Experimental Determination Op the Specific Heats of Sodium

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EXPERIMENTAL DETERMINATION OP THE SPECIFIC HEATS OF SODIUM,

COBALT, MANGANESE, AND COBALT-IRON BELOW 1°K.

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By ROGER EDGAR GAUMER, B. S.

The Ohio State University 1959

Approved by

Adviser Department of Hiysics and Astronomy ACKNOWLEDGEMENTS

I wish to acknowledge my debt to Dr. C. V. Heer for his continued support and guidance throughout the course of these investigations.

Dr. R. A. Erickson should be acknowledged for a variety of invaluable suggestions concerning experimental procedures. To my fellow graduate student, Mr. David Murray, I extend sincere thanks for all manner of help. Finally, I wish to express gratitude to my wife, Suzanne Morrison Gaumer, for unfailing moral support during these lean years.

This work was supported in part by funds granted to The Ohio

State University by the Research Foundation and by a contract between the Air Force Office of Scientific Research and The Ohio State University Research Foundation.

ii TABLE OF CONTENTS

ACKNOWLEDGMENTS...... * » . . . LIST OF TABLES ......

LIST OF ILLUSTRATIONS ...... INTRODUCTION ......

Chapter I. THEORY OF THE SPECIFIC HEAT OF SOLIDS . . .

Lattice Specific Heat Electronic Specific Heats in Metals

II. EXPERIMENTAL TECHNIQUES ...... Cooling Above 1°K. Cooling Below 1°K. Apparatus Thermometry Vapor Pressure Thermometry Sampe Preparation

III. EXPERIMENTAL RESULTS...... General Methods Thermometer Calibration Copper Sodium Cobalt Manganese Cobalt-Iron Alloy

IV. INTERPRETATION OF R E S U L T S ......

Section 1t Copper and Sodium Section 2* Cobalt, Cobalt-Iron Alloy, and Manganese

LIST OF REFERENCES LIST OP TABLES

Table Page

1* Uanganese Deviation Curve D a t a ...... 61

2« Data Tablet Copper Heat Capacity •••••••• 64

3* Data Tablet Sodium Heat Capacity •••*«••• 66

4* Bata Tablet Cobalt Specific Heat • • 74

5* Data Tablet Manganese Specific H e a t ...... 77

6« Data Tablet Cobalt-Iron Alloy Heat Capacity • • • • 84

7« Specific Heats of Some Transition Metals•••••• 98

iv LIST OP ILLUSTRATIONS

Figure Page

1. Einstein Theory of Lattice Specific Heat ...... 8 2. Debye Theory of Lattice Specific H e a t ...... 13 3. Born-Von Karman Theory of Lattice Specific Heat ...» 1?

4* magnetic Refrigerator ...... 23

5« Magnetic Refrigerator Engine and Sample Area ...... 24

6. Superconducting Thermal Switch ...... 33 7* Thermal Switching R a t i o ...... 36 8. Magnetic Calibration Curve ...... 40

9. Carbon Resistor Calibration Curve ...... 43 10* Sodium Distillation Apparatus ...... 46 11. Sodium Sample Assembly...... 47

12. Cobalt Sample Assembly ...... 49 13* Manganese Sample Ass e m b l y...... 50

14* Cobalt-Iron Alloy Sample Assembly...... 52

15* Typical Heating C y c l e ...... 56 16. Heat Capacity of C o p p e r ...... 63 17. Specific Heat of Sodium M e t a l ...... 71

18. Specific Heat of C o b a l t ...... 73 19* C/T Versus T^ for Manganese...... 81

20. Specific Heat of Manganese...... 83 21* Cobalt-Iron Alloy Heat Capacity ...... 87

v INTRODUCTION

Experimental Investigations of the specific heat of solids at low

temperatures have constituted a sizable portion of cryogenic research during the last half century. The first liquefaction of helium

occurred at Leiden, under Kammerlingh-Onnes, on July 10, 1908.^

Cryogenic research may well be said to date from this historic event.

It is of interest to note that virtually all progress in the under

standing of the mechanisms which contribute to the specific heat of

solids also dates from the early part of this century. The specific heat of a solid at ordinary or "room” temperature is

almost entirely due to the lattice vibrations. However, below 1°K.

the specific heat of a solid is predominantly due to the energy of

the electrons. As a consequence, experimental data concerning the

specific heats of solids below 1°K. are of considerable value in

increasing our understanding of the electronic and atomic domains.

The present state of the theory of specific heats of solids will

be reviewed in the first section of this dissertation. It will become clear that there exist several uncertainties in the present theory,

despite the great advances made since 1910. It was with the hope of contributing information which would be of value in the construction

of a completely satisfactory theory of specific heats that the inves

tigations to be described here were begun.

1 2

The specific heat of sodium metal is of interest for several reasons. The structure of sodium is such that it would theoretically 2 approach very closely the free-electron model of Sommerfeld. Owing to the relatively simple electronic configuration of sodium, there has been extensive theoretical investigation of the hand structure, and experimental information would he useful in assessing the applic ability of the various theoretical treatments. Finally, previous measurements of the specific heat of sodium have indicated the poss ible presence of unexplained anomolies in the specific heat. Such anomolies, if verified, would require a modification of the present theory. For these reasons the specific heat of pure sodium metal has been measured. The results are given in Chapter III. Another area where there is need for experimental information on heat capacities is that of the transition metals. In these elements

the electronic configuration is rather complex, and this complexity has led to the development of band theory. Measurements on these metals at low temperatures serve to give a direct indication of the

filling of the d-bands. Another impetus for the study of these metals has been the observation of a specific-heat contribution at

low temperatures that is attributed to a hyperfine coupling between

the nucleus and the electronic magnetic moment of the atom. The

existence of a specific-heat contribution attributable to hyperfine

coupling was originally indicated by the nuclear orientation^ produced

in cobalt crystals by a hyperfine interaction. The hyperfine-coupling

specific heat in cobalt metal has been measured by Heer and Erickson^

between 0.6 and 3.0°K., and their results suggest a progressive increase in specific heat as the temperature is decreased to 0.01°K.

For these reasons the specific heats of cobalt metal* manganese metal* and cm alloy of cobalt-iron have been measured in the temperature

region 0.4-l»5°K * These results are also given in Chapter III.

All of these investigations have been carried out in a magnetic

refrigerator* the principles of which have been described pre- S 7 viously. * * Details of the particular refrigerator used have been a given by Stroud. The general oalorimetrio procedures and details of

experimental modifications are given in Chapter II. Also to be dis

cussed in Chapter II is the subject of thermometry, which is one of

the more important areas of calorimetry below 1°K.

Results of the measurements are given in Chapter III* and these

results are discussed and analysed in Chapter XV. CHAPTER I

THEORY OF THE SPECIFIC HEAT OF SOLIDS

Solids may "be described as an assembly of parallepipeds which form a three-dimensional lattice. Each parallelepiped, or unit cell, contains an identical arrangement of atoms. Specific heat is thermodynamically defined as that quantity of heat necessary to raise the temperature of a unit mass of the sub stance by unity. Cp is the symbol for specific heat at constant pressure. In the temperature region 0.0-4.0°K.-, the difference between Cy and C^ is negligible for all practical purposes. In low- temperature work it is customary to use the symbol C without a subscript to indicate the specific heat} i.e., the quantity of heat necessary to increase the temperature of one mole of the substance by 1°K. As a consequence of the definition of specific heat, an application of the first law of thermodynamics yields the result that the specific heat of a substance refers to the rate of change of internal energy of the system with change in temperature. Thus,

(1 .1 )

(1.2)

4 5 at low temperatures. It is then clear that the specific heat of a solid depends entirely upon the internal energy of the solid and so upon the manner in which temperature changes affect the internal

energy* The internal energy of a solid may be divided into two main cate gories. One category is that of internal energy owing to vibrations

of the atoms in their lattice, while the other type of internal energy is that associated with the electronic structure of the individual

atoms. The lattice component is dominant at ordinary temperatures, while the electronic component becomes relatively sizable only at

very low temperatures where the lattice vibrational energy is quite

small*

Lattice Specific Heat

The theory of the lattice specific heat of solids has been the

subject of extensive investigation in recent times. This section will endeavour to outline the development of the theory and to indicate the salient features of each development. In 1819 Dulong and Petit^ made the empirical observation that the

product of specific heat and atomic weight was approximately constant

at a value

c v = 3 R ^ ^ ~°/c d-3) This empirical law found a justification in classical theory by assum ing that each atom in a crystal is associated with a definite lattice

point and vibrates about its equilibrium point because of the forces 6 exerted by neighboring atoms. The vibrations of the atoms are assumed to be simple harmonic. The energy of a simple harmonic oscillator of frequency V ’is, in one dimension

The average energy of the oscillator in thermal equilibrium may be shown to be

£ (1.5) independent of frequency. The total internal energy of a system of N such oscillators in a three-dimensional solid will be

(1.6) and the Dulong-Petit value follows.

During the rest of the nineteenth centuryf a sizable quantity of experimental data on the specific heat of solids was collected and evaluated in terms of the Dulong-Petit law. It became evident that the specific heat of most solids was not a constant but rather was a function of temperature. Above room temperature, most specific heats increased slightly; below room temperature the specific heats de creased rather markedly* The inadequacy of classical theory in explaining the specific heats of solids gave impetus to the first attempt to apply the new quantum theory to this problem. In 1907

Einstein^ developed a quantum theory of specific heat.

Einstein assumed that a solid consisted of atoms which were free to vibrate id the lattice with a frequency but that these vibra- 11 tions were governed by the quantum hypothesis of Planck. This amounts to postulating that each atom is equivalent to three mutually

perpendicular Planck oscillators, with each oscillator being capable 7 of assuming discrete energy values nh y^t where n is an integer and h is Planck's constant (h - 6.624 x 10-^ erg-sec.). Based on these quantum principles, the mean energy of an atomic oscillator, instead of being the classical value of kT, becomes Planck's expression

(1.7) The total internal energy of the solid is

u - ' since a three-dimensional Isotropic oscillator is postulated with N oscillators per mole of a monatomic solid. Differentiating, the lat tice atomic heat is

Figure 1 shows the behavior of according to (1 *9)* The obvious chief difference between classical theory and the Einstein theory is the introduction of a temperature dependence in the lattice atonic heat. Only in the high-temperature limit, kT > > h V^does the specific heat approach the classical value of 3B. In the low-temperature limit, Equation (1.9) indicates that should approach zero exponentially. The low-temperature limit of Einstein's specific heat formulation, that C^— > 0 as T — ^0, satisfies the requirements of Nernst's postu late of the third law of thermodynamics. One form of Nemst's postu late is (1.10) o I caloriee/mole-°K 0 6 5 / , 02 , 0,fy 0,3 0,2 0,1

iue1 Seii etacrigt h hoyo isen (after Einstein) Specific heataccording tothe theoryof Einstein.Figure1. \ \ \ \ o Q / / O I J 1 ’ I 9 / t t t * / / ✓ s £J * kT \ f 07 , 0,9 0,8 0,7 Ofi j i 1 i 0 ______0 _ _ _ _ — ~ 1t 0 oo 9 from which the conclusion that 0 as T — follows*

Although Einstein's work represented a major advance in the undeio- standing of the specific heats of solids, the agreement with experi mental data was qualitative only. The primary difficulty with Einstein's simple theory was the assumption of a single oscillator frequency rather than a continuous frequency spectrum. Einstein 12 himself pointed out a better approximation to the specific heat of a solid should be obtained using a spectrum of frequencies. We have seen that the application of the basic principles of quantum mechanics leads to an expression for the internal energy of a crystalline solid of the form

It is assumed that the quantized lattice vibrations, or phonons, obey Bose-Einstein statistics. If the number of permitted lattice waves per mole whose frequencies lie between P *and P* + dP* 1b g(y')cL)^ the total number of vibrations permitted per unit volume will be

Clearly this number cannot exceed 3N, the number of parti cles times the three polarizations. Consequently, there must exist some maximum frequency ^ m , such that

- 3 y Y (’-is) The distribution function g(V^,also referred to as the vibration spectrum, thus becomes of paramount importance in the development of more refined theories of the specific heat of solids. There have been two major contributions to the theory of lattice specific heat since 13 1 * 14 Einstein's original work. The theories of Debye ■ and Born-Von Karman were published almost simultaneously in 1913* Since the Debye theory 10 is the Bimpler of the two, it will he discussed first.

Debye*s theory of lattice specific heat replaces the single oscillator frequency K by a frequency spectrum. The atoms in a solid are assumed to be coupled and therefore to vibrate collectively. The solid is assumed to be isotropic, homogeneous, elastic, and infinite in extent. If the solid is assumed to be a continuous medium, it is poss ible to calculate the number of standing waves which can be fitted into the solid and which have frequencies in the range between ¥ and y* + d/^l Debye finally uses the fact that a solid is a discrete lat tice rather than a continuum, so that the total possible number of normal modes of vibration may be determined, and consequently the total number of states in the frequency interval from to dX^ may be evaluated.

The determination of the number density g ( ) ^ ) consists essentially of finding the number of standing waves which will fit into a cubical box of volume V, up to a certain frequency maximum uicljw Remembering that to each wave propagation vector there corresponds one longitudinal and two transverse modes of vibration, it may be shown that

3 ^= V zZ [■£ (1.13) where v and v are the longitudinal and transverse velocities of wave 1 t propagation, and V is the atomic volume. Now g ( > 0 is evaluated in terms of octant spherical shells enclosed between neighboring spherical surfaces of radii i^and ¥* + This

*Details of this process are elaborated in a review paper by DeLaun&y. ^ 11 gives

$Or)eLy'=vrV(itf+ i ^ ) Y ^ Y ' ( 1 ., 4) and substituting (1.14) into (1*13) gives

9 / V fy{Y)dy'= ~ y^ y • (1 *15) Substituting (1*15) into (1.11) we have the energy U

9A/h U - y c l ^ J ' • C1*16) In order to facilitate evaluation of (1.16), sane new variables are now introduced: Debye characteristic temperature equals

< % = (1.17)

(1.18) and Equation (1.16) becomes

1 n * * % , V = 9X %r £ -#nrp . <...»> This expression is first evaluated at the high-temperature limit T > & . In this region T —> x —>0, and (ex - 1) — >x. Thus, the integrandD approaches JC x 2dx. The integral then approaches the limiting value — J - ^) , and the energy expression becomes

3 ^ ’ V s ^ \ 3 Z/-~ 97? (^) i -~3/f,?7, (1.20)

77 >&a * ^ cu = 37? J (i.zi) and the high-temperature limiting value is seen to be in accordance with 12 classical theory, approaching the Dulong-Petit value of 3B for speoifio heat*

At very low temperatures, > > T, the upper limit of integra tion in (1*19) approaches *o and the integrand consequently approaches a oonstant value which is independent of temperature.

x 3 u = 9 X © r r . (1 .2 2) This expression may be evaluated by expansion of the integrand in infinite series, the result being

7 7 v ~~0 >> 7* ; W - I*** ( 1-23) 7 j r o and~~

~~& „ > > T 7~~

Ct-~ s \ &o) ~ / ? v v L&oJ (1,24) For intermediate temperatures, Qjj ^ T, it is necessary to obtain from Debye's formulation by rather extensive mathematio manipulation* Tables of the Debye function are available, one such being given by Beattie,*^ Figure 2 contains a plot of lattioe speoifio heat based upon the Debye theory.

~~Perhaps the most striking feature of the Debye theory is the~~

~~prediction that <=<. at low temperatures* This T^ variation of~~

~~the speoifio heat has been confirmed by experiment, and has beoome a standard form for part of the low temperature variations of~~

~~speoifio heat of solids* Cy/( 3k) 0-2 0-4 0-6 0-8 10 0 Thespecific according heat Debye’stheory function ato of as~~

~~0-2~~

~~0-4~~

~~0-6~~

~~Figure 2 Figure Tl@ 0*8~~

~~1-0~~

~~1-2~~

~~1-4 T/Q. 14~~

~~The agreement of the Debye theory with experiment has been, on tha whole, fairly good, considering that the derivation ia based upon assumptions which few, if any, real solids satisfy.~~

~~The third power law should be a valid representation of tha speoifio heat of solid norsnetallic materials in tha temperature~~

~~7± _l . range 0 <- /s. , to or so* It has, however, been found impossible to represent the experimental data by a single Debye characteristic temperature throughout the temperature spectrum.~~

~~This is presumably a reflection of the degree of departure of real solids from the assumed qualities of the Debye theory.~~

Debye's theory of speoifio heats is based upon the assumptions that a solid is isotropic and continuous. It is the opinion of several authors that the agreement between debye's theory and the experimental data is surprisingly good inasmuch as most real solids exhibit rather considerable deviations from the model used by Debye. (See, for example,

Boberts and Miller.)^ 1 * 18 ®ie aim of the theory developed by B o m and Ton Kerman was the same as that of Debye i to find a suitable expression for the spectrum of frequencies. The chief difference between • i the theories ia that B o m and Ton Karmen endeavor to c a l c u l a t e the frequency speotrum by means of a detailed consideration of the lattioe modes of vibration for the particular crystal structure under investigation. This latter approach, although considerably 15 more involved., should, predict the specific heat of a particular

~~solid with an accuracy appreciably superior to the Debye theory.~~

~~Host of the reoent work in the theory of lattioe atomio heats is~~

~~• « based upon the approach of B o m and Von Kaiman.~~

~~The details of this method are too involved to be given~~

~~here* In principle, it is assumed that the solid is composed~~

~~of atoms vibrating in a crystal lattioe and that the atoms~~

~~vibrate as a collection in certain normal modes* The model~~

~~is that of a large number of ooupled oscillators, the one~~

~~dimensional picture being a simple alternation of springs and masses. The atoms are presumed to vibrate under the influence~~

~~of Hooke' s law type foroes, the nature and magnitude of these~~

~~forces being determined by the interatomic a pacings of the~~

particular orystal and the velocities of wave propagation*

~~The propagation velocities, in a crystalline solid, are dependent~~

~~upon the direction of propagation. Also, the waves are not, in~~

~~general, either purely transverse or purely longitudinal.~~

~~The normal mode frequencies may be calculated from the~~

secular determinant of the normal coordinate transformation matrix*

~~The determinant is of order 3Z, whore Z is the number of atoms~~

~~per unit cell* The distribution function gCV7) must be evaluated~~

~~by approximation methods. In the simple case of a one-dimensional i « lattioe, B o m and Von Karman, sorfcing in momentum space, showed~~

~~that the high temperature speoifio heat could be evaluated without~~

constructing a particular frequency spectrum. The extension of 16 this msthod to the three-dimensional case Is quite difficult*

However) since the three-dimensional case is the one of praotical importance) there has been considerable progress in this area in 19 recent years. Blackman in particular has done much valuable work in this area. A recent review artiole encompassing the major portions of the work done in applying the lattioe approach is given by Born and Huang. 20 It appears that the Debye theory begins to fail at

T <. O' jj, and, according to Blackman,^ the true region must lie below T ^ . A careful delineation of the principles and methods 5° » i 15 of Boro-Von Karman approach has been given by DeLaunay. Figure 3

~~shows the manner in which varies with the temperature) aooording~~

~~to the detailed calculations of Leighton 21 for face-oentered-cubio~~

~~crystals• In oonoluding the section on the theories of lattioe atomio heats of solids) same general observations are in order. For one thing) it~~

~~is now known that the speoifio heat is a function of temperature. It~~

~~is also clear that) for low temperatures) this function is cubic in~~

~~temperature. Although there is some doubt as to what constitutes a~~

~~low enough temperature) the temperature range below 1°K. is unambigu~~

~~ously suitable for the pure metals. Consequently) for the purposes of~~

~~this discussion the Debye theory of speoifio heats should be an~~

~~adequate model with whioh to compare experimental results.~~

~~In actuality* the lattioe contribution to the speoifio heat of~~

~~metals is comparatively minor for T 1°K. She electronic contribu~~

~~tion to be discussed in the next seotion is of paramount Importance. Silver~~

~~Theory~~

~~o 20 no so 80 100 T ° K — *• Fifur* 3, Variation of Detgre S' with Toapexattfre* (after Leighton) 18~~

~~Electronic Speoifio Heats in Metals~~

The basic model for the free—eleotron theory of metals was sug- 22 jested by Crude* It is assumed that a metal contains free electrons in thermal equilibrium with the atoms of the solid* The potential energy of the free electrons is presumed constant within the interior of the metal* The total internal energy ascribed to the free electrons is found from the product of the number of electrons per unit volume and the average energy of an electron. It is supposed that a discrete set of energy levels is available to the electrons, and the essential problem in the determination of internal energy is the evaluation of the number of electrons whose energies lie between £ and ^ + dt*

~~The evaluation of the number of states with energy between £ and~~

£ + d £ depends upon the type of statistics employed. Classical theory employed the Boltzmann statistics, while modem quantum theory utilises the Fermi-Dirac statistical distribution Auction*

~~Classical theory would give the electronic speoifio heat to be~~

~~C e = % / ? (1-25) where n^ is the number of free eleotrons per atom* Consequently, the~~

~~total speoifio heat would be~~

C = C e + <:t = % ^'R-b3-R ' O - * ) Sinoe ^ 1 to aooount for the eleotrioal properties of a metal, this means that the total speoifio heat for a metal should be o of the order gR. This is contrary to experience, and another instance

~~of the inability of classic theory to adequately explain speoifio~~

heats of solids. Just as Einstein aooounted for the variation of 19 lattioe speoifio heats "by introducing the Bose-Binstein statistics, 23 Sommerfeld aooounted for variations in the electronic heat capacity by applying Fermi-Dirac statistics. The Fermi-Dirac statistics are appropriate for a discussion of electrons due to their asymmetric wave functions and half—integral spins.

~~The Fermi—Dirao distribution function is~~

f ^ oc c ; = ( e fc ^ - l ) ' d.*T) where the normalizing oonstant is the chemical potential of the electrons, frequently called the Fermi energy level of the electron distribution. It can be shown by methods analogous to those utilized in developing the Debye theory that the number of electrons per cubic centimeter n is given by

~~c t o {„ < - . * and evaluation of the distribution function in the limiting oases leads to~~

~~^ r Y'7r( y/z J ^ d ^ <'' ' S9) €> Solving the above for ^~~

~~^ * B S . 2 C (1 .30) w h . r . V i . molar rolum. and n i. th« mnbar of Mfr.oN o l . o t r o n . ft per atom. 20~~

~~At ordinary temperatures the electronic speoifio heat is swamped by the lattioe contribution. At low temperatures the speoifio heat is given by~~

~~( i . » ) for a particular metal} then, the basic theoretioal expression for the atomio heat of a metal is~~

~~(1.32)~~

It is clear from the above that there will exist some temperature low enough so that the electronic contribution becomes large in comparison to the lattioe contribution. In general, this temperature is ^ 1 ° K .

The simple free electron model used thus far is evidently an over-simplification. The assumption that the electrons move in a oonstant potential field is not realised in practice, nor will the electrons be completely free in praotioal oases*

The band theory of metals has been developed in order to include the details of the lattioe structure in the theory* The electronic wave functions are developed from atomio wave functions surrounding the lattioe ions* Adjacent ions have overlapping wave functions, oausing discrete atomio energy levels to spread out into bands of levels in a crystal. The band theory has been extensively reviewed by Mott,^ Seits,^ and others.

The details of the band theory will be discussed later insofar as they apply to the metals discussed here* Some other types of 21 specialised theory, such as Schottky-type anomolies and spin wave energies will also he disoussed in referenoe to the specific heats of the transition metals.

As a result of the very haalo theory developed so far, it should be clear that the low temperature specific heat of ordinary metals can usually be represented as the sum of a lattioe and an electronic contribution

(1.32) Certain modifications of this general representation are usually necessary for a particular metal. These modifications will be disoussed in Chapter IT, in connection with the conclusions whioh may be drawn from these measurements. CHAPTER II

~~EXPERIMENTAL TECHNIQUES~~

In making the measurements described in this paper, a number of highly specialized experimental techniques and procedures have been developed. It is the purpose of this section to describe the standard cyrogenic techniques rather generally and to describe certain of the more unusual techniques developed in this laboratory in considerable detail. The reason for this approach is to attempt to give future workers in the field some of the information which otherwise must re sult from a trial and error process. For reference in this section, Figures 4 and 5 are given. Figure 4 is a photograph of the magnetic refrigerator* Figure 5 is a schematic drawing of the experimental area within the magnetic refrigerator.

~~Cooling Above 1 ° K .~~

The first area to be discussed is that of cooling from room temper- o ature to 1 K. This is a common process in present-day cryogenic research and will be dealt with quite briefly. The process consists essentially of precooling an outer dewar vessel (see Figure 4) with liquid nitrogen and then, using air as the exchange gas, cooling the

~~inner helium dewar to nitrogen temperature. It is important to prevent any trace of helium gas from entering the vacuum space of the inner~~

~~(helium) dewar during the process of admitting and pumping out the air~~

~~22 Figure 4. Magnetic Refrigerator 24~~

Figure 5. Magnetic Refrigerator Engine and Sample Area 25 exchange gas* the thermal conductivity of helium gas in an order of magnitude greater than that of air. After the inner dewar is at nitro gen temperature, the transfer of liquid helium into the dewar may he effected. The liquid helium to be transferred is normally in a 50- liter storage vessel. The helium liquid is transferred from the storage vessel into the inner dewar of the magnetic refrigerator through a vacuum-jacks ted transfer siphon by means of a slight 1 psi) over pressure of pure helium gas. The transfer-siphon vacuum jacket should be evacuated prior to a transfer. Care must be taken that there is no binding of the siphon at any point. Helium gas should be blown through the siphon immediately prior to a transfer so as to minimize the possibility of air solidification and consequent blockage of the siphon. Extreme care should be exercised to prevent any air from entering the helium storage vessel} the opening into this vessel is quite small and easily blocked by solid air. Should this passage become blocked, it must be opened immediately. A sharp steel rod should be available for this purpose. Before a transfer is started, it should be known positively that the bottom of the transfer siphon is at least a few inches beneath the liquid level in the storage vessel. If the liquid level should drop below the siphon tip during a transfer, there will be considerable wastage of helium before it becomes evident that this situation exists.

That end of the siphon which is in the helium dewar of the refrigerator should be down as far as possible} this will result in the most efficient utilization of the gas phase cooling in the initial stages of transfer. 26

~~While the above transfer is going on, there should be a small quantity of purified (by means of a liquid N^-charcoal trap) helium gas present in the sample assembly.~~

~~The initial pressure of helium exchange gas should be 10 to 100~~

^£Hg. More may be admitted if necessary. By means of the exchange gas the sample, engine, salts, etc. will be gradually lowered in tem perature from nitrogen temperature to helium temperature. When the helium dewar is finally filled with liquid, the sample assembly should come in to thermal equilibrium at 4«2°K.; this may be verified by noting whether the resistance thermometers have stopped changing in value.

Reduction of temperature from 4*2°K. is accomplished by decreasing the pressure in the liquid-helium dewar vessel. This reduction of pressure is brought about by a conventional vacuum system, i.e., forepump and oil diffusion pump. At the present time, the forepump ia a large, high-capacity Kinney mechanical pump} the diffusion pump is a Consolidated Vacuum Corporation MB-200-021, using butyl phthalate as the pumping fluid. The heater for the diffusion pump has been modi fied and, at this writing, is a 220-volt household-range unit. When this heater is operated at 175 volts, the resultant vacuum has been observed to be 100^/4. Hg., corresponding to a temperature of 0.98°K*

A small step that seems to be of considerable practical value is the practice of loosely blanketing the diffusion-pump heater and boiler with aluminum foil which reflects most of the radiant energy back into ihe boiler. Although these details may appear trivial at first glance, the lowest temperature attainable by pumping on the liquid helium is of 27 the utmost importance since the operating cycle of the magnetic refrigerator "begins at this point.

~~Work is presently under way to modify the helium-hath vacuum system. The large Kinney pump is being replaced by a smaller Stokes mechanical pump in series with a Consolidated Vacuum Corporation~~

KB300-09* 7500-watt oil diffusion pump. The smaller MB-200 booster will remain in the system, although it will be connected to the large booster through four-inch copper pipe rather than by means of the flexible vacuum hose presently in use. It seems probable that these

improvements will make possible the attainment of vacuums of the order of 30 to 40^-Hg., corresponding to a temperature significantly less them 1°K. In turn, this will result in a considerably improved refrigerator cycle for reasons which will be discussed later. Although it

~~is possible to reach temperatures as lovt as 0.8°K. by these methods, the amount of equipment necessary becomes formidable. Furthermore,~~

~~0.9°K. is adequate for operation of the magnetic refrigerator.~~

~~Cooling Below 1°K.~~

~~The cooling below 1°K. is accomplished by means of the adiabatic~~

~~demagnetization of a paramagnetic salt. A device which employs the~~

~~above principle and also provides for the continuous extraction of heat~~

~~from the sample space is called a magnetic refrigerator. The funda mental features of the magnetic refrigerator have been comprehensively~~

~~described in the literature. 5 * 6 *'7~~

~~Many details of the particular magnetic refrigerator employed in 0 this research are given by Stroud. 28~~

In order to avoid useless repetition, no effort will be made to cover the same ground covered in References 5-8. A very general dis cussion of operational principles is given below, chiefly so that the reader will be able to appreciate the significance of the various refinements of technique which have been made during the course of these measurements. An apparatus capable of continuously removing heat and maintain ing a constant low temperature is extremely useful in specific heat determinations. If the heat were not continuously being extracted from the sample, the warm-up times would be prohibitively short for any reasonable rate of heat input to the sample. Continuous heat removal is accomplished through the use of thermal "valves" that may be opened or closed as required by the application of an external magnetic field.

These valves (see Figure 5) are links of lead which are superconducting in the absence of a magnetic field and normal in the presence of a magnetic field greater than about 800 gauss. Lead in the supercon ducting state is a much better thermal insulator than lead in the normal state. Thus, the effect of applying an external magnetic field to the valve is to make the valve conduct heat. The following cycle of operation refers to Figure 5* (l) with the bottom valve open and the top valve closed, the refrigerant salt is magnetized, the resultant heat of magnetization flowing through the bottom valve into the helium bath; (2) with both valves closed, the magnetic field is reduced to a low value, and the salt is cooled as ~a result of the adiabatic demag^- netization; (3) with the top valve open and the bottom valve closed, the demagnetization is continued to its lowest point, and the reservoir

~~salt and sample are cooled; and (4) with both valves closed, the salt 29 is magnetized to a point coinciding with the starting temperature for the ayole.~~

The refrigerator engine oonsists of the working salt, the reservoir salt, the two lead links, and connecting pieces. Details of Q construction have been described by Stroud. However, certain changes have been made in the apparatus in attempts to increase the over-all effectiveness•

In the most recent experiments, the salt, manganese ammonium sulphate, instead of being packed or pressed into the brass shell is grown in place on the copper—fin system. The salt is consequently in good oontaot both with the fin and with itself, leading to an improve ment in over-all thermal conductivity. In some oases the red fiber connectors have been replaced by thin-walled stainless steel tubing for increased strength— the thermal conductivity remaining about the same. Lock washers have been added at all screw connections to pre vent the high thermal impedances which would arise if the connections were loosened by vibration.

~~Perhaps the most critical feature of the engine is the design of the lead links. There is inevitable back flow of heat since, even~~

~~in the superconducting state, a finite amount of heat is transported.~~

~~The controlling factor in link design is the area—to—length ratio}~~

~~this ratio has been varied since the first design due to the necessity~~

~~of trying to choose the optimum link design for the specific applica~~

~~tion. If the back flow through the top link is large, it is reflected~~

~~as an approximately sinusoidal ripple in the temperature of the sample. 30~~

~~Both thermal—valve magnets and the refrigerant magnet have been replaced. The present measured values of field strength are~~

~~8,250 gauss for the refrigerant magnet and 830 and 790 gauss for the upper and lower switching magnets respectively. The magnets are oil-cooled.~~

The oyclio operation of the magnets is automatically controlled by a system of motor-driven rheostats, relays, etc., and has been Q fully desoribed by Stroud. The total cycle time is two minutes, which appears to be approximately correct. Auxiliary air cooling of the rheostats Improves their operation considerably.

In those instances where the refrigerator has failed to work properly, the trouble has usually been found to lie in the para magnetic salt itself. More work is needed in the development of stable salt pills with improved thermal characteristics.

~~The lowest temperature attainable with the magnetic~~

~~refrigerator is dependent upon a variety of factors. These faotors are of two general natures! heat—extraction rate and heat-input~~

rate. One factor very important to the heat-extract ion rate has already been mentioned— the design of the thermal valves. The other major factor in determining heat-extraotion rate is the paramagnetic

~~salt. The entropy-temperature curves for a particular salt determine~~

~~the extraction rate and ultimate temperature. The temperatures~~

~~reached in the experiments discussed here were 0.3°K« to 0.4°K»,~~

~~which was considered adequate for determining the specifio heat~~

~~parameters ^ a n d 6^. However, considerable effort was made to~~

reduce the heat-input rate to the sample. Three new radiation 31 shields were added inside the pumping tube to decrease heat input due to radiant energy. The electrical leads entering the sample space were replaced and great care taken to insure intermediate contact with the helium bath so that the amount of heat entering the sample by conduction was small. As a final precaution, the helium exchange gas was pumped out for hours prior to taking a set of measurements to reduce gas-phase conduction to a negligible value. Sample temperature records indicate a total heat leak into the sample of no more than a few ergs per second, which should not affect the refrigerator operation. ThiB concludes the discussion of cooling procedure. As mentioned previously, the details of these processes are described in detail by Stroud. 8

Apparatus The magnetic refrigerator itself has the function of providing a low temperature environment in which measurements may be made* Considerable auxiliary equipment is necessary to perform the experiments. Some of this equipment, such as the helium-bath vacuum system has been discussed elsewhere in this paper. In this section certain other elements of the apparatus will be considered. The high vacuum system consists of a mechanical forepump

Q and two diffusion pumps in series. Details are given by Stroud. The only modification of this equipment has been the installation of a dropping resistor in the power supply to the Consolidated Vacuum Corporation 1ICF60-01 diffusion pump so that it operates at 32

~~its rated voltage of 9 0 volts rather than at the 110-volt supply.~~

~~The primary function of the high vaoinra system is to aahieve thermal~~

~~isolation of the sample during oalorimetrie measurements. The system~~

~~seems quite satisfactory for this purpose as judged from the pressure~~

~~( 10mm. Hg.) and from the observed heat leak into the sample. _7 In observed pressure of 10 ran. Hg. at the gauge corresponds to a~~

~~pressure of 10"*^° mm. Hg. at the sample.~~

~~A differential manometer is used to measure the vapor pressure~~

~~of the helium bath. The two fluids are mercury and butyl phthalate.~~

~~Although the theoretical density ratio at 20°C. is 12.944* it is wise~~

~~to determine this ratio experimentally from time to time. Uanometrio~~

~~readings have been faoilitated by the addition of mirrors behind the~~

~~liquid columns • With the resultant reduotion in reading error due~~

~~to parallax, it seams probable that readings may be made accurately~~

~~to ~ 0.03 am. oil. This corresponds to an error of approximately~~

~~8 x 10“3 degrees Kelvin at 1.1 °K.~~

~~Various eleotrioal circuits are necessary for oalorimetrie~~

~~observations. Heat is added to the sample electrically, and the~~

~~temperatures are determined by variations in resistance of carbon~~

~~radio resistors. Standard potentiometrio techniques are used through- g out and are desoribed by Stroud. A standard direct current in~~

~~ductance bridge circuit is used to measure the susceptibility of~~

~~the paramagnetic salt.~~

A major piece of apparatus which has been developed for 5 28 27 these measurements is a so-oalled "superconducting* switch * *

~~(see Figure 6). The purpose of this switch is to provide themnal 33~~

~~- No. 18 COPPER WIRE T O SAMPLE' TO RESERVOIR~~

~~SILVER SOLDER JOINT 1/8* STAINLESS STEEL TUBE~~

~~LEAD RIBBON, A /L -3* IO~*c« SOFT SOLDERED TO COPPER WIRES SILVER SOLDER JOINT~~

~~BRASS CUP a .010 NYLON LINER~~

~~SWITCHING COIL .0 0 2 COLUMBIUM WIRE H ■ 1211 I omim~~

Figure 6. Superconducting Thermel Switch end Superconducting Solenoid 34 isolation or, alternatively, thermal oontaot between the sample and the reservoir. The switoh is oloaed, allowing the sample to oool to reservoir temperature, after the exahange gas has been pumped away.

When the sample has reached reservoir temperature, the switoh is opened and the sample is thermally isolated, and oalorimetrie measure ments may be started. The operating principle of the switoh is the

~~same as that of the thermal "valves'* in the magnetic refrigerator; namely, that the link either does or does not conduct heat, dependent~~

~~on the application of an external magnetio field.~~

~~The novel feature of this switoh is that the magnetio field~~

~~is supplied by a zero—power ooil* The external field is supplied by a small ooil wound of .003-inah niobium wire which is itself a super~~

~~conductor. Since the ooil is superoonduoting, there is no power~~

~~dissipation in the bath. This power dissipation would be a serious~~

~~drawback for a normal metallic ooil because of its effect of increas~~

~~ing the evaporation rate of the helium bath and thus decreasing the~~

~~time available for experimentation. The superoonduoting switoh is~~

~~designed to be a sturdy unit which may be salvaged and used again~~

~~each time the sample assembly has to be taken apart. The pure lead~~

~~link has an it! ratio of 3-08 x 10-4 cm. J for the superoonduoting~~

~~ooil, the magnetio field at the center of the ooil is given by~~

~~H (gauss) - 1211 I (amperes) (2.1)~~

~~with the field at the ends of the ooil dropping off to 8396 of this~~

value. When a sufficiently large current is passed through the super- 35 conducting coil, the induced magnetio field will exceed the threshold field of the niobium itself and result in the ooil going into the normal state. This value of current was experimen tally determined to be 2.1 amperes.

In practice, the operation of this unit has been very satisfactory. Figure 7 is a reproduction of an actual chart strip record. The observed switching ratio at 1°K. is about 6s 1, and becomes very high at 1°K.

The use of this switch has made it possible to begin a series of heating cycles at reservoir temperature and make measurements to 1°K. without the heat leak through the link becoming exoessively large. In concluding, it should be mentioned that the link itself need not be lead? indium, tin, sine, and aluminum are all possibili ties and each has particular advantages and disadvantages.

~~Thermometry~~

The thermometers used in these measurements are of three typest vapor pressure, magnetio, and carbon resistance. Vapor pressure is, of course, the primary thermometer, and the other two are calibrated against vapor pressure in the region 4*2-1.0°K.

~~Below 1°K. the magnetio thermometer is used as the standard, and the carbon resistors are calibrated against it.~~

~~Vapor Pressure Thermometry. Vapor pressure measurements in the range 4.2-1.0°k are standard procedure at this time, and these measurements have been carried out in accordance with The 36~~

~~10 00 3 0 90 •o 1*0~~

~~too~~

~~1 9~~

Figure 7* Thermal Switching Ratio Magnetic Thermometry* Magnetio temperatures are determined by a ballistic method employing an induotance bridge. Absolute tempera— tures are determined in accordance with deKlerk"^ from the basio relationship between galvanometer deflection and temperature whioh is derived in Stroud's thesis. 8

The relationship used is (2.2) where d is ballistic galvanometer deflection in centimeters, a and b are constants, and T* is the magnetio temperature. A plot of experimental d versus l/T thus provides a relationship for deter mining T* for any given deflection. In practice, the calibration of the magnetio thermometer usually extends from 4*0 to 1.0 °K.» and T* values as low as 0«3°K. are calculated from observed values of the deflection. It is important to realise that I* is by no means equivalent to the absolute thermodynamic temperature T. A relation ship between T and T* is neoessary. This relationship is customarily developed in two steps. The first step is to determine from T*. This oorreotion accounts for the influence of the shape of the salt specimen on the internal magnetio field. Kurti and Simon^ first recognised the necessity for this oorreotion. Following deKlerk;^0, for temperatures below 1°K., (2.3) 38 where

~~A = l&ir-6) c/v <2-4> for a cylinder whose axis of rotation Is parallel to the direction~~

~~of magnetization~~

t ■ T r f -/)] e --V / - (% F <“ ) where a - the diameter of ellipsoid and o - the length of ellipsoid. “2 °K/co C/V is the Curie constant per unit volume, given as 2 x 10 1 by Kurti and Simon31 and by Cooke. 32 for manganese ammonium sulphate.

~~The salt sample used in this work was a cylinder with a/e Cr.1 /7 (2*7)~~

~~•° e = \) H y s ~ o <2 -8>~~

~~fluid t = o. (2. 9 )~~

~~A - (%77-- 0. V/jr) (o. oao) = 0.67J't>A (2.10) The above would be correct for a solid single crystal of density~~

~~1.83 gn./oo* Actually, a collection of smaller crystals is used~~

~~with a resultant density in the ellipsoid proportional to the filling~~

~~factor ^ • In our case this filling factor was measured as 0*435> so~~

~~we have finally A - (.07J-) (.¥Sir) = O. 0 33 VC. (2.11) and T* - T * + 0.033 °X, (2-12> 39~~

The Intrinsic susceptibility of the salt is influenced by the presence of a magnetio field. Stark splitting of the energy levels, hyperfine splitting due to nuclear interactions, and ionic magnetic and exchange interactions all occur. Although theories exist for the above, direct experimental deteimlnations for manganese ammonium sulphate have been perfoxmed by Cooke. 32 The result of the above magnetic interactions is, of course, that the true thermodynamic temperature differs from T*Lorenta • A graph of this relationship is given in de Klerk's article in the Eandbuoh der Phyaik.^0 It turns out that in the region 0.3-0.6°K. the magnitude of the correction is such that

~~r - 7 * ± V ° A ,/ (2.13) so that~~

~~T~ ?T*r ^ - oa.Y= T*+.033 OS.Y (2.14) or~~

77 - 7’’*' + 0.009 °K. (z-’S) which is a final expression making it possible to determine the true thermodynamic temperature from the deflections of a ballistic galvanometer. It should perhaps be stressed that the last oorreotion above may well be £ 10j£. A sample of a magnetio calibration curve is shown in Figure 8 . The magnetio temperature measurements may now be used to calibrate the carbon resistors as seoondaxy— truly tertiary— standards.

~~Carbon radio resistors have been found to possess a negative MANGANESE~~

50 BALLISTIC GAUVANOMETER DEFLECTION VS. T*

~~d-cm~~

~~20 7-16-59~~

~~l/T -° K Figure 8* Magnetic Thermometer Calibration 41~~

~~temperature coefficient of eleotrio&l resistance. The carton is~~

~~formed of minute orystallites which are semiconductors. The theory of semiconductors predicts a negative temperature coefficient of electrical resistance on the basis of the application of Permi-Dirac~~

~~statistics to the carriers which are postulated as being occasional~~

~~electrons loosely bonded to lattice positions.~~

~~The exact manner in which resistance varies with temperature~~

in a carbon resistor is critically dependent upon purity and mechanical history* It also appears that the E-T relationship is affeoted by thermal cyoling. Various brands of resistors have been used in cryogenic research in the past* JU. 1 en-Bra dlsy 2*7 -A- to

~~270 have been studied by Clement* Dolecek* and Logan.^ These~~

~~resistors have excellent stability characteristics* but the absolute~~

~~values of resistance become unpleasantly high below 0 *3° £• IRC~~

~~270 * 1/2-watt resistors were employed early in this program* but~~

~~appeared to suffer from two deficienciest first* the effeot of~~

~~cyoling between helium temperature and room temperature was notice~~

~~able | and seoond* repeated observations indicated a marked departure~~

~~of the calibration curve from linearity as the 0 *3-0 *4°K* temperature~~

region was approached*

The type of resistor used in recent work is a Speer 470JI*

~~1/2 watt* Considerable information on the characteristics of Speer 34 resistors has been published by Hicol* The resistors used were~~

~~kindly supplied by Dr* Niool* now with the A.D.Little Company* The~~

~~resistance of these units la approximately 1000-A- at 4*2°K*» 42~~

2000-ft- at 1.0°K., and 3500 JV. at 0 .5°K. which is a suitable range of resistance to cover with standard potentiometrio techniques* It was found that these resistors were somewhat current sensitive, although the precise cause remains unknown. A calculation, after

Bennan, 35 of the temperature rise in the resistor due to power dissipation indioates that this effect is insufficient to account for observed resistance variations. The resistors were operated at a power of about 10~^ watt, the measuring current being maintained constant throughout an experiment. The current souroe for the sample theimometer contained a voltage divider circuit and precision potentiometer to insure current stability. The protective layer was not removed from these resistors. The resistors were inserted in tight-fitting copper jackets which were, in turn, screwed directly into the sample. The calibration equation used for the Speer resistors was

~~“ ///J77 (2.16) where the constant A and B were determined by plotting log^B versus~~

^ log1 OR/T over the temperature interval 4.2—1 .0°K. Below 1°K., as previously mentioned, the resistors were brought into agreement with the magnetically indicated temperatures. A typical resistor calibration curve is shown in Figure 9*

~~Sample Preparation~~

~~It is clear from a consideration of the theory of specific heats of metals that the purity and metallogxaphio history of a specimen 43~~

~~MANGANESE SAMPLE THERMOMETER~~

~~3.25 NEW SPEER CARBON RESISTOR 1/2 WATT, 470 OHM~~

~~3.20~~

~~3.15~~

~~310~~

~~3.05~~

~~3.00~~

7-16-.59 2.96 1.3 1.4 1.5 1.6 1.6 I *«QioR N T Figure 9* Carbon Resistor Calibration Curve 44 influence the specific heat very considerably. In the preparation of the four samples discussed here, every effort was made to secure the highest purity possible. Also whenever possible, the heat treat ment of the metal was suoh as to avoid any strains or unstable ciystallographio states which might give rise to a low—temperature speoific-heat ancmoly.

~~Sodium. Primarily because of its well-known chemical activity, the preparation of the sodium sample presented considerable difficulty.~~

Because of this chemical activity, it was necessary that the sodium be enclosed in a container during oalorimetrie investigations. A thin- wall oopper cylinder waB accordingly designed and constructed. After inserting the heater and the theimometer in the copper cylinders, the heat capacity of the assembly was measured in the temperature range

0.35-1 *0°K. The heater consisted of 80 -/!■ of manganin resistance wire wound on a nylon core and inserted in a tightly fitting oopper jacket. The themometer was mounted in the same way and was one of the Speer 4 7 0 71., 1/2-watt resistors mentioned previously. The mass of the container was 134 grams. After determining the specific heat of the container, it was decided to fill the container with jure sodium metal by a direot distillation process. Uuch valuable inform ation regarding the handling and preparation of sodium metal is contained in the Liquid Metals Handbook, Na-HaJC Supplement. ^ ACS reagent-grade sodium of stated purity > 99*99^ *as obtained. All handling of the sodium was aoocmplished under a blanket of pure argon 45 gas* When it was necessary that the sodium he stored for a period of time, it waa placed in a container of toluene* No appreciable surface contamination of the metal was observed*

The distillation was effected in a stainless steel apparatus enclosed inside a bell jar. Figure 10 is a drawing of the apparatus* A vacuum of less than 1 JUL Hg* was maintained at all times. The still consisted of a boiler, oondensor, and receiver. All heating was accomplished by means of electrioal heating tape which had been thoroughly out gassed. The stainless steel components were surrounded with aluminum foil to prevent overheating of the bell jar by radiation*

The distillation was carried out at a temperature of about 250°C. As the liquid sodium condensed, it ran out of a hole in the bottom of the oondensor directly into the oopper cylinder previously mentioned.

After the oopper cylinder was filled with sodiixn, the temperature of the sample was reduced to room temperature at the rate of 3-5°C. per hour in order to thoroughly anneal the specimen* The sodium metal

(64*6 grams) was deposited directly on the heater and thermometer assemblies, thus insuring exoellent thermal contact* After filling, the sodium was sealed off by plaoing a oopper disk over the filling hole* The sample assembly was then rigidly mounted inside a fiber tube, and oalorimetrie measurements begun* Figure 11 is a drawing of the sodium sample assembly*

~~Cobalt. The cobalt metal was kindly supplied by the African~~

~~Metals Corporation in the form of a solid cylindrical ingot of 11*36 SODIUM DISTILLATION APPARATUS~~

~~TANTALUM FOIL C -HEATER TAPE~~

~~HEATER TAPE 6 ARGON SUPPLY THERMOCOUPLE LEADS TO HIGH VACUUM A BASEPLATE SYSTEM B BOILER C CONDENSER D RECEIVER E BELL OAR F RADIATION SHIELDING Figure 10. Sodium listillation Apparatus 47~~

~~SAMPLE ASSEMBLY SODIUM~~

A COPPER ROO J SODIUM SAMPLE B BOTTOM CAP K THERMOMETER C SAMPLE SHIELD ASSEMBLY D TOP CAP L HEATER E FIBER TUBE M GROUND SCREW F COPPER MOUNTING N SUPERCONDUCTING COLLAR LEADS 6 MOUNTING SCREWS(3) 0 WIRE THERMAL H COPPER SEAL-OFF DISC CONTACTOR I COPPER CONTAINER Figure 11. Sample Assembly Sodium 48 moles. This cylindrical sample was the same one used by Heer and

~~Erickson^ in previous specific heat measurements. Dimensions are~~

~~indicated in a photograph of the cobalt sample (Figure 12). Chemioal analysis of the sample indicates a purity of 99*9& the principal~~

~~impurity being iron. X-ray studies carried out by Dr. A.Austin~~

~~of Battelle Memorial Institute revealed that this sample was of hexagonal close-packed structure.~~

~~The sample preparation was quite simple in this case. Holes were drilled and tapped for a heater, carbon resistance thermometer,~~

~~and ground screw in one end of the specimen. The other end was tapped~~

~~for three 6/32-in oh screws which attached the cobalt specimen to a~~

~~copper collar. The copper collar was then press fit into a fiber~~

~~tube, thus insuring a rigid sample mount with minimum heat leak due~~

~~to vibration. A view of the sample mount was given in Figure 11.~~

~~This method of mounting the sample was used, with minor modifications,~~

~~for all the specimens.~~

~~Manganese. The manganese metal was supplied in the form of~~

~~chips by the Bureau of Mines in Bartlesville, Oklahoma. The manganese~~

~~chips were of stated purity > 99*9£* The chips were the result of an electrodeposition process. The ohips were then oast in the form~~

~~of a solid cylinder 1-7/16-inches in diameter and 1-5/16 inches in~~

~~length by Dr. O.W.F.Rengstorff of Battelle Memorial Institute.~~

~~X—ray studies showed the manganese to be alpha—phase large body-~~

~~centered-cubic crystal structure. The mass of manganese in the~~

~~cylinder was 3.10 moles. Owing to the difficulty of machining 49~~

~~Figure 12. Cobalt Sample Assembly Sample Assembly 51 manganese metal, it was neoessary to design a oopper fixture to~~

~~hold the sample. A photograph of this fixture was 146.5 grams.~~

~~The fixture and sample were then mounted as one unit inside a fiber~~

~~tube as in Figure 11. Heater, thermometer, and ground—sorew~~

~~assemblies were screwed in place in deep wells in the oopper fixture,~~

~~with heater and thermometer in direct contact with the manganese~~

~~sample. The degree to which this arrangement succeeded in establish~~

~~ing thermal contact will be discussed in the section on experimental~~

~~results•~~

~~Iron-Cobalt Alloy. An alloy of 6*5 weight per cent of iron,~~

~~93*5 weight per cent of cobalt was prepared using cobalt pellets of~~

~~purity 99*9& kindly supplied by Dr. F.HJforrall of Battelle Memorial~~

~~Institute, and iron rod of 99*999^ purity, supplied by A.DJlaokay, Inc..~~

~~The specimen was prepared by Mr. George Economy of The Ohio State~~

~~University Metallurgical Engineering Department. Preparation of the~~

~~sample was accomplished by means of electrical induotion heating in~~

~~vacua. This technique insures thorough mixing of the constituents~~

~~by the formation of large currents in the metal. The weight of the~~

~~finished sample was 36.97 grams. A photograph of this sample assembly~~

~~is given in Figure 14* According to the phase diagram in Cobalt and 37 Its Alloys. 1 this alloy should be beta cobalt, and, indeed, x-ray~~

~~studies revealed the structure to be face-centered—oubio consistent~~

~~with the lattice parameters of beta cobalt. A hole was drilled and~~

~~tapped through the alloy sample, and a cylinder of oopper was screwed~~

~~tightly into the alloy. The measuring assemblies were attached as for 52~~

~~Figure 14* Cobalt—Iron Alloy Sample Assembly 53 the manganese sample) and then press fit into a fiber tube* The total oopper correction for the attached parts amounted to 39*6 grams* CHAPTER I I I~~

~~EXPERIMENTAL RESULTS~~

~~General Methods~~

The experimental determination of specific heat was accomplished by measuring the temperature increase which resulted from the addition of a known quantity of heat to the sample. A Q was supplied electric ally through a heating ooil which was in intimate contact with the sample • The quantity of heat A ft is

= (3 .1 ) where A Q is given in mill!Joules when the potential drop E is in volts, the current I in milliamperes, and the time t in seconds. The potential drop was measured with a Rubicon TJype-B potentiometer

(Catalogue No. 2780). The current I was measured with the same potentiometer by measuring the potential drop across a 10-/L standard resistor (Rubicon No. 88779)* The time was measured with an electric timer (Precision Scientific Company) that read to 0.1 second. Since the external leads to the heater coil were of superconducting niobium wire, it was insured that the measured potential drop was wholly that due to Joule heating in the sample. The estimated accuracy of both

~~the potential drop and current determinations is 0.001^, while the accuracy of time determination was no worse than 0.37&. Consequently,~~

54 55 the accuracy of determination of A Q is certainly far less than 1^6, and mayt In fact, he oonsidered a negligible source of experimental error. The experimental determination of A t , the temperature change of the sample, presented more difficulty. A typical experimental heating cycle is shown In Figure 15* Specific heat determinations were made from the different rates at which the sample cooled when sup plied with joule heat. This technique has been fUlly described by

Logan, et al 38 and by Clement and Quinnell. 39 Using the change-in- slope technique, it is obvious from Figure 15 that the aocuracy of determination of the rate of change of temperature caused by incomplete thermal isolation was a faotor in the over-all accuracy of speoific- heat determination. In most cases, however, it was readily possible to determine the residual cooling slope to 1^. The dashed lines in

~~Figure 15 represent extrapolations of the residual oooling rates~~

(AB and CD) before and after adding heat to the sample, while the heavy horizontal lines BO represent the total temperature change during the heating interval. The average temperature during the cycle occurs at the intersection F of the heating slope line BC and the temperature- ohange line. The quantity which is actually measured on the chart strip is the potential drop across the sample thermometer. The chart strip is on a Brown self-balanolng reoording potentiometer (Y153X11),

~~0-100 microvolts full scale. The potential drop across the sample~~

~~resistor is first fed to a Rubicon potentiometer of the same type as~~

~~that used for measuring heater current. The Hubioon potentiometer is balanced to the nearest 100 microvolts, and the unbalance then read~~

on the Brown potentiometer. The determination of T, the average Figure 15. Epical Heating Cycle 57 temperature during a cycle, and A T, the effective change in sample temperature, must be made from measured values of, respectively, 5 and AB . Since the measuring current is maintained constant, the quan tity directly related to temperature is the resistance R of the carbon resistor. The relationship between temperature and resistance was previously given as

~~7 7 = * ^ - (3'2) where a and b are constants determined by plotting log^^B versus~~

y iQfl^ 7 The constant b is the interoept of the curve and can be evaluated to an accuracy of about 1?6 by plotting calibration data in the 4-l°K* region. The resistance can be measured quite accurately,

~~just as in determinations of A Q. Consequently, almost the entire~~

~~error in a temperature determination lies in the constant a which is~~

~~found from the slope of the calibration curve. This slope, unfortu nately, is not completely constant below 1°K. and seems to deviate~~

~~from linearity rather markedly in the region of 0 .3°K. The slope of~~

~~the calibration curve in this temperature region is found from corre~~

~~lations with the magnetic temperature T*. Although every effort was made to determine the slope constant with the utmost accuracy, it is~~

~~felt that these values may have an error of 5?6, and this error will be~~

~~directly refleoted in A T and, consequently, in specifio heats. A~~

~~summation of the above statements indicates the fact, well known to~~

~~the cryogenic researcher, that the chief difficulty in measurements~~

~~below 1°K. is temperature determination. The accuracy of these results~~

~~at the lowest temperature reached is accordingly approximately 58~~

Analysis of the data seems to 'validate this viaw] the scatter of the data being, for the most part, about 5 to Of oourse, the number of data points taken tends to improve the statistical validity of the results. It should be mentioned that the temperatures of the helium bath and the sample shield are monitored throughout all measurements, and these temperatures are utilized for oross-oorrelation purposes in establishing the calibration curve for the sample thermometer. The above auxiliary temperature measurements are made also with Speer

~~carbon resistors and the electrical circuitry is similar.~~

~~Thermometer Calibration~~

~~This section will be devoted to a description of the thermometer~~

~~calibration prooess. Examples of typical curves are given in Figures~~

~~8 and 9« This set of curves happens to apply to the work on manganese, but is completely representative. Simultaneous equilibrium temperature~~

~~determinations were made at three points between 4 *2°K. and the X -~~

~~point of helium, and six points were taken between the ^ -point and~~

~~1.0°I£. The points were spaoed so as to be approximately equidistant~~

~~on the scale for determination of magnetic temperature. Approximately~~

~~ten calibration points were used for all the different experiments] more points than this out down the time for aotual speoifio heat mea~~

~~surements. The fundamental temperature determination was based on the~~

~~helium—bath vapor pressure, using the ^ 5 5 soale. The deflections of~~

~~the ballistic galvanometer in the susoeptibility bridge were measured~~

~~and plotted versus reciprocal temperature. The resistances of the~~

carbon thermometers were simultaneously measured. One oarbon 59 thermometer wae in the sample, one in the sample shield, and one immersed in the helium hath. The hath thermometer and sample-shield thermometer, in addition to providing vital information with regard to magnetic refrigerator operation, are used to provide auxiliary o information for the correlation points helow 1 K. After the calibra tion proper has heen completed, the temperature of the system is lowered. Vapor-pressure readings are no longer meaningful, hut simultaneous determinations hy the magnetic and resistance thermo meters enable one to extend the calibrated range of the sample thermo meter into region of measurements without having to rely upon a linear extrapolation of the calibration curve. The basic calibration equation is determined from vapor-pressure data. It is, to repeat,

The constants a and b are obtained from a plot of loglf)R versus T o g nT 3 ^ * The constant b is the intercept of this curve, and may be determined easily from the curve to an accuracy of better than 0.3%.

The constant a, however, is equal to the square of the slope of the calibration curve, and can be expected to vary somewhat with decreasing temperature. Following Keesom and Pearlman,^ the value of a may be calculated from the basic resistance—temperature formula, assuming b is known. The correlation points mentioned above suffice to establish such a deviation curve, T being measured magnetically. The smoothed a (T) curve resulting from this process may be used then for low- temperature interpolation. It was standard procedure to take about five such correlation points below 1°K. and, using the deviation curve so established, thus determine the explicit form of the temperature- 60 resistance relationship for the sample thermometer in a particular temperature range. The manner of variation in slope parameter a was a decrease between 0.5°K* and 0.3°K. The reason for this fairly marked decrease in slope is not presently understood. The decrease of the slope does appear to be systematic, however. Power dissipation in the resistor itself seems an unlikely cause as, first, the input power was ^10“11 watt and, secondly, varying the input power by a factor of 100 had no apparent effect upon the systematic slope decrease. One thing is quite clear from these measurements! if specific heat determina tions are based on a linear extrapolation of a carbon-resistor temperature-resistance curve, there is a possibility of serious error in the results. The 4—1°K. calibration curve is merely a basis for intelligently establishing a useful deviation curve. It is obvious that the accuracy of these results are limited by the calibration pro cedure, and it is equally obvious that more work needs to be done in determining and explaining the temperature variation of resistance of commercial carbon radio resistors in the range 1.0-0.1°K. Table 1 gives the data for the deviation curve applicable to the calibration discussed in this section.

~~Copper~~

It was necessary to determine the heat capacity of the copper con tainer which was later to be filled with sodium. The specific heat of copper is presently quite well known, and these measurements were undertaken for two reasons other than determination of the specific heat. One reason was to establish the validity of the experimental 61

~~TABLE 1 MANGANESE DEVIATION CURVE DATA~~

a 4.0 0.0978 1.0 0.0978 0.8 0.0970 0.5 0.0926 0.4 0.0845 0.3 0.08 (a)Sample temperature - T - t where b * 2 .70786. technique. The second and major reason was to determine the heat capacity of this particular copper container so that it could he sub tracted from the heat capacity of the composite copper-sodium sample. In order that the copper represent only a small fraction of the com posite heat capacity, it was necessary to keep the mass of copper as low as possible. The mass of copper actually measured was 134 grams, or 2.11 mole. During these measurements, thermal contact between sample and sample shield was established through a 10-centimeter A —A length of 20-gauge copper wire - 5*18 x 10~ cm.). The heat capa city determinations were carried out over a relatively short span of temperature because existing specific-heat formulations seemed com pletely adequate for extrapolation purposes. Data pertinent to speci fic heat determinations are given in Table 2. A representative set of calibration curve data for thermometry was given in the section pre vious to this. In this section, the calibration constants for the sample thermometer are listed on the data sheet. The specific heat of the copper is graphically represented in Figure 16. 62

~~Legend, for Figure 16. Heat Capacity of Copper~~

~~Least-aquarea data fit) *0.73 millijoule/mole-°K. Experimental Data of March 2 0 , 1959 • Results of Corak et al^ (1954)~~

~~Results of J.Bayne^ (1955)~~

~~Mass of oopper measured. * 2.11 mole 64~~

~~TABLE 2~~

~~MATA® COFFER HEAT CAPACIIT~~

~~o, A C , T , W A 1. millijoule Cycle microjoule °K. millidegree %~~

~~Data of March 20. 1959~~

~~— -- ~~~

~~2 17.06 ,408 22.9 .146~~

~~3 9.40 .406 14.8 .635 4 8.94 .316 14.3 .625~~

~~5 7.41 .362 12.3 .602 6 6.41 .366 11.0 .583~~

~~1 13.12 .310 20.2 .649~~

~~8 14.55 .311 21.8 .661~~

~~9 9.76 .311 14.9 .655 10 9.48 .393 12.9 .135 1,(o) M - - -~~

~~12 10.41 .509 13.8 .159 13 31.52 .529 38.4 .911 14 41.18 .548 50.3 .938~~

~~15 46.80 .544 51.5 .909~~

6J Ufa (a)por sample thermometer T - - -- *

~~Magnetization cycle altered during beating. 65~~

~~Sodium~~

The mass of sodium which was distilled into the ahove copper container was 64*6 grams, or 2.81 moles. The measurements were carried out using the same copper-wire thermal connector as that employed in the copper measurements. The heat capacity of the composite sample was determined in the range 0.4-3.0°K. The total heat capacity turned out to he expressible in the general form

c = V T +■ . (3.3) Since it was known that part of the total heat capacity was due to the oopper container, and since the heat capacity of copper is known to he of the same form as the ahove, it was possible to express the total heat capacity as

(3-4) and, by subtracting the copper contribution either graphically or analytically, the specific heat of the sodium metal could be determined. In order to make this copper correction, our experimentally determined value of the electronic coefficinet was used in conjunction with the average of recent experimental values of to establish a total o copper correction. This correction for the 2.11 moles of copper was

= /.i S’ T +o. 0-5) Before making the copper correction, the usable raw data for total heat capacity (39 points) was subjected to a least squares analysis, follow- A ^ ing the methods outlined by Yardley Beers. This analysis assumed 66

~~C 2 that a plot of versus T would result in astraight line. The~~

~~analysis gave the composite heat capacity as~~

~~S'. 3 ? -h /.SO 7* * 0 .6) and, subtracting the copper correction, gives r = 3,7/ T + L3&> t 3 {3>7) No, A or, per mole~~

The data for sodium are listed in Table 3, while the results are graphically represented in Figure 17*

~~TABLE 3 DATA TABLE* SODIUM HEAT CAPACITY~~

~~- (a} C* A Q , ?, A T , millijoule Cyole microJoule millidegree ®K.~~

~~Data of April 25, 1959~~

~~1<*>~~

~~a (0) __r~~

~~3 473.26 .883 82.99 5.703 4 205.00 .721 46.28 4.429~~

~~5 135.85 .649 34.50 3.937~~

~~6 95.66 .579 27.93 3.425~~

~~7 49-79 .517 .16.42 3.032 8 49.05 .493 17.52 2.800 67~~

TABLE 3— Continued*

~~A Q» T , ^ A T, milli.joule Cycle microjoule °K. millidegree ^K.~~

~~9 28.59 .454 10.63 2.690~~

~~10(d ) — — ——~~

~~11 11.95 .417 • 4.791 2.494~~

~~12 361.39 .908 58.43 6.185 13 223.38 .828 42.16 5.298~~

~~14 199.99 .799 35.74 ■5.596~~

~~15 217-59 .758 46.00 4.730 16 161.74 .706 35-60 4.453~~

~~17 118.45 .633 30.42 3.894 18 58.24 .588 17.51 3.326~~

~~19(e) — — — —~~

~~20 74.02 .536 21.47 3.488 21 439*63 1.001 63.77 6.894~~

~~22 351.89 .976 52.44 6.710~~

~~23 307.79 • 956 48.64 6.328 24 254.96 .936 41.32 6.170~~

~~25 225.56 .918 38.98 5.787 26 223.01 .892 33.98 6.563~~

~~27 192.25 .879 32.31 5.950~~

~~28 179.95 .866 30.02 5.994 68~~

~~TABLE 3— Continued.~~

~~0 , A Q T, A T, millijoule Cycle miorojoule OK. millidegree °K.~~

~~Bata of April 2.6 13 3 9 101 5,348.2 1*935 249*6 21.427 102 6,449*6 1 .989 276.4 23.336~~

~~103 5,611.2 1.837 289.5 19.380 104 3,548.3 1.648 232.8 15.244 105 3,476.6 1.520 267.1 13.018~~

~~106 968.4 1.264 99.68 9*715~~

~~1 0 7 ^ —— — —~~

~~108 264.9 1.009 38.27 6.922~~

~~109 39*66 *515 12.71 3.120 110 35*40 *492 13*26 2.682~~

~~111 22.96 *456 9.35 2.456~~

~~n 2(e) — — — —~~

~~113 13.41 .422 5*41 2.479 114 10.06 *403 4.08 2.466~~

~~115 10.67 .408 4.57 2.335 116 9*16 .411 3.95 2.319~~

~~117 13.41 .415 5.59 2.408 69~~

~~FOOTNOTES FOR TABLE 3~~

~~uo, f r 9//SO v (a^For sample thermometer~~

~~(^Insufficient heat, vfcl 'Faulty timing. (^Switching error. (^Refrigerator cycle faulty.~~

~~(f W f scale on strip chart. (s)wrong heater switch thrown. 70~~

~~Legend, for Figure 17* Specific Heat of Sodium Metal~~

~~Data of April 25 -26, 1959 Leaat squares fit to above data~~

~~Results of LJ4.Roberts^ (1957) Electronic texmj ^ - 1*32 millijoule/mole-0!!2~~

~~• — • Lattice texm; ^ - O .485 millijoule/mole-°K^ / / / / / / / / / / JL ✓ / / /~~

~~|___ i 1 15 2.0 25 Figure 17. T-°K Specific Heat of Sodium Metal 72~~

~~Cobalt~~

~~The experimental results for cohalt metal are given in Figure 18.~~

Data are listed in Table 4» Since the data for cobalt were taken at various times and under a variety of experimental conditions, it is necessary to elaborate on the information in the data table. For these runs, a IBC-270_/>. , ]£-watt resistor was used as the sample thermometer. Information as to calibration procedures for these resis- 45 tors is given by Erickson and Heer. The equation used for thermometry was

J < r a T - f - a. • The sample and sample shield were connected via a copper wire, 4 _ . —A — * 8.6 x 10 cm. The specific heats listed in Table 4 for this run of L June 2, 1958, are those calculated using the end points of the heating cycles to deduce £iT. The specific-heat values for each point were cal culated also by the technique of adding the cooling and heating slopes and determining C from the relationship

C = Sr S f - ^ / i p • (3.10) ~3 x / 7 Z £ The values plotted in Figure 18 for this run are the average values of specific heat resulting from these two separate determinations. The thermometric constants were b « 3*919 and a « 0.138. w - / / to - /

~~/ TO ' / i / / « /~~

~~C/T / / 90 - / / / / 40 - / / ■ / / / 30~~

~~/ 00 0 « # > to - O O-t-99 -WUMiWLUCS , ° o T-ll-W -MCM9C ttLUU • / / « t ! H - non / • 10*9049 -SLOK * IH1-M -9U0K n - ■/ 'S /~~

~~I___ |_ ■ ‘ » 1 1 * 19 30 Figure 18. Cobalt Specific Heat 74~~

~~TABLE 4 DATA TABLE: COBALT SPECIFIC HEAT~~

~~4L Qr C* micro joule T, A T , mllli.loule mole °K. millidegree mole- K.~~

~~Data of June 2, 1958~~

~~1 201.0 .612 19.0 11.1~~

~~2 205.6 .624 18.0 12.6~~

~~3 103.6 .625 8.1 11.6~~

~~4 329.4 .637 26.0 12.3 5 192.0 • 646 16.0 12.6~~

~~6 241.7 .641 20.1 12.1~~

~~7 246.3 .565 17.6 14.0 8 184.0 .578 13.3 13.8~~

~~9 164.6 .586 12.0 13.7~~

~~10 208.6 .593 15.3 13.6~~

~~11 149.5 .599 11.2 13.4~~

~~12 161.8 .604 12.4 13.1~~

~~13 353.0 6.25 26.1 13.5 VO ■vt 0 14 227.0 . 18.2 12.5~~

~~15 356.1 .655 29.5 12.1 16 142.8 .670 12.1 11.8~~

~~17 207.5 .625 16.3 12.7 298.7 .640 25.8 11.6 75~~

~~TABLE 4— Continued.~~

~~Q» _ C t micro joule T, T, milli.joule Cycle mole °K. millidegree mole-uK.~~

~~20 183.9 .669 13.9 13.2~~

~~Data of July .11^1358~~

~~1 358.2 .327 14.1 25.4~~

~~2 433.0 .360 18.1 23.9~~

~~Data of‘ September 5. 1958~~

~~1 207 .336 8.8 23.5~~

~~2 178 -349 7.4 23.9~~

~~3 202 .370 6.0 33.7 4 204 .402 11 .2 18.2~~

~~5 221 • 446 12.1 18.3~~

~~6 199 .489 14.5 13-7 7 201 .530 12.2 16.5 8 242 .583 19.6 12.4~~

~~9 247 .637 18.7 13.2 10 204 .696 20.0 10.2~~

~~11 266 .756 27.1 9.8~~

~~12 236 .815 26.6 8.9~~

~~13 305 .898 30.8 9.9~~

~~14 678 .999 64.2 10.6 7 6~~

~~TABLE 4— Continued.~~

~~AQ, c, microjoule T, AT, millijoule Cycle mole °K. millidegree mole—°K.~~

~~Bata of October 28, 1958~~

~~1 201 *548 15*3 13*1 2 202 *552 14*8 13*6~~

~~3 199 *565 15*1 13*2~~

~~Bata of November 11, 1958~~

~~1 266 *485 18.8 14*2 2 228 *498 15*3 15.0~~

~~3 197 .510 14*6 13*5~~

~~a'Time of heating unrecorded.~~

The data of July 11, 1958, were obtained under the same conditions as those of June 2, 1958, except that the sample thermometer constants were now a * 3*949 and b - 0 .125* The helium supply, unfortunately, ran out during this run, and only two heating cycles were completed.

The data of September 5r 195$» were taken under the same conditions as the other runs listed thus far, with the sample thermometer constants being a « 3*927 and b ■ 0.123* A new link of copper wire was inserted for the run of October 28, A i “•958, ^ ■ 6.9 1 10 ^ cm. A new IRC resistor was used, with constants a m 3.950 and b ■ 0.123* For the last sat of data, November 11, 1958, the thermometer constants were a • 3*947 and b - 0.116. 77

~~Manganese~~

~~The data for manganese are given in Table 5* Speer resistors were used. Temperatures were computed using the sample thermometer relation~~

o. o r ?8i> / - (3.11) in conjunction with the deviation chart given in Table 1. The sample and sample shield were linked by the superconducting switch. In order to obtain the C values listed in Table 5» the correction for the heat capacity of copper must be subtracted from the total heat capacity obtained by dividing £ Q by A T. The copper correction for the 2.304 moles used in the assembly was

~~c = /.?3sr-bo./vs7’i (3.12) following the data of Friedberg et al. 46 Data plots are given in Figures 19 and 20.~~

~~TABLE 5 DATA TABLE1 MANGANESE SPECIFIC HEAT~~

~~_ C , AQ» T, A T , mil li joule Cycle microjoule °K. millidegree mole-wK.~~

~~Data of July 16, 1959~~

~~1 437 .901 11 38.03 2 364 .862 11 31.50~~

~~3 636 .649 21 29.13 78~~

~~TABLE 5— Continued.~~

~~C, AQ, T, A T, mlllljoule Cycle microjoule °K. mlllidegree mole~°K.~~

~~4 824 .647 27 23.35 5<>) — — — —~~

~~6 19,581 3.086 134 136.50~~

~~7 23,490 2-570 188 118.05 8 19,972 2.171 197 96.14~~

~~9 11,765 2.081 123 90.74 10 6,768 1.851 83 77.40~~

~~11 5,046 1.749 66 72.64~~

~~12 2,489 1.636 35 67.64~~

~~13 2,302 1.548 33 66.54~~

~~14 1,864 1.453 31 57.17~~

~~15 2,673 1.419 44 57.88 16 2,069 1.364 35 56.38~~

~~17 2,878 1.240 57 48.06 18 1,929 1.132 39 47.29~~

~~19 1,250 .975 32 37.24~~

~~Lata of July 18, 1959~~

~~1 8,639 2.111 86 95.43 2 4,148 1.381 70 56.47 79~~

~~TABLE 5— Continued.~~

~~C, A Q , T, A T , milli Joule microjoule °K. millidegree mole-°K.~~

~~Data of August 7, 195 9 1 1,422 • 770 39.78 34.35 2 1,780 .768 48.67 35.16~~

~~3 2,017 .737 58.54 33.10 4 1,426 .703 38.11 34.23~~

~~5 2,076 .689 60.16 33.26~~

~~6 1,424 .635 4'1.23 31.04~~

~~7 1,485 .670 38.44 34.54 8 1,427 .696 42.96 31.96~~

~~9 1,609 .733 42.17 33.19 10 1,513 .767 36.80 35.50 11 1,601 .803 47.90 33.99 CD . 12 1,779 W UJ 49.78 34.19~~

~~13 1,559 .730 49.39 30.23 14 1,550 .737 42.91 34.79~~

~~15 1,791 .776 57.69 29.62 16 1,841 .828 60.01 29.16~~

~~17 1,425 .675 50.1 27.22 1,426 .715 44.8 30.54~~

~~1,424 .715 47.4 28.74~~

~~1,425 .742 44.8 30.45 TABLE 5— Continued~~

~~c, AQ> T A T, millijoule Cycle microjoule °K. millidegree mole-^K.~~

~~21 1,779 .77 6 54.5 31.22 22 1,781 .812 55*0 30.86~~

~~23 2,076 .843 57.5 34.55~~

~~24 2,076 .871 54.7 36.34~~

~~25 2,076 .885 53-5 37.16~~

~~26 1,425 .640 51.1 26.73~~

~~^^Insufficient heat.~~

~~Cobalt-Iron Alloy~~

~~The specific heat of the alloy was determined using a Speer resis tor with the calibration equation~~

~~rjr _ 0.0f'7O J'J*7,o • ~ moy )-* ° * 13) In the process of temperature determination, a deviation table of the~~

~~type shown in Table 1 was employed. The copper correction is given in~~

~~the data table, and was computed in the same fashion as that employed~~

~~for the manganese sample* Sample and sample shield were linked ther~~

~~mally through the superconducting switch as in previous work.~~

~~The total mass of the alloy was 38*70 grams. The constituents~~

~~were 6.55$ by weight of iron and 93*45/6 by weight of cobalt. In terms MANGANESE HEAT CAPACITY~~

~~C/T 40 MILLIJOULE~~

~~• 9-7-59 o 7-16*99~~

~~20 2.0 3.0 4.0 5.0 6.0 t 2-° k£ Figure 19. 82~~

~~Legend for Figure 20. Specific Heat of Hanganese~~

~~• • • Lata of July 16, July 18, August 7 (1959) — • • • Least-squares data representation — • • • Spin-wave term, BT^~~

~~— • • • Hyperfine term, Atf™2~~

— • • • Electronic term,)^ T *

~~MANGANESE~~

~~T-°K Figure 20. Specific Heat of Manganese 84 of atomic per cent, this corresponds to 6 .87$ iron and 93•'13/6 cobalt.~~

The gram equivalent weight for this alloy is 58*7 grams, from which it follows that 0.66 mole of the alloy was measured. The heat capacity of this alloy is tabulated in Table 6, while Figure 21 illustrates the specific-heat relationship for this alloy.

~~TABLE 6 BATA TABLE* COBALT-IRON ALLOY HEAT CAPACITY~~

~~— ®» A Q, T, AT, millijoule Cycle microjoule °K. millidegree °K.~~

~~Bata of August 30. 1959~~

~~1 295-6 .463 24.8 11.70~~

~~2 353.1 .47 36.8 9.40 3 294.1 .478 27.4 10.52 4 302.0 .484 25.5 11.63~~

~~5 317.3 .494 32.1 9.67 6 324.0 .500 29.2 10.89~~

~~7 346.6 .505 31.3 10.85~~

~~8 320.2 .511 36.2 8.63~~

~~9 322.9 .515 37.7 8.34 10 323.1 .517 38.16 8.15~~

~~11 343.4 .529 32.6 10.30~~

~~12 354.6 .554 35.7 9.69~~

~~13 312.0 .571 33.7 9.02~~

~~14 451.2 .583 51.2 8.56 85~~

~~TABLE 6— Continued.~~

~~C, A Q , T, ^ T, milli.joule Cycle microjoule °K. millidegree °K.~~

~~15 337.5 .590 42.7 7.65 16 484.7 .599 53.7 8.77 17 479.4 .619 58.7 7.97~~

~~18 563.1 .625 64.9 8.41 19 606.3 .643 70.8 8.28 20 715.3 .657 84.2 8.21 21 628.7 .701 76.7 7.89~~

~~22 736.0 .721 88.6 7.99~~

~~23 863.5 .742 113.8 7.27 24 772.9 .757 98.0 7*57 25 1,116.0 .759 145.2 7.36~~

~~26 1,127.9 .813 151.0 7.12 27 1,407.9 .859 190.8 7.01 28 1,450.0 .896 201.7 6.81~~

~~29 1,825.7 .890 252.9 6.84~~

~~30 1,913.7 .962 270.7 6.66 31 1,740.5 .965 254.8 6.60~~

~~32 1,325.3 • 967 190.1 6.53 33 1,640.6 .990 235.1 6.53 34 2,427.2 1.119 363.8 6.15~~

~~35 1 ,009.6 1.141 135-3 6.93 86~~

~~TABLE 6— Continued.~~

~~_ C, Q» T, T, mi Hi.joule Cycle microjoule °K. millidegree °K.~~

~~36 1,789.1 1.203 239-5 6.90~~

~~37 2,844.7 1.311 371.1 7.05 Fe-Co A lloy~~

~~2 3 4 5 6 7 8 9 10 T *_°K-»~~

~~Figure 21. Iron-Cobalt *-Uoy Heat Capacity CHAPTER IV~~

~~INTERPRETATION OF RESULTS~~

~~The results of experimental specific heat measurements on~~

~~the various metals were stated in the previous chapter. It remains~~

~~to interpret these results, both as to their quality and as to their meaning. This chapter is divided into two sections. Section 1 is~~

~~devoted to a discussion of the specific heats of copper and sodium,~~

~~while in Section 2 the speoifio heat results for cobalt, manganese,~~

~~and the cobalt-iron alloy are interpreted. In each seotion it is~~

~~neoessary to introduoe some refinements of the basic theory given~~

~~in Chapter I in order to explain the experimental results. The~~

~~The band theory of electronic speoifio heats is introduced in~~

~~Seotion 1, and is found adequate to explain the observed specific~~

~~heats of copper and sodiun. In Seotion 2 the theories of the~~

~~hyperfine and spin-wave contributions to specific heat are intro~~

~~duced, and the observed speoifio heats of manganese, cobalt, and~~

~~the oobalt-iron alloy are interpreted.~~

~~Seotion It Copper and Sodium Copper. The measurements for copper, as was mentioned~~

~~earlier, covered only the temperature range from 0.35°K* to~~

0.65°K. Consequently, the experimental data can only be expeoted 88 89 to represent the linear term in speoifio heat. A least squares analysis of the data gives V - 0.80 milli joule/mole— ■°K. The average scatter of the data was 5-7#* No appreciable experimental error is attributable to any cause other than errors in thenncmetry.

~~In the determination of Y , however, an error arose through failure to subtract the cubic, or lattioe, contribution to the heat capacity.~~

This correction would result in the average diminution of If by 3*67#* and a consequent value of Y • 0.78 milli joule/mole—°K., throughout the temperature range studied. This value of may be compared with other reoent experimental values of V*

~~Esteimann et al4® .69~~

~~Kok and Keesem4*^ .73~~

~~Bayne42 .72 Corak et al41 *92.~~

~~Since no special effort was mads to provide high-purity copper, this degree of agreement appears satisfactory.~~

~~Very little need be said about the theoretical implications of the results on copper. This metal has been studied quite exhaustively in the past, both experimentally and theoretically.~~

The observed experimental values of the results on copper. This metal has been studied quite exhaustively in the past, both experi mentally and theoretically. The observed experimental values of 21 are in excellent agreement with the calculation of Leighton, who based his work on a theoretically determined vibration spectrum i i following the Born—Ton Kantian method. Band theory calculations for 49 the electronic contribution have been carried out by Jones and by 90 50 Krutter, who both, obtain 0.7 * In excellent agreement with

~~experiment. It seems clear that the speoifio heat of copper is of~~

~~the form~~

~~c = v r + fr (4.1)~~

~~Sodium. The speoifio heat of sodium was given in the~~

~~preceding chapter as~~

~~C = /.32T -tO.HS 7** (4.2;~~

~~and the method of data analysis was also given. In the least-~~

~~a qua re s analysis, it was determined that the standard deviation~~

~~from the mean of the linear term was approximately and that of~~

~~the cubic coefficient was about 2?&. The data scatter may be seen~~

~~from the curve to be of the order of In addition to the least-~~

~~squares analytical analysis, the data were analysed graphically.~~

~~If the specific heat oan be represented as the sum of a linear and~~

~~a cubic term, then one can write c4> = v y- p t7 * (4.3) and a plot of C/T versus T^ should be a straight line whose intercept~~

~~is y and the slope of which is ^ • This analysis was oarried out,~~

~~the results being in accord with those given by the analytical pro~~

~~cedure. The real reason for carrying out the graphical analysis is~~

~~to insure that the speoifio heat is of the form assumed. These~~

~~results may be compared with three other experimental determinations~~

~~of the speoifio heat of sodium metal. 91 Eayne^ measured the specific heat of sodium between 1.0°K.^o~~

0.3°K., hut was unable to analyze his data because of an anomolous behaviour at 0.87°K* No indication of such an anomoly was observed in our work, although this temperature region was monitored with particular care. It should be mentioned that the direct observation of oooling rate on the strip ahart of a potentiometer would seem to be an excellent way to detect the presence of an anomoly in speoifio heat, since one would surely expect a discontinuity in the oooling rate at this point. A A LJt. Roberts has published measurements on sodium extending from 1 «5°K. to 20°K. His values are Y » 1-37# ■ 158°K.

~~Comparison with our values of Y -1.37 and <9^ - 159°K. indicate very nice agreement.~~

~~The final comparison with experiment is with that reported by~~

Phillips ^ at the 14th Calorimetry Conference. These measurements were from 0 .25°K* to 1 °K., and the values reported in a recent private oonmunioation were Y' • 1.45» ■ 156°K. It is evident that, while the & £ values are in reasonable agreement, the 10£ difference in values probably is too great a discrepancy to ascribe to experimental error. However, since there was no stated value of experimental accuracy in the work reported by Phillips, this point cannot be

~~pursued.~~

~~A suamaxy of the experimental work on sodium indicates a~~

~~degree of agreement that is unusual for any metal. It seems virtually~~

~~certain that the speoifio heat of pure sodium metal may be written as 92~~

~~c= r r + er * ( 4 4 ) where~~

~~y ~ y (4 .5 ) and~~

~~&-0 ~ /S' & ° / T t (4.6)~~

~~A comparison of the experimental values of Y and % with the values resulting from various theoretioal models is of interest. Two comparisons may be made with regard to the lattice speoifio heat.~~

Blackman, 52 using the technique of B o m —Von Hannan, 11 has calculated the for sodium and obtains - 147°K« He employed an approxi- & o & o mate formula for the oase where (C^ ” C^g) / C11 is TOZ3r small. (The C values are velocities of propagation of sound waves along the indicated crystal axes.) Bhatia*^ gives a value of - 148°K. 1 • as a result of a calculation of the Bom-Von Karman type, but employing three force constants rather than the customary two (for a similar calculation employing two force constants, see B . U . T 5 4 ).

~~All these theoretioal works have the distinction of~~

~~predicting a low temperature maximum in the speoifio heat somewhere~~

~~in the neighborhood of 5°K.j however, no such maximum has been~~

~~observed by Roberts,^ by Hill, Skaith, and Parkinson,^ or in these~~

~~experiments. Indeed the substance of all recent observations has~~

~~been that the speoifio heat of sodium exhibits a regular decrease~~

with decreasing temperature. All the theoretioal work is based on 93 the assumption that the crystalline form of sodiun is body-oente red- cubic, and the reoent observation by Barrett 56 of a partial trans formation to hexagonal-oloae-packed structure at 5°K* mould undoubtedly affect these calculations. The actual elastic constants of the crystals are used in the theoretioal calculations of specific heat, and a change in lattice parameters will directly influence the predicted specific heat.

There are numerous opportunities to compare the measured electronic specific heat with theory. The first step in making such comparisons is to calculate what V should have been, on the basis of the Soumerfeld free-eleotron theory. Using the form c«- 0./3* v » t 7' v , r and making the usual assumption that nfc, the number of free electrons per atom, is unity, one obtains

The above numerical value is obtained using the density of sodium at nitrogen temperature. It is customary to discuss the degree of departure of the actual electronic energy level distribution from the free-eleotron model by the use of an "effective mass", where m* "•■. The XAv — V&1U6value uUxnSturns OvlXout TOto DObe 1.2. 1 «6« This XAXB 19is wOrOXmerely in y iroo n another way of saying that the observed 0 is 1.2 times the Jr calculated on the basis of a free-eleotron hypothesis. To understand the implications of the above effective mass ratio, it will be necessary to outline the pertinent features of the band theory. 94

~~Band Theory. The free-eleotron. theory aesumea that the~~

~~potential energy of the electron is a constant, with energy levels~~

~~given simply by~~

~~This is an evident oversimplification, since a real electron must move in a field of varying potential, that potential being determined~~

~~by eleetron-lattioe and eleotron—eleotron interactions. The determina~~

~~tion of the energy levels for an eleotron, therefore, is a problem in~~

~~determining the potential field, and must be treated by an approxi mation process. The Sommerfeld free-eleotron theory may be regarded~~

~~as a first approximation to the solution of this problem. The next~~

~~approximation results from assuming that the potential due to the~~

~~electrons la constant, while using the potential due to the periodic~~

~~lattice as the potential energy texm of the Sohrodinger equation.~~

~~The tight-binding approximation uses the atomic wave functions for~~

~~the lattioe ions to synthesise wave funotions for an eleotron in a~~

~~crystal. The wave functions for adjaoent ions will overlap with the~~

~~result that the discrete atomic levels will spread out into bands of~~

~~levels in the crystal. The amount of spreading of the funotions~~

~~determines the width of the bands, so narrow bands will result from~~

~~the core electrons, but the valence electrons will give rise to bands~~

~~so broad that overlapping of the bands may oocur. If two adjacent~~

~~bands do not overlap, there will be a forbidden energy region between~~

~~them. In a metal, however, the Fermi level must be within a band. 95~~

It has been seen that speoifio heat is dependent upon the density of energy levels, and now it appears that this density of levels depends upon the eleotron-lattioe interaction. Now the rate of ahange of eleotron momentun in a oxystal depends on the eleotron- * lattice interaction, and the concept of an effective mass m was adopted to indicate the degree to which the aforementioned inters action affected the potential energy of the eleotron. Phrased in another way, m*/m should represent the degree of accuracy of the free eleotron theory as compared to elementary band theory, including eleotron-lattioe interactions. Bardeen 57 has calculated the band structure for sodium and finds that an effective mass ratio m*/m »

~~0.95 leads to agreement with his calculated band structure.~~

There appear to be two important modifications of the basic band theory as applied to sodium. The first modification, reoently 55 discussed by Buckingham and Saha froth, deals with the corrections introduced by taking into account the lattice vibrations and their

~~resultant effect on the eleotron-lattioe interaction energy. The~~

~~conclusion is that in the region T 0°K. the speoifio heat should~~

~~increase by a factor (1 f over its unperturbed value. Evaluation of the factor -f- is unfortunately impossible in closed form, and must be Judged from experiment. This point will be discussed again~~

~~in the concluding remarks for this seotion.~~

~~The seoond modification of basic band theory is the 58 "collective eleotron" approach of Bohm and Pinesm These authors discuss the effect of eleotron-eleotron interactions, which are ig~~

nored in simple band theory. The inter-eleotronio interactions are 96 divided into two partst short-range and long-range. The long-range intemotions are viewed as a type of plasma oscillation similar to organized modes) this effect is second order in comparison with the short—range effeots. These latter effects are oonoemed with the interactions of free electrons with a screened potential of the form

Taking the eleotron-eleotron interactions into account, Pines 59 finds a value of 0.82 for the ratio of his calculated electronic specific heat coefficient| as compared to the value calculated on the free- eleotron model.

A recapitulation of all the above is that the Sacmerfeld free-eleotron model predicts V - 1.095 milliJoule/mole-°K^, and the band theory modification of Bardeen reduces this value to Y >1.04*

~~Taking eleotron-eleotron interactions into aooount, further reduces this value to / ■ 0.89* Only ifthe factor in the work of~~

~~Buckingham and Sohafroth 55 on the electron-vibrating lattice inter-~~

e obtained. This seems a reasonable assumption in the light of the statements of these authors. Actuallyt it is apparently somewhat naive to predLiot numerioal values of the various correction factors when the experimental values are of perhaps accuracy, and the theoretioal values are individually in doubt by about 20J&. Two statements may be made in suunaxy. First, it seems that there exists no irreconcilable difference between present experiment and present theory) the chief difficulties are the lack of precise values for 97 some correction constants. Secondly, the electronic coefficient of specific heat at low temperatures is appreciably larger than that calculated on the basis of the free-electron theory, which implies a greater free energy for the electron system.

~~Section 2t Cobalt. Cobalt-Iron Alloy, and Manganese~~

~~Comparison of Results. All of the specific heat expressions given in this section will be of the general form~~

or n * 3. The first term represents the electronic contri bution, the cubic term represents the lattice contribution, the term in _2 T is attributed to hyperfine coupling, and the last term is postulated as a spin-wave contribution to the specific heat. Table 7 lists the experimental results for pure cobalt, the face-centered-cubic cobalt- iron alloy, and manganese metal. The results for cobalt show that the lattice contribution to specific heat is quite small in the liquid-helium temperature range. The lattice contribution is s'* 5$ of the total heat capacity at 4°K«, and 0.002$ of the total heat capacity at 0.5°K. The measurements of Duykaerte^ in the liquid-hydrogen temperature range give <9jj * 443°K. Because the lattice contribution is known to be negligible below 1°K., our data analysis assumes that - 2. c “ yT + A T (4.14) or

~~(4.15) 98~~

~~TABLE 7 SPECIFIC HEATS OF SOME TRANSITION METALS~~

~~Tempera ture Specific Heat Range, Parameters °K. Metal Measured By * e A B~~

~~0 .35-1.0 Cobalt 5.1 — 3.1 — Gaumer~~

~~0.35-0.7 5.6 — 4.8 — Arp, Edmonds, and Petersen^®~~

~~0.6-4.2 4.7 .022 3.3 — Heer and Erickson^ 61 2-18 5.0 .022 — — Duykaerts~~

~~0.5-1.3 Co-Fe 7.1 3.2 _ Gaumer Alloy~~

~~0.35-0.7 4.7 — 4.6 — Arp, Edmonds, and Petersen^~~

~~— 0.5-3.2 Manganese 12.1 0.5 0.4 Gaumer co (e ) 1-4 11.8 ——— Zimmerman and Sato0<:~~

~~11-20 11.8 .032 — — Booth, Hoare, and Murphy^~~

(a)Result of indirect measurements. and by plotting C/T versus 1/T^ the specific-heat parameters y and A may be obtained* V being the intercept of the resultant straight line and A the slope of the line. Figure 18 is such a plot for cobalt. Although this method is sound enough theoretically, the scatter of the data on such a plot is considerable. A 5$ error in thermometer calibration is reflected as a 10$ error in C/T and a 15$ error in T~^. In the temperature range of interest, a 5$ error in temperature may be 0.020°K., which is possible if not probable. Consequently, the only 99 reasonable hope for accurate determination of values of the low- temperature specific-heat parameters would seem to lie in the statisti cal weight of a large number of data points. Even so, the possibility of a systematic error in temperature measurement cannot be ignored. It is evident from Table 7 that there is fair agreement in the experimental values for V » but that the values of the hyperfine constant A disagree by nearly 50$* The cause of this disagreement is not presently under stood. Specific-heat measurements in the temperature range 0.01-0.40°K. will be required in order to establish the values of V and A with more certainty. Although a spin-wave contribution to the specific heat of cobalt may exist, it cannot be observed experimentally because this 3/2 T contribution is masked by one or the other of the three known terms in all temperature regions. Only two measurements exist for the cobalt-iron alloy. Although the alloys differed slightly as to the percentages of the constituents, both were of face-centered-cubic crystal structure. The method of data analysis employed btv us was the same as that given above for cobalt. Our expression for specific heat is the result of a least-squares analysis of 35 data points. The maximum scatter of our data is indicated in Figure 19» and is no worse than 10$ in the lowest temperature region. Consequently, it seems unlikely that the disparity between our results and those of Arp, 60 Edmonds, and Petersen can be attributed to our experimental error.

It also seems unlikely that impurity effects or metallographic history could explain the discrepancy in results, as other research indicates that the electronic configuration is not readily affected. A complete set of specific-heat measurements extending from 0.1° to 4*2°K. would 100 undoubtedly clarify the existing situation.

~~There exist no other low-temperature measurements of the specific~~

heat of manganese. The only specific-heat parameter which may "be com pared with the results of other workers is V, the electron coefficient. Satisfactory agreement of the )f values for manganese is indicated in 63 Table 7* Booth, Hoare, and Murphy found the Debye characteristic temperature to be <9^ « 392°K. in the liquid-hydrogen temperature range.

~~a 64 This value for was determined also by Armstrong and Grayson-Smith rs. O at still higher temperatures. Ueing * 392 K., the lattice contri~~

~~bution turns out to be 0.032 T^ millijoule/mole-°fC., so the lattice terra is relatively unimportant even at 3°K., the highest temperature~~

~~where data was taken in these measurements.~~

~~Two curves were given in the previous chapter to illustrate the~~

~~specific-heat results for manganese. The C versus T curve indicates~~

~~the general course of the low-temperature specific heat, while the C/T 2 versus T curve was drawn only to provide a graphic illustration of the~~

~~presence of a low-temperature anomoly of the Schottky type.~~

~~Since the hyperfine contribution is definitely indicated on a c /t 2 versus T plot, one course in the data analysis is to assume~~

~~C = V 7~ + A T ~ (4.16) from 0.5° to 3*0°K., and find the best fit for the curve drawn through~~

~~all the experimental points. This gives~~

~~c = (A9 i /) 7- * (.* i.a)7- Another course of analysis is suggested by the indication of slope on~~

the C/T versus T term. It 1b known that this slope cannot be due to 101 the usual cubic lattice term, but a possible source of additional specific heat, which is also cubic, is the spin-wave exchange contribu tion, mentioned in the discussion of cobalt. Assuming the presence of a cubic contribution of some sort, the analysis was carried out in two parts. First, a least-squares fit was found, assuming (4.18) and next, in the higher temperatures regions,

/, ; c - v t ^ 5 r 3 . {4. 19) The temperature limits for analysis were chosen so that the contribution being ignored constituted less than 1$ of the total specific heat. The expression resulting from this type of analysis was

* ^ C=/£*/7r+'0»S~ T9 + Y 7” . (4.20) The control of purity and crystalline phase for manganese is 63 rather difficult and was mentioned only by Booth, Hoare, and Murphy, who used spectroscopically pure phase manganese. The degree of agreement of the results (11.8 versus 12.1) is perhaps a reflection of the similarities of the two samples. Neither the hyperfine constant A nor the spin—wave constant B can be compared with other experimental results, as there are none. The reason that the representation of the data which includes a

~~BT^ term was chosen was largely that this form represents a better fit to the experimental data than the simpler representation. If one~~

simply draws a straight line on a C versus T plot through all the data points, the results are that all the high-temperature points fall above 1 0 2 the line and all the low-temperature points helow the line, if the hyperfine contribution is considered* This type of data scatter would seem to indicate two separate sources of systematic error, one occuring in the high—temperature region and another of reversed sign operating in the low-temperature region. While this is not impossible, it is highly unlikely. Systematic errors arising from thermometry would surely cause either too high or too low an apparent specific heat consistently throughout this short temperature range.

~~Specific Heat Theory for Magnetic Materials~~

The basic theory of the specific heats of solids was presented in Chapter 1. In the first section of this chapter it was necessary to introduce the band theory of electronic specific heat in order to adeq uately explain the experimental results for copper end sodium. It now becomes necessary to introduce two new topics in order to explain the observed specific heats of the transition metals measured by us. The materials which were measured were either ferromagnetic or anti-ferro- magneticj since the internal energy of a magnetic material is rather more complex than for a non—magnetic material the theory of the spec ific heats of magnetic materials must necessarily include considerations additional to the basic theory. The first consideration to be dis cussed is that which has previously been referred to as the hyperfine coupling contribution to specific heat. The second topic of interest in specific heat theory is the spin-wave contribution. It will be nec essary to utilize both these specialized theories in addition to the band theory of metals in order to explain the observed specific heats of the transition elements in even a qualitative fashion. 103

Hyperfine Interaction. In a ferromagnetic substance at a temp erature which is small compared to the Curie temperature, the spins or orbital moments of the "ferromagnetic" electrons are all parallel with in a magnetic domain* The same statement may be made for anti-ferroma©- netic materials below their NeAl temperatures, except that the spin alignment is anti— parallel* As a consequence of this complete elect ronic polarization each nucleus finds itself in a magnetic field which is of the same magnitude and direction throughout a domain. A nucleus with magnetic moment ^C^and nuclear spin quantum number I will have

21+1 orientations with respect to the magnetic field. These orient ations will have energies differing from each other by^t^H^/21, and the distribution of the nuclear spin orientations among these energy levels is assumed to be in accord with Boltzmann statistics. This implies that the distribution is governed by a factor^^H^/2IkT. The magnetic field which the nuclei experience is denoted by Uniform distribution prevails at high temperatures (kT»^H^/2l)

~~and the nuclear spins are oriented at random. At temperatures such~~

~~that kT ar/t iLjy /2I those orientations corresponding to the lower energy levels become preferential and eventually, at very low temp~~

eratures (kT< ‘A V /2l), there will be complete nuclear polarization. Whenever there is a redistribution among energy levels there will exist a corresponding specific heat contribution. Heer and Erickson^ have

~~considered this problem in analogy with that of an ion in a strong~~

crystalline field of axial symmetry, which is the situation in the cobalt salts. They give as an expression for the specific heat c / _ J lz+/)A'S* } 104 where S is the electronic spin and A is the coefficient of dipole

~~interaction between the nucleus and the electrons.~~

~~W. Marshall J gives a complete discussion of the hyperfine inter~~

~~action in terms of the effective magnetic field acting on the nucleus.~~

~~The expression given for the nuclear specific heat caus d by the hyperfine coupling is~~

~~> where gn is the nuclear g-factor. Marshall assumes that the effective~~

~~field acting on the nucleus is composed of three parts! the local magnetic field at the nucleus, an effective magnetic field which acts~~

~~through the contact interaction with the 4s electrons, and an effective~~

~~field contribution arising from the interaction of the nucleus with the~~

~~electrons of the same atom.~~

~~The type of interaction discussed here was discovered in studies~~

~~of the optical spectra of free atoms. The interaction is here displayed~~

~~in a hyperfine structure of the spectral lines, hence the name for this~~

specific-heat contribution. The nuclear polarization discussed here may be observed not only in low-temperature specific-heat measurements, but also in the anisotropy of )f -ray emission at low temperatures. The

~~first experimental indication of the hyperfine coupling in cobalt was~~

~~found in low-temperature measurements of Y — ray anisotropy.^*^~~

~~Spin-Wave Interaction. Spin-wave theory predicts the appearance~~

~~of a magnetic specific-heat contribution which is proportional to T^/^~~

~~for an anti-ferromagnetic. Spin-wave theory is appropriate for a dis~~

cussion of low-temperature phenomena where the magnetization differs 105 only slightly from that at absolute zero* Modem spin—wave theory is based on approximating the magnetic spin system for a crystal by a system of quantum mechanical harmonic oscillators. The T3/2 contribution was predicted by spin-wave theory, notably 68 by Bloch in 1932. A recent review article by Van Kranendonk and 69 Van Vleck describes the model used and formulates expressions for specific heats of ferromagnetic and anti-ferromagnetic materials. In simple terms, this contribution is presumed to arise because, at very low temperature, quantum mechanics predicts that a small but finite number of electrons will have spins anti-parallel to the field. For each such electron there will be a spin-wave and associated energy. The number of energy states per unit volume is proportional to T 3/2' . The internal energy of the system is then dependent on T x T 3/2' , and finally the specific heat is dependent on T3^ . Mott and Jones'^ discuss this theory and give a numberical expression for specific heat as

~~C = / . / * 7T ^ . (4.23) '7o where O* is the magnetization at temperature T and is the satura tion magnetization. The magnetization at low temperatures, according to Rayne and Kemp,*^ is given by~~

~~nickel obtaining~~

c - 0 . 0 9 7’"^* r-rx • (4*25) Although the value of A for cobalt is not currently available, it seems 106 reasonable that the values for nickel and for cohalt would he of the same order of magnitude* If the ahove assumption is justified, the specific-heat contribution for cobalt caused by spin-wave function could not have been detected in the work described here.

~~More complete expressions for the spin-wave specific-heat contribution are given by Van Kranendonk and Van Vleck. 69 ' For a ferromagnetic element at low temperatures, the specific heat is given as~~

/fi //3 ) (4.2 6) for a simple cubic lattice. For body-centered-cubic and face-centered- cubic lattices the expression above is to be multiplied by a factor of 1/2 or 1/4j respectively. For an anti-ferromagnetic substance, the expression given is

~~(4.27) for the simple cubic case. This result was obtained also by Kouvel and 72 Brooks. The factor J in the above expression is the exchange inte gral and may be written as~~

~~v T - /Z ^S Q S * /), (4.28) follovring Kit tel. Using Equation (4.28) inEquation (4.26), one obtains for the simple cubic case of a ferromagnetic~~

~~, (4 .2 9 ) where T is the Curie temperature and z is the number of nearest c neighboring atoms. For an anti-ferromagnetic, the expression for the simple cubic case becomes 107~~

( 4 . 3 0 ) where Tn is the Neel temperature. An extensive discussion of the subject of this section appears *1A in a very recent supplement to the Journal of Applied Physics. The implication of the theory developed to date is that there should be a

spin-wave contribution to specific heats of ferromagnetic and anti ferromagnetic materials at temperatures below their Curie or Neel points. Furthermore, the specific heat of a ferromagnetic should follow the

~~Block law, while an anti-ferromagnetic should follow a law. There is, of course, a serious question as to the degree of applic ability of the theory to a real anti-ferromagnetic or ferromagnetic.~~

~~It must be stressed that the spin-wave specific-heat contribution~~

is strictly a low-temperature effect for T/Tc ^ 1• As Stoner, 76 77 Wohlfarth, and Nagamiya et al have pointed out, an essential pre mise of the Block spin-wave treatment is the complete parallelism of all spins at T » 0°K. Indications are that moBt real ferromagnetics

and anti—ferromagnetics would exhibit sense degree of departure from complete spin alignment. However, Wohlfarth shows that the theory should be qualitatively correct for very low temperatures T O f 1°K. The above theory is interpreted as indicating that a T^ spin-wave contribution to the specific heat of magnetic materials is entirely possible in the liquid-helium temperature range, and that the contri

~~bution should drop rapidly toward zero with increasing temperature.~~

~~The effect of such a mechanism would be to slightly alter the value of 108~~

~~Debye temperature measured at higher temperatures, such as 10°K. This hypothesized mechanism should be subject to direct experimental veri fication.~~

~~Discussion of Results~~

~~Cobalt. The low temperature specific heat of cobalt may be~~

~~represented as c/t? — y T7 + A 7** • (4.31) o In the region of 1 K. the lattice term in the specific heat is negli gible. Any spin-wave contribution to the specific heat is unimportant~~

~~in comparison to the electronic term. The spin—wave term would be~~

~~expected to be quite small, since this contribution is proportional~~

~~to (T/Tc) ^ , and the Curie temperature for cobalt is 1400 K. The electronic specific heat parameter for cobalt is quite high in relation~~

~~to other elements, although not notably high for a transition metal.~~

~~This high electronic specific heat is explained by a generalization~~

of the band theory of metals. The transition metals are characterized by a partially filled 3d band. The density of states at the Fermi level would be expected to be quite high, since the d-bands are more narrow than s—bands. This is because the spatial extension of the d—band wave function is appreciably less than that of a e—band wave function. For a ferromagnetic substance such as cobalt the d-band is presumed to split into two sub-bands, corresponding to the two spin

~~directions* The lower energy band fills first, and those valence~~

~~electrons in excess of five are distributed between the s-band and the~~

higher energy sub-level of the d-band. Since the lower level can make 109 no contribution to the heat capacity, the observed values of electronic specific heat must be doubled before a calculation of density-of-states C/N nA can be made* Krutter and Slater have calculated the shape of the

3d band. Although no calculation seems to have been made for the elect ronic term in cobalt metal, Fletcher^ has calculated this value for nickel, basing his work on a "tight—binding” approximation and using the d-band shapes mentioned above* The agreement with experiment is excellent. It may be presumed that, at least qualitatively, the elect ronic coefficient of specific heat for ferromagnetic transition metals, and cobalt in particular, are adequately explained by the band theory. The hyperfine contribution to the specific heat of cobalt is of such a magnitude as to indicate an effective magnetic field at the nucleus of the order of 185 kilogauss. This value is in excellent agreement with the values of effective field found by experimental measurements of the V -ray anisotropy. Below 0.85°K. the hyperfine specific heat contribution is predominant, and indications are that the hyperfine component will continue to increase with decreasing _p temperature in a proportionality to T until a temperature of some where around 0.1 °K. is reached. From the point of view of an experimentalist, the present state of the theory suggests two logical directions for further research} an extension of hyperfine specific heat measurements to much lower temperatures in order to discover the course of the specific heat

increase, and a series of experiments on ferromagnetic elements alloyed with both other ferromagnetic and non-magnetic elements. This last should serve to assess the validity of the theoretical model employed. 110

~~Cobalt—Iron Alloy. The low—temperature specific heat of the face— centered-cubic alloy is of the same general form as that of cobalt~~

(Equation 4*30). The effect of alloying with iron was to increase the electronic term by nearly 50/6, while the hyperfine contribution was gd g-j changed very little. Marshall and Kurti have made predictions as

~~to the effect of alloying on the hyperfine specific heat. Weiss and ForrerSo have found that the atomic moment of cobalt is increased from~~

~~11.0 to 11.5 magnetons when alloyed with 9# iron. It seems unwise to base any far-reaching conclusions on the fact that the hyperfine~~

~~specific heat remained relatively constant for pure cobalt and for the alloy. Present theory is inadequate to explain exactly what changes in~~

~~the band structure may be expected, and the situation is further com plicated by the change in crystal structure from hexagonal-close-~~

packed to face-centered-cubic. The increase in the electronic specific heat is presumably due either to the change in crystal structure or to a modification of the shape of the d-band. Much more information is required as to the

~~specific heats of ferromagnetic alloys before the theory can be put into final form. The fact is that the addition of only 656 of iron to~~

~~cobalt has resulted in nearly a 50# increase in electronic specific~~

~~heat. Further work along this line would surely seem indicated.~~

~~Manganese. The specific heat of manganese in the range 0.3-3°K.~~

~~is of the form~~

~~= y r + A T ~ e + • <*•*> The electronic coefficient for manganese is extremely high ( » 12.1). 111~~

This may be accounted for in a very general way by the same type of band theory argument given for cobalt, but the agreement with theory is still qualitative*

It is reasonable that the basic ideas of the theory of hyperfine specific heats is applicable to anti-ferromagnetics just as to ferro magnetic substances. The hyperfine contribution to the specific heat of manganese is only as large as that which would be expected for

1/5 mole of cobalt. If this small hyperfine term is interpreted as indicating a small effective magnetic field at the nucleus, it becomes essential to assume some mechanism by which the component magnetic fields are either shielded from the nucleus or are partially cancelled.

~~Cancellation seems a likely possibility for an anti-ferromagnetic such as manganese.~~

The term in BT^ iB tentatively attributed to a spin-wave contribu- 72 tion to specific heat. Kouvel and Brooks have calculated that the coefficient of the spin-wave T^ term involves a considerably larger proportionality constant than the familiar Dehye T^ term. This conclu sion is borne out by these measurements! the ratio of the spin-wave contribution to the lattice contribution being of the order of 17*1• The magnitude of the proportionality constant for the spin-wave term would be expected to be inversely proportional to the cube of the Keel temper ature, which is only -v# 100°K. for manganese. There exist three recent instances in which experimentalists have attributed a portion of the observed specific heat to a spin-wave term. Kurti and Safrata®^ represent thiB term for terbium as 20.8 T^/^ millijoule units. Kouvel^ obtains a term in the specific heat of a magnetite 112 crystal of 1.32 T3/2 millijoule units. Rayne and Kemp^ calculate a contribution of 0.1 T 3/2 for nickel. Using Equation (4.29) to calcu late the spin-wave term for nickel, a value of 0.2 T 3/2 is obtained, which tends to validate the theoretical formulation. It is rather difficult to use Equation (4.30) to calculate the spin-wave specific heat of manganese because of the complexity of the crystal structure of •C-manganese. The crystal structure of manganese is basically body- centered-cubic, but the unit cell contains 58 atoms of four different crystallographic types.It is thus difficult to assign a value to z, the number of nearest neighbors. However, if Equation (4.30) is evaluated in terms of z, using the values S * 0.25 from the work of 8S 1 o Shull and Wilkinson J and a Neel temperature of 100 K. it is possible to obtain agreement with the experimentally determined value of B if z, the number of nearest neighbors, is set equal to 32. This order of magnitude for the number of nearest neighbors in (^-manganese does not seem unreasonable.

~~Conclusions~~

It was mentioned in the introduction that low-temperature specific heat measurements might be expected to furnish information of consider able value in increasing our understanding of the atomic structure of matter. One conclusion may certainly be drawn from these measurements! the above expectation has been amply justified. As the temperature is lowered, the macroscopic causes of specific heat give way to the micro scopic. The lattice contribution becomes relatively trivial at helium temperatures and, in the region below 1°K., even the specific heat 113 S attributable to electronic energy begins to become relatively small.

~~Finally, as indicated most clearly by the transition metals, the pre dominant contribution to specific heat comes from the nucleus itself.~~

It is unfortunate that the lowest temperatures reached in these measurements were such that the nuclear contributions to specific heat were only beginning to become clear. Future work in the determination of specific heats of metals will undoubtedly extend to 0.01°K., perhaps through the use of a two-stage demagnetization technique.

These measurements have served to clarify the relatively simple band structure of sodium. The measurements have confirmed the presense of a hyperfine specific-heat contribution in the transition metals, where the unfilled 4d band introduces complexities into the theory.

The measurements for manganese suggest that still another form of specific-heat contribution is present at low temperatures, and consider ation of the spin-wave model for this last contribution may be expected to further our understanding of the structure of matter. It is clear that our knowledge of the low-temperature specific heats of metals is increasing rapidlyj it is equally clear that many experiments remain to be done. LIST OF REFERENCES

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~~7. Heer, Barnes, and Daunt, Fhys. Rev. £1_, 412 (1953).~~

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~~9* P. L. Dulong and A. T. Petit, Ann. Chim. et Fhys. J_0, 395 (1 Si9)-~~

10. A* Einstein, Ann. Fhysik 22, 180 (1907)*

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~~85. C. G. Shull and M. K. Wilkinson, Revs. Mod. Phys. 2£, 100 (1953). AUTOBIOGRAPHY~~

I, Roger Edgar Gaumer, was "bom in Zanesville, Ohio, April 6, 1926. I received my secondary school education in the public schools of Zanesville, Ohio, and Newark, Ohio. My undergraduate training took place at Antioch College, where I received the Bachelor of Science degree in 1948. After a period of time which was spent in working as a research physicist in industrial and governmental laboratories, and also in the Armed Services, I began my graduate training at The University of Florida in 1952. In 1953 I entered the graduate school of The Ohio State University and have been in residence continuously since that time. I have held positions as graduate assistant, teaching assistant, research technician, and research fellow while completing the requirements for the degree Doctor of Philosophy.

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