a corre- but the chain is now of finite length and simply sponding sequence of estimates (taking 7= 1.250 runs from the origin to the lattice point at r (these and using the accurately known critical point) is: being the only points at which an odd number of bonds may now meet). Appreciable assistance r] - 0.0696, 0.0684, 0.0621, 0.0583, 0.0580, (d = 3). in the calculations may be gained from the known series for the susceptibility and from available numerical data on self-avoiding lattice walks. These values seem to be converging to a limit in the From the series for G{r) one may form the moments vicinity of 0.058 to 0.060 and studies of related series lead to a similar result. We may conclude with fair confidence that

~~{T) = r*G(r)dr, txs p j 7]=0.059±0.006, i^=0.644±0.002, (d=3). (50)~~

~~s or = rT(r) = r r £ 2 (^ ). (47) The results for the B.C.C. and F.C.C. lattice are r r quite similar although less precise as the series~~

are shorter. The value (50) for r) is consistent with Of course the zeroth moment is just proportional to the rather natural conjecture r\ = Vig = 0.06250 the susceptibility itself while the second moment 20 (implying v = /3i = 0.64516 . . .) but the true value is directly related to the initial slopes of plots of the might well be lower, say, t> = Vis = 0.05555 (implying 2 inverse scattering versus k' (and to the so-called 9 v — /i4 = 0.64285 . . .). (It is interesting to note Debye "persistence length," see ref. 3). By using that the values of the analogous exponent for the (40) and (43) we find immediately that self-avoiding walk or excluded volume problem are 2 j]' = /9 and Vis in two and three dimensions, 4 -" P-iiT) « M2 /k ~ ll(T- Tc )^y. (48) respectively.)

One may also use the series for G(r) to estimate Consequently estimates of the critical behavior of Gc(r) = Gc(x, y, z) directly for small values of r fM2 (T) will yield estimates of v and 17. For the simple although this cannot be very accurate owing to the cubic the following terms have been obtained singularity of G(r) as a function of T which is simi- (while one more is available for the plane square lar to that in i.e., the energy roughly {T—Tc ) log lattice) (T— Tc ). For the nearest lattice points to the origin we estimate: P> = 6v + 72w 2 + 582i/3 + 4032t;4 + 25542y5

~~(0, 0, 1) 0.328, (0, 0, 2) 0.160, (0, 0, 3) 0.11; + 153000?;" + 880422*; 7 + 4920576v8 + 26879670v9~~

~~+ 144230088^°+ .... (49) (0, 1, 1) 0.206, (0, 1, 2) 0.137; (1, 1, 1) 0.163.~~

113 These values are quite consistent with (31) and (50) 106 yield inde- i.e., with Gc(r) ~ 1/r , but they do not an pendent estimate of r\. For the Heisenberg model fewer terms can be calculated and their behavior is less regular especially for low spin values. The most favorable case is the F.C.C. lattice for S = °° where we obtain the sequence of estimates:

~~v = 0.786, 0.760, 0.724, 0.710, 0.702.~~

From this and other sequences we feel able to estimate k*

FIGURE 2. Plot if inverse scattering to be expected with a posi- t} = 0.075 ±0.035, v = 0.692 ±0.012. (51) tive value of r) (larger than that estimated for the three-dimensional Ising or Heisenberg models) illustrating the difference between the apparent linear intercept f0(T) and the true The values of r\ and v (as those of y) do not appear intercept £(D-l/x(0). to depend significantly, if at all, on the value of the 4 spin (or on the lattice structure).

(8) It seems clear from these calculations that tj speakers but perhaps I should finish by reviewing has a positive value for three-dimensional Ising very briefly a few relevant experiments which, I models and for the Heisenberg models. The value think, give some indications of small deviations from of 7) is, however, rather small although it is interest- the Ornstein-Zernike theory. These deviations are ing to note that it is apparently slightly larger for in the expected direction although, I should stress, the Heisenberg model. (This is in accord with the problem is still very much "in the air"! the larger deviation of the exponent y from its Walker and Keating in their recent neutron classical value of unity.) I feel 17 might be some- scattering experiments on the binary alloy /3- what larger still in a continuum fluid with mole- brass [17] observed that when T approached Tc cules of a nonsimple shape where "excluded volume the intensity of the superlattice line (which is effects" would be expected to play a larger part. analogous to Kt in a fluid system) increased more

In as much as 17 is positive we must expect a plot rapidly than expected on the classical theories. of the inverse scattering intensity versus k 2 to be Their observations were, in fact, consistent with curved at and (by continuity) near Tc for small k y=1.25. They also observed a slight narrowing of as shown by (33) and (42). Since the expected the line relative to a Lorentzian line shape near Tc value of r\ is so small, however, this curvature (even which would be expected from (42) with a small if present) might not be very evident. In fact, positive value of rj. More recent experiments on the main effect (as may be verified by plotting curves /3-brass by Dr. Als-Nielson and Dr. Dietrich will according to (42) for small values of tj) is to cause be reported tomorrow. r the scattering plots "to hang up" so that the inter- Jacrot's original neutron scattering experiments 1 — cepts appear not to approach zero as 7 »7V, and on iron yielded, on reanalysis, the estimates y = 1.30 even at Tc the curve appears to extrapolate linearly to 1.33 for the behavior of l/x(0) near Tc . This is 2 with A to a nonzero value. These effects are in agreement with direct measurements of the sus- indicated in figure 2 where, however, a relatively ceptibility as has already been reported (and with large value of 17 has been assumed for the sake of the Heisenberg model predictions). Passell and clarity. coworkers have also found a value of y — 1.33 from What do the experiments say? Well, we will their more recent experiments on iron. They seem, hear the latest situation from some of the following in addition, to find evidence that the intercepts of the scattering curve are "hung up" in a way con-

*Note added in Proof. However, recent work on the susceptibility for 5= \ indicates sistent with a small value of 17; but this is very that the value of y is probably appreciably larger than the S — °° value y — 1.33 (al- tentative and I will leave Dr. Passell himself to de- though rj may not change much); see G. A. Baker. H. E. Gilbert. J. Eve. and G. S. Rushbrooke. Physics Letters. 20, 146 (19661. scribe his very careful and difficult experiments.

~~114 9~~

~~Thomas and Schmidt [18] have made x-ray experi- the inverse scattering plots at temperatures very~~

~~ments on argon in the critical region; they observed close to critical. I do not want to anticipate Dr.~~

~~no deviations from Ornstein-Zernike theory! Chu's lecture later this afternoon but let me say~~

~~However, they did not measure along an isochore that the value of rj that seems to fit his excellent~~

~~nor did they go very close to the critical point. data, namely rj — 0.35, is much larger than our~~

~~It is to be hoped that these experiments will be simple theoretical models would suggest. Maybe~~

~~extended both with x rays and with light, since the this is an effect of steric hindrance due to the shape~~

~~behavior of (presumably) simple fluids like xenon of his molecules — I do not know! Whatever the~~

and argon is obviously of cardinal interest. answer, it is clear that there remains much scope for The most extensive critical scattering measure- experimental and theoretical work before our under- ments have been made on binary fluid systems standing of these subtle but fundamental questions where in most cases one or both species are com- will be complete. paratively complicated organic molecules. This makes me. as a theoretician, somewhat apprehen- References

sive when it comes to applying the (relatively) [1] See the associated notes on "Critical Point Singularities." simple theories we have been discussing! Pro- these Proceedings page 21. fessor Brumberger will review this field in detail [2] L. S. Ornstein and F. Zernike, Proc. Acad. Sci. Amsterdam 17, 793 (1914): Physik. Z. 19, 134 (1918). later this afternoon so let me just sketch some of the [3] M. E. Fisher. J. Math. Phys. 5, 944 (1964): reprinted in The recent work. Professor Debye and his collabora- Equilibrium Theory of Classical Fluids, eds. H. L. Frisch and J. L. Lebowitz (W. A. Benjamin. Inc. 1964). tors working in particular with polystyrene-cyclo- [4] R. J. Elliott and W. Marshall. Rev. Mod. Phys. 30, 75 (1958). Fixman. hexane solutions have found little or no evidence [5] This has been pointed out by others, e.g.. in M. Adv. in Chemical Phys. 4, 175 (1964), although it seems not of classical theories — even as deviations from the to be generally appreciated. S. Green. Chem. Phvs. 33, 1403 (1960). regards the value of y for x(0) which appears to be [6] M. J. [7] F. H. Stillinger and H. L. Frisch. Physica 27 , 751 (1961). quite close to unity. However, the experiments [8] R. Abe, Progr. Theoret. Phys. 33, 600 (1965). Soviet Phys. JETP are quite difficult to perform, especially near T [9] A. Z. Patashinskii and V. L. Pokrovskii, c , 19, 677 (1964). and Mclntyre. Wims. and Green, in experiments on [10] L. Onsager. Phys. Rev. 65, 117 (1944). ithe same system, have indeed found deviations from [11] L. P. Kadanoff. to be published. [12] B. Kaufman and L. Onsager. Phys. Rev. 76, 1244 (1949). the classical expectations close to Tr roughly of [13] M. E. Fisher, Physica 25, 521 (1959). Math. Phys. 63 (1961). the sort to be expected with a positive value of 17. [14| C. Domb and M. F. Sykes. J. 2, [15] G. A. Baker. Jr.. Phys. Rev. 124, 768 (1961). Brady and Frisch [19] have also observed somewhat [16] See the figure due to R. J. Burford and M. E. Fisher in Proc. similar nonclassical anomalies in the system Inc. Conf. on Magnetism. Nottingham. Sept. 1964 (Phys. Soc. London) page 81. perfluoroheptane in iso-octane. [17] C. B. Walker and D. T. Keating. Phys. Rev. 130, 1726 (1963). 2506 More recently Dr. Chu in a series of very careful [18] J. E. Thomas and P. W. Schmidt. J. Chem. Phys. 39, (1963). «-dodecane in dichloroethyl ether experiments on [19] G. W. Brady and H. L. Frisch, J. Chem. Phys. 35, 2234 has found significant deviations from linearity in (1961); 37, 1514 (1962).

~~115 209-493 O-66— a~~

~~Scattering of Light and X-rays from Critically Opalescent Systems H. Brumberger~~

~~Syracuse University, Syracuse, N.Y.~~

Introduction The review will attempt to survey, briefly, the] experimental literature dealing with light and x-ray scattering from one- and two-component systems near critical points, beginning in 1950 and conclud- The experimental investigation of critical phe- ing in March 1965. Magnetic phenomena will not nomena has become an actively pursued area of be considered, but attention confined to gas-liquid, research in recent years for two major reasons — liquid-liquid, and solid-solid critical points. The renewal of interest in the statistical-mechanical review will not be completely exhaustive, but theory of these phenomena (for reviews see, for attempt to compare the most important results instance, references 1—5), and a considerable in- critically, and to extract any common observational crease of experimental sophistication coupled with features. Some criteria for comparison will be a better understanding of the experimentally signi- discussed below. The systems reported on are) ficant variables. summarized in tables 1—3.

~~Table 1. One-component fluid systems~~

~~a System Technique References / (A)~~

Argon X ray 6 (1942) 7 (1951) 8 (1963) 5.4 near transition. Neon X ray 9 (1962) Nitrogen X ray 10 (1950) 11 (1964) 6.0 near transition. Ethylene Light 12 (1950) Carbon dioxide Light 13 (1960) Methane X ray 14 (1957) Ethane X ray 14 (1957) Light 15 (1950) Propane X ray 14 (1957) Butane X ray 14 (1957) Ether X ray 16 (1960) Benzene X ray 16 (1960)

~~Debye "range of intermolecular forces.~~

~~116 Table 2. Fluid binary systems~~

~~a System Technique References / (A) reriiuorometnyl- cyclohexane-~~

carbon tetrachloride Light 1/ (195U) 14. / Laroon disulfide- A ray lo (i95o) memanoi Light ZU (1954) in nc<;\ Phenol-water A ray 19 (1950) Light ZU (1954) lsobutyric acid-water Light ZUon (1954)1 1 o c a \ T irrVit 90 HQ^Zh| r nenoi-nepiane j_iigni zu ^iyo4 ) Methanol-cyclohexane Light ZUon (1954)noc/n Light Zl (196Z) 12.7 T ' L. Light 22 (1963) on / 1 n c /i \ Methanol-hexane Light ZU (1954) Nitrobenzene- isopentane Light ZUon (1954)nnc/i\ on nnc/i\ Aniline-cyclohexane Light ZU (1954) oi /in/;o\ Light Zl (1962) ~ u Light 25 (1965—1964) on nnc/i\ 1 nethylamine- water Light ZU (1954) p, p -Uichloroethyl-

ether-n-decane Light Z41 (1V041 ) ~ lo l-wciane-pernuoro- heptane Av ray Zooc. (1V01)no/tit A ray ZO (1901) v A ra) Li (iyoz) A ray 2o (1904) U 1 2, 6-Dimethylpyndine- water Light Z9 (1904) Q 7 Nitrobenzene-n-heptane X ray ouo.n (iyoo)nOA^ O.D 0.1 nQA/1\ 1 Light 51 (1904 ) Pertiuorotributyl- amine-isopentane TLighta s r Lk 4- oL (1904> IOC.1Z.O i /i n A ray 52 (1964) 14.U Polystyrene-cyclohexane Light oo (190U) 54 (196U) r rom Z /—06 or 55 (190Z)noAO'i depending on 50 (190Z) average molecular oi0.7 (iyoz)nOA9\ weight ooo.q (ivou)no^n^ v 07/ A ray OV (.IVODJ m — 14.700^ and Light p\ /8'-Dichloroethyl- ether-rc-dodecane Light 4U (1965)

~~a Debye "range of intermolecular forces." 117 Table 3. Solid systems~~

~~System Technique References~~

~~a-fi Quartz Light 41 (1957) Aluminum-zinc X ray 42 (1958)~~

~~1 2 General Criteria 2. i~ {s) versus s curves obtained at different~~

temperature distances AT{AT=T—TC ) from the The observations reported in the tabulated pub- critical temperature should be essentially parallel. lications must be subjected to critical examination 3. The extrapolated zero-angle intercepts of these

~~from two viewpoints: (1) Are the reported data curves should vary linearly with AT and go to zero~~

reliable — i.e., were proper precautions taken to at TV. insure purity of materials, precise temperature Further analysis of the approximations in the control, etc., and were other experimental factors O.-Z. theory [4] indicates the possibility that the 2 kept to a minimum or corrected for? These might i~ 1 (s) versus s plot might show downward curvature 2 include interfacial reflections, dust, inconstancy near T^> T, for small values of s , and that the -I of light source, cell geometry, multiple scattering, i (0) versus AT plot might no longer be linear. refraction corrections, beam divergence, mono- Miinster and coworkers [43, 44] have recently -1 chromaticity, etc.. for light-scattering experiments; shown that a more rigorous result for i , on the and collimation errors, background corrections, basis of a modernized version of the O.-Z. theory, is

absorption corrections, monochromaticity. etc. for 2 2 1 k + s l x-ray experiments. (2) Are data sufficiently i- (s)< the 4npA (Di extensive (i.e., do they extend over a sufficient 4npA range of angles and temperatures), do they confirm theoretically predicted behavior, and does the col- k•— where > 0 as T^> Tr . p is an "'effective density." lective body of observations indicate some trends A is a constant. (1) yields the O.-Z. relation as ani of sufficiently general nature to rule out the pos- asymptotic approximation for small s. and would, sibility that they are experimental artifacts? This l predict long-range curvature in the i~ (s) versus\ is a particularly important point for critical phe- 2 s plot. nomena, which apparently are subject to a variety Debye. in recent publications [32]. has also* of experimental effects not usually observed under stressed the fact that his original theory [1] is anj noncritical conditions (for instance, gravitational approximation taking only the average square of; effects). the local gradient of density or composition flue-j Most of the recent theoretical treatments of criti- tuations into account, and that inclusion of higher- cal opalescence have been based on an extended, order terms in the series expansion for the inverse! and' sometimes reformulated. Ornstein-Zernike intensity would introduce curvature into O.-Z. plotsj (O-Z) Theory, and on various lattice gas calcula- 2 when carried over a sufficient range of s . Never- tions [3. 4, 5]. Direct experimental tests of the theless, the approximate Debye theory is useful in scattering behavior predicted by classical O-Z correlating experimental observations, and yields theory as reformulated by Debye are the characteristic parameters / and L. defined by 1. The inverse of the relative scattered intensity

~~should vary linearly 2 1 with s , where s = 4-n-A- sin (6/2), A. is the wavelength f4-TTi~G(r)i-(lr in the medium, 9 the scattering L 2 = (persistence length) 2 2 (2) angle. /47r/- 6'( r)dr~~

~~118 2 / tions on 2 argon and nitrogen are in general agree- and L = (3) 1 ment with the classical O.-Z. theory. The zero- angle inverse intensity is linear with AT for both~~

where t = T]Tc . I. the "range of intermolecular vapor and liquid over several degrees near the forces." is predicted to be constant for O.-Z. be- critical temperature Tc . A meaningful /-value havior; where this parameter has been reported, 5 A near Tc — is obtained in both cases. No its value is given in tables 1-3. deviations from the Ornstein-Zernike plot— i~\s) 2 versus s — are observed at small angles: this may be because the smallest AT was 0.05 °K and the Systems One-Component smallest scattering angle about 2xl0~ 3 radians (MoKa radiation for argon, CuKa for nitrogen). generally One observation which turns out to be All three of the light-scattering measurements — true is that a differential of about 10 years be- carbon dioxide [13], ethane [15], and ethylene [12] — tween publications dealing with the same system were made at scattering angles of 90° only. In the is frequently good reason to discard the earlier case of carbon dioxide, the critical temperature work. was only approached to within ~ 0.2°, and the For argon and nitrogen, several x-ray studies are authors report generally good agreement with Ray- available in the 1950-1965 period. For argon, the leigh's equation except nearest Tc , where the 1951 paper by Graham and Lund is based on [7] exponent of k is 3.2 instead of 4. The experiments earlier data obtained by Eisenstein and much on ethane and ethylene which were very precise Gingrich which were taken to a minimum scat- [6] as far as temperature was concerned (±0.001 °C) angle of 2° only. correlation function is tering A but much less precise with respect to pressure obtained by direct Fourier inversion of the extrap- control psi), indicated four regions: (1) T > T (±1 c , olated scattering curves and used to calculate 4 k~ law holds, (2) a region where intensity increased, X-dependence decreased, within about a degree

~~2 2 above T (3) a reversal in intensity with close (r )=L ( , approach to Tc (ascribed to multiple scattering and~~

absorption), (4) and a rapid, erratic increase (as- 6-8.5 yielding values of A, which would indicate cribed to condensation droplets) which showed that measurements were not extended to sufficiently hysteresis phenomena on temperature reversal. small angles. In view of this situation, only the The authors conclude that the O.-Z. theory, which recent work on argon bv Schmidt and Thomas [8] predicts a change in wavelength dependence pro- can be used to test the O.-Z. predictions. -2 portional to A. , does not apply at temperatures

For the x-ray investigations of nitrogen [10], below the temperature of maximum scattering, i.e., paraffins [14], ether [16], benzene [16], and neon very near Tc . Nevertheless, they calculate "cluster [9] somewhat similar remarks apply. In the Rus- diameters"' from the O.-Z. intensity expression insufficient sian paper on ether and benzene [16], (for the region where this applies) to be of the order experimental information is given: it is difficult to of 1000 molecular diameters closest to Tr . While credit the observation of several x-ray diffraction it is not possible to translate these observations maxima within 2 deg 6 for benzene, and to accept directly into the language of this paper, it appears the authors' analysis of their data as indicating that the O.-Z. theory yields meaningful param- only. density fluctuations of certain discrete sizes eters—comparable to values for L found from the For the other systems, the measurements were Debye treatment — within a 1-deg region of Tc , but not angles or extended to sufficiently small ATs. not in the immediate neighborhood of the critical nor were collimation corrections applied. Inter- point. pretation of these data was generally attempted by The general conclusion for the one-component means of a Guinier approximation (assuming systems evaluated is that the x-ray observations spherical inhomogeneities) and invariably yielded show good agreement with O.-Z. theory, the light- very small radii (3-14 A) for the inhomogeneities. scattering observations reasonable agreement ex-

The recent nitrogen work of Schmidt and Thomas cept very near Tc , where results are seriously [11] must remain the major source of useful data afflicted by multiple-scattering and turbidity for this system. Schmidt and Thomas' observa- problems.

1 19 The possibility of gravitational effects was exam- Replotting of the Quantie data for the aniline- ined by Mason and coworkers [45-49] , who report cyclohexane (strong) and isobutyric acid-water definite effects in one-component but not in two- (weak) systems shows the following: for isobutyric component systems. These are probably not seri- acid-water at 0.02 runs and 0.07 °C from Tc , nearly ous for the above cases. parallel O.-Z. fines are obtained, both showing a slight downward curvature at small angles. For aniline-cyclohexane (at 0.04 and 0.10 °C), parallel Fluid Two-Component Systems but curved O.-Z. lines result, with upward curvature! at both smallest and largest angles. The high- Because of considerable difficulties in controlling angle behavior seems fairly typical for strongly! pressures precisely for one-component systems near opalescent systems and is also observed for nitro-i the critical point, more experiments have been done benzene-heptane, etc. with two-component systems, which show analogous It is interesting to compare these results to the! behavior near the extrema of the consolute curve. light-scattering observations of Debye and co-, The temperature variable is more easily control- workers [21, 22] for cyclohexane-aniline and meth- lable, but purification of the samples becomes anol-cyclohexane. The latter system, according harder. to Quantie, exhibits weak opalescence and similar, Most of the work on two-component systems has light-scattering behavior to isobutyric acid-water. ( been done by light-scattering, and for several Slightly different concentrations and critical tem| systems a series of recent publications is available. peratures are reported by the two investigators: The x-ray observations by Kirsh and Mokhov on it is probable that the purity of Debye's samples isj the carbon disulfide-methanol and phenol-water better. In the methanol-cyclohexane case, I systems [18, 19] neither approach Tc sufficiently "normal" O.-Z. plot is found within 0.68 °C from Tc closely nor extend to sufficiently small scattering There are no obvious deviations from linearity angles. It is difficult to understand the series of except a slight upward drift at small and large five small-angle maxima reported within a scatter- angles. O.-Z. curves are nearly parallel. These ing angle of one degree at temperatures 22.5 and measurements were only extended to a scattering 28.5 for the °C from Tc CS 2 -MeOH system. angle of 30°, however. The zero angle intercepts: An extensive series of binaries investigated was somewhat doubtful because of the limited lowl by Chow Quantie using light-scattering. [20] angle range, are linear with AT". For cyclohexane! Results fell generally into two categories systems — aniline, the Debye group reports nearly horizontai showing strong opalescence, a wavelength exponent O.-Z. curves bending upward substantially at higf between 3.2 and 4.3, and a maximum scattered scattering angles and slightly at small ones, anc becoming more erratic at larger A7"s. Intercept: intensity at about y (CS 2 -MeOH,

~~N0 2 -isopentane,~~

~~maxima (MeOH-cyclohexane, isobutyric acid-H 2 0, experiments by Mclntyre and Wims [23] have~~

MeOH-C H 14 Et N-H 0). It is likely that the several and interesting discrepancies 6 , 3 2 brought new angular intensity distributions for the strongly to light for the cyclohexane-aniline system. A opalescent systems are distorted in the forward definite downward curvature of the O.-Z. plots is direction by multiple scattering effects; this is observed at AT < 0.01 °C and at small scattering borne out by Quantie's observation that even the angles, using sample cells of thickness down t

account for multiple scattering were made by scattering in this instance) may introduce a host o I Quantie. new ones in the neighborhood of the critical point! 120 Among the remaining fluid binary systems necessitates particularly accurate intensity meas- e investigated by means of light-scattering alone are urements and collimation corrections. er perfluoromethylcyclohexane-carbon tetrachloride In the case of nitrobenzene-n-heptane, while an

l [Zimm, 17], 2, 6-dimethylpyridine-water [Pancirov overlap has still not been achieved, the change in and Brumberger, 29], rc-decane-/3, /3'-dichloroethyl slope with Ar for the O.-Z. curves based on x-ray

ether and rc-dodecane-/3, /3'-dichloroethylether intensities is several times greater than for [Chu, 24, 40]. Of these, two [17 and 24] apparently light-scattering curves in the same temperature show "normal" O.-Z. behavior, the latter to within interval; further, an /-value nearly three times as

~~0.04 °C of Tc . The 2, 6-dimethylpyridine-water large is obtained from the light-scattering results.~~

system can largely be described by the Ornstein- These observations imply that there is long-range Zernike-Debye theory, but shows typical upward curvature in the O.-Z. plot and that zero-angle x-ray deviations from O.-Z. linearity at high scattering intensity extrapolations are likely to be most

~~angles and the beginnings of a downward deviation unreliable unless the light-scattering range is sub-~~

at small angles and near Tc (Ar=0.02 °C). The stantially overlapped. dodecane-dichloroethyl ether system, on which par- The perfluorotributylamine-isopentane measure- ticularly careful measurements were made by Chu ments reported show just adjacent curves which

very near Tc (AT — 0.002 °C), and at several wave- are not, however, taken at exactly the same tem- lengths, shows a gentle concave-downward curva- peratures. The x-ray results alone show a distinct ier. ture of O.-Z. lines _1 2 the over the entire range of s. curvature in i versus s ; a downward curvature at 2 2 It is significant that these results cover a consider- large s and no curvature at small s are indicated, 15 able range of s and that multiple scattering effects Ar=0.08 °C. Light and x-ray data yield a con- 1S were minimized by choosing a system of small An sistent picture in terms of the Debye parameter I. and superimposing scattering curves for several Here also, the high-angle behavior is not un-

~~f wavelengths. It is not, of course, unexpected that expected. a large-angle deviation of the O.-Z. plot from line- Where the use of critical opalescence data of~~

~~arity appears, since the O.-Z. correlation function polymeric systems is concerned, the author feels is an asymptotic approximation for large r and that some caution is indicated because interpre-~~

~~should therefore hold best at small s. tation of the observations is rendered highly complex~~

|||S. An x-ray investigation exclusively has been made by the considerable range of molecular weights in by Brady and coworkers on the perfluoroheptane- even the most carefully fractionated polymer i-octane system [25-28]. Here also, a downward sample, and by the anisotropy of the molecules deviation of O.-Z. curves is found at the smallest themselves. Nevertheless, Debye and coworkers scattering angles, the O.-Z. curves typically showing have found that the light-scattering observations

jan inverted S-shape; it must, however, be noted agree well with the Ornstein-Zernike-Debye theory. ^ that the curves were obtained no closer than 0.16 Mclntyre, Wims, and Green, on the other hand, fen

' I C from the separation temperature at the composi- report that the extension of measurements to , " lit . ii. jtion used, which was hot the critical composition. small angles reveals a severe drop-off of O.-Z. _1 recent i values versus More precision experiments on this system curves near Tc ; extrapolated (0)

1 ""fare reported to be in progress [50]. A7 deviate upward from linearity. Chu has 13,e The remaining fluid binaries to be discussed achieved a composite light- and x-ray scattering probably represent the most important experimental O.-Z. curve which apparently shows no small- ^ set, since composite light and x-ray scattering data angle deviations but does drop below the O.-Z. ts is fare available for them, and they thus extend over a line at large s. Here, again, each curve is a com- >nn fvery much greater range of s-values. Included are posite of data taken over a range of temperatures 1 '"'nitrobenzene-n-heptane [Pancirov, Farrar, and as large as 0.32 °C. All of the polystyrene observa-

ffl Brumberger, 30, 31], perfluorotributylamine- tions are strongly dependent on the average molec- '"'isopentane [Debye, Caulfield, and Bashaw, 32], ular weight of the dissolved fraction. ^'and polystyrene-cyclohexane [Debye et al., Mclntyre Solid Systems let al., Eskin, Chu, 33-39]. Experimental difficulties jiiooi are considerable in trying to overlap the two regions. Phenomena analogous to those observed in fluid P^This must primarily be done by extending the x-ray systems appear in the neighborhood of certain ill measurements to extremely small angles, which phase-transitions in the solid state.

121 The aluminum-zinc system exhibits greatly rc-dodecany — /3,/3'dichloroethyl ether system, for enhanced small-angle x-ray scattering at tempera- which long-range O.-Z. curvature is reported for tures above the separation temperature of the AT -0.002 °C [40] and where multiple scattering solid solution phase, 351.5 ±0.4 °C. A careful is apparently negligible, and those for cyclohexane- study by Minister et al. [42, 43, 44], indicates that aniline [23] indicating downward small-angle curva- a treatment of the data is feasible by means of an ture of the O.-Z. plots for AT < 0.01 °C. extension of the Ornstein-Zernike theory to eq (1). Small-angle x-ray measurements alone are less

l 2 Here, again, curvature in the i~ (s) versus s plot conclusive; the small-angle drop-off reported for is observed. isooctane-perfluoroheptane is in the range where The available light-scattering evidence for the collimation corrections become very significant. a-(3 transition of quartz [41] is limited to measure- In any case only one or two experimental points

77 -4 are shown lying below the O.-Z. curves. Zero-angle ments at — and was found to obey the X law. extrapolations are almost certainly misleading here. Partial depolarization (about 6%) of the scattered Additional evidence that the theoretical predic- light is reported. The presence of an opalescent tions of the O.-Z. theory are inadequate comes strip of finite width between the two phases is taken from the combined light-scattering and x-ray scat- as evidence that the transition is not first-order. tering data of references 30-39. All of these report

curvature over a large range of s, but the observations are not sufficient to permit general conclusions General Conclusions to be drawn. The total weight of evidence indicates that there It is clear that a search for deviations from the are real deviations from the Ornstein-Zernike pic- predictions of the classical Ornstein-Zernike-Debye ture, and that these are in the direction suggested theory must for the moment center on observations by newer theories. What is uncertain is the pre- dealing with binary systems. There is no clear cise nature of the deviations. This suggests quite deviation apparent in any of the one-component clearly what needs to be done: studies so far undertaken. This is unfortunate, 1. Future experiments should be a composite of since our theoretical understanding is greater for light- and x-ray scattering to cover the maximum the simpler systems. Conversely, it is not unex- possible range in 5. To this end, small-angle light- pected that deviations from classical behavior occur scattering studies should be included, and a range in systems of greater complexity, where the scat- of wavelengths should be used. With the avail- tering involves not just the intermolecular poten- ability of laser sources, much greater control overi tial e(r) , but tn, e-22. and 612 [1]. the monochromaticity of the radiation and the scat-! For binary systems investigated by one tech- tering geometry can be exercised. nique only, the results on the whole are inconclu- 2. The critical point must be approached more sive. Nearly all the light-scattering observations closely (AT ~ 0.001°) though this may introduce new; have certain features in common: reasonably good difficulties which are not now apparent. fit to the O.-Z. curves is found, but deviations appear 3. A serious attempt (theoretical and experi- at largest and smallest scattering angles. Those mental) needs to be made to account for multiple at large angles are usually upward, the others are scattering, since this problem will be aggravated as sometimes downward for temperatures quite near the critical point is approached more closely.

Tc , but do not appear regularly; this may be a func- 4. It is likely that, at least for some time, the most tion of the minimum scattering angle accessible precise experiments will be done with binary sys- in each experiment. Extrapolated zero-angle inter- tems. It would be of great value to concentrate cepts are proportional to AT for some systems but on relatively few of these which display deviations not for others and in any case rather suspect since and also are good experimental systems, and to! few small-angle light-scattering data exist. The test them with a much greater variety of experi-i observed anomalies unfortunately often occur at mental approaches — scattering, transport proper- the limits of experimental accessibility, and un- ties, etc. It is quite apparent that many of the, doubtedly are at least to some extent rendered phenomena occurring near critical points are poorlyjl uncertain by multiple scattering effects. Perhaps understood and others still unknown. In spite ofjj the best light-scattering results are those for the the concentration on binary systems, precision! 122 experiments on one-component systems of spheri- [21] Debye, P., Chu, B., and Kaufmann, H., J. Chem. Phys. 36, 3378(1962). cal molecules are still desirable. It is to be ex- [22] Chu. B.. J. Phys. Chem. 67, 1969 (1963). pected, however, that deviations from classical [23] Mclntyre. D., Wims, A., private communication. [24] Chu. B., J. Chem. Phys. 41, 226 (1964). behavior will be harder to find, and will occur in [25] Brady. G. W., and Petz, J. I., ibid. 34, 332 (1961). regions where the experimental difficulties are [26] Brady, G. W., and Frisch, H. L., ibid. 35, 2234 (1961). [27] Frisch, H. L., and Brady, G. W., ibid. 37, 1514 (1962). very great. [281 Brady, G. W., ibid. 40, 2747 (1964). [29] Pancirov, R., and Brumberger, H., J. Am. Chem. Soc. 86, 3562(1964). Bibliography [30] Brumberger, H., and Farrar, W. C, Proc. Interdisciplinary Conf. Electromag. Scatt. p. 403 (Pergamon Press, New York, 1963).

P., Chem. Phys. 3 , 680 ( 1959). [1] Debye, J. 1 [31] Brumberger. H., and Pancirov, R., J. Phys. Chem. 69,4312 [2] Kac, M., Uhlenbeck, G. E., and Hemmer. P. C, J. Math. (1965). 216 (1963). Phys. 4, [32] Debye. P.. Caulfield, D.. and Bashaw, J., J. Chem. Phys. 41, [3] Fixman, M., Adv. in Chem. Phys. VI, (1964). 3051 (1964). Fisher, M. E., J. Math. Phys. 5, 944 (1964). [4] [33] Debye, P.. Coll. H.. and Woermann, D.. ibid. 32 , 939 (1960). article "Critical Fluctuations," Fluctuation [5] Miinster, A., on [34] Debye, P.. Coll, H.. and Woermann, D., ibid. 33, 1746 (1960). E. ed., (Aca- Phenomena in Solids, p. 180, R. Burgess, [35] Debye, P., Woermann, D., and Chu, B., ibid. 36, 851 (1962). demic Press, N.Y. 1965). [36] Debye, P., Chu, B., and Woermann. D., ibid. 36, 1803 (1962). Gingrich, N. S., Phys. Rev. 62 , 261 [6] Eisenstein, A., and [37] Mclntyre, D., Wims, A., and Green, M. S., ibid. 37, 3019 (1942). (1962). Phys. 1380 [7] Graham, W., and Lund, L. H., J. Chem. 19, [38] Eskin, V. E., Vysokomolekul. Soedin. 2, 1049 (1960). (1951). [39] Chu. B., J. Chem. Phys. 42, 426 (1965). P. ibid. 2506 (1963). [8] Thomas, j. E., and Schmidt, W., 39, [40] Chu, B., Paper presented at APS Meeting, Kansas City, ibid. 392 (1962). [9] Stirpe, D., and Tompson, C. W., 36 , March 1965. (1950). [10] Wild, R. L., ibid, 18, 1627 [41] Yakovlev, I. A., and Velichkina, T. S., Uspekhi Fiz. Nauk. [11] Thomas, J. E., and Schmidt, P. w\, in press. 63,411 (1957). Phys. 18, Tl2j Cataldi, fl. A., ancTDrickamer, H. G., J. Chem. [42] Miinster, A., and Sagel, K.. Mol. Phys. 1, 23 (1958). 650(1950). [43] Miinster, A., and Schneeweiss, ch., Z. physik. Chem. (NF) Kriticheskiye Yav- [13] Skripov, V. P., and Kolpakov, Yu. D., 37,353(1963). v Rastvorakh, Moskov. Gosudarst. leniya i Flyuktuatsii [44] Miinster. A., and Schneeweiss, ch., ibid. 37, 369 (1963). Soveshchaniya, Moscow, 1960: p. 126. Univ., Trudy T45] Murray, F. E., and Mason, S. G., Can. J. Chem. 30 , 550 [14] Buttrey, J. W.,J. Chem. Phys. 26, 1378(1957). (1952). ibid. 655 (1950). [15] Babb, A. L., and Drickamer, H. G., 18, [46] Whiteway, S. G., and Mason, S. G., ibid. 31, 569 (1953). 81. [16] Mokhov, N. V., and Labkovskii, Ya. M., ref. 13, p. [47] Murray, F. E.. and Mason, S. G., ibid. 33, 1399 (1955). (1950). [17] Zimm, B. H., J. Phys. and Coll. Chem. 54, 1306 [48] Murray, F. E., and Mason, S. G., ibid. 33, 1408 (1955). [18] Kirsh, T. V., Zh. Khim. Fiz. 32, 1116 (1958). [49] Murray, F. E., and Mason, S. G., ibid. 36, 415 (1958). [19] Mokhov, N. V., and Kirsh, T. V., ibid. 30, 1319 (1956). [50] Mclntyre. D., Wims, A., and Brady, G. W., Proc. Conf. (1954). [20] Chow, Quantie, Proc. Roy. Soc. 224A, 90 Small-Angle X-Ray Scatt. Syracuse, 1965; to be published.

Experiments on the Critical Opalescence of Binary Liquid Mixtures: Elastic Scattering*

~~B. Chu~~

~~University of Kansas, Lawrence, Kans.~~

~~The purpose of this paper is two-fold. Firstly, character, will clear up some of the confusion~~

it tries to provide a careful review of many of the which many readers inevitably get in trying to in- pertinent experimental studies on the elastic terpret scattering data on critical opalescence scattering of visible light as well as on the small- without investigating the experimental details. angle x-ray scattering of critical binary liquid mix- Secondly, preliminary work on visible light scatter-

~~tures. It is hoped that the review portion of the ing of a binary fluid mixture, n-dodecane-/3,/3'~~

~~article, although in the main argumentative in dichloroethyl ether, at small values of s/A. (5 = 2~~

sin (9/2, A = wavelength in the medium) and very close to its critical mixing point does demonstrate •Elastic scattering here refers to the unresolved scattered intensity in the absence of a spectrometer or an interferometer. This differs from inelastic scattering where a breakdown of the Ornstein-Zernike [1] and the the frequency spectrum of the scattered light has been resolved into central. Brillouin, Debye theories. Further and Raman [2] theoretical work components by means of optical beat frequency technique, interferometry and spectrometry. becomes necessary.

123 Experimental Validity of Elastic Scat- the horizontal curves in reference 8 of figure 3 tering Measurements may partially be the result of an over correction on the alinement of the light scattering photometer not easy to perform Accurate experiments are as indicated by the slightly negative slope of LUDOX point and the interpretation can near the critical in the same figure [3]. as be confused by many experimental effects, such It seems that further investigation on the system multiple scattering, impurities, and temperature aniline-cyclohexane becomes necessary. Such fluctuations. Qualitatively, the observed data on experiments have been performed at Kansas with to confirm the classical binary fluid mixtures seem better instruments [3, 6, 9]. We have measured one finds theories [1, 2]. On closer examination, the critical opalescence of the aniline-cyclohexane data pub- numerous "anomalies," especially from system at three concentrations, 39.6, 40.6, 41.6 vol. one of fished in the past 10 years or so. Perhaps, percent of aniline. The corresponding phase "anomalies" the reasons for us to find so many separation temperatures are respectively 29.89g, deviations from the is because we expect to see 29.89^, 29.8% which are about 0.5° above the values theory on the basis of simple analysis in classical reported in the literature [10]. A typical plot of theoretical limitations and approxi- 2 terms of its reciprocal relative scattered intensity versus sin 9/2 to say, the reported deviations mations. Needless at — 1. a k() 578 m/x is shown in figure For such tend to coincide with some of the theoretical expectations and vice versa. However, I am reasonably certain that many observed data are tinged with

experimental effects [3]. It would be desirable to have more accurate and extensive data to sub- 39.61 vol. % Aniline in Cyclohexane stantiate my suspicions. Indeed, several systems have been and others are being reinvestigated. For the time being, I am only able to provide mostly indirect experimental evidence. Reviews on both pre- theory [4, 5] and experiment [3, 6] have been sented elsewhere. Data are quoted for illustrative purposes only.

1. Maximum scattered intensity (light scattering): For the system aniline-cyclohexane, Chow [7] observed a pronounced peak at a scattering angle 9 of about 90° from a plot of scattered intensity versus 6 at intermediate temperature distances from the critical solution temperature (T—Tc 40 61 vol. % Aniline in Cyclohexane < 0.3°). Debye, Chu, and Kaufmann [8] reinvestigated the system and concluded from measurements at one concentration that the maximum scattered intensity reported by Chow was an experi-

~~mental effect. It should be noted that, in critical opalescence studies, scattering measurements at one concentration seldom provide definitive~~

~~is even more so for the system answers. This AT = 03I aniline-cyclohexane partly because aniline cannot~~

be purified easily and it undergoes photochemical decomposition; and partly because the system scat- 01 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9 strongly near its critical mixing ters light very ! sin (f) point. As the shape of the coexistence curve changes markedly in the presence of impurities, FIGURE 1. Plot of reciprocal relative excess scattered intensity it is not certain whether the results reported by 0 versus sin*2 Debye, Chu and Kaufmann [8] was indeed measured 39.6, and 40.6, vol. percent aniline in cyclohexane. Observed phase separation at the critical solution concentration. Furthermore, temperature= T„= 29.89 °C. / excess (6, T) = 1(6, T)-/(0, T-Tp ~ 10°). X„=578 m/i.

124 appreciable with cylindrical cells of about 8 mm in 3. Attenuation correction (light scattering): For inside diameter even at relatively large temperature very intense opalescence, the extinction of both distances. For this reason, we used an incident incident and scattered light as the beam goes wavelength of 578 m/u, and performed our measure- through the critical fluid mixture has to be taken into

ments not very close to the critical solution tempera- account [6]. Multiple scattering usually changes ture so that the effects of multiple scattering were the shape of the scattering curves, such as an up- reduced. The results illustrate that the scattering ward sweep in an OZD plot. Even in the absence behavior of this binary fluid system is similar to of multiple scattering, attenuation correction be-

~~that of other binary liquid mixtures [3, 6]. It is comes important especially when we want to know quite certain that both the maximum scattered the temperature dependence of the extrapolated~~

~~intensity and the lack of dissymmetry [8] for this zero-angle scattered intensity over large tempera- system are experimental artifacts. The Debye ture ranges.~~

~~/-parameter was estimated to be in the same range According to the Debye theory, [2] we may assume~~

~~as those of other binary liquid mixtures [3, 6]. that in the visible region, the scattered intensity / at 2. Upward curvature {light scattering): Virtually constant wavelength has the form:~~

~~all binary fluid mixtures in the strongly opalescent = A(T) + B sin 2 region exhibit upward curvatures at large angles in j 1 (1) a plot of reciprocal scattered intensity versus~~

while, in fact, we have for the measured scattered (or sin 2 912). Such a plot is sometimes re- ^jj intensity I* after correction for effective scattering ferred to as an OZD plot. Closer inspection shows volume: that the cylindrical scattering cells are either so

2 large (e.g., 10 mm diam) that multiple scattering A(T) + B sin (2) becomes appreciable or so small (e.g., 2 mm diam) that reflection and refraction problems are no longer F'(T) is an experimental correction factor based avoidable. The only exception was Zimm's classi- on the temperature dependence of the observed cal work [11] in which a different geometry for his extinction coefficient and the size of the scattering photometer was used. The most convincing test cell. We have assumed that F'(T) is independent for multiple scattering consists of a tedious proof the scattering angle. In an OZD plot, we should cedure involving collimation and detector slits of not get parallel straight lines corresponding to various cross sections, scattering cells of different different temperature distances since, according to diameters and geometry, and composite scattering eq (2), the measured slope is F'(T) • B. Instead, a curves by means of both light and small-angle small increase in slope may be detected for strongly x-ray scattering 13]. With those parameters, [12, opalescent systems as the critical solution tempera- especially from a composite curve of light and small- ture is approached. This could be one of the rea- angle x-ray scattering of polystyrene in cyclohexane sons which may contribute appreciably the changing [13], we have shown that the scattering curves con- slopes reported by R. Pancirov and H. Brumberger tinue from the visible light to the x-ray region [15] and by D. Mclntyre, A. Wims, and M. S. Green without abrupt bends. Similarly, the same con- [16] since neither articles have discussed this ex- clusion can be drawn from a superimposition of the perimental effect. The same correction was also scattering curves using three wavelengths (Ao ignored in Zimm's work [6, 111. = region 365, 436, and 578 m^t) in the visible [3, 4. Downward bend in an OZD plot (light scat- to 6]. Therefore, the upward sweep has be an tering): Eskin [17] found a downward bend at experimental effect. In addition, the scattering relatively large temperature distances from the due to fluctuations to differently seems behave phase separation temperature (7^) for the system from that due to particles, as a binary fluid mixture polystyrene-cyclohexane away from the critical could show an in plot upward curvature an OZD solution concentration, while Mclntyre, et al. [16], indicating the presence of multiple scattering found a downward bend at only small temperature while a LUDOX (colloidal silica) solution of com- distances from Tp . Furthermore, the pronounced parable scattering power may reveal only a hori- downward dips reported by Mclntyre, et al., appear zontal straight line in the same plot [14]. only after the presence of upward sweeps for which

~~125 multiple scattering is responsible. With unaccount- to zero angle by means of fight scattering where the~~

able effects due to polydispersity, it should be s ° — 10~ 4 -1 values of go down to about 1 X A . The emphasized that only binary liquid mixtures of A. small molecules or one component systems are validity of this extrapolation by means of small- appropriate if we are interested in testing theories. angle x-ray scattering often becomes questionable On the other hand, polymer fluid mixtures are of at small temperature distances from the critical interest in terms of the Debye /-parameter which is solution temperature. It is my belief that the ab- related to the cohesive energy densities. We have normally small I (~ 2.6 A) for the system nitroben- shown that, in an OZD plot, at very small tempera- zene-n-heptane [19] is in error. ture distances, say T— Tp — 0.002°, the system 7. Constant intensity (small-angle x-ray scatter- rc-dodecane and fi, ft' dichloroethyl ether, [18] indeed ing): The usual critical opalescence behavior clearly exhibits a gentle downward curvature virtually contradicts the constant intensity results of G. W. extending over the whole range of s/\. More ex- Brady and H. L. Frisch [20]. The same system, tensive studies are still in progress. ra-perfluoro-heptane and isooctane, has been rein- 5. Temperature dependence of extrapolated zero- vestigated by fight- and small-angle x-ray scatter- angle scattered intensity: Plots of reciprocal extrap- ing [21]. Our light scattering data, although par- olated zero-angle scattered intensity versus tem- tially qualitative in nature because of multiple perature are slightly concave upwards and tend to scattering, showed no region of constant intensity. level off or to extrapolate to a nonzero value at Tc . Instead, they were found to be similar to the be- For a binary liquid mixture, it is very difficult to havior observed in other binary liquid mixtures. obtain this extrapolation in view of the fact that the measured scattered intensity depends not only on Plan of Attack the attenuation correction factor but it is also a sum of two terms, one due to concentration fluctu- the it ations, and the other due to density fluctuations. Once we know difficulties, takes a little When we want to compare the observed data with planning before we should start our experiments. In principle, want to select a critical binary theory, it is the concentration term that is of interest. we fluid The density term is usually, but not always, negli- mixture which satisfies the following con- gible over large temperature ranges. A plot of the ditions: reciprocal extrapolated zero-angle scattered in- 1. Purification: Both components could be purified by preparative gas chromatography. tensity due to concentration fluctuations, | 2. Stability: Both components are stable and do versus temperature shows that the straight line not undergo photo-chemical reactions. behavior, predicted by the approximate Debye 3. Temperature control and measurements: The phase separation temperature by means of visual theory, holds over short temperature ranges [6]. observation could to about 0.001° It is conceivable that deviations from the form be determined without excessive patience. The same tempera-

. exist over large ranges of tempera- j (n =C(T—Tp) ture limit applies to its controls (proportional tem- / f(o) perature controller) Mueller ture, i.e., over decades of the temperature distance and measurements (G-2 bridge). Therefore, it is reasonable to aim at a from Tp . We have identified the maximum phase 0.001°. separation temperature on the coexistence curve as (T—Tp) value of We also try to select a the critical solution temperature. Under these system which has a critical mixing temperature conditions, a correction term for density fluctua- near the room temperature. tions, in addition to the attenuation correction, 4. Multiple scattering: For strongly opalescent systems, multiple scattering invariably becomes the should be taken into account. So far, all the re- difficult to overcome. j ported data on the temperature dependence of most serious and obstacle extrapolated zero-angle scattering, except for the As we have mentioned earlier, one way is to reduce the sample cell size and the effective scattering system n-decane and dichloroethyl ether [6], have ignored both corrections. volume. A cleverer approach (in my opinion) is 6. Extrapolation to zero-angle from OZD plots to select a system which is not strongly opalescent. (small angle x-ray scattering): It may be justified Even then, we are limited to cylindrical cells no to extrapolate the reciprocal of scattered intensity larger than about 6-10 mm in diameter. A small

~~126 value of An, which implies a small (—], is a suit-~~

able criterion. The symbols n and c refer to the index of refraction and the concentration of the fluid mixture respectively. 5. Large electron density differences: We want a system which scatters x rays as strongly as possible.

~~It will be helpful if the two components possess a large difference in their electron densities.~~

6. "Gravitational" effect: In one component systems, the effect of gravity very near the critical point has to be taken into account. This effect is negligible for critical binary fluid mixtures where the scattering due to concentration fluctuations is predominant. Since, in many respects, the behavior of two-component systems is closely FIGURE 2. A schematic diagram of the modified constant temperature bath and slit systems in the light scattering pho- of simple fluids, analogous to the condensation tometer. we choose to begin our investigation on two- component systems. One-component systems are Further insulation on the thermostat-tank inside much more satisfying in terms of theory, although the light scattering photometer was necessary. The they are experimentally more difficult. modifications are shown in figure 2. We favored In view of the above conditions which I have the immersion tank technique because the bath imposed, we start looking for a two-component had a reasonably large heat capacity, and the system in which both components can be purified immersion oil, which matched the index of refrac- by preparative gas chromatography, and are stable; tion of glass, reduced both refraction and reflection has a convenient critical solution temperature, a difficulties of a small light scattering cell. The bath small difference in the index of refraction of the temperature was controlled to better than 0.0005°. two components, and a large difference in their All metal parts in the tank were insulated, and there electron densities. Fortunately, we have found not was a metallic shield around the light scattering one two-component system but a homologous series cell to insure uniformity of temperature over of saturated paraffins as one of the two components the whole fluid mixture. The phase separation

~~with /3, /3'-dichloroethyl ether or chlorex as the temperature of the sample was measured in a other component. The paraffin-chlorex systems separate constant temperature bath and inside the~~

satisfy all of the imposed conditions. In addition, photometer by means of a telescope. The observed the homologous series provides us a series of sys- phase separation temperatures agreed to within tems with increasingly smaller refractive index 0.001° for each liquid mixture. differences as the molecular weights of the alkanes Results and Discussion increase. Therefore, it is within reason to expect that experimental data on critical binary liquid The Debye theory provides an excellent repre- mixtures very close to the critical point by means of sentation of experimental facts in the intermediate light scattering are accessible with existing tech- temperature distances over large ranges of s/k. niques. We may also take advantages of the homol- Even deviations from the straight fine behavior ogous series for molecular interaction studies in in plots of reciprocal scattered intensity versus terms of the Debye theory. 2 2 (s/A) (or (d/k) ), by means of small-angle scattering Experimental Details of x rays, has been interpreted in terms of the Debye theory [12, 13, 22]. On the other hand, deviations

Detailed descriptions of the light scattering pho- from the straight line behavior in an OZD plot at small values of s/k and very close to the critical tometer, [9] purification of materials, preparation solution exist Figure 3 shows of solutions, phase separation temperature determi- temperature do [18]. a plot of reciprocal scattered intensity versus nations are discussed elsewhere [6, 18]. Only (s/k) 2 using a cylindrical cell of approximately 6 the changes will be specified. mm

127 1 I I I I I I 1 I I I I I — — I ————————— — plot at small values of s/A. and at very small tem- 40.6 wt% n-DODECANE IN 0,?- DICHLOROETHYL ETHER perature distances from the phase separation tem-

T = 33.50 °C p 2 perature. However, the magnitude of the curvature CYLINDRICAL CELL DIAMETER = 6.1 mm is yet uncertain as the measured scattered intensity has to be corrected for both attenuation and density fluctuations. Furthermore, we do not know the

~~range of s/A. in which any of the modified theories [5]~~

~~are expected to hold. According to the equation [5] '~ - 2 2 7,/ -, = D0 [k + (s/A.) ] where D0 , k, and i) are~~

~~assumed to be three adjustable parameters, J] is~~

~~estimated to be in the range 0.2-0.3 when T—Tp~~

~~is small. It is expected that r\ should be smaller after the intensities have been corrected for attenuation and density fluctuations. The temperature dependence of the extrapolated zero-angle~~

~~scattered intensity is sketched in figure 4. It — is quite certain that — is approximately propor-~~

~~tional to the temperature distance. Deviations do exist and seem to concave upwards. On the other~~

hand, there is perhaps an uncertainty on the critical solution concentration of about 1—2 wt percent in our preliminary measurements of the coexistence 20 curve for the n-dodecane-chlorex system. Further ; 8 (SA) X I0 A" scattering experiments with several concentrations

FIGURE 3. Plot of reciprocal relative scattered intensity near and at the critical solution concentration are in progress. We must also remember that, in an versus HjJ OZD plot, extrapolation to zero scattering angle Values of scattered intensities are denoted by hollow squares, solid circles and hollow circles for values K n of 365, 436, and 578 m/x respectively. is permissible only in the absence of very long

range correlation, i.e., no abrupt change in curva- in inside diameter. The curve with AT — 0.002° ture in the extrapolated region is present. For has been reproduced, and results from cells of this reason, measurements at even lower angles different diameters and incident light of different by means of "long-wavelength" light scattering wavelengths agree. A careful examination ol fig- become desirable. ure 3 reveals that the curvature at AT= 0.046° is almost as large as that at AT= 0.002°. This could partially be attributed to the fact that only the scattered intensities measured at A. = 3023 A had been corrected for dust and stray light, while measurements from the other two wavelengths

(A. = 2516 A, 4027 A) were not corrected for these effects. Note that the three hollow circles for the curve at AT— 0.046° for the smallest values of 2 (s/A.) were much lower than the general trends of the other points belonging to the same curve.

When the system is more strongly opalescent, contributions due to dust and stray light become negligible as shown by the first few hollow circles at small values of s/A (A = 4027 A) for the curve at AT = 0.002°. We believe that, for the system FIGURE 4. Schematic variation of the inverse extrapolated zero- angle scattering due to concentration fluctuations estimated n-dodecane and /3,/3'-dichloroethyl ether, there from preliminary results on the system n-dodecane and /3, f3' exists a gentle downward curvature in an OZD dichloroethyl ether.

128 This work was supported by the National Science Since we have seen that the exponent 17 is positive but small relative to unity this leads to a rather remarkable conflict with Foundation (GP-3922). The author wishes to the requirement that this combination should vary as r~'' as men- thank W. P. Kao and J. S. Lin who performed tioned above. It is rigorously known, of course, that the specific heat for an Ising model in two dimensions diverges logarith- the measurements. mically. I think the situation in three dimensions is in a state References of flux. One would have at worse a small positive index a, perhaps of the order of V10. In three dimensions, therefore, there could be some slight deviation of the index d from 3 for S. Ornstein and F. Zernike, Proc. Acad. Sci. Amsterdam [1] L. this quadruplet distribution function long range behavior, but 17, 793 (1914): Physik. Z. 19, 134 (1918): ibid. 27, 761 it certainly would not become equal to the long range behavior (1926). of the square of the pair correlation function. [2] P. Debye, in Non-Crystalline Solids, ed. by V. D. Frechette E. Fisher: I (John Wiley & Sons, Inc., New York, 1960) p. 1: J. Chem. M. May make one comment on that? What Dr. Phys. 31, 680 (1959): in Electromagnetic Scattering, Stillinger said is, of course, correct. If I look in the Ising model ed. by M. Kerker (The Macmillan Company, New York, at saS\, where 0 and 1 are nearest neighbors, and at s,s, +8, where r r 1963) p. 393. and + 8 are near neighbors, then the behavior of (s„s,SrS r +&) — (soSi) for large r should tell [3] B. Chu, J. Am. Chem. Soc. 86, 3557 (1964). (srSr+s) me how the specific heat diverges. [4] M. Fixman, Advan. Chem. Phys. 6, 175 (1964). John Stephenson, a student of mine, has actually evaluated this [5] M. E. Fisher, J. Math. Phys. 5, 944 (1964). correlation function explicitly which is not too difficult to in [6] B. Chu, J. Chem. Phys. 41, 226 (1964). do the Ising model [1]. He finds in fact, as one 2Kr 2 Chow, Proc. Roy. Soc. (London) A2 2 90 (1954). might anticipate, that the behavior is as e' lr in this case. [7] Q. 4, — — [8] P. Debye, B. Chu, and H. Kaufmann, J. Chem. Phys. 36, His correction factor [1 + (>4 (kt)] satisfies Q4 (x) * o as x * o and 3373(1962). — 1 + Q.i(.x) > 2 as x—» ». Once you can express the problem in [9] B. Chu, Rev. Sci. Instr. 35, 1201 (1964). [10] O. K. Rice and R. W. Rowden, J. Chem. Phys. 19, 1423 terms of the long range behavior and you have a good theory for (1951): D. Atack and O. K. Rice, Discussions Faraday the long range behavior, it is fine. However, as Dr. Stillinger Soc. 15, 210 (1953), J. Chem. Phys. 22, 382 (1954): pointed out again, although formally all the information is con- F. R. Meeks, R. Gopal, and O. K. Rice, J. Phys. Chem. 63, tained in this quantity (indeed one could also get the pair correla- 992(1959). tion function from it), it is a different set of asymptotic limits [11] B. Zimm, J. Phys. Colloid Chem. 54, 1306 (1950). that are required. [12] B. Chu, J. Chem. Phys. 42, 2293 (1965). G. S. Rushbrooke: May I make a comment which is due to Dr. [13] B. Chu, J. Chem. Phys. 42, 426 (1965). Widom? It concerns the things that Dr. Fisher talked about, [14] P. Debye, H. Kaufmann, K. Kleboth, and B. Chu, Trans. but below the critical temperature rather than above. Widom Kansas Acad. Sci. 66, 260 (1963). simply takes the equation which relates the compressibility to [15] R. Pancirov and H. Brumberger, J. Am. Chem. Soc. 86, the density fluctuations. Suppose the system is separated into 3562(1964). two phases. Then in a small volume of dimensions of 1/k, the [16] D. Mclntyre, A. Wims, and M. S. Green, J. Chem. Phys. 37, range of the correlation function, we ought to have fluctuations in 3019(1962). the density equal to the width of the phase boundary. Now, [17] V. E. Eskin, Vysokomolekul. Soedin. 2, 1049 (1960): V. N. if one just introduces this idea, one obtains the relation: Tsvetkov, V. E. Eskin, and V. S. Skazka, Ukr. Fiz. Zh.

7 , 923 (1962). y'=v'd-2p (1) [18] B. Chu and W. P. Kao, J. Chem. Phys. 42, 2608 (1965). [19] H. Brumberger and W. C. Farrar, in Electromagnetic Scat- where y' is the exponent of the susceptibility below the critical tering, ed. by M. Kerker, (The Macmillan Company, New point, v' the exponent of k (T) below the critical point, /3 the York, 1963), 403. p. exponent for the phase boundary and d the number of dimensions. [20] G. W. Brady and H. L. Frisch, J. Chem. Phys. 35, 2234 This works exactly for the Ising model in two dimensions. In (1961). three dimensions it depends what you take for v' . If you take [21] B. Chu, M. Pallesen, W. P. Kao, D. E. Andrews and P. W. /3 and 8 as given by Fisher and introduce the relation y' = /3(8 — 1) Schmidt. J. Chem. Phys. 43, 2950 (1965). into (1), you get exactly v' =0.646 below the critical point, equal [22] P. Debye, D. Caulfield, and J. Bashaw, J. Chem. Phys. 41, to the value for v above the critical point as given by Dr. Fisher. 3051 (1964). So for the Ising problem it looks as if these ideas which work above the critical point might also work below the critical point.

~~Discussion I would like to ask Dr. Fisher whether he thinks this is reasonable.~~

The other point I want to ask Dr. Fisher is how far he thinks F. H. Stillinger: Dr. Fisher has very lucidly pointed out that the the conclusions about exponential decay depend on short range long range behavior of the critical pair correlation function forces. This is established for the Ising model. However, doesn't tell one very much about specific heat anomalies. I quite a lot of people argue that the direct correlation function would like to point out that there is, however, an analogous goes off like the pair potential. This is not a very strong argu- quantity which gives direct information on the specific heat ment. However, when the pair potential and consequently anomalies by virtue of its long range behavior in the critical C(r) decays as l/r", you cannot have the direct correlation function region. It is well known that the specific heat C, is a fluctuation decaying more rapidly than the total correlation function, if the of potential energy. If one writes out the fluctuation formula Ornstein-Zernike theory is true. in terms of spin distribution functions for an Ising model, one encounters a combination amounting to a quadruplet distri- M. E. Fisher: Well, that last question is one which I suppose I bution function for two pairs of spins, the spins in each pair have to answer since it was asked. However, I think it is a rather being close together, minus the product of the two pair correla- difficult question! tion functions. The implication of a logarithmically divergent If one thinks of Coulomb forces, then of course, they do not sense unless has specific heat is that the critical point long range distance behavior make one some counter-charge. But then we d have the phenomena of screening and the correlations are of this combination would have to be as r~ , where d is the dimen- presumed to decay more rapidly. This to sionality. Here the long range distance r is the distance between seems be a property the centers of the two pairs. of the fact that Coulomb forces obey the Poisson equation. This is, I think, of it. If one goes ahead naively and attempts to evaluate this com- one way interpreting But if you try and bination of distribution functions using a Kirkwood superposi- work the arguments which would give you that result with a force which is like a force tion approximation, or something roughly equivalent to it, one Coulomb and even has counter- charges, but which little 1/r, arrives at the conclusion that the long range distance behavior decays a faster than you would not gel out should vary as the square of the critical pair correlation function. complete screening. Now in as far as the Van der Waals

129 forces can be thought of as originating through the electromagnetic interactions of matter, it seems to me still possible, in principle, that you have some overall screening effects and that you still do have an exponential damping. But if you just take a completely radially symmetric fixed 1/r6 potential without

any corrections, then I agree you cannot expect pure exponential damping since the direct correlation function would presumably go down in this way. So I am thinking here of models which have a potential that cuts off either exponentially or goes strictly

~~to zero, if one really wants to prove it. But I would think that~~

~~for real systems it is something of an open question.~~

W. Marshall: Dr. Fisher mentioned that the old theories can give very misleading results about the specific heat because you essentially take the long range behavior and extrapolate that into small distances. This, of course, is correct but one must be rather careful about the precise meaning. It is not a necessary consequence of the theories that you must get the wrong answer. In particular in the Elliott-Marshall theory [2] you get a difference equation for the correlation function and that difference equation happens to be valid at small distances as well as large distances. Therefore, that particular theory just reproduces the usual Bethe-Peierls result for the energy.

P. Debye: I would like to go back to some experimental facts. For instance the theoretician says, "Now I am at the critical temperature." Well, I do not know the critical temperature. Nobody can determine what the critical temperature really is. So how is there going to be an experiment at the critical temperature? Any experiment is a little bit away from the critical temperature and I cannot even define how far it is away from the critical temperature.

~~M. E. Fisher: Well, for example, some of the experiments dis-~~

~~cussed yesterday on the magnetic systems give you Tc very accurately.~~

~~P. Debye: I don't know of an equivalent experiment in liquids.~~

M. E. Fisher: Well, liquids are more difficult. The coexistence curves, I think, will tie down the critical points sufficiently ac- Figure 2 curately. Within a few millidegrees, to which some experiments

~~seem to pin it down, I would not worry about the differences.~~

P. Debye: I would like that the theoretical people tell me when I know approximately the answer. However, if you ask me what I am so and so far away from the critical point, then my curve the interaction between two polymers is which go through

~~should look so and so. each other, I do not know the answer and I must do the experi-~~

2 fi I would like to make two remarks in connection with the paper ment. Figure 1 shows I~' as a function of (s/X) X 10 . One of Dr. Brumberger. sees small deviations of a straight line if you go to large values of One remark is that I am not so pessimistic as Dr. Brumberger. slk. And if one really wants deviations which give the value for If one wants to find deviations which appear in the immediate the fourth moment of interaction, one has to go to considerably neighborhood of the critical point, one should not use x rays larger values of s/X as shown in figure 2. From these data one because the relevant variable is not s = 2 sin (0/2), but s/X. Thus can obtain the second moment and the fourth moment but not when X is very small, one must go to extremely small angles 6 the sixth moment. The maximum value of s/X possible with light 10" 4 -1 in order to obtain some information. Thus if one wants to find is 5 X A ; the x-ray measurements covered the range from deviations in the neighborhood of the critical point, one has to 2.6 X 10" 3 A" 1 to 2.5 X 10" 2 A"'. Over this range the scattered use light where s/X can be small for reasonable values of 9. intensity changed by a factor of 56. In the paper of Dr. Chu, it is shown that there are probably deviations from the Ornstein-Zernike theory near the critical point. However, if one goes to large values of s/X, one should also find Dr. Brumberger made a nice remark when he said that other deviations from a straight line in an Ornstein-Zernike plot. The experiments of another kind should be done. It has been said straight line is only a first approximation. So if one measures for and also Uhlenbeck made this remark at the beginning of this large values of s/X, one obtains information on the second moment conference that in the neighborhood of the critical point the of interaction and, if one goes farther, on the fourth and sixth system can be disturbed rather easily from the outside. In

moment of the interaction. Thus from the molecular point of this con ction I remark that we studied the effect of an electrical view it is also interesting to go to large values of s/X and there one field on the critical opalescence of a binary liquid mixture.

has to use x rays. For the details we refer to the publication [3]. Also here it is It is really true, as Dr. Brumberger said, that you should apparent that the Ornstein-Zernike theory is only correct if one connect the light experiments and the x-ray experiments with is not too near to the critical point. each other. This has been done at several places and we have So I want to emphasize that one has a straight line I~ l as a /"' 2 done it too. For this purpose, I present two curves for function of (s/X) over a large distance, but if you go to very large 2 as a function of (s/X) in figure 1 and figure 2 for a polymer, which values of s/X, you get deviations as has to be expected. If you Dr. Brumberger does not like. I do like polymers, as one obtains go to small values, which you can only do with light, then you information about their interaction, which cannot be predicted can expect to find the things which are the subject of this after- easily. If you ask me what the interaction of two argon atoms is. noon.

130 above Tc . They are sometimes straight, sometimes a little W . Marshall: Dr. Brumberger explained the difficulties that you frequently get with light and x rays. I would like to emphasize erratic, and even seem to have a little s shape. the fact that in looking at these systems many of the difficulties are very much eliminated if you did the experiment with neutrons. B. Chu: The data on the aniline-cyclohexane system as I have This is a technique about which many people who have worked shown in figure 1 of my paper demonstrate the experimental traditionally in this field do not think of because it is something difficulties which one encounters in critical opalescence studies Indeed Egelstaff and Enderby [4] have already verified new. whenever multiple scattering becomes appreciable. These the structure factor of simple liquid metals as given by the that measurements on such a strongly opalescent system (aniline- x rays is not quite correct. The correction is quite important cyclohexane) were performed at relatively large temperature in the region of small s/\. I would support very strongly, as distances {AT = 0.2° to 1°) and at a longer wave length (\0 =578 nuz Dr. Brumberger said, that neutron experiments also must be rather than the usual blue or green lines of a mercury arc) so done. It would be nice to do it on the same material which that the effect of multiple scattering remains negligible. On the means that almost all systems which Dr. Brumberger discussed other hand, the wiggles in figure 3 of my paper refer to the scat- have to be ruled out because they contain hydrogen and they all tering of a weakly opalescent system, namely, ra-dodecane and have to be deuterated. /3,^-dichlorodiethylether. Furthermore, the plot is reciprocal 2 scattered intensity versus (s/X) , and thus, the experimental M. S. Green: I would like to ask Dr. Marshall: What are the small- "fluctuations" in the data to have been magnified. est values of k that you can reach with neutrons as compared with seem light and witlucrays?

D. Mclntyre: But even 0.1 °C away from Tc ? W. Marshall: That depends on which experimentalist you talk to. It is important to go from one to the other and get them into competition. The usual kind of A-vectors that are easy are those B. Chu: Yes. The critical scattered intensity for the system, is /3,/3-dichlorodiethylether 0.1° [ covered by x rays. But there no reason why one should not re-dodecane and at away from Tc build a special small angle scattering apparatus like that which is quite weak, especially when we have used very narrow slit Lowde or Jacrot have used [5]. Then you can get very small systems. However, our main interest here was to try to deter-

A-values. I think Jacrot can probably tell us exactly what it mine the shape of the scattering curve very near the critical would be. mixing point, not at temperature distances 0.1° away from the critical mixing point. In fact, such wiggles disappear whenever

~~:! -1 B. Jacrot: 1 think you can go to 10~ A or maybe slightly lower. we increase the sensitivity of our detection scheme.~~

W . Marshall: Thus then the neutron experiments would be as good as light experiments in this respect. D. Mclntyre: Examining your papers, I find that you suggest that you have corrected in some manner for the multiple scat- A. Arroll: I want to ask Dr. Debye what is meant by the critical tering. Is this so? We have always found that what we thought region. When the susceptibility or compressibility obeys a to be multiple scattering in this range of AT was exhibited as a 4 = /3 power law already for ( T— T, )jT,- 0. 1, should you not say convexity to the abscissa at the ends of a conventional O-Z plot that you are already in the critical region when the temperature of a light scattering curve. In your systems I should think that is 7V/10 away from the critical? In discussing possible devia- you would get the same behavior. tions from the Ornstein-Zernike theory you consider what happens at temperatures hundredths or thousandths of a degree-

away from the critical temperature. However you did not men- B. Chu: No. I do not believe that with our experimental setup, tion what is found for the power law of 7~'(0) which is inversely multiple scattering is appreciable for this weakly opalescent proportional to the compressibility: I~'{0) means the inverse system. We have tried to correct for attenuation and to avoid extrapolated zero angle intensity. multiple scattering. I do agree with you that whenever multiple

~~scattering does become appreciable, it invariably destroys~~

~~P. Debye: It is hard to find a deviation from linearity for /~'(0) the linear behavior in an O-Z-D plot. as a function of T, because you are not certain that the extrapo-~~

~~lation is correct. D. Mclntyre: In one of your curves it appeared.~~

H. Brumberger: The behavior of I~'{0) as a function of tempera- , B. Chu: This is not multiple scattering. ture does deviate from linearity in many cases over a wider temperature region. D. Mclntyre: What is it?

M. S. Green: I think we have a real problem here. Although I I don't know whether Dr. Debye would agree, I believe that light , scattering experiments do in fact show a curvature for /~'(0) as a B. Chu: I do not believe that the curvature in figure 3 is due to j function of temperature many degrees away from the critical multiple scattering because we have performed our experiments temperature. On the other hand, any deviations from the with cylindrical cells of two different diameters, and the shape of Ornstein-Zernike theory (one should also say Debye theory) do such curves can be superimposed. This may not be fool proof. also hope to carry our studies further with more variations not occur unless the temperature is within 0.01 °K from the We critical temperature. in the geometry of our optical system. I would also like to make j Perhaps this means there are two critical another remark. It is difficult to determine the behavior of regions. I don't know the answer, but it is a problem which 7~'(0) as a function temperature. thing, most investi- needs to be resolved. Perhaps it is related to the fact that the of For one logarithmic singularity of the specific heat does not show up gators have neglected correcting for the attenuation of light as it until one a a critical goes through the sample cell. Since the turbidity of the sample j comes within hundredth of degree of the depends strongly on temperature as we approach the critical point, although the behavior of C v is nonclassical in a much temperature, a plot of the extrapolated reciprocal angle wider region.B zero " is scattered intensity (without attenuation corrections) versus e D. Mclntyre: I see no essential disagreement between Dr. Chu's temperature does not represent the true behavior of 7"'(0) versus new results for the cyclohexane-aniline system and our results T. Now, all I can say is that the curvature in a plot of I~'(0) a with the same size cell in the same temperature range. I per- versus T for our system is very small if there is any. As far as I e ! sonally believe on the basis of our data that the plot of /~'(0) can see, it is very straight as shown in figure 4 of my paper. |U against AT is gradually curved upwards over the whole tempera- Yet, we must remember that I remarked on its nearly straight * ture range. Now I would like to enter a slight quibble. I am line behavior over a temperature range of about only 0.3°. The a little surprised at the form of your uncorrected scattering nonlinear behavior over a whole temperature range (say 10°) curves. There are some wiggles in your curves at 0.1 to 0.2 °C remains to be examined.

~~131~~

209-493 0-66— 10 D. Mclntyre: I think that there is actuaEy experimental con- that you are studying terribly long range effects. A light scat- currence in the fact that when measurements are made at small tering experiment is sensitive to distances between 4000 A anc

A7*s and low angles a downward curvature in the O-Z plot, 10,000 A and the temperature differences T—Tc , which corres- such as yours, will be found. So I think that the scattering sponds to such ranges are expected to be in the millidegree range. experiments made by various people are in agreement on this If one were to look in the intermediate range between light and point. x rays, I would guess one would see curvature in a range T—Tc of one degree. In the x-ray results one sees curvature in the 10 deg range. Thus I think you have to look in the temperature P. Debye: I think everybody is now convinced that there is range that corresponds to the range of the wavelength used in curvature in the immediate neighborhood of the critical point the scattering experiment. Unless the experiments show other and that the curvature is in the expected direction. wise, this interpretation answers the question of the size of the A. Isihara: Could you explain why the/"" 1 versus (s/\)2 plot curves critical range. downwards for small values of s/A? j P. Debye: It one wants to investigate something characteristic M. S. Green: 1 would like to comment on this question. As Dr. | giver Fisher emphasized, the essential approximation of the Ornstein- for the immediate neighborhood of the critical point for a angle 0, one has to go to longer and longer wavelengths Because Zernike theory is that the direct correlation function is short the characteristic variable is not s, but s/A. ranged which means it has a finite second moment. The second P. Anderson: I am not talking about s/A, but I talking about moment of the direct correlation function is essentially the slope am 2 a physical range, in particular, about the of the versus (s/A) curve. The fact that this curve turns correspondence between ranges in temperature and ranges in If downwards means that the second moment is increasing. The space. you were 1 from critical truly significant prediction would be that the second moment of deg away the point, you would find curvature^al the direct correlation function not only increases but becomes much shorter wavelengths. infinite at the critical point. Nobody can actually show that, but this seems to be indicated by the experiments. P. Debye: No.

M. E. Fisher: The value of tj estimated by Dr. Chu is much bigger M. E. Fisher: I will attempt not to add anything to the confusion! than I would have expected. As far as I am concerned there First of all, I think it is a question of pure semantics. Unless are still some puzzles, because the curve of /"'(()) versus tem- you have some theory of the critical point you cannot talk about perature does not go to zero if you extrapolate to the critical the critical region except in terms of deviations from this theory. temperature. I think if you had done P-V-T measurements you We are talking in terms of the Ornstein-Zernike theory here, would have observed that the fluctuations-of-composition- where there is a ^-dependence and a temperature dependence compressibility would go up to infinity while your /~ l (0) still to be considered. Now, if we look at the magnetic case, there I seems to stick to a small finite value. Maybe there is some is no doubt that for the susceptibility deviations can be seen reason why the real critical temperature is lower, but from what very far away from the critical point. I find it puzzling that one Dr. Chu has said, I don't see it! does not seem to see these deviations in the binary system. As I far as one does not see these deviations here, one has not really | A. Arrott: Returning to my previous question, it seems that for understood the difference between these two systems. a binary mixture the critical region is much smaller than for a However, looking at the ^-dependence, it is perfectly possible magnetic system. Why is there such an order of magnitude that you can have an Ornstein-Zernike /r-dependence absolutely j difference when you consider the critical region for a binary mix- correct right up to the critical point even if the thermal behavior ture and that for a magnetic system? is not correct. The index tj I find theoretically is very small.

Because 17 is small you would expect, I think, to see deviations only in a small region. So I think to talk about the critical region E. Helfand: I believe the size of the critical region is related to is not too helpful. What Dr. Debye said is correct, namely, the range of the forces. in the This conclusion emerges study that when one is talking about some deviation one must attach of systems with long range forces. The shape of the particles to it a magnitude and a range of temperatures where it is to b< in the binary systems about which we are speaking cause may observed, which can be different for one phenomenon compare( these systems to be less in corresponding states with the Ising to another. When these ranges differ from one system to anothe or Heisenberg models than one fluids or component magnets. for the same phenomenon, we have a real challenge.

I M. S. Green: I don't think your answer is correct. I don't think Dr. Sette: I want to point out that the results of sound absorption really we have a narrow critical region in a binary system. The measurements for a liquid vapor critical system and a binary: ( phenomena which Dr. Chu showed take place within indeed a mixture are different. In a one-component system the absorp- / range of one hundredth of a degree. However, there are other tion increases sharply only in a temperature range of one degree | phenomena in the two phase region which occur over a much from the critical temperature. In the case of a binary mixture , wider temperature range. For instance, the 113 7' (AT) power law the absorption increases rapidly starting at T— c = 30 °C. for the phase boundary curve, which is a nonclassical phenome- This indicates that the two systems behave in a different way. non, is just as much true for these systems as it is for a magnetic I believe that a comparison between the magnetic system and I system or for a one component fluid. I guess the question of the the binary system should be made with some precaution. 4 /3 power law for the compressibility is still a matter of controversy, but in my opinion (and I think Dr. Mclntyre agrees with me), there are deviations from the straight line for the compressibility as a function of temperature in a much larger temperature range than we are talking about here. References

E. Helfand: The degree of the coexistence curve is not as clearly related to the distance parameter of the correlations as are [1] J. Stephenson, preprint, Ising-model spin correlations o: phenomena above the transition such as the compressibility the triangular lattice. II. Fourth order correlations, to b (or its analogue for mixtures or magnets) and the critical scat- published. tering curves. [2] R. J. Elliott and W. Marshall, Rev. Mod. Phys. 30, 1 (1958 [3] P. Debye and K. Kleboth, J. Chem. Phys. 42, 3155 (1965). [41 P. A. Egelstaff and J. E. Enderby, to be published. P. W. Anderson: To answer the question of Dr. Arrott, is not the [5] B. Jacrot and T. Riste, Thermal Neutron Scattering, ch. VI, reason why the critical region appears to be so terribly narrow, ed. P. A. Egelstaff, (Academic Press, 1965).

~~132 SCATTERING: INELASTIC~~

~~Chairman: K. S. Singwi~~

~~133~~

~~Critical Scattering of Neutrons by Ferromagnets~~

~~W. Marshall~~

~~Atomic Energy Research Establishment~~

~~Harwell, Great Britain~~

~~1. Introduction where~~

Recently a number of authors have discussed ^(K, (o)=-^- r dt y eiK R^(Sg(0)S|(0). (2.3) the theory of critical phenomena in magnetic materials and produced results differing from the conventional and simple theory as described in many Only the terms with a and /3 equal contribute to textbooks [1-4]. Some predictions of the new (2.2) if the sample has only scalar interactions be- theories have already been verified experimentally. tween the spins. Throughout this paper we shall Neutron scattering techniques can be used to give assume this, i.e., ignore dipolar interactions and a thorough investigation of critical phenomena anisotropic exchange. and this paper discusses the theory needed in the Define interpretation of the experiments. at' k In section 2 the cross section formulae are S(K, t) =Je &(t) (2.4) written down in various forms and it is shown how the cross section is related to the generalized wave then vector dependent susceptibility. In section 3 special attention is given to the time dependence xx * x x f (K, fo)=A; f dte-m (S (-K, 0)S {K, t)) of fluctuations near the critical point and it is ZttN J- x shown that the conventional theory fails. A modi- (2.5) fied theory is described in section 4. Because of the difficulty of the problem this theory is only with similar definitions for the y and z-directions. qualitative but it points to the existence of damped Below the critical temperature the correlation spin waves, or some similar phenomena, even above function in (2.3) has a nonzero value at infinite this gives the Bragg scattering contribu- Tc . Section 5 mentions two other experiments time and indicating the existence of spin waves above the tion &f(K, a)). critical temperature. Section 6 gives conclusions.

~~y^(K, o)=i 8(6>). (2.6) 2. Formulae and Definitions~~

The remainder is the "diffuse" contribution The neutron has initial wave vector k and initial energy E. It is scattered into a state of wave vector k' and energy £". The wave vector change and energy change are, respectively: = dte->«>{(S«( - K, O)S0(K, t)> J* K=k-k -(S«(-K)>(SW)}. (2.7) = = 2 ~ 2 (2.1) na> E-E' (h l2m0W k' ). We notice that

~~The cross section is then da>ytf(K, w)=- {~~

~~(2.2) } (2.8)~~

~~135 and in the limit A^O this is proportional to the and mean square fluctuation in magnetization which, by elementary thermodynamics, is proportional to the magnetic susceptibility. Hence (2.15)~~

lim da&ff(K, a) = kTx*ls?n* (2.9) By definition /°^(K. 0) is unity and hence, from K^U J -x (1.14), where the susceptibility tensor xali is defined in the do)F a^{K, &>)=!. usual way. For the correct choice of principle i (2.16) 3 zz xx yy axes, x"' has two values x an d X ~X below zz=z xx T,-, and value, yy T,. one X X ~X - above Therefore The form of (2.9) suggests that we seek a more (2.13) gives general relation, valid at all K, by introducing the x AT f 1 — quantity x^K) which describes the response to a a/3 % X (K)= dor field of wave vector K. The idea of this is well known but, so far as the author is aware, the first w)-y«f(K, formulation for an explicitly neutron scattering {^(K, 6>) Bragg}- (2.17) problem was given in a recent paper by Mori and

Kawasaki [5]. Following the latter paper we introduce a relaxation function This is an important relation because it is the generalization of (2.9) and because it unambiguously relates the observed cross sections to a quantity

ali a of theoretical interest. Whether or not we under- R {K, t)= I dk(e"*S (- K, 0)e-^(K. t)) Jo stand the full a> dependence of the cross section we can deduce xa/3 (K) provided the experimental -/3- (2.10) results are known with sufficient accuracy to evalu-

~~ate the integral on the right-hand side of (2.17).~~

~~Frequently it is more convenient to measure the ; It can then be shown that total count rate at a fixed scattering angle instead~~

~~j of the full 5^(K,~~

~~~ the integral in (2.17) should be performed at fixedt = e iatRa t) (2J1) i^^\y ^ K not at fixed scattering angle. The error introduced by this is small if the incident neutron energy~~

is large compared with the energy changes, 7f(o. and also that Second, the count rate is not weighted with the) nwti factor {1 — e~ } l~H(i>{5 and therefore we require a X TO = ^/^(K, 0). (2.12) that iTui kT. If this last condition is satisfied the factor becomes unity and the experiments give; X^(K) directly. Hence The molecular field theory for a ferromagnet

~~above Tc predicts x)-y*(K, a>) Bragg~~

~~= ^^F^K^) (2.13) rx(K) = Z2(K) = (2.18 f-,r^) X X- x 2 r?(KT + q ) where diere~~

* F^(K, w) = y- [ dte-**y*»(K, t) (2.14) " 77" J — x K=r+q (2.19

136 K . and t is the nearest reciprocal lattice vector, fore it appears that in the theory of Elliott and Marshall a diffusive-like behavior is built into the x = .. (2.20) perpendicular correlation function (S (0)S${t)) X, ^/AS(S+l)/3/,T( We will return to this point in the next section. r? gives a measure of the range of the exchange interaction 3. The Inelasticity of the Scattering

~~i .-, it To consider this 1- problem we concentrate atten- (2.21) XX 6 tion on F (K, co). Above Tc this is identical to 2^ ZZ F' '(K., co) and therefore gives a complete descrip-~~

tion of the co-dependence, while below Tc it gives the single spin wave scattering. We are not able and Ki is an inverse correlation range defined above XX to calculate F (K, co) rigorously but we can get 7V by some information about the moments of it. We first Xc T-Tc note that (2.22) x"(0) 7V xx X j- R (K, 0 = <[S*(-K, 0), S (K, t)]>. (3.1) Other simple theories of ferromagnets give results t 1 qualitatively similar. These predictions have been tested by Passell Hence et al. by neutron experiments on metallic iron. [6], a 2 X They conclude that the general form (2.18) is correct S*(-K,0), S (K, t) dtf within the limits of experimental error but that rffcf

~~not have a linear variation with , as does T—Tc and predicted by (2.22) but instead d xx x -R (K,t) = S'(-K, 0),£^S (K, t) 2 yl* l T 1 ~ (T-Tc ioxT-Tc (2.23) dt (3.2)~~

This 4/3 law is in agreement with recent theoretical It is to XX work by Domb [1] and by Gammell, Marshall, and easy show that F (K, co) is even in w so only the even are nonzero. Morgan [2] who found a 4/3 law for the divergence moments Then Z2 of x (0) using the mathematical technique of suc- 2 2 xx cessive Pade approximants. (w )=|" da)(x) F (K, co)

~~Below Tr the molecular field theory again gives the general form (2.18) but with |r-/~(K, o~~

~~2 Tc -T rV = for T~~

B (3 ' 3) -3^»i£*"< -?>L «ix =0 (2.24) and similarly Because this simple theory gives the incorrect power law above Tc we would expect it also to be <«,<> incorrect below Tr but, to date, no experimental results of sufficient accuracy are available below

Tc . The theory due to Elliott and Marshall [7] If we now assume a nearest neighbor model these gives results virtually identical with (2.18) to (2.24) two expressions can be evaluated. We find, using with one important difference, they predict a non- (3.2) and (3.3), zero k\x below Tr . The precise reason why the theories differ in this way is not yet understood; non- 0)2 = < > SZS - 2 (7 " yK)(SyS + « } (3 5) interacting spin waves also give k x = 0 and there- aff(K) ° "

~~137 where matic effect, shows that long wavelength fluctuations are slow. (3.6) At infinite temperature the discussion is straight- P forward. From (2.18) and (2.22) we get~~

-> Tr l)/3/,T= 4 X(K) Xr IT=^^S(S + x„ (3.9) The expression for (to ) is very long and will not be repeated here. It is of a similar form to (3.5) with XX where we use to stand for the a denominator \ (1Q and a numerator involving x<> Curie susceptibility. correlation functions between four spins which are The short range correlation functions appearing in close together. This numerator is also propor- the moment formulae are also inversely proportional to tional to (yo — jh•), as is (3.5). T and a tedious calculation gives the From these remarks we can now reconstruct the results general character of the moments (to") as a function 2 <&> S(S+ of n. >^m 3 l) (y0 -yK) (3.10) 0 (a) (a) ) is unity. 2 4 (b) (to ), (to ) and all higher moments are in- 4 xx (. (3.8) still valid because this term is very small.

We know that the specific heat has a weak singu- In the forward direction we notice, since both 2 4 2 4 (to ) and (to ) are proportional to K , that (to ) is larity (probably logarithmic) at Tc , and hence e(T) 2 2 much larger than (to . It follows that although is continuous, but with an infinite slope, at Tc . ) This is a weak singularity compared to the diver- the half width at half height of F(K, to) must be

small the shape must have large wings to it, i.e. gence in x(0)- this essential difference is because any short range correlation function can have only F(K, to) must be something like a cutoff Lorentzian weak singularities whereas x(K) involves long range We therefore assume correlation functions. We conclude that effect

~~(d) produces numerical factors slowly varying with T. 1~~

F(K, to) for I col s 2 2 < Effect (b), the thermodynamic slowing down of 7Tto + r (3.12 fluctuations, shows that the scattering becomes more nearly elastic near Tc . Effect (c), the kine- =o for \to\ > s

138 where The general expression for < w 4 > can be used to give 2 3 l/2 77 r ) i 4 2 2 < to > = <(o > at T= 0. (3.21) and The only distribution which satisfies (3.21) is a 1/2 S-function. Hence (3.20) and (3.21) together give s = (3.14) 2 <" > XX F (K, oj) = ^ 8(a-u h ) + h 5(oj + co a ) rr = 0] (3.22) second and fourth moments are chosen to give the where correctly. Substituting from (3.10) and (3.11) gives, 2 to leading order in K , u)K = 2JS{yo — yK)lh: (3.23)

S(S+1) 1/2 Ta=^|(7()-7a) (3.15) is the correct spin wave frequency. As the tem- {/• - l-3/16S(S+l)} perature is raised the spin waves get damped and the S-functions of (3.22) become broadened into XX cutoff Lorentzians. i.e., the singularities of F (K., w) 4J 1 2 s=-^r [5(5+l)] 1 ' 2 [/— l-S/ieStS+l)] ' move slightly off the real axis. (3.16) As the temperature is raised still further the spin wave interactions result in an energy renormaliza- where r is the number of nearest neighbors. If we tion. It is well known that at low temperatures this write energy renormalization follows a T5l ~ law. Up to,

say 0.87V, it is probably reasonable to use (3.5) 2 TA = AA atT=* (3.17) directly to give a rough estimate of the spin wave energy. Hence, very roughly, and use molecular field theory to give J in terms of evaluate A. For Fe we find (ok ~ 2 " S*S?, 2 fc , we can now J (7o 7a ) { + >V' lt

~~for 0 < T < 0.8 Tr . (3.24) 2m0A/£= 19 atT=* (3.18)~~

In the neighborhood of T, we expect [The factors on the left-hand side of (3.18) are convenient and give a dimensionless number.] (3.18) 5(S+1) is only a rough estimate because we have used a < SuS" > ~ < SI'S' > (3.25) 3(r-l)" nearest neighbor Heisenberg model to represent metallic Fe and furthermore we used molecular Henc< field theory to relate J to Tr . Nevertheless it should be of the correct order of magnitude. It A K2 (3.26) should be noted that the only significant singular- where ities of (3.12) are poles on the imaginary axis, at

= . 1/2 w ±/F - A A ~(2.A/V/6fi){2S(S+l)/3(r-l)} (3.27) At low temperatures the discussion is also straight forward. In a ferromagnet the last factor of (3.5) Substituting for J as before we find for Fe [S=l, 2 field theory and becomes S and both molecular r — 8] the result spin wave theory give 2m*A\n ~ 30. (3.28) (K) = (3.19) 2J(y„-yA ) This last estimate is rough because we have made

11 ence no real attempt to calculate the renormalized oj h within, say. a factor of 2. Nevertheless within < 2 2 S 2 atT = 0 (3.20) w >^4J (>-yA W this kind of uncertainty it should be satisfactory 139 up to about 0.87V and beyond that temperature expression like (3.17) with the spin wave interaction problem is too difficult to discuss here. 2moAtft ~ 11.4 [Experiment T~ 7V]. (3.31) We now notice by comparing (3.17) and (3.28) that the separation between the spin wave peaks at This is about two orders of magnitude larger than about 0.87V is roughly the same as the width of (3.30) the scattering curve at infinite temperature. The (2) T K is approximately independent of tempera- shape of F •"'(!£, o>) is very different at the two tem- ture. This is in contrast with (3.30) which varies peratures but the scale of the dependence on o» is sensitively with T because of the r\i<\ dependence.

2 roughly the same. (3) T A is proportional to K within experimental to error. term proportional K*, as predicted by \ rj No We now turn to a discussion of F- (K, oj) near (3.30), is observed. TV. Several authors [5, 9, 10] have discussed this We are forced to conclude that the conventional or a similar problem using various theories; all theory as we have just summarized it, fails sub- of these theories are, however, basically the same stantially. Therefore, in the next section, we look as that which we will now summarize. Because of for an improvement. the slow T dependence of effect (d), near but just above TV we may estimate moments as

~~2 2 4. Theoretical Discussion for T ~ Tr >k,t~ A ^ Xo/x(K,r) (3.29)~~

4 4 < oj > k,t ~ < a) > A>x Xo/x(K, T) The discussion of section 3 has shown that the inelasticity near TV is hard to understand and there- J fore in this section shall discuss the qualitative where Xo is the Curie susceptibility. By comparison we features which it now appears a good theory of the with the discussion of F(K, oj) at infinite tempera- 2 critical region ture we see this has the effect of replacing K by must have. 2 4 2 We first note that the calculation for infinite tem- K Xo/x(K, T). The ratio < to >/< or > is now

perature the spin calculation at Ij even larger than at infinite temperature because (3.18), wave about 0.8 TV, i.e., (3.28), and experiment near TV, (3.31), X(K, T) is large near TV. In the absence of any all give results roughly in agreement as regards other information about F(K, oj) we assume it is a x,r cutoff Lorentzian just as at infinite temperature; the dependence of F (K, oj) on oj. This suggests xx that oj oj) in fact by analogy to (3.17) we get immediately the scale of F (K, does not change significantly from 0.8 TV up to infinite temperature but that the shape of F xjc (K.,oj) changes consider- j rK =^X%FAK*ii(Ki + **)TclT. (3.30) ably from two distinct spin wave peaks below TV j into a single Lorentzian at infinite temperature, j calculations rely upon calcula- From either (1.22) or (1.23) we see that t\k\ is But which moment information small near TV and in typical neutron experiments tions are quite unable to give sensitive therefore the as described the observations are made at angles such that on shapes and theory 2 2 in section 3 is suspect. K ~ k . Thus, for an experimental situation such examine more carefully the behavior as that used by Passell et al. [6], or Jacrot et al. We now (2.18) gives [11], the expression (3.30) is some two orders of of (3.29). Using magnitude smaller than the appropriate expression 2 2 2 < co > = < oj > (r? k\ + r\K ) TJT (4.1) j for infinite temperature, (3.17). Hence the theoreti- a, r a. x cal prediction is that there is a very substantial i 2 < 0>*>K.T= k. x (r? k\ + riK ) TJT (4.2) narrowing of the scattered neutron distribution as j

5 2 at infinite temperature,! the temperature is reduced towards Tc . (4.1) is proportional to K ' 2 2 However, in both the experiments by Passell because < oj >k.* is proportional to K but as the et al., and by Jacrot et al., none of the features of temperature is lowered (4.1) becomes smaller and j (3.30) are observed. In particular the experiments becomes proportional to K4 at temperatures suf- 2 suggest: ficiently close to Tc that k\< K . If we assume XX (1) The order of magnitude of Y K is given by an that F (K, co) remains a cutoff Lorentzian we are

140 immediately led to (3.30). But the above behavior from the origin. The conventional theory, (3.30), is quite consistent with a gradual change of shape near Tc tells us that the two significant poles have XX at infinite 2 in F (K, to) from a Lorentzian tempera- moved along the imaginary axis to± i A/C xo/x(K", T) XX ture to one which is dominated by two peaks at and the other singularities of F (K., to) remain 2 2 o) = ±AK for temperatures such that k\ < K . at a long distance from the origin. However, the

These peaks would make a negligible contribution neutron experiments insist that this is not correct, 4 themselves to < to > and the higher moments. there must be singularities of some type distributed Such peaks, if they existed, would have the obvious at distances ~ AK2 from the origin; the poles interpretation of quasi-spin waves above Tc . predicted by (3.30) may "or may not exist but if However we notice that a rectangular distribution they do exist they must have a weight small com- 2 o) = equally well ~ 2 bounded by ±AK would serve pared to those at distances AK . XX of (K, to) is special in- as a dominant distribution near T< - We also recall The form F below Tc of xx 2Z XX that, above T F {K,(o) and F (K, to) are iden- terest. Well below Tc we know that F (K, to) c , tical and we cannot possibly associate single spin is dominated by two spin wave peaks centered at wave processes with F"(K,a>). Nevertheless ±(i)k, where coa is the renormalized spin wave the general point remains valid, it is quite consistent energy, and with a width dependent on the spin with (3.32) and (3.33) for any curve whether double wave lifetime. As the temperature is raised does XX peaked or not, to be sharply confined to limits F (K., to) retain this form or does a distinct dif- 2 (o = ±AK provided it also has weak "tails" which fusive mode appear in addition? We may specu- 4 can give large values to < oj > and higher moments. late that an analogy to the classical fluid may exist. This argument can be summarized as follows: A classical fluid, at small K, may be described by

(i) The moment calculations are sensitive to the the usual equations of hydrodynamics. In the tails of the distribution F xx (K.,oj) whereas the neu- absence of any dissipative effects, viscosity or tron experiments measure only the central portion thermal conductivity, the relaxation function XX of Fxx (K,a>). [equivalent to F (K, to)] has two 8-function peaks

2 c. (ii) This central portion has a width, ~ A K at ± csK where s is the velocity of sound. The , roughly independent of T from 0.8 Tc to infinite introduction of viscosity or thermal conductivity temperature but with a shape varying from a double damps these sound waves and simultaneously a peak (representing spin waves) at the lower limit diffusive mode also appears in the relaxation func- to a single peak at high T. This central portion tion. In other words, in the complex to plane, as always contains the major part of the area of soon as the sound wave poles move off the real axis, F xx (K,co). a new pole appears on the imaginary axis. In suit- XX (Hi) The tails of the distribution F (K, to) are able cases therefore, the relaxation function as sensitive to T and vary so as to give agreement with measured by experiment as a function of real to, the moment calculations. They are difficult to can have three maxima, one at to = 0 and two sym- observe experimentally. metrically placed either side at the appropriate zz xx (iv) Above T F (K,o>) is identical to F (K,o)) renormalized frequencies 13]. In the magnetic c , [12, but below Tc does not show distinct spin wave peaks. system the same phenomena might take place

However the co-scale is roughly similar. as the spin waves are damped. This possibility The above conclusions appear to be dictated by is intimately linked with the problem of the energy the experimental results which were obtained for renormalization of spin waves very close to Tc and the general range K~~K\. For either K

K > Ki the neutron experiments give no informa- particular discussion we shall not pursue it further tion and therefore it is possible that in these regions other than to remark that it is also connected with the to dependence is qualitatively different. the behavior of k\x below Tc (this problem was

The clearest way of describing the situation is in mentioned in sec. 2). XX terms of the singularities of F (K, to) in the com- The discussion of this section is most clearly plex to plane. We have already remarked that summarized in figure 1 which shows the positions

J ' r the high temperature theory, (3.17), gives two of the singularities of F (K, to) in the complex to 2 significant poles at ±iAK ; the other singularities plane according to the conventional theory and also

? -rx of r (K, to), [which are needed to describe the two possibilities which would be consistent with cutoff procedure for |to|>s], are distant ~~ Jjfi the neutron results. In this figure singularities

~~141 -X X- K X- o K~~

~~) :~~

~~X X X x x X T~0 8TC X X X X X X~~

~~: t~~

~~"spin wave" singularities remain within a distance ! 2 2 2 IAK r (k + K, )t /T 2 , c - K FROM ORIGIN - v ^ A r \~~

~~/ \~~

~~1 / - /y— J AK 2~~

~~2 : LAK 2 2 , t AK > 'lAK~~

~~T » T c~~

~~: > c ; :~~

x FIGURE 1. Possible singularities o/F *(K, &>) in the complex co-plane. Figure la shows the results of conventional theory. Figure lb indicates the qualitative nature of the singularities of the analogy to hydrodynamics holds. Figure lc indicates an alternative possibility in which poles do not collapse to the origin at 7V. In all diagrams the singularities at distances — Jlfl from the origin are ignored. which are a long distance —Jlfi from the origin concerned ferromagnets we would naturally ex- are not shown. Figure la gives the conventional pect to carry over some results to antiferromagnets. theory, figure lb the theory we would expect by In particular if spin waves exist above Tc in a ferro- analogy to hydrodynamics and figure lc the theory magnet we would expect them to exist above the we might expect if no "diffusive" mode appears Need temperature of an antiferromagnet. It is

al. below Tc . At the present time we are not able therefore worth noting that Murthy et [15], to tell whether possibility, (b) or (c) is correct. have observed spin waves in MnO above the Need temperature. As we would expect, these spin 5. Further Experiments waves peaks are broad. MnO is an unusual antiferromagnet in that TN ~ 120 °K but 0 ~ 600 °K; In this section we shall list briefly other experi- therefore at the Need temperature we expect an ments which give information on the form of exceptional amount of short range order to exist. XX F (K, w). In the presence of this large short range order we (a) In experiments on Fe.j04, Riste [14] observed would expect the spin waves to be better defined that spin waves existed above Tc and plotted their than in the typical antiferromagnet and therefore lifetime as a function of temperature. His results in figures lb and lc the "spin wave" singularities can be taken as experimental evidence in favor would approximate closer to simple spin wave poles. of the existence of spin waves above T,.. Unfor- This is probably why these neutron experiments tunately his experiments did not cover a wide observed them whereas experiments on other enough range of scattering surfaces to distinguish antiferromagnets, for example, Turberfield et al. between figures lb and lc. [16], on MnF2 do not see distinct spin waves above (b) Although all the discussion of this paper has

142 6. Conclusions [3] M. E. Fisher and M. F. Sykes, Physiea 28, 939 (1962). [4] M. F. Sykes and M. E. Fisher, Physiea 28, 919 (1962). [5] H. Mori and K. Kawasaki, Prog. Theor. Phys. 25,723(1962). The theory of neutron scattering from ferromag- [6] L. Passell, K. Blinowshi, P. Nielsen and T. Brun, Proc. International Conference on Magnetism. 1 nets has been reviewed and it is shown how the [7] R. Elliott and W. Marshall, Rev. Mod. Phys. 30, 75 (1958). experiments give the wavelength dependent sus- [8] P. G. de Gennes, J. Phys. Chem. Solids 4, 223 (1958). [9] L. Van Hove, Phys. Rev. 95, 1374 (1954). ceptibility unambiguously. The experiments show [10] P. G. de Gennes and J. Villain, J. Phys. Chem. Solids 13, 10 (1960). that the conventional theory of the time dependence 2 [11] B. Jacrot, J. Kostantinovic, G. Perette and D. Cribier. It of the fluctuations near Tc , is incorrect. is too [12] L. D. Landau and E. M. Lifshitz. Fluid Mechanics, Per- gamon Press. difficult to construct an alternative theory imme- [13] W. Marshall, Lectures on Critical Phenomena, A.E.R.E.-L. diately but to explain the results it is necessary 144 (1964). Supplement Bill, that some remnant of "spin-wave motion" remains [14] T. Riste, Journal Phys. Soc. Japan 17, 60 (1962). Proc. of International Conference on Mag-

above Tc . Two possibilities are described in terms netism and Crystallography. G. Venkataraman, K. Usha Denig, B. A. of the singularities of the relaxation function in [15] N. S. Satya Murthy, Dasannacharya and P. K. Iyengar, Symposium on In- the complex co-plane. elastic Neutron Scattering, Bombav, 1964. [16] K. C. Turberfield. A. Okazaki, and R. W. H. Stevenson, Proc. Phys. Soc. 85,743(1965). 7. References

[1] C. Domb and M. F. Sykes, Phys. Rev. 128, 168, (1962). ' Nottingham 1964. 99. Marshall and L. Morgan, Proc. 2 [2] J. Gammel, W. Roy. Soc. Symposium on Inelastic Scattering of Neutrons in Solids and Liquids, Chalk River, 275, 257 (1963). 1962.

~~Critical Magnetic Scattering of Neutrons in Iron~~

~~L. Passell~~

~~Brookhaven National Laboratory, Upton, N.Y.~~

~~Abstract~~

~~Measurements of the angular and energy dis- and strength respectively of the spin correlations),~~

)~ l - 30±0 - 04 tributions of 4.28 A neutrons scattered at small is found to vary as (T— Tc . Near the Curie angles from iron at temperatures above the Curie temperature there was sufficient intensity to meas- temperature are described. The results are inter- ure the ratio of the coefficients of the K4 and K 2 preted in terms of Van Hove's theory of critical terms of the angular distribution. The value of 2 magnetic scattering and yield information on the this ratio, 29 A , is related to the existence of long range of spin correlations and the dynamics of the range couplings within the spin system. Details spin ordering process. For the, dimensionless of certain recent modifications of the theory of parameter, 2m A/7?, which describes the time de- critical systems are discussed and compared with pendence of the spin fluctuations, we obtain the the experimental results. value 11.0 ±0.6 at T-Tc = 2 and 18 °C. The zero This work is described in detail in a paper by L. field magnetic susceptibility, as determined by Passell, K. Blinowski, T. Brun and P. Nielsen, the parameters r (which represent the range Phys. Rev. 1886 (1965). Ki and x 139A,

~~143 Critical Neutron Scattering From Beta-Brass~~

~~O. W. Dietrich and J. Als-Nielsen~~

~~Research Establishment Riso, Roskilde, Denmark~~

~~Introduction (2) the absolute value and the temperature dependence of the SRO correlation range, and~~

Beta-brass, which is an approx. 50 percent to 50 (3) some conclusions concerning the shape of percent alloy of Cu and Zn, forms at low tempera- the pair correlation function. tures an ordered B.C.C. lattice. A B.C.C. lattice can be thought of as composed of two S.C. lattices — Cross Section so-called superlattices, denoted respectively A in occupation of lattice sites are and B — displaced half a cube diagonal with respect The changes believed to be so slow that a static approximation is to each other. Perfect order means that all Cu rigidly valid in deriving the relevant cross section. atoms are at A sites, all Zn atoms at B sites. The order gradually decreases with increasing tempera- The cross section for a scattering vector #c close to a superlattice point t in the reciprocal space is ture; at the critical temperature Tc (Tc — 468 °C) the average occupation of say an A site is entirely da bV LR0 2 8(k random, i.e., the long range order (abbr. LRO) has dCt

~~disappeared. Above Tc , however, local order still exists: If a certain A site is occupied by say a Zn + Er pcf (R)e-« r (1) atom there will be an excess probability over ran-~~

domness that the nearest site — a B site — is occupied by a Cu atom and so on. This local order or short ampli- range order is described by a pair correlation func- a and b denote respectively the scattering tion (abbr. pcf.) tudes for Cu and Zn nuclei, R is a direct lattice 2W In a diffraction experiment the LRO will give vector and the Debye-Waller factor e~ accounts rise to narrow diffraction peaks (Bragg scattering for the influence of the thermal vibrations on elastic peaks), whereas the SRO will give rise to broader scattering. of 2 and less intense peaks (critical scattering) provided For T < Tc the temperature dependence (LRO) intensities. For that the radiation is scattered sufficiently different is deduced from the Bragg peak only from a Cu atom and a Zn atom. This criterion T > Tc this kind of scattering disappears and make neutrons more suitable than x rays in case critical scattering remains. It will later be shown of /3-brass. Walker and Keating have previously K R that the Ornstein-Zernike pcf ~ e~ ^ /R fits the reported on a neutron scattering experiment from A excellently, and with this a /3-brass crystal [1], which was furthermore critical scattering data enriched with Cu65 to enhance the difference in shape of the pcf the critical scattering cross sec- scattering amplitudes. Due to rapid Zn-evaporation tion becomes: from their crystal, it is however not possible to draw detailed quantitative conclusions from this (da 1 1 cnt (2) 2 2 experiment. From our present experiment, where \dCl, r1 \k — t| + k a /3-brass single crystal with normal isotopic abun- dance was used, the following can be deduced:

~~(1) The temperature dependence of LRO in the Ki is called the inverse correlation range, n the close vicinity of T strength parameter. c ,~~

~~144 Ideal Situation in Reciprocal Space~~

~~(3 - Brass Single X-tal Sphere 1" Diameter Encapsuled 1 : 2 Arm Scan Rock Scan in Steel~~

~~Be(002) Real Situation in Reciprocal Space~~

~~M(T-T0 ) J(K-XJ~~

~~Figure 1. Experimental setup in the (1, 0, 0) reflection. The angle between vertex and (0, 0, 1) is approx. 12°. whereby the occurrence of multiple scattering is prohibited.~~

The Experimental Setup FIGURE 2. Upper part: Wavevectors in reciprocal space corresponding to figure 1 (idealized resolution). Lower part: Definitions of distribution function with finite Monochromatic neutrons were extracted from the resolution. The contour of J(k-k0 ) indicates f.i. a horizontal cut in the surface of half widths. polyenergetic reactor beam by (0,0,2) — Bragg reflection in a Be single crystal (mosaic spread

5' s.d.). Higher order neutrons were filtered out of a mechanical velocity selector (see by means to the spectrometer arm. Two different scan types fig. 1). rather long wavelength of 2.70 was A A through a superlattice reflection in reciprocal chosen to safely avoid parasitic multiple Bragg space is sketched in figure 2. reflections in the /3-brass single crystal, which in the (1,0,0) reflection was oriented with (0,0,1) approx. 12° from vertex. Also the instrumental resolution could be made quite narrow at this Instrumental Resolution wavelength, which is essential for the interpretation of the critical scattering as shown below. Due to instrumental resolution the endpoint of The reactor beam was only collimated by the beam the scattering vector k is smeared, which is in- tube dimensions, the monochromatic beam was dicated the distribution by function J Ko ( k). In cylindrically collimated (30 mm diam per m) and the .limited region in reciprocal space, where the scattered beam was horizontally collimated J K<) { k) will be used in the calculations below it is by Soller slits (3.5 mm per m). The monochro- assumed that J is only dependent on the difference matic beam intensity was 2.5 -105 neutrons/sec. — i.e., — k K0 , JKQ ( k) =J( k Ko). Also the end- The counting time was mastered from a preset point of the f-vector is smeared due to the mosaic count on a monochromatic beam monitor (abbr. spread, indicated by the distribution ' function BMc in figures). M (t~ t0 ) (see fig. 2). The intensity 7(k0 , to) with the The spherical crystal (1 in. diam) was tightly spectrometer and crystal set at the average values encapsuled in a thin stainless steel container K0 and to of the scattering vector and T-vector re- to avoid Zn evaporation. A 5 percent variation spectively is found as the 6-dimensional convolution of the Zn content changes T 14 °C, but in the six- c [2] integral: months duration of the experiment Tc remained constant within a few tenth's of a degree. The /(Ko, to) = j" jj( k - k0)-M{t -to) crystal was mounted in an oven of sufficiently wide diameter so neutrons scattered from irradiated •ct(k, t) d K(h, (3) parts of the walls of the heating coil could not possibly hit the BF.s-detector. The crystal table could da where the cross section (k, t) has been abbrevi- be turned around a horizontal and a vertical axis; -j^ the latter could also be coupled in the ratio 1:2 ated to ct(k, t).

~~145 t~~

~~For Bragg scattering ct(k, t) — 8( K — t) and~~

~~Critical : = 4.4° Scan T-Tc~~

~~o Measured intensities — It — KO To) = J ( Ko) M(T T0 / Bragg* j ) J 22-10 Lorentzian, X, = 4. 50 *10~2 A" 1~~

~~8(k -r)dKdr. (4) 20 Resolution~~

~~18 Folded Lorentzian, best fit For critical scattering ct(k, t) = (r(fc — t) = 16~~

~~cr(e)S( k -r-€)de CD and N 10~~

~~/ (K0 — r0 )= (T(e)de k - K0 )M(T —ro) D crit I I I 7(~~

~~/ <_> 6~~

~~4- •8(k — r — e)d Kdr 2- Background k*-(ko-€))M{t-to) 0~~

~~S(k* — T)d K*c?r Crystal Setting A~~

Figure 3. 4.4° . — — A rock-scan through (1, 0, 0) above Tc /crit( Ko —T () ) — / Bragg( «o « T„)or(e)f/c (5) j The excellent fit of an Ornstein-Zernike correlation function is apparent.

above Tc . The circular points are measured The measured critical scattering is thus the intensities as function of crystal setting. The 3-dimensional convolution integral of a measurable narrow curve is the corresponding normalized o\ function 7B ratw ( Ko —rQ ) and the cross section Bragg reflection peak, measured below TP , and and an exact convolution can be performed without the dashed curve is the Lorentzian (2) for the best any assumptions of the shape of the distribution K\ -value. These curves are drawn upon a con- functions J{k — k0 ) and M(t— t0 ), apart from the stant background, which was measured at several already mentioned approximation J k „(k) — J(k— Ko). points far from the superlattic point (1,0,0). A

scan at a high temperature (70° above Tc ) showed the same constant background, when corrected Analysis of Critical Diffraction for the remaining small amount of critical scatter- Peaks ing. This justifies to some extent a constant background subtraction, which was used throughout In practice, the unfolding calculation was per- the analysis of all critical data. formed in the following way with the aid of a It is seen that the Ornstein-Zernike pair cor- digital computer: relation function fits the data excellently. The Three mutual perpendicular scans through the scatter of measured points around the folded

~~Bragg peak were measured below Tc and ap- Lorentzian (solid line) is random and in statistical proximated by three normalized Gaussians. The agreement with the uncertainties on the count rates.~~

~~product of these was used as 7B ragg (k 0 — 0 ). In f.i. a rock scan measurement the computer calcu-~~

lated the ratio between ar(Ko,r0) from (2) and Best Fit for Different Correlation Functions / (ko, to) from (5) for different directions of t 0 and for a certain value of Ki. For this value 2.0* 5.7* 12.1' of Ki the sum of weighted, squared deviations Pair Correlation Function Ornstein e-X,R -2 between Xi 2.38 110" 2 /(ko,t0 ) and the measured intensities Zernike Ft was calculated, and by iteration in Ki -values this 2.38-10" 2 -2 10" 2 10"2 e -XiR Xi 4.54 .10 5 . 12 - 6.82« Fisher sum was finally minimized. V 0.0000 0.0000 0.0000 0.0000

2 2 10" 2 An example is demonstrated in figure 3, where a Xi 2.38-10" 4.54-I0" 5. 12 « 2 6.82. 10" Hart 6 6 5 6 3 1.00-10 1.08-10 2.97-10 2.32. 10 scan perpendicular to (1, 0, 0) was measured 4.4 °C

~~146 1~~

~~Temperature Dependence of LRO different parts of the crystal. The currents in and SRO the heating coils in the oven were adjusted until all curves intersected the abscissa at the same~~

Theoretically (LRO), which is proportional thermocouple voltage, and it was then concluded to the Bragg peak intensity, is proportional to that temperature gradients could be neglected. 3 = = (Tr-Tf with 0 5/16 0.3125 for the Ising Figure 5 shows similar curves for the (1,0,0) Model [5]. reflection. (3 is here found to be 0.27 which Figure 4 shows the peak intensity from the differs significantly from the value 0.325 obtained

(1,1,1) reflection as function of the temperature from the (1, 1, 1) reflection. This might be due to a 10° from below the critical temperature to a little different degree of extinction in the two cases. above this point. When the intensities are plotted The extinction correction will be relatively larger as function of AT = Tc — T'm a double log plot, it is for big values of AT than for small values, since the seen that a power law is valid near Tr and that cross section and thereby the beam attenuation

(3 = 0.325. through the crystal increases with increasing values The intersection of the long-range order curve of AT. The extinction correction will therefore with the abscissa gives the thermocouple voltage qualitatively have the effect of increasing the slope when the crystal is in the critical state. The tem- of the line in the double log plot. At present we perature dependence of all the parameters to be have not investigated this problem in detail, and discussed is expressed by the temperature differ- the figures show uncorrected Bragg peak intensities. ence AT and this is measured as the corresponding The tail of the curve for temperatures above the difference in thermocouple voltage multiplied with critical point is the peak intensity of the critical the scaling factor 0.103 °K//xV. The uncertainty scattering, and the difference in order of magnitude of the intersection contributes the main part of from the Bragg peak intensity should be noted. the uncertainty of 0.1 °K on AT. A possible temperature gradient can be found by measuring similar curves with reduced beam hitting

~~Counts Counts 10 6 BMc 30.000 5x10 5 BMc -6x10 3 5 ^^.^ 20.000 A 3~~

~~•2 10.000~~

~~1 \ 1uV-0.103*C~~

~~i -0.103° . v.- luV C 3800 3850 3900 pV Temperature 3800 3850 3900 uV Temperature — i Counts Counts 5x105 BMc 106BMc 5x10 3~~

~~• 1~~

~~.3 .5 1.0 2.0 5.0 10.0°K 5 10 20 K - —»- Temperature T T ) Temperature (Tc -T) ( r~~

~~Bragg peak intensity versus temperature Figure 5. Bragg peak intensity versus temperature for the (/, /, 1) reflection. (1, 0, 0,) reflection.~~

~~147~~

209-493 0-66— 1 Also the temperature dependence of the SRO has The difference between peak intensities for the been calculated in the Ising Model. For a ferro- (1, 0, 0) and the (1, 1, 1) reflections is due to dif- magnet it has been found that the susceptibility ferent Debye-Waller factors. The measured ratio

(T— T~\y is seen to be 1.28 and, using Chipmann's data for X obeys a power law being proportional to the Debye temperature around 750 °K [6], the ratio with y equal to 1.25 [5]. It is a well known fact of the Debye-Waller factors is 1.32. that the susceptibility x ls related to the pair cor- The weighted average value of the power y is relation function through (x ^ 2k pcf (/?)). 1.11 ±0.01, which should be compared with the

On the other hand the peak intensity 70 for T > Tc theoretical value of 1.25. is also proportional to this sum as seen from the Figure 7 shows the values of K\ obtained at dif- cross section formula (1). Recalling the close ferent temperatures in different scan types through analogy of the order-disorder transition in alloys different superlattice points in the reciprocal and magnetic materials, it is therefore expected that lattice. It is worthwhile to note the consistency 1 \-y /"J — f of the results. Each point represents one diffrac- h —Tfr^) y can therefore be determined tion peak, which takes, on an average, 48 hrs of from the measured temperature dependence of the measuring time. peak intensity, independently of any shape that For comparison with theory, the value of K\ at the pair correlation function might have, provided 75° above Tc calculated from the theory of Elliott of course that the resolution correction is correct. and Marshall is also shown.

Figure 6 shows the peak intensities of critical The peak intensity 70 is with the Ornstein- 2 scattering from the (1, 0, 0) and from the (1, 1, 1) Zernike pcf proportional to • Ki)~ . reflections for temperatures up to about 35° above r, is theoretically expected to decrease slightly

Tc . The intensities are corrected for reduction with increasing temperature [7]. This implies due to resolution and the curves shown in the lower that twice the slope of the least square fitted line part of the figure show the corrections. The peak to the Kj-values is an upper limit of the parameter value reduction amounts to as much as a factor of y already discussed. It is concluded therefore 2 at the lowest temperatures and since the corrected that y is less than 1.18 ±0.02, in consistency with intensities for both reflections obey the same power the value found from the peak value runs. law, this gives confidence in the calculations on On the other hand, the measured values of resolution corrections. Ki together with the experimental result that

~~Peak Intensities 1 Correlation Range M<~~

~~Counts~~

~~1.113* 0.009 o Rock scan(U).O) L i| . 2000 20~~

~~Rock scand.1.1 ) - 10 1000~~

*<1.18»0.03

~~~ 2JD0 200 - 1.50 > £ ' 1.00 1 1 ' ' ' ' ' I " 3 Jzk 10 20 50X10"3 "S^-~~

~~5 10 5 10 20 50*K T-T r 20 50'K T-Tc Temperature Temperature~~

FlGURE 6. Critical peak intensities, corrected resolution for FIGURE 7. Inverse correlation ranges k, versus temperature. reduction, for (1, 0, 0) and (1, 1, I) reflections. Each point represents the result of a scan as shown in figure 3.

~~148 )~~

~~erties of the crystal have changed during the long T-Tc\~ • -2~~

~~temperature dependence of r : . In a linear approxi- Future experiments on other /3-brass crystals might shed fight on the problem. mation with rf = rfc (l — a—jrV it was found that a~~

was equal to 3.6±1.2. which is compared with the Conclusion temperature dependence of ri given by Elliott and Marshall who found that a — 1.2. [7], The following conclusions can be drawn from the After completion of these measurements it was present experiment: concluded that the most accurate information could (1) The Ornstein-Zernike correlation function is be obtained from rock-scans through the (1, 0, 0) in excellent agreement with the data. A complete reflection. A series of such scans were therefore Fourier analysis to obtain all SRO parameters is measured, but the inverse correlation ranges were not possible due to the very limited regions of the now found to be 20 percent lower than the previous reciprocal space where the critical scattering is data over the entire temperature region as shown sufficiently intense for experimental access. on figure 8. (2) The absolute value of the short range order The (1, 1, 1) reflection data, which was obtained correlation range is roughly in agreement with at the end of the previous run, tends toward the theory, but could possibly be an individual prop- data of the latest (1, 0, 0) reflection run. The crit- erty of a given crystal. ical temperature, as well as the Bragg peak inten- (3) The temperature dependence of LRO is not sities, were still exactly reproducible. Also the in contradiction to the latest Ising model results, temperature dependences of Ki and Io as well as but definitely outrules the results of older theories Io(k\) are in agreement with the previous data. = It must therefore be concluded that some prop- ^f.i. Bethe-Peierls theory, P ~^j-

~~(4) The temperature dependence of SRO is in slight but definite disagreement with the Ising model in the nearest-neighbor interaction approxi-~~

~~i 10* so Counts Vi 2x107 mation.~~

~~JT r 1.10 i 0.03 Note added in proof. Later measurements have~~

~~20- indicated that the /3-brass crystal contains a few percentages of y-phase. Recent measurements on~~

~~o =Rock scan (1.0.0) -2 10-10 a pure /3-brass crystal gave an exponent in the tem- A =1:2 scan(I.O.O) perature law for the susceptibility of y= 1.25 ± .02.~~

~~X ^5 References~~

~~1726 [1] Walker, C. B., and Keating, D. T., Phys. Rev. 130, (1963).~~

~~[2] Hansen, M., Constitution of Binary Alloys (McGraw-Hill 20 5'o-l6- 3'{'I=Ic Book Co., New York., N.Y., 1958). [3] Fisher, M. E., Physica 28, 172 (1962).~~

5 10 20 50°K T-T r [4] Hart, E. W., J. Chem. Phys. 34, 1471 (1961). Temperature [5] Essam, I. M., and Fisher, M. E., J. Chem. Phys. 38 , 802 (1963). Arsenal, [6] Chipman. D. R.. MRL Report No. 67, Watertown FIGURE 8. Later results of critical scattering around the (1,0,0) Massachusetts (1959). reflection. Phys. 30, 1 [7] Elliott, R. I., and Marshall, W., Rev. Mod. These results are not consistent with the results in figure 6 and figure 7. Note, (1958). however, that 7V, /3, y, and /o(k,) are the same.

149 The Spectrum of Light Inelastically Scattered by a Fluid Near Its Critical Point*

~~N. C. Ford, Jr., 1 and G. B. Benedek~~

~~Massachusetts Institute of Technology, Cambridge, Mass.~~

If a light beam passes through a perfectly homo- from the illuminated region is given by geneous medium we expect no scattering of the light in directions other than the forward direction. E(R.K,t) This conclusion results from the fact that waves (1) scattered from each point in a homogeneous medium interfere destructively in every direction where: other than the forward direction. Of course, if = the wavelength of the radiation is of the order of K k -k (2) the interatomic distances then on this scale the is the medium will be nonhomogeneous and scattering and where E () amplitude of the incident plane can be expected to occur in conformity with the wave, oio is the angular frequency of the incident Bragg conditions. However, for light waves the radiation, sirup is the angle between the direction Bragg equations cannot be satisfied. of polarization of the incident light and the direction

Despite this argument there is considerable light of k'. 8e(K, t) is the fluctuation of the dielectric scattering even from the most highly purified liquids. constant having the wave vector equal to the scat- This scattering grows stronger as one approaches tering vector K. Equation (1) signifies that scatter- the liquid-gas critical point. In fact, in this critical ing in the direction k' is produced by a fluctuation in dielectric constant vector = k' — k. region the scattering is so impressive that it is having wave K called "critical opalescence." Von Smoluchowski This corresponds classically to a Bragg reflection off the fluctuation having wave vector K. Quantum [1] was the first to suggest that in a real liquid the density is not perfectly uniform but that over mechanically eq (1) and (2) signify that constructive regions of the order of the wavelength of light there interference of the scattered wavelets occurs only were fluctuations in the density of the liquid. At when the momentum is conserved between the each instant of time, he suggested, the liquid is not photons and the scattering fluctuation. the intensity of the scat- of a single density, but instead there is a distribu- In order to determine tion of density around the average density. The tered light Einstein then went on to calculate the scattered field: light is then scattered off the fluctuation in the mean squared density around the average. 4 Stimulated by Von Smoluchowski's suggestion, <|E ( fi.K,„p>= £S ^,2^(| 8,(K,„|^ Einstein [2] undertook a quantitative calculation (^) of the scattering of light from the fluctuations in a liquid with special attention to the situation near The fluctuation in the dielectric constant can be density the critical point. If we take the wave vector of taken as arising out of the fluctuation of the the incident light beam as k, and that of the scat- 8p viz: tered beam as k', then Einstein was able to show de that the electric field scattered a distance R away 8e(K, t) 8p(K, t) (4)

''This research was supported by the Advanced Research Projects Agency under and Einstein evaluated the mean squared amplitude Contract SD 90. 1 Present address: University of Massachusetts. Amherst. Mass. of the fluctuation in the density for K = 0 using the

~~150 Boltzmann principle which weights a fluctuation scattered light depends upon the form of the time~~

~~by the work required to produce it. In this way dependence of the fluctuation in the dielectric Einstein found for K = 0 constant. In fact, the spectrum is obtained if the~~

~~time correlation function (E(K, t'+r) • E*(HL, t))~~

~~is known [8]. This correlation function is related (5) B 7 to the temporal fluctuations in the scattered field by~~

where V is the volume of the illuminated region, <£(K, t + r) £*(K, f)> p0 is the mean density, BT is the isothermal bulk modulus, kT is Boltzmann's constant times the = E(K,t + r) E*(K, t)dt (7) absolute temperature. The phenomenon of crit- 1™ 2T J_, ical opalescence follows simply from (5) since as in 7 — where we have dropped the distance R the desig- T > Tc the isothermal bulk modulus goes to zero, nation of the functional dependence in E. The or perhaps more strikingly, the compressibility 2 correlation function for the scattered field is re- grows very large indeed and hence (|8e(0, t)\ ) lated to the correlation function for the fluctuations gets very big. Physically the enormous increase in the dielectric constant by in the scattering results because large density fluctuations become energetically very easy to (E(K, t + r) produce since the compressibility is very large. The large density fluctuations in turn produce large^ in the dielectric constant and hence 3 fluctuations ~^(27r) , (8) a great deal of light scattering. The work of Einstein was later extended by Orn- and the spectrum of the scattered light is given by stein and Zernike [3]. They noted that eq (5) was

a good approximation for 8e(K, t) provided that one ' 1 iui T T • *< t))e — S(K, w') f < E < K - ' + > E K - dT (9) was not too near Tc . As 7 » Tc , however, it be- =7T- Ztt j 2 comes necessary to evaluate (|8e(K, t)\ ) for K ^ 0. intensity the is just Their result, which has also been obtained in other and the total under spectrum S

~~ways [4, 5], has the form for small K 2 (\E(K, t)\ ) = s(K(o')d(o' . (10)~~

~~3 2 (277) <|8e(K, t)\ ) (6) B T + (aK)'< The question thus arises, what is the time correlation function for the fluctuations in the dielectric~~

~~where a is a length of the order of the interatomic constant? To answer this we must have some~~

distance. Thus, the scattered light intensity grows model for the origin of the fluctuations in e. Bril- extremely large only in the forward direction K = 0. louin pointed out that one such origin was to be For other directions the scattered intensity will found in the thermally excited sound waves which not in fact diverge. continuously travel through the medium. These Following the work of Einstein, and Ornstein sound waves are fluctuations in pressure at constant and Zernike, much experimental effort was devoted entropy. The pressure fluctuations in turn modu- to a study of the temperature dependence and the late the dielectric constant and so produce scatter- K dependence of the scattered light intensity. ing. To calculate the modulation of the dielectric Work continues to the present day and in fact is constant produced by the thermally excited sound being reviewed at the present conference. Despite waves, we observe that: this activity, there exists no experimental infor-

mation on the spectrum of the scattered light in - (Se(K, t))s (8p(K, t))s (11) dp/s^) the critical region of a fluid. The first person to investigate theoretically the spectrum of light scattered from a liquid or solid where 8p(K, t) is the Fourier amplitude for the pressure was L. Brillouin. His first investigation [6] of this fluctuation in the sound wave having wave subject was in 1914. This was followed by a more vector K. Since the sound wave travels as a wave with frequency complete analysis in 1922 [7]. The spectrum of the ±fl(K) and has a finite lifetime

151 1/HK) we may expect that the correlation function of the central components and the Brillouin doublets fluctuations is for the pressure was explained by Landau and Placzek [12, 4]. They pointed out that the fluctuations in the dielectric ±,n,K,T = 2 rr <8p(K, t + T)8p*(K, t)> (\8p(K, 0| )e e- . constant are produced not only by the adiabatic (12) pressure fluctuations (the sound waves) but also by the isobaric entropy fluctuations. Putting the matter very explicitly, the dielectric constant can From this it follows at once that the spectrum of be regarded as a function of the entropy (s) and the scattered radiation is given using equations the pressure (p) and fluctuations in either of these 12, 11, 9, 8, by quantities will cause fluctuations in the dielectric

~~2 2 constant. Thus , 9 /Wv sin~~

8e(K, t) = (&(K, t)) + (8p(K, t)) . (15) (^j p ^ s 2 (27r)3(^ <|8p(K, t)\ ) The second term on the right hand side of eq (15) was considered by Brillouin. Landau and Placzek U E(K) 1 stressed the importance of the first term. In 2 2 7rl[a) '-(wo±n(K))] + r (K)J order to obtain the time dependence of the entropy (13) fluctuations they argued that the fluctuations in entropy were proportional to those of temperature Thus the spectrum of light scattered by the sound through waves in liquid consists of a doublet. Each com-

ponent of the doublet is shifted away from the in- T(8s)p = cp(8T)p . (16) cident frequency cu by an amount ±fi(K). This 0 But the time dependence of a temperature fluctua- shift is simply the frequency of the sound wave tion is governed by the Fourier heat flow equation which was responsible for the scattering. The viz: Brillouin doublets can be regarded as a Doppler shift in the frequency of the incident light as a 2 4sr(r,f) = 4v SHr, t) (17) result of a reflection off the moving sound waves. at c* Or equivalently the condition where c* is the specific heat per unit volume, A. is

~~the thermal conductivity, 8T(r, t) is the temperature~~

w'=aj„±a(K) (14) fluctuation at the point r and time t. To obtain the time dependence of the temperature fluctuations can be looked upon as a conservation of energy with wave vector K, we compute the Fourier trans- between the incident photon, the emergent photon, form of eq (17) and find that and the scattering phonon. The plus sign corre- = sponds to phonon annihilation and the negative sign 4- 8T(K, ;)= 4r^ 8T(K, t). (18) at to phonon creation.

~~The existence of the Brillouin doublets for light Thus the temperature fluctuations do not propa-~~

scattered by a liquid was first shown by the experi- gate, they are diffusion like in their decay, and a fluctuation will die away exponentially with time. ments of Gross in 1930 [9, 10]. This subject has since expanded considerably and the experimental Thus we have that the correlation function for the information on liquids has been well reviewed by entropy fluctuation will have the form

I. Fabelinskii [11]. The experiments showed, 2 r even in the most carefully purified liquids, that the (8s(K, t + T)8s*(K, i)) = (\8s(K, t)\ )e- ^ spectrum also contained fight whose frequency was (19a) equal to that of the incident radiation. That is, where in addition to the Brillouin doublets, there is a central component in the spectrum of the scat- re(K) = (19b) tered fight. The existence and relative magnitude

~~152 c~~

~~Since the entropy fluctuations are independent that the line width of the conventional light sources~~

~~of the pressure fluctuations we may calculate the were so broad that its spectrum spread out well spectrum of light scattered by the entropy fluc- beyond the Brillouin doublets thus preventing~~

~~tuations alone and we obtain using 8, 9, 15, and 19 resolution of the spectrum near the critical point. that The invention of laser light sources with high~~

~~(S(K, o/))entropy intensity and very narrow spectral width has made fluctuation it possible to study [13, 14] the spectrum of light~~

~~2 scattered from a normal liquid with much greater top f sin if resolution than was previously possible. In view of 'p the information contained in the Brillouin compo-~~

~~] Te (K) nents we undertook a- study of the spectrum of X 2 (20) tt l(o)'-ft)0) +ri:(K; light scattered from the fluid sulfur hexafluoride~~

(SF 6 ) in the vicinity of the critical point Tc = 45.55 Since the entropy fluctuations do not propagate, °C. The critical pressure and density of this fluid they appear in the spectrum with the same fre- = of nearly spherical molecules is pc 37.11 atm and quency as the incident light frequency. The width 3 p c = 0.730 g/cm . We used as a spectrograph a of this central component gives the lifetime for high resolution Fabry-Perot interferometer which the entropy fluctuation with vector wave K. could resolve lines spaced as closely at 50 Mc/s. The entire spectrum of the scattered light is The expected splitting of the Brillouin components the sum of the spectrum due to the sound waves was about 500 Mc/s. The light source was a single and the entropy fluctuations, i.e., moded, helium neon laser, whose frequency was constant to within about 10 Mc/s. We searched S(K, ))entropy + (s(K, Co)) sound (21) fluctuation waves. along the critical isochore over a temperature range

0.1 °K < T- Tc < 20 °K without finding any resolved 2 2 By evaluating <|8s(K, t)\ )/( |6>(K, t)\ ) for K~0 Brillouin doublets. While this experiment must Landau and Placzek could show that the intensity be repeated with greater sensitivity, we feel that in the central component is greater than that in the negative result may indicate that the widths of the Brillouin components by the ratio (c,, — r )lc r . the Brillouin doublets in this critical region are As we can see from eqs (13) and (20), the spectrum much broader than the displacement of each fine of the light scattered by a liquid contains a great from the central component. In other words, the deal of information of importance in the critical sound wave lifetimes for sound waves with a fre-

region. In particular we should mention the follow- quency of ~ 500 Mc/s is actually shorter than the ing points. The width of the Brillouin components period of the wave. These considerations are 2 provides the sound wave lifetimes as a function of supported by an co extrapolation of measurements temperature and wavelength. The spectral posi- of the low frequency sound wave attenuation in

~~tion gives the sound velocity as a function of wave- SF6 near the critical point [15]. As a result of~~

~~length and temperature. Note that since the these observations it appears that the most strik-~~

~~adiabatic bulk modulus approaches zero as T—> T, ing feature of the spectrum is the central compo- the Brillouin doublets should move in toward the nent which grows in intensity and gets extremely~~

center as one approaches Tv . The intensity of the narrow as Tc - Brillouin components gets large like the adiabatic Very recently Alpert, Yeh, and Lipworth [16] compressibility. The central component contains have reported the observation of the width of the the following information. The width gives the central component in the spectrum of light scat- lifetime for decay of the nonpropagating fluctuations tered from a critical mixture of cyclohexane and of the entropy. The precise shape of the central aniline. They used an elaborate "homodyne spec- component gives the time dependence of the cor- trometer" [17] which analyzed the beat between relation function. intensity — The gives (cp c v ) the scattered light and part of the incident light. as a function of the temperature. We report here the first measurements of the Despite the information contained in the spec- width of the central component for light scattered trum, until very recently, there have been no obser- from a pure fluid, SF 6 , near its critical point. We vations of the spectrum of light scattered near the use for this study the very simple detection scheme

~~critical point of a fluid. The reason for this is shown in figure 1. 153 from the entropy fluctuations is given by Laser \s -$4 <£(K,n 'entropy Photomultiplier~~

~~cooy sin~~

~~ttW^I) iioQt ( 2 (8s(K,r) )„e- 23)~~

~~With this time dependence of E(K., t) it follows that the second term on the right-hand side of eq~~

~~(22) is zero and the correlation function for the~~

~~output of the phototube is determined entirely by~~

~~the correlation function of the scattered field viz:~~

~~Re(T) — (E (K, t +t)E* (K, t) > entropy~~

~~2 2 sin~~

~~-i«o' -r.(K)T (\8s(K,t)\ e (24)~~

~~where we have used the Landau- Placzek assump-~~

tion, eq (19), to obtain the form of the correla- The light scattered in the angular range between tion function for the entropy fluctuations. Thus 6 and 0 + A.6 is collected by a lens and focused the correlation function for the output of the photo- onto the surface of a phototube. The phototube tube has the form: current is proportional to the square of the inci-

2 2 dent electric field. Thus the fluctuations in the (\E(K,t +t)| |£(K,0| > output current of the phototube gives the fluctua- = 2 2 tions in E(t)E*(t). This fluctuating current is ana- (/U0)) +\Re(T)\ (25) lyzed into its spectrum using a spectrum analyzer this fluctuation, i.e., the spec- and the spectrum is displayed on a recorder. and the spectrum of trum of the input to the spectrum analyzer has the The spectrum, as obtained by this "self beat" form S^K, a.*):' or square law detection system, gives the spec- 2 X trum of the fluctuations in |£"(/)| - We may Sb(K,»)=^- f (E(K,t+r) relate this spectrum to the spectrum of E(t) by making use of the fact that if E(t) is a Gaussian +il» T random variable, the correlation function for E{t)\- E*(K,t))e dT (26) j is related to the time dependence of E(t) by the -1 equation = Rm \8(a>) + TT

~~2Te (K) 2 2 2 2 (27) <|£(K, r+r)| |£(K, f)| > = ((|£(K, 0| )) 2 2 (cu + (2r,(K))~~

+ (E(K, f + T)£(K, t)) <£*(K, t+r)E*(K, t)> The first term on the right side of eq (27) corresponds to the d-c photocurrent. This d-c current + \(E(K,t + r)EUK,t))\ 2 (22) does not enter the spectrum analyzer as it is removed with an RC blocking filter as shown in where we have explicitly written K the scattering figure 1. Thus the power spectrum of the current vector in the expression for the correlation func- fluctuations entering the spectrum analyzer is tions to emphasize that these functions depend on given by the second term on the right side of eq the scattering vector. We now use the fact that (27). The spectrum analyzer uses a linear full the time dependence £(K, t) for the scattering wave rectifier as its detector element. As a result,

~~154 it can be shown [8] that the output current I(a>) as a~~

~~function of frequency a) is directly proportional to~~

~~the square root of the input power spectrum. /(o») T = 45. 317 °C is displayed on the recorder, and from the discussion~~

~~above we see that I{co) is given by: (A-), = 200c/s XV~~

~~^^Cl<> 9 = 8.7°~~

~~1/2 I I I I 1 RAO) 2T P (K) /( cj)— (const) \(af2 2 tt^ +(2T (,(K)) : (28)~~

~~We can therefore use eq (28) to analyze the shape of the I(co) spectrum and thereby determine the~~

~~decay rates of the entropy fluctuations re (K). Of~~

course, if the output spectrum I(oj) does not have the form predicted by the Landau theory we can obtain from 1(a)) information on the form of the time dependence of the correlation function for the entropy fluctuation. We show in figure 2 the spectrum analyzer output I(

~~from the critical point, and near the critical isochore. The scattering angle was about 8.7°. We Frequency (c/s~~

estimate that the half width at half height of this FIGURE 3. The spectral distribution of light scattered from SF6 near its critical point for different scattering angles. spectrum is about 350 c/s. This measurement is Observe t he broadening at the larger scattering angles.

remarkable in that it shows that lasers now make it possible to detect shifts in the frequency of the scattered light as small as a few hundred cycles per second out of the incident light frequency of 5 X 10 14 c/s. In figure 3 we show similar spectra taken at fixed temperature for different scattering angles to check the K 2 dependence of the half width of the line. The results of this investigation are shown in figure 4 for two different temperatures. Within the error limits imposed by these early measurements we see that the width of the scattered spectral line is indeed proportional to K 2 In figure 5 we show the temperature dependence of the half width for fixed scattering angle in order

~~to examine the temperature dependence of A./c*.~~

~~2 sin (9/2)~~

~~FIGURE 4. The dependence of the half width of the scattered spectrum on the square of the scattering vector K = kn sin 6/2 for two different temperatures near the critical point.~~

~~Finally we may inquire as to whether the value of 500 1000 1500 r (K) as deduced from 1(a)) and eq (28) is in agree- Frequency (c/s) r ment with the formula of Landau. Along with~~

~~FIGURE 2. The spectral distribution of the light scattered by J. V. Sengers [18] we have estimated the tempera-~~

the nonpropagating entropy fluctuations in SF6 very near the ture dependence of A. and c* for SF B at the critical critical point. Note that the width of this spectral distribution is only a few hundred eycles per isochore and thus obtain a theoretical estimate for second. The incident light frequency is 5 X 10 14 c/s. The resolving power of the

~~spectrometer (f/Af) is about 1 X 10'-. the temperature dependence of TP(K). This esti-~~

~~155 cycles per seconds out of the incident light frequency of 5 X 10 14 c/s. By studying the detailed~~

shape of the spectrum and its temperature dependence we can obtain information on the time and temperature dependence of the correlation function for the entropy fluctuations in the critical region. Finally, we have presented the first measurements of

~~these fluctuations in a pure fluid (SF6) near its critical point.~~

~~References~~

M. Von Smoluc-howski. Ann. d. Fhys. 25, 205 (1908). A. Einstein. Ann. d. Physik 38, 1275 (1910). L. F. Ornstein and F. Zernike. Proc. Acad. Sci. Amster- dam 17, 793 (1914-1915). L. Landau and E. M. Lifshitz. Electrodynamics of Contin- FIGURE 5. The temperature dependence of the half width of the uous Media, pp. 393ff. (Addison Wesley. Reading. Massa- scattered spectrum lifiht from SF<; a density near the of for chusetts. 1960). critical isorhore. M. E. Fisher. J. Math. Phys. 5, 944 (1964). The critical point is al 45.55 °C. L. Brillouin. Comptes Rendus de l'Academie des Sciences 158, 1331 (1914). L. Brillouin. Ann. de Physique 17, 88 (1922). mate is in a good qualitative agreement with the W. B. Davenport and W. Root. An Introduction to the The- ory of Random Signals and Noise (McGraw-Hill Book Co.. present observations [19]. New York. N.Y., 1958). In conclusion we wish to stress the following [9 Gross. Nature 126, 201. 400. 603 (1930).

[10 Gross. Zeit. f. Phys. 63, 685 (1930). points. First, that the present experiments demon- [11 Fabelinskii. Usp. Fiz. Nauk 63, 355 (1957). strate that it is possible to study the spectral dis- [12 Landau and G. Placzek. Phvsik Z. Sowjetunion 5, 172 (1934). tribution of the light scattered from entropy fluctu- [13 Benedek. J. B. Lastovka. K. Fritsch. and T. Greytak. ations near the critical point. These studies can be .O.S.A. 54, 1284 (1964). carried out using a very simple "self beating" or [14 Y. Chiao and B. P. Stoichieff. J.O.S.A. 54, 1286 (1964). [15 G. Schneider. Can. J. Chem. 29 , 243 (1951). "square law" spectrometer whose operation has [16 s. Alpert. Y. Yeh. and E. Lipworth. Phys. Rev. Letters been described. The spectrometer, which uses a 14, 486 (1965). [17 Z. Cummins. N. Knable and Y. Yeh. Phvs. Rev. Letters laser beam as a light source, permits a measurement 12, 150 (1964). communication. of the spectral distribution in the scattered light [18 V. Senders, private [19 N. C. Ford and G. B. Benedek, Phys. Rev. Letters 15, 649 which is as narrow as a few tens or hundreds of (1965).

156 Time-Dependent Concentration Fluctuations Near the Critical Temperature*

~~S. S. Alpert~~

~~Columbia University, New York, N.Y.~~

The title of this talk which is printed in the con- of concentration and temperature at constant pres- ference program as "Time-Dependent Density sure. It is this central unshifted fine which my Fluctuations Near the Critical Temperature" colleagues, Dr. Y. Yeh and Dr. E. Lipworth, and is incorrect and should be altered to read "Time- myself have experimentally observed, using a laser Dependent Concentration Fluctuations Near the homodyne spectrometer of high resolving power. Critical Temperature." This is readily apparent Before discussing the results of our experiment, when one considers the differential element of I should like to digress for a few minutes and to scattered energy given by the expression sketch for you the general functioning of the laser homodyne spectrometer developed at Columbia

~~dEsc dfl University by Cummins, Knable, and Yeh [3] in — OC B(6, A). 4 Io A the early part of 1964.~~

The laser homodyne spectrometer [3] has a re- 14 Here dEsc is the differential element of scattered solving power of one part in 10 and is capable of energy which is normalized to the incident intensity scanning the spectral profile of the central compo-

~~/0 ; dCl is the element of solid angle into which the nent of the scattered triplet. The essential func-~~

~~light of wavelength A is scattered. The factor tional components of this spectrometer are the~~

5(0, A) gives the dependence of the scattered energy laser light source, a divided optical path, the ex- on the scattering angle 6 and on the wavelength A. ternally driven Bragg tanks, and the detection ap-

It is seen that the scattered intensity depends on paratus and associated data-processing electronic the change of the dielectric constant e of the me- equipment. (See figs. 1 and 2). These functional dium with respect to both the density p and the components are presented in outlined form below: concentration c. For a binary system such as 1. Laser source. As previously noted in this re- aniline-cyclohexane, which is the system studied, port it was desired to observe the spectral profile there is reason to believe that the mean square of the central component of the scattered light fluctuation of concentration 8c 2 becomes very large triplet. This profile was expected to be on the

as the critical temperature is approached. This order of a few hundred cycles per second. Clearly point will be discussed later in this talk. However, the spectrometer fight source must be inherently

since the compressibility of the binary system is more narrow than the desired spectrum. The nonsingular in the region of the critical tempera- only available light source fulfilling this require- ture, the mean square of the density fluctuation ment at the present time is the He-Ne laser. Jaseja remains finite and small: hence it is reasonable [4] has shown that under highly controlled condi-

to assume that it is the concentration fluctuations tions and for short times, the fine width of a single which are responsible for the light scattered from mode of the He-Ne laser is less than 1 c/s. This binary systems near the critical temperature. was the reason underlying the use of a He-Ne laser In the preceding talk, Professor Benedek clearly capable of a power output of about 15 mW with an outlined the nature of the light scattered from a pure intermodular spacing of 60 Mc/s. liquid system. The scattered light from a binary 2. Divided optical path. Although the frequency mixture is a triplet where the outer two components, output of the He-Ne laser is instantaneously quite the Brillouin doublet [1], result from thermally narrow, over any reasonable time interval of ob- excited acoustic modes and where the central servation (~ 1 sec) the fine width is broadened unshifted component [2] results from fluctuations to the bandpass of the optical cavity which is about 10 kc/s. This broadening results from thermal *This work was supported in part by the Army Research Office — Durham under vibration effects on the mirrors and also any Contract DA-31-124-ARO-D-296. and in part by the Joint Services Electronics

~~Program under Contract DA-28-043 AMC-00099 (E). microphonic effects present. It thus becomes~~

~~157 COLLIMATING LENS 4. Detector and data handling electronics. The 6328 A He - Ns LASER 12-Mc/s "beat" falls on an S-20 photocathode and~~

~~appears in the photomultiplier output current (fig.~~

~~2). The photomultiplier output which contains the spectral-profile information centered about a fre-~~

~~SAMPLE CELL quency of 12 Mc/s is fed into a radio receiver tuned~~

~~to that frequency. The resulting i.f. signal is fur- BRAGG SSB MODULATOR ther "mixed" in two sequential stages so that the~~

~~resulting signal is greatly reduced in frequency.~~

~~The last mixer is provided with a mechanically scanned local oscillator input. The output of the 6328 .~~

NARROW BAND 1 FILTER final mixing stage is passed through a narrow-band amplifier centered at a frequency of 400 c/s. Since FIGURE 1. Schematic arrangement of optical components of laser homodyne spectrometer. the narrow-band amplifier will only amplify in this range and since the local oscillator input to the final

mixer is swept, the final output will yield a frequency profile of the original 12-Mc/s beat. The instru- PHOTO MULTIPLIER S-20 mental bandwidth of such a system was 13 c/s. ANTENNA The system studied in the present experiment was input n^- a cyclohexane-aniline mixture, 53 percent cyclo- 2.5 Mc COMMUNICATIONS XTAL OSC RECEIVER STRIP hexane and 47 percent aniline by weight. The CHART RECORDER mixture was contained in a cylindrical glass cell in a constant-temperature enclosure, the tempera- 501,150 c/s XTAL OSC. ture of which could be adjusted and held constant

~~to about 1 X 10~ 3 °C over extended periods of time.~~

PASS characteristic spectra in 3. BAND Some are shown figure FILTER NARROWBAND LOW AMPS are typical of the the LEVEL These spectra observed with LOW PASS MIXER FILTER laser homodyne spectrometer; I might add that the word "typical" is not used here in the sense that SWEPT L.O. 1550 ± 200 c/s Miss America is the "typical" American girl, but to indicate the ordinary performance of our spec- FIGURE 2. Block diagram electronic equipment used in of trometer. The spectra shown in figure 3 are normal- laser homodyne spectrometer. ized to give the same magnitude on our final recorder. In actuality the signals become less intense essential to avoid this cavity broadening by a "beating" or heterodyning technique. This is accom- separate plished by dividing the laser light into two H h50 optical beams. One beam falls directly on the photocathode detector, together with light from the other beam which is first scattered off to sample being studied.

3. Bragg tanks. The heterodyning technique suggested under the previous heading is difficult T-T,=.0695°C to detect because the resulting "beats" occur at ultraviolet frequencies and at dc. For practical detection a "beat" should occur at radio frequen- id) cies. The "beat" can be made to occur at radio frequencies by shifting the frequency of each arm of

~~the optical path by known amounts. This was done T-T,=.I54°C T-T,=.400°G~~

by the use of two Bragg tanks [5] filled with water observed at a scattering and acoustically driven with quartz transducers at FIGURE 3. Characteristic spectra angle of 20.5° and at varying temperatures. 18 Mc/s and at 30 Mc/s. Signals are normalized to give recorder traces of approximately the same magnitude.

~~158 T~~

Figure 4. The halfwidth at half-intensity versus T — c . Included in observed halfwidths is an instrumental error of 13 c/s which can he directl) subtracted out. Note that at T—T,* the extrapolated halfwidth does not vanish.

2 + 2 2 (»-»0) r (c/.) as the temperature above critical is increased, as can be deduced from figure 3, by the increasing FIGURE 5. Log-Log plot for Lorentzian fit. presence of noise superposed on the signal. As Since it is assumed that the sample approached its critical temperature from S(K, «l = above, a definite narrowing of the width of the

Kly\ . the log of is plotted versus Inji (» — vo)-+y*. Here y is the halfwidth at half- scattered profile was observed as shown in figure 4 v) intensity. where the scattering angle was 20.5° from the forward direction; T, was defined as the temperature at which a meniscus was first seen to appear visually. It is to be noted that at T= Tc , the halfwidth of the scattered spectrum is 38 c/s. This includes a 13-c/s instrumental line width which can be directly subtracted out if the scattered spectrum is Lorentzian. which seems to be

~~3mm SAMPLE CELL the case. Thus for T= T, . a half width of 25 c/s T-T, =.0I4"C was observed for the corrected spectrum.~~

A question to be answered is what is the line shape of the observed spectra. At first glance it 0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 2 2 I0 XSIN -«- would seem (see fig. 3) that the lines are Lorentzian. | The curves do. indeed, match an assumption of 2 being Lorentzian as is shown in figure 5. Such FIGURE 6. Plot of Av versus sin {6/2) for a 3-mm sample cell at T — Tc = 0.014 °C. an assumption would predict a concentration fluctuation decay of the form V^c2 e~ t,T which would correspond to a 7 = 0.013 sec for our sample at To indicate that our overall results are reasonable, the critical temperature. I would like to invoke the following argument The scattering 20° angle 6 was varied between based on published work of Landau and Lifshitz [6]. and 40° and the width at half intensity was observed. From hydrodynamical considerations these authors The results are shown in figure 6 where it is seen deduce the following coupled equations for a binary a system: that ^ sin 2 -. The intercept corresponding to 0 = 0 represents the instrumental line width y8c = Dfo*8c + v 2 sr which can be directly subtracted out. j (1)

159 and quantitative, studies of the spectral profile of scattered fight such as we have performed can be used as a new experimental basis with which to * at cp \dcJp.T at construct theoretical models of the nature of liquids. where D is the diffusion constant kr is the thermal- diffusion ratio, x ,s tne thermal diffusivity, and ix References is a suitably defined chemical potential. The other [1] L. Brillouin, Ann. Phys. (Paris) 17, 38 (1922). symbols have been previously defined. Landau [2] E. Gross, Nature 126, 201 (1930). and Lifshitz [7] have also shown that for a binary [3] H. F. Cummins, N. Knable, and Y. Yeh, Phys. Rev. Letters 12, 150 (1964). mixture, [4] T. S. Jaseja, A. Javan, and C. H. Townes, Phys. Rev. Letters 10, 165 (1963). H. F. Cummins and N. Knable, Proc. — AT [5] IEEE 51, 1246 (1963). [6] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison- 8c2 = (2) Wesley Publishing Co., Inc., Reading, Mass., 1959). [7] L. D. Landau and E. M. Lifshitz, Statistical Physics (Addi- \dc Jp.T. son-Wesley Publishing Co., Inc., Reading, Mass., 1958). [8] R. Mountain, J. Res. NBS 69A (Phys. and Chem.), No. 6, 523 (1965). As the critical temperature is approached, the chemical potential and its first and second derivatives with respect to concentration vanish. Thus

2 near the critical temperature 8c goes to a l arge 2 2 Discussion value and since 8T remains nonsingular 8c > 8T . Under these simplifying assumptions derive we B. Jacrot: I would like to comment on Dr. Marshall's explanation from eqs (1) the single diffusive expression of the inelastic neutron scattering by assuming that the spin waves persist even close to the Curie point. Indeed the line width which I observed experimentally is in'good agreement with the contribution estimated from spin waves as shown by Dr. 2 t-8c = Z)V 8c. Marshall. At this time I think it is just a coincidence, but the at explanation may well be correct.

The solution of this equation is E. Callen: I am uncomfortable with the application of the beautiful analysis of Dr. Marshall to metals, because I see no evidence of a large line width mechanism, the complex susceptibility 1 arising from excitation of electrons among Landau states. Sc = - exp (47rDf)3/2 Wt W . Marshall: I certainly agree with you. that we must be cautious in applying the theory to metals, which, of course, is the only which has a Fourier transform of the form thing I have done. I would answer your remark in the following way. What I am really trying to say is that as soon as you stop damping the spin waves a diffusive mode will appear on the = 2DK2 imaginary axis. There are various ways in which you could T(K, a>) describe this. You may prefer to describe it in terms of spin 2 2 2 (3) (DK oj ) + waves and the damping of spin waves. That is perfectly satisfactory. But we have also to look, in this particular formula- where K is -^-sin0/2. Hence we would, indeed, tion, at what this pole is on the imaginary axis, and on the effect of this diffusive motion. In the 'case of a liquid we know that the sound waves are damped by thermal conductivity and vis- expect a Lorentzian line shape at a fixed scattering cosity. Similarly we can explain our results in terms of the 2 angle and also a line width dependence on sin 0/2 damping of spin waves. But as soon as we put in the thermal conductivity, a third pole appears in the expression on the at fixed temperature. imaginary axis. And that pole becomes largest in the proximity Dr. Raymond Mountain [8] of the National Bu of the critical temperature.

reau of Standards has solved the eqs (1) using H. B. Callen: A simple analogy may make this diffusive mode a Fourier and Laplace transform techniques and gets little less mysterious. The excitations of a noninteracting Fermi the same results for the line shape in expression (3). gas at very low temperatures are oscillations of the Fermi surface which are coherent excitations of the whole system which occur In closing I would like to point out that in the at high frequency. I think they are the analog of the spin waves. past, researchers have only been able to measure When the temperature becomes large enough so that the relaxa- scattered intensities. We now have available tion term becomes appreciable these high frequency excitations disappear, being replaced by one-particle excitations (at very techniques as demonstrated in this paper of meas- low frequencies). These one-particle excitations seem to cor- uring spectral profiles. Hopefully, as our tech- respond to the diffusive mode. In the transition region the one- particle and the collective mode may coexist, although the col- niques become more refined and our results more lective oscillations of the Fermi surface are then damped.

~~160 R. Nathans presented measurements of neutron scattering in However, the results of Jacrot do not allow a detailed comparison~~

antiferromagnetic potassium manganese fluoride [1J. with the theory. I think it would be very interesting to go also to higher temperatures. Perhaps, it is necessary to work with a higher precision for this purpose. B. Jacrot: In connection with the work of Dr. Nathans. I want to remark that the inelasticity for an antiferromagnetic system like manganese fluoride should be ten or twelve times larger than for L. Passell: May I make a brief comment on that? When we iron due to the absence of kinematieal slowing down. This has tried to do an energy analysis, we ran out of intensity very rapidly been confirmed qualitatively by some unpublished results ob- as we went up in temperature. With the facilities we had in tained a few years ago at Saclay. New experiments are now in Denmark, we could never hope to get much at temperatures progress. higher than 30 °K above the Curie temperature. We do not have a sufficient intensity for that purpose. On the other hand,

I believe that in Saclay you do have the possibility to cover a R. Nathans: Well, that may be possible. We have only seen larger temperature range, because you have a cold neutron the data of Turberfield in manganese fluoride, which indicate a source. could not see any difference between the value of small inelasticitv in the critical region. We the parameter A at 2 "K above the Curie temperature and that at 18 °K above the Curie temperature. It may be that there is a M. S. Green: Dr. Passell. why did you expect to observe elastic small change of A with temperature, which begins to occur at scattering before you started the experiment? higher temperatures. Would you care to comment on this, Dr. Jacrot?

L. Passell: Because the parameter A which describes the time dependence of the spin fluctuations should be inversely pro- B. Jacrot: I agree that it is extremely difficult to observe any portional to the susceptibility as indicated by Dr. Marshall. effect of the temperature. In the curve which we first published, This was the argument which Van Hove used in his original we showed a temperature effect for A. However, that were formulation of the theory. On this basis we assumed that we just crude data and after a careful analysis with the computer, were going to have essentially quasi-elastic scattering at the it turned out that A is practically the same at the Curie point ordering temperature. and at 30 °K above the Curie point. This means that we have to carry out experiments with much higher accuracy. It is very doubtful that this is possible. In order to observe the variation M. S. Green: Did not Van Hove predict a finite inelastic scattering of A with temperature it is better to do the experiments on nickel for finite ^-values? where the effect is bigger.

L. Passell: Yes, that is right. But that would give only a small correction. P. Debye: In connection with the paper of Dr. Ford and Dr. Bene- dek, I want to remark that already 30 years ago the spectral distribution of light scattered in liquids was photographed by First, if W. Marshall: May I comment on this point? you use Ramm using an echelon [4]. I think this work should not be for- what I call the conventional theory and which Van Hove gave, gotten in any historical review. , then of course the scattering is inelastic. And' *h is gives you s the difference between say a neutron experiment and a N.M.R. experiment. However, when you ask theoretically how inelastic P. Debye gave also theoretical arguments as to why the line the scattering is, you discover that so far as neutrons are con- width of light scattered by a binary mixture in the vicinity of cerned the spectral distribution looks like a sharp 8-function. the critical point, as observed by Dr. Alpert and coworkers, So if that were true, we could say that for experimental purposes should the scattering is as elastic as we could possibly want it to be. 2 2 a. increase proportional to 5 = 4 sin {012) with increasing scattering angle 6; b. increase proportional to the temperature distance T—T,. Secondly, Dr. Passell polite in describing the was much more The substance of this discussion remark has been published in situation than I would Jacrot's have been. We knew about Phys. Rev Letters 14, 483 (1965). results and that he had found the scattering to be inelastic, but we did not believe it could be true. So we felt that there

probably was some experimental flaw in these experiments; I line W . Marshall: I wonder why the width observed by Dr. now apologize to Dr. Jacrot for thinking that to be the case. Alpert does not go to zero when the temperature approaches I also like to make another point. It is true, of course, thai the critical temperature. we would expect proportionally very much greater inelasticity in an antiferromagnet than in a ferromagnet, simply because we do not have the kinematic narrowing. However, we also S. S. Alpert: There is a possibility that were not exactly at the have the fact that the Curie temperature of iron is 1042 °K com- we pared to 80 °K for manganese fluoride, so that the higher ex- critical point. changes energies in iron compensate the difference to a certain extent.

W . Marshall: That is one suggestion which I was going to make. However. I want to bring up another suggestion. We are used J. Villain: As a theoretician, J would be very interested in the to the assumption that the scattering from a particular molecule temperature dependence of the inelasticity parameter A. In can be taken as a constant. We associate with each molecule tact, de Gennes 12] has predicted a thermodynamic slowing down a given scattering power which we allow to vary with angle. which must be added to the kinematic slowing down. It is pre- However, suppose that the scattering power of each molecule is dicted that the line width close to the critical temperature is slightly dependent on its neighboring molecules. Then the would smaller than at the temperatures higher than the Curievtem- scattering power depend on short range order. You may perature. I think that the original treatment of de Gennes and then get a term which is in fact not divergent. This might be myself [2] is not correct and that this thermodynamic slowing something that would have to be taken into account in these very

down is perhaps weaker, but it must exist. Our present delicate experiments. It is really a question; I do not know theory [3] to account for this effect is in good argeement with whether it is sensible or whether the orders of magnitude are the results of Jacrot for the temperature dependence of A. right.

~~161 s~~

Commenting on the neutron scattering from /i-brass. I would wave vectors we are talking about. If we indeed go to higher like to congratulate Dr. Dietrich and Dr. Als-Nielsen on two facts: /-values in these systems which seems possible by pushing the (1) They did some very beautiful experiments which other laser technique further, and certainly when we use neutrons people have attempted and have failed to do. where we have higher /-values, it is very important to modify (2) They were very good politicians because on the last day of the discussion of the fluid to include rigidity. At the higher fre- their experiments, they managed to destroy their crystal, which quencies the liquid is just as rigid as the solid. In EgelstafT was probably the only good crystal of /3-brass in the world. We neutron experiments on liquids already you can see that you must may have to wait a long time before we get any better results. include rigidity to get the right results. Thus the rigidity of the

I would like to make also some comments which are stimu- liquid will affect the shape of the curve for the higher /-values. lated by Dr. Benedek's talk. He commented and in my talk I Whether the laser will reach that region, has to be found out in referred to the fact that as soon as the sound waves get damped the future. a diffusive type mode appears. Some of you may think that sound waves are not only damped by the thermal conductivity but also by the viscosity, which, of course, is true. You may References then ask why the viscosity did not appear in the formula for the line width of the central line. The reason for that is that the fl] R. Nathans, M. Cooper and A. Arrott, Proc. Conference or viscosity and thermal conductivity appear quite differently in Magnetism, San Francisco, 1965, to be published. the hydrodynamic equations. Indeed there is another mode [2] P. G. de Gennes and J. Villain, J. Phys. Chem. Solids 13, 10 which appears as soon as you introduce damping through vis (I960). cosity. but it is a mode which stays on the imaginary axis and [3] J. Villain. J. Phys. 26,405 (1965). never comes down to the origin even at the critical point. There- [4] W. Ramm, Physik Z. 35, 111, 756 (1943); E. H. L. Meyer and fore, it does not produce any appreciable effects at the kind of W. Ramm, Physik Z. 33, 270 (1933).

~~162 TRANSPORT AND RELAXATION PHENOMENA~~

~~Chairman: R. W. Zwanzig~~

~~163 209-493 0-66-12 [~~

~~.L. Behavior of Viscosity and Thermal Conductivity of Fluids Near the Critical Point~~

~~J. V. Sengers~~

~~Institute for Basic Standards, National Bureau of Standards, Washington, D.C.~~

~~1. Introduction irreversible fluxes associated with the thermody-~~

namic forces by the linear laws. This is the most This paper reviews the experimental information serious difficulty encountered in measuring the on the behavior of the viscosity and thermal con- thermal conductivity and will be further discussed ductivity of fluids near the critical point. Section in section 4. As is well known, equilibrium or a

2 presents some general remarks on experimental steady state is approached very slowly by a physical complications associated with the measurement system near a critical point. Consequently, some of these properties in the critical region. Ex- authors have observed hysteresis effects in their perimental work on the viscosity of one component experimental observations of the viscosity [2, 6, 50]. fluids near the liquid-vapor critical point is reviewed A prerequisite for the study of the behavior of in section 3. The thermal conductivity of one any physical property in the critical region is a component fluids is considered in section 4. The knowledge of the values of the appropriate thermo- behavior of the viscosity and thermal conductivity dynamic variables at which the property is meas- in binary liquid mixtures near the critical mixing ured. Near the liquid-vapor critical point the tem- point is discussed in sections 5 and 6, respectively. perature and density and near the critical mixing Section 7 is a summary of conclusions based on point of a binary liquid mixture the temperature and these experimental results. concentration should be known. If a P-V-T re-

~~lationship is known, one can calculate the density 2. Some General Remarks from the measured pressure and temperature. Close to the critical point, where the isotherms~~

In addition to the usual difficulties associated become very flat, this procedure becomes inac- with experiments in the critical region, measure- curate. This inaccuracy is greatly increased by ments of the transport coefficients are hampered uncertainties in the absolute value of the tempera- by the fact that they are nonequilibrium quantities. tures at which either the transport coefficients or Most accurate measurements have been carried the isotherm data were measured. Uncertainties out by introducing macroscopic gradients into the in the temperature also hamper a comparison with system. In order to obtain the transport coeffi- other properties near the critical point. Such a cients the hydrodynamic equations must be solved comparison for a binary liquid mixture is com- under certain imposed boundary conditions. plicated by the fact that the critical parameters Exact solutions of these equations are known only are very sensitive to impurities. for fluids with constant properties in idealized If one measures the transport coefficients as a geometrical situations. However, in the critical function of pressure and temperature, care should region a number of physical properties like local be taken to work at small enough pressure and density and specific heat vary appreciably with temperature intervals as otherwise the whole critical position as a result of the gradients. This com- region is easily overlooked. In experimental work plicates the analysis considerably. Close to the on transport coefficients this has led to considerable critical point even the validity of the Navier-Stokes confusion, which will be discussed further in sec-

~~equations [41] and of the linear laws has been tion 4 in connection with thermal conductivity~~

questioned [38, 42, 59]. measurements. It is equally important for the If in a fluid near the liquid-vapor critical point study of any other property near the liquid-vapor the temperature is not uniform, gravity can easily critical point.

~~generate convection currents and convective Another procedure consists in following an iso-~~

~~heat flow which must be distinguished from the choric method where the temperature is varied~~

165 while the total amount of gas is kept constant. it allows the measurement of the viscosity at a However, this method also has its pitfalls. First, certain horizontal level in the fluid and conse-

if one lowers the temperature at constant density, quently, in principle, at a certain local density. phase transition occurs at a certain temperature In the capillary flow method the density varies along

which is equal to the critical temperature at the the capillary. critical density. Below this transition tempera- An example of the "normal" behavior of the vis- ture one measures either in the vapor or in the cosity outside the critical region is shown in figure

liquid, depending on the experimental arrangement. 1; the viscosity coefficient of argon [4, 7] at three As a consequence the density, at which the proper- temperatures, high relative to the critical tempera-

ties under consideration actually are measured, ture {tc = — 122.3 °C), as a function of density. One does change rapidly as a function of temperature, amagat unit of density is the density at 0 °C and although the density averaged over vapor and liquid 1 atm. 1 At a given temperature the viscosity co-

phase is kept constant. The temperature at which efficient increases monotonically with the density; 2 2 this transition occurs should be determined ac- (dr]/dp)r and (d r}jdp )T are both positive. The

curately. Moreover, even above this temperature temperature derivative (d7]/dT) p is positive up to a a large density gradient is developed close to the density well beyond the critical. At very high den-

critical temperature so that the local density at a sities (dr)ldT)p becomes negative [3, 8], but this is certain position in the vessel is still not independent not relevant for our present discussion. Some-

of the temperature. Ultimately, close to the liquid- times the excess viscosity 17 — 170, where 170 is the vapor critical point a knowledge of the local value of viscosity coefficient in the dilute state at the given

~~the density is required. temperature, is considered to be only a function of~~

The influence of critical phenomena is not re- the density. This is, however, not true in general. stricted to the immediate vicinity of the critical Most measurements of the viscosity and thermal point. For instance, an anomaly in the specific conductivity in the critical region have been carried

heat can be detected at temperatures considerably out for carbon dioxide because of its easily acces- higher than the critical temperature. In this paper sible critical temperature. 2 The discussion will the qualitative properties of viscosity and thermal therefore be mainly based on the phenomena ob- conductivity regarding the existence of similar served for CO^. The few investigations on other anomalies in the critical region will be discussed. substances sustain the conclusions obtained for CO2 However, the precise mathematical character of The experimental investigation of the viscosity of such anomalies can only be obtained from measure- CO2 in the supercritical region started with the work ments very close to the critical point. In view of of WARBURG and VON BABO in 1882 using a cap- the difficulties mentioned above an analysis of the illary flow method [15]. Even that early these latter kind is not yet possible for the transport authors were aware that knowledge of the density is coefficients. essential and were careful to determine the density as well as pressure and temperature. They realized 3. The Viscosity of a One Component that the pressure difference along the capillary Fluid Near the Liquid-Vapor Criti- should be kept small and applied corrections for cal Point the variation of the density along the capillary. In an illuminating discussion of this experimental work

Accurate measurements of the viscosity of com- M. BRILLOUIN [1] remarked that these correc- pressed gases have been mainly carried out either tions are practically canceled by the variation of the by a transpiration method or by the oscillating disk viscosity associated with the variation of the density. of and von Babo at 32.6 °C method. In the first method the gas is caused to The data Warburg flow through a capillary and the viscosity is ob-

1 3 tained from the flow rate. In the latter method the One amagat unit of density for argon corresponds to 0.0017834 g/cm . 2 In this review the values given by Micheis et ai. [60], are used as critical parameters I damping of an oscillating disk due to the viscosity ofCOz :

3 of the surrounding medium is determined. Both (, =31.04 °C. P, = 72.85 atm. pr = 0.468 g/cm = 236 amagat. methods have been used to measure the viscosity An alternative possible set of critical values was given by Schmidt and Traube [65]:[

~~3 in the critical region. Close to the critical point (,. = 31.00 °C. />,. = 72.86 atm. pr = 0.468 g/cm = 236 amagat.~~

~~3 the oscillating disk method has the advantage that One amagat unit of density for CO* corresponds to 0.0019764 g/cm .~~

166 the critical temperature). Although the absolute accuracy of these old measurements may be limited, ARGON the relative precision is sufficiently good to warrant O 75°C A 25°C this conclusion. • -50°C After some inconclusive work on the subject by

~~PHILLIPS [11], CLARK [2], and SCHROER and~~

~~BECKER [13], an experimental investigation of the viscosity near the critical point of CO2 was under-~~

~~taken by NALDRETT and MAASS [10] using an~~

oscillating disk [6]. The critical region was approached by the isochoric method. One viscosity isotherm at 31.14 °C was also determined. The P- 5 authors conclude that the viscosity coefficient shows a small anomalous increase close to the critical point. At the critical density this anomalous in-

~~crease is about 1 percent at 32.0 °C and about 10 percent at 31.1 °C. The results confirm the conclusion of Warburg and von Babo that at 32.6 °C any~~

anomaly is smaller than 0.5 percent. The authors did not consider the influence of the variation of the 300 400 500 600 700 DENSITY. AMAGAT density as a function of the height in the fluid. A cell with a height of about 13 cm was used, the r IGLiRE 1. The viscosity of argon as a function of density at oscillating disk being located in the lower end of temperatures high relative to the critical. the vessel, and an estimate from the density versus height relationship published in the literature [65,67] shows that this effect could have caused an apparent increase of the same order as the anomaly observed. Therefore, the conclusion that the viscosity ex- C02 O 32.6°C WARBURG a VON BABO hibits an anomalous increase near the critical point tc = 31.04 °C is not justified. fl = 236 AMAGAT A pronounced anomalous increase in a wider temperature range and with the viscosity increased by a factor 2 at 31.1 °C was reported by MICHELS, BOT-

3- ZEN, and SCHUURMAN [8]. Their data were obtained with a capillary flow method which was known to be accurate outside the critical region. As pointed out by the authors, the method is not reliable in the critical region due to the large density gradient in the capillary resulting from the use of

~~100 200 300 400 a pressure difference of about 0.5 atm. Never-~~

~~DENSITY , AMAGAT theless, they conjectured that the anomalous~~

behavior observed was essentially real [8, 9]. FIGURE 2. The viscosity of CO , at 32.6 °C according to Warburg and von Babo. In order to resolve the discrepancy between these results, new measurements of the viscosity of CO2 are shown in figure 2. They also measured vis- were carried out by KESTIN, WHITELAW, and cosity isotherms at 35.0 and 40 °C which have not ZIEN using an oscillating disk method [5]. They ap- been included in figure 2 since the temperature proached the critical region along different iso- effect turned out to be small. Since the viscosity chores as did Naldrett and Maass. Their results isotherms at 35 and 40 °C are nearly parallel to the show convincingly that any anomalous increase is one at 32.6 °C. the authors conclude that the vis- much smaller than that reported by Michels et al. cosity does not show any anomaly at supercritical Close to the critical point they find an anomalous temperatures down to 32.6 °C (about 1.6 °C above increase of the same order of magnitude as that

167 reported by Naldrett and Maass. The data ob- Close to the critical temperature they find an in- tained by Kestin et al., at 31.1 °C and at 34.1 °C crease of 1 to 2 percent on one isotherm, which are shown in figure 3a, b as a function of density. is of the order of the measuring accuracy in this For comparison the behavior of the viscosity iso- difficult region. therm at 50 °C is indicated and data obtained by From the experimental work described, we can

~~Naldrett and Maass and a few data of Michels et al., draw the following two conclusions. outside the critical region are included in the figure. 1. Measurements of the viscosity of CO2 with~~

~~Recently the viscosity of CO2 was also measured the oscillating disk method [5, 10] as well as meas- by BARUA and ROSS with a capillary flow method urements with the capillary flow method [12, 15]~~

[12]. By using a horizontal capillary and extrapolat- have shown that the viscosity coefficient does not ing to zero flow rate [4], they were able to avoid the show any anomalous behavior at supercritical difficulties inherent in the viscometer of Michels temperatures down to a temperature (T—.TC )ITC et al. The analysis of their experimental data has = 1 percent from the critical. not yet been completed, yet it seems that their 2. The significance of the small anomalous in- results are in agreement with those of Kestin et al. crease at temperatures closer to the critical is not clear at present. A correction should be sub- 8 tracted from the data of Naldrett and Maass in this region which in magnitude might be compar- co 2 J able to the anomaly reported. Using the same a NALDRETT, MAASS did not find an anom- . t=3l.l°C ° MICHELS. BOTZEN, SCHUURMAN apparatus Mason and Maass • KESTIN, WHITELAW. ZIEN / alous increase of the viscosity of ethylene near the

~~t = 50 °C critical point, although the data are somewhat /// / preliminary. It is interesting that the difference between the data of Kestin et al., and the uncor-~~

~~rected data of Naldrett and Maass is also of the~~

same order as the anomaly. Kestin et al., have discussed the possible influence on their measurements of the variation of the density with height. 2 Because their oscillating disk is located at a position somewhat higher than the center, the varia-

~~(a) tion of density leads to a correction in the opposite E~~

~~, 1 ! direction from that to be applied to the data of Naldrett and Maass and therefore tends to enlarge~~

1 1 1 the anomalous effect. The authors point out that C02 their data become less reliable in this difficult region and they are taking the attitude that their MICHELS BOTZEN SCHUURMAN - (° • , t 34 l°C - " V KESTIN, WHITELAW, ZIEN / data prove only the absence of the pronounced t=50°C anomaly suggested by Michels et al., and that any

anomaly is smaller than 20 percent. The reality of the small anomaly can be decided definitely J? 4 only if the local value of the density is determined simultaneously and the effect of a variable density on the interpretation of the viscosity measurements — is carefully analyzed. 2 - This picture of the viscosity in the critical region is sustained by the work on other substances. STAR-, LING, EAKIN, DOLAN, and ELLINGTON have

~~(b) measured the viscosity of ethane, propane and n-~~

1 .! 1 1 butane with a capillary flow method [14]. Although 100 200 300 400 DENSITY. AMAGAT the authors did not analyze their measurements as aj function of density, they have measured at suffi-l FIGURES 3a. b. The viscosity 0/ CO; iis (i function of density near the critical temperature. ciently small pressure intervals to conclude that

168 the viscosity does not show a pronounced anomaly from the critical temperature the density is only in the critical region. However, near the critical 185 amagat at the critical pressure, whereas the point neither the density at which the viscosity critical density is 236 amagat. Consequently was determined nor the precision of the values for extrapolation of data at constant pressure as a func- the viscosity coefficient is sufficiently well known tion of temperature easily overlooks the critical to decide whether a small increase, as observed region entirely and therefore does not reveal any by Kestin et al., exists. anomaly in the critical region. Along an isotherm

Recently DILLER reported that the viscosity of the density is similarly sensitive to a small change parahydrogen does not show any anomaly near the in pressure. critical point [3]. Measurements at temperatures Another procedure sometimes used to obtain a for which T— Tc is ten times smaller than in the thermal conductivity value associated with the case of CO2 are required, because the critical critical point is based on the assumption that the temperature of parahydrogen is only 32.976 °K. — A. is excess thermal conductivity \ ( >, where Xo

Consequently, only the measurements at one iso- the thermal conductivity at 1 atm at the tempera- therm (T— 33.000 °K) are relevant for our present ture considered, is only dependent on the density discussion. Indeed, the data on this isotherm do and not on the temperature. However, this rela- not show any appreciable anomaly near the critical tion actually has been verified outside the critical density. However, this conclusion is based only region only and therefore cannot be used legitimately on a few experimental points for which calcula- in the critical region. "Critical thermal conductiv- tions of the density from the P-V-T data are diffi- ity values," obtained in this way, are often used as cult. An analysis of the viscosity data on adjacent reduction parameters for correlation of the thermal isotherms shows that the precision is only a few conductivity of different substances [66]. It must percent. Therefore, no decisive conclusion can be kept in mind that these reduction parameters be drawn from this work regarding the existence of a do not necessarily represent the actual thermal con- small anomaly. ductivity near the critical point. Similar procedures have been used to obtain critical values 4. The Thermal Conductivity of a One of the viscosity. The reduction parameters for Component Fluid Near the Liquid- the viscosity obtained in this way are closer to the Vapor Critical Point actual viscosity in the critical region, because any anomaly in the viscosity near the critical point is of the Outside the critical region the behavior small. With few exceptions, values for the trans- to that of the thermal conductivity is very similar port coefficients in the critical region which are viscosity described in the previous section. How- based on data outside the critical region are not the ever, there exists a serious controversy in discussed in this review. literature whether the thermal conductivity, con- Two methods for the measurement of the ther- trary to the viscosity, exhibits a pronounced anomaly mal conductivity of compressed gases have had in the critical region. An introductory survey wide acceptance, the concentric cylinder and the on the subject has been presented by Ziebland [37]. parallel plate method. In the first the fluid is en- The thermal conductivity is sometimes studied closed between two coaxial cylinders, either in either as a function of pressure at constant tempera- horizontal or in vertical position, and the heat flow ture or as a function of temperature at constant through the gas layer is determined as a function pressure, without considering the dependence on of the temperature difference between the cylinders. the density. The region around the critical density In the latter method the fluid is contained between is restricted to very narrow pressure and tempera- two horizontal plates and the heat flowing from the ture intervals in this procedure, because both upper plate to the lower plate is determined as a (c)pj()p)T and (tipjtiT),, become infinite at the critical function of the temperature difference. A special point. Measurements at the critical pressure are case of the concentric cylinder method is the hot not equivalent to measurements near the critical wire method. This method is not suitable for the density unless the temperature is equal to the critical region because the heat transfer analysis critical temperature within a tenth of a degree. is complicated, too large temperature differences For instance, considering the density dependence are usually involved and, most importantly, con- along an isobar, in CO-2 at a temperature 0.15 °C vection cannot be avoided.

169 Papers reporting an anomalous increase of the Ar=0 from the experimental data obtained for X' thermal conductivity in the critical region are gen- using various values of AT. In the critical region erally considered with much skepticism due to the contribution of convection becomes very large the high probability of convection. An exact the- and such an extrapolation can diminish the accuracy

oretical analysis of convection is very complicated considerably even when care is taken that only and has been carried out only in very special cases. values for \' corresponding to the same average We do not go into the details here, but remark that density and average temperature are used. In the presence of convection can be detected by some cases, such as a horizontal fluid layer heated varying the thickness of the gas layer or the tem- from below, convection only seems to start if the perature difference across this layer. The heat Rayleigh number becomes larger than a certain transfer coefficient or so-called apparent thermal critical value [64]. This has also been reported conductivity coefficient, which includes a contribu- for a concentric cylindrical layer in horizontal tion from convection, depends on the geometry of position [23, 58]. When this instability occurs, the apparatus and of the boundary conditions extrapolation to AT=0 is not legitimate and can involved. lead to overcorrections. Heat transferred by convection is usually related The controversy over the existence of a thermal to the Rayleigh number R defined as conductivity anomaly started in 1934. KARDOS attempted to measure the thermal conductivity 2 3 R = gap cp d ATIk7] (1) of CO> using a hot wire method [21]. He found a tremendous increase of heat transfer in the critical where g is the gravitation constant, a the expansion region, which was later recognized as an effect of coefficient, p the density, cp the specific heat at convection [28]. SELLSCHOPP measured the constant pressure, d the thickness of the gas layer, thermal conductivity of CO2 with two concentric

Ar the temperature difference, X the thermal con- cylinders in horizontal position [29]. He observed ductivity and n the viscosity coefficient. In the an increase in the apparent thermal conductivity critical region the Rayleigh number becomes near the critical density at supercritical tempera- extremely large [26, 30]. By the use of many tures, but his measurements were also influenced by approximations and assuming that the convection convection. The whole effect was attributed to is small, the error introduced by convection can convection and it was assumed, without experi- be derived explicitly: mental justification however, that in the critical region the thermal conductivity was only dependent

X' — X d on the density and not on the temperature [18, 29]. —— = C/? 7 sine (2) TIMROT and OSKOLKOVA [33] reported that the thermal conductivity of CO2 did not show any where A.' is the apparent thermal conductivity co- anomaly in the supercritical region, but their efficient as measured, / the length of the gas layer, measurements obtained with a hot wire method e the angle between the temperature gradient and were actually carried out too far away from the the vertical, and C a proportionality constant. In critical point. order to variation obtain (2) one neglects the with COMINGS et al., measured the thermal conduc- temperature of the fluid properties other than the tivity of several gases using two concentric cylinders density, linearizes the hydrodynamic equations, in horizontal position. Anomalies in the super- assumes a spatially uniform temperature gradient critical region were reported for ethane, propane, and butane [22] but the reality of the effect could and considers the limiting solution for y^l- not be established again due to the possibility of Without these approximations the error may well convection. An anomalous increase of the thermal depend on d and AT in a more complicated way. conductivity of argon at 0 °C, originally reported The absence of convection is verified, nevertheless, by the present author among others, later turned

if the measured thermal conductivity coefficient out to be entirely caused by convection [24]. at a given density and temperature does not change The first critical study of the existence of m under sufficient variation of AT or d. thermal conductivity anomaly was carried out by If is convection present, one sometimes tries GUILDNER [19]. He measured the thermal con- to obtain the correct value of X by extrapolating to ductivity of CO2 using a vertical cylindrical gas

~~170 layer with a thickness of 0.68 mm. Convection 26, 30]. Contrary to the concentric cylinder~~

could not be avoided in this apparatus so that A. method the horizontal flat layer method has the was determined from measurements at various advantage that the angle between temperature temperature differences by extrapolation to AT gradient and the vertical can be made very small to = 0 at constant average temperature. He con- suppress convection. The experimental results cluded that the thermal conductivity itself exhibits obtained for the thermal conductivity as a function a pronounced anomaly. This anomalous increase of density are presented in figure 4. The thermal is already 30 percent at 40 °C at the critical density. conductivity shows indeed a pronounced anomaly He approached the critical temperature to within at the critical point. An anomalous increase can

1 °C. Although at 75 and 40 °C the influence of be detected as much as 20 °C above the critical convection was small, closer to the critical tempera- temperature which becomes 250 percent at 1 °C ture the influence of convection became large dimin- above the critical temperature. Convection was ishing the accuracy of the data. avoided by the use of a small distance between On the other hand, VARGAFTIK stated that the the plates (d=0A mm) and small temperature thermal conductivity did not exhibit an anomaly differences (AT varied from 0.03 to 0.25 °C). The in the critical region [34]. This assertion was based absence of convection was verified experimentally; on experimental wotk of SHINGAREV, which unfor- in the whole critical region the thermal conduc- tunately, has not been made available and therefore tivity values measured were independent of the cannot be critically examined. According to the temperature difference used. Of course, one discussion given by Vargaftik, these conclusions should compare measurements with different AT, were drawn after applying corrections for convec- but at the same average density and temperature. tion by extrapolation to zero temperature difference. For the details we refer to the previous publica-

It is clear that measurements in the critical region tion. The reality of the anomaly was independently free of convection are highly desirable. For this pur- confirmed by a few measurements with a larger pose a detailed study of the thermal conductivity of plate distance. An attempt was made to include iso- C02 was carried out by MICHELS, SENGERS, and a measurement of the thermal conductivity VAN DER GULIK using a parallel plate method [25, therm at 31.2 °C using even smaller temperature differences. These results are shown in figure 5. Obviously the values at this temperature are less

~~CO2 25.1 °C + 30.0°G © 31.2°C A 32.1°C cmsec°C ® 34.8°C • 3I.200°C O 40.1°C 7 O 31 .217 °c • 49.6°C C 31 .27 °c X 75.I°C~~

~~A 32. I °C~~

~~400 Am~~

~~FIGURE 4. The thermal conductivity of CO> as a function of Figure 5. The thermal conductivity of CO > at 31.2 and 32.1 °C density. as a function of density.~~

171 accurate than those of figure 4, since the experiments using a hot wire apparatus at various tempera- mental verification of the absence of convection tures, but at constant pressure. As mentioned becomes more difficult due to the inaccuracy in before data at constant pressure as a function of the density. temperature do not approach smoothly the critical Due to the variation of temperature, the density density. Consequently, the measurements were variation in the pressure vessel is very compli- not carried out near the critical density. The in- cated. It is therefore difficult to approach the terpretation of Abas-Zade that he followed the critical region by the isochoric method. The coexistence line up to the critical temperature density must be calculated from the experimental is erroneous. pressure and temperature, which may introduce Very careful thermal conductivity measurements systematic errors at each isotherm. Thus the were carried out by AMIRKHANOV and ADAMOV detailed behavior along the critical isochore. par- using both a horizontal parallel plate method and a ticularly the character of the divergence of the vertical concentric cylinder method [17], Using the thermal conductivity, cannot accurately be de- parallel plate apparatus, a series of measurements duced from the data. was carried out with plate distances of 0.3 mm and

It is notable that at 75 and 40 °C. where the of 0.6 mm, and the temperature difference was amount of convection in the apparatus of Guildner varied from 0.07 to 1 °C. With the smaller plate was reasonably small, there is a satisfactory agree- distance the data turned out to be independent of ment with the data of Guildner. confirming the AT", which guarantees absence of convection. anomaly of 30 percent at 40 °C. It turned out, Data obtained with the larger plate distance and however, that closer to the critical temperature also those obtained using the concentric cylinder the anomalous increase is considerably smaller apparatus were dependent on Ar showing the than that reported by Guildner. presence of convection. The results obtained with The fact that the thermal conductivity measured the smallest value of AT agreed for all 3 series of was independent of the temperature difference measurements. The values obtained for the used not only guarantees the absence of con- thermal conductivity therefore seem to be very vection, but also verifies the validity of the law of reliable. As has been pointed out repeatedly

Fourier in the temperature range considered. The before, it is equally important to verify whether validity of the law of Fourier in the critical region the measurements were carried out sufficiently was questioned by SIMON and ECKERT [32], From close to the critical density. Unfortunately, the interferometric studies of laminar free convection authors do not describe their experimental pro- near a vertical wall a quantity k was derived, which cedure. They do not report whether and how either they identified with the thermal conductivity co- pressure or density was determined. In addition, efficient of the fluid. This quantity k turned out although they measure the temperature difference to increase with increase of heat rate and it was AT carefully, no device was presented for meas- conjectured that the anomaly in the thermal con- uring the absolute average temperature of the gas ductivity would disappear in the limit of zero heat layer, which obviously differs from the temperature rate. However, an analysis of their data shows of the surrounding thermostat. It is not clear at that this quantity k remained dependent on the which temperatures and densities the thermal con- heat rate outside the critical region as well, where ductivity actually was determined. If the measure- the validity of the law of Fourier cannot reasonably ments were carried out at the coexistence line, one be questioned. Therefore, this quantity k should would expect a large variation of the density when not be identified with the thermal conductivity the temperature difference was varied from 0.1 coefficient. to 1.0 °C, at the liquid side even leading to evapora- The existence of an anomaly in the thermal con- tion of the liquid. On the contrary, their results ductivity near the critical point was denied by indicate that even up to a few hundredths of a the density Abas-Zade [16| and by Amirkhanov and Adamov degree below the critical temperature did not varied from 0.1 to [17]. These authors do not base this assertion on change when AT was measurements in the supercritical region, but on 1 °C. Thus it seems that the measurements were data obtained at temperatures below the critical not carried out near the critical density. which they claim are associated with the coexist- From the data given in figure 4 it is evident that ence fine. ABAS-ZADE performed his measure- the first indication of the existence of the thermal 172 conductivity anomaly can be noticed from the A similar result is obtained [31] if one analyzes occurrence of intersections of the thermal con- the thermal conductivity data obtained by IKEN- ductivity isotherms as a function of density at BERRY and RICE for argon [20]. However, since temperatures considerably higher than the critical these authors did not adequately examine the ab- temperature. It is interesting therefore to study sence of convection, we cannot rely on their results as a function of density data given in the literature as decisive. somewhat remote from the critical point. As an In view of these results NEEDHAM and ZIEB- example, we considered the experimental data ob- LAND studied the thermal conductivity of ammonia tained by ZIEBLAND and BURTON [35] for the X in more detail [27, 37] than before [36]. The data of nitrogen. These data were originally plotted were obtained using an apparatus consisting of along isobars and consequently did not reveal any two concentric cylinders with a gap of 0.2 mm. anomaly. However, if the data are plotted as a The results obtained at two supercritical tempera- 4 function of density the thermal conductivity iso- tures are shown as a function of density in figure 7. therms shown in figure 6 are obtained. 3 A pro- An anomalous increase can be noticed near the nounced increase in the thermal conductivity can critical density even at 157.1 °C. The absence be noticed at — 140 °C (7 °C above the critical tem- of convection was verified by measurements with perature). This observation is based on only two different values of A7\ At 138.8 °C the thermal experimental points, but the crucial experimental conductivity goes through a maximum. The densities were estimated by Needham and Ziebland value at —140 °C, showing an anomalous increase from P-V—T data published in the literature. of 35 percent, was obtained with two different values However, at 138.8 °C these P-V—T data are vir- of A7\ which excludes the possibility of convection. tually unknown, so that the densities cannot be This is a clear illustration of how one can be misled calculated with sufficient accuracy to conclude if the density dependence of X is not considered that the maximum does not occur at the critical explicitly. density.

3 3 One amagat unit of N 2 corresponds to 0.0012502 g/cm . We conclude that the existence of a pronounced anomaly in the thermal conductivity has been verified by convection free measurements obtained both with a parallel plate method and with a concentric cylinder method.

~~NITROGEN 4 3 One amagat unit of NH3 corresponds to 0.0007714 g/cm .~~

~~Ol I I ! I I O IOO 200 300 400 500~~

~~density, amagat DENSITY . AMAGAT~~

~~Figure 6. The thermal conductivity of nitrogen at — 125 and FIGURE 7. The thermal conductivity of ammonia as a function — 140 °C as a function of density. of density.~~

~~173 —~~

~~Table 1. Comparison of the thermal conductivity with the specific heat along the critical isochore of CO2 (pc — 236 amagat)~~

~~k/c v k/Cp t A/X75 v (klCi^TiV* / ^ V/ 4 O (klCr^T^V / ^y/t 0~~

~~°c 75 1.0 1.0 1.0 50 1.1 1.0 0.5 40 1.3 1.0 .3~~

~~1 34 8 1 6 J. • X1 I 32.1 2.5 1.3 m 31.2 6.1 1.6 .02~~

~~An interesting question is whether the anomaly 1 1 1 1 1~~

~~in the thermal conductivity is related to that in co 2 the specific heat. For this reason the variation of T = 31 04 °C 100 c = 2 36 AMAGAT X of CO) as a function of temperature at the critical pc~~

~~5 1 k = 10 cm" isochore is compared with the variation of c, and~~

~~cp in table 1. For c,- the experimental data [62] are supplemented with values calculated from the P-V-T data at temperatures where no experimental~~

data for cv are available [61]. The values for cv —- are obtained by adding the calculated difference -11= -_IL!£— ? < /^--^__50JC__^-^^^^ 25 °C cp — c r to the values for c v obtained as mentioned

above. It is evident that c,, increases much faster \\\34.8°C///' in the critical region than A., so that the ratio A/c,,, V\32l°C/y decreases by two orders of magnitude when the temperature is varied from 75 to 31.2 °C. On the \/3l.2 0C other hand the variation of k/c r is an order of mag-

~~nitude smaller than that of X itself. Very close to 001~~

~~the critical point cv as well as a is not accurately~~

~~known. A relationship between the anomaly in 1 100 200 300 400 500 600 X and c, is well possible. In view of the recent DENSITY, AMAGAT~~

~~work on the cv anomaly, it is natural to ask whether~~

A diverges logarithmically. Such questions about FIGURE 8. Expected line width of the central component of s 1 light scattered by CO > for k = 10 cmT . the detailed behavior cannot be decided at present

~~and further experimental research is required. According to the classical theory the line width of~~

the central component of the spectrum of the corresponds to light from a He-Ne laser scattered i scattered light is determined by the thermal dif- through an angle of 60°. Because c diverges much f) | fusivity X/pcp [57, 63]. The line width to be ex- faster than A, the thermal diffusivity [31] and con-

~~j pected for CO> from the static measurements of A sequently the line width of the central component is presented in figure 8 for k—10 5 cm, which decreases rapidly in the critical region.~~

174 5. The Viscosity of a Binary Liquid Mixture Near the Critical Mixing Point Many binary liquid mixtures exhibit a critical mixing point near room temperature. The viscosity of a number of different mixtures of this type has been studied. There are not enough data obtained by different authors for the same substance. Moreover, the critical parameters are very sensitive to the purity of the components. Although some attention has been given to the influence of a third component on the viscosity in the critical mixing region [43, 47, 50, 51, 53], the influence of any impurity on this anomaly is

~~not clear. It seems that the effect depends on the binary system as well as on the character of the impurity. Most viscosity measurements for binary liquid~~

~~mixtures have been carried out with some version 40 80~~

~~VOLUME % i -C H AT 25 °C of the capillary flow method. We have not made a 8 I8~~

critical analysis of the experimental procedures FIGURE 9. The viscosity of the mixture isooctane-perfluoroheptane used by the various authors. However, there is as a function of concentration according to Reed and Taylor [46]. sufficient agreement to establish the existence of an anomaly in the viscosity near the critical SEMENCHENKO and ZORINA investigated the mixing point for systems with either an upper viscosity of the systems triethylamine-water and

or lower critical solution point [39, 40. 43-53]. nitrophenol-hexane [49]. The first mixture has In the beginning of the century, an anomalous a lower critical mixing temperature. For both increase of the viscosity near the critical mixing systems an anomalous increase of the viscosity point was reported by various authors, among others up to 20 percent was found. The anomalous

~~by FRIEDLANDER [43] for the systems isobutyric phenomenon only occurred at temperatures within~~

~~acid-water and phenol-water, by SCARPA [48] for 1.0 or 1.5 °C from the critical and in a concentration~~

~~phenol-water, by DRAPIER [40] for nitrobenzene- range of 10 mole percent. hexane and cyclohexane-aniline, and by TSAKA- Recent measurements of the viscosity of the~~

LOTOS [52] for isobutyric acid-water, trimethyl- nitrobenzene-isooctane mixture obtained by PINGS ethylene-aniline, triethylamine-water and nicotine- et al., confirmed that for this system also a viscosity water. anomaly occurs in a very small temperature range For simple binary liquid mixtures the anomalous [45]. No anomaly could be detected at 4.5 °C above increase of the viscosity is of the order of 15 to the critical temperature, but an anomalous in- 25 percent at the critical mixing point. crease of 5 to 7 percent was noticed at 0.45 °C REED and TAYLOR observed an anomalous in- above the critical temperature. crease in the viscosity of the mixtures isooctane-per- We conclude that the viscosity does show an fluorocyclic oxide, n-hexane-perfluorocyclic oxide, anomalous increase near the critical mixing point, isooctane-perfluoroheptane [46]. The effect was but for a number of simple mixtures the anomalous most pronounced in the mixture isooctane-per- behavior occurs only at temperatures within 2 °C fluoroheptane; their results for this system are of the critical temperature. shown in figure 9. The viscosity exhibits an anomalous increase up to 25 percent. According 6. Thermal Conductivity of a Binary to these data the anomalous behavior can be de- Liquid Mixture Near the Critical tected as far away as 10 °C from the critical tem- Mixing Point perature. However, other authors found that for some systems the anomaly occurs only at tempera- Experimental investigations of the thermal con- tures much closer to the critical. ductivity of binary liquid mixtures near the critical

175 mixing point are very scarce. This is surprising The most extensive study was carried out by since the experimental complications are somewhat GERTS and FILIPPOV, who measured the thermal less difficult than those encountered in the meas- conductivity of a number of binary liquids using a urement of the thermal conductivity near the hot wire apparatus [54]. Measurements of the liquid-vapor critical point. thermal conductivity of the system nitrobenzene-rc- hexane were carried out with four different temperature differences: 0.01, 0.05, 0.15, and 0.3 °C and

found to be independent of this temperature difference. In addition the systems nitrobenzene-rc- heptane, methanol-n-hexane and triethylamine- water were investigated. The results obtained by Gerts and Filippov for the nitrobenzene-/<-hexane mixture are shown in figure 10 and those for the

nitrobenzene-rc-heptane system in figure 11. It is evident that, contrary to the behavior near the liquid-vapor critical point, the thermal conductivity of these binary liquid mixtures does not show any pronounced anomaly near the critical mixing point. Similar results were obtained for the other two systems mentioned. An anomaly even as small as that found for the viscosity of binary mixtures seems to be absent. The data were not obtained at enough different concentrations to 3.00 - study the behavior in detail and the results cannot be plotted as a function of concentration as was

~~done for the viscosity in figure 9. The absence of 19.5 20.0 2 0.5 a pronounced anomaly near the critical mixing TEMPERATURE, °C point seems to be established. This difference~~

FIGURE 10. The thermal conductivity of the system nitrobenzene- between the behavior of the thermal conductivity n-hexane according to Gerts and Filippov [54]. near the liquid-vapor critical point and that near (63.0 wi % GiR-.NO-.) the critical mixing point was noticed before by

I SKRIPOV [56]. Additional experimental data are C H N0 -n-C H 6 5 2 7 l6 highly desirable to verify this conclusion and to study smaller details. On the other hand an anomalous increase of the thermal conductivity was reported by OSIPOVA for

the phenol-water system [55]. The data were obtained with a parallel plate method. The uncertainty in the temperature difference and the scatter of the experimental data suggest that these results should be considered with some reservations. No experimental verification for the absence of convection was presented. 5.9 — O

~~7. Summary~~

The experimental information on the viscosity 19.0 19.5 20.0 TEMPERATURE, °C and thermal conductivity in the critical region can be summarized as follows: FIGURE 11. The thermal conductivity of the system nitrobenzene- 1. Measurements of the viscosity with the oscil- I n-heptane according to Gerts and Filippov [54].

~~(Curve 1: wt <7, : 62 C s H-,N02 curve 2: 52.3 wt % C 6 H 5 N0 2 .) lating disk method as well as with the capillary flow~~

176 method have shown that the viscosity does not [8] A. Michels, A. Botzen and W. Schuurman, Physica 23 (1957) . 95. show any anomaly near the liquid-vapor critical [9] A. Michels, Proc. Intern. Symposium Transport Processes in Stat. point if the temperature is not within (T—TC)TC Mech., Brussels, 1956, p. 365. [10] S. N. Naldrett and O. Maass, Can. J. Research 18B (1940). = 1 percent of the critical temperature. Whether 322. an anomalous behavior of the viscosity occurs at [11] P. Phillips, Proc. Roy. Soc. (London) A87 (1912), 48. [12] J. Ross, private communication. temperatures closer to the critical point is not clear [13] E. Schroer and G. Becker, Z. physik. Chem. A173 (1935), at present. If such an anomaly does occur it is 178. [14] K. E. Starling, B. E. Eakin, J. P. Dolan and R. T. Ellington, probably smaller than 20 percent. Progress Intern. Research Thermodynamic Transport 2. Measurement with the parallel plate method Properties, A.S.M.E., Princeton, 1962, p. 530. K. E. Starling, Critical region viscosity behavior of ethane, as well as with the concentric cylinder method propane and ^-butane, Master thesis, Institute of Gas have shown that the thermal conductivity exhibits Technology, Chicago, 1960. [15] E. Warburg and L. von Babo, Ann. Phys. Chem. 253 (1882), a pronounced anomaly near the liquid-vapor critical 390. point. Further experimental work is required to study the precise character of the anomaly and to Thermal Conductivity of

~~investigate whether the anomaly of A. is related to One-Component Fluids~~

specific c . the anomaly of the heat v [16] A. K. Abas-Zade. Doklady Akad. Nauk SSSR 68 (1949) 665; 99 (1954) 227. A. K. Abas-Zade and 3. The viscosity of binary liquid mixtures shows A. M. Amuras- lanov. Zh. Fiz. Khim. 31 (1957) 1459. an anomalous increase near the critical mixing [17] Kh. I. Amirkhanov and A. P. Adamov. Teploenergetika 10, No. 7 (1963) 77: Primenenie Ultraakustiki point. For simple systems this increase is between k Issle- dovan Veschestva 18 (1963) 65. Kh. I. Amirkhanov. 15 percent and 25 percent at the critical point. A. P. Adamov and L. N. Levina. Teplo i Massoperenos. Pervoe Vses. Svesch.. For a number of liquid mixtures, however, this Minsk 1 (1961) 105. [18] E. Borovik. Zh. eksp. teor. Fiz. U.S.S.R. 19 (1949) 561. phenomenon is only observed at temperatures [19] L. A. Guildner. Proc. Nat. Acad. Sci.. 44 (1958) 1149: J. Res. NBS 66A (Plus, and Chem.) No. 4. (1962) 333. 341. i within 2 °C of the critical temperature. [20] L. D. Ikenberry and S. A. Rice. J. Chem. Phys. 39 (1963) 4. It seems evident that the thermal conduc- 1561. tivity of a number of binary liquid systems does [21] A. Kardos. Z. gesamte Kalte-Ind. 41 (1934) 1. 29: Forsch. Gebiete Ingenieurwesens 5 (1934) 14. not show an appreciable anomaly near the critical [22] F. R. Kramer and E. W. Comings, J. Chem. Eng. Data 5 mixing point. (1960) 462. D. E. Leng and E. W. Comings, Ind. En- : Additional experimental information Chem. 49 (1957 ) 2042. J. M. Lenoir. W. A. Junk and is highly desirable, since this assertion can only E. W. Comings. Chem. Eng. Progress 49 (1953) 539. be based on the results of one experimental [23] J. M. Lenoir and E. W. Comings. Chem. Eng. Progress 47 (1951) 223. investigation. [24| A. Michels. A. Botzen. A. S. Friedman and J. V. Sengers. Physica 22 (1956) 121.

A. Michels, J. V. Sengers and L. J. M. van de Klundert, Physica 29 (1963) 149. [25] A. Michels. J. V. Sengers and P. S. van der Gulik. Physica The author is indebted to Dr. H. Ziebland who 28 (1962) 1201. 1216. presented the thermal conductivity data for am- [26] A. Michels and J. V. Sengers. Physica 28 (1962) 1238. [27] D. P. Needham and H. Ziebland, Int. J. Heat Mass Transfer monia shown in figure 7. The author also ac- 8 (1965) 1387. knowledges the stimulating interest of several [28] R. Plank and O. Walger, Forsch. Gebiete Ingenieurwesens 5 (1934) 289. colleagues at the National Bureau of Standards [29] W. Sellschopp, Forsch. Gebiete Ingenieurwesens 5 and in particular that of Dr. M. S. Green. (1934) 162. [30] J. V. Sengers, Thermal conductivity measurements at r elevated gas densities including the critical region, thesis, References van der Waals Laboratory, Amsterdam, 1962. J. V. Sengers and A. Michels, Progress Intern. Research Thermodynamic and Transport Properties. A.S.M.E., Viscosity of One-Component Fluids Princeton, 1962, p. 434. [31] J. V. Sengers, Int. J. Heat Mass Transfer 8 (1965) 1103. [1] M. Brillouin, Legons sur la viscosite des liquides et des [32] H. S. Simon and E. R. G. Eckert. Int. J. Heat Mass Transfer gaz. I., pp. 184-195 (Gauthier-Villars, Paris, 1907). 6 (1963) 681. [2] A. L. Clark, Trans. Roy. Soc. Canada III 9 (1915) 43; 18 [33] D. L. Timrot and V. G. Oskolkova, Izvestia V.T.I. No. 4 (1924) 329; Chem. Revs. 23 (1938) 1. (1949). [3] D. E. Diller, J. Chem. Phys. 42 (1965), 2089. [34] N. B. Vargaftik. Proc. joint Conference Thermodynamic [4] G. P. Flynn, R. V. Hanks, N. A. Lemaire and J. Ross, J. Transport Properties Fluids. London, 1957, p. 211. A Chem. Phys. 38(1963), 154. [35] H. Ziebland and J. T. A. Burton, Brit. J. Appl. Phys. 9 [5] J. Kestin, J. H. Whitelaw and T. F. Zien, Physica 30 (1964), (1958) 52.

(l 161. [36] H. Ziebland and D. P. Needham, Progress Intern. Research [6] S. G. Mason and O. Maass, Can. J. Research 18B (1940), Thermodynamic and Transport Properties, A.S.M.E., 128. Princeton, 1962, p. 441.

~~I- [7] A. Michels, A. Botzen and W. Sch uurman, Physica 20 [37] H. Ziebland, N.P.L. Thermal Conductivity Conference, (1954), 1141. Teddington, England 1964.~~

177 Viscosity of Binary Liquid Mixtures [55] V. A. Osipova, Doklady Akad. Nauk Azerbaidzhan S.S.R. 13 (1957) 3. [38] W. Botch and M. Fixman, J. Chem. Phys. 36 (1962) 310. [56] V. P. Skripov, Conference Critical Phenomena and Fluctu- [39] P. Debye, B. Chu and D. Woermann, J. Polymer Science ations in Solutions, Akad. Nauk SSSR, Moscow, 1960, Al (1963) 249. p. 117. [40] P. Drapier, Bull. Acad. Belg. CI. Sci. 1911, p. 621. [41] M. Fixman, J. Chem. Phys. 36 (1962) 310. [42] M. Fixman, Adv. Chem. Phys. 6 (1964) 175. [43] J. Friedlander, Z. physik. Chem. 38 (1901) 385. Miscellaneous [44] W. Ostwald and H. Malss, KoU. Z. 63 (1933) 61. [45] C. J. Pings, private communication. [57] N. C. Ford and G. B. Benedek, Conference Phenomena [46] T. M. Reed III and T. E. Taylor, J. Phys. Chem. 63 (1959) Neighborhood Critical Points, N.B.S., Washington, D.C., 58. 1965. [47] V. Rothmund, Z. physik. Chem. 63 (1908) 54. [58] H. Kraussold, Forsch. Gebiete Ingenieurwesens 5 (1934) [48] O. Scarpa, Nuovo Cimento (5) 6 (1903) 277; J. Chim. Phys. 186.

2 (1904) 447. [519] I. R. Krichevskii, N. E. Khazanova and L. S. Lesnevskaya, [49] V. K. Semenchenko and E. L. Zorina, Doklady Akad. Nauk Inzh.-Fiz. Zh., Akad. Nauk SSR 5 (1962) 101. SSSR 73 (1950) 331; 80 (1951) 903; Zh. Fiz. Khim. 26 [60] A. Michels, B. Blaisse and C. Michels, Proc. Roy. Soc. (1952) 520. (London) A160 (1937) 358. [50] V. K. Semenchenko and E. L. Zorina, Doklady Akad. Nauk [61] A. Michels and S. R. de Groot, Appl. Scient. Research Al SSSR 84 (1952) 1191. (1948)94. [51] V. P. Solomko and S. M. Smiyun, Dopovidi Akad. Nauk [62] A. Michels and J. C. Strijland, Physica 18 (1952) 613. Ukrains'koi RSR 1961, 649. [63] R. D. Mountain, Rev. Mod. Phys. 38 (1966) 205. (1909) 397; [52] D. E. Tsakalatos, Bull. Soc. Chim. France [4] 5 [64] A. Pellew and R. V. Southwell, Proc. Roy. Soc. (London) Z. Physik. Chem. 68 (1910) 32. A176 (1940) 312. [53] E. L. Zorina and V. K. Semenchenko, Zh. Fiz. Khim. 33 [65] E. H. W. Schmidt and K. Traube, Progress Intern. Research (1959) 523, 961. Thermodynamic Transport Properties, A.S.M.E., Prince- ton, 1962, p. 193. Thermal Conductivity of Binary [66] L. I. Stiel and G. Thodos, Progress Intern. Research Thermodynamic Transport Properties, A.S.M.E., Prince- Liquid Mixtures ton, 1962, p. 352. [67] S. A. Ulybin and S. P. Malyshenko, Advances Thermo- [54] L. G. Gerts and L. P. Filippov, Zh. Fiz. Khim. 30 (1956) physical Properties at Extreme Temperatures and Pres- 2424. sures, A.S.M.E., Purdue Univ., 1965, p. 68.

~~Nuclear Magnetic Resonance Measurements Near the Critical Point of Ethane~~

~~M. Bloom 1 2~~

~~Harvard University, Cambridge, Mass.~~

In this talk I shall discuss the application of nuclear magnetic resonance (NMR) techniques to the measurement of two transport properties of fluids near the critical point. The two transport properties we shall discuss are the nuclear spin- lattice relaxation time T\ and the sell-diffusion constant D of the fluid. Both of these observables T are easy to measure using the NMR pulse tech- 0 2T nique of E. L. Hahn [1]. The Hahn pulse technique: A simple version of FlGURE 1. Drawing of an oscilloscope display amplitude of the detected r-f signal as a function of time for the two-pulse the NMR pulse technique is illustrated sche- nuclear magnetic resonance experiment. Following the pulses at times 0 and t indicated by shaded regions are induction in matically figure 1. Suppose that a time inde- signals of maximum amplitude A(0) and A{t) respectively. The echo has a maximum amplitude A{2t).

1 Alfred P. Sloan Fellow. pendent magnetic field 2 Permanent address: Department of Physios, University of British Columbia, Vancouver, British Columbia, Canada. On sabbatical leave 1964-65 while holding = a John Simon Guggenheim Memorial Fellowship. H H t> k + G z k-GA (1)

~~178 is applied to the nuclear spin system such that the tional degrees of freedom of the system so that one gradient G is small so that can usually write~~

~~Gx, Gz < Ho (2) ^Ra + Rb+Rc (5) 1 1~~

is the relaxation to di- over the entire volume of the sample, i and k Ra rate due intramolecular are unit vectors in the x- and z-directions respec- pole-dipole and/or electric quadrupolar interactions tively. The spin system is assumed to be in equilib- [6]: i.e., those interactions which transform as rium before the first of the two r-f pulses applied F2m (fl,), where fl, is the orientation of a vector

fixed in the i. at times t = 0 and t = r respectively. Assume that molecule R B is the relaxation rate the r-f pulses are sufficiently intense and that their due to dipole-dipole interactions between nuclear frequency is sufficiently close to the Larmor spins on different molecules, i.e., those interactions 3 which vary as Y-> m(0.ij)l r ^, where ry = (ry, fly) is frequency oio = yHo, where y is the nuclear gyro- magnetic ratio, that the lengths of the pulses can the vector joining a pair of spins on different mole- be chosen such that the z-component of magnetiza- cules [7]. Rc is the relaxation rate due to the inter- 77 action between the nuclear spins and the rotational tion is rotated by an angle - by each of the pulses. angular momentum J,- of the same molecule [8] Then, if the z-component of the nuclear magnetiza- (the spin-rotation interaction).

relaxes towards its equilibrium value ex- 1 tion Of the three terms contributing to Tr in eq (5), ponentially with a time constant 7\, the maximum a molecular theory is available only for Rb [7], amplitudes of the detected nuclear induction signals which is expressed in terms of Fourier transforms ,4(0) and A(t) following the pulses are related by [1] of the correlation functions

~~,WT dvdr e g(r, r t) — — dr A(0)-A(t) T J(q))= , p p-3 exp (3) A(0) Ti (6)~~

~~where g(r, r', t) is a time dependent pair distri-~~

If the x and y components of the nuclear mag- bution function for the system; g{r, r', t) drdr' is netization relax towards their equilibrium value of the probability that the vector joining a pair of zero exponentially with a common time constant molecules is between r and r + rfr initially and if the spins in the Ti and the spatial motion of sample between r' and r' + dr' at a time r. For a system solution the dif- is adequately described by the to of identical nuclear spins R B is expressed in terms fusion equation, the maximum amplitude of the of J(o)o) and J{2u>o). Note that typical values of

8 1 "spin-echo" at time 2t is given by [2] O)o are ojo ^ 10 sec^ which is an extremely low 2t_2 frequency insofar as the dynamical motions of a 2 A(2t) exp y GW (4) fluids are concerned. The other important point T2 3 to notice about eq (6) is that J (to) gets its major

Equation (4) correctly gives relative values of A(2t) contribution from pairs of molecules which are in terms of To, G, D and r for any given angles of very close together, because of the r~ 3 dependence rotation by the r-f pulses [3]. For systems with of the dipole-dipole interaction. strong NMR signals, it is easy to measure T\ and Physically, one can say that R B gives a measure D to an accuracy of ± 5 percent and with care an of a specific type of inelastic scattering cross sec- accuracy of ± 1 percent can be attained. tion corresponding to collisions between pairs of Note that the self-diffusion constant measured molecules in which the total nuclear Zeeman energy here is the "spin self-diffusion constant." The is changed. The low frequency o>o corresponds to interpretation of this type of self-diffusion constant the fact that the energy exchanged between the has been given by Emery [4] and by Fukuda and spin system and the translational degrees of free- 7 Kubo [5]. dom is very small (< 10" eV). Interpretation of T\ measurements: For most There are no detailed molecular theories for simple systems there are three main types of inter- Ra and R c in dense fluids, except for liquid H 2 [9], actions which enable the nuclear spin system to and this has inhibited the interpretation of the many exchange energy with the translational and rota- Ti measurements already performed on liquids

~~179~~

209-493 0-66— 13 in molecular systems. Ra and R ( are related to sample was in thermodynamic equilibrium near low-frequency Fourier transforms of correlation the critical point. He usually kept the sample at functions of Y-z„,{fli(t)) and J,-(f). Since the inter- constant temperature for 24 hrs before making T\ actions responsible for molecular reorientation are measurements and then made a second set of meas- the anisotropic intermolecular interactions, the urements after waiting for an additional 24 hrs. correlation functions of Y2m (Cli(t)) and J,-(f) are each The temperature gradient across the sample was related to correlation functions of these (short less than 0.004 °C. range) anisotropic interactions in a manner which As may be seen from figure 2, Noble's critically has been defined for one system [9], that of H 2 . loaded samples were in vertically mounted glass

It is very likely, in the light of the theory for H 2 , tubes and he made measurements on two regions, that R A and Rc can be expressed in terms of Fourier transforms of correlation functions similar to that 3 of eq (6), for interactions of shorter range than r~ . The behavior of T\ near the critical point; From the preceding discussion, we can see that the quan-

~~tity of interest in the interpretation of T\ is g(r, r', t) COPPER CYLINDER at short range, i.e., for r, r' gE Ro, the range of the miermolecular interactions. Defining j(r, r', w) by~~

~~the quantity of interest in the low-frequency limit~~

~~is y(r, r', 0), i.e., the area under the time dependent~~

pair distribution function for r, r' £; R 0 . This prob- VAPOR COIL lem is analogous to the calculation of the area under the correlation function for relative orientations of neighboring spins near the ferromagnetic Curie temperature. The interpretation of T\ near the critical point may be compared with the inelastic 3" scattering of neutrons, which is dominated by the long range correlations. Unlike the short range

~~parts of ,Mr, r', r) of interest here, the long range parts can be studied explicitly near the critical point by the methods of macroscopic fluctuation~~

~~theory [10]. Experimental results for T\: The simplest type~~

of system in which to investigate T x near the critical LIQUID COIL point is a system such as He ;i in which Ra = R( =0. However, because of the smallness of Rh near the critical point, special precautions would have to be taken to eliminate spin relaxation due to collisions with the walls of the container and such a study has

~~not yet been carried out. For all the systems studied near the critical point thus far Ra + Rc ^> Rb- The first study of 7\ near the critical point was~~

~~conducted on SF 6 by Schwartz [11]. Since then several substances have been studied including 6 mm SAMPLE HOLDER benzene and some of its derivatives [12], HC1~~

~~and HBr [13], and ethane [14] (C 2 H6 ). The results obtained are qualitatively similar for all the cases~~

~~studied, but the study of ethane by Noble is the only FIGURE 2. Drawing of sample holder used by J. D. Noble show one in which care was taken to ensure that the ing location of r-f coils.~~

~~180 . I~~

one in the lower half and the other in the upper a wide range of temperatures [14]. Figure 4 shows half of the tube so that none of his measurements T\ for the sample in each of the two coils in the actually corresponds to the critical point. The vicinity of 7*c- amplitude of the NMR signal for each of the coils Experimental results for the diffusion constant D: is proportional to the density of the sample within Noble has also measured the diffusion constant in the coil. The effect of gravity on the ratio of densi- ethane with approximately 1 percent 0 2 impurity ties in the two coils PlIpv, corresponding to the ratio added [14, 16] (see fig. 3) The O2 was added to of the liquid to vapor densities below the critical reduce T\ and thereby enable more accurate results temperature Tc, is important as may be seen from to be obtained for D since the experiment could be figure 3. pdpv in the pure sample is compared with repeated more often. The measurements of D as those obtained by other techniques [15]. The a function of temperature near Tc are shown in measurements in the impure sample will be used figure 5. The estimated maximum error in the later in interpreting the measurements of D made measurements is ±5 percent including systematic in that sample.

It should be noted that for the critically loaded samples, the meniscus separating the liquid and solid phases below Tc is observed to remain in a fixed position midway up the sample tube in the region of temperature not too far below Tc- Thus, in this temperature range, one can say that pt + pv 555 2pc independent of temperature, where 3 pc = 0.207 g/cm is the critical density. Therefore, the measurement of PlIpv is sufficient to determine the densities p/, and pv in the region of temperature in which the meniscus is constant. It was found that for the region T < Tc, the value of Ti obtained for the liquid was^ always longer than that obtained for the vapor in equilibrium with the 10 t_—l—I—I—!—I——l-J—I—1—1—I—1—1—— 0 10 20 30 40 50 60 liquid. This indicates that the time required for TEMPERATURE CO molecular reorientation in the liquid is always Figure 4. Ti as a function temperature in a pure sample shorter than for the less dense gas. The results of of ethane which was critically loaded and mounted vertically are qualitatively similar to those obtained for other Values are shown for the region of temperature near Tc for nuclei in the upper and lower parts of the sample tube respectively. systems [11, 12, 13] and have been obtained over

~~A 18 A VAPOR SELF -DIFFUSION IN ETHANE~~

~~A o a a~~

•

~~• • • • LIQUID COIL LIQUID VAPOR COIL~~

~~, i , i , , J , i , . . , i_ T 31 32 c 33 10 20 30 40 50 60 TEMPERATURE CO TEMPERATURE PC)~~

Figure 3. The ratio pjpv of the average density of the ethane within the lower coil to the average density within the upper FIGURE 5. Self-diffusion constant D as a function of tempera- coil as measured by the relative amplitudes of the NMR signals ture in ethane with approximately 1 percent 0> added as an using each of the coils. impurity. Measurements are shown lor pure ethane and for ethane with 1 percent 0 2 added. The sample was critically loaded and mounted vertically. Values of D are shown The measurements for pure ethane are in good agreement with those of Palmer (13]. for nuclei in the upper and lower parts of the sample tube respectively.

~~181 TEMPERATURE <°K) L//> Experimental 100 150 200 250 300 350 v 5.2~~

~~4.8~~

~~o Dota taken from reference 6/ 4.4 • present work~~

~~| 4.0- E u i o Tjj o VAPOR~~

~~o'- < fi • LIQUID •q 3.2- 1.15 cji A -CRITICAL SAMPLES x NON «. -12 ! =8.02xl0 2 2.8 / SOLID LINE ^29 c 1.10-~~

~~2.0~~

~~25 27 29 31 33 35 37 39 TEMPERATURE CK) 1.6~~

7. FIGURE Plot of Dp versus temperature nearTc . Measured values of p/./pr are also shown. The samples used were loaded to the critical density. Measurements of D„ based on the slopes of \ID versus p plots at 32 and 40 °C using noncritically loaded samples are also shown. The dotted line at

31.9 °C labeled 7Y is an upper limit for the critical temperature since it is the lowest temperature at which p/./pr = I, within experimental error. Figure 6. Plot of\n (Dp) versus In T. The points below 0 °C are based on the smooth curve drawn through the diffusion constant data of reference 17. of figures 6 and 7 is that whereas, for systems not too close to the region of the critical point Dp is errors, while the measurements of p/Jpv are accu- a function of temperature only, the diffusion con- rate to ±2 percent. stant decreases anomalously near the critical point.

Interpretation: Experiments performed at 32 In figures 6 and 7, TV for the impure sample and 40 °C in noncritically loaded samples showed used has been nominally given as 31.9 °C as that °c p" 1 °K. actually represents D in the range pc/2 < p < 2pc . Thus compared with 32.32 This

it Dp is a function of temperature only in this density an upper limit for Tt since is the temperature at range far from the critical point. A plot of In (Dp) which pijp\=\.0 within experimental error. versus In T from just above the melting point at Since we do not know the equation of state for the

~~89.9 to 333 °K approximately 30 °K above the criti- impure sample, we cannot take into account the~~

cal point, is shown in figure 6. The results of influence of gravity in a reliable fashion, but an Gaven, Stockmayer, and Waugh [17] far from the examination of figure 3 indicates that the true critical point are included. The resulting straight value of 7c (or region of 7V) is probably close to the

2 - 9±0A line gives Dp T . This is an empirical temperature at which Dp goes through its mini- relationship which does not necessarily hold for mum value. other substances. Figure 7 gives a plot of Dp As far as we know these are the first measure versus T in the temperature region within a few ments of a self diffusion constant near the critical degrees of 7c. A comparison with the results of point. Dr. B. Jacrot has, however, drawn our figure 6, as given by the solid line of figure 7, shows attention to some published measurements of the

that Dp goes through a pronounced minimum for the diffusion constant of Iodine in CGv near the critical ! liquid near TV with its minimum value about 20 point [18], where it is found that the diffusion con-I percent below the solid line. A similar but smaller stant is practically zero. This is consistent with| effect is observed for the vapor but, unlike the data the anomalous decrease observed by Noble for[ for the liquid, we cannot say definitely that the Dp for ethane near 7V as shown by figures 6 and 7. vapor minimum is not due to a systematic error, Obviously, it would be desirable to make furtherl which we estimate to be less than 5 percent. A measurements of D near the critical point for single measurement of D for a pure sample at 24 °C, samples having better geometry (flat, horizontal having an error of ± 10 percent agrees with D for samples) and a higher degree of purity, e.g., SFb- the impure sample. would be a good system to study since the relaxa-i

~~The interpretation which we propose for the data tion time T\ of pure SFh is reasonably short [ll].l~~

182 References 1 10] L. van Hove. Phys. Rev. 95, 249 (1954). (11] J. Schwartz, Ph. D. Thesis, Harvard University, (1958), unpublished. [1] E. L. Hahn, Phys. Rev. 80, 580 (1950). D. K. Green Powles, Proc. Phys. Soc. 87 [2] H. Y. Carr and E. M. Purcell. Phys. Rev. 94, 630 (1954). [12] and J. G. 85, [3] B. H. Muller and M. Bloom, Can. J. Phys. 38, 1318 (1960). (1965). S. Blicharski K. Krynicki, Phys. Pol. 409 [4] V. J. Emery, Phys. Rev. 133, A661 (1964). [13] J. and Acta 22, K. Krynicki Phys. Letters [5] E. Fukuda and R. Kubo. Prog. Theor. Phys. 29, 621 (1963). (1962); and J. G. Powles, 4, (1963). [6] A. Abragam. The Principles of Nuclear Magnetism, pp. 298- 260 300. 313-315 (Oxford University Press, 1961). [14] J. D. Noble, Ph. D. Thesis, University of British Columbia [7] H. C. Torrey, Phys. Rev. 92, 962 (1953): M. Eisenstadt (1965). and A. G. Redfield, Phys. Rev. 132, 635 (1963): reference [15] H. B. Palmer, J. Chem. Phys. 22, 625 (1954). 6, pp. 300-305, 458-466: I. Oppenheim and M. Bloom, [16] J. D. Noble and M. Bloom, Phys. Rev. Letters 14, 250 Can. J. Phys. 39, 845 (1961). (1965). .1. V. W. H. S. .1. Chem. [8] P. S. Hubbard. Phys. Rev. 131, 1155 (1963). [17] Gaven. Stockmayer and J. Waugh, [9] T. Moriya and K. Motizuki, Prog. Theor. Phys. 18, 183 Phys. 37, 1188, (1962). (1957): T. Moriya, Prog. Theor. Phys. 18, 567 (1957): [18] I. R. Krichevskii, N. E. Khazanova and L. R. Linshitz, (in M. Bloom and I. Oppenheim. Can. J. Phys. 41. 1580 Dnklady Acad. Nauk. SSSR 141, 397 (1961) Russian). (1963).

~~Ultrasonic Investigation of Fluid System in the Neighborhood of Critical Points~~

~~D. Sette~~

~~Instituto di Fisica, Facolta' Ingegneria, University of Rome, Rome, Italy~~

~~1. Introduction procedure will require some repetition. We consider first the case of liquid-vapor critical~~

It is the intention of this paper to examine the media. research that has been so far carried out in fluid critical systems by means of sound waves. Since 2. Liquid-Vapor Critical Systems 1940 several authors have attempted to study the propagation of sound waves in order to find infor- 1 Table I, although it is not complete, indicates the nature the of these mation on and behavior the substances which have been the object of the particular media: various indications and sug- more extensive experiments. Together with pure gestions have been obtained, which have con- liquids, few binary or ternary mixtures have been tributed to a better definition of the problems investigated in the hquid-vapor critical region. concerning critical systems. The solution of these The experiments have been performed either at a problems is however far from complete. single frequency or in a rather limited frequency The quantities which are usually measured are range. Moreover velocity and absorption coeffi- the velocity and the sound absorption coefficient: cient determinations have usually not been made on they may be determined as functions of tempera- the same system. ture, frequency and, in the case of mixtures, of Figure 1 gives the results of Tielsch and Tanne- composition. It must be observed that the infor- berger [5] for the sound velocity in gaseous CO2 mation available at present for the systems in which as a function of pressure at constant temperature. sound measurement have been performed, is only In the examined frequency range (0.4-1.2 Mc/s) partial because it usually refers to only one of these and in the limits of error (0.2%) no dispersion has quantities (i.e., velocity or absorption coefficient) been found. The same conclusion was reached by determined in a limited range of one of the vari- Parbrook and Richardson [3] in the range 0.5-2 ables, temperature, frequency, and composition. Mc/s. The velocity goes through a minimum which Strong similarities exist between liquid-vapor becomes sharper and sharper as the temperature and liquid-liquid critical systems as well as some approaches the critical one: the minimum velocity dissimilarities. It is preferable for a clearer ex- position to review separately the work made on 1 In particular much research of Russian authors is included in thesis not easily each ol the two types of systems, though this available: some results are reported in quotation [9).

~~183 Table I. Sound Propagation in Liquids-Vapor Critical Systems~~

~~Substance Temperature Frequency Quantity Author measured~~

~~Helium 5.1994 °K 1718 mm Hg 7V-0.03 p-40 1 Mcjs Chase-William- 7V + 0.04 °K p c — 35 mm Hg son-Tisza [1]. Xenon 16.74 °C 58.2 atm 15-19 °C 0.25-1.25 Chynoweth-~~

~~Schneider [2]. Carbon dioxide 31.4 °C •73 atm 19-38 °C up to 100 atm 0.5-2 Parbrook-~~

~~Richardson [3]. 28-38 °C 5-98 atm 0.270 Herget [4). 25-48 °C 35-120 atm 0.410 Tielsch-~~

Tannaberger [5|. Ethylene 9.7 °C 50.9 atm 7-18.7 °C up to 100 atm 0.5-2 Parbrook- 0.27-0.6 Richardson [3j. 9.7-23 °C 35-75 atm 0.27-0.6 Herget [4]. Hydrogen 51.4 °C 81.6 atm 38-62 °C 1.9 Breazeole [6]. chloride 52-62 °C 1-9 Breazeole.

N 2 0 Nitrous 36.5 °C 71.7 32-37 °C 55-150 atm 0.96 Noury [7]. oxide n-hexane 234.8 29.5 atm along the saturation curve 2-3 Nozdrev [8]. rc-heptane 266.8 °C 26.8 atm 2-3 Nozdrev. Methyl acetate 233.7 °C 46.3 atm along the saturation curve 2-3 Nozdrev. 5-13 Kal'ianov [10]. Nozdrev [9]. Ethyl acetate 250 °C 37.8 atm along the saturation curve 2-3 Nozdrev [8]. 5-9 Nozdrev- Sobolev [11]. atm Propyl acetate 276.2 °C 32.9 along the saturation curve 2-3 Nozdrev [8]. Methyl alcohol 240-240.5 78.7 atm along the saturation curve 2-3 Nozdrev. Ethyl alcohol 243.1 63.1 atm along the saturation curve 2-3 Nozdrev. Isobuthyl 283.4 along the saturation curve 2-3 Nozdrev. alcohol Sulphur hexa- 45.6 °C 36 atm 0.6 c, a Schneider [12]. fluoride

Water 374 °C 217.5 atm. along the saturation curve Nozdrev- Osadchii- Rubtsov [13]. Benzene 239 °C for 10% along the saturation curve Nozdrev- methyl alcohol benzene Taraitova [14]. Benzene methyl- 240-320 °C along the saturation curve Grechkin- alcohol-toluenq Nozdrev [15].

184 measured at 31 °C is about one half of the value in normal conditions. Figures 2 and 3 give Schneider's results (600 kc/s) on velocity and absorption in sulfur hexafluoride, a nonpolar compound with spherically sym-

metric molecules [12]: the data are in this case available for the liquid phase as well as for the vapor 2 on both sides of critical temperature {Tc ). The velocity in the liquid and in the gas seems, in the limits of experimental errors, to reach a common minimum at a temperature which coincides with

Tc ; the absorption coefficient in both phases in- 3 70 80 90 creases very rapidly approaching Tc . The last PRESSURE (ATM) circumstance has been observed in all systems examined and the various researchers have not FIGURE 1. Sound velocity in C0» versus pressure, at constant temperature succeeded in making a measurement at Tc . (H. Tielsh and H. Tanneberger [5]). Figures 4 and 5 give the results of Chynoweth

~~and Schneider in xenon [2]. The absorption co-~~

~~efficient was measured at 250 kc/s and it shows a behavior similar to that observed in sulfur hexa- 200~~

~~.LIQUID PHASE fluoride when T approaches Tc . The velocity instead was measured in the range 250 kc/s- 1.250 5 180 Mc/s: a dispersion has been detected whose magni-~~

>- tude is largely higher than the experimental error t 160 o o SUPERCRITICAL GAS (0.2-^-0.7%). We will discuss the inferences of _l UJ > this result shortly. At present we call attention o 140 z to the fact that Chynoweth and Schneider have not o <" 120 noticed any tendency of velocity towards a discontinuity or a sudden drop to zero when T approaches

~~- Until recently, this 44 46 48 Tc has been a general con- TEMPERATURE, °C clusion of experiments. Nozdrev [8], for instance, who has made velocity measurements using light~~

~~FIGURE 2. Sound velocity for the liquid and the vapor phase diffraction by sound waves, claims to have been~~

~~of sulfur hexafluoride (W . G. Schneider [12]). able to visually follow the continuous change of sound velocity through the critical point. The fact that the velocity of sound remains finite~~

~~at the critical point is in agreement with standard~~

~~thermodynamics [16]. In this state, the derivative of pressure respect to volume at constant tempera- LIQUID PHASE ( dP\ =0. ,0.4 ture vanishes: | — I Consequently, the spe- av~~

~~VAPOUR PHASE cific heat at constant pressure {Cp ) is infinite while~~

Cv , the adiabatic compressibility and the sound velocity stay finite. o0.Z It may be of some interest, at this point, to mention some results that have been obtained by Noz- °° Si r, drev [8, 9] who has performed extensive velocity

0.0 2 in figure correspond lo measurements in 42 44 46 50 The full and broken lines above Tc 3 the respectively near the sound source and far from it. TEMPERATURE, °C supercritical gas performed 3 The velocity and absorption curves for liquid and vapor as obtained by Schneider

cross at about 0.6 °C below Tc . This fact, which has no simple explanation, has not Figure 3. Sound absorption for the liquid and the vapor been found in any other case although some systems have been purposely investi- phase of sulfur hexafluoride (W. G. Schneider [12]). gated [8].

185 associations may persist to the critical \ up point. Moreover, in the absence of dispersion and in the limits of experimental errors, Nozdrev establishes an empirical simple relation between the arith- metical average of sound velocities in the gas

~~(c,) and in the liquid (c/,) and (T—Tr )~~

~~= ^^ cCT +a(Tc -T) (1)~~

~~being c cr the common velocity at Tc and a a constant. This relation would be valid in the range between~~

~~Tc and a temperature 15 to 30 °C below. The standard thermodynamical treatment of critical state from which the conclusion on C,„~~

C v , and c have been deduced, derives from some T EMPERATURE (°C) assumptions on the behavior of thermodynamical functions 4 Fi(;i'RE 4. Sound velocity versus temperature in xenon (A. G. near the critical point: in particular Chynqweth and W. G. Schneider [2 J). it is assumed that the thermodynamical quantities (as function of V and T) do not show mathematical singularities along the curve which limits the region in which the substance can not exist as a homogeneous phase either in stable or in metastable equilibrium. Only in such case the above-

~~mentioned limit curve is determined by (tt~) =0 at \BvJT the critical point alone. The truth of this treatment has been seriously questioned recently. Some experimental results~~

on C v in argon [17] and in oxygen [18] have been obtained which indicate a logarithmic temperature dependence of such a specific heat near the critical point. If this behavior remains unchanged up to

~~the critical point, C,- and the adiabatic compressi-~~

~~bility would be °° and c = 0 at T=TC [19]. To obtain further information, Chase, William-~~

son, and Tisza [1] have performed sound velocity measurements in helium much more carefully 15 0 16.0 17.0 18.0 19,0 and precisely than used in previous investigation TEMPERATURE i'C) near the critical point. The temperature was reg- 4 3 FIGURE 5. Absorption per wavelength (a -A) versus temperature ulated to ± 10~ °K (instead of ±2.10 °K in Chyno-j in xenon (A. G. Chynoweth and W. G. Schneider [2]). weth and Schneider measurements in Xenon), \ the pressure to ±0.1 mm Hg; the velocity measure- measurement in critical fluids along the saturation ments were performed with a phase sensitive curve. He has found, in the limits of error of his method having a resolution of 0.01 percent (the instrument, that the velocity variation along the error in the Chynoweth-Schneider setup was ±0.2 saturation curve in both vapor and liquid near the — 0.7%). Figure 6 gives the results obtained fori critical temperature can fairly well be expressed the velocity both as a function of temperature (atj by using van der Waals equation of state in the case a pressure />=1718±1 mm Hg practically coinci-l of organic unassociated substances. This conclusion is not true for the liquid phase of associated liquids (e.g., ethyl alcohol) which proves that ' Quotation [16] p. 262.

186 from zero. The possibility of a leveling off at a much lower but finite value has not yet been excluded. The behavior of velocity, C, and adiabatic compressibility at the critical point remains a very interesting and exciting problem, which may clarity in depth the nature of systems in the critical state:

~~it requires further experimental as well as theoretical research. Closely related to the nature of liquid-vapor~~

~~critical systems is also the enormous absorption that these media show. Two mechanisms have been proposed: 5 scattering and relaxation.~~

~~A. G. Chynoweth and W. G. Schneider [2] have studied sound propagation in xenon in the hope of~~

finding out which is the more important dissipative process. On the basis of an approximate calculation of scattering losses produced by inhomogeneities due to density fluctuations in the medium, they conclude that these losses do not appear sufficiently -40 -30 -20 -10 0 10 20 30 40 large to explain the experiment. They believe P-Pc (mmHg) therefore that, as Schneider had already suggested

FIGURE 6. Sound velocity as a function of temperature (a) and in the case of sulfur hexafluoride, the more impor- in the critical region helium (C. E. Chase. R. C. pressure (b) of tant dissipative mechanism is structural relaxation Williamson, and L. Tisza [1]). caused by sound-induced perturbation of equilibrium among molecular aggregates (clusters) present (at dent with pc ) and as a function of pressure tem- in the medium. The sound dispersion found in perature 5.200 ±0.0002 °K practically coincident xenon (fig. 4) gives support to this explanation of with rc = 5.1994 °K). The velocity decreases rap- high absorption. The relaxation phenomena would idly approaching the critical point. High absorption be characterized by a distribution of relaxation has not allowed measurements over a region of times. about 2.10~ 3 °K or 2.5 mm Hg. Different conclusions were reached by H. D. The fast decrease of c approaching the critical Parbrook and E. G. Richardson [3] studying CO> point in the range of temperature or pressure and by M. A. Breazeale [6] in hydrogen chloride. examined makes Chase, Williamson, and Tisza The last author was unable to detect dispersion in think the limit zero for c at Tc to be possible. The the range 1 to 3 Mc/s; the absorption results could adiabatic compressibility Ks as a function at T or p be interpreted as produced by relaxation processes has been calculated making suitable approxima- with a wide distribution of relaxation times, but it tions for density: in the range examined, Ks seems seems also possible to explain them on the basis to have a logarithmic singularity similar to that of a scattering mechanism if correlations between found for C v in argon and oxygen. density fluctuations in adjacent volumes are taken This recent research, having questioned the into account. To evaluate the scattering losses, standard thermodynamical treatment of critical Breazeale, as already Chynoweth and Schneider states, opens a new exciting field of investigation. had indicated, chooses the Libermann-Chernow It must be observed, however, that the experimental [20, 21] formula, which was developed to deal with results so far obtained, although they have caused sound absorption caused by temperature inhomoge- serious doubts on the exactness of traditional ther- neities in the ocean. In the calculation of this modynamical treatment have not yet definitely formula the validity of an exponential autocorrela- proved it wrong. The measurement of c in helium for instance, shows a fast decrease of velocity

5 -3 It seems that losses due to finite amplitude id* waves cannot have importance, as when T approaches Tc , but at a separation of 10 °K experiments have shown that the signal detected by a sound receiver depends expo- from Tc the velocity has still a value quite different nentially on the source-receiver distance.

187 tion function is assumed. The absorption coefficient from the Libermann-Chernow curve, which essen- 4 i> due to scattering by inhomogeneities of diameter tially corresponds to a dependence of a from , a is could be explained by using an adequate correlation between density fluctuations in neighboring k4 a 3 ot : volumes. Breazeale is therefore of the opinion s 2 (2) 1 + 4a F that, while relaxation may be important at lower

frequencies, scattering is most likely the main where k=~z~ and — is the fractional change of source of absorption in the range 1 to 9 Mc/s. k c sound velocity. In order to make a numerical 4 calculation a was chosen 10" cm, i.e., approxi- 3. Mixtures of Partially Soluble mately 103 molecular diameters as suggested by Liquids light scattering in various substances near the critical point; moreover, the value 0.25 was used for Table II indicates the few systems which have Ac been studied in some way in the critical region of Figure 7 gives Breazeale's experimental results c solubility. Almost all research has been limited at various temperatures; the flags indicate the limits in each system to only one mixture with a composi- of errors. These results seem to depend upon the tion close to that of the maximum (or minimum) of square of the frequency (dotted lines). The solid the solubility curve. line on the same figure gives as calculated from (2) Figure 8 is relative to a mixture having 47.6 wt

Ac percent of n-hexane in aniline [2]. The velocity with above mentioned values of a and — : the correct c was measured at 600 kc/s. The values of c in order of magnitude of a is obtained; according to the two separate phases approach each other as

~~Breazeale the deviation of the experimental curves T tends to Tc . On this basis, Chynoweth and Schneider formulated the tentative conclusions that the velocity in the three phases gradually~~

tends to the same value as T approaches Tc and that no anomalous behavior is present. A dif- 52°C ferent indication, however, emerges from the work of M. Cevolani and S. Petralia [23] in the aniline- cyclohexane mixture (49 wt % of cyclohexane)

~~(fig. 9).~~

~~More research is. needed to determine the exact behavior of velocity in the three phases in the~~

immediate proximity of Tc ; the use of apparatus today available with smaller errors than those so far used in this kind of experiment should allow precise information. No great effort, so far, has been made to investigate dispersion in the critical region of solubility and the few indications available [26] do not allow to reach any conclusion on the subject. A common experimental finding is the high attenuation of sound waves in the critical region. Figure 10 [22] shows the typical behavior of the absorption coefficient per wavelength (a k) as a function of temperature through the critical region

of solubility: it refers to the triethylamine-water 4 6 8 10 wt of amine) at 600 kc/s. Figure FREQUENCY (Mc/s) system (44. 6 % a . — lor mixtures n- Figure 7. Absorption coefficient versus frequency at different 11 [25] gives the parameter temperature in the super critical region of hydrogen chloride composition; (M. A. Breazeale [6]). hexane-nitrobenzene as a function of

~~188 TABLE II. Sound Propagation in Mixtures of Partially Soluble Liquids~~

~~System Composition Tc Quantity Temperature Frequency Author measured range range~~

°C °C Mc/s Aniline 47.6% w rc-hex 68.3 c, a 55-74 0.6 Chynoweth- rc-hexane Schneider [22]. Aniline 51% w an. c, a 15-55 3 Cevolani- cyclohexane Petralia [23]. Do 20-63% mole an 30.7 c, a scattering 29-34 1.5-5 Brown- Richardson [24]. Nitrobenzene 0-100 21.02 °C a 25 8 Sette [25]. n-hexane 53.2 w% nitrob. 23.2 23-28 1-9 Alfrey- Schneider [24]. Water- 44.6 w% am. 17.9 c, a 10-28 0. 6 Chynoweth- triethyla- Schneider [22]. mine 44.6 w% am. a 15 7-54 Sette [25]. 34 w% am. 19.5 10-20 1-9 Alfrey- Schneider [24]. Water- 34 w% Ph. 66 c, a 51-80 3 Cevolani- phonol Petralia [23].

~~—i 1 1 1 I I u I I L 56 58 60 6 2 64 66 68 70 72 74 TEMPERATURE CO~~

~~FIGURE 8. Sound velocity through the critical region of the aniline-hexane system~~

~~(A. G. Chynoweth and W . G. Schneider [2]).~~

~~189 —~~

FIGURE 9. Sound velocity through the critical region of the FIGURE 10. Sound absorption per wavelength (oc-k) versus aniline-cyclohexane system (M. Cevolani and S. Petralia [23]). temperature for the water-triethylamine system {A. G.

~~Chynoweth and W . G. Schneider [22]).~~

~~aou n-HEXANE - NITROBENZENE~~

~~600 absorption~~

~~ility~~

~~200 — () T~~

~~0.2 0.4 0.6 0.8 1.0 Mol. fraction nitrobenzene~~

~~FIGURE 11. Sound absorption versus composition and coexistence curve for n hexane- nitrobenzene mixtures (D. Sette [25]).~~

the experiment was performed at 25 °C, i.e., at a indicates an essential difference between the temperature 4 °C higher than the consolute tem- two types of systems. perature. The comparison of the results in figures Various attempts have been made to find out 10 and 11 with those obtained in the liquid-vapor what the process responsible for the largest part critical region (figs. 3 and 5) shows that in the criti- of losses in these systems is. The dissipative cal solubility case the processes which give rise causes which have been examined are viscosity, to additional losses operate in a wider range of relaxation, and scattering. Let us briefly review temperature around the critical one. This fact the research made on this subject.

~~190 P~~

140 3.1. Viscosity and Thermal Conductivity 1» / iTE:r TRIETHYLAr^ INE Different from the case of liquid-vapor critical CHt NC WE rH -sCHNEIC ER media, the viscosity coefficient (17) of mixtures 100 formed by partially soluble liquids undergoes a large increase when temperature approaches the critical one. However, the classical losses due to this fact, i.e., calculated with Stokes' formula for an homogeneous medium, would be still inadequate to explain the experiment: the total viscosity absorption coefficient would amount to 20 a few per mille of the observed value. ** 1 1 1 I 1 I I I — —U 1 I I l_l r* —«

The fluid however is not homogenous. R. Lucas 0.1 I 10 100 v (Mc/s) [27] has calculated the sound absorption coefficient when density fluctuations are present in the liquid. FIGURE 12. a/v 2 versus frequency far a water-triethylamine Lucas assumes that the density fluctuations are mixture at 15 °C (D. Sette [25]). time independent and isotropic and that the sound velocity does not vary along the direction of propagation. Moreover he maintains Stokes' relation erally found in these kinds of mixtures; figure 12 between the two viscosity coefficients. The ab- [25] gives some results for a mixture water-triethyl- sorption coefficient then results amine with 44.6 wt percent of amine. Unfortu-

ca nately, however, a similar decrease of — with 2 a$ = Av + B (3) v frequency can be expected also in the case of other where dissipative effects, mainly of the relaxation type. A calculation using the Lucas' formula has been carried out for the water triethylamine mixture

3 for which data on the thermodynamic behavior 3 poc \ tJo / of the system are available. Rough approximations were introduced; the results seem to indicate that, though the viscosity losses in the y PoC T Jo \dxj however inhomogenous medium are higher than those given by Stokes' formula and may become of some and po the average density; p the local density; importance at low frequencies, they are still in- m = —; s, the condensation, given by p = po (1 + s); sufficient to explain the largest part of the observed P attenuation. di—dxdydz the volume element of the liquid. A Another cause of losses is the existence of thermal and B depend upon the properties of the liquid conductivity. The corresponding contribution to and the density fluctuations in it, but they are absorption coefficient in normal liquids is much frequency-independent. As a consequence of the smaller than the viscosity term. Although thermal presence of density fluctuations, the expression conductivity determinations in critical region are of the absorption coefficient has two corrective missing in the literature, it seems unlikely that terms for Stokes' expression thermal conductivity substantially contributes to the large absorption found in the critical media. 8772 V ar> = — ~ (4) 3 poC* 3.2. Scattering

The first corrective term has the same frequency de- A rough description of the medium is obtained pendence of the Stokes' term, the second one is con- by thinking of the system as formed by a mother phase in which clusters of various sizes and com- stant with frequency. The parameter — therefore positions are dispersed. The presence of in- decreases with frequency. Such behavior is gen- homogeneities may lead, as in the case of liquid-

191 vapor critical systems, to scattering of sound 22. S °/s ANILINE 33.3 °/0 ANILINE energy. On the importance of this process the i o 1° opinions do not entirely agree. X 28. • 29.2° The dimensions of individual clusters as deter- o 30.0° 9° fl 3 I • mined by light scattering experiments are of the 33. 1° -6 order of wavelength of visible light (0, 5T0 m), i.e., 0-6 much smaller than the sound wavelength (~ 10" 3 m at 1 Mc/s) in the frequency range of experiments: 0.4 if the clusters acted as individual scatters the contribution of scattering to sound energy loss would Q.2 be very small. It is however necessary to assume, as in the case of liquid-vapor critical systems, that there actually are strong correlations between 43.8 °/ ANILINE 53-9 °/0 ANILINE fluctuations in adjacent volumes. If one considers c 7 29.3° x 29-7° theory exponential 0 Liebermann's in which an 08 a 30.S • 30.9° o 31.8° o 33.3° autorrelation is assumed one finds a dependence 3° • 33. fi 36-3° X 34.7° of a on frequency which is not much different from 0.6 proportionality to the fourth power of v, i.e., 0-4 should increase with frequency. This is contrary to some experiments where this parameter 02 has been found decreasing within the range examined: figure 12 refers to water-triethylamine mixtures and similar results have been obtained in nitrobenzene-n-hexane (5 to 95 Mc/s). Liebermann's theory therefore does not seem FIGURE 13. Angular distribution of scattered sound at 5 Mc/s satisfactory for an explanation of the experimental in aniline-cyclohexane mixtures (A. E. Brown and E. G. Rich- results, and it is at the present difficult to see ardson [26]). whether by using an adequate correlation function, it is possible to explain the entire increase of ab- TABLE m sorption in the critical mixture by means of /= = 1 -5 Mc/s 2-5 Mc/s 5-OMc/b scattering. Temper- ature °c a. Some experiments however have definitely shown c a c a <*• c in a critical mix- the existence of scattered energy 22-5% aniline goes through it. Recently A. E. ture when sound 34 0 057 0 011 1262 0-145 0-11 1260 and E. G. Richardson [26] have shown this 33 0 070 0 030 1262 0 15 011 1260 Brown 32 0 080 0 035 1262 015 011 1262 effect in aniline-cyclohexane mixtures. These 31 0 097 0-023 1262 0155 011 1264 30-5 0115 0 023 1262 017 012 1266 authors have measured the energy received by a 30 0140 0011 1262 0185 014 1268 29-25 0 105 0 06 1262 0-26 0145 1270 probe whose axis forms an angle 6 with the sound 33-3% beam radiated from the source. At 1.5 and 2.5 aniline 34 0042 0006 1281 009 0-055 1282 0-23 0-20 1288 Mc/s a broadening (4^-5° at 2.5 Mc/s) of the beam 33 0 045 0 007 1284 0 105 0 055 1282 0-23 0-20 1291 32 0 057 0 011 1288 0130 0 055 1289 0-25 017 1288 radiated has been observed when the source is in 31 0 060 002 1204 0160 0 055 1289 0-40 017 1294 30-5 0 057 0-007 1206 0 132 0 055 1296 0-25 0-20 1297 the critical mixture. At 5 Mc/s, while the intensity 43-8% falls to zero (first minimum) at an angle 6 of 1.5° if aniline 34 0 050 0035 1312 0 080 0 063 1318 0-24 017 1310 the medium does not contain inhomogeneities when 33 0 057 0 035 1317 0 086 0 063 1319 0-27 0-20 1316 32 0 092 0-035 1323 0-130 0 070 1322 0-30 015 1322 the radiation takes place in a critical mixture one 31 0 083 0 04 1323 0160 0 046 1319 0-38 0 05 1333 30-5 0 077 0 052 1323 0140 0 046 1318 0-32 0 03 1342 obtains patterns of relative intensities at various 53-9% angles 6 as those shown in figure 13. From the aniline 34 0 040 0 035 1355 0 070 0 052 1360 014 0 023 1366 position of maxima and minima the authors cal- 33 0 040 0 035 1355 0 075 0 046 1360 016 0 023 1361 32 0 040 0 035 1356 0 087 0 043 1361 018 0 023 1360 culate dimensions of effective scatters, which are 31 0 055 0 030 1359 014 0 058 1371 0-23 0 036 1369 30-5 0 054 0 030 1360 0115 0 050 1375 0 23 0 029 136S of the order of magnitude of 100 to 400/i, (compared 30 0-080 0 023 1360 0 092 0 058 1379 0-26 0 036 1376 with \ = 500|U,). Table III gives the experimental

~~192 absorption coefficient (a) and the coefficient that water-triethylamine mixture the experimental~~

~~Brown and Richardson determine as due to scat- results (fig. 12) when interpreted on the basis of the~~

tering (as) for mixtures of the composition examined relaxation theory, clearly show this fact; a very and at the frequencies used. These results in- rough estimation of the range of relaxation frequen- dicate a substantial contribution of scattering to cies indicates that it extends at least from 2 to 30 the total losses: in this case, scattering appears to Mc/s. be the most important dissipative process. The M. Fixman [28, 29, 30] has recently considered authors observe however that, in the limits of ex- in some detail the manner in which the charac- perimental error, scattering does not seem satis- teristic fluctuations of critical mixtures may cause factory to explain the totality of losses experimen- a frequency dependence of one or more of the tally found. Using the roughly approximate thermodynamical quantities which are of impor- procedure of subtracting the scattering absorption tance in sound propagation; he too has reached the coefficient from the experimental one, an unex- conclusion that the specific heat is the quantity plained part remains which has the same order of most hkely to be involved in the observed behavior. magnitude as the absorption coefficient in pure In an early attempt [28], he considered a fluctuating

17 2 1 liquids. ( — = 50 X 10~ sec cm" in aniline and heat capacity associated with density fluctuations; V adjacent volumes of the liquid reach different tem- 17 \ 200 10 in cyclohexane I. peratures when passed through by a sound wave: sound energy absorption would appear as a conse- 3.3. Relaxation Processes quence of the heat flow in the equalization of temperature gradients. Such a process however can Various researchers have thought that relaxation produce a much smaller absorption than observed processes of various kinds may be involved in the experimentally and therefore can not be responsible propagation of sound waves in critical mixtures and for the high losses observed. The dissipative could be responsible for the large absorption experi- effect just mentioned is described in the nonlinear mentally found. equation of motion and energy transport by a term The existence of clusters of various compositions linear both in the temperature variation produced dispersed in a mother phase induces the presence by the sound wave and in the density or composition of a great number of equihbria which may be per- fluctuations; in Fixman's early treatment other turbed by the temperature variations due to the terms which are either quadratic or of higher order sound wave. The kinetics of the system is to be in local fluctuations, were neglected. described by reactions which take place either In a later study Fixman has found the way to among clusters of different kinds or between clus- handle the quadratic terms and he was able to show ters and molecules of the mother phase to generate that they are strongly coupled with local tempera- clusters having a new composition. The situation tures; these terms originate an anomalous static is similar to the one found in binary mixtures of heat capacity in agreement with experiment [31] the water-alcohol type when a maximum of the and a frequency dependent heat capacity which absorption coefficient versus composition is present: could be responsible for the sound absorption in such case, most of the absorption increase seems increase in the critical region. In this calculation, connected with equihbria which exist among various the composition is expressed as a function nolecular associations of the two components, and of position by means of a Fourier are altered by temperature changes produced by series or integral, with ound waves. suppression of all Fourier components having a

wave number (k) greater than some k . It The relaxation processes which arise are of the max is then possible to obtain a fluctuating ame kind as those found in liquids where a low entropy density (85) associated with critical ate chemical equilibrium is present and can be composition fluctuations which is perturbed escribed by means of a specific heat function of by the temperature variation produced requency. Owing to the great number of equihbria by sound and gives rise to an excess heat capacity per resent among clusters of various sizes and compo- unit volume itions, a very broad distribution of relaxation times

~~s to be expected. This indication is in agreement ith experimental findings: in the case of the~~

~~193 T~~

In the expression of (85) enters the radial distribution function g(r) at time t expressed by means of the Fourier components up to kmax . The dynamics of the long range part of g(r) is assumed to satisfy the diffusion equation

2 2 2 2 ! [K V g-V V J=/ ^] (6) being h a diffusion constant which may be related to more usual parameters and k the reciprocal of an "indirect" correlation length. For a mixture at the critical composition k can be expressed by means of a short-range correlation length /

~~7 _ j~~

~~T-Tc~~

2 Fixman's analysis led him to conclude that k is 14. FIGURE Absorption per wavelength versus (T — c ) for strongly coupled with local temperature and that aniline-n hexane critical Solid line theoretical, dashed line experimental. IM. Fixman (29j). this circumstance can be the reason for both the increase of static heat capacity and the appearance of a frequency dependence of specific heat. There- variation with temperature could be explained by 2 fore he substitutes k with Fixman's theory. The theory should be tested for the variation of absorption coefficient with

«+($« (8 ) frequency. Fixman's calculation predicts a very small velocity dispersion and this result could be in and supposes 8T to be sinusoidal function of time. agreement with experiment because no dispersion Fixman obtains for the complex dynamic heat has been so far detected in the limits of errors. capacity per unit volume an expression and fur- The variation of velocity with temperature indicated nishes a numerical integration of the real and by the theory (3% for a temperature increase of imaginary part close to the critical point. 6 5 °C above the critical point) is about twice the This expression can be used to calculate the change observed. Fixman's theory, which could complex speed of sound from which velocity and be extended to a liquid-vapor critical system sup- absorption are derived. The comparison of these ports the idea that relaxation processes may be an theoretical values with experiment requires the important, if not the most important, dissipation knowledge of: (1) the correlation distance 1; (2) a process in the critical systems. friction constant, /3, which enters in the expression of the diffusion constant; (3) the specific heat per

mole in the absence of fluctuation C° ; (4) the ratio p C° 4. Conclusions —11 of specific heat y0 = in the same conditions. Cg The results of the comparison between theory and At the present status of research, most of the and experiment in the case of the mixture n-hexane- high absorption found in liquid-vapor and liquid- aniline (47.6 wt % of hexane) studied by Chynoweth liquid critical systems is most likely due to one or and Schneider are indicated in figure 14. C° both of the two processes: (1) scattering by inhomo- and yo were approximately evaluated by the Author geneities due to density or composition fluctuations, 13 -1 while /3 = 9.10 sec and / = 4 A were assumed (2) relaxation in the cluster equilibria perturbed by as reasonable values. It appears that the numerical the temperature variations induced by sound waves. values of the observed absorption as well as its It is not possible, at present, to go much further in specifying what the most important process may be

~~"The function has been tabulated by Kendig, Bigelow, Edmunds, and Pings [32]. in particular cases.~~

194 7 point of view, It is to be mentioned at this point that recently configurations. To support this A. S. Sliwinski and A. E. Brown [33] have stressed Sliwinski and Brown have considered some other the importance of a strong difference existing be- experiments which indicate the different ways of tween critical mixtures and liquid-vapor critical interaction of sound waves with a liquid-liquid systems. In the first case, in fact, the equalization critical mixture and a liquid-vapor critical system. of composition fluctuations must proceed through a The striations which can be observed with a diffusion mechanism which is particularly slow Schlieren setup in a medium at rest are not sig- near the critical point; instead, the merging of two nificantly changed when sound is applied in the or more density fluctuations to form a uniform liquid-liquid case, while they are broken in liquid- region in a liquid-vapor critical medium does not vapor systems as though a pulverization of assem- require diffusion. Such a difference could play blies had taken place. A similar effect is evident pro- an important role when the sound wavelength is when one follows the light diffraction patterns either smaller or of the same magnitude than the duced by a sound wave in the medium with time linear dimension of assemblies: this is the normal after sound application. In figure 15 the light dif- case when ultrasonic waves interact with critical fraction pattern in an aniline-cyclohexane mixture systems. The sound wave produces a variation of at 31 °C, in carbon dioxide at 3.5 °C and 72 atm, the pressure and temperature of assemblies around and in ethylene at 9.6 °C and 52 atm are shown at values which are near the critical ones: in the case V12 sec intervals from the instant when sound (1 of liquid-liquid system, as a consequence of the Mc/s) is switched on (arrow). While the patterns slowness of diffusion in the short time in which the remain unchanged in the liquid-liquid critical mix- interface between assemblies disappears, the pas- ture: they rapidly become blurred in the liquid- sage of sound does not strongly affect the form and vapor case. size of assemblies; in the case of liquid-vapor the j assemblies lose their identity when the surface

I 7 tension disappears and they may reappear suc- The existence of substantia] differences between assemblies in the two types of critical systems is also shown by the temperature dependences of a experimentally

~~cessively in a form quite different from the original found (figs. 3. 5. and 101. jj~~

~~in m~~

~~'S.~~

~~P- in m~~

~~>y s. in~~

209-493 O-66— 14 It would seem therefore that sound has a tre- 5. References mendous effect on the structure of assemblies in the liquid vapor case while it does not practically [1] C. E. Chase, R. C. Williamson, and L. Tisza, Phys. Rev. affect the configuration of clusters in critical mix- Letters 13, 467 (1964). [2] A. G. Chynoweth and W. G. Schneider, J. Chem. Phys. 20, tures. Sliwinski and Brown think that this could 1777 (1952). be an indication of the fact that scattering is par- [3] H. D. Parbrook and E. G. Richardson, Proc. Phys. Soc. (London) B65, 437 (1952). ticularly important in critical mixtures while it [4] C. M. Herget, J. Chem. Phys. 8, 537 (1940). must be less significant in liquid vapor media: in [5] H. Tielsch and H. Tanneberger, Z. Physik. 137, 256 (1954). [6] M. A. Breazeale, J. Chem. Phys. 36, 2530 (1962). the last case structural relaxation effects would [7] J. Noury, Compt. Rend. Acad. Sci. Paris 233, 516 (1951). be the main cause of absorption. [8] V. F. Nozdrev, Soviet Phys. Acoustics 1, 249 (1955). V. F. Nozdrev. Application of Ultrasonics in Molecular The survey of the work so far carried out in [9] Physics (Gordon and Breach, New York, 1963). critical systems has shown how the problems [10] B. I. Kal'ianov and V. F. Nozdrev, Soviet Phys. Acoustics 4, 198 (1958). associated with the propagation of sound waves [11] V. F. Nozdrev and V. D. Sobolev, Soviet Phys. Acoustics 2, in critical media have not yet been solved. At 408 (1956). [12] W. G. Schneider, Canadian J. Chem. 29, 243 (1951). present, they seem to be posed in clearer terms [13] V. F. Nozdrev, A. P. Osadchii, and A. S. Rubtsov, Soviet than some years ago, but much theoretical and Phys. Acoustics 7, 305 (1962). [14] V. F. Nozdrev and G. D. Tarantova, Soviet Phys. Acoustics experimental research is needed. The important 7, 402 (1962). areas of research are essentially two: (a) the low- [15] V. I. Grechkin and V. F. Nozdrev, Soviet Phys. Acoustics 9, 304 (1964). frequency sound velocity at the critical point; [16] L. D. Landau and E. M. Lifshitz, Statistical Physics (Per- (b) the analysis of absorption and possibly, of gamon Press, 1959). M. I. Beyatskii, A. V. Voronel, and V. G. Gusak, Soviet dispersion of sound waves. The progress of [17] Phys. JEPT 16, 517 (1963). experimental techniques should today allow more [18] A. V. Voronel, Yu. R. Chashkin, V. A. Popov, and V. G. Simkin, Soviet Phys. JEPT 18, 568 (1964). precise determinations of sound velocity and of [19] C. N. Yang, and C. P. Yang, Phys. Rev. Letters 13, 303 the absorption coefficient: moreover, it is impor- (1964). [20] L. Liebermann, J. Acoust. Soc. Am. 23, 563 (1951). tant to extend the ranges of temperature, frequency [21] L. Chernov, Wave Propagation in a Random Medium and composition in which the measurements are (McGraw-Hill Book Co., New York, 1960). [22] A. G. Chynoweth and W. G. Schneider, J. Chem. Phys. 19, made. From the theoretical side, we expect 1566 (1951)'. progress in the thermodynamical treatment of [23] M. Cevolani and S. Petralia, Atti Accad. Naz. Lincei 2, 674 (1952). critical systems, in scattering theories adequately [24] G. F. Alfrey and W. G. Schneider; Disc. Far. Soc. 15, considering correlation between fluctuation in 218 (1953). [25] D. Sette, Nuovo Cimento 1, 800 (1955). adjacent volumes, in separating the contributions [26] A. E. Brown and E. G. Richardson, Phys. Magazine 4, 705 to absorption due to relaxation and scattering proc- (1959). [27] R. Lucas, J. Phys. Rad. 8, 41 (1937). esses when they are of comparable importance. [28] M. Fixman, J. Chem. Phys. 33, 1363 (1960). The profit of these researches will surely be re- [29] M. Fixman, J. Chem. Phys. 36, 1961 (1962). [30] M. Fixman, Advance in Chemical Physics Vol. VI (Inter- warding in establishing the characteristics and the science Publ, New York, N.Y., 1964). behavior of critical systems since sound propaga- [31] M. Fixman, J. Chem. Phys. 36, 1957 (1962). [32] A. P. Kendig, R. H. Bigelow, P. D. Edmonds, C. J. Pings, tion is so intimately connected with the particular J. Chem. Phys. 40, 1451 (1964). nature of these media. [33] A. S. Sliwinski, A. E. Brown, Acustica 14, 280 (1964).

~~196 Ultrasonic Investigation of Helium Near its Critical Point~~

C. E. Chase* and R. C. Williamson**

~~Massachusetts Institute of Technology, Cambridge, Mass.~~

~~1 1 5 i 1 The recent discovery of a logarithmic singularity 1 1 1 i i i \> '~~

~~in the specific heat at constant volume C\ of fluids P= 1718 ± 1 MM HG _ 1 10 at the critical point [1-3] has stimulated renewed \ 105 interest in this region. A consequence of this~~

o INCREASING T h singularity that was soon demonstrated [4, 5] _J00 o • DECREASING T \ is that the adiabatic compressibility k must like- LU s w 95 wise be singular and the velocity of sound, given by

~~2 >- u =l/pKs , must go to zero. We have accordingly 90 4~~

at 1 1 1 1 1 1 1 in frequency 1 1 measured the sound velocity He a 8 85 _i 40-30 -20 -10 0 10 20 30 40 of 1 Mc/s in the critical region. Some of the results LlI 3 10 (T-TC )(°K) > t 10 of these are reported herein. measurements T = 5.200 ± 0.002 °K Figure 1 shows the velocity as a function of tem- 105 perature along an isobar and as a function of pres- 100 sure along an isotherm passing approximately through the critical point. These measurements 95 reported previously (together have been [5, 6] 90 o INCREASING P \ X with a description of the experimental techniques, • DECREASING P \ 5

i i i i l i i i at i i 1 85L_i i l I si 1_ which will not be repeated here). However, we -40 -30 -20 -10 0 10 20 30 40 (MM HG) have subsequently found that, as a result of slight P-PC hysteresis and lack of reproducibility in the pressure FlGl'RE 1. Sound velocity as a function of temperature (

~~a logarithmic singularity in this quantity at the crit-~~

*National Magnet Laboratory, supported by the U.S. Air Force Office of Scientific ical point. This fact is of interest as an experi- Research.

~~"Department of Physics. mental result independent of any theory. However,~~

~~197 in order to a more meaningful comparison make proach to within 110 mdeg of Tc [7, 10]. However,~~

with the results of specific-heat measurements and eq (1) can be criticized on other grounds: e.g., the with theory, we would like to know the adiabatic coexistence curve is actually much more nearly compressibility. This requires knowledge of the symmetric in the density than in the volume (for density, which has never been measured close to He 4 departures from perfect symmetry in density 4 the critical point in He ; the only measurements in amount to only 2% per degree Kelvin). This this region are those of Edwards and Woodbury difficulty was noted by Edwards and Woodbury

[7] along the coexistence curve, which . xtend to [7], who eliminated the spurious volume symmetry within 36 mdeg of Tc in the liquid and 50 mdeg in of eq (1) and obtained agreement with their meas- the vapor. It is therefore necessary to find an ured orthobaric densities within 110 mdeg of Tc by appropriate equation of state from which values adding two higher terms to the expansion, obtaining of the density can be calculated. Inasmuch as classical treatments such as the van 2 2 - V) = . ( dPId r At + Bv' + Ctv' + Dt (3) der Waals equation are only approximate, and more recent approaches have not yet succeeded in There are, however, further difficulties that are yielding an equation of state suitable for our pur- not eliminated, but rather compounded, by this poses, we have resorted to the technique of expand- procedure. This can be shown by integrating ing in power series about the critical point. This eqs (1) and (3) to obtain families of isotherms. procedure has to be used with caution because of For T < Tc , the resulting temperature-dependent the nonanalytic character of the thermodynamic constant of integration can be chosen to yield the observed relation along the vapor-pressure functions at the critical point [8]; nevertheless we P—T believe that the following developments could be curve; above Tc it is plausible to choose this con- used as a point of departure to be corrected by a stant so that the critical isochore lies on the extrap- singular function becoming significant in the im- olated vapor pressure curve. The results of such mediate neighborhood of the critical point. In a calculation are shown in figures 2 and 3. The particular, this procedure has shed light on the calculated isotherms are observed to cross in proper choice of thermodynamic variables in the both cases; moreover this crossing is worse (in critical region, and has disclosed previously un- the sense of occurring closer to the critical point) reported inconsistencies in the standard treatment for the Edwards-Woodbury expansion than for that of the critical point given by Landau and Lifshitz of Landau and Lifshitz. The anomalies are so

[9] and modified by Edwards and Woodbury [7]. pronounced that they could scarcely be removed Landau and Lifshitz expand the slope of an iso- by any reasonable adjustment of the parameters thermal as a power series in t — T— T and v' = expan- c V— V T , or the constants A-D. These P , V c c , c c according to the equation sions are thus unsuitable for quantitative calculations. N-\d 2 F/dV2 2 The way out of this difficulty is found by a simple ) = -(dP/dV) = . . T , x T At + Bv' +. interchange of the role of the variables and V in (1) N eq (1). This leads to a description in terms of the I where A and B are positive constants. This pro- conjugate pair of variables /x, (where p. is the p J cedure leads to a coexistence curve symmetrical chemical potential), as opposed to the traditionally 1 about the critical point when plotted as a function preferred pair — P, V, and in fact represents a I of the volume: nontrivial change in the definition of the thermo-

~~dynamic system. The equation which is the analog y 2 v' (; =-vi=i-SAt/Byi . (2) of eq (1) is now~~

2 2 ' 2 = . . V-Hci . The fact that the exponent of the coexistence curve Flf)N ) T v (d/xldp)r=at + ftP +. is 1/2, as is nearly always the case with such (4) expansions, does not present a problem in the p' = — present case; in contrast to the behavior of most where p pc , and the slope of the isothermals other substances (for which an exponent of 1/3 is given by 4 is observed), He obeys the 1/2-power law experi- 2 mentally from the above-mentioned closest ap- (c)P/c)p) r =p(c)pLlf)p) T = p(at + /3p' ). (5)

~~198 3 -10 t/Tc~~

~~Figure 4. Graph {p' and (p' 2 against — t/T of LlpQ Y cJpc ) c from data of reference 7. 2 2 Equation (6) predicts a single straight line of slope 3a/^3 + (p'Jpc) Oip'Jpc) -~~

~~By development exactly parallel to that of reference~~

~~9, we now find that the coexistence curve is symmetric in the density:~~

FIGURE 2. Isotherms and coexistence curve calculated from the 1/2 p; = = <-3a;//3) . (6) 4 -PG Landau-Lifshitz expansion (eq (1)) with A=7.95 X 10 dyn 5 s mol cm' deg-', B = 62 dyn moP cm'", (dP/dT) s „, = 1.7 X 10 dyn cm'2 deg~', P = 2.288 X 10e dyn cmT2 T = 5.200 °K, 2 c , c To evaluate a and /3 we note that a plot of p' and 3 L Vc = 57.628 cm mol''. P'q against — t should he on a single straight hne through the origin of slope 3a//3. Such a plot,

~~based on the data of Edwards and Woodbury [7], is shown in figure 4, from which we find a/j8 = (1.03 ±0.02) X 10~ 4 mole 2 /cm 6 °K. Note that the range~~

~~of agreement with experiment is as wide as that achieved by Edwards and Woodbury with their~~

~~four-parameter expansion [11]. As in reference~~

[7], fi can be evaluated approximately in terms of the measured density and isothermal compressibility [12] at 5 °K, yielding 0 = (6.86 ± 1.2) X 1013 10 4 dyne-cm /mol . The resulting isotherms in the

fx, p plane are shown in figure 5. In this case the temperature-dependent integration constant that determines the separation of the isotherms was evaluated from the relation Ndp=VdP — SdT, using entropy values along the critical isopycnal derived by numerical integration of the specific-

heat data of Moldover and Little [3]. Isotherms in the P, p plane, calculated with the same assumptions used in deriving figures 2 and 3, are shown in figure 6. In neither case are inconsistencies evident. Although this expansion was derived for purely

practical reasons, we believe it has significance FIGURE 3. Isotherms and coexistence curve calculated from the beyond mere curve-fitting. In particular, the great 3 Edwards-Woodbury expansion (eq (3)), with c = 8J6x 10 2 8 5 5 2 improvement resulting from the proper choice of dyn mol cm' deg~' , D = 7.78 X 10 dyn mol cm' deg' , A, B, (dP/dTW, P T as in 2. c , c , V c figure variables, and the natural way in which the sym-

~~199 T PP~~

Figure 5. Isotherms and coexistence curve in the p., p plane calculated from the density expansion (eq (4)), with a=7.06 4 l: 4 X10" = < dyn mo/-- cm defC' , j8 6.86 X 10 dyn mot cm'", Figure 7. Adiabatic compressibility (a) as a function of log other parameters as in figure 2. |T (b) as a I | and function log |P in the critical — c of — c | Fur method of calculating spacing of isotherms see text. region, calculated from the results of figure 1 and the density expansion (eq (5) and fig. 6).

~~velocity data to yield k. , which is plotted s logarith- !| mically as a function of \ or T \ in figure 7. T— c |P— c |~~

~~This figure is similar to one previously given, ' j~~

~~[5, 6] but the differences resulting from use of our 1 improved expansion for the calculation of the density are significant. Results in the liquidlike~~

~~modification (P > Pc or T < Tc) are virtually unchanged, and the presence of a logarithmic singu- |j~~

~~larity in k.s over more than a decade is clearly indicated. However, in the gaslike modification 1]~~

~~k.s now appears to be almost constant, and if a I~~

singularity is present at all, it must only show up P?P within C the last 4 or 5 mdeg (or millimeters) away from the critical point. Figure 6. Isotherms and coexistence curve in the P, p plane Velocity measurements have also been made calculated from the density expansion (eq (5)),- all parameters

~~as in figure 5. along a number of isotherms above Tc . Some of~~

these measurements are shown in figure 8. These i metry of the problem enters when these variables curves all show pronounced minima as a function are used, appear to us important. These features of pressure. It is perhaps important to point out I j should survive even in a more sophisticated treat- that all these minima are rounded rather than cusp- ment that includes a proper handling of the shaped, even within about 10 mdeg of Tc (the closest J singularity. point at which the velocity could be observed con- I Values of the density along the critical isobar tinuously through the minimum). The fact that j and isotherm, calculated in the same way as those the locus of these minima in the P, T plane falls plotted in figure 6, have been combined with our approximately on the extrapolated vapor pressure

~~200 T-Tc (°K)~~

~~1 a* I I II I i I I I 1 1700 1800 1900 2000 2100 P(MM HG)~~

~~FIGURE 8. Velocity as a function of pressure along isotherms above the critical temperature.~~

~~2000 FlGl RE 10. Plot of u%i,i against T/C\ from data of reference 3.~~

1 800 In this approximation a plot of u 2 versus TjCy should 1600 2 HG) 2 be a straight line of slope Pc (dPldT) at . This is 5 by the solid line in figure 10; the dotted line 5 1400 shown Q- is a correction for the nonzero value of (dPldp)r

~~1200 derived from our density expansion (eq 5). The~~

~~circles represent our values of u^ in at various tem- 1000 peratures plotted against the specific-heat measure- 800 ments of Moldover and Little [3]; use of u min is~~

consistent with the assumption (BP I dT)y — (dP / dT) sat 4 2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 T(°K) and the fact that the observed minima he on the extrapolated vapor pressure curve; i.e., we are FIGURE 9. Locus of velocity minima along the isotherms of figure 8 in the P. T plane. assuming that the velocity minima he along the Full line: vapor pressure curve. + critical point. line p = pc - The agreement of these points with Yang and Yang's prediction, plus the small cor- curve, as is shown in figure 9, allows us to reach rection term (dPldp)r, is reasonably good, although some further conclusions. According to Yang all the points he about 2 to 5 percent too high [13]. I and Yang [4], the velocity of sound along the critical We ascribe this small systematic difference to isopycnal above Tc should be given by errors either in the choice of (dP/dT)sat , in the meas- u2 = p- 2 ured values of Cy, or in the temperature scales of (dPldT)l t (TICv), (7) the two experiments.

where Cy is taken along the same isopycnal and Since eq (7) ought to become more accurate as the critical point is approached, we can now esti- (dP/dT) sat is the slope of the saturated vapor pres- mate how low the velocity ought to fall along iso- sure curve just below Tc . Starting from the ther-

modynamic relation therms very close to Tc . As can be seen with the aid of the temperature scale at the top of figure 10, (dP/dV), = (dPldV) T -(dPldT)UTICv), (8) the minimum velocity along an isotherm just 10~ 4 °K i above Tc would be —55 m/sec, about 2/3 of it ; is easy to see that eq (7) can be obtained from the the lowest value we have observed. Thus, even in exact expression an ideal experiment in which measurements could

~~2 be made very close to the critical point in spite of u = (dPIdph + PcHdPldmTICv) (9) the large attenuation there, it would hardly be~~

~~by neglecting (dP/dph (which vanishes as T-^>TC ) possible to observe much lower values of u than = that (dPjdT)y ) (dP/dT^jTKT^- and assuming ,(t>tc we have done.~~

201 The authors are extremely grateful to Laszlo [6] C. E. Chase and R. C. Williamson. Proe. 9th Intl. Conf. on Low Temp. Phys., Columbus. Ohio. 1964 (to be published). Tisza for suggesting this experiment, for playing [7] M. H. Edwards and W. C. Woodbury. Phys. Rev. 129, a major part in the derivation of the density expan- 1911 (1963). [8| L. Tisza, Phase Transformations in Solids (John Wiley & sion and the elucidation of its significance, and for Sons, Inc., New York, N.Y., 1951). p. 15; M. E. Fisher, J. continued helpful advice and encouragement Math. Phys. 5, 944 (1964).

[9] L. D. Landau and E. M. Lifshitz, 259 ff. Statistical throughout the course of this work. Most of the p. Physics, (Pergamon Press, New York. 1958). numerical computations were carried out at the [10) The deviations from a 1/3-power law observed by Edwards and Woodbury appear to be much too large to ascribe M.I.T. Computation Center. to experimental error. If these deviations are real, they might be attributed to quantum corrections to the law References and Notes of corresponding states (J. de Boer, Physica 14, 139, 149 (1948); J. de Boer and R. J. Lunbeck. Physica 14, 509 (1948)), and similar behavior ought to be observed in 3 M. I. Bagatskii. A. V. Voronel', and V. G. Gusak, Zh. [11 He . Such appears to be the case (see paper by R. H. Eksperim. i. Teur. Fiz. 43 , 728 (1962) [translation: Soviet Sherman in this conference). Phys.-JETP 16,517(1963)]. [11] The line in the figure actually has a finite intercept of +4.6 [2] A. V. Voronel". Yu R. Chashkin, V. A. Popov, and V. G. millidegrees on the abcissa. This may indicate flatten- Simkin. ibid. 45, 828 (1963) [translation: Soviet Phys.- ing of the coexistence curve close to the critical point, JETP 18,568(1964)]. or may be the result of errors in the measured temper- [3[ M. Moldover and W. A. Little. Proc. 9th Intl. Conf. on atures in reference 7 or the assignment of Tc . Phys., Ohio. 1964 (to be published). Low Temp. Columbus. [12] M. H. Edwards and W. C. Woodbury, Can. J. Phys. 39, [4] C. N. Yang and C. P. Yang. Phys. Rev. Letters 13, 303 1833 (1961). (1964). [13] Note that if the velocity minima did not fall exactly on the C. E. Chase. R. C. Williamson, and L. Tisza, Phys. Rev. [5] line p = pc the agreement would be worse, for a smaller Letters 13, 467 (1964). value of (/ is required to fit the theoretical curve.

~~Ultrasonic Investigation of the Order-Disorder Transition in Ammonium Chloride~~

~~C. W. Garland and R. Renard~~

~~Massachusetts Institute of Technology, Cambridge, Mass.~~

~~Introduction of the order-disorder type, involving the relative orientations of the tetrahedral ammonium ions;~~

The lambda transition in solid ammonium chloride it was assumed that in both the ordered and the was discovered by Simon [1] from heat-capacity disordered states the ammonium ions liberate measurements. An x-ray analysis by Simon and about an equilibrium position. This idea has been von Simson [2] showed that ammonium chloride confirmed by many subsequent experiments. has a CsCl-type cubic structure both above and In particular, Levy and Peterson [9] have carried below the critical temperature; thus the transition out a neutron-diffraction determination of the hydro- could not be due to a change in lattice type. Hettich gen positions in both the low- and high-temperature

[3J found that ammonium chloride is weakly piezo- modification, and Purcell [10] has analyzed the nmr electric in the low-temperature modification, data to obtain the relaxation time for an NH£ ion whereas it is not piezoelectric above the critical to move from one orientation in its cubic cell to the temperature. From this he concluded that the other orientation. Both these investigations show tetrahedral ammonium ions are oriented parallel that free rotation does not occur. in the low-temperature modification. Menzies Indeed, in many respects NH 4 C1 is an ideal and Mills [4] came to the same conclusion on the crystal for studying cooperative order-disorder basis of the Raman effect. Pauling [5] suggested phenomena. The ordering is completely analogous that the transition was related to the onset of free to that for a simple-cubic ferromagnet in a zero rotation of the ammonium ions, and quantitative external field. The change in interaction energy refinements of this idea were carried through by between parallel and antiparallel NH| ions is

Fowler [6] and by Kirkwood [7]. In opposition to almost completely due to octopole-octopole terms this, Frenkel [8] interpreted the transition as being between nearest and next-nearest NHJ" neighbors

202 [11], and thus NH 4 C1 is quite a good example of an measured at five fixed temperatures spaced between Ising lattice. Furthermore, the ordering process 250 and 308 °K. Thus, we have very precise should have little effect on the dynamics of such velocity data (±0.005%) on both the ordered and an ionic lattice. disordered phases. This experimental data will wish to discuss here a variety We of ultrasonic be described elsewhere [13] in detail, but is pre- measurements which have been made on single- sented here in a graphical form. crystal ammonium chloride. Longitudinal and All the results given below are in terms of the transverse (shear) acoustic velocities have been variation of one of the three elastic constants Cn, — measured [12, 13] over a wide range of frequencies C44, and C =(cn Ci2)/2. Third-order elastic con- (5 to 60 Mc/s), temperatures (150 to 320 °K) and stants are not used, and at high pressures the pressures (0 to 12 kbar). The attenuation of longi- quantities cn, C44, and C are in a sense "effective" tudinal ultrasonic waves has been measured [14] elastic constants. The constants Cu and C44 were

~~at 1 atm for frequencies between 5 and 55 Mc/s calculated from the experimental longitudinal and~~

and temperatures from 200 to 270 °K. In all these transverse velocities (U) in the [100] direction with = = investigations, special emphasis was given to the the relations pUf cu and pUf c44 ; the constant C "anomalous" behavior near the lambda transition. was calculated from pUf> — C", where Ur is the veloc- As shown in figure 1, the transition temperature is ity in the [110] direction of a shear wave polarized a fairly strong function of pressure, increasing from perpendicular to the [001] axis.

~~- 242 °K at 1 atm to ~ 308 °K at 10 kbar. (Note the pronounced curvature of the lambda line and Experimental Results the hysteresis observed at low pressures.) At a~~

constant pressure of 1 atm, the ultrasonic velocities The variations of the shear constants C44 and C" were originally measured by a pulse-echo method and of the compressional constant cn with pressure as a function of frequency and temperature, and at five different temperatures are given in figures

~~no dispersion was observed [12]. Recently, the McSkimin pulse-superposition method has been used at a fixed frequency of 20 Mc/s. The tem-~~

~~perature dependence at 1 atm has been restudied, and the pressure dependence of the velocities were~~

~~12.000~~

~~10,000 3000 6000 9000 12.000 PRESSURE (bar) - 8000 FIGURE 2. Dependence of c 44 on pressure at Ti=250, T> = 265 = T:l = 280. J, 295. and T5 = 308 °K. UJ § 6000~~

~~4000 —~~

~~2000~~

250 270 290 310 330 T (°K) 3000 6000 9000 12.000 FIGURE 1. Phase diagram for NH 4 CI (the high-pressure, low- temperature field corresponds to the ordered phase). PRESSURE (bar) The solid data points were obtained from the abrupt "break" in the ultrasonic shear velocities at the A transition point (see figs. 2 and 3). The vertical bars indicate static FIGURE 3. Dependence of C on pressure at various tempera- measurements of Bridgeman [15|. The light lines numbered 1 through 7 represent various isochores. tures (see fig. 2 legend).

~~203 3 0 _J 3000 6000 9000 12,000 A 8 PRESSURE (bar) $ e #~~

8 FIGURE 4. Dependence of Cu on pressure at various tempera- ! tures (see fig. 2 legend). Dashed portions of the curves indicate regions where data are less accurate or are 3.50 missing due to high attenuation.

~~100 150 200 250 300~~

~~FIGURE 7. Variation of Cw with temperature.~~

~~1 circles Curve A: data at 1 atm (solid circles = pulse superposition method [ 3 J. open = corrected pulse-echo data [12]). Curve B: calculated curve at constant volume~~

~~3 1 to the volume at 1 atm and J/2 =34.15 cm mole' ; corresponds to at 280 °K and 191.25 °K (marked by cross).~~

~~2, 3, and 4. Data points are not shown on these figures since a variety of corrections must be applied to the data, which requires some smoothing. Each~~

isotherm is based on at least 30 experimental points 150 200 250 300 350 TOO which show very little scatter. Note that C44 is much more sensitive to pressure than C" but that FIGURE 5. Variation of C44 with temperature at 1 atm [open

~~circles) and at V, =34.00, \, = 34.15. V3 = 34.27, V 4 = 34.43, the character of the anomalous variation near the~~

: 1 V = 34.51, V„ = 34.77, V = 34.93 cm < mole' (crosses). 5 7 lambda line is very similar for both these shear The transition temperature is known for each of these volumes, and thus the location of the change in slope for each of the constant-volume lines is fixed. constants. Indeed, there is a marked reduction in the rapid variation just above the transition as

~~the lambda line is crossed at higher pressures.~~

~~This same reduction in anomalous behavior is also seen in the cn data. The adiabatic compressibility s (3 can be calculated from~~

~~1.52 1/^=3 (c» + 2c«) = cii-|c' (1)~~

~~z 150 s and it is clear that /3 goes through a sharp maximum at a lambda point. Experimental data points are shown in figures 5,~~

~~6, and 7 for the constants C44. C , and Cn as functions~~

of temperature at 1 atm. In addition, the variation 200 250 t(*k) of C44 and C with temperature is given for seven different constant volumes, and the behavior of FIGURE 6. Variation of C with temperature at 1 atm (open is also shown. circles) and at various constant volumes (see fig. 5 legend). Cn at a single constant volume

204 These constant-volume curves were obtained by Further experimental work on attenuation is cur- determining the pressure required at a given tem- rently in progress. attenuation near the perature to give the desired volume (Bridgeman's Since there is no excess point for waves, it is possible to follow data [15] and our own were used) and then reading lambda shear the appropriate elastic constant values from the behavior of the shear velocities through the a small figures 2 to 4. transition region. Both c44 and C showed For the shear constants, one can immediately hysteresis, the magnitude of which is indicated see that most of the temperature change near 7\ in the insert on figure 1. We have not shown data since equilib- at 1 atm is due to the rapid change in volume in points in this region on figures 5 and 6 the transition region. Our data permit us to com- rium was very difficult to achieve (slow changes in pletely analyze the effects of volume changes and the shear velocities occurred even after 45 min). temperature changes so that one can see the effect The presence of this hysteresis and the very abrupt of ordering at a constant volume and temperature. change in shear velocities near Tk are significant For longitudinal waves, volume changes do have a properties of our measurements at 1 atm. (Similar significant effect on the velocity, but the anomalous indications of hystersis and abrupt changes near dip in Cn is essentially due to ordering effects. In 7\ are seen from volume measurements and cooling figure 7, the shift in the minimum between curves A curves.) and B is merely a result of the value chosen for the Discussion constant volume; the important difference between these curves is their shape near the minimum. Equilibrium data. Because of the formal anal- For shear waves there is no indication of excess ogy between the ordering of NH4 C1 and of a ferro- ultrasonic attenuation near the lambda line, but magnet in the absence of an external field, it is for longitudinal waves there is a very sharp maxi- appropriate to consider the mechanical behavior mum in the attenuation coefficient a at the transi- of an Ising lattice. If the coupling between the tion point. Quantitative measurements of a for lattice system and the "spin" system is sufficiently longitudinal waves along [100] have been made at weak, an Ising lattice will become unstable in the 1 atm as a function of frequency and temperature immediate vicinity of its critical point and a first- [14]. The results are shown in figure 8 for 15, order transition will occur. Furthermore, hysteresis 25, 45, and 55 Mc/s. Data have also been taken - will be expected in the critical region. This in- at 5 Mc/s, where a rather large (0.5 dB cm '), stability has been demonstrated for a two-dimen- temperature-independent background attenuation sional square lattice [16, 17], where there is an is present (presumably due to beam-spread losses). analytic solution to the Ising problem. The im-

~~portant conditions for this instability are (1) weak~~

~~lattice-spin coupling, (2) a finite compressibility~~

~~for the disordered lattice, and (3) an infinite C v for the "spin" system at the critical temperature. The~~

~~effect will be large only when (a) the lattice is soft,~~

~~(b) the critical temperature is large, and (c) dJ/dV is large (where J is the usual Ising interaction energy). Such an instability has been proposed~~

~~previously for other models [18, 19] and discussed~~

~~in general thermodynamic terms by Rice [20], but little attention seems to have been paid to this~~

problem. It would appear that NH4 C1 is a much more appropriate system for observing these effects than most ferromagnetic solids would be. 200 220 240 260 280 T,"K To illustrate the instability and the possible hysteresis in order-disorder transitions, let us look

FIGURE 8. Attenuation ultrasonic coefficient a at various at the two-dimensional case [16]. Figure 9 shows frequencies. Data obtained on two samples with different path lengths were in excellent agreement a plot of pi and — p

~~205 between 4 and 5 on heating or 4' and 3' on cooling is only metastable. Although the shape of the pi curves will be different for a three-dimensional~~

~~Ising lattice, the instability will still occur if Cv~~

~~(and thus dp,/dV)^^ a t the critical point. Thus~~

the hysteresis described above can still occur. As seen in figures 2, 3, 5, and 6, the shear constants of NH4CI are sensitive functions of volume. Thus the experimental hysteresis in Cu and C can indicate the presence of the proposed hysteresis in V. Note that experimentally the hysteresis decreases for a transition higher up along the

~~lambda line (i.e., at some finite external pressure).~~

FIGURE 9. Plot of p/ and — pdi versus the area cr for a two- This would be expected from our analysis of the dimensional Ising lattice. Ising lattice since the only difference between work- The family of curves pi were calculated at seven evenly spaced temperatures from T\ up to 7V The lamily of lines— pai were drawn to represent a disordered lattice ing at zero pressure and at some large constant pres- with typical compressibility and thermal expansion coefficients. The encircled numbers 1 and 7 indicate the spin and lattice isotherms at 7*1 and 7V sure is in the lattice stiffness. For high external pressures, the slope of — pai will be greater and the maximum width AT of the hysteresis will be less. Moreover, a semiquantitative estimate of the mag- Pdi is that of the completely disordered lattice, nitude of AT at 1 atm is in reasonable agreement and pi represents the attractive (negative) pressure with experiment. Completely analogous considera- due to the ordering of an Ising spin system. An tions can be developed for an instability and hyster- intersection of the two appropriate isotherms esis on compressing an Ising lattice at constant (pi and —pdi) will correspond to = and p p, + p,u=0 temperature. An elaboration of the effects of will give an equilibrium area cr if the stability con- hysteresis and instability on measurements at 2 2 dition {c) A/cta > 0) is satisfied. The stability either constant pressure or constant volume will condition requires that the slope of — p,n be greater be given in detail elsewhere [16]. In addition, our than that of p,; e.g., the system is unstable at a shear data strongly suggest a sluggish first-order point such as 4". Now consider the change in transition in the critical region. For points taken area with T for a system under zero external stress at temperatures more than 1 °K away from 7\ and (p — 0). As the temperature increases from T\ at pressures more than 100 bar away from p\, to 7 5, cr can increase continuously cr, from to o-5 equilibrium was achieved within about 15 min after 1 (points 1 to 5 on fig. but as 7 —> 9) T5 from below the the temperature or pressure was adjusted. In 2 2 system becomes unstable (d = A/da 0) at point 5 the immediate vicinity of a critical point, very slow and there must be a first-order change in area from changes in velocity were still observed 45 min o"5 to cr.y. further heating, cr On increases con- after the temperature or pressure was adjusted, tinuously from cr to . ov 7 However, on cooling as would be expected in a metastable region. Also

from T7 to T3 the area can decrease smoothly from the changes in C44 and C in this narrow hysteresis cr 7 to cr3 '. As T^>Tz from above the instability region are extremely abrupt, even when compared occurs at 3' point and there is a first-order change with the very rapid variations observed in the

from cr ' to cr . Below 3 3 T3 , a decreases smoothly ordered phase near the transition line. on cooling. Thus, there can be a hysteresis loop Leaving aside the important but relatively narrow near the critical point with a first-order jump in range where instability may occur, it is possible cr at T5 on heating and a first-order drop in cr at to discuss the effect of ordering at constant volume

T3 on cooling. The values T3 and T5 determine on the elastic constants in terms of an Ising model. the maximum width of this loop since the system We have carried out a general stress-strain analysis becomes mechanically unstable at points 5 and 3'. of the two-dimensional Ising lattice (using Onsager's Naturally, there is a temperature T4 for which the analytic solution for the case J^J') and have ob- free energy at point 4 equals that at point 4'; tained explicit formulae for the isothermal stiff-

thermodynamic equilibrium would give a first- nesses Cu, C44, and C = (cu — Ci2)/2. The details order transition at T4 and no hysteresis. The region will be reported elsewhere [21], but the essential 2Q6 features can be seen from the simpler formula for Dynamical data. In the past few years there T the isothermal volume compressibility fS : has been increasing interest in the theory of the dynamical properties of substances near a lambda point. Landau and Khalatnikov [24] were the first = = C/(0, H) W (w)

by others [25, 26] and a somewhat simplified result where C/(0, H) and Ui(0, H) are the Ising heat ca- for the relaxation time t for long-range ordering pacity and internal energy at zero field as a function can be given in the form of H =J/kT. The subscript dl denotes the appropriate quantity for a completely disordered lattice T+--=A+>-l\T-Tc \ (3) (for which C/ and Ui equal zero). The elastic con- 2 stants dj are given by {d A/dXjdxj)r where the x's where + and — denote temperatures above and are the appropriate strains. Thus the expression + below Tr . The phenomenological constants A for Cn will have a form similar to eq (2), and the and A~ depend on the substance involved but have dominant term in determining the shape of near + Cu a fixed ratio of A /A~ — 2. A more general statis- T\ will involve C/. For = {c u — c v>)/2 one re- C tical-mechanical treatment of this problem has quires a difference between two such expressions, been given by Kikuchi [27]. For a temperature and the singular part involving C/ will cancel out. range close to the lambda point one can show that final expression for is in its The C complicated Kikuchi's results for the long-range relaxation times dependence on V and T, but it does predict a smooth are still of the form given in eq (3). However, the variation through the transition region. For c44 the constants A + and A~ now depend on microscopic term containing disappears identically since Ci transition probabilities and the ratio A + /A~ near dJldy = 0 (y is the change in angle during a c44 Tc is ~ 1.6 rather than 2. 2 2 shear), but d Jldy # 0 and the shape of C44 near An essential feature of our ammonium chloride

7a is determined by Ui. In general terms these data which requires explanation is the fact that the predictions are all fulfilled by our NH4CI data as attenuation a has its maximum at T\. One would functions of temperature at constant volume. In expect a to vanish at 7\ if t becomes infinite there. the case of Cn one must take account of the sig- However, the theories mentioned above deal with nificant difference between the adiabatic and the Ttv, the relaxation time for ordering at constant isothermal constants near 7\, but even so the shape temperature and volume. The experimental con- m Cn is strikingly related to the shape of the heat ditions for measuring attenuation are adiabatic capacity curve. In addition, the shape of c 44 is rather than isothermal, and although the process is what one would expect from an internal energy irreversible one can consider it to be essentially curve. No detailed discussion of C will be given isentropic for small sound amplitudes. In this here, but in crude terms the effect of ordering is case the appropriate relaxation time is Tsv, which very small for this shear mode and this indicates can be related to Ttv by that the "anomalous" variations in Cw and Cr> almost completely compensate each other. J_ J_ = + M7-Y (4) A final remark about the equilibrium properties of NH 4 C1 can be made in connection with the

Pippard equations. It is known that these phe- where A: is a rate constant, C, is the specific heat at nomenological relations are fairly well obeyed [12] infinite frequency ("frozen" specific heat), and £ is in NH 4 C1, but a generalization of them in terms of a progress variable characterizing the internal (non- stress-strain variables [22] shows that there are equilibrium) state of the system at a given V and T. 2 some unexplained difficulties. We have recently The detailed behavior of (dS/d^) is not known, discussed the Pippard equations [23] on the basis but it seems likely that this quantity varies only of a three-dimensional Ising model and can now slowly with temperature near 7\. We shall assume 2 give a better insight into the reasons for their below that the quantity kT(dS/r)^) IC r has a con- behavior. stant value (call it B) over the temperature range of

~~207 interest. With this assumption and eqs (3) and (4), • 5 Mc (a -0.53 db/cm) we find that O 15 Mc (a -0.15 db/cm) A 25 Mc a~~

(5) TSV for the relaxation time associated with long-range ordering, where AT=\T— T\\ and the constant A has different values above and below 7\. Accord- ing to eq (5), TS v will have a finite maximum value

~~at Tx and if this value is reasonably small (i.e., 2 2 co T~~

~~FIGURE 10. Plot of ar/a versus temperature, close to the lambda For a mechanism involving a single relaxation point at p = 1 atm.~~

Data a! 45 and 55 Mc/s were not available over this temperature range. Note t hat time, we have the usual expression the a values tor 5 and 15 Mc/s have been corrected for a temperature-independent background attenuation (beam-spread losses).

quite what one would expect [14] and new experi- (6) 2 mental work along these lines is in prog- Wcu 1 + o) t|, currently ress. Also if NH4CI undergoes a first-order transition very close to the "lambda point," then Unfortunately the variation of (eft — c§i)/2Ucp with one would expect r to undergo a discontinuous temperature is not known, but it is unlikely that change between two finite values. This this could be an important factor in determining may supersede the explanation given the temperature dependence of a. One would above for the behavior of a probably expect only a small and slowly varying [29]. dependence of eft — c?i on temperature. We shall References assume (as did Chase in his analysis of helium [28]) [1] F. Simon, Ann. Physik 68, 4 (1922). that (eft — c?i)/2{/cn has a constant value through- [2] F. Simon and C. von Simson, Naturwiss. 14, 880 (1926). out the transition region. We shall also assume [3] A. Hettich, Z. Physik. Chem. A168, 353 (1934). 2 [4] A. C. Menzies and H. R. Mills, Proc. Roy. Soc. A148, 407 (and can subsequently justify) that o» r| <^ 1 for all r (1935). temperatures near T\. With these assumptions [5] L. Pauling, Phys. Rev. 36, 430 (1930). [6] R. H. Fowler, Proc. Roy. Soc. A149, 1 (1935). and eq (5), we can rewrite eq (6) in the form [7] J. Kirkwood, J. Chem. Phys. 8, 205 (1940). [8] J. Frenkel, Acta Physicochimica 3, 23 (1935). [9] H. A. Levy and S. W. Peterson, Phys. Rev. 86, 766 (1952). or 2Ucu 1 2Uc u /AT ' [10] E. M. Purcell, Physica 17, 282 (1951). = li ' (7) a ?x-c°n\A [11] C. W. Garland and J. S. Jones, J. Chem. Phys. 41, 1165

(1964) . [12] C. W. Garland and J. S. Jones, J. Chem. Phys. 39, 2874 2 If all our assumptions are valid, then oj /a should (1963). [13] C. W. Garland, and R. Renard, J. Chem. Phys. 44, 1130 be independent of frequency and should vary lin- (1966). early with the temperature on both sides of 7\. [14] C. W. Garland and J. S. Jones, J. Chem. Phys. 42, 4194

(1965) . 2 2 a> in units of Figure 10 shows a plot of /a, cps [15] P. W. Bridgeman, Phys. Rev. 38, 182 (1931). _1 N cm, versus temperature for all of our data close [16] C. W. Garland and R. Renard, J. Chem. Phys. 44, 1120

(1966) . to T\. Various details concerning the correction [17] C. Domb, J. Chem. Phys. 25, 783 (1956). for background attenuation and the justification [18] C. P. Bean and D. S. Rodbell, Phys. Rev. 126, 104 (1962). [19] D. C. Mattis and T. D. Schultz, Phys. Rev. 129, 175 (1963). 2 2 of oj t 1 are given elsewhere but in a gen- < [14], [20] O. K. Rice, J. Chem. Phys. 22, 1535 (1954). eral way the data shown in figure 10 seem to agree [21] R. Renard and C. W. Garland. Phys. Rev. J. Chem. Phys. 44,1125(1966). 2 with value co /a at its eq (7). The of minimum [22] C. W. Garland, J. Chem. Phys. 41, 1005 (1964). ut [23] C. W. Garland and R. Renard, J. Chem. Phys. (in press). (at ~ 242.0 °K) is 0.52 X I0 : the slope above TK [24] L. D. Landau and I. M. Khalatnikov, Akad. Nauk. S.S.S.R. 16 1B is 0.46 X 10 and the slope below 7\ is 0.66 X 10 . 96, 469 (1954). These slopes give a ratio A + /A~ = 1.43, as compared [25] I. A. Yakovlev and T. S. Velichkina, Sov. Phys. Uspekhi 63, 552 (1957). to ~ 1.6 expected on the basis of Kikuchi's model. [26] T. Tanaka. P. H. E. Meijer, and J. H. Barry, J. Chem. Phys. When the analysis is extended to data obtained 37, 1397 (1962). [27] R. Kikuchi, Ann. Phys. (N.Y.) 10, 127 (I960). at higher frequencies and at temperatures further [28] C. E. Chase, Phys. Fluids 1, 193 (1958). C. Garland and C. F. Yarnell, Chem. Phys. (in press). away from 7\, the deviations from eq (7) are not [29] W. J.

~~208 Discussion R. Zwanzig: If you analyze these correlation function formulas for fluids which have internal degrees of freedom interacting~~

weakly with the translational degrees, it turns out that the shear observations on E. H. W . Schmidt presented some experimental viscosity does not show any particular effect from the internal critical point of empha- convective heat transfer near the CO> [1) degrees of freedom. On the other hand the thermal conductivity the difficulties which arise in measuring the thermal con- sizing shows some effect and the bulk viscosity even a bigger effect [2]. ductivity in the critical region. Thus there exists a counter example to Dr. Green's suggestion that the correlation function formulas should show roughly the same behavior. J. J. M. Beenakker: I have a problem concerning the dimensions of the apparatus in which one tries to measure the transport E. Helfand: I think the anomaly in the thermal conductivity properties near the critical point. As soon as one expects that arises from the thermodynamic force involved. Any molecular any anomaly in viscosity or thermal conductivity is related to model would speak in terms of energy transport. However, the long range correlations or fluctuations, care has to be taken the thermal conductivity is related to a temperature gradient. that the dimensions are large compared to the magnitude of the Therefore to go from the energy gradient to the temperature correlation length. gradient one has to introduce the specific heat. This is similar to the case of the diffusion coefficient where one has to transform a gradient of the chemical potential into a concentration gradient. M. E. Fisher: It seems to me there is no need to worry about For this reasion a connection between the anomaly in the thermal the dimensions relative to the correlation length because all conductivity and in the C, is not surprising. the light scattering and neutron scattering experiments show that even when you have what you call long range correlations,

they are still of the order of 1000 A when you come very close H. Zwanzig: 1 would suggest to the experimentalists to study to the critical point. And that is for the pair correlation function. the dependence of the friction on the frequency with the oscil- Most of these transport coefficients involve higher order correla- lating disk viscometer as well as observing slow oscillations. tion functions which one would expect to die out more rapidly. If there were nonlocal effects involving a penetration length Thus although in principle one could get difficulties very close to and a correlation length, one would perhaps be able to see from the critical point, I don't think one has to worry about this in the frequency dependence of the friction whether any peculiar practice. behavior would occur.

J. Kestin: One can regard all these experiments as having been J. J. M. Beenakker: In the case of an oscillating disk viscometer performed at zero frequency. When one wants to go to higher one has to consider a penetration depth between oscillating plates frequencies, one must consider the possibility that hydrodynamic and the gas layer. As in the critical region the density is chang- instabilities will be encountered. ing rapidly with height, I wonder whether you can make a state- As Dr. Sengers pointed out. I think one should measure the ment of the absence of an anomaly in the viscosity close to the two quantities: density and viscosity or density and thermal critical point. conductivity, simultaneously. This does not seem to be impossible. J. V. Sengers: That was one of the reasons why I made the statement that any anomaly in the viscosity is absent at temperatures more than 1 percent away from the critical temperature. At N. J. Trappeniers presented experimental data for the spin-

present I would rather leave open the question of the behavior lattice relaxation time oi deuterated methanes [3]. of the viscosity closer to the critical point. N. J. Trappeniers: We have also made diffusion measurements in the critical region, but I did not present jhose. because one J. S. Rowlinson: I would like to raise the question whether the has to be extremely careful to find the density. I think that one C r of a binary mixture has an anomaly near the critical mixing cannot say much, if one does not have accurate density measure- point. There is certainly an anomaly in C „ but as far as I know t ments. In the critical region the density varies extremely there is no anomaly in C, in this case. rapidly. Therefore I would like to ask Dr. Bloom the following: You make your measurements of the diffusion coefficients with an accuracy of about 5 percent in the critical region. If you J. V. Sengers: I don't know. I looked for experimental data for C, near the critical mixing point of binary mixtures in the litera- would make an error in density of 5 percent or even 10 percent, is easily if not dald, ture, but I did not find any. which made one does have accurate P-V-T would it not be possible to accumulate the errors by plotting the product Dp so that the scattering in your points thus obtained Pings: C. J. Dr. Sengers, did you not also run your experiments would explain the anomaly which you think to detect in the with two different distances between the plates? You seem to critical region? get the same results which proves the absence of an effect from convection. M, Bloom: I agree with you that it is dangerous. We have made density measurements in three different samples (these were J. V, Sengers: At the isotherms close to the critical temperature the impure samples) and I showed the ratio pi.lpv of the liquid (31.2, 32.1, 34.8 °C) measurements were taken using only one and gas densities. These were obtained by making measure-

plate distance, because it is difficult to do those measurements ments at the top end of the tube and at the bottom end of the tube, with a larger plate distance. However, the anomalous behavior while the density gradient was in the middle of the tube. We is already present at higher temperatures where measurements measured the ratio pi.lpv to within 2 percent. The diffusion with two different plate distances were indeed carried out, which coefficient was measured within approximately 2 percent. An give an additional proof of the reality of the anomaly. error of 5 percent has been assigned to D to take into account systematic errors and that is a conservative estimate. The anomalous decrease in Dp which we observed is about 20 per- M. S. Green: I want to point out that the difference in behavior cent. But I agree with you that it is dangerous. I think one between viscosity and thermal conductivity is theoretically very should do measurements with a flat horizontal sample to elim- surprising. Both the viscosity and the thermal conductivitv inate gravitational effects. can be expressed as the time average correlation of relevant fluxes. As the flux for the thermal conductivity and the flux

for the viscosity have a similar form, it is hard to imagine why C. J. Pings presented experimental data for the sound absorption there would be any difference in behavior. in the nitrobenzene-iso-octane system [4].

209 L. Tisza: To Drs. Chase and Williamson should go the credit for O. K. Rice: In connection with the paper of Dr. Garland and their experiments. However, I have to assume the responsi- Dr. Renard, I might say that about 12 years ago I wrote a paper bility for the prediction that the sound velocity tends to zero as in which I showed by purely thermodynamical means that for the critical point is approached. The argument is based on a compressible lattice the transition will change into a first- thermostatics and can be applied only to the real part of the sound order transition [6|. velocity in the low frequency limit. Since the absorption of not decrease, one can expect that the sound velocity sound need C. W. Garland: We agree that this can be proved in a phenom- would drop until the signal is lost. This seems to be the experi- enological way, but we considered that doing it for an explicit mental finding. model, in which the shape of the curve is given analytically by the Onsager solution, provided an additional way of looking at the problem with which people who like models can perhaps R. Zwanzig: I would like to make one remark about the general feel more confident. interpretation of ultrasonic data. Many years ago Dr. Tisza pointed out that ultrasonic absorption data could be explained formally by the introduction of a volume viscosity or bulk vis- M. E. Fisher: In this connection I would also like to draw atten- cosity into the hydrodynamic equations [5]. However, as far as tion to the work of Domb {7J, Bean and Rodbell [81 and Mattis

1 know, nobody ever does this in connection with ultrasonic and Schultz [9] who reach similar conclusions to those first relaxation, especially near the critical point. There is one good discussed by Dr. Rice but from a more microscopic view reason for this, namely that there is hardly any way of meas- point. It seems to me that the general validity of all these uring the bulk viscosity. The only way you do it nowadays is to treatments is open to doubt in respect of the conclusions drawn measure the sound absorption, so that it seems redundant. At about the changed nature of the transition point. The physical present, however, there are theoretical expressions for the bulk reason is the neglect of the fluctuations in the lattice expansion viscosity (derived first, in fact, by Dr. Green) and there is at least and the consequent "renormalization" of the original interaction a handle by which one can investigate theoretically what the responsible for the phase transition. This difficulty can be seen magnitude of the bulk viscosity ought to be. Very little is known explicitly in the case of a linear chain model which never has a about this either, but one does not have as yet a good reason to phase transition (as shown by an exact solution) but for which disregard the classical hydrodynamics with the bulk viscosity. the treatments referred to would predict a first order transition for certain parameter values. This has been demonstrated

by Mattis and Schultz [9] (although the arguments they give as K. F. Herzfeld: The situation is not so simple. It is completely to why this might not matter in three dimensions are correct, of course, that one can formally describe the sound unconvincing). absorption via a bulk viscosity. In a polyatomic gas the bulk viscosity is arising mainly from the relaxation of the internal for of degrees of freedom to their equilibrium distribution. In liquids, N. J. Trappeniers presented experimental data the C- however, there are more processes contributing to a bulk vis- ammonium chloride as a function of pressure along the \-line. cosity. At low frequencies one can describe the relaxation of the different processes with one "bulk viscosity constant." However, at higher frequencies one has to take into account the fact that the different processes relax with different relaxation times and show resonances at different frequencies. Conse- References quently the bulk viscosity is dependent on the frequencies and to analyze the different processes involved. The situa- one has [1] E. Schmidt, Int. J. Heat Man Transfer 1, 92 (1960). is analogous to that regarding the dielectric constant which tion [2] R. Zwanzig, J. Chem. Phys. 43, 714 (1965). be considered constant at low frequencies. only can [3] N. J. Trappeniers. C. J. Gerritsma, P. H. Oosting, Physica 31, 202 (1965); Phys. Letters 16, 44 (1965). A. V. Letters R. Zwanzig: That is right. The only reason why I raised the [4] C. J. Pings and Anantaraman, Phys. Rev. 14, point is the fact that there are presumably rigorous statistical me- 781 (1965). chanical expressions for the frequency dependent viscosity. [5] L. Tisza, Phys. Rev. 61, 531 (1942). The question arises whether the ultrasonic absorption can be [6) O. K. Rice, J. Chem. Phys. 22, 1535 (1954). described by these formulas or whether the ultrasonic absorption [7] C. Domb, J. Chem. Phys. 25 , 783 (1956). is due to something more complicated like scattering by density [8] C. P. Bean and D. S. Rodbell, Phys. Rev. 126, 104 (1962). fluctuations. [9J D. C. Mattis and T. D. Schultz, Phys. Rev. 129, 175 (1963).

~~210 SUPPLEMENTARY SESSION~~

~~21 1~~

~~209-493 0-66— 15~~

~~Effect of Gravity on the Equilibrium Mass Distribution of Two-Component Liquid Systems. The Observation of Reversed Diffusion Flow~~

~~H. L. Lorentzen and B. B. Hansen~~

~~University of Oslo, Oslo, Norway~~

At the conference the similarity between the producing diffusion. Therefore, when the gradient coexistence curves of one-component and two-compo- of the chemical potential is zero, also the diffusion nent systems has been stressed. The similarities coefficient will be zero.

between the two systems have to be extended to Bj0rn Bergsnov Hansen [1] and I determined the equilibrium mass distributions in vertical the diffusion coefficient at critical density as func- containers. tion of the temperature distance from the critical

In one-component systems it is well known that for the two systems: succinonitrile-ethanol and close to the critical point the mass distributions at triethylamine-water. We found the diffusion co- equilibrium are represented by sigmoid curves in efficients to be close to linear functions of the temperature distance from critical, the height versus density diagrams. This is caused AT—T—TC , by the effect of gravity. The same applies to the and approaching zero at the critical temperature. two-component systems. Thus in a vertical con- Determinations of the diffusion coefficients as tainer filled with mixture at critical conditions, function of the composition difference from critical

homogeneous mixtures do not represent equilibrium. Ac = c — c , as well as the temperature difference Due to the effect of gravity — also in the two-compo- from critical AT has shown the following relation- nent systems — equilibria in the critical region are ship to be basically correct. represented by sigmoid curves in the height versus n composition diagrams. (See fig. 1.) = k>AT+ \k2Ac (2) D \ In the experimental part of this paper will be shown exposures of curves of mass distribution, where k\ and k> are constants (k-z not the same on which spontaneously developed in two-component both sides of the critical composition) and n is liquid systems. They are representing equilibrium close to 3. mass distributions. One of the observations which support this is the observation of transport of material through the layers of the equilibrium distributions. The transport was towards greater composition differences.

It will be shown in three different ways that the equilibrium mass distribution in the critical region cannot be the homogeneous mixtures, but distributions represented by sigmoid curves in the height versus composition diagram. The developments are founded on the low values of the gradient of

~~the chemical potential (d/x/dc) in the critical region. According to classical thermodynamical view the~~

~~gradient of the chemical potential is zero at the critical point:~~

Composition Furthermore, according to modern views the gra- FIGURE 1. Schematic drawing of equilibrium mass distributions dient of chemical potential is the virtual force in the height versus composition diagram.

~~213 1~~

This is in agreement with theoretical considera- In a binary mixture the equilibrium condition tions which Professor Joseph Mayer had pointed will be governed by the equation out to me [6]. dx_ Mi —pVt When the diffusion coefficient approaches zero g (3) dh dfJL\ldx at the critical point the thernutl diffusion ratio approaches infinite. In a binary system a temperature where x is the mole fraction gradient leads to a flow which will ultimately sep- n\/n\ + n>, h is the height, is the molecular arate the liquid in two layers. However, by this M weight, p is the density of the mixture, V is the partial flow a composition gradient is built up and conse- molar volume, \ \ = {dV/dn, the quently a mass diffusion flow will counteract the chemical potential and g the acceleration of gravity. effect of the temperature gradient. At the When in the critical] region dfx/dx is very "Endzustand" the two effects are in balance. small, dx/dh will be great. Reversed diffusion flow. It was mentioned in At the critical point the diffusion coefficient ap- j the introduction that diffusion flow towards greater proaches zero and thus the composition gradient j

composition ! dc differences has been observed. Fick's will approach infinite. —jjj law is founded on the observation that at normal ! conditions the Studies of the thermodiffusions in the critical Endzustande of the diffusion process is that of region of the system aniline-cyclohexane here have a homogeneous mixture. When, however, homogeneous mixtures do not represent shown the Endzustande to be sigmoid curves with ||

equilibrium, the diffusion process will i'( the greatest gradients at critical composition. not go towards homogeneity, but towards the equilibrium \ The gradient increases where the temperature j condition. of the critical composition approaches the critical, Fick's law then may have to be rewritten in the and the gradient may well become infinite at this form: temperature. When this is so, it is obvious that, when the temperature gradient diminishes towards dc dc\° = : zero constant) and the Endzustande goes J -D* (4) (T= ~dh dh towards that of an equilibrium, the mass distributions at this equilibrium will also be sigmoid curves where h is the direction of the diffusion flow in a in the height versus composition diagram. vertical container, dcjdh is the concentration In a binary mixture the molecules of the compo- gradient of the liquid with respect to the height; nents will have different buoyancy relative to the dc/dh approaches the equilibrium distribution mixture. Therefore, the molecules of one of the characterized for the temperature and composition components will have a tendency to move upwards in question by the gradient (dc/dh)0 at equilibrium. and the molecules of the other to move downwards I dc fdc\"\ The term \rrr\ of eq (4) will be negative in the liquid. Consequently, a composition gradient yj^— j

will develop and an inhomogeneous equilibrium when (dc/dh)° is greater than dc/dh. The flow./ is distribution will be established, whereby these then negative, and we will observe a "reversed tendencies are balanced by diffusion flow towards diffusion flow." homogeneity. Reversed diffusion flow can also be observed

~~Because of the overwhelming influence of the independent of the effect of gravity. If a mixture~~

diffusion flow such effect has not been observed is left at a temperature of the coexistence curve at normal conditions. In the critical region, how- the mixture may be split in two phases divided by

ever, where the diffusion coefficients are very small, a surface. This is schematically shown on figure 2. the effect will have to be remarkable, and at the The lines AB and B'A' indicate equilibrium phases.

critical point the effect will result in an equilibrium When now the temperature is changed in the di- distribution of an infinitely great composition rection away from the critical temperature, then gradient. the compositions of the phases at the new tempera- In a liquid the potential energy increases with the ture will be outside the compositions corresponding height. In liquid mixtures — at equilibrium — the to the former temperature (dotted lines CD and difference of potential energy of the layers at the D'C) various heights will be balanced by a change of the Generally the formation of the new phases will be composition of the mixtures with height. introduced by fog formation. In some cases, 214 The compositions of the mixture at the various heights inside the container were determined by

means of an integral optical system [3]. This allowed visual observations and photographic re- cordings, which actually are diagrams with the +> height in the container as ordinate and the optical density as abscissa. In the region of the critical solution points, one observes that the mass transports towards equilibrium distributions are using the formation of fog as vehicle. Thereby droplets of compositions at densities different from that of the mother 1 liquid are formed and move by buoyancy upwards or downwards in the liquid mixture. After a period of time the fog has disappeared and a new mass i A' Composition distribution can.be observed. When the temperature of a nearly homogeneous FIGURE 2. A schematic drawing of the mass distribution at mixture of a composition close to the critical, and I two temperatures Ti and T>.

T, is the temperature closest to I he critical, and the lines AB and A'B' indicate the at a temperature outside the coexistence curve, is two phases at equilibrium at 7Y The temperature difference between T, and T± is very small, and the dulled lines DC and D'C indicate the equilibrium distribution changed to one of the temperatures of the co- 1 at T,.

When the temperature is from 7*, to changed T± no fog is formed. A diffusion How existence curve, the material of the sample is then of material passing the surface DD' will be observed as mass distribution curves of the form AD and A'D'. ( brought to a state of unattainable instability. Two phases must then develop and they do so probably instantaneously by the formation of fog. however, when the phases remain "undercooled," Figure 3 shows an example whereby a nearly the formation of fog does not take place, and a homogeneous mixture of succinonitrile-ethanol was diffusion flow of material will take place. The cooled to about 50 mdeg below the critical. When diffusion flow is crossing the surface border and the fog had disappeared the composition of both is a "reversed diffusion flow" towards greater dif- phases were inhomogeneous and had the form of ferences in composition. two arms of a sigmoid curve. The two arched curves at the surface together with the straight lines at greater distance from the surface show the distributions observed after a period of time at the new temperature.

~~I have not been able to make an exact determination of the form of the arched curves. The form appears, however, to be close to the error function curve observed by free diffusion.~~

~~Experimental Procedures~~

The average compositions of the mixtures were close to the critical compositions. The liquid mixtures were sealed in all-glass containers— ca. 15 mm high and of about 5 mm 2 cross section area. The liquid mixture almost co ipletely filled the container. Figure 3. System: Succinonitrile-ethanol. The container was placed in a thermostat which Ordinate: Height in container. allowed the temperature to be controlled with an Abscissa: Composition. Straight vertical and horizontal lines are orientation accuracy better than ± 1 mdeg [2]. The tempera- lines. ture was determined by means of a platinum resist- Figure 3 shows the composition which appeared when the near homogeneous mixture of succinonitrile-ethanol had been cooled from a temperature above the critical down ance thermometer. to a temperature of about 50 mdeg below the critical and the fog had disappeared.

215 Figure 3 is representative for a great number of This surprising reformation of the mass distribu- observations also with other liquid systems which tion was then systematically studied, and Bj0rn have been studied here. Bergsnov Hansen and I subsequently found a tem-

Figure 3, however, does not represent the sole perature treatment of the sample — which was sealed form which the phases may show when the fog dis- inside an all-glass cell — whereby we could reproduce appears. In some cases only one of the phases may the phenomenon ad libitum. The temperature of be smoothly sigmoid and the other closer to homo- the thermostat was set at a temperature below the

geneous. This is explained by convection flow in critical (outside of the coexistence curve). The this phase during the establishment. cell was then removed from the thermostat, warmed When the temperature of the mixture is not to room temperature and cooled to about 10° below brought to the temperature of the coexistence curve, the critical and carefully tilted. Thereby a certain a redistribution of the material can not be enforced. concentration gradient was established in the liquid

~~It happened, however, that in a mixture of tri- mixture. (Typical is the lower curve of fig. 5.) ethylamine-water the reestablishment took place The container was then reinserted in the thermostat~~

also at a temperature outside those of the coexist- and it was observed that the composition curve re- ence curve and in the temperature range reaching mained unbroken. from the critical temperature to about 100 mdeg After periods of time of different duration at outside the coexistence curve. constant temperature a belt of fog was spontane- The system triethylamine-water was studied at ously formed at varying levels in the liquid (in the lower critical solution point of the system. At all cases but one in the upper middle section of a lower solution point the temperatures outside the the liquid column). The belt was at first fractions coexistence curve are below the critical tempera- of a millimeter thick and gradually developed up- ture, and the temperatures of the coexistence curve wards as well as downwards in the liquid. above. Figure 4a shows a photograph made after the During the general study of this system the fog had appeared. The fog consisted of two layers sample was left for one night at a temperature out- separated by a sharp boundary. The upper layer side the coexistence curve. Next day the arbitrary of fog entirely veiled this part of the liquid. mass distribution curve was found to have been The lower layer of fog allowed the fight to pass partly reformed into a smoothly formed arched and disclosed that the mixture at these levels had curve in the diagram height versus composition. attained nearly homogeneous compositions. The nature of this fog, however, was such that light was scattered, and at these levels a background illumination was observed. 30 min later the fog had disappeared and, where the upper layer of fog had been, the curve had attained a smooth arched shape leading downwards to a kink from which the curve by a near-vertical section continued into the original curve. (Fig. 4b.) The smooth arched curves represent layers with defined composition gradients. The gradient increases towards critical composition, and must be regarded as a section of an equilibrium distribution \>—A curve. i% As mentioned the fog started as a very thin belt, broader with time. In one series of FIGURE 4. System: Triethylamine-water. which grew Ordinate: Height in the container, scale shown on experiments the temperature was immediately the right side of the figure. Abscissa: Composition of the mixture, scale unit lowered just as the fog appeared. Thereby the below photograph 4. a. fog disappeared, and a comparison between the Exposure 4.a shows ihe originally straight curve partly disrupted by layers of log however, allows the light to pass and Below this layer is another layer of fog. which, curve, before and after the fog disappeared, showed produces background illumination at these heights. Exposure 4.b shows the composition curve after the fog had disappeared. The that the fog was first formed where the original curve has attained a regular arched form where the upper layer of fog had been. When making the exposure a. the horizontal orientation lines were not illuminated. gradient and the gradient of the developed arched The photographically recorded composition curves do not represent the entire column of the liquid mixture. curve were equal (fig. 5) [5]. 216 When enough time had gone, the smooth arched curve grew past the critical composition and passed a point of inflexion. From this point on the composition gradient decreased with increasing distance from critical composition. Maximum gradient was found at — or very close to — critical composition. When the smooth arched curve was observed to grow in both directions this naturally was caused by an increase in the height of these layers upwards as well as downwards. This was a consequence of a transport of material through these layers, which transport was a diffusion flow towards increasing differences in composition, "reversed diffusion flow." Figure 6 and figure 7 show exposures from two series in which the curves mentioned above had been established. (See legend.) In the series of figure 6 the temperature was lowered in steps (away from the critical). When the temperature

FIGURE 5. System, ordinate, and abscissa as on figure 4, how- changed, a fine fog veiled the regular arched sec- ever, different scales. tion of the curve for a shorter period of time, and Two photographs are mounted together by dislocation along the ordinate. The lower is the original composition curve before the development of fog. The upper curve shows the composition after the liquid mixture had been cooled and the log had thus disappeared. Visual observation had disclosed that the formation of fog started at composition close to the left vertical orientation line, where the exposures show that the gradients of the original and the developed regular arched curves are the same.

The latter observation explains why the belt of fog appeared at compositions somewhat different from critical composition (in all cases but one at triethylamine richer mixtures). The smooth arched curves which appeared after the fog had disappeared generally did not reach to the critical composition. When the system was left undisturbed at constant temperature after the fog had disappeared, the smooth arched curve Figure 6. System, ordinate, abscissa and scales as on figure 4 grew in both directions. The form of the originally Figure 6.a shows the regular arched curve developed In both sides of the critical composition dining 10 hr al constant temperature. 41.5 mdeg below the critical. formed curve remained unchanged. It was, how- The temperature was then lowered to S3 mdeg below critical, and figure 6.b shows the composition curve after 7 hr at this temperature. The composition gradients had ever, observed slowly to change its height in the decreased in the layers of the smooth arched composition curve. A comparison of the "lower arm" of the curve does indicate that a diffusion towards greater composi- container. tion differences has taken place (towards a continuation of the regular arched curve).

~~+ 2 mdeg +3 mdeg + 5 mdeg + 17 mdeg (exp. 25) (exp. 26) (exp. 27) (exp. 28)~~

FIGURE 7. System, ordinate, abscissa and scales as on figure 4. Pive photographs of the same series with stepwise increasing temperatures, mounted in successive sequence of time. The opening of the concentration curve in figure 7a is due to the optics of the apparatus. In figures 7b-e two phases are formed.

217 in this section the form of the curve changed component towards the bottom of the container. basically by an increase of the composition gradients In the same way the known sigmoid form of the mass dc/dh. distribution curves in the one-component systems I Figure 7 shows exposures of a series with step- can be regarded as a consequence of the fall of wise increase of temperature. Here too a faint the molecules balanced by a diffusion upwards. fog was formed as in the series above. At the tem- This leads to the conclusion that the diffusion coeffi- peratures of the exposures 7a and 7b the smooth cients also of the one-component systems are very j arched section of the curve changed basically at small in the critical region and will apply to an equa-

~~each step by a decrease of the gradients dc/dh at tion analogous to eq (2).~~

the various compositions. Hirschfelder [4] and Prigogine [5] have pointed out At the step 7b the critical temperature is passed that diffusion measurements are of fundamental | and the concentration curve is broken close to importance for the study of liquid mixtures on the critical composition. molecular level.

~~Exposure 7d is taken at a temperature 17 mdeg By experimental determination of the equilibrium~~

~~above critical. The left part of the curve is arched mass distributions in mixtures, preferably in fields I~~

basically in the same way as shown on figures 1 of increased acceleration, one should have an ! and 3. The upper part of the right side leading to additional method for the study of liquid mixtures — the surface dividing the phases is nearly vertical. not only in the region of the critical solution points, The reason for this may be a convection flow in but also for the study of liquid mixtures in general. ) the lower phase during the establishment of this phase. End Remarks

The reestablishment of the mixtures were ob- In the beginning of this paper it was said that the served in 1957. Professor Joseph Mayer kindly effect of gravity added to the analogy which is known J reviewed the reports from the experiments and I to exist between the binary and the one-component showed that the form of the new established curves 1 systems. This may be further illuminated:

chemical poten- j in were related to the gradient of the 1. The presence of a temperature gradient tial and consequently to the diffusion coefficient as U the binary systems leads to the Soret effect. In function of the temperature and composition. I I the one-component systems a temperature gradient am very much in debt to Professor Mayer, who will lead to mass distribution curves which will thereby gave me the clue to the problems. constitute isobars in the temperature-density I also express my thanks to Professor Odd it must however be re- diagram. In both cases j Hassel for his interest and assistance. Further- membered that the effect of the temperature gra- more, I express my thanks to Professor E. G. D. dient in the experiments will come in addition to Cohen and Dr. J. M. H. Levelt Sengers, who have the effect of gravity. one- reviewed my paper. It may also be mentioned that when in the component systems the isobars are experimentally References known, and also the gradient dp/dT at the critical density, the isotherms can be constructed. A simi- [1] Hans Ludvig Lorentzen and Bjorn Bergsnov Hansen, Acta lar relationship may be expected to hold in the Chem. Scand. 12, 139 (1958). Hans Ludvig Lorentzen, Acta Chem. Scand. 7, 1335(1953). binary systems between the Soret effect and the [2] [3] Hans Ludvig Lorentzen, Acta Chem. Scand. 9, 1724(1953). effect of gravity. [4] Hirschfelder. Curtiss, Bird, Molecular Theory of Gases and Liquids, p. 539 (John Wiley & Sons, Inc., New York, N.Y., 2. It said that the effect of gravity in the was 1964). of (North- binary systems was a necessary consequence [5] Prigogine, The Molecular Theory of Solutions, p. 51 Holland Publ., 1957). the small diffusion coefficient in the critical region, [6] Professor Joseph Mayer, private correspondence (Chicago, j caused by the fall of the molecules of the heavier June 21, 1957).

~~218 ( < <~~

~~Ferro- and Antiferro magnetism of Dilute Ising Model~~

~~S. Katsura and B. Tsujiyama~~

~~Tohoku University, Sendai, Japan~~

~~1. Introduction Quenched random dilute magnetism has been in-~~

vestigated by Behringer [4], Sato, Arrott and The properties of alloys of magnetic substance Kikuchi [5], Brout [1], Rushbrooke and Morgan [6], and nonmagnetic substance are investigated experi- Elliott and Heap [7], and Abe |8j. In experiments mentally and theoretically as problems of dilute ideal randomness in Brout's sense is not realized 1 magnetism. Here we consider Ising spins of S = /2 an actual "randomness" depends on the history as a magnetic substance and the energy of the of the preparation of samples. Experimental situ- system is assumed to be given by a sum of exchange ation lies, more or less, between two idealized limit- energies between nearest neighbor Ising spins, ing cases. Hence it seems worthwhile to compare positive or negative, corresponding to ferro- and physical properties of those two systems for exactly antiferromagnetic coupling. That is, the inter- solvable cases. action energy between nonmagnetic ions and that On the other hand, Syozi [9] investigated a two- between nonmagnetic ions and Ising spins are dimensional decorated lattice model of dilute mag- neglected. This restriction can easily removed. be netism, where Ising spins locate on lattice points The calculation based on the inclusion of the inter- of regular two-dimensional lattice, and nonmag- action energy of nonmagnetic ions will be carried netic atoms or Ising spins locate on middle points out in future. between nearest neighbor lattice points (fig. 1). The idealization of the preparation of the system He obtained exactly thermal properties of two- can be classified into two limiting cases. We dimensional lattices by reducing its partition func- denote the spin average by )., and the configura- ( tion to Onsager's partition function. We can get tion average by -. ( ) ( thermal and magnetic properties of the one-dimen- The free energy of the system which was heated at sional Syozi's model since the original Ising model infinitely high temperature and was cooled infi- of a one-dimensional system is known in a mag- nitely rapidly up to temperature T is given by netic field. We will also compare properties of Syozi's model with above mentioned two cases. FQ =-kT(ln(Z)s) c . (1.1)

In this system the true thermal equilibrium is not realized and it is a free energy of an idealized quenched system in which ions are frozen randomly at their positions. 1.1.1 On the other hand, if the temperature and ex- r t ternal parameters are varied infinitely slowly and

~~< » i > < » O true thermal equilibrium is realized, the free energy of the system is given by > • > < ) > 0- — • < — — • •~~

~~FA =^ kT In (Z)s, c . (1.2) < t i o~~

Although the interaction energy of the nonmag- ) 9- — ) •— —•— netic ions are to be taken into account, the system corresponds to an idealized annealed system. The distinction between these two cases has been FIGUREm1. Syozi's decorated lattice model of dilute Ising model. discussed by Brout [1], Mazo [2], and Morita [3]. O Isinj- spins, • Ising spins or nonmagnetic atoms.

~~219 ) =~~

~~2. Annealed System: Formulation by By solving D from (2.3) as a function of p and C, Generating Function Method D = D(K, p, Q (2.4)~~

In this and next sections, it will be shown that the problem of the annealed system is formulated can be derived. D corresponds to chemical po- by introducing a generating function in case of tential of spins divided by kT in an alloy of non- Ising spins, and that in the case of a one-dimen- magnetic atoms and spins. When (2.4) is sional problem evaluation of the generating function substituted into is reduced to an eigenvalue problem [10]. More- D) - E= E(K - c (2 5) over that eigenvalue equation is solved exactly ~i {^dir) and the free energy, the energy and the susceptibility are obtained. and

A crystal lattice whose lattice point is named i = 1.2 .... N is considered. On each lattice M = m =M(K, C, D), (2.6) (~^) k d point there is either an Ising spin of S = V2 or a nonmagnetic atom. If p, = 1, 0 or — 1 is the state energy E and magnetization M can be expressed as of each lattice point, the total energy of this system functions of K, p, and C. This problem is equiva- is given by lent to the Ising model of S = 1 with anisotropy. = E ~ -m%"2 p,.. 9 X ^3 (2. 1

is factor, Here m magnetic moment (=\ gfxB , g'-g 3. Annealed System — Continued. (Mb- Bohr magnetron) of spin, £f is external magnetic Eigenvalue Problem in the Case of field. J > 0 corresponds to ferromagnetic inter- One-Dimensional System action and J < 0 antiferromagnetic interaction. ^ (0) The formulation in 2 leads to an eigenvalue prob- denotes summations over all pairs of neighboring lem in the case of one dimensional system. Gen- lattice points. The second term is Zeeman energy. erating function Z is given by p = 1 or — 1 corresponds to Ising spin (up or down) and = to nonmagnetic respectively. p 0 a atom, = N Z TrV , (3.1) Instead of considering the problem of the system in which concentration of Ising spins is given by p i v=v\iwy v,vywy\ (3.2) (that of nonmagnetic atoms by 1 — p), the generating function Z with parameter D is calculated. under a periodic boundary condition. Here

~~±1 '° (2-2) ^; vA\ 111 V2 = \ wnere~~

~~2k? kT = Vz \ 1 (3.3)~~

~~Since p| = 1 for spin states and 0 for nonmagnetic atoms, If the maximum eigenvalue of V=V^WpVxV^WT~~

~~d In . 1 Z _ \ __ _ /js * s ^maxi ^ can De expressed in terms of Amax in the ? PK Y) C) (2 3) P ' /" ' N dD ' limit of 7V-> oo~~

~~220 ,~~

~~S lim Z" =K (3.4) we have~~

~~S.7'Sr>rS,S 2 = H<°>+H<»~~

The explicit expression for \max (K, C, p) which is a solution of a cubic equation is much too compli- (H<® 0 0 o mi cated to be differentiated analytically. It is suf- = o m o o m (3.8) ficient, however, to derive X exactly only up to max o n§ m) o the second order in powers of C since we are interested mainly in the energy and the susceptibility at zero field. From the symmetry of this matrix, where 6 is determined in such a way that H\V van- 2 the eigenvalue at C — 0 and the coefficient of C in ishes. Hm is an even function of C and HiU an the expansion of Amax can be obtained simply by odd function. solving a quadratic equation. V can be written

(0) as a sum of even function and odd function of C. In order to obtain \max in case of C — 0, // is regarded as an unperturbed term and H (1) as a perturbation. From second order perturbation theory k2dP ch C. a ch — k~ 2 d2 (first order perturbation term vanishes) = —c —c V d ch , 1, d ch ) >w=M°i + (3.9) r 2 2 k-'d , ach^, k cP ch C

~~Taking C — 0 into consideration, H^, H\i§, and~~

~~2 2 Hty are expanded with respect to C, k d sh C, d sh |,~~

~~2 01 4 H^} = k°° + C k + 0(C ), (3.10) + d sh ^, 0, -dsh (3.5)~~

~~2 2 2 n 4 0, -dsh k d sh C 5- = C \ + 0(C )- (3.11) H<$-H<®~~

~~2 The term of 0(C ) does not appear from the third where and higher order terms of the perturbation.~~

~~Thus the maximum eigenvalue of (3.2) is obtained eD~~

~~pip — l)(k- + k--)- (** + A-*-4)p(p-l)-l-(2p-l) V2<2 - A* - A- *)p(p- 1)+1 0~~

~~V2 (3.6) (3.12) 0 The energy E and the susceptibility \ m case 2 of <$f=0 is expressed in terms of d~ .~~

~~1 and 1 A- + A:- NJ' 4k-k~ l~~

~~4, + k~ 2 -d- X 1 + /sin 0, / 2 2 2 2 2 \ (k + k- -d~ ) + 8d- .~~

~~S-i = \ cos 0, (3.7)~~

~~\ 0, (3.13)~~

~~221 + p 2~~

~~ATX = 2(X01 + X") Expanding the product we have Nm- A 00~~

~~1 (Z) = h'V-C s ch^C £ X X {flMYjlvIM')~~

A 4 + l-d" 2 (A 2 -2) + A 2 2 2 2 2 2 V(A' + A- -J- ) + 8rf- ffl n) m n X (pi'" Vj >)fikfik' • p4 th i<:th C. (4.3) 4] (A 2 + A- 2 -d- 2 )(A4 ^-j+4rf- 2 F 4A- 4 + 2rf" 2 + 2 2 2 2 2 Carrying out the summation with respect to p,=± 1, V(A- + A- -rf- ) +8rf" we have 3A- 2 - A 2 2 2 2 2 2 2 + d~ + V(k + k- -d- ) +8d-

N (Z) s = 2 - ch^ch^C X Number of graphs (3.14) 2 2 [ »i n which have m bonds and of which the sum of the number of end points and odd junctions is d~ 2 in (3.13) and (3.14) is regarded as given by m n n in that configuration] X th Kth C. (4.4) (3.12) in terms of p. Substituting p—l into (3.13) and (3.14), well-known results on the Ising model

When we take the logarithm and divide by N, only E 1 u v ATx (3.15) Wm 2 the term proportional to N in the bracket survives. Restricting our interest in th°C and in th2 are reproduced. C terms in the case of one dimension, we find that the average number of graphs which have m bonds and two

~~m + 1 end points is Np , since the probability of finding an Ising spin at each lattice point is p. Taking the 4. Quenched System configurational average by ( } c , we have~~

~~Next we consider the quenched system in one- dimension. The partition function for a given con-~~

~~figuration of nonmagnetic atoms is given by [n = 0, m = 0] = l,~~

~~[n = 0, to=1, 2, 3, . . .]=0 (Z) = exp \k s £ £ p.m + C£ pj (4.1) +1 [ii = m] = Np"' < 2, )c ,~~

~~(Ns )c = Np, = chPKcWiC ^ n~~

~~M N ch Kch sC 1 II ence X f + ^i^AK + 2 E~~

~~2 . = (ln(Z) s > ( /Vp lnchA: + /VplnchC + /Vp log~~

~~2 m+1 m 2 + ln l+N^th Cp th K (4.5) X(^ Lt / )(^;, ,)th /^+. . . pj J j~~

~~- z x(l 2 lim ( In ( Z ) s > f = plog2 + p lnchK + plnchC + S^-thC + 2SAt^th C+ . . . .) (4.2) AT->o=/V~~

~~Here = ±l, in contrast to the annealed system. + 1 i 2 4 p + ^p'" th" ^th C + 0(th C). (4.6)~~

Ns is the number of Ising spins. M is the number of bonds connecting Ising spins. The summation Thus the energy is given by only the first term ^ in (4.1) is carried out for only Ising spin pairs. W of the series and the series for the susceptibility 222 I can be summed, giving 5. Numerical Results and Discussions

j a(ln(Z),s ), ._ NJ E thK > (4.7) Now we compare the numerical results of three 2 dK — -TP~ cases, an annealed system, a quenched system, and Syozi's model. ^{i +22;p-th-jc} (4.8) Figure 2 shows the energy in zero magnetic — — field. The energy of Syozi's model for p is (p \ ) 2 pthX ^ m Np 1 + X tanh K for \ < p < 1. Figure 3 shows the inverse kT 1-pthK susceptibility of ferromagnetic case. Figure 4 shows the susceptibility of an antiferromagnetic 2 kT/ I J case in the annealed system. Figure 5 is that for the quenched system and figure 6 shows that for Syozi's model. As the temperature goes to zero,

the first tends to zero while the second and the third tends to infinity. Figure 7 shows the concentration dependence of the antiferromagnetic susceptibility of the quenched system and the annealed system. In the above results the difference in the low temperature susceptibility between the annealed system and the quenched system in the case of

~~antiferromagnetic interaction is most drastic. In the quenched system the susceptibility at the concentration p is obtained by the arithmetic mean of each configuration corresponding to the same~~

~~I JI"X /Nn~~

~~FIGURE 2. Energy in zero magnetic field. Solid line: the annealed system. Dotted line: the quenc hed system (0 S p S 1 1. Dashed line: Syozi's model (-J ^ p 1).~~

~~Annealed System~~

~~Ouenched System~~

~~Syozi~~

~~5 2kT/ J~~

FIGURE 3. Inverse susceptibility in the ferromagnetic case. FIGURE 4. Susceptibility Solid line: the annealed system. Dashed line: the quenched system. Long dashed of an antiferromagnetic case in the line: Syozi's model. annealed system.

~~223 2 "X IJI X/Nm IJ I /n~~

224 concentration p, and the paramagnetic configura- IBM 7090 in UNICON (University contribution tion gives the dominant contribution compared with of Japan IBM). the antiferromagnetic configuration. On the other hand, in the annealed system, the energy of the 6. References paramagnetic configuration is high compared with [1] R. Brout, Phys. Rev. 115,824(1959). the antiferromagnetic configuration, and hence it [2] R. Mazo. J. Chem. Phys. 39, 1224(1963). dies and the susceptibility tends to zero. [3] T. Morita, J. Math. Phys. 5, 1401 (1964). [4] R. E. Behringer, J. Chem. Phys. 26^ 1504 (1957).

~~[5] H. Sato, A. Arrott, and R. Kikuchi. J. Phys. Chem. Solids. 10, 19(1959).~~

[6] G. S. Rushbrooke and D. J. Morgan, Mol. Phys. 4, 1 (1961): ibid. 4, 291 (1961); G. S. Rushbrooke. J. Math. Phys. 5, The authors wish to thank for valuable discussions 1106 (1964). R. J. Elliott and B. R. Heap. Proc. Roy. Soc. A265, 264 Professor R. Kubo, Professor M. E. Fisher, Dr. [7] (1962). S. Inawashiro, and Dr. T. Kawasaki. The numeri- [8) R. Abe, Prog. Theor. Phys. 3 1, 412 (1964).

I. Syozi, Prog. Theoret. Phys. 34, 189 ( 1965). cal calculation was carried out using NEAC 2230 [9] [10] H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252. 263 in the computing center of Tohoku University and (1941).

~~The Statistical Mechanics of a Single Polymer Chain~~

~~S. F. Edwards~~

~~University of Manchester, Manchester, Great Britain~~

It is clear from the papers presented at this con- for the probability that the end points are at 0, r erence that near critical points thermodynamic when the length is L. This has the form unctions and correlation functions take on forms hich reflect only a very small amount of the de- ailed structure of the systems considered, and re rather manifestations of the three dimensionality

~~f space and the short range of interactions. The onfigurational statistics of polymers shares this~~

~~eature and is therefore interesting as an indication~~

~~f the critical behavior be generated how may (1.2) athematically: in fact the problem is one way of pproximating to the Ising lattice as was mentioned~~

y Professor Domb [1]. In this talk I shall develop he theory applied to polymers themselves however, nd comment on their critical behavior, since hether the analysis will really be useful for bulk ystems remains to be seen. A polymer without interaction will have the well-known distribution For the case of excluded volume i.e., a repulsion ' 3Iz (2ttLI\- / 3r 2 \ , between elements of the polymer, p has no simple p(r,L) = exp(-—) (1.1) (—) analytic form, but salient features are shown in

~~225 y~~

~~the diagram (schematic) (1.1) including the case of hard exclusions in which~~

~~u |R,„-R,,| < a (2.4)~~

~~= 0 otherwise.~~

~~A lattice polymer, can also be included if the defini-~~

~~2 3 6/5 tion (1.3) is suitably modified, but it is again a com- P d raL plication not a simplification. The probability (1.3) that starting at the origin, the polymer's nth fink~~

~~will end at r, is given by~~

~~P(r, L) - R„)8(r-~~

~~. ^9/5 (2.5)~~

where L, the length, is nl. A method of evaluating p for a slightly simpler problem, that of the infinite polymers, has been given by the author (reference Expressions will also be obtained for the number of hereafter referred to as I) in a paper which configurations allowed to the polymer. [2], develops a physical picture of what is going on and translates the physical picture into mathematics. 2. Mathematical Formulation of the A more formal but precise method of approach is Problem given in an appendix to I, and in this paper, after reviewing the physical arguments, it is this formal Firstly the problem is to be stated mathematically development which will be given and extended to and brought down to its simplest form. Suppose give a more complete theory of the single finite the polymer consists of n identical molecules freely polymer. hinged in a chain, and there is a potential U be- Firstly it must be explained that the fact that tween molecules. If the length of one molecule th there are n integrations in (1.1) is a nuisance; it is is /, and the coordinate of the end of the m molecule of finding a particular easier to consider a continuous function R(L') where is Rm , then the probability 0 < L' < L and integrate over all functions R(L'). configuration P(Ri, .... R n) is given by The constraint FJ p(R;, Rm) is the expression (R/ R;+,) P(R, R»)=^n^ ' originally considered by Weiner when he introduced functional integration and can be expressed as exp U{R,„-Ri) (2.1) ^ £ 3Z 3R w=t=; JV exp dL' (2.6) ( — dU

~~where Jf is the normalization so that, putting V= Ul~ 2~~

~~R,i = r PYl(PR = l (2.2) ' j p(r, L)=JT [ d(path R{L')) J R(0) = 0~~

~~and p is the constraint 3/ BR exp dL'~~

~~p(Ri, Ri+1) = 8(|Ri-Ri+ i|-*)- (2.3)~~

1 dL' (Note since normalizations are constantly occurring V R(L')-R(L")) (2.7) ~2kT Jo /o dL" • they will all be denoted by JV.) Of course, one can put structure into the molecules making U an integral over the molecule and one can put in bond flexibilities, but the basic problem is already in At this point a parametric representation can be

~~226~~

! employed. It is well known that case one has to split V into a part with positive Fourier coefficients and a part with negative Fourier X coefficients e^-^i-dx | e-^-dx. (2.8) and parameterize them separately. | i This will be equivalent to \ becoming complex and the functional integration being over its real This can be generalized to and imaginary parts independently. Although physical forces always do have attractions and exp ( — OiAijaj/2^ 2 repulsions it is well known from the study of the Ursell-Mayer cluster expansion, that provided the — = I I exp ( / ajXj XiAjj?Xkl2 ^ ^j? system remains of low density the effects of inter- \ / J-xJ-x j f. ft particle potentials can be simulated by pseudo potentials which will give rise to the correct virial g-s^- ncfec, (2.9) coefficient. Thus if the excluded volume is defined by where is the inverse matrix to y4, and hence to = the functional form V j [1-exp (-U/kT)]d?r (2.14) exp a(a)A (ap)a(p)dad(3) (-j the effective pseudo potential is v8{r), whose in-

~~verse is v~ l 8(r). It will be assumed now that v~~

~~i a(a)x{a)da {Yldx) exp is positive, i.e., the mean effect of the forces is a repulsion. Then finally~~

~~x(a)A- l (aP)x(f3)dadl3~~

~~exp x{a)A- l ((x(3)x{fi)dadp Yldx L (K{L'))dL' 2 (S)(PS 8 I X X X8R (2.10) 2v~~

~~1 (8R for the integral over all paths). where A is the inverse operator to A, (2.15)~~

1 = - j A- (a/3)A(/3y)dl3 8(a y). (2.11) The theorem states that the probability p is that of diffusion in the presence of a potential ix (a Markov This theorem allows one to write process), averaged over all potentials with a

~~Gaussian weight. It is a well-known theorem else-~~

~~' " ' " v(R{L - R{L )]dL dL ~2^rr ) where in physics. Since the evaluation of the functional integral over 8R is equivalent to solving~~

~~r the diffusion equation, one can obtain the final = Jt exp (R(L'))dL' \ 8\ x form~~

~~AT p(r,L) = jrj 8 G{r,L, s 1 X [xiWdxWx (2.16) d r(PSx{r)V- (r-s) ls) (2.12) ~ | | X where~~

~~wh ere 2 3 r([X ]) = exp(--| X d s (2.17)~~

~~V-\rs)V(sr')(Ps = 8(r-r'), / (2.13)~~

2 / (r)l' r. / r ,v/ ^-^V + X x and 8\ represents the integral over all functions. j (2.18) _1 cor- It is supposed that X i s positive definite, j A and Jf~Jf(JS) is defined from pd r— 1. responding to V being repulsive. If this is not the

~~227~~

~~209-493 O-66— 16 3. The Solution where contains the angular derivatives. It will turn out that the steepest descent function is~~

~~Two problems have presented themselves in spherically symmetric, so the angular terms will~~

~~(2.16). Firstly one has to get G from (2.18) and be ignored for the present and treated as if it~~

~~were E) r also if it r then do the integral over X- It will be argued that~~

If there were a very strong self-repulsion, the poly- l/6iV /2 iL- (E + Xr m ds (3.6) mer would shoot out in a straight line and the aver- 2 W age would be taken over the orientation of this 2 -1 i.e., (47rr . These to straight line, would give ) and functionally with respect x give upper and lower bounds on the form of the mean polymer density so that one expects the x 1 l2 s

~~in fact it will turn out to go like -jjx. and \ and 6 r2 r* Therefore one may write also It is convenient to Fourier transform G with~~

~~2 respect to L I p- iE — — x( s )ds. (3.8) V Jo~~

~~2 = iE-^V + ix [ X ]) S(r) (3.1) = i Clearly X X< - E) (3.9) and to seek a solution in the form and E = E{i\ L). (3.10)~~

~~Make these equations clearer by introducing G = e*/r. (3.2)~~

~~5 / L \ n1 ' 6 Then 3 V^y 2 16/~~

~~2 3 2 2/3 1 \ ' /6\'/ v+^w-ig-my-o ,3.3, (3.11) 8tt // V/~~

~~228 p(r L)=JT(L) exp [-r^g, e)J (3.18)~~

~~- 5'3 = ^(L) exp [ r^Lr- )] (3.19) s' = s/r~~

~~/ = yl "(£) exp [ — Z(r, L)]. (3.20) Then one has to determine e in terms of £ where~~

~~sav. 1 2'3 5 f ds's' z (3.12) The function e is sketched in the introduction. ^ 4 / 3 1 ' 2 3 (tf + es' ) J,, Its maximum comes at dZ/dr=0, i.e.,~~

~~4 3 1 2 ?? = / )- / aza£, f §z dx(s) (^ + e5' (3.13) 0 = ^£^ + d£ dr J 8x(s) dr~~

~~1 2 3 £ = f f ds's' l #(s'). (3.14) J Jo (3.21)~~

~~The e, 5' relation is plotted below Since Z is stationary with respect to E and x ^is gives~~

~~1/2 ^xVmRHy) (£ + x(o») (3.22)~~

~~(3.15)~~

~~i.e.,~~

~~s'=o \l/3 / 6 „ \2/3 I X(r=r )- m 2 3 (3.23) 477/ / >i'~~

Near the peak the — Special cases are e = 0, #= 1, (£=1); e = 00, A 1/3 '4/ 3 2/ 3 = -oo, = - 3 . t, = im i.e., L = l-j rf,{ X constant. (3.24) ^e-^s'- ; e # e5 _ Thi g physi 5 cally sensible, solution has # positive and leads to a relation like And the distribution looks like

~~13 JT{L) exp Lf(r- sl3 (IY ^ 2 g1 J r t v (3.25)~~

~~(3.16) where / and g are constants derived from (3.23), (3.13), and (3.14). For the right £ is small, e large~~

~~1/2 so that £ 9? e" .~~

~~(3.26)~~

This implies a distribution bA'exp (— cr72L) where One may now see that neglecting fluctuations b and c are constants. Towards the origin p be- for the moment comes very small, and one is in the welbknown

~~trouble of trying to fit a tail of a function by working p(r, L) = .#"(L) exp (E + m ds [0r x) away from its maximum. This will be considered in detail in section 4.~~

~~2 3 The fluctuations_i.e., the remaining quadratic iEL — 7- I x ^ s (3.17) integrals over E — E and x~X have not yet been~~

~~229 considered. Unlike steepest descent methods in the origin. The local effects should be adequately a single variable, these are of great consequence handled by the conventional methods and they~~

~~and indeed are the substance of conventional field really will involve the fine structure of V, and if one~~

theory. To go straight ahead expanding to order attempts to use the pseudo potential one gets 2 X is inadequate as can be seen physically. Con- divergences. To avoid these one must write sider what one expects the addition of one more

molecule to the polymer to involve. Say for the J J p(r,EAX]) = -e (331) moment one confined oneself to steps on a close packed lattice. There would be at most eleven where sites to go on to i.e., 12, less the one it is at. But

~~it must have wandered around the neighborhood 2 (V + V,4VM(r) = -x(r) before it arrived at the present site, so one expects~~

~~a further reduction bringing the number of avail- 2 (V + 2V,4V)£(r, r') = 8(r-r') able sites to /A<11. In addition the long range~~

effect reduces it further to fji(l+f(n)) (where 2 f(n) (V 2V//V)C(r, r r = + 2 , 3 ) Vfi(r, r2 )V£(r, r3 ) will turn out to be (log n)/5n). If one were not 2 using the steepest descent method by normal field I 1 A 2 theory one would argue, after Dyson, that by v 477-r studying the perturbation series, that (using (G) and so on. for (2.16) unnormalized) The interference between x an d the fluctuations is analogous to the Lamb shift and a proof is needed that this does not interfere with the 2 (iE -- A + 2(/b, E)) (G(k, E)) = 1 (3.27) functional forms discovered so far. Though this

appears to be so by explicit calculation, I am trying to obtain a general proof on the lines of the Dyson where in the first approximation treatment of electrodynamics. ~ |F(k j)| l(k,E)= [ V; (3.28) 4. The Inner Region

but it is difficult to get in general. If (G) is con- One can see why the inner region has been so 2 / A' \ far inaccessible. The solution has a peak near sidered to — /' have a pole i.e., ( iE — + 20 (G) = 1 i/5 — r' all the probability lies in the J L and almost and 2 is expanded around it peak. This means that though the chance of the polymer getting back close to the origin is small,

~~it is much larger than that suggested by the solu-~~

~~(3.29) tion (3.5). That solution suggests that in order to~~

~~be still near r ~ 0 as L—* °o one must have stayed~~

all the time near r ~ 0. Clearly what will happen <±±Mlp+-*!L.\So -_1 E- (G) is that one will go out into the high probability region and then walk back to the inner region. (3.30) 1 + 2, = Since under the parametric integral sign one has a Here ^' ^ S 3 "renormanzec^" ste length, T+2~ ^ P Markov process one may always break up G and the 2o/(l + 22) factor represents the changed number of available sites as it expected. (Since G(r, L, [ X ]) loL gives a e~ factor in (G) it will not appear in p). = — The conventional field theory fails completely to j G(r — s, L M, [x])G(s, M. [xW's (4.1)

~~j describe the long range effects and it is these which 2 determine the r : L relation. As has been shown Note there is no integral over M and the left-hand~~

the present theory suggests that these long range side should be independent of M. To modify (4.1) I effects are simulated by a potential centered at to be M independent one can average along the 230 whole length to give than one point and get a better value for this con-

stant, but the n~ 915 would still come the same. 1 ['' = ± dM f dh Having sketched out the entire function one L) 0 can now normalize and get the height of the peak £-17/10 w hich clearly dominates the rest of the 8 G(r-s, ])G(s, M, (4.2) X L-M[X [ XW 2 j function. By evaluating r one finds a series, the

-1/10 1/3 7/1 expansion parameter being (L ir / °) which Now average using the JWKB for the G must be small for the above analysis to be valid. remarked 1 Finally, the number of walks can be = j'_dM d?s Sxlr-sl-'lsl" dEij dEi p j- j j j upon since it is an interesting combinatorial point

even if it has little bearing on polymers. As has been noted, the dominant feature will be the local effect of the effective phase space available per 2 from )1/2 ~ molecule. The long range effect will come + (y-)' {El + * iEAL ~ M) ~ iElM T the unnormalized Green function which is

~~(4.3) 3 7^exp(-j" (L-g^rir } (4. 10)~~

~~Integrating over r one gets the leading term to be~~

~~1 ' 5 v @(|a — s|<|r — s\) p/'n . + (4.4) 2 2 1/2 T |«- 5 (£, | + x)~~

~~d?a . 5. The Collapse of Polymers~~

~~\ / / J|a-s|<|r-s|| 5 — af What happens when the virial coefficient changes (4.5) sign? 2 Instead of the repulsion pushing the polymer out faster than the random configurations,~~

2 A (4.6) it will now be pulled in. For v large and attractive I J|a|<|s| a the polymer will condense into a solid ball, a state easily obtained experimentally even for a single In particular to get back to r = 0 one has chain. But for v near zero a new semicondensed state appears possible, and passage through the '6i\ 1/2 f0(|a|<|s|) 112 (E-z + x)- Flory temperature (i.e., v — 0) will produce a phase

change for a very long chain [3]. The physical picture will be of the polymer repeatedly passing e(la-sl<| g |) + (4.7) through the same region of space so that again a \a — sr polymer density is set up which now traps the polymer. Consider this problem from the point of view -if of I i.e., study the probability that, after a chain

~~length L'{0~~

~~[Z,]). As in I (where the present q is called p) e(|a- s |<| s |) _ + (£i+x) 1/2 (4.8) one can argue that the polymer field can be repre- a — s sented by approximating~~

~~I won't write out the details from here on for the~~

~~answer is defined dimensionally and must be~~

1 This value has been conjectured by Fisher, Faraday Society Discussion 25, p. 200. 2 915 Dr. J. Mazur has pointed out to me that t =0 may be much more complicated than p(0, L)°c L~ i.e., n- 9/5 (4.9) is generally supposed. The discussion of this point due to Flory assumes that chains

~~with r = 0 are Gaussian, but a single chain is a distinctly nonuniform entity having a~~

high "polymer density" at its center of gravity, lending to a low value at the end points. The details of the integration serve to fix the con- So one could have a 'tlL)" effect from the middle to the ends. The following discussion

~~stant. Presumably one could break up G at more may need modifying in the light of this comment.~~

~~231~~

~~1 Thus one is led to the eq (5.6) in which C now V(R{L')-R(L"))dL'dL" Jof'TJo takes the role usually taken by the eigenvalue,~~

~~and since q will not depend on L' , q can be identified~~

~~= V(R(L')-s)q(s, L")dL'dL". (3.32) with it:~~

l -V 2 + w(q(r)-q 2 ))q(r)=0 But q{r,L') can now be argued to be independent of L' since the self trapped polymer will repeatedly q(r)d3 go through every point of its density; it will, of r=l course, stil\ be a function of L. Since V is short 2 3 range one may now approximate bv writing q (r)d r. (5.6)

~~V(R(L')-S)q(S, L")dL'dL" By redefining r—r'w/l one brings the equation into the final form w L CL q(R(L'),L")dL'dL" (5.1) LiJ JO JO~~

L q(R(L'))dL' rather like the Schrbdinger equation with an at- 2 tractive potential and eigenvalue q . The solution 2 where w is Lvl~ . This now gives a functional has . to be obtained numerically and clearly will integral in the Markov form so that one can break have the form the functional integral up to L at an intermediate 3 3 L' and define = / / \ q Q(rPlvL)(^ , (5.5) j q{r',L') = JT{L') j exp -y dL Jo( [W 'J j where Q is a dimensionless function. Clearly in this case

~~w q(R{L") ) dR (5.2) 2 2 2 6 r a v L t~ (5.6) I o~~

~~which is quite different both from the L 6/5 and the Einstein Law, r 2 ^ LI. The distribution changes which satisfies abruptly as~~

~~v) = V2+ ^»-C(L'))(? = §(r). (5.3) 0^v 0^v (0. (df7 "!~~

~~Ignoring C for the moment, the resulting differential equation will have eigenfunctions E„.~~

~~L _ V2 + w r e£ " = 0 (5.4) I have been greatly helped by discussions with (a77 6 ^ >)<7» Professor G. Gee, Professor C. Domb, Professor~~

~~M. E. Fisher and Dr. J. Mazur. V 2 + wq(r)^g(r, L) = d(r)8(L') (JT~Z 6. References~~

= (O)e^' ®(L'). (5.5) to latest is in g(r, L) 2 q n {r)qt [1] Reference the numerical work contained Domb, C, Gillis, J.. and Wilmers, G., Proc. Phys. Soc. 85 , 625 (1965); and in a paper by J. Mazur to be published. The There will be positive and negative eigenvalues. relation to this problem as a run-in to the Ising model is discussed in Professor Domb's contribution to the As L' —> oo only the positive ones survive and unless conference. the largest positive eigenvalue tends also to zero [2] Edwards, S. F., Proc. Phys. Soc. 85, 613 (1965). [3] The standard text particularly with reference to v — 0 is the whole expression will diverge i.e., this eigen- Flory, P. J., Principles of Polymer Chemistry, Cornell value is brought to zero by the normalization. University (1953).

~~232 Some Comments on Techniques of Modern Low-Temperature Calorimetry~~

~~I. M. Firth~~

~~University of St. Andrews, St. Andrews, Scotland~~

Introduction heat capacity measured. Temperature was measured by a platinum resistance thermometer the Precision calorimetry at low temperatures started wire of which also acted as the electrical heating in 1929 when a sensitive phosphor-bronze ther- element for the calorimeter. Measurements with mometer was made by Keesom and Van Den Ende this arrangement were uncertain even with the most

~~[1]. This led shortly to the discovery of the heat careful calibration of the resistance, because of capacity connected with the superconductive irregular changes in the resistance of the platinum~~

~~phase transition [2] and also the electronic heat wire heating the calorimeter. Nernst avoided~~

capacity in metals [3] was discovered and found to this disadvantage and at the same time made his be in reasonable agreement with the theoretical method more simple, by measuring the variations predictions. Recently with the use of carbon of temperature resulting from electrical heating by

~~resistance thermometers [4] and a-c bridge lock-in using a thermocouple [9]. One junction was fixed~~

amplifier methods of measurement [5], a new era to the calorimeter and the other lay on a lead block of precision has been entered. This has allowed also in the vacuum chamber, figure 1. The heat the more exciting phase transitions at low tempera- capacity of lead being large, the temperature of the tures to be checked. This era has come oppor- block remained nearly constant and any small drift tunely for at a recent meeting in Washington on took place slowly and uniformily so that the tem- "Phenomena in the Neighborhood of Critical perature of that junction could be corrected. A

Points" [6] it was emphasized that to obtain agree- copper shield that enclosed the calorimeter was ment with theoretical predictions very great care soldered to the lead block and this refinement was a and precision must be taken in experimental meas- great improvement on the open calorimeter.

urements, not only of heat capacities, but of all Nernst could measure temperature to 0.1 °K and properties related to such critical phenomena: specific heat to ~ 2 percent. e.g., magnetization, thermal conductivity, optical It is easy to see from this arrangement how adi- and neutron diffraction. abatic methods developed. The thermal insulation

~~In considering the present situation in calorim- of this device can never be perfect so long as there~~

etry, it is useful to review the way in which the main requirements of low temperature calorimetry Pb were introduced. Although Gaede [7] used a metal calorimeter isolated somewhat from its surroundings it was Nernst [8] who recognized a few years later in 1909, that on account of surroundings there was usually a temperature gradient inside a calorimeter which would prevent making accurate, or often meaningful, measurements. To obviate this he placed the calorimeter in a vacuum realizing that, radiation being small at low temperatures, it was possible to obtain excellent thermal isolation and reduce the above error to a minimum. Of course, it was necessary to change the temperature of the calorimeter and to do this he used an exchange gas, noting in the course of his work the drawback of remnant gas on the thermal isola- FIGURE 1. Calorimeter of Nernst showing the lead block, tion and, absorbed gas on the calorimeter on the copper shield, heaters, and measuring thermocouple.

233 is a temperature difference between the calorimeter This gives more uncertainty about the temperature and the shield. But, by heating the shield so that of the sample for it will always be less than that the temperature difference is zero at all times, i.e., measured for the calorimeter. Corrections for there is no voltage across the thermocouple, there overheating can be applied but extrapolation of the will be no heat input to the calorimeter by radiation temperature drift curves on both sides of the heat or by conduction along its supporting threads. All pulse are more uncertain. It should be stated, heat that is applied to the calorimeter through the however, that where the extraneous heat input to heater then goes completely towards raising its the calorimeter is large only the pulse method may temperature and no corrections have to be made for be useful. heat losses: a correction has to be added because It is usual to measure temperatures in the of the joule heating in the heater leads but it has range 0.3 to 5 °K with an a-c bridge, using a high been shown rigorously that half the heat generated gain lock-in amplifier -as detector, and a carbon rein the leads goes into the calorimeter [10]. In sistor as the temperature sensitive arm of the adiabatic schemes a second thermometer is re- bridge [4, 5], Commercial carbon radio resistors, quired to measure the temperature of the calorim- especially made to be temperature insensitive at eter. room temperature were found some years ago to be most usefully sensitive at low temperature. Bridges 7 Methods of this type can detect changes of 10~ °K, that is 0.02fi in 46 kfi when the resistor is chosen with care. Nernst used a constant heat input to his calorimeter but since then intermittent heating has often In the advanced scheme of Schiffman et al. [12], been used in adiabatic calorimetry. However, during continuous heating the bridge is repeatedly recently in accurate work on some phase transi- unbalanced by very small steps in a sense that the bridge returns to balance as the heating progresses. tions e.g.. Buckingham and Fairbank [11] on the 4 The time taken between successive balances is superfluid transition in He , Schiffman et al. [12], electronic counters. on superconducting transitions, a preference has measured by The temperature is small been shown again for constant heating methods. jump between balances and so the heat capacity is related linearly to the input There are two reasons for this preference. In power and the is measuring the input of heat during a pulse where time intervals. There no doubt about extrapolating the heating curve as there is in the pulse the pulse may be only a few seconds in duration, for this is an inherent feature of this the current through, and the voltage across, the method, method. heater are difficult to measure accurately. The The same method can be used to evaluate the power is often of the order of a few ergs per second extraneous heat input to the calorimeter. Heating which means that currents of a few microamps is stopped and the temperature drift observed in and voltages of a few millivolts have to be measured the same way. It is a simple matter to examine with devices that have a short response time. the stray heat input at different temperatures in Secondly, after the pulse, the recorded temperature the range under study. More consistent data on will not settle down for some time, and in any case the heat leak is also obtained by repeating measure- the temperature immediately after the pulse is ments with varying electrical heat inputs knowing difficult to ascertain. It is important to know what that the heat capacity remains constant, so obtain- this temperature is after the pulse and how it is ing the heat leak from a difference equation, figure 2. drifting with time so that meaningful extrapolations can be made to the middle of the heating pulses to HEAT LEAK DETERMINATION measure the temperature increment caused by the / HEAT CAPACITY DETERMINATION pulse. Schiffman [13] has observed that in the Qa same temperature range, from pulse to pulse, the Hh ,,

Ql , , II II calculated heat capacities of the calorimeter may li ii ii not be consistent. Where measurements on solids are carried out, "overheating" may easily occur TEMPERATURE (as Parkinson and Quarrington [14] have described) FIGURE 2. Heat capacity and leak determined at two different because of the small thermal conduction of the solid. heat inputs.

234 With careful attention to the points that follow in percent and they respond to changes in temperature this discussion, leaks as low as 1-10 erg per minute, in 50 msec. The temperature controlling system 10~ 7 a temperature sensitivity of 10"6 to °K and must be smooth and accurate in operation, or as

~~an accuracy in measuring the heat capacity of 0.05 Nernst emphasized [9]. the heat leak will fluctuate percent can be obtained. randomly. In certain applications a temperature~~

An interesting experimental feature of the con- guard ring from which the calorimeter is suspended tinuous heating method can be noted here. If the may be sufficient to reduce heat leaks: all electrical continuous record of the output of the bridge is leads and other connections to the calorimeter are, simultaneously differentiated, a direct recording of course locked to it. is obtained that is. to first approximation, proportional to the heat capacity. This is so because RF Effects over small increments of temperature both the It was not realized until the report of Ambler temperature response of the carbon resistance and Plumb [16] that there is often considerable thermometer and the response of the off-balance eddy current heating in calorimeters caused by voltage of the bridge, with respect to the resistance commercial radio transmitters, particularly those thermometer arm. are practically linear. Directly in the television bands. Although the situation monitoring the heat capacity in this way is often in the U.K. is not quite as serious as that in the U.S., useful in observing a singularity in the temperature because of the greater restriction on the number of dependence of the heat capacity that characterizes T.V. channels, there will be very few laboratories so many critical point phenomena. now that are not within 50 miles of some transmitter. The best possible radiation shield is one that Television transmitters have a very small vertical encloses the calorimeter within the vacuum space. beam angle and generally have some horizontal The temperature of the shield must be monitored, directivity. Consequently they have a high aerial compared to that of the calorimeter and adjusted gain so that the effective radiated power may be to it by a controlled heater in adiabatic schemes. as much as 500 kW. At a distance of 10 miles For temperatures above 20 °K the comparison is the rf radiation density would be, therefore, about conveniently done by two thermocouples. Below 4 _1 2 2 10 erg sec m , and at 50 miles, as much as 4 10 this temperature comparison by temperature sen- _1 -2 erg sec m . In a poorly screened system, with sitive resistors is difficult because it would be most many earth loops, if only a small part of this is difficult to make a. matched pair: in the case of picked up by the equipment connected to the commercial carbon resistors, choosing a matched calorimeter there would be an overwhelming con- pair would be almost impossible. In this tem- tribution to the otherwise small extraneous heat perature range, therefore, the shield very often is input to the calorimeter. held at some mean temperature close to that being It would be possible to enclose the whole ap- investigated. This is a perfectly satisfactory tech- paratus in an rf shielded cage but this is incon- nique when the range of measurement is small, venient and expensive. In preference, other so often the case in studies on critical phenomena. precautions should be taken: as much equipment as In fact, where the bath temperature can be adjusted, possible should be operated from battery supplies; and adequately controlled, the shield may be dis- leads should be screened with a solid, continuous posed of completely, the walls of the vacuum cham- conductor — normal braided screening is effective ber acting as the "adiabatic" shield. This omission in keeping rf fields in. but is quite useless in exclud- restricts the versatility of the apparatus for it is ing them; necessary removable joints in the leads difficult to adjust liquid helium to above 4.2 °K, should be made by connectors that give complete because the vapor pressure is higher than 1 atm, screening and force the connector pins together or to just above 2.2 °K where boiling takes place with great pressure. vigorously.

~~Hence for greatest versatility an independent Vibrations shield should be used. Germanium thermometers~~

[15] are very useful in measuring the temperature This is only one of the extraneous heat inputs of the shield for they can be calibrated under that can be avoided with careful experimental plan- repeated cycling to room temperature to 0.05 ning. Others arise from vibrations of the whole ap-

235 or of the calorimeter its paratus on suspension be recorded. Often constantan wire is used as the isolation the the system. Good of cryostat from heater element but in the range 1 to 5 °K this wire laboratory can be achieved by using a massive shows a large variation in resistance. The exter- frame mounted on supports whose period of oscil- nal heater current control circuitry can be chosen lation is very low. Coil springs with a dampening so that the power input is least sensitive to this cushion of crimped knitted stainless steel mesh change in resistance. is good but an air or water hydraulic suspension will give a very long period, the pressure of the Accuracy fluid being adjusted accordingly. Where the cryostat bath is pumped to reduce its temperature, To obtain the highest long term accuracy, the the periodic fluctuation in gas pressure in the important parts of the measuring equipment must pumping line caused by the periodic pumping action lie placed in a thermostatically controlled, room of rotary pumps, can be eliminated by using a large temperature bath. The a-c bridge for temperature baffle volume or a system of plumbing that causes measurement and the heater supply and control destructive interference of the periodic pressure circuit are the most important. Remembering fluctuations at the cryostat head. Vibration along 6 that a change of 1 part in 10 in the calorimeter the pipes must be removed before the cryostat thermometer is to be measured, the other arms of suspension by massive clamping or ingenious bel- the a-c bridge must he stable to this amount. Con- lows systems. Isolation not only reduces the sidering the temperature coefficient of standard acoustical heat input but removes the experi- resistances to be 0.001 percent per °C. a local menter's dependence on the random noisy doings of change in laboratory temperature of 0.1 °C will be his colleagues! sufficient to reduce the required accuracy. The It is customary now to use mechanical heat accuracy is also hampered by the seeming impos- switches instead of low pressure exchange gas sibility of finding a switch for the other arms of to cool the calorimeter and to make adjustments the bridge that does have a reproducible contact of its temperature. To achieve good thermal con- resistance on repeated cycling. tact over the switch at low temperatures the pres- The accuracy of the whole experiment depends sure between calorimeter and bath has to be large largely on the accuracy of calibration of the ther- even though the contact surfaces are coated with mometer. There have been many suggestions as indium, which remains somewhat plastic even at to how the original equation of Clement and Quinell low temperatures. Raising the calorimeter by the [4], representing the temperature response of supporting threads does not generally produce carbon resistors, can be improved. The more sufficient contact force and upon release gives rise to notable of these have been by Hoare [17] and wild pendulum oscillations of the calorimeter. For Moody and Rhodes [18| who introduced an expres- an average calorimeter swinging over 1° these sion involving a large number of adjustable param- oscillations add 10 2 ergs to the calorimeter and equilibrium may not be established for many eters in the form ;j,= ^w/„(ln R)". n = 0. 1. 2. . . . minutes. The use of a metal bellows piston to This expression allows easy recalculation to lift the calorimeter on to the heat switch enables high accuracy of all the parameters after each a large contact force to be exerted and allows con- calibration, and removes the necessity to use cor- trolled slow release on to the suspension threads rection graphs. Having an exact description of without causing oscillations. the temperature response facilities subsequent Heating conversion of resistance to temperature in calculations to determine the heat capacity. As only a few ergs per second are required in In this expression the best values for the heating the calorimeter during heat capacity meas- coefficients are determined by minimizing urements, a mercury battery is a very convenient 1 1 77 where T, is the absolute source of stable heater current. The current and 1 voltage to the heater can be monitored, or if the temperature obtained from accurate vapor pressure heater resistance can be reliably calibrated in the measurements. Tm the temperature given by the working temperature range, only the current need above expression from measurements of resistance

~~236 and N the number of pairs of T and R used in the the energy input can be monitored by repeatedly calibration. Minimizing this quantity leads to recording the current to the calorimeter heater:~~

the linear simultaneous equations for a, and does the time small temperature jumps t between known not give undue emphasis to any region in the tem- is easily available. The temperature drift, caused

perature range. Moody and Rhodes found it by extraneous heat input, can be recorded similarly, unnecessary to go beyond n — 3 because more as can the thermometer calibration information, terms than this did not reduce significantly the except here the bath temperature would be recorded rms deviation between experimental and calculated against a definite balance condition of the a-c bridge. values of temperature. The computer programme could be in three parts. An important part of this calibration proceedure Firstly, the calibration calculation of the carbon is that a computer was programmed to perform the resistance thermometer; secondly, the heater input, calculations. The rms deviation was calculated extraneous heat input and temperatures of the calo- and calibration points that were above a preset rimeter would be found; and thirdly, the specific value, for a given value of u, were rejected: the re- heat would be computed. After taking experi- sponse parameters were then recalibrated until mental results for some 12 hr, the final results should the allowed deviation was not exceeded. be obtained in about 30 min of computer time.

~~This is to be compared with 30 to 40 hr by manual Automatic Evaluation of Heat calculations. Capacity To take full advantage of this modification of calorimetric practice both techniques and data~~

There can be few workers now who calculate recording proceedings have to be carefully con- manually the heat capacity from experimental sidered. Precautions must be taken to ensure that necessary is in data: the repetitive calculation involved can be information not thrown away the process of recording and that sufficient are better done, in less time, by computer. With a checks included in well designed method of undertaking the experi- the whole process, from apparatus to ment, data can be recorded automatically so reduc- final results, so that meaningful results are obtained. ing the overall time to perform the calculations, and the errors in recording large amounts of data. References In the continuous or intermittent techniques the methods of recording data automatically are dif- [1) W. H. Keesom and J. N. Van Den Ende. Leiden Comm. No. 203C (1929). ferent, intermittent method being more difficult. [2] W. H. Keesom and P. H. Van Laer, Physica 5, 193 (1938). Here, for each heating period the duration, and [3] J. A. Kok and W. H. Keesom, Physica 1, 175 (1934). J. R. Clement and E. H. Quinell. Phys. Rev. 92, 258 (1953). voltages to the heater are recorded to give the energy [4] [5] C. F. Kellers The specific heat of liquid helium near the input and so must be the values of pairs of tempera- lambda point. PhD Thesis. Duke University (1960). [6] Phenomena in the Neighborhood of Critical Points. NBS ture and time during the drift periods before and Misc. Publ. 273. after the heating pulse. Care must be taken how- [7] W. Gaede. Physik. Zeitscher. 4, 105 (1902). [8| A. Eucken, Physik. Zeitscher. 10, 586 (1910); W. Nernst. ever, not to record any overheating for only those Ann. d Physik 36, 395 (1911). parts of the drift curve that are linear will allow a [9] W. Nernst. Zeitscher. f. Elektrochein. 20, 357 (1914). [10] W. Mercouroff, Cryogenics 3, 171 (1963). computer to be programmed to extrapolate to the [11] M. J. Buckingham and W. M. Fairbank. Prog. Low Temp. middle of the heating pulse. It is clear from the Phys. 3, 80 (1961). [12J C. A. Schiffman, J. F. Cochran, M. Garber and G. Pearsall. discussion of Parkinson and Quarrington [14] that Rev. Mod. Phys. 36, 127 (1961). this purposeful neglect of important data cannot [13] C. A. Schiffman, private communication. [14] D. H. Parkinson and J. E. Quarrington, Proc. Phys. Soc. A lead to results accuracy. However, auto- of great 67,569(1964). matic schemes using intermittent methods have [15] F. J. Low, Advan. Cryog. Eng. 7, 514 (1961). [16] E. A. Ambler and H. Plumb, Rev. Sci. Instr. 31, 656 been devised [19]. (1962). From what has been said already about the tech- [17] F. E. Hoare, Proc. Phys. Soc. B. 68, 388 (1955). [18] D. E. Moody and P. Rhodes, Cryogenics 3, 77 (1963). niques of Schiffman et al. [12], automatic methods [19] H. .). Blythe. T. J. Harvey. F. E. Hoare. and D. E. Moody. could be used here to obtain meaningful records: Cryogenics 4, 28 (1964).

~~237~~

List of Participants Can . H. Y. Rutgers University New Brunswick. N.J. Alperin, H. A. Caulfield. D. F. Brookhaven National Laboratory Catholic University N.Y. Upton, Washington. D.C. Alpert, S. S. Chase. C. E. Columbia Radiation Laboratory National Magnetic Laboratory Columbia Universit) Massachusetts Institute of Technology York. N.Y. New Cambridge. Mass. Als-Nielsen, J. Chestnut, D. Atomic Energy Commission Shell Development Company Research Establishment Risii Houston, Tex. Roskilde. Denmark. Chu. B. Anderson. P. W. University of Kansas Bell Telephone Laboratories Lawrence. Kans. Murray Hill. N.J. Cohen, E. G. D. Andrews. D. Rockefeller Institute University of Missouri New York. N.Y. Columbia. Mo. Cooper, Martin J. Arrott. A. Brandeis University Ford Motor Scientific Laboratory Waltham, Mass. Dearborn. Mich. Cooper. Martyn .1. Baer. S. Brookhaven National Laboratory Yeshiva University Upton, N.Y. New York, N.Y. Craig. P. P. Barieau. R. E. Brookhaven National Laboratory U.S. Bureau of Mines Upton. N.Y. Amarillo. Tex. Davis. B. W. Barry, J. H. University of North Carolina University Ohio Chapel Hill. N.C. Athens, Ohio. Debye. P. Bashaw. J. Cornell University Cornell University Ithaca. N.Y. Ithaca, N.Y. Dietrich. 0. W. Baur, M. E. Atomic Energy Commission University of California Research Establishment Risii Los Angeles. Calif. Roskilde. Denmark. Beckett, C. W. ' Dotnb. C. National Bureau of Standards University of London King"s College Washington, D.C. London. England. Beenakker. J. J. M. Kamerlingh Onnes Laboratory De Dominicis, C. Leiden. The Netherlands. Commissariat a I'Energie Atomique Nucleaires Saclay Benedek. G. B. Centre d'Etudes de

. France. Massachusetts Institute of Technology Sacla> Cambridge. Mass. Dorfman. J. R. Belts, D. D. University of Maryland University of Toronto College Park. Md. Toronto, Ont., Canada. Douglass. D. H.. Jr. Bienenstock. A. University of Chicago Harvard University Chicago. 111. Cambridge, Mass. Edwards. M. H. Bloom. M. Royal Military College Kingston. University of British Columbia Ontario Vancouver. B.C., Canada. Canada Blume. M. Edwards. S. F. Brookhaven National Laboratory University of Manchester Manchester. England Upton , N.Y. Emery, V. Boyd, M. E. J. Brookhaven National Laboratory National Bureau of Standards Washington. D.C. Upton, N.Y. Fairbank, Bru m berger, H. W. M. Syracuse University Stanford University Syracuse. N.Y. Stanford, Calif. Farrell. R. Buckingham. M. J. Catholic University University of Western Australia Washington. D.C. Nedlands W. A., Australia. Felderhof. B. U. Buff. F. P. Instituut voor Theoretische Fysica University of Rochester der Rijksuniversiteil Rochester, N.Y. Utrecht, The Netherlands. Firth. I. Callen. E. M. Naval Ordnance Laboratory University of St. Andrews White Oaks, Md. St. Andrews. Scotland. Callen, H. B. Fisher. M. E. University of Pennsylvania University of London King's College Philadelphia, Pa. London, England 239 Fixman. M. Ikenberry, L. D. Yale University University of North Carolina New Haven. Conn. Chapel Hill, N.C. Ford. N. C. Isihara, A. University of Massachusetts State University of New York Amherst, Mass. Buffalo, N.Y. Friedberg, S. A. Jackel, R. Carnegie Institute of Technology Office of Naval Research Pittsburgh. Pa. Washington, D.C. Jackson, J. L. Gammel, J. L. Texas A and M College Howard University College Station. Tex. Washington, D.C. Gammon, R. Jacrot. B. Johns Hopkins University Commissariat a l'Energie Atomique Baltimore, Md. Centre d'Etudes Nucleaires de Saclay Saclay, France. Garcia-Colin, L. S. Direccion del Reactor Nuclear Jalickee, J. Comision Nacional de Energi'a Nuclear Catholic University er Washington, Insurgentes Sur 1079, l piso D.C. Katsura, Mexico 18, D. F., Mexico. S. Garland, C. W. Tohoku University Massachusetts Institute of Technology Sendai. Japan. Cambridge, Mass. Kasteleyn, P. W. Gilmer, G. H. Instituut-Lorentz der Rijksuniversiteit Washington and Lee University Leiden. The Netherlands. Kawasaki, Lexington, Va. K. Massachusetts Institute Green. M. S. of Technology Cambridge, Mass. National Bureau ot Standards Washington, D.C. Kawatra, M. P. Grindlay, J. Courant Institute of Mathematical Scie National Research Council New York. N.Y. Ottawa, Ont., Canada. Keating, D. T. Guth, E. Brookhaven National Laboratory Oak Ridge National Laboratory Upton, N.Y. Oak Ridge, Tenn. Keen, B. E. Guttman, L. Yale University Argonne National Laboratory New Haven, Conn. Argonne. 111. Kellers, C. F.. Jr. Hammel. E. F. Wells College Los Alamos Scientific Laboratory Aurora, N.Y. Los Alamos, N. Mex. Kerr, E. Heims, S. P. Los Alamos Scientific Laboratory Brandeis University Los Alamos, N. Mex. Waltham. Mass. Kestin, J. Helfand, E. Brown University Bell Telephone Laboratories Providence, R.I. Murray Hill, N.J. Kierstead. H. A. Heller. P. Argonne National Laboratory Brandeis University Argonne, 111. Waltham, Mass. Kikuchi. R. Herzfeld. K. F. Hughes Research Laboratory Catholic University Malibu. Calif. Washington, D.C. Klein. M. Hill, R. M. National Bureau of Standards Lockheed Research Laboratories Washington. D.C. Palo Alto. Calif. Klein, M.l Hilsenrath, J. Case Institute ot Technology National Bureau of Standards Cleveland. Ohio Washington, D.C. Knobier. C. M. Hiroike, K. University of California University of Chicago Los Angeles. Calif. Chicago. 111. Kouvel. J. S. Hohenberg. P. C. General Electric Research Laboratory Bell Telephone Laboratories Schenectady, N.Y. Murray Hill, N.J. Langer, J. S. Horwitz, G. Carnegie Institute of Technology Yeshiva University Pittsburgh. Pa. New York, N.Y. Larsen, S. Y. Hudson. R. P. National Bureau of Standards National Bureau of Standards Washington. D.C. Washington, D.C. Hugenholtz. N. Leff, H. S. Case Institute of Technology Instituut voor Theoretische Fysica der Rijksuniversiteit Ohio Groningen, The Netherlands. Cleveland, L. C. Hynne. F. Levitt. University of Copenhagen Georgetown University Copenhagen. Denmark. Washington. D.C. 240 Lieb. E. Neighbours. J. R. Yeshiva University U.S. Naval Postgraduate School New York, N.Y. Monterey, Calif. Lipsicas, M. Nielsen, P. Brookhaven National Laboratory University of Copenhagen Upton. N.Y. Copenhagen, Denmark. Livesay. B. R. Nishimura. H. Georgia Institute of Technology Catholic University Atlanta, Ga. Washington. D.C.

Lorentzen, H. L. Oppenheim, I. Universitetets Kemiske Institutt Massachusetts Institute of Technology Blindern-Oslo, Norway. Cambridge. Mass. Manchester. F. D. Papoular, M. University of Toronto University of California Toronto, Ont., Canada. Los Angeles. Calif. f Marcus. P. M. Parette. G. International Business Machines Corporation Commissariat a l'Energie Atomique Watson Research Center Centre d'Etudes Nucleaires de Saclay Yorktown Heights. N.Y. Saclay, France. Marshall. W. Parks. R. D. Atomic Energy Research Establishment University of Rochester Theoretical Physics Division Rochester, N.Y. Harwell, England. Passell. L. Martin. P. C. Brookhaven National Laboratory Harvard University Upton, N.Y. Cambridge, Mass. Piccirelli. R. A. Marvin. R. S. National Bureau of Standards National Bureau of Standards Washington, D.C. Washington, D.C. Pings, C. J. Matsubara, T. California Institute of Technology Kyoto University Pasadena, Calif. Kyoto, Japan. Potter, R. L. Mclntyre, D. American Cyanamid Company National Bureau of Standards Stamford Research Laboratory Washington, D.C. Stamford, Conn. Melrose, J. C. Rao, N. G. Socony Mobil Oil Co. Catholic University Field Research Laboratory Washington, D.C. Dallas, Tex. Renard. R.

Michels. A. M. J. F. Massachusetts Institute of Technology Van der Waals Laboratorium Cambridge. Mass. Universiteit van Amsterdam Ricci, P. Amsterdam-C. The Netherlands. Universita di Roma Misawa, S. Rome, Italy. Massachusetts Institute of Technology Rice, O. K. Cambridge, Mass. University of North Carolina Moldover, M. R. Chapel Hill, N.C. Stanford University Roder, H. M. Stanford, Calif. National Bureau of Standards Montroll, E. W. Boulder, Colo. Institute for Defense Analysis Ross. J. Washington, D.C. Brown University Moorjani, K. Providence, R. I. Catholic University Rowlinson, J. S. Washington, D.C. Imperial College Morita. T. London, England. Catholic University Rubin, R. J. Washington, D.C. National Bureau of Standards Moruzzi, V. L. Washington, D.C. International Business Machines Incorporated Rudnick, I. Watson Research Center University of California Yorktown Heights, N.Y. Los Angeles, Calif. Mountain, R. D. Ruelle. D. National Bureau of Standards Institut des Hautes Etudes Scientifique Washington, D.C. Bures-Sur-Yvette (S. et O.). France. Muenster. A. Rushbrooke, G. S. University of Frankfurt University of New Castle Frankfurt. Germany. New-Castle-on-the-Tyne, England. Nagel, J. F. Schmidt. E. H. W. , Yale University Technische Hochschule Miinchen New Haven, Conn, Munich. Germany. i Naiditch, S. Schmidt. P. Unified Science Associates, Inc. University of Missouri Pasadena, Calif. Columbia, Mo.

! Nathans, R. Schultz. T. D. Brookhaven National Laboratory New York University Upton, N.Y. University Heights. Bronx. N.Y. Schwartz, L. H. Ulrich, D. V. Northwestern University Bridgewater College Evanston, 111. Bridgewater. Va. Scott, R. L. van Kampen, N. G. University of California Instituut voor Theoretische Fysica der Rijksuniversiteit Los Angeles, Calif. Utrecht, The Netherlands, van Leeuwen, J. M. J. Sengers, J. M. H. (Mrs.) Instituut voor Theoretische Fysica der National Bureau of Standards Katholieke Universiteil Nijmegen, The Netherlands. Washington, D.C. Verlet. L. Sengers, V. J. Universite de Paris National Bureau of Standards Orsay. France Washington, D.C. Villain, J. Sette, D. Commissariat a l'Energie Atomicme Universita di Roma Centre d'Etudes Nucleaires de Saclay Rome, Italy Saclay, France Sherman, R. H. Walker, C. B. Los Alamos Scientific Laboratory U.S. Army Materials Research Agency Los Alamos, N. Mex. Watertown, Mass. Shirane, G. Walker. J. F., Jr. Brookhaven National Laboratory New York University Upton, N.Y. Bronx, N.Y. Sikora, P. T. Walker. L. R. North Carolina College Bell Telephone Laboratories Durham, N.C. Murray Hill, N.J. Singwi, K. S. Webb. W. W. Argonne National Laboratory Cornell University

~~Argonne, 111. Ithaca, N.Y.~~

Smart, J. S. Weber. L. A. International Business Machines Corporation National Bureau of Standards Watson Research Center Boulder, Colo. Yorktown Heights, N.Y. Weinstock, J. Stillinger, F. H. National Bureau of Standards Bell Telephone Laboratories Boulder, Colo. Murray Hill, N.J. White, J. A. Storer, R. G. University of Maryland Courant Institute of Mathematical Sciences College Park, Md. New York, N.Y. Wilkinson, M. K. Oak Ridge National Laboratory Stout, J. W. University of Chicago Oak Ridge, Tenn. Wims, A. M. Chicago, 111. National Tanaka, T. Bureau of Standards Washington. Catholic University D.C. Washington, D.C. Wolf, W. P. Yale University Taylor. T. R. New Haven, Conn. University of Missouri Yamamoto, T. Columbia, Mo. Kyoto University Teaney, D. T. Kyoto, Japan. International Business Machines Corporation Yang, C. N. Watson Research Center Princeton University Yorktown Heights, N.Y. Princeton, N.J. Temperley, H. N. V. Yeh, Y. United Kingdom Atomic Energy Authority Columbia Radiation Laboratory Aldermaston, England. Columbia University Tisza. L. New York, N.Y. Massachusetts Institute of Technology Younglove, B. A. Cambridge, Mass. National Bureau of Standards Trappeniers, N. J. Boulder. Colo. Van der Waals Laboratorium Zuber, N. Universiteit van Amsterdam General Electric Company Amsterdam-C, The Netherlands. Schenectady, N.Y. Lhlenbeck, G. E. Zwanzig. R. W. Rockefeller Institute National Bureau of Standards New York. N.Y. Washington. D.C.

~~209-493 U.S. GOVERNMENT PRINTING OFFICE : 1966 OL—~~

~~242~~