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•••••• 89 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.