Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1970 Measurement of the volume dependence of the heat capacity of solid neon Robert Quentin Fugate Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Condensed Matter Physics Commons Recommended Citation Fugate, Robert Quentin, "Measurement of the volume dependence of the heat capacity of solid neon " (1970). Retrospective Theses and Dissertations. 4837. https://lib.dr.iastate.edu/rtd/4837 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. I 71-14,225 FUGATE, Robert Quentin, 1943- MEASUREMENT OF THE VOLUME DEPENDENCE OF THE HEAT CAPACITY OF SOLID NEON. Iowa State University, Ph.D., 1970 Physics, solid state University Microfilms, A XEROX Company, Ann Arbor, Michigan MEASUREMENT OF THE VOLUME DEPENDENCE OF THE HEAT CAPACITY OF SOLID NEON by Robert Quentin Fugate A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Solid State Physics Approved ; Signature was redacted for privacy. Signature was redacted for privacy. or Department Signature was redacted for privacy. D n of ^aduate College Iowa State University Of Science and Technology Ames. Iowa 1970 PLEASE NOTE; Some pages have small and indistinct type. Filmed as received. University Microfilms I I TASLE OF CONTENTS CHAPTER 1. INTRODUCTION 1 Two-Body Potentials 3 Lattice Dynamical Models 5 Debye Model 9 Volume Effects 13 CHAPTER II. EXPERIMENTAL APPARATUS 19 Dewar and Cryostat 19 General features 19 Construction details 21 High Pressure Calorimeter 28 High Pressure System 31 Gas Recovery System 35 Electronics 37 Thermometer circuit 37 Bomb heater circuit 39 Neon Gas 4l CHAPTER III. EXPERIMENTAL PROCEDURES 43 Qualitative 43 Thermometry 44 Addenda Heat Capacity Measurements 46 Neon Sample Solidification 47 Dead Volume Effects and Sample Mass Determination 48 Specific Procedures in Data Collection 49 I I I Page CHAPTER IV. RESULTS 56 Data Anal ys is 5& Addenda Results 59 Solid Neon Results 60 CHAPTER V. DISCUSSION 75 T = 0 K Limit 75 T > 0 K Results 76 Comparison with Theory and Other Data 80 Concluding Remarks 83 FIGURES 85 BIBLIOGRAPHY 115 ACKNOWLEDGEMENTS 11 9 APPENDIX A 120 GR638 Thermometer Fit Constants 120 APPENDIX B 121 Individual Data Points for the Addenda and Neon Samples Ne4-Ne8 121 1 CHAPTER 1. INTRODUCTION The solidified noble gases (neon, argon, krypton, and xenon) form an important class of solids. They are characterized by closed-shell atoms in a cubic-close-packed structure, low melting points, and weak binding. They also represent a very interesting case where the quantum mechanical zero-point motion of the atoms has a very important effect on the elastic and thermal properties of the solid. Table 1, for instance, shows that the zero-point energy varies from 32% of the cohesive energy for neon to about 3% for xenon. Thus, zero-point energy plays an im­ portant role in the Rare Gas Solids (RGS) because of the weak interatomic forces. The zero-point energy of an ionic solid, for instance, may con­ stitute only a fraction of one percent of the cohesive energy which is of the order of 10 ev/atom (compared with 0.026 ev/atom for neon). As such, the RGS are nearly the ideal case for the comparison with experimentally determined thermodynamic properties of realistic theoretical calculations of the effects of zero-point motion on the lattice properties. Historically, the body of theoretical knowledge concerning the RGS has far exceeded experimental results. However, this situation has changed drastically during the past five years with systematic investi­ gations of heat capacity, neutron scattering, thermal expansion, com­ pressibility, and other properties being reported. The results of these experiments indicate a need for more sophisticated theoretical models and inHeeH have encouraaed theorists to improve their earlier, less complex ideas. 2 Table 1. Some properties of the Rare Gas Solids^ Nearest Parameters in the Zero- Neighbor MLJ potential (NN Cohesive Poi nt D istance interactions only) Energy Energy Sol id ao (%) a (A) c (ev) (ev/atom) (ev/atom) Ne 3.13 3.04 0.006 0.026 0.008 Ar 3.76 3.71 0.020 0.108 0.010 Kr 4.01 3.97 0.027 0.155 0.008 Xe 4.35 4.32 0.038 0.223 0.007 ^Taken from Reference k. The major trends in these developments are outlined in a series of review articles, in 1957 Dobbs and Jones (1) and in 1964 Pollack (2) gave extensive references to theoretical and experimental work. Boato (3) also has written a review article but with relatively few references. Morton (4) and Werthamer (5) recently have reviewed the current theo­ retical approaches used for the RGS. The present work represents a high precision experimental investi­ gation of the volume and temperature dependence of the heat capacity of solid neon. The types of considerations which are needed to interpret their significance are outlined below. In particular, solid neon interesting because of the relatively large size of the zero-point is CwCr^y 55 COi'ûparôd WtLn citc cnci yy y^iciuic i/f wiiiun icsuiL^ in large amplitude atomic vibrations even at absolute zero. The vibrations. 3 though of large amplitude (approximately 6% of the nearest neighbor dis­ tance), r-e considerably smaller than for helium and hence allow for more precise theoretical calculations. Two-Body Potentials Theoretical models of the RGS are made computationally tractable by the assumption of a relatively simple two-body potential function com­ posed of long range attractive and short range repulsive terms. A widely used starting point for the attractive part of the potential is the Van der Waal s r ^ potential which arises from the dipole-dipole interactions between the fluctuating charge distributions of the atoms. The repulsive term arises from the Pauli exclusion principle and its exact mathematical form is not well founded but is probably an exponential of the form exp(-Àr). The Mie-Lennard-Jones (MLJ) potential, which is one of the most popular of several ad-hoc potentials used to describe RGS, has the form (1-1) where r is the separation between two atoms, e is the depth of the potential well at the minimum, a is the location of the minimum, and n is a number of the order of 12. Values of a and e for the MLJ potential are listed In Table 1. The repulsive term is represented explicitly as an exponential in the Buckingham potential. V(r) = Gxp [-n(: - 1)] - Ç (7) i, (1-2) 4 where n Is 12 or 13. The Morse potential contains three explicit param­ eters and is written V(r) = Cjexp [-2c - 1)] -2 exp [-c (^ - 1)]j ^ (1-3) In both of these potentials € and a have the same meaning as for the MLJ potent ial. There has been much discussion concerning the values to be used for the potential parameters o and e. First principles calculations (6-9) have not yielded definitive results for the electronic interactions be­ tween rare-gas atoms in a lattice. It thus has been customary to obtain the parameters from either gas phase data such as transport properties, viscosity, diffusivity, the second virial coefficient, and atomic beam scattering (10-14) or from solid state properties-primarily the 0 K, P=0 lattice spacing, sublimation energy and bulk modulus in the case of the Morse potential (15-20). Although the gas phase data may give more realistic values for a true two body potential, the parameters derived from solid state data give better results when used in lattice dynam­ ical theories (21,22), since they represent an average of the properties of the atoms in the lattice. Other potential functions (such as the Munn-Smith and Polynomial potentials) are used in RGS calculations (4), but the MLJ and the Buck­ ingham are the most popular. There seems to be no definite advantage to either the MLJ or Buckingham potential as far as accounting for experi- mAnfal rACilf in a sinnifiranflv snnerinr fashinn. Indeed, both DO- tentials have their advocates (23,24) and both have been widely questioned 5 on the grounds that a simple two-body potential may be unreasonable in a discussion of the RGS. In fact, the role of many-body forces in the RGS is very much undecided with little experimental evidence to sub­ stantiate or disprove theoretical predictions (25,26,4). If there are appreciable many-body forces in the RGS then the ad-hoc two body param­ eters must compensate for these in some manner in order to adequately describe the crystal properties. Some information about the crystal potential can be deduced from pressure effects on the solid state properties. Changes in the volume require that the atomic motions be affected by different portions of the potential function thus yielding indirect information about its shape. Lattice Dynamical Models Modern lattice dynamical theories (27-32) are based on the ideas in­ troduced by Born and von Ka'rma'n in 1912. They considered the solid as a set of N (number of atoms) coupled harmonic oscillators having 3N normal modes of vibration. The allowed frequencies of vibration are the solu­ tion of a set of 3N coupled simultaneous equations in which the inter­ atomic potential is expanded in a Taylor series to second order (the harmonic approximation) in the atomic displacements.