ABSTRACT EXPERIMENTAL AND THEORETICAL STUDIES IN Low TEMPERATURE PHYSICS: SUBLIMATION AND VAPOR PRESSURE 0F Ar36 AND THE COMBINED EFFECTS OF PHONON ATTENUATION AND IMPEDANCE MATCHING ON KAPITZA RESISTANCE By Jon Lynn Opsal The sublimation and vapor pressure of Ar36 were measured in the temperature range 23.752-87.375 K. Pressures below 1 Torr were measured with a McLeod gauge and corrected for effects of thermal transpiration and mercury streaming. The estimated accuracy of these pressure measure- ments ranges from 12 near 1 Torr to 102 near 10-5 Torr. Above 1 Torr a calibrated Bourdon gauge was used to give pressures to 19.03 Torr. Temperatures were measured to :9 mK with a N.B.S.-calibrated Pt re- sistance thermometer. A liquid helium'bath was used throughout the temperature range for which the experiment was done. The data were fit to theoretical sublimation pressure curves to obtain values for the static lattice energy, lattice vibrational energy, and geometric mean frequency of the phonon spectrum. The data were also compared with theoretical calculations of others based on an anharmonic self- consistent phonon theory. Equivalent sublimation-pressure data on normal Ar are compared with our Ar36 data in the temperature range 62.315-84.503 K. This comparison yields vapor-pressure ratios which are in reasonable agreement with theory and other experiments. Thermo- dynamic properties of normal Ar and of Ar36 calculated from these data are also compared. It is found that several of the differences in properties may be qualitatively understood in terms of the increased 0 gm 60 W \1 Jon Lynn Opsal zero-point energy of Ar36 compared to normal Ar. The Kapitza resistance, RK(T)’ is calculated for a Cu-He4 inter— face using the acoustic mismatch theory of Khalatnikov and of Mazo and Onsager. Included in the calculation are the effects of: phonon attenuation in the copper as well as impedance matching due to the in- creased He density near the interface. We calculate RK(T) for several values of attenuation in the Cu; detailed calculations are displayed for the case of equal attenuations for longitudinal and transverse waves in the solid. Also considered are different attenuation profiles in the Cu. We use a density profile, for the He, calculated from compress- ibility data and the van der Waals attractive force between He and the Cu substrate. Our model includes the effect of a solid layer of He at the copper surface, i.e. in such a He layer both longitudinal and trans- verse waves are allowed. Included in our calculations are the effects of different density profiles for the He. The calculations indicate that for suitable choices of the physical parameters, the theoretical results for RK(T) agree in magnitude as well as T dependence with the experimental data. Unfortunately, the important physical properties have not yet been experimentally determined and a definitive test of this theory must await such data. EXPERIMENTAL AND THEORETICAL STUDIES IN Low TEMPERATURE PHYSICS: SUELIMATION AND VAPOR PRESSURE 0F Ar36 AND THE COMBINED EFFECTS OF PHONON ATTENUATION AND IMPEDANCE MATCHING ON KAPITZA RESISTANCE 13? Jon Lynn Opsal A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1974 ACKNOWLEDGMENTS I should like to express my deep appreciation to Professor G.L. Pollack, who suggested both the experiment and theoretical problem, for his helpful supervision throughout the course of this research. I would also like to thank David Christen and Thomas Milbrodt for their assistance in doing the experiments and for their many helpful discussions and suggestions during the course of the theoretical calculation. Special thanks are also due to Professor R.D. Spence for several conversations and much valuable advice relating to the theo- retical calculation. Finally, I would like to acknowledge the financial support of the United States Atomic Energy Commission and the National Science Foundation. 11 Chapter TABLE OF CONTENTS I 0 INTRODUCTION 0 O O O O I O O O O O O O O O O O O O A. B. Discussion of Sublimation and Vapor Pressure . Description of the Kapitza Resistance Effect II. THE EXPERIMENT ON SUBLIMKTION AND VAPOR PRESSURE . A. B. C. D. Theory . . . . . . . . . . . . . . . . . . . Experimental Method. . . . . . . . . . . . . 1. Temperature Measurement. . . . . . . . . 2. Pressure Measurement . . . . . . . . . . 3. Gas Sample Analysis. . . . . . . . . . . Results of the Experiment. . . . . . . . . . 1. Discussion of Eguation_ 16. . . . . . . 2. Obtaining,Thermodynamic‘ Properties From the Data. . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . 1. Behavior of Lattice Vibrational Ener . 2. Comparing’Properties_ of Ar36 and Ar . 3. Comparison with the Work of Others . . . III. THE THEORETICAL CALCULATION OF KAPITZA RESISTANCE. A. B. C. D. The Model. . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . 1. Elastic and Viscoelastic Materials . . . 2. The Heat Flux and gapitza Resistance . . Numerical Method . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . 1. Effects Due £2.Phonon Attenuation. . . . 2. Effects g£_lncluding Impedance Matching, IV. SUMMARY AND CONCLUSIONS. . . . . . . . . . . . . . A. B. Appendices The Sublimation and Vapor Pressure Experiment The Calculation of Kapitza Resistance. . . . A. DECOMPOSITION OF THE DISPLACEMENT FIELD. . . . . . B 0 THE ENERGY flux 0 O O O O O O O O O O O O O O O O 0 iii Page NH 11 12 14 17 17 21 25 27 30 30 34 36 36 39 39 44 45 45 47 69 69 7O 72 74 Appendices C. THE TRANSMISSION COEFFICIENTS. KAPITZA RESISTANCE PROGRAM . ADDITIONAL FIGURES OF TR . TABULATION OF CALCULATED RKT3 VALUES LIST OF REFERENCES. iv Page 76 80 94 108 114 Table D1 F1 F2 F3 F4 F5 LIST OF TABLES Experimental Sublimation- and vapor-pressure and temperature data. . Typical values of various terms which appear in Equation 16. The static lattice energy, E0, and geometric mean of the lattice vibrational spectrum, w , calculated from our data using Equation 16, are sfiown in the second and fourth columns, respectively. column shows the values of the heat of sublimation calculated from our data using Equation 5. Lattice vibrational energies calculated from Equa- tion 20 are shown in the second column. Theoretically predicted values from Equation 22 are shown in the third column. Comparison of sublimation pressures of Ar36 and Ar Data for ArN in the temperature range 62.315- 74.681 K are taken from Reference 1. ture range 75.557-84.503 K ArN of Christen and Opsal37 are used. Kapitza resistance, RK, in the form RKT3, as a function of phonon attenuation in the copper. List of the FORTRAN program, KR and its subroutines, used to calculate the Kapitza resistance. Kapitza resistance, , in the form RKT3, as a function of phonon attenuation in the copper. Calculated RKT3 values Calculated RKT3 values Calculated RKT3 values Calculated RKT3 values used in Figure 8. used in Figures 9-11. used in Figures 12 and 13. for Figures 14-17. The third For the tempera- the unpublished data on Page 19 24 26 28 31 46 80 108 110 111 112 113 LIST OF FIGURES Figure 10 11 A sketch of the apparatus used for this experiment. Sublimation pressure of Ar36 as a function Of temperature. Plot of £n(PT1/2) versus l/T for data ranging from T - 68.902 K, P - 44.47 Torr to T - 83.639 K, P - 510.82 Torr. A system of solid and liquid layers typical of those used in the calculations. Calculated energy transmission coefficient, TR, at three frequencies as a function of angle of incidence. The frequencies are expiessed in terms of temperatures defined by w - 3.7 x 10 1T (sec-1). Both longitudinal and transverse waves are included in the first, most dense He layer. TR calculated at three frequencies. Only longitudinal waves are included in the first He layer. TR calculated at three frequencies. Only longitudinal waves are included in the first He layer. R T3 calculated as a function of temperature T. For t e lower curve both longitudinal and transverse waves are included in the first helium layer, while only longitudinal waves in the first helium layer are used to calculate the upper curve. R T3 calculated for two attenuations in the copper. e first helium layer is considered to be a solid; i.e., both longitudinal and transverse are used. RKT3 calculated for two attenuations with a helium density factor F - 1.4. T3 calculated for two attenuations with a helium density factor F I 1.5. vi Page 13 18 22 42 48 49 52 57 58 59 6O Figure Page 12 T3 calculated for two attenuations and a helium density factor F - 1.4. Transverse waves are in- cluded only in the first, most dense, helium layer. 61 13 RKT3 calculated for two attenuations and a helium density factor F - 1.5. Transverse waves are in- cluded only in the first, most dense, helium layer. 62 14 RKT3 calculated using an attenuation profile in the copper V - 0.25 fog a layer which extends from the copper surface 20 A into the copper, and V - 0.12 beyond the layer. The individual points are experi— mental values; A - Reference 21, + - Reference 51, Y‘— Reference 52, x - Reference 55. 65 15 RKT3 calculated using an attenuation profile in the copp r: V - 0.30 for 20 X, V . 0.12 for the next 250 , and V - 0.6 beyond the layers. The individual points are experimental values; A - Reference 21, + - Reference 51, U‘- Reference 52, x - Reference 55. 66 16 T3 calculated using an attenuation profile in the copper: V - 0.42 for 25 R, V - 0.12 for the next 250 X, and V - 0.06 beyond the layers. The individual points are experimental values; A - Reference 21, + - Reference 51, n-— Reference 52, x - Reference 55. 67 17 RKT3 calculated using an attenuation profile in the copp r: V I 0.50 for 25 R, V - 0.12 for the next 100 , and V = 0.06 beyond the layers. The individual points are experimental values; A - Reference 21, + ~ Reference 51, [1' Reference 52, x - Reference 55. 68 El TR calculated at frequencies corresponding to the temperatures; 1.6, 1.75, 2.0 and 3.0 K. Only longi- tudinal waves are included in the first He layer. 95 E2 TR calculated at frequencies corresponding to the temperatures; 0.1, 1.0, 1.25 and 1.4 K. 96 E3 TR calculated at frequencies corresponding to the temperatures 1.43, 1.44, 1.45, and 1.46 K. 97 E4 TR calculated at frequencies corresponding to the temperatures; 1.47, 1.48, 1.49, and 1.5 K. 98 E5 TR calculated at frequencies corresponding to the temperatures; 1.55, 1.6, 1.75, and 2.0 K. 99 E6 TR calculated at frequencies corresponding to the temperatures; 2.25, 2.5, 2.75, and 3.0 K. 100 vii Figure Page E7 TR calculated at frequencies corresponding to the temperatures 3.25, 3.5, 3.75, and 4.0 K. 101 E8 TR calculated for a small attenuation, V - 0.01, at frequences corresponding to the temperatures; 0.1, 1.0, 1.25, and 1.4 K. 102 E9 TR calculated for a small attenuation, V - 0.01, at frequencies corresponding to the temperatures; 1.43, 1.44, 1.45, and 1.46 K. 103 E10 TR calculated for a small attenuation, V - 0.01, at frequencies corresponding to the temperatures; 1.47, 1.48, 1.49, and 1.5 K. 104 E11 TR calculated for a small attenuation, V - 0.01, at frequencies corresponding to the temperatures; 1.55, 1.6, 1.75, and 2.0 K. 105 E12 TR calculated for a small attenuation, V - 0.01, at frequencies corresponding to the temperatures; 2.25, 2.5, 2.75, and 3.0 K. 106 E13 TR calculated for a small attenuation, V - 0.1, at frequencies corresponding to the temperatures; 3.25, 3.5, 3.75, and 4.0 K. 107 viii I. INTRODUCTION There are many areas for fundamental theoretical and experimental research in low temperature physics. This thesis deals with two dif- ferent phenomena which occur at low temperatures: The first is an experiment to measure the sublimation and vapor pressure of the argon isotope, Ar36, in the temperature range 23.752-87.375 K. In the second, we make a theoretical investigation of the thermal boundary resistance which occurs at the interface between two different materials when heat flows across the interface. This effect is commonly known as Kapitza resistance and is most pronounced between liquid He4 (or He3) and a dense, elastic solid, for example a metal at very low temperatures, T s 1 K. The sublimation and vapor pressure experiment is discussed in Chapter II of this thesis and the theoretical calculation of Kapitza resistance is the topic of Chapter III. A. Discussion of Sublimation and Vapor Pressure Properties of the rare-gas solids have long been of interest be- cause the interatomic forces are weak, short-ranged, and relatively well understood. To a good approximation these forces can be repre- sented by two-body central forces, and comparatively simple theories can therefore be used to predict properties of the rare-gas solids. In particular sublimation pressure is calculable from lattice-dynamical theories and consequently can be used as a test of such theories. The 2 purpose of this experiment is to provide an accurate table of sublima- tion-pressure data for Ar36 which extend over several orders Of magni- tude. From these data we calculate the static lattice energy, heat of sublimation, vibrational energy, and geometric mean frequency of the phonon spectrum. Since the sublimation pressure of normal argon (ArN) has been measured previously with the same apparatus,1’2 we compare the sublimation pressure of ArN with that of Ar36 and discuss the observed differences. we also discuss the Observed differences in some of the calculated properties mentioned above. Vapor and sublimation pressures of ArN have recently been mea- sured in the range 75-85.2 K by Chen, Aziz, and Lim.3 Lee, Fuks, and 4 Bigeleisen have, also recently, made differential measurements comr N and Ar36. Their data, paring vapor and sublimation pressures of Ar in the form of P(ArN) and P(Ar36) - P(ArN), extend from 62-102 K. Earlier measurements on the argon isotopes have been made by Clusius and co-workers5 in the range 84-88 K, and by Boato and co-dworkers6'8 in the ranges 84-119 and 72-83.7 K. The vapor-pressure ratios for isotopes of solid Ne and Ar have been calculated by Klein, Blizard, and Goldman9 using the improved self- consistent phonon scheme of Goldman, Horton, and Klein.10 B. Description of the Kapitza Resistance Effect The thermal boundary resistance, RK’ between liquid helium and a solid is known as Kapitza resistance, and is defined by RK - A AT/Q (cm? K/W) . (1) In Equation 1, O is the heat flow, A is the area of the interface, and AT is the temperature discontinuity across the interface. Kapitza11 discovered the phenomenon in 1941 while investigating the thermal 3 conductivity of superfluid He“. He observed a temperature discontinuity, AT, between a heated metal surface (Cu) and the liquid helium of ap- proximately 2 mK per mW/cmz of heat flux, OIA (between 1.6 K and the lambda point). That is RK = 2 cm2 mK/mw (- 2 cm2 K/W) in the tempera- ture range 1.6-2.172 K. A possible explanation of the phenomenon was first put forth by Khalatnikovlz’13 in 1952 and later, but independently, by Mazo and 13,14 Each of these authors treated the heat flow across the Onsager. liquid-solid interface as a thermal distribution of quantized elastic waves and calculated the transmission coefficient for waves incident at the interface using classical elastic waves. Since the velocity Of sound in a solid may be an order Of magnitude or more larger than in liquid helium, and the density of a solid may be 1-2 orders of magnitude larger than that of liquid helium, there will be a large acoustic mis- match and the flow of phonon energy across the liquid-solid interface will therefore be impeded. Rx's calculated using this acoustic mis- match theory have a T.3 temperature dependence (due to assuming a Debye density of states for the phonon modes). However, experimentally ob- served RK's are typically an order of magnitude or more smaller than the theoretical values and have temperature dependences ranging from T"2 to T‘4.5.13 In an attempt to resolve some of these discrepancies between theory and experiment, several alternative theories have been proposed, of which we shall mention the ones closely related to our work. Challis, Dransfeld, and Wilks15 modified the original acoustic mismatch theory of Khalatnikov and of Mazo and Onsager by taking into account the im- proved matching due to the increased density of liquid He4 near solid 4 surfaces.16 They suggested that the improved acoustic matching would result in an increase in the energy flux across the liquid-solid inter- face and therefore decrease RK' Although RK calculated with their model has a temperature dependence of T4"2 above 1 K, we feel that the im— proved agreement with experiment in the magnitude of RK is significant. Concurrent with our present calculations of Kapitza resistance, there has been other work”.20 dealing directly with the effects of phonon attenuation in the solid, in particular copper, on RK' The basic assumption used in the models proposed by these other workers (and by us) is that within 1000 X or so of the solid surface there is a region in which the phonons are more strongly attenuated than in the bulk solid. Kuang Wey-Yen21 in analyzing his experimental RK data for several materials considered the effects of an amorphous layer (the Beilby layer) which is formed on the surface of a cold worked metal. Such a deformed layer has a thickness of about 50-100 2 and it is estimated21 that the transition to the bulk polycrystalline properties occurs over a distance of 103-104 3. His analysis considered the possibility of the elastic constants being lower (hence lower sound velocities) in this amorphous region, which of course would improve the acoustic matching for the system and thereby decrease R While we K' agree completely that the effects associated with a thin deformed layer beneath the surface of the solid (in particular phonon attenuation) are significant with respect to decreasing RK, we feel that the impedance matching effects associated with the dense helium region at the liquid- solid interface should be included. The purpose of the third chapter of this thesis then is to demon- strate that the combined effects of impedance matching and phonon 5 attenuation in the solid result in calculated Kapitza resistances which agree quite well with experimental data. We first describe the model which we use and then the theory required to obtain the desired quantity, RK' Following this we give a brief description of the numerical tech- niques used and give the accuracy which we expect from such calculations. Finally, we present our results for a variety of parameters and make some comparisons with experimental data. II. THE EXPERIMENT ON SUBLIMATION AND VAPOR PRESSURE A. Theory The condition for a multi-phase thermodynamic system to be in thermal equilibrium is that the specific Gibbs free energies for each of the phases in thermal equilibrium be equal. For a solid-vapor system (or liquid-vapor system) this is expressed as G[Vapor] - G[solid(or liquid)] . (2) The calculations of G for each phase in Equation 2 may involve assump- tions about the system which neglect properties unique to the boundary separating the two phases. This problem can be circumvented however, by treating the interface as a third phase, separate from the bulk materials on either side of the interface. That is, thermal equilibrium for a three-phase system implies equilibrium between any two phases [in particular the solid(or liquid) and vapor phases] and Equation 2 is therefore applicable in general. Applying classical thermodynamics to a solid-vapor system (or liquid-vapor system) in thermal equilibrium yields the Clausius- Clapeyron equation which may be‘written22 d(£nP) . -L . d(1/T) R[l - P(vc - 3)]RT] (3) In Equation 3 P is the pressure, T is the temperature, L is the heat of sublimation, R is the gas constant, and vc is the molar volume of the condensed phase, i.e., the solid or liquid. The second virial 7 coefficient, 8, is defined by the gaseous equation of state: Pvg . RT(1 + B/vg) , (4) in which v8 is the molar volume of the gas. Over a sufficiently narrow temperature interval Equation 3 can be integrated to give the classical vapor-pressure equation: ZnP-A/T+E, (5) where A - -L . (6) R[1 - P(vc - B/RT] The heat of sublimation L may then be calculated by fitting sub- limation-pressure data to Equation 5. If data for a large temperature range are available, then the temperature dependence of L may be cal- culated by successively applying this technique to adjacent narrow temperature intervals. Using statistical mechanics to calculate the Gibbs free energy of the gaseous phase, a somewhat different vapor-pressure equation can be obtained. The classical partition Function, Q, for a canonical en- semble of N particles is defined by23 Q - (I/h3Nszr3N dp3N exp(-BH) . (7) In Equation 7 the variables of intergration, r and p , denote compo- i i nents of the canonical position and momentum coordinates, respectively. The variable H is the total hamiltonian for the system, h is Planck's constant, and B is defined by B - (kBT)-1 where kB is Boltzmann's con— stant and T is the temperature. For a system of N particles in a volume V, Equation 7 can be written Q - (I/N!) (kaT/Znh2)3N/2 f dr3N exp(-8 Z vij) , (3) i [ .2. (1/Uj)2“] -- a/T + b . (13> n=l j=l where a - Eo/R (14) and b 315mg + (l/2)£n[(m/2U)3(l/kB)] . (15) In these equations, E0 is the static lattice energy and g8 is the Gibbs free energy of mono-vacancy formation. The 3N lattice fre- quencies, wi’ determine the geometric mean frequency of the lattice frequencies, mg’ and the 2nth moment of the lattice frequencies, . The remaining symbols are the same as those defined in Equations 3 and 8. If one uses the Debye approximation for and truncates the infinite series in Equation 13 at the 4th order term, Equation 13 10 becomes; £n(PT1/2 ) + exp(-g8/kBT) - P(vc - B)/RT - (3/40)(OD/T)2 + (l/2240)(OD/T)4 - a/T + b . (16) In Equation 16, OD is the characteristic Debye temperature of the solid. The static lattice energy, E0, represents the binding energy of the solid and can therefore be calculated by summing, over all lattice sites, the potential of pair interaction evaluated at the mean separation dis- tance for the atoms. It should be mentioned at this point, that Equation 13 was ob- tained through the use of an expansion which is valid only for high temperatures, T 3 40 K. A difficulty with this theory then, is that in the range where it is most easily used (for example Equation 16 is valid for T 3 58 K) phonon-phonon interactions are important and should not be neglected. In fact, in analyzing our experimental data using Equation 16, we do obtain results which suggest that the quasi-harmonic lattice theory is not valid for such high temperatures. The improved self-consistent phonon theory of Goldman, Horton, and Klein10 for the Helmoltz free energy of the anharmonic crystal does include phonon-phonon interactions and appears to be a valid theory for argon even at temperatures near the triple point (T = 84 K). In their theory, anharmonic effects are included in two ways: The change in the potential with temperature, which is consistent with temperature de- pendent changes in the frequency spectrum and lattice spacing (due to anharmonicity) is considered in the self-consistent portion of their theory. This results in the so-called "effective potential". The phonons thus obtained are said to be renormalized and have infinite lifetimes. Using this newly calculated "effective potential", they then ll calculate the third order terms which describe interactions between renormalized phonons. This is the "improved" part of their theory. It is interesting to note that the self-consistent theory resembles the quasi-harmonic theory in that anharmonicity results in a tempera- ture dependence in the crystal volume and frequency spectrum. How- ever, the quasi-harmonic theory does not consider that the potential between any pair of atoms in the crystal lattice is altered by the motions of all the neighboring atoms. In the self-consistent theory, this effect is taken into account by replacing the pair potential with the thermally averaged pair potential. The self-consistent nature of the theory becomes apparent at this point: The thermally averaged potential is calculated using the phonon distribution function and summing over all phonon modes. However, the phonon modes are them- selves determined by this thermally averaged potential. Such a cal- culation must then be iterated until a self consistent set of results is obtained. The improved self-consistent theory of Goldman, Horton, and Klein10 has been used to calculate vapor pressure ratios of isotopes of Ne and Ar by Klein, Blizard, and Goldman.9 Their calculations have been carried out for both (13, 6) and (12, 6) potentials of the Lennard- Jones type using nearest neighbor interactions. we use their calculated values of the Helmoltz free energies of Ar36 and Arao to make compari- sons of our sublimation-pressure data and vapor-pressure ratios with theory. B. Experimental Method A detailed description of the apparatus and experimental tech- nique used is given in Reference 1. A schematic representation of the 12 apparatus is shown in Figure 1. There are, however, some important modifications which were made for this experiment. With the gas-sample container open to the apparatus, the pres- sure of the gas sample at room temperature was about 180 Torr. Since this is below the triple-point pressure of Ar, it was necessary to con- dense the sample directly from the vapor to the solid beginning at about 76 K. The volume of the solid formed was approximately 0.15 cm3 at 50 K. After cooling to 30 K, volatile impurities were removed by repeated fractional distillation. The sample temperature was controlled manually with a lO-turn, 5000 - Q potentiometer in series with a regulated power supply which supplied current to the heater on the copper block. By carefully controlling the amount of He exchange gas, the heater current required was minimized. This was done to insure that no thermal gradients would be set up along the copper block. For temperatures above 55 K, the sample temperature could in this manner be controlled during the course of a single measurement to i_1 mK, as judged by the response of the Bourdon gauge. Temperature control below 55 K was within the limits of the thermometer accuracy, 1:3 mK. To insure that thermal equilibrium had been established such control was generally maintained for l/2tn 1. Temperature Measurement The sample temperature was measured using a Pt resistance ther- mometer imbedded in the Cu block as shown in Figure 1. The thermometer was a four-lead Model 8164 Leeds and Northrup capsule-type and a calibration of the thermometer was supplied by the National Bureau of Standards. A small amount of vacuum grease was used to enhance the thermal contact between the thermometer and the Cu block. To further 13 3.30 :OIII .38.... 5.52.3? eons—Eu ciaomIIL use .2353 .23: £52.37 .33.... .85 asses $5) 02:5 23:... o... .o-eom 22285.... no 335252.... E goo-o :0 1: A sketch of the apparatus used for this experiment. Figure 14 insure that the thermometer and Cu block were at the same temperature, the thermometer leads were thermally anchored to the Cu block. A single-potentiometer technique was used to measure thermometer resistance and hence the sample temperature. Two of the leads were connected to a DC constant current source of 2 mA. The remaining two leads were run to a Leeds and Northrup K-S potentiometer to measure the potential drop across the thermometer. To account for thermal emfs along the thermometer leads, all resistance measurements were repeated with the direction of the thermometer current reversed. Although the sensi- tivity of the measurements using this technique is approximately 0.5 mK, the estimated accuracy 13;: 3 mK. This is due primarily to the time factor involved in making each measurement. That is, during the course of a single measurement, the thermal emfs (as well as the sample temperature) can change slightly. These effects combined with the uncertainties in the thermometer cali- bration Q: l mK) give the estimated accuracy of temperature measurement, ‘: 3 mK. 2. Pressure Measurement The sublimation and vapor pressures of Ar36 extend over several orders of magnitude for the temperature range used in this experiment. For this reason it was necessary to use two techniques covering two pressure ranges to measure pressure. For pressures above 1 Torr, a calibrated Bourdon gauge was used. The particular instrument was a Texas Instruments Model 142 quartz spiral Bourdon gauge. A single Bourdon tube, which covered pressures from 0-1000 Torr, was sufficient for this experiment. For our Bourdon gauge, a photocell detector is used to detect the deflection of the Bourdon 15 tube by locating a beam Of light reflected from a mirror attached to the Bourdon Tube. A counter which divides the full scale deflection into 300,000 counts is attached to the photocell detector. For a Bourdon tube with a pressure range of 1000 Torr then, the corresponding sensitivity of the gauge is about 3 microns. A mercury manometer read with a Wild cathetometer was used to calibrate the Bourdon gauge. With the cathetometer, the mercury level in the manometer could be measured to 110.02 mm. Corrections were applied to these readings to account for the capillary depression of the mercury meniscus.26 Corrections were also made for the thermal expansion of mercury (readings were corrected to the 0° C density of mercury) and for the local deviation from.the standard gravitational acceleration (the standard value is g - 980.350 cmMsecz, while at Michigan State University g - 980.665 cm/secz). The correction for the thermal expansion of mercury was the largest correction (f 0.52) but was nevertheless an accurate enough correction so as to not intro- duce any new errors. NO errors were introduced by the gravitational correction since it, too, is known to a high degree of precision. The correction for the depression of the mercury meniscus was a small correction (5 0.06 Torr) but was also the most uncertain. From the tables in Reference 26 it was clear that we could expect as much as a 152 error in the meniscus correction. That is, an additional error of i 0.01 Torr is introduced by the meniscus depression effect. As judged by our calibration curve for the Bourdon gauge, the accuracy of our pressure measurements could be given as 1:0.03 Torr. This accuracy is consistent with the above mentioned uncertainty in the meniscus correction combined with the errors associated with the cathetometer readings. 16 To measure pressures below 1 Torr, a Consolidated Vacuum Corpora- tion Type GM+100-A MbLeod gauge was used. A liquid nitrogen cold trap separated the McLeod gauge from the sample chamber and thus prevented any mercury contamination of the sample. Because of this cold trap, however, it was necessary to correct the McLeod gauge readings for mercury streaming.27 Mercury vapor will condense in the cold trap; therefore, there will be a flow of mercury vapor from the McLeod gauge to the cold trap which will then result in a reduced pressure in the MeLeod gauge. Data and equations given in Reference 27 were used to make this correction, which was always less than 102 of the MCLeOd gauge reading. 4 It was also necessary to correct for the effects of thermomolec- ular flow.28’29 When the mean free path of molecules in a tube (con- necting two chambers at different temperatures) is comparable to or larger than the diameter of the tube, a pressure gradient is observed along the length of the tube. The pressure will be higher in the warmer chamber. As the mean free path becomes shorter, the pressure difference decreases and eventually goes to zero. This effect is most pronounced at low temperatures and low pressures. An empirical relation and experimental data given in Reference 28 were used to make this correc- tion. At the lowest temperatures and pressures measured, this correc— tion lowered the measured pressure by as much as a factor Of 3. At temperatures near 50 K and pressures of 0.1-1.0 Torr, the correction was less than 22 of the measured pressure. The correction became negligible at higher temperatures and pressures. s} ‘1 17 3. Gas Sample Analysis The gas sample was purchased from the Mbund Laboratory and pro- vided with a mass-spectrometer analysis. Amounts by mole fraction of each constituent were given as: Ar greater than 99.52, N2 less than 0.42, Hz and 02 each less than 0.12; of the Ar, more than 99.92 was Ar36. Our own mass-spectrometer analysis, performed by the Chemistry Department at Michigan State University upon completion of the experi- ment, yielded by mole fraction the following impurity concentrations: N2 less than 0.12, 0 less than 0.022, and H less than 0.00042. 2 2 C. Results of the Experiment The experimental sublimation- and vapor-pressure and temperature data are shown in Table 1 and plotted in Figure 2. At the lower temp- eratures, say below 35 K, the observed pressures begin to level off and the scatter in the data increases. The leveling off is due primarily to the presence of those volatile impurities which could not be removed from the sample. Another limiting factor, and one which contributes to the scatter in the data, is that these pressures were the lowest which could be measured with our McLeod gauge. In order to obtain a measure of the amount of scatter in the data shown in Table l, we used a truncated form.of Equation 16: £n(PT1/2) - a/T + b , (17) where the parameters a and b are the same as those defined in Equations 14 and 15, respectively. We first fit the data by the method of least squares to Equation 17 over two temperature ranges in the solid. The calculated coefficients are a - -988.088, b - 20.2338 for the temperature range 35.479 - 60.183 K, and a - -994.1123, b - 20.33541 for the temperature range 60.326 - 83.760 K. We used Equation 5 to represent I03 .. I I I l I I I l T l I [/I: r- ." -1 D: — 0.. —I P ’.0 d P . d A W -.- -: t - .° : '2 r .. c. ' ,- T a lw :- .O 1 Lu ’ " (I: b . .— 5), IO" .— ' -: cn : ' 2 m o a: - : .1 “- IO" .— - -: 2 b O. «- SE! b '- 2 IDA;- ' 1: g : : as - ' . 2;; lc;“::- U. '1: : .0 E -5 a... -: '° E" .' : L. ’ - '0'. 1 l 1 J_ 1 l l l J l l l 1 20 3O . 4O 50 60 70 80 90 Figure 2: TEMPERATURE T(K) Sublimation pressure of Ar36 as a function of temperature. 19 TABLE 1 Experimental sublimation- and vapor-pressure and temperature data. Temperature Pressure P-Pca c IP"Pealcl T (K) P (Torr) (Torr P 23.752 2.8 x 10‘6 24.768 6.5 x 10-6 25.898 9.9 x 10'6 28.576 1.1 x 10'-5 29.324 1.4 x 10‘5 + 1.4 x 10-5 1.00 30.637 1.8 x 10"5 + 1.7 x 10‘5 0.94 32.121 1.8 x 10"5 + 1.3 x 10’5 0.72 33.305 3.4 x 10-5 + 2.0 x 10'5 0.59 33.968 3.8 x 10-5 + 1.4 x 10-5 0.37 35.479 8.5 x 10-5 + 0.2 x 10'5 0.024 36.113 1.3 x 10-4 - 0.4 x 10-5 0.031 36.945 2.9 x 10-4 + 0.5 x 10’4 0.172 38.152 6.3 x 10-4 + 0.7 x 10-4 0.111 39.276 1.02x 10-3 - 0.14x 10'3 0.137 39.774 1.44x 10'3 - 0.14x 10‘3 0.097 41.430 3.85x 10‘3 - 0.33x 10'3 0.086 41.972 5.50x 10'3 - 0.15x 10’3 0.027 42.951 9.26x 10‘3 - 0.29x 10"3 0.031 44.212 0.0172 - 0.0009 0.0523 44.493 0.0206 - 0.0002 0.0097 45.489 0.0327 - 0.0008 0.0245 46.933 0.0621 - 0.0022 0.0354 47.940 0.107 + 0.008 0.0748 48.962 0.157 + 0.006 0.0382 50.453 0.272 + 0.002 0.0074 51.009 0.339 + 0.007 0.0206 51.930 0.470 + 0.006 0.0128 52.937 0.670 + 0.010 0.0149 53.928 0.928 + 0.007 0.0075 54.956 1.31 + 0.03 0.0229 55.987 1.80 + 0.02 0.0111 57.005 2.40 0.00 0.0000 57.935 3.17 + 0.02 0.0063 58.889 4.08 - 0.04 0.0098 60.152 5.76 - 0.05 0.0087 60.183 5.80 - 0.04 0.0069 60.326 6.07 - 0.02 0.0033 61.118 7.47 - 0.02 0.0027 62.315 10.15 + 0.01 0.0010 62.904 11.71 - 0.01 0.0009 64.139 15.76 + 0.03 0.0019 64.887 18.70 0.00 0.0000 66.633 27.59 + 0.02 0.0007 66.956 29.54 - 0.01 0.0003 67.927 36.26 - 0.02 0.0006 68.902 44.47 + 0.16 0.0036 70.154 56.90 + 0.09 0.0016 Table 1 (cont'd.) 20 Temperature Pressure P-Pca c 13:3n31g_ T (K) P (Torr) (Torr) P 71.471 73.29 + 0.22 0.0030 72.356 86.07 - 0.02 0.00023 73.510 105.96 - 0.01 0.00009 74.681 130.30 + 0.32 0.00246 75.557 150.62 — 0.16 0.00106 76.826 185.95 + 0.12 0.00065 78.167 230.12 + 0.10 0.00043 78.702 249.91 - 0.03 0.00012 79.612 286.81 - 0.30 0.00105 80.403 323.29 0.25 0.00077 80.885 346.06 - 0.64 0.00185 81.439 375.37 - 0.28 0.00075 81.440 375.46 - 0.24 0.00064 81.878 399.37 - 0.61 0.00153 82.324 426.06 + 0.05 0.00012 82.668 446.42 - 0.61 0.00137 83.241 484.02 + 0.08 0.00017 83.639 510.82 - 0.20 0.00039 83.760 519.62 + 0.13 0.00025 83.991 533.57 - 0.25 0.00047 84.045 537.13 0.00 0.00000 84.503 565.80 - 0.01 0.00002 85.047 601.60 + 0.16 0.00027 85.492 632.02 + 0.13 0.00021 85.998 668.16 + 0.20 0.00030 86.482 704.22 + 0.25 0.00036 87.031 746.65 + 0.01 0.00001 87.244 763.73 0.00 0.00000 87.316 769.43 - 0.15 0.00019 87.375 774.06 - 0.33 0.00043 ex 91? 21 the data in the liquid range 83.991 - 87.375 K.with calculated para- meters A - -806.7794 and B - 15.88561. The third and fourth columns Of Table l are based on these parameters applied to the appropriate tempera- ture ranges. Notice that the calculation of pressure, with parameters a and b for the lowest temperature range, has been extended to 29.324 K on Table 1. This illustrates that the sublimation pressures begin to level off at the lower temperatures. For the most part the deviations of the actual pressures from the corresponding equations is what one would expect with scatter of i 3 mK and i 0.03 Torr. Figure 3 is a plot of £71(PT1/2 ) versus l/T for data ranging from T - 68.902 K, P - 44.47 Torr to T - 83.639 K, P - 510.82 Torr. The straight line is the least squares fit, of the data shown, to Equation 17 with calculated parameters a 8 -992.9865 and b - 20.31851. A measure of the "goodness of fit" of Equation 17 to the data, is the average fractional deviation IAP/PI E I(Pexp - Pcalc)/Pexp|ave’ For 8V8 the set of points in Figure 3, IAP/PI - 0.00087, which is too small ave to be observed in the figure. 1. Discussion 2; Equation 16_ Sublimation-pressure data can be used to determine several prop- erties of Ar36 by comparison with theory. We did this by fitting P(T) data of Table l to the theoretically derived Equation 16. Further dis- cussion of this equation is now in order to make clear how this was done and what it means. The second term in Equation 16 expresses the effect of vacancy formation on vapor pressure. It can be written as exp(-g3/kBT) - exp(8/kB)exp (-h/kBT), where s and h are, respectively, the entropy and enthalpy of mono-vacancy formation. This term is equal to the fractional nIA 22 3-5 IIIIIIIIIIIIIIHIIIIIIIII _ l I Lllll l 6.0 " ‘ LELlLllllllllllllJllllllJ IZO l2.5 l3.0 I15 I40 I45 RECIPROCAL TEMPERATURE 1" (10'3 K") Figure 3: Plot of £n(PT1/2) versus l/T for data ranging from T=68.902 K, P = 44.47 Torr to T - 83.639 K, P - 510.82 Torr. 23 vacancy concentration. We take the value of s/kB for Ar36 to be that suggested by other workers30 for normal argon, s/kB = 2.0, since no value for the isotope is available. A suitable value for the para- 31 meter h/kB is then determined from measurements which give the vacancy concentration of normal argon at the triple-point as exp(-g8/kBTtr) f 0.131. This gives h/kB 2 724.57 K. The vacancy formation term, as a function of temperature, is at most then exp(-g8/kBT) - exp(2) exp(-724.57/T). This is the form used for our analysis; some typical values are given in Table 2. For temperatures near the triple point, 80.403 - 83.760 K, one effect of including this vacancy correction term is to decrease the parameter a of Equation 16 by 0.079%. This effect becomes smaller at lower temperatures and in particular, for the temperature range 58.889 - 66.633 K, is only 0.0052. A somewhat larger effect is found in the third term of Equation 16, -P(vc - B)/RT. This term combines the effects of finite crystal volume and gas imperfection. Values for the molar volume, vc, are Obtained from Dobbs and Jones.32 The second virial coefficient, 8, is calcu- lated according to Hirschfelder, Curtiss, and Byrd.33 Some representa- tive magnitudes of vc, B, and -P(vc - B)/RT are given in Table 2. In the temperature range 80.403 - 83.760 K, the inclusion of the finite crystal volume-gas imperfection term increases the parameter a of Equation 16 by 1.611. At the lower temperatures, 58.889 - 66.633 K, the effect on a due to this term is reduced to 0.982. I The 4th and 5th terms of Equation 16, -(3/40)(OD/T)2 and (l/2240)(0D/T)4, respectively, are the leading terms of the infinite series contained in Equation 13. To estimate these terms we used for the Debye temperature of Ar36, 0D36 - 88.02 K. This Debye temperature 24 TABLE 2 mm mucouomomv mm moooummomo Hm monouomumn on oooouomomm Typical values of various terms which appear in Equation 16. «tea a m.m ammo.o mmuo.o an.mmm nn.w~ mloa x m.H mmo.mm «10H x «.0 ammo.o msHo.o we.mm~ we.e~ «tea a o.m mow.om «lea x o.n mmoo.o oeao.o m~.No~ om.e~ «lea n ¢.n ~o~.mm euoa N N.m moa.o mica u c.» H¢.mm~ nH.q~ cued x m.m oam.m~ mica x N.H «NH.o mica x ~.m om.nmm om.m~ «tea a o.~ Nom.mo mica x n.H n¢H.o mica x m.H «H.wom mn.m~ mica x «.5 «oo.~c mica x ~.~ on.o «lea x m.m um.an< am.m~ mica x c.m mmm.mn a o AOHQM\naoMI.AoHomwmaoM. m efia\noVaoe-\Hv NAH\ evaoe\nv am\xm . >ve mI u> Aamx\ unease Axes sowuoouuoo vusoaoeuwooo ouaoao> p.osowuoouuoo mooauomuuoo uaoao> ouuawm Hmauw> umaoa moaoom> ousumuoaame HoOfinmsooalasuooso loowuoomuoqaw mmu vacuum Hmumauo 25 for Ar36 was calculated from that obtained by Morrison and co-workers34 for ArN, ODN - 83.5 K, assuming that On varies inversely as the square root of the mass. Values for the fourth and fifth terms of Equation 16 at various temperatures are given in Table 2. Their effects on the parameter a of Equation 16 are as follows: For the temperature range 80.403 - 83.760 K, -(3/40) (OD/T)2 decreases a by 1.402 and (1/2240)(OD/T)4 increases a by 0.027%. For the temperature range 58.889 - 66.633 K, -(3/40)(OD/T)2 decreases a by 1.752 and (l/2240)(0D/T)4 increases a by 0.064%. For intermediate temperatures these effects change monotonically. The next higher order term in the infinite series in Equation 13 can be approximated as (-l/180740)(0D/T)6. For temperatures greater than 58 K, its magnitude is less than 2% of the term in T-a, and is therefore negligible. 2. Obtaining Thermodynamic Prqperties From the Data We fit P(T) data of Table l to Equation 16 by the method of least squares in the following manner: Beginning at T - 58.889 K, P - 4.08 Torr, the first 10 points in order of increasing temperature and pres- sure are fit to Equation 16. Then the first point is dropped, the 11th point is added and another 10-point fit is made. This procedure is repeated until the last data point to be used, T - 83.991 K, P - 533.57 Torr, is included in the fit. In all, 25 intervals were fit to the data. For each interval of fit, the average of the inverse temperature, , is calculated. The temperature corresponding to that average, ‘1, is used to Obtain the temperature dependence of the parameters a and b for that interval; these values for each data interval are shown in the first column of Table 3. 26 TABLE 3 The static lattice energy, E , and geometric mean of the lattice vibra- tional spectrum, w , calculaged from our data using Equation 16, are shown in the secon and fourth columns, respectively. The third column shows the values of the heat of sublimation calculated from our data using Equation 5. Static Heat Geometric lattice of mean Temperature energy sublimation frequency T(K) -Eo(ca1/mole) L(cal/mole) 108(1012 sec-1) 62.068 2022.05 1923.63 7.681 62.866 2018.64 1919.80 7.610 63.627 2014.79 1915.55 7.533 64.490 2011.89 1912.18 7.478 65.470 2009.94 1909.63 7.440 66.502 2007.77 1906.80 7.400 67.502 2006.00 1904.32 7.367 68.563 2003.08 1900.61 7.314 69.614 2003.05 1899.76 7.313 70.685 1999.69 1895.51 7.254 71.694 1998.21 1893.06 7.229 72.812 1995.18 1888.94 7.179 73.896 1991.60 1884.07 7.121 74.978 1990.00 1881.15 7.095 76.014 1988.55 1878.32 7.072 76.966 1987.40 1875.66 7.054 77.891 1984.36 1871.11 7.008 78.703 1981.45 1866.66 6.965 79.438 1981.45 1865.20 6.964 80.131 1977.64 1860.35 6.909 80.726 1975.36 1857.26 6.876 81.237 1976.33 1857.33 6.890 81.735 1975.75 1856.15 6.882 82.153 1974.80 1854.48 6.869 82.513 1970.31 1849.12 6.805 fa :0; dat. 27 In the manner just described we obtain, as functions of tempera- ture, the static lattice energy, E - a R, and the geometric mean fre- O quency, m8 - [(2n/m)k31/3]1/2 exp(b/3). The first two columns of Table 3 show the values our analysis gives for T and E0, respectively. Values obtained for 018 are shown in the last column of Table 3. Sublimation-pressure data can also be used to obtain the tempera- ture dependence of the heat of sublimation L. To do this, the same P(T) data used to obtain Eo and m8 are fit to Equation 5 over succes- sive temperature intervals in the manner described above. Values for L resulting from this analysis are shown in the third column of Table 3. The lattice vibrational energy, Evib’ can now be calculated according to the thermodynamic relation, L - Ug - Uc + P(v8 - vc) . (18) In this equation U8 and Uc are, respectively, the internal energies of the gas and crystal, and v8 and vc are the molar volumes discussed previously. Now UC can be expressed as, UC - Eo + Evib’ which leads to: Evib u -Eo - L + P(v8 - vc) + Ug . (19) To order of the second virial coefficient, Equation 19 can be rewritten as: Evib - -Eo - L - P(vc - B) + (5/2)RT - (RT2/v8)(dB/dT) . (20) Values for B are the same as those used in the calculations of Eo and L. The results of these calculations for Evib are shown in the second colum of Table 4. D. Discussion In Table 3 the static lattice energy, EO(T), derived from our data, appears as an increasing function of temperature. Qualitatively, 28 TABLE 4 Lattice vibrational energies calculated from Equation 20 are shown in the second column. Theoretically predicted values from Equation 22 are shown in the third column. Temperature Lattice Vibrational Energy (experiment) (Debye theory) T(K) Ev1b(ca1/mole) Ev1b(ca1/mole) 62.068 406.46 406.37 62.866 410.78 410.66 63.627 414.89 414.81 64.490 419.56 419.51 65.470 424.91 424.86 66.502 430.54 430.48 67.502 436.05 435.94 68.563 441.89 441.78 69.614 447.68 447.58 70.685 453.59 453.50 71.694 459.24 459.07 72.812 465.46 465.24 73.896 471.65 471.26 74.978 477.80 477.30 76.014 483.72 483.09 76.966 489.34 488.40 77.891 494.76 493.57 78.703 499.68 498.10 79.438 504.13 502.21 80.131 507.95 506.09 80.726 511.10 509.44 81.237 513.98 512.33 81.735 516.48 515.14 82.153 518.77 517.50 82.513 520.98 519.53 29 one expects this behavior since the crystal expands with increasing temperature. However, a quantitative comparison between the EO(T) values for Ar36, in Table 3, and the theoretical EO(T) presents numerous difficulties. In principle one could calculate EO(T), in the effective two—body approximation, from the Lennard-Jones 6-12 potential and lattice parameter data, ao(T), for Ar36. However, ao(T) for Ar36 has apparently not yet been determined so that one is forced to use ao(T) obtained from.measurements on ArN. If this is done and a lattice sum calculation for an fcc lattice taken over all neighbors is made, the general result for EO(T) is given, for example in Kittel35 as 800:) - 2€[12.131(ovr2-/ao(T))12 - 14.454(o/2/ao(T))6] . (21) Using lattice parameter data as given by Dobbs and Jones32 and energy and distance potential parameters, respectively as e/kB- 119.4 K and o - 3.40 2, obtained by Zucker,36 we find the resulting EO(T) to be higher than our experimental values by about 30 cal/mole. That is, our data show that the Ar36 solid is more tightly bound than predicted by this theory. There are two principal sources of this disagreement. The first is simply that ao(T) for Ar36 is different from ao(T) for ArN. However, since ao(T) for Ar36 is expected to be larger than that for ArN, making this correction would still further increase the theoretical EO(T). As will be seen later though, estimates of this increase are small, less than 2 cal/mole. we believe that the main source of dis- agreement between theory and experiment is due to the effects of anharmonicity on the interatomic potential and therefore also on the lattice dynamics for the system. 30 It is interesting to note that if one mistakenly excludes the terms, -(3/40)(0D/T)2 and (1/2240)(0D/T)4, from Equation 16 and makes the empirical fit to the data with the equation truncated in this way, then the resultant values of EO(T) coincidentally agree remarkably well with the effective two-body theory. It is also interesting to note that excluding the term-P(vc - B)/RT from Equation 16 signifi- cantly alters the behavior of EO(T). Instead of increasing monotoni- cally with T, EO(T) reaches a maximum at T - 75 K and thereafter is a decreasing function of T. 1. Behavior g£_Lattice Vibrational Eneggy The lattice vibrational energy, in the Debye theory, is given by23: Evib - (9/8)R0D + 3RT D(OD/T) , (22) where Y vcy) - (303) I dx 1:3 (ex - 1r1. 0 Values for Evib’ based on this theory and using 0D36 - 88.02 K, are shown in the third column of Table 4. The agreement with the experi- mental results in the second column is quite good. 2. ‘ggmparing_Properties 2;.Ar36 and ArN we compared sublimation—pressure data shown in Table l for Ar36 with that given in Reference 1 for ArN and found the sublimation pres- sure of Ar36 to be consistently higher than that of ArN. This difference in sublimation pressure is small, ranging from about 1.4% at T - 62 K to approximately 0.62 at T - 83 K. In Table 5 we compare P(T) data of Ar36 with that of ArN in the form P36 - PN. For temperatures in the range 62-75 K.we used the PN data given in Reference 1. In the temperature range 75-84 K we used 31 TABLE 5 Comparison of sublimation pressures of Ar36 and ArN. Data for ArN in the temperature range 62.315-74.681 K are taken from Reference 1. For the temperature range 75.557-84.503 K, the unpublished data on ArN of Christen and Opsal37 are used. Temperature Pressure Present work Data of difference Reference 4 T(K) P36-PN (Tarp) 102(P36-PN)/PN 102(P35-PN)/PN 62.315 0.14 1.399 1.466 62.904 0.17 1.473 1.434 64.139 0.22 1.416 1.373 64.887 0.23 1.245 1.323 66.633 0.35 1.285 1.255 66.956 0.33 1.130 1.240 67.927 0.37 1.031 1.207 68.902 0.64 1.460 1.167 70.154 0.71 1.264 1.121 71.471 1.01 1.397 1.078 72.356 0.91 1.069 1.051 73.510 1.20 1.145 1.020 74.681 1.46 1.133 0.983 75.557 1.12 0.749 0.960 76.826 1.69 0.917 0.926 78.167 2.04 0.894 0.895 78.702 2.12 0.856 0.880 79.612 2.26 0.794 0.869 80.403 3.19 0.997 0.840 80.885 2.54 0.739 0.833 81.439 3.19 0.857 0.823 81.440 3.22 0.865 0.823 81.878 3.08 0.777 0.811 82.324 3.84 0.909 0.801 82.668 3.35 0.756 0.803 83.241 4.26 0.888 0.787 83.639 4.15 0.819 0.803 83.760 4.58 0.889 0.736 83.991 3.92 0.740 0.651 84.045 4.18 0.784 0.649 84.503 4.34 0.773 0.638 32 newer, improved, data taken independently on the same apparatus by Christen and Opsal.37 It was found by these workers, in a reexamina- tion of the ArN data of Reference 1, that in the range of 75—84 K the earlier data were about 0.32 (i.e. 0.5 to 1 Torr) too high due to uncontrolled temperature gradients in the Cu block. These temperature gradients were due to excessive heat input into the stainless-steel inlet tube. In the work of Christen and Opsal on ArN and in the present experiment, these temperature gradients and their effects were eliminated. Table 5 shows that P36 is still consistently higher than PN; the improved data increase the difference at 83 K to 0.892. In an attempt to quantitatively explain this difference we fit the ArN data of Reference lto Equation 16 in the same manner as des- cribed for Ar36. We obtained values for the geometric mean frequency, mg, of ArN which when multiplied by the ratio of the square roots of the masses, J40736, averaged 0.5% higher than those shown in the fourth column of Table 3 for Ar36. The significance of this small difference will be discussed below. The values obtained for the static lattice energies of Ar36 and ArN are very close and one cannot say with any certainty just how much they do differ. Part of the difficulty is that the AtN data gave a comparatively irregular EO(T). We did, however, calculate the 36 _ N) N 36 _ 3 EO )ave’ with the result (E0 E0 ave average difference (Eo 1.72 cal/mole. A rough approximation of this difference may be obtained from theory in the following way: The lattice sum calculations men- tioned earlier predict for the EO(T) curve (of ArN) an average slope, AEo/AT - 2.10 cal/mole K, in the temperature range 62.068-82.315 K. For this same temperature range, the Debye theory gives the average 33 N . slope for the Evib(T) curve (of Ar ), AEvib/AT 5.57 cal/mole K. Using these values then, an estimate of the dependence of static lattice energy on vibrational energy is found to be, AEo/AEvib - 0.378. The Debye theory further predicts the average difference in the vibra- tional energies Of Ar36 and ArN for this temperature range to be, 36 40 _ (Evib Evib)ave 3.15 cal/mole. Based on this discussion, we expect then an average difference in static lattice energies, (E036 - E040)ave= 1.19 cal/mole. We believe that this compares favorably with the ob- served value, considering all the approximations we have made. 36 Although our analysis of Ar and ArN data does not yield a conclusive explanation of the observed difference in sublimation pressure, we do believe it to be consistent with the following explanation: The zero-point vibrational energy, E2, of a solid in the harmonic approximation is simply a sum over frequencies, 3N Ez - (1/2) X Hmi. Since the interatomic potentials Of Ar36 and ArN i=1 are the same, the effective force constants which determine the m 36 are i the same. Thus, the zero-point vibrational energy of Ar should be higher than that of ArN by the mass dependent factor, 740/36. This means that Ar36 will be more anharmonic and therefore expand more with increasing temperature. That is, a higher zero-point energy implies a higher static lattice energy which in turn explains the higher sub- limation pressure. Of course, a point is reached where effects of anharmonicity in the interatomic potential become important. Qualitatively, from the shape of the interatomic potential, one expects the effective force constants, which determine the frequency spectrum, to decrease as anharmonicity increases. Our analysis reflects this in two ways. The 34 geometric mean frequency, m8, of Ar36 is a decreasing function Of temperature as shown in Table 3. Secondly, the geometric mean of Ar36 averages about 0.5% less than J40736 times that of ArN, that is [(m2/40736 - w26)/w:6]ave - 0.52. This also is consistent with Equa- tion 16. For example at T - 72 K, the observed difference in sublima- tion pressure, (P36 - P40)/P40 = 12. Assuming the static lattice energies to differ by E36 - Eg - 1.2 cal/mole, and assuming the dif- ference in Debye temperatures, 036 - OE - 4.52 K, the difference in geometric means must, according to Equation 16, compare as: (#7073? ml; - Ugh/035 - 0.3:. 3. Comparison with the Wbrk gf_0thers we compared our Ar36 data with that of Lee, Fuks, and Bigeleisen,4 (LFB) and found our pressures to be lower than theirs for temperatures below 73.5 K and higher for temperatures above 73.5 K. These dif- ferences range from about -0.6 Torr at 63 K to + 2.2 Torr at 83 K. However, LFB's temperature measurements are based on the ArN vapor— pressure data of Flubacher, Leadbetter, and Morrison.34 we therefore compared ArN data from Table 5 with that of Flubacher, Leadbetter, and Mbrrison and found the differences to be similar to those des- cribed above for Ar36. We have found no explanations for these differences but have been able to conclude that the measurements made on ArN and Ar36 in our laboratory are consistent with one another. To make a comparison of our sublimation-pressure data with an anharmonic crystal theory, we used Equation 12 and the Helmoltz free energy as calculated with a (13,6) potential by Klein and co- workers.9’38 Our experimental pressures were higher than those predicted by Equation 12 throughout the temperature range 60-80 K. 35 These differences ranged from 32 at 60 K to 1.6% at 80 K. In order to compare our vapor-pressure ratios with theory and other experiments, we made a least squares fit of the values shown in Table 5 to the equation: AP/PN - c/T2 + d/T , (23) where AP - P36 - PN. Equation 23 is an approximate form of Bigeleisen's39 expression for the natural-logarithm isotope separa- tion factor, a, given by (no - [l - P40(vc - B)/RT] £n(P36/P40). The calculated coefficients are c - 57.87 and d - -0.014. Using these values for c and d in Equation 23 we find that our vapor-pressure ratios are about 42 higher than LFB's in the temperature range 62-83 K. In the fourth column of Table 5, we have included LFB's vapor-pressure ratios which have been interpolated to our temperatures. We also calculated [no and compared our results with the theoretical predictions of Klein g£_§l,, 9 as given on Figure 4 of Reference 9. In comparison with theirs, assuming a (13,6) potential, our values are lower by less than 12 at 60 K, essentially equal at 65 K, and about 3% higher at 80 K. th III. THE THEORETICAL CALCULATION OF KAPITZA RESISTANCE A. The Model The model we propose begins as follows: The copper has a thin de- formed layer21 (i.e., perhaps a region of high dislocation density40141) beneath the surface, which will absorb the evanescent waves generated by a large fraction of the phonons incident from the liquid helium. These are phonons which normally, in the absence of such an absorbing layer, would be totally reflected. For the Cu—He‘ system under con- sideration this corresponds to phonons incident at angles greater than ~6° (i.e., the critical angle of reflection associated with transverse waves in the solid.) Since energy will be therefore dissipated in the deformed layer, there will naturally arise the question: What happens to the dissipated energy? While we cannot answer this completely, we can, however, make a plausible argument. If this energy goes into phonon modes, then the energy may be assumed, for the most part, to remain in the solid. That is, since the probability of reflection from the solid to liquid interface is quite high (2 0.99), the scattering of phonons back into the liquid helium can be neglected. As part of our model we consider the density of helium.near the CurHea interface to increase as the copper surface is approached. In order to calculate this density profile as a function of distance from the interface, we proceed in the manner of Challis g£_al,15 we assume that the van der Waals force between a helium atom.and the copper 36 37 substrate varies as d‘4, where d is the distance from the substrate. The potential energy of a single helium atom can then be written as @(d) - -od‘3. For the constant a we take the value originally given by Schiff,"2 (which was used in Reference 15) a - 5.3 x 10'37 erg cm3. Other estimates of o, for metals, have been made by Sabisky and Anderson43 using the Lifshitz theory of the van der Waals force. For a metal with a plasma frequency, mp = 10 eV, (for Cu, mp = 10 eV) they obtain a - 2.14 x 10"37 erg cm3 for d :_10 2. However, for distances as great as d = 20 X, the change in a is not more than 1%. The poten- tial ¢(d) - -od‘3 can therefore be used with a constant, since in our calculations d never exceeds 15 X. At first it might appear that the large difference in the two values of the potential parameter might be important. As will be seen later, however, the effect on the density profile, with respect to our calculations, is not so important. Treat- ing the helium as a continuum, we associate with a potential gradient, Y0, a corresponding pressure gradient, VP; for which (o/m)Y¢ ' -YP - (24) In Equation 24, p is the density of the helium as a function of dis- tance from the copper substrate and m is the mass of a helium atom. Equation 24 is equivalent to the integral equation P(d) (a/m)d"3 - I dP/D(P) . (25) In order to solve Equation 25 f5imP(d) and hence p(d), we use experi- mental data‘“""’7 which give liquid and solid helium densities as a function of pressure extrapolated to 0 K. Our calculated density jprofile, using a - 5.3 x 10"37 erg cm3, has a discontinuity in 0 at 0 <1 2 8 A which corresponds to the liquid-solid transition. For the potential parameter, a - 2.14 x 10"37 erg cm3, the liquid-solid 38 transition occurs at d = 6 A. Since the zero-point motion of the atoms is so large for helium, it is unlikely that the transition will be so well defined. In order to account for this and at the same time to consider that the potential parameter a may be too large, we have smoothed out the calculated density profile. The smoothed profile which we then use has the analytic form p(d) - puecl + a/xP) . (26) where a - 9.95, b - 1.79, and one is density of bulk liquid helium. The density profile described by Equation 26 gives a value for p(d) corresponding to the density of solid helium at d = 7 2; that is 00 X) - 0.189 gm/cm3. we should mention at this point that in our calculations we consider the effects of changing the density profile in a number of ways. Our first approach is to consider the density as a constant for distances, d, within 7 X of the copper substrate. In this way we first see the effects of including transverse waves in the dense helium region. Next we consider cases in which the density attains a higher value, becoming constant at distances of 6, 5, and 4 X from the sub- strate. Finally we consider the effects of actually changing the shape of the profile in the region where the density is increasing (for example; 7 §_d §_15 X). we should like to point out now that the latter has only a slight effect on our calculated RK values. It is perhaps appropriate at this point to describe a portion of the calculation. In order to calculate the energy flux associated with the propagation of waves through a mediumnwith a continuously varying density, we subdivide that medium into thin layers. The density at the midpoint of each layer is equal to the real density at that point 39 in the medium. We make the layers thin enough so that none ever satisfies the 1/4 wavelength condition for optimum transmission.“8 B. Theory We first consider the propagation of waves through a system of solid and liquid layers. In order to establish the conventions used in our calculations, we present a brief description of the elastic theory of solids and liquids. More mathematical details of the theory are given in Appendices A, B and C. 1. Elastic and Viscoelastic Materials For an isotropic, elastic, solidl'8 the stress-strain relations can be expressed in the simple form T13 - u[(dSi/3xj) + (BSj/3x1)]+ A511(88k/3xk) . (27) In Equation 27, TU is the stress tensor, A and u are the Lame parameters, Si is the displacement field, (l/2)[(BSi/3x ) + (BSj/Bx1)] is the strain 1 tensor, and 011 is the unit tensor. Summation is implied over repeated indices. The equations of motion for the system are given by p(azsi/atz) - aTij/ij , (28) where p is the density of the material. Because isotropy is assumed, a solution of Equation 28 can be decomposed into two fields; a curl-free field plus a divergence-free field. Therefore, we can choose to work 'with a velocity field, 3 (with cartesian components defined by vi-IBSth) *which can be obtained from a scalar potential, O, and a vector potential, 15. These potentials satisfy wave equations 9(329/3t2) - (A + 2n)V2¢ . <29) p(az-Ulatz) - qu-d: . (30) Rfltich have planedwave solutions. For waves propagating in the xz-plane 40 then, these potentials are given explicitly by: + + + + + 0 - [A exp(ik-r) + B exp(-ik-r)]exp(-imt) and w ' (0,0,0) where w - [C exp(iZ-E) + D exp(—i:-;)]exp(-imt). The corresponding total 4 + 1 1 + velocity field is given by, v - vi + vt, where vZ - YO, vt - Y x w, and the subscripts l and t refer to longitudinal and transverse waves, respectively. For the case of a non-viscous fluid, Equations 27, 28 and 29 apply with the shear modulus, 0, set equal to zero. When viscosity is present49 the shear modulus is pure imaginary, u - -imn and Equations 27, 28, 29 and 30 are used. 0 is the coefficient of viscosity of the fluid. Finally when the material is a visco-elastic solid,49 the elastic con- stants A and u are replaced by A - im[§-(2/3)n] and u-imn, respectively, where n and 5 are coefficients of viscosity. Equations 27, 28, 29 and 30 are used for this case also. The cases for which the elastic constants are complex result in plane waves which are attenuated, that is the wave vectors are complex. We therefore conSider the effects of phonon attenuation in our calcula- tions by using complex wave vectors in the absorbing material. For example, for a longitudinal wave vector, k, we then have R - (m/c£)(l + 1V£) , (31) ‘where VZ and cl are, respectively, the loss factor and velocity for longi- tudinal waves. Similarly, for a transverse wave vector, K, we have K - (tn/ct) (1 + in) , (32) where Vt and c are, respectively, the loss factor and velocity for t transverse waves. At this point we should mention that since the com- ;ilex elastic constants are restricted to lie in the 4th quadrant of the complex plane,"9 the loss factors can only have values between 0 and 1 (1.809 0 :- V£’t _<_ 1). 41 Restricting ourselves, once again, to waves propagating in the xz-plane, we consider the boundary conditions49 (based on Equation 28) which are appropriate to the cases encountered in our calculations. For a non-viscous liquid-liquid interface, the conditions are that the normal components of the velocity field and stress tensor be con- tinuous. Since we will always be dealing with plane boundaries in the xy-plane, this requires that sz, T , and v2 be continuous. Continuity 22 of sz is trivially satisfied since u - 0 on both sides of the inter- face. For a liquid-solid interface, we then have the same conditions (continuity of sz, Tzz’ and vz with sz - 0), however, all three must be used, since there are now three unknown wave amplitudes to determine. At a solid-solid interface, we must have continuity of sz, Tzz’ vx, and vz, and there are (in this case) four unknown wave amplitudes to solve for. In Figure 4, we have a diagram of a system of solid and liquid layers which is typical of those used in our calculations. The labeling of the x and z axes indicates the positive directions along those axes. The boundary of the semi-infinite solid is at z - 20 E 0 and the semi- infinite liquid has its boundary at z - 2“. In between the semi- infinite solid and semi-infinite liquid is a system of n layers, some JLiquid and some solid. The subscripted variables, 21, along the bottom (1f the figure, indicate the values of 2 which bound each layer. Also shown are the resulting waves which propagate through the layers when a WNIVE is incident from the liquid at z - zn. The solid layers support both: longitudinal and transverse waves (indicated by k1 and K1, respec- tivefily), while in the liquid layers, there are only longitudinal waves (indicated by k1). The angles shown from the normal (91 and vi) 42 Zn-I X< jFigure 4: A system of solid and liquid layers typical of those used in the calculations. 43 correspond to real angles only when the corresponding wave vectors are real (no attenuation). When attenuation is present, the real angles have to be determined from the real parts of the wave vectors. The dashed line in the middle of the figure along the z-axis is used to imply the presence of the n-2 layers not shown. To calculate the propagation of energy through the system of layers shOwn in Figure 4, we employ the method given in Reference 48. For the solid layers we relate the stress and velocity components of the mth layer to those in the (mrl)st layer by applying the boundary conditions appropriate at solid—solid interfaces. Since there are some differences between our results and those given in Reference 48, (most likely misprints) we show our results in Appendix C of this thesis. For the liquid layers, we use the concept of the input impedance of a layer as discussed in Reference 48. Results of these calculations are also shown in Appendix C. For the entire system of solid and liquid layers, we then calculate the transmitted wave amplitudes in the solid at z - 0 and the reflected wave amplitude in the liquid at z - zn, fior unit amplitude of the incident wave. Knowing these amplitudes we then.calculate the energy transmission coefficient, TR, as a function (If frequency, m, and angle of incidence, 0 E 6n+1. From Equations 27 auui 28, it follows that the cartesian components of the energy flux, + P, at any point in the medium are given by (‘9 P1 = (1/2) Re(-Tijv3) , (33) and therefore the energy transmission coefficient, TR, is given by TR [Re( szvx) + Re( Tzzvz)]/mpn+1Re(kz)n+1 . (34) Our reason for calculating TR at both ends of the system of layers is to provide a check on our calculations. That is, since energy must be 44 conserved, in the absence of attenuation, TR must be the same at both ends of the layered system. In our subsequent calculations, conserva— tion of energy was always satisfied where required. 2. The Heat Flux and Kapitza Resistance To calculate the Kapitza resistance, we proceed along the lines of the acoustic mismatch theories of Khalatnikov12 and of Mazo and 0nsager.14 We quantize the incident energy flux, Pz(1n°), according to Pz(1n°) - (hm/Q)°He case. 0 is the volume, cHe is the sound velocity in bulk liquid helium, and m is the angular frequency of the incident wave. The transmitted flux, P2, is then given by Pz- (hm/0)TR (m,0)cHecose. Assuming a Debye density of states for the incident phonons, we then calculate the total heat flux, W(T), as a function of temperature, T, w U/Z W(T) - (h/4U2cng) I dm m3 n(m,T) I d6 TR(m,e) sine cose . (35) 0 0 In Equation 35 n(m,T) is the Bose-Einstein distribution function, n(m,T) - [exp(fim/kBT) - 1]'1, kB is Boltzmann's constant, and h is Planck's constant divided by 2D. The Kapitza resistance is defined by RK-1 a dW/dT, and using Equation 35 we then obtain “ U/Z RK-l 8 (kB/h)4(h/4"ZCH§)T3 I dx x4ex(ex ‘ 1)-2 I de TR(k Tx/h,0)sin0cose. 0 0 B (36) Equation 36 is the expression we use to calculate RK' In the next section we describe the numerical method used to obtain values for RK as a function of temperature. C. Numerical Method In order to calculate RK using Equation 36, we must evaluate the double integral 1(T) ' I dx x4 (ex - l)-2 g(kBTx/fi) . (37) 0 45 where 11 g(kBTx/h) - I de TR(kBTx/h,0) sine cos6 . (38) 0 Remembering that m - kBTx/h, we first form an array of points, g(m1), over a range of m values suitable to the temperature range for which Rk is to be evaluated. To obtain the points in this array, we use an adaptive numerical integration scheme which varies the increment size, A6, according to the amount of curvature in the integrand. we then evaluate g(kBTx/h) for arbitrary values of x and T by linearly inter- polating between the appropriate points in the array 8(wi)' The same integration scheme is then employed to perform the integral over x in Equation 37 and thereby obtain I(T). For temperatures T 5 1.0 K, we expect better than 12 accuracy in our integrations. Above 1.0 K, the accuracy will decrease to perhaps as much as 52 at T - 2.0 K. The computer program used to perform these integrations is shown in Appendix D. D. Discussion 1. Effects Due £2_Phonon Attenuation In Table 6 we show results of our initial calculations in which only the effects of phonon attenuation in the copper are included, i.e., impedance matching is omitted. A more complete tabulation of these results is given in Table F1. Only a single attenuation variable, V, is shown, since we have taken the loss factors, Vi and Vt, [defined in Equations 31 and 32] as equal. That is, V E Vz - Vt. This is a reasonable choice for the following reason: In making these calculations we found that for a given transverse wave attenuation, Vt’ the calculated R.K values were relatively insensitive to changes in 46 TABLE 6 Kapitza resistance, RK’ in the form RKT3’ as a function of phonon at- tenuation in the copper. Loss Factor Kapitza Resistance x (Temperature)3 . R‘KT3 V mmz K‘WL 0.0 1270 0.001 ~540 0.002 501.9 0.01 401.8 0.02 334.3 0.06 201.4 0.10 145.0 0.12 127.2 0.15 107.6 0.20 85.8 0.25 71.5 0.30 61.4 0.40 48.1 0.50 39.6 0.70 29.7 1.0 22.0 47 the longitudinal wave attenuation, Vz. Also, the relationship be- tween VB and Vt is not yet well understood. Acoustic experiments50 on water-stainless steel interfaces do, however, show VB and Vt to be the same order of magnitude (actually, V : ZVK) for ultrasonic fre- t quencies up to about 100 MHz. The loss factor, V, is further assumed to be independent of frequency and temperature. Under these assumptions, still neglecting impedance matching, RKT3 is independent of temperature. Effects of frequency dependence in V on RK are discussed in Reference 17. For the density and sound velocities in copper we use; pCu - 8.93 gm/cm3, ez - 5.0 x 105 cm/sec, and et - 2.3 x 105 cm/sec. In the bulk liquid helium we use; pHe - 0.145 gm/cm3 and cHe - 2.38 x 104 cm/sec. we can use the results shown on Table 6 to determine the range of values of V which should be used for our subsequent calculations. The smallest values of V that give RK values which agree with Kapitza resistance experiments at low temperatures, lie in the range 0.10-0.12. Anderson, Connolly, and Wheatley51 measured Kapitza resistances and obtained: RKT3 - 134 cm2K4/W at T - 0.06 K and RKT3 - 127 cmZKa/W 52 we at T - 0.08 K. However, from RK values measured by Zinov'eva find; RKT3 = 60 cmZKA/W at T - 0.08 K and from Table 6 this corresponds to a loss factor V = 0.30. At higher temperatures, somewhat higher attenuations are required to fit this theory to experiment. For our subsequent calculations then, we use values of V in the range V - 0.12-0.50. 2. Effects 2; Including Impedance Matching_ we shall now discuss the results when the effects of impedance matching are included in the theory. In Figures 5 and 6 we show the 48 10°_ I I l I [Till]!!! _ d 1041— '7 at P ‘ .1— -- c— *z" a ”T: 1.1st ‘ E10-'2- : .K 1 ‘L ,_ q 8; 4* T: 0.1K E s t ‘ 810's: .1 ... __ I g . 4“ k- F' I 10'“: ., 10-5 1 l I I IIIIIIIIIII 0.0 OS 1.0 1.5 RNGLE 0F INCIDENCE teed} Figure 5: Calculated energy transmission coefficient, TR, at three frequencies as a function of angle of incidence. The fre- quencies are expressed in terms of temperatures defined by . m = 3.7 x 1011T (sec'l). Both longitudinal and transverse waves are included in the first, most dense He layer. 49 10° A I I l I TIIIIIIIIIh r d ,_ V= 0.12 .3 -1 _ 10 _ .1 95 e .4 5.. .2, TI: 1.49m ‘ 310‘2b T: 1.0 i u. c—t ‘L P 8 I- I: 0.1K —I U .3. 7 7 $104: j 2 - as e 10"“: 10.5, I L I Ll L 111 II 0.0 0.5 1.0 1.5 MOLE a" INCIDENCE (rod) Figure 6: TR calculated at three frequencies. Only longitudinal waves are included in the first He layer. 50 calculated energy transmission coefficient, TR, at three frequencies. The frequencies are expressed in terms of temperatures for which m3 n(m,T) is a maximum; i.e., m - 3.7 x 1011T (sec-1). The axis cor- responding to the angle of incidence is expressed in radians and ranges from 0.0-1.6 rad. The point where all the curves in Figure 5 tend toward zero is n/2 rad - 90°. we have made this axis non-linear to bring out the structure in TR at small angles of incidence; that is, the axis is expanded for small angles. For the density profile we use Equation 26 and the parameters, a and b, as given below Equation 26. It is convenient to introduce a variable, F, which defines the maximum - F 0 When the distance from O1 He. the copper surface is such that the helium density is equal to 01, density, 01, of the helium by: the density is taken to be constant from that point to the surface. For the calculated curves in Figures 5 and 6, F - 1.3 and the corre- sponding distance over which the density is constant is 7 A. Since F - 1.3 corresponds to the lowest density for which He4 is a solid, the effects of including transverse waves in the dense helium region should be considered for values of F.: 1.3. For calculating the curves shown in Figure 5 then,transverse waves are included in the solid helium layer. For comparison, the curves shown in Figure 6 are calculated with the solid helium layer treated as a liquid in the sense that transverse waves are excluded. The sound velocity in liquid helium, a linear function of density, is taken from the experimental work Of Abraham and co-workers.53 In the.solid helium we use the sound velocity, c for longitudinal waves, £9 13180 a linear function of density, as measured by Vignos and Fairbank.54 51 We then choose ct - (1/2)c£ as a suitable average for the sound velocity of transverse waves. To illustrate more clearly the structure of the transmission coefficient, TR, as a function of angle of incidence, we show in Figure 7 TR calculated for a small attenuation, V - 0.01. The other parameters used for calculating the curves shown in Figure 7 are the same as those used for Figure 6 (i.e., only longitudinal waves are included in the first, most dense He layer). For the lowest frequency shown (corresponding to T - 0.1 K) the TR curve in Figure 7 is almost identical to the curve one would obtain if the impedance matching effects were neglected. That is, the impedance matching region has very little effect at this lowest frequency, since the wavelength in bulk He cor- responding to T - 0.1 K (A = 400 X) is much longer than the thickness of the impedance matching region (d = 15 A). The first dip in this curve corresponds to the critical angle of incidence associated with longitudinal waves in the Cu; ecrl - sin"1 (cue/oz) - 0.0475 rad. When no absorption is present, longitudinal waves in the Cu, for angles of incidence beyond this critical angle, travel along the interface and are exponentially at- tenuated inward from the interface. This attenuation should not be confused with the attenuation resulting from absorption. When the attenuation of a wave is along the wavefront (that is, perpendicular to the direction of propagation) there is no absorption of energy. Energy absorption in a wave is present only when the wave has some attenuation along its direction of propagation. waves which are attenuated laterally are called evanescent waves and in the absence of absorption carry energy only along the interface. That is, evanescent waves will 52 10° I I I I [IIIIIIIII I 1 IO" " j I -4 95 - a I E " T: 1.49 K j 31073 - T: 1.0' d It _, ._ In. g '- T: 001 ‘ , d 5 " I .. _ :1 1 . . I— P' ._ .q 10 :3 3: F' -1 L ‘4 10-5 1 l I l l L\\ 0.0 0.5 1.0 1.5 RNGLE 0F INCIDENCE (rod) Figure 7: TR calculated at three frequencies. Only longitudinal waves are included in the first He layer. 53 give no contribution to the transmission of energy from the liquid to the solid. This is clear since kz [- (m/c£)cos6£] becomes pure imaginary when sinez [- (czlcne)sin6] is greater than 1 and the energy flux through the interface is proportional to the real part of kz. In- cluding absorption in the solid (as we have done in Equations 31 and 32) results in k2 values which always have a real part; i.e., energy is transmitted through the interface for all angles Of incidence. The second dip in this curve corresponds to the critical angle of incidence associated with transverse waves in the Cu; 6 - sin-1(cHe/ct) - 0.1035 rad. The transverse waves are then also crt evanescent for angles of incidence greater than 6 Just beyond the crt' transverse critical angle is the Rayleigh peak at an angle of incidence called the Rayleigh angle; 0 - 0.110 for this Cu—He4 system. For the R small attenuation used in Figure 7 the peak is quite narrow and well defined. It correSponds to the angle of incidence for which the longi- tudinal and transverse evanescent waves are propagating in exactly the same direction. As V approaches zero the width of the Rayleigh peak approaches zero and its contribution to TR thus goes to zero. As V becomes larger (see Figure 6 for example) the Rayleigh peak broadens and much of the other structure also becomes less pronounced. There 18 an additional small peak for angles of incidence near II/2 (890°). This peak represents the strong coupling of waves at grazing incidence from the liquid to the transverse waves in the solid. For the higher frequencies shown in Figure 7, this peak does not appear for reasons whichwillbe explained later. The effects of including transverse waves in the solid helium layer are qUite evident when the curves in Figures 5 and 6 are compared. 54 Associated with the longitudinal waves in the first, i.e., most dense, He layer there is a critical angle of incidence. In Figure 6 only longitudinal waves are allowed in this layer, and therefore for waves incident from the bulk at angles greater than this critical angle, only evanescent waves exist in the layer. The critical angle is very close to the point where the three curves in Figure 6 appear to intersect. In this supercritical region, the penetration depth for evanescent waves is proportional to the wavelength of the evanescent waves. Therefore, as the frequency increases the transmission coefficient in the supercritical region will decrease. By comparison, in Figure 5, the critical angle for longitudinal waves is not even apparent, since the transverse waves seem to play the dominant role in determining TR. This is not surpris- ing since the critical angle for transverse waves is very near n/2. The considerable enhancement of TR at the frequency corresponding to T - 1.49 K is due solely to the transverse waves in the first helium layer. In Appendix E, we show a more complete set of figures in order to illustrate the frequency dependence of the transmission coefficient TR. The TR curves in Figure El combined with those in Figure 6 span the fre- quency range corresponding to 0.1—3.0 K, for the case of no transverse waves in the He. These TR curves show that the penetration depth of the evanescent longitudinal waves in the first, most dense He layer de- cneases with increasing frequency. For angles of incidence near 0.5 rad, a peak begins to appear in the T - 3.0 K curve of Figure El. This corresponds to a strong coupling of the longitudinal waves in the first He layer with the transverse waves in the Cu. This peak is analogous t0 the peak which occurs near n/Z rad in the T = 0.1 K curve of Figure 6. 55 The curves in Figures E2-E7 show the transmission coefficient, TR, for frequencies corresponding to the temperature range 0.1-4.0 K. As the wavelength is decreased (frequency is increased) we see that the significant enhancement of TR occurs at increasing angles of incidence. This behavior is consistent with a condition for optimizing trans- mission of waves through a 1ayer48; A/4 - d case, where A, d and 6 are respectively, the wavelength, thickness, and angle of transmission for the layer. Applying this condition to the transverse waves in the first He layer yields results which are consistent with the behavior of the TR curves in Figures E2-E7. If one views the transmission of waves through a layer in terms of multiple reflections within the layer, the optimum transmission occurs when the waves undergo con- structive interference. Comparing the T - 3.0 K curve of Figure E6 with that of Figure E1, we can see the results of destructive and constructive interference of both types of waves in the first He layer. As an example, for angles of incidence beyond about 0.5 rad, the T - 3.0 K TR curve of Figure E6 is significantly less than the lower frequency TR curves shown in Figures E5-E6. we can only at- tribute this to important interference effects which must occur be- tween the additional evanescent longitudinal waves which accompany the multiple reflections of transverse waves within the layer. With the exception of the strong peak near 0.5 rad, the T - 3.0 K curve 0f Figure E6 looks very much like the T - 3.0 K curve of Figure E1. The strong peak, of course, is due to constructive interference of the transverse waves in the first He layer; and as the wavelength is decreased, this peak occurs at increasing angles of incidence. F1sures E2-E7 show this dependence quite clearly. 56 When comparing the TR curves of Figures E1 and E6 one must remember that the longitudinal sound velocities in the first He layer, in each case, are not the same. For the calculated TR curves of Figure E1, experimental sound velocity data for liquid He4 are used, while for Figure E6 experimental sound velocity data for solid He4 in the first He layer are used. The critical angle for longitudinal waves in the first He layer will be less when the solid Heb data are used (Figure 4 E6) than when the liquid He data are used (Figure El) since the longitudinal sound velocity in solid He“ is greater than in liquid Heé (for equal densities). In Figures E8-El3 we show calculated TR curves using a small attenuation, V - 0.01, and with all parameters identical to those used for Figures E2-E7. The purpose of these is to enable one to identify the structure of the TR curve as a function of angle of incidence and frequency. For example, in Figure E10 we can see the contributions to TR associated with the two critical angles, ecrl - 0.0475 rad and ecrt = 0.1035 rad, the Rayleigh peak at 6R - 0.110 rad, and the peak associated with the constructive interference of transverse waves in the first He layer (which occurs at increasing angles of incidence 88 the frequency is increased). We show calculated Kapitza resistance values as a function of temperature (in the form RKTB) in Figures 8-17. The parameters used in calculating the lower RKT3 curve in Figure 8 are the same as those used.for Figure 5. Similarly, the upper RKT3 curve in Figure 8 is Calculated using the same parameters as those used for Figure 6. The relatively higher transmission coefficients associated with including transverse waves in the solid He layer, as shown in Figure 5, 57 200 I IIIIIII] ITIIIIIIIIII_ = 1.3 1: .1 A 3 _ '4 s'\ ,3} 501- — i 3 n h- — '— 02‘ r- —-J - 4 10 I LJLlllll I IIIIIIIJLIII 0.01 1.0 2.0 0.1 TEMPERRTURE T(K) Figure 8: RKT3 calculated as a function of temperature T. For the lower curve both longitudinal and transverse waves are included in the first helium layer, while only longitudinal waves in the first helium layer are used to calculate the upper curve. 58 200 I I IIIIII] I II IIIIIIIII_4 v: 0.12 F: 1.3 1 P- -# lOOr- -‘ "v: 0.25 ‘ 5 _ a-\ g: 50- I 0 6‘ r— .— 62‘ r — 10 L 11111111 I JJllLLllllll 0.01 0.1 1.0 2.0 TEMPERRTURE T (K) Figure 9: T3 calculated for two attenuations in the copper. The f rst helium layer is considered to be a solid; i.e., both longitudinal and transverse are used. 59 200 I II HIII] I I IIIIIIIIIII; v: 0.12 F=1.t1 j r- d - -1 100— —‘ T;v= 0.25 " 3 1- 35‘ 50'- fl 0 &r - g... 02‘ r.- r— -—J 10 1 11111111 1 111111111111 0.01 0.1 1.0 2.0 TEMPERRTURE T (K) Figure 10: '1‘3 calculated for two attenuations with a helium density factor F - 1.4. 60 200‘ r I I IIITTI I ITTI1IT]III_ :O,12 F: 1.5 : P d 1— —1 100'—' -— — 4 ”v=o.2s " 3 P- -' ? gt 50— § 0 a?! .— .— 02‘ 1— —-1 10 1 11111111 1 111111111111 0.01 1.0 2.0 0. l TEMPERRTURE T (K) Figure 11: R '1'3 calculated for two attenuations with a helium density f§ctor F - 1.5. 61 I l 1IIIT] I I ITIIIIITIT_ I F: 1.11 : V’ I 1001—- -—.1 1' 1 __ V: 0.25 J 23 . 4 =- «i‘ 501- ‘J E O «U r— .— w -1 - A 10 1 1 1111111 1 L11111111111 0.01 0.1 1.0 2.0 TEMPERRTURE T(K) Figure 12: T3 calculated for two attenuations and a helium density factor F = 1.4. Transverse waves are included only in the first, most dense, helium layer. 62 200 I IIIIIII T FIIITHIIIIS —1 —-1 F: 1.5 : _1 1.- Cl! 1001- — ‘v= 0.25 I I ‘ II- -—1 '5 _. \ a .3.‘ 50- I 3 a b .— 02‘ P 1" — 10 1 11111111 1 111111111111 0.01 0.1 1.0 2.0 TEMPERRTURE T (K) Ffiigure 13: T3 calculated for two attenuations and a helium density factor F - 1.5. Transverse waves are included only in the first, most dense, helium layer. 63 correspond, naturally, to a lower RK‘ The effect of including trans- verse waves in the solid helium layer is seen to be significant over a wide temperature range. Figure 9 shows the effect on RKT3 of using a higher attenuation, V - 0.25, for the case of transverse waves allowed. The effect of this increased attenuation is to lower the RK’ although the overall T dependence is still the same. In Figures 10 and 11 we compare RKT3 values for two different attenuations in each figure but with higher values for the parameter F. In Figure 10, the distance from the copper substrate for which the helium density is constant (91 - 1.4 one) is about 6 2. Similarly, in Figure 11 the distance for which the helium density is constant (pl . 1.5 pHe) is about 5.3 X. we see that the basic shape of the RKT3 curve in either case has not changed by comparison with Figure 9. The RKT3 minimum, for a given value of V, is lower and occurs at a higher temperature when F is increased. we observed this trend to hold, for increasing F, on up to F - 2.0. The physical reason for this trend is that higher values for the density of the first He layer give higher sound velocities and this corresponds to longer characteristic wave- lengths. In Figures 12 and 13 we show calculated RKT3 values using (with one exception) the same parameters, respectively, as for the curves shown in Figures 10 and 11. For the curves shown in Figures 12 and 13 transverse waves are included only in the first, most dense, He layer - 6 K and in Figure 13 for (in Figure 12 for p - 1.4 pHe’ d 1 1 o p - 1.5 9. d1 - 5.3 A) whereas, in Figures 10 and 11, transverse 1 He’ 0 ‘waves are allowed in the dense He region for p 3_l.3 pHe’ d §_7 A. Including transverse waves over a longer distance in the dense He 64 region improves the impedance matching. This can be seen by comparing the RKT3 curves in Figures 10 and 11 to those in Figures 12 and 13, respectively. In order to test how sensitive our results were to the details of the density profile we investigated the effect of changes in the analytic form of p(d) given in Equation 26. We observed only small changes resulted in the calculated values of RKT3. That is, the two important factors in the density profile are the maximum values of the density, and the distance over which the density is constant. In Figures 14, 15, 16 and 17 we show RKT3 values for attenuations in the copper which vary with distance from the capper surface. This is what one expects for a real solid. We also show on these figures some experimental data21’51’52’55 to facilitate comparisons. In Figure 14, V in the Cu has the following simple dependence: V = 0.25 for a layer which extends from the capper surface 20 X into the copper, and V - 0.12 beyond the layer. For Figure 15: V - 0.30 for 20 K, V - 0.12 for the next 250 X, and V . 0.06 beyond the layers. For Figure 16: V - 0.42 for 25 X, V - 0.12 for the next 250 X, and V - 0.06 beyond the layers. Finally, for Figure 17: V - 0.50 for 25 2, V - 0.12 for the next 100 X, and V - 0.06 beyond the layers. The calculated RKT3 values used in Figures 8—17 are also tabu- lated in Appendix F. 65 :KKJ__ I 1’ *1 I 1'] ll] 1 1 1* l l T 11] 1'1 I_‘ b O . F: 103 j 1— —-1 r- -« wm— — _. .1 1-. —1 3 "" —1 ,\ “a; so- I 3 n - y— a!‘ .9 + .e r— . -J 5 x11 0 "‘ I 10 1 11111111 1 111111111111 0.01 0.1 1.0 2.0 TEMPERRTURE T(K) Figure 14: R T3 calculated using an attenuation profile in the copper: VK=OO.25 for a layer which extends from the copper surface 20 A into the copper, and V - 0.12 beyond the layer. The individual points are experimental values; ‘ — Reference 21, + — Reference 51, ca- Reference 52, x - Reference 55. 66 200 I I ITIrIIr I I ITIIHITIIP : I P — C =ls3 Z; 1— —-1 100— fi p- -1 P Id 53 P’ .1 a} gf 50" '— 8 5' r 4 .— 112‘ _ .1 ‘0 .. r- . -1 ‘ 1 o "x‘1 ‘ 10 1 11111111 1 111111111111 0.01 1.0 2.0 o I 1 TEMPERRTURE T (K) Figure 15: RKT3 calculated sing an attenuation profile in the copper: V a 0.30 for 20 , V - 0.12 for the next 250 A, and V = 0.06 beyond the layers. The individual points are experimental values; A - Reference 21, + - Reference 51, °'- Reference 52, x - Reference 55. 67 200: I I IIIIII] I I I ITIITI VIII- E a _ F: 1.3 1 1- -1 r -1 100t—' “ r d 1— -# E; P’ .1 \ ’ gr 501- 0 E 0 0T, .— '— m“ 1 1 +— . “I! 101 L 11111111 1111111111111 0.01 0.1 1.0 2.0 TEHPERRTURE T (K) Figure 16: RKT3 calculated sing an attenuation profile n the copper: V - 0.42 for 25 , V . 0.12 for the next 250 , and V - 0.06 beyond the layers. The individual points are experimental values; A - Reference 21, + - Reference 51, n - Reference 52, x - Reference 55. 68 200 I I IIIIIII I I IFIIIIIIIIL — -J I I P’ F: 1.33 I: 1 I'- -1 F d 100'- - _ .1 r— O -4 3 1— n —-I :I'\ n O a! 501' o - E 0 I": l— 5... \‘ °‘ 1 P 1 . x 10 1 1 1111111 1 111111111111 0.01 1.0 2.0 0.1 TEMPERRTURE T(K) Figure 17: RKT3 calculated sing an attenuation profile 6n the copper: V a 0.50 for 25 , V - 0.12 for the next 100 A, and V 8 0.06 beyond the layers. The individual points are experimental values; A - Reference 21, + - Reference 51, t1 - Reference 52, x - Reference 55. IV. SUMMARY AND CONCLUSIONS A. The Sublimation and Vapor Pressure Experiment We have measured the sublimation and vapor pressure of Ar36 over a wide temperature range in the solid and for temperatures in the liquid up to the boiling point. These measurements were reproducible for two different runs to the extent that it was not possible to distinguish the two sets of data. The effects of impurities are most evident (see Figure 2) at the lowest temperatures for which data were taken. These effects, however, represent the presence of non-condensable impurities in the vapor which at the higher temperatures have only a negligible effect on the ob- served pressures. The presence of condensable impurities (i.e., N2 and 02) have a significant effect only for temperatures in the liquid range, T 2 84 K, and even then the effect on vapor pressure is small; these impurities raise the vapor pressure by less than 0.1% (~ 1 Torr at 84 K). The most important result of our experiment is that our sublima- tion-pressure data are in good agreement with the predictions of the improved self-consistent phonon theory of Goldman, Horton and Klein.9’10 Our Ar36 data further show, when compared with equivalent data on ArN, the sublimation and vapor pressure of Ar36 to be consistently higher than that of ArN. This isotopic effect on sublimation pressure is commonly referred to as the vapor-pressure ratio between Ar36 and Ar”. 69 70 Our vapor-pressure ratios are also in good agreement with the predic- tions of the improved self-consistent theory referred to above. More exact knowledge of the thermodynamic properties of rare-gas solids and of isotopic effects could be determined if other kinds of experimental data were available. Some particularly useful kinds of experiments, for example, would be specific heat and thermal conductivity measurements on normal and isotopically pure rare-gas solids. B. The Calculation of Kapitza Resistance We calculated the Kapitza resistance, RK(T), over a range of temperatures (0.01-2.0 K) for which experimental data are available. In addition to considering the effects of phonon attenuation in the Cu, we have included the impedance matching effects associated with a dense 4 interface. He region of varying density at the Cu-He As we have seen from our initial calculations, neither of these effects, when considered alone, is sufficient to explain the experi- mentally observed RK values. By considering only phonon attenuation effects in the Cu, we can obtain RK values which agree in magnitude with experimental RK values at low temperatures; T s 0.5 K. We can, furthermore, obtain the experimentally observed temperature dependence, at these lower temperatures, by allowing the attenuation to vary with distance from the copper surface. It is not possible, however, to obtain agreement with experiment in the magnitude or T dependence of R.K for T 3 0.5 K without also considering the effects of impedance matching. We believe we have shown the importance of considering impedance matching effects in conjunction with phonon attenuation effects on Kapitza resistance. It is clear that the magnitude and temperature 71 dependent features of experimentally observed Kapitza resistances are included in the results of our model. It is also clear, however, that much more experimental information than is currently available, is needed about the detailed structure of the copper and helium in the vicinity of the interface. APPENDICES APPENDIX A: DECOMPOSITION OF THE DISPLACEMENT FIELD In order to decompose the displacement field into longitudinal and transverse components, we first express Equation 28 in the vector form: + + + p(BZS/Btz) - uv2s + (AI-2p) yy-s . (Al) + Next, we consider the displacement field 8 as composed of two parts: + -> + S - SK + St , (A2) such that + Y x Sz = 0 (A3) and y-Et - o . (A4) _) Equation A3 defines the longitudinal component, 5 and Equation A4 f Using Equations A3, A4 and + defines the transverse component, 8 t O the vector identity + + + y x (y x A) - -V2A + yy-A , (A5) we can express Equation Al in the form: -> + + + p(azsgecZ) + p(stzlatz) - uvzst + (A + 2p) stz . (A6) It is clear then, that solutions of Equation Al can be obtained by solving the equations: . + p(BZSK/Btz) = (A + 2n) v23z (A7) and + -> p(BZSt/atz) = “V23: . (A8) 72 73 It is sometimes more convenient to work with the velocity fields, :2 5 a§z/3c, and 3t 3 BSt/at. This is particularly true when one wishes to calculate the transmission and reflection of energy at a boundary. Using Equations A7 and A8 we obtain the equations for 3, and 3t: p(aZQ/acz) - (A + 21,) v23, , (A9) and p(323t/ac2) - pv2$t . (A10) At this point we define the velocity potentials, ¢ and $,by the equations: Vi = Y¢ . (All) and ..) 3t = v x I» . (A12) Since Y-(Y x E) = 0 and Y x (Y¢) - O, we then obtain, using Equations A5, and A9-A12, the desired wave equations: p(azwatz) - (A + 2n) v2¢ . (A13) and _) .+ p(azw/Btz) = uV2w . (A14) Equations A13 and A14 are, respectively, Equations 29 and 30 of Chapter III. APPENDIX B: THE ENERGY FLUX In this section we shall consider the prOpagation of energy in an anisotropic, Viscoelastic material. That is, we consider the general form for the stress tensor T13 49: Tij - Cijkmgkm_+ nijkmvkm ’ (Bl) where 3km = (1/2)[(BSk/3xm) + (BSm/axk)] (32) and vkm - Bakm/at . (B3) In Equation Bl, Cijkm is the elastic tensor and "ijkm is the viscosity tensor. Equation 32 defines the strain tensor in the same way as it is defined for Equation 27, so that the equations of motion for this system are given by Equation 28: p(BZSi/Btz) - BTij/axj The components of the elastic tensor, Cijkm' have the following sym- (28) metry properties49: Cijkm ‘ Cjikm " Cjimk " Cijmk " Ckmij ' kaij ' kaji " Ckmji - (34) This is known as Voigt symmetry and is a consequence of two assumptions: The strain tensor is symmetric (no local rotations) and the existence of the elastic potential (analogous to the Hooke's law assumption for stretching in one dimension; i.e., assuming the potential is a quadratic function of strain). 74 75 The components of the viscosity tensor, ”ijkm’ also have this Voigt symmetry. The key assumption in this case is that the rate at which energy is absorbed is quadratic in the time derivative of strain vkm. In terms of the velocity field v1 (defined in Appendix A), Equation 28 may be written p(BVJ/at) - BTij/Bxi . (BS) Multiplying Equation BS by v5 and summing over repeated indices gives 8 at[(1/2)pvjvj] - (aTij/3x1)vj . (B6) Using Equation Bl and the symmetry relations expressed in Equation B4, we then obtain the conservation equation: 3 The left side of Equation 37 is the time derivative of the energy density, and the second term on the right side of Equation B7 is the rate at which energy is absorbed. For this reason, the term -T1jvj is taken to be the energy flux. Assuming time dependent solutions (of Equation B5) having the iwt simple harmonic form, e‘ , we find the time averaged energy flux, _). P, with components given by Equation 33: APPENDIX C: THE TRANSMISSION COEFFICIENTS Applying the appropriate boundary conditions to a system of solid layers, the components of stress and velocity in the mth layer (at z - 2m) are related to those in the (m—l)st layer (at z = zmll) by the following set of equations,48 ti”) - a1?) egm-l) (c1) m) where the 5% are defined by, .{w . -153). an» - .gm Egan . -Tm xz ’ and 52m) = vim). The coefficients, ai?), in Equation C1 are given by: a(m) = cosZ P + 2 i 2 11 7m cos m s n Ym cost sf?) = (prm/Km)81nYm cosZymlcost - cost] afg) = i[sin27m sinQm - (kx/kmz)c082ym sian] a(m) = -iw [(l/k )coszz sinP + (l/K )sin22 in ] 14 pm mz Ym m m2 Ym 8 Qm a(m) = (k /w )[cosP - osQ ] 21 x 0m m c m aég) = Zainzym cost + cosZym cost a§§> = (-i/wpm)[(kx2/kmz)sian + sz sian] 3&2) a 1[sin2ym sian — (kx/kmz)c082ym sian] 76 77 fig?) I ikx[(1/sz)c082Ym sin m.” (2kmz/Km?)sian] agg) I -iwpm[4kmz/Km2)sinzym sinPm + (l/sz)c0522ym sian] a5?) I Zsinzym cost + COBZYm cost a5?) I (prm/Km)sinym cosZym[cost - cost] a[(1111) . (—1/wpm)[kmzsian + (klesz)Sian] aé‘z") = ikx[(1/1 tanPj)] . (c3) The phase angle, P1, is defined in the same way as for the solid layers, and Z3 is the characteristic impedance of the jth layer; 23 I ij/ka. Recalling that the waves in the jth layer are given by the scalar potential, ikaz -1kaz i(kxx-wt) (11.1 = [A18 + Bje ]e , we then have the following relations; n iP AM+llAn+l a (pn+1/pm+1) II 1[(z§1n) + Zj)/(Z§1n) + Zj+1)]e ‘1 (C4) jIM+ a (in) _ (in) BM+llAM+1 (2M zn+1)/(ZM. + ZM+1) (C5) R E Bn+1/An+l " (2111“) " Zn+1)/ (2511“) + Zn“) (C6) Equations Cl-C6 enable us to calculate the energy transmission, TR, at the boundary of the semi-infinite solid (2 I 0) and at the boundary of the semi-infinite liquid (2 I zn), At 2 I zn we have TR - 1 - IRI2 (C7) At 2 I 0 the expressions are much more complicated, but we can describe the procedure used. In the semi-infinite solid, the velocity potentials are given by, ikozz 1(kxx-wt) iKozz i(kxx-wt) ¢o ' (Ace )e and $0 I (Coe )e (M) Since the shear stress component, sz , is zero at the liquid—solid interface, we can obtain a relationship between the wave amplitudes, 79 A0 and Co’ by using Equation CZ. Furthermore, using Equations C4 and C5 we can calculate any of the other stress and velocity components at the liquid-solid interface. A0 and CO can both then be determined. Knowing the wave amplitudes, A0 and Co’ we then calculate the energy transmission coefficient at the boundary of the semi—infinite solid according to Equation 34. APPENDIX D: KAPITZA RESISTANCE PROGRAM TABLE D1 List of the FORTRAN program, KR and its subroutines, used to calculate the Kapitza resistance. 100 PROGRAM KR(OUTPUT) COMMON PO,CO,VO,P1,C1,C1T,Vl,VlT,PS,CSL,CST,VSL,VST COMMON/Y/VI,W,N COMMON /z/ THO,DTH COMMON/XY/ERRORl,DTH1,DTH2,LAST,LAST1,LAST2 COMMON /xz/ IFLAG , JFLAG ,KFLAG COMMON/xzv/LFLAG,XMAxv,Nv COMMON/TOCH/A COMMON/TOARRAY/TM,TRW,DTRW,TRWS,DTRWS,NMAX,DELTA,ALPHA COMMON/LAYERS/NINF,NSUP COMMON/TOK/T,DT,NOFT,DX,NT,B,ERRORA,ERRORF,TMIN,TMAX COMMON/DENSE/AX,BX,XMIN,XMAX COMMON/SAVE/PHMAX,QHMAX COMMON/VINS/PIS,V18,XMINS COMMON/VINH/PIH DIMENSION TM(200),TRW(200),DTRW(200),TRWS(200),DTRWS(200) DIMENSION STHO(64),CTHO(64),TR(64),TL(64) FORMAT(5F15.5) PHMAX=50. QHMAX=50. ALPHA=.1 DX=3.83 F=2. G=.16 XMIN=3.6O XMAX=10. BX=ALOG(G)/ALOG(XMIN/XMAX) AXI(F-1.)*(XMIN**BX) T=.4 $ TMIN=.377 $ TMAX=8. $ DT=.1 s NOFT=20 $ DELTA=.05 F=1.4 $ XMIN=6. $ XMAxv=6. $ Nv=1 $ NSUP=16 $ KFLAGIZ ALPHAIALPHA*(XMAX-XMIN+4.)/(XMAX-XMAXV+4.) NINFIXMAX-XMAXV+5 IFLAG=1 JFLAG=1 LFLAG=1 ERRORlI.01 $ ERRORA=.005 $ ERRORF=.01 A=r353825 $ B=7.846685E8 P0=.145 $ CO=23800. P1=F*PO $ P1H=P1 8O 100 101 81 Table D1 (cont3.) C1=-72437.+637606.*P1 $ ClT=.5*Cl IF(KFLAG.EQ.1)C1=-44173.+468509.*P1 PS=8.93 $ CSL=SOOOOO. $ CSTIZBOOOO. VOIO. VIIO. VSL=.12 V1T=V1 $ VST=VSL LAST1=6 LAST2=6 PIS=P1H $ VIS=V1 $ XMINS=XMIN PISIPS $ Cl=CSL $ C1T=CST $ VIS=.25 $ XMINS=20. XMINIO. CALL KAPITZA END SUBROUTINE KAPITZA COMMON PO,C0,VO,P1,C1,C1T,Vl,VlT,PS,CSL,CST,VSL,VST COMMON/XY/ERRORI,DTHl,DTH2,LAST,LAST1,LAST2 COMMON/TOARRAY/TM,TRW,DTRW,TRWS,DTRWS,NMAX,DELTA,ALPHA COMMON/TOK/T,DT,NOFT,DX,NT,B,ERRORA,ERRORF,TMIN,TMAX DIMENSION TM(200),TRW(200),DTRW(200),TRWS(200),DTRWS(200) FORMAT(5E15.S) FORMAT(1IS,5E15.5) THMIASIN(1.) DTHlIASIN(CO/SCL) LASTITHM/DTHI DTHlITHM/LAST DTH2I(THM-LAST1*DTH1)/LAST2 CALL INITIAL CALL ARRAY(TMIN,TMAX) NM=NMAx DO 2 I=1,NM PRINT 101,1,TM(I),TRW(I),DTRW(I),TRWS(I),DTRWS(I) TIT-DT DO 1 I=1,NOFT T=T+DT CALL SUMX(T,DX,ERRORA,ERRORF,FLUXT,FLUXTS) T3=T**3 C2=CO**2 RKI=B*FLUXT/C2 RKT3=1./RKI RKIS=B*FLUXTS/C2 RKTS3=1,/RKIS RKIRKT3/T3 RKSIRKTS3/T3 PRINT 100,T,RK,RKT3 PRINT 100,RKS,RKTS3 CONTINUE RETURN END 82 Table D1 (cont'd.) SUBROUTINE CHOICE(X,T,TRWC,TRWCS) COMMON/TOCH/A COMMON/TOARRAY/TM,TRW,DTRW,TRWS,DTRWS,NMAX,DELTA,ALPHA DIMENSION TM(200),TRW(200),DTRW(200),TRWS(200),DTRWS(200) T1=A*T*X IF(T1.LT.TM(1))GO TO 1 I=1.+ALOC(T1/TM(1))/ALOC(1.+DELTA) IF(I.GE.NMAX)GO TO 3 TRWCITRW(I)+(T1-TM(I))*DTRW(I) TRWCSITRWS(I)+(Tl-TM(I))*DTRWS(I) GO TO 2 3 TRWCITRW(NMAX) TRWCSITRWS(NMAX) GO TO 2 1 Tch=TRw(1) TRWCS=TRWS(1) 2 CONTINUE 100 FORMAT(115,5E15.5) RETURN END SUBROUTINE INITIAL COMMON PO,CO,V0,P1,Cl,ClT,Vl,VlT,PS,CSL,CST,VSL,VST COMMON/X/AKO,AK02,AK12,AKSL2,AKST,AKST2 COMMON/VINS/PIS,VlS,XMINS TYPE COMPLEX AKO,AK02,AK12,AKSL2,AKST,AKST2 v11=v1 v1=v13 AK12=(1.+(0.,1.)*V1)/C1 AK12=Ax12**2 AKO=(1.+(O.,1.)*VO)/CO AK02=AKO**2 AKSL2=(1.+(0.,1.)*VSL)/CSL AKSL2=AK5L2**2 AKST=(1.+(O.,1.)*VST)/CST AKSTZIAKST**2 v1=v11 RETURN END SUBROUTINE CONST(M,STHO,CTHO,TR,TL) COMMON PO,CO,VO,P1,C1,CIT,v1,v1T,PS,CSL,CST,VSL,VST COMMON/X/AKO,AK02,AK12,AKSL2,AKST,AKST2 COMMON/Y/VI,W,N COMMON /z/ THO,DTH COMMON /xz/ IFLAG,JFLAG,KFLAG COMMON/DENSE/AX,BX,XMIN,XMAX COMMON/VECTOR/AKSLZ,AKSTZ,AKX,AKX2,AKIZ COMMON/TRICF/52CST,C2CST COMMON/SAVE/PHMAX,QHMAX 100 16 ll 12 18 19 TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE TYPE COMPLEX COMPLEX COMPLEX COMPLEX COMPLEX COMPLEX COMPLEX COMPLEX COMPLEX 83 Table D1 (cont'd.) AKO,AK02,AK12,AKSL2,AKST,AKST2 AKX,AKX2,AKOZ,AKlZ,AKSLZ,AKSTZ SGST,SGST2,SZGST,SZGST2,CGST,CZGST,CZGST2 20,21,ZSL,ZST,2N,PH,TANPH AKI,AKIZ,ZI,R EXPH,AL,SINPH,COSPH R2,AT,V22,FLUXSL,FLUXST,FLUXI szz U,V,ATAL,ZN1,W4 DIMENSION 0(4),V(4) DIMENSION S'I'HO(M) ,CTHO (M) ,TR(M) ,TL(M) FORMAT(5E15.5) DO 1 K=1,M THOITHO+DTH STHO(K)=SIN(THO) CTHO(K)=COS(THO) AKXIAKO*STHO(K) AKX2=AKX**2 AKOZICSQRT(AK02-AKX2) AKIZICSQRT(AK12-AKX2) AKSLZICSQRT(AKSL2-AKX2) IF(AKSLZ.EQ.(O.,O.))GO TO 3 AKSTZICSQRT(AKST2-AKX2) SGST=AKX/AKST CGSTIAKSTZ/AKST SZGST=2.*SGST*CGST SGST2=SGST**2 C2GST=1.-2.*SGST2 CZCST2=CZCST**2 SZGST2=l.-CZGST2 ZOIPO/AKOZ RZO=REAL(ZO) GO TO (16,17),KFLAG CONTINUE ZSLIPS/AKSLZ IF(AKSTZ.EQ.(O.,O.))GO TO 11 ZST=PS/AKSTZ ZNIZSL*CZGST2+ZST*SZGST2 GO To 12 ZNIZSL CONTINUE RZN+REAL(ZN) To To (6,7),IFLAG CONTINUE IF(AKIZ.EQ.(O.,O.))GO TO 18 ZlIPl/AKlZ GO TO 19 Zl=(1.E50,1.E50) CONTINUE) ALI2.*ZSL/(ZN+ZI) ZI=ZI PH=10.*XMIN*W*AKIZ IF(AIMAG(PH).GT.PHMAX)PH=(1.,O.)*REAL(PH)+(O.,1.)*PHMAX 13 14 17 20 21 22 23 24 84 Table D1 (cont'd.) IF(AIMAC(PH).GT.PHMAX)PH=(1.,0.)*REAL(PH)+(O.,l.)*PHMAX COSPHICCOS(PH) IF(COSPH.EQ.(0.,O.))GO TO 13 TANPHICSIN(PH)/COSPH ZNIZl*(ZN-(O.,1.)*Z1*TANPH)’(Zl—(O.,1.)*ZN*TANPH) GO TO 14 ZNIZI*ZI/ZN CONTINUE GO TO 20 XMIN1=XMIN CALL SOLIDUS(ZN1,ATAL,U,V,W4) ZNIZNI ZI=ZN CONTINUE DO 2 LI1,N DI=(XMAX+4.-XMIN)/N D=DI*(2.*L-1.)/2. PI=P0*(1.+AX/((D+XMIN)**BX)) CII-44l73.+468509.*PI GO TO (9,10),JFLAG VI=V1*(PI/PO-l.) CONTINUE AKI=(1.-(0.,1.)*VI)*(CI**2) AKIIl./AKI AKIZICSQRT(AKI-AKX2) IF(AKIZ.EQ.(O.,O.))GO TO 21 z1=21 ZIIPI/AKIZ GO TO 22 z1=21 ZI=(1.E50,1.ESO) CONTINUE IF(KFLAG.EQ.2.AND.L.EQ.1)GO TO 23 EXPHICEXP((O.,1.)*PH) AL=AL*EXPH*(ZN+ZI)/(ZN+ZI) GO TO 24 AL=(1.,0.) P2=PI R2=(ZN-ZI)/(ZN+ZI) AKZZIAKIZ CONTINUE PH=10.*W*AKIZ*DI IF(AIMAG(PH).GT.PHMAX)PH=(1.,0.)*REAL(PH)+(0.,1.)*PHMAX COSPH=CCOS(PH) IF(COSPH.EQ.(O.,O.))GO TO 15 TANPH=CSIN(PH)/COSPH ZNIZI*(ZN-(0.,l.)*ZI*TANPH)/(ZI-(0.,1.)*ZN*TANPH) GO TO 2 15 ZN=ZI*ZI/ZN CONTINUE EXPH=CEXP((O.,1.)*PH) ALIAL*EXPH*(ZN+ZI)/(ZN+ZO) 7 5 25 27 26 85 Table D1 (cont'd.) GO TO 4 CONTINUE ALI2.*ZSL/(ZN+ZO) GO TO 4 R=(1.,o.) AL=(O.,0.) GO TO 5 CONTINUE R=(ZN-ZO)/(ZN+ZO) ALIAL*P0/PS CONTINUE IF(KFLAG.EQ.2)GO TO 25 TR(K)=1.-R*CONJG(R) ALIAL*AKSLZ/AKOZ TL(K)=AL*CONJG(AL)*RZN/RZO GO TO 26 TR(K)=1.-R*CONJG(R) XMIN=XMIN1 IF(W4.EQ.(O.,0.))GO T0 27 AL=AL*Ps/P2 v22=(O.,1.)*AX22*AL*(1.-R2) ALIVZZ/W4 AT=AL*ATAL FLUXSL=(U(1)*AL+V(l)*AT)*CONJG(U(4)*AL+V(4)*AT) FLUXSTI(U(3)*AL+V(3)*AT)*CONJG(U(2)*AL+V(2)*AT) FLUXI=P0*CONJG(AKOZ) TL(K)=REAL(FLUXSL+FLUXST)/REAL(FLUXI) GO TO 26 TL(K)=TR(K) CONTINUE CONTINUE RETURN END SUBROUTINE SOLIDUS(ZN1,ATAL,U,V,W4) COMMON PO,CO,VO,P1,Cl,ClT,V1,VlT,PS,CSL,CST,VSL,VST COMMON/X/AKO,AK02,AK12,AKSL2,AKST,AKST2 COMMON/Y/VI,W,N COMMON/VECTOR/AKSLZ,AKSTZ,AKX,AKX2,AKlZ COMMON/TRIGF/SZGST,CZGST COMMON/DENSE/AX,BX,XMIN,XMAX COMMON/XZV/LFLAG,XMAXV,NV COMMON/SAVE/PHMAX,QHMAX COMMON/VINs/PIS,VIS,XMINS TYPE COMPLEX AKO,AK02,AK12,AKSL2,AKST,AKST2 TYPE COMPLEX AKSLZ,AKSTZ,AKX,AKX2,AK12 TYPE COMPLEX SZGST,CZGST,SGI,CGl,SGlZ,SZGl,SZGlZ,C2G1,CZGlZ TYPE COMPLEX 2N1,ATAL,ANUM,ADEN,AK1T,AKlTZ,AKITZ TYPE COMPLEX QH,PH,SINQH,COSQH,SINPH,COSPH,SINQHB,SINPHA TYPE COMPLEX A,U,V,W1,W4 DIMENSION A(4,4),U(4),V(4) U! 13 86 Table D1 (cont'd.) P11=P1 s P1=Pls $ V1T1IV1T s v1T=v13 $ XMIN1=XMIN $ XMIN=XMINS AK1T=(1.+(O.,l.)*V1T)/ClT AK1T2IAK1T**2 AKlTZICSQRT(AK1T2-AKX2) SGlIAKX/AKIT CGlIAKlTZ/AKlT SGlZISGl**2 CZGl=1.-2.*SG12 SZGl=2.*SGI*CGl c2c12=czc1**2 SZGlZ=1.-C2612 PH=10.*W*AKlZ*XMIN IF(AIMAG(PH).GT.PHMAX)PH=(1.,0.)*REAL(PH)+(O.,l.)*PHMAX SINPHICSIN(PH) COSPH=CCOS(PR) IF(PH.EQ.(O.,O.))GO TO 5 SINPHAISINPH/AKIZ GO TO 6 SINPHA=10.*W*XMIN*(1.,0.) CONTINUE QH=10.*W*AK1TZ*XMIN IF(AIMAG(QH).GT.QHMAX)QH=(1.,0.)*REAL(QH)+(O.,1.)*QHMAX SINQHICSIN(QH) COSQHICCOS(QH) IF(QH.EQ.(O.,O.))GO TO 7 SlNQHB=SINQH/AK1TZ GO TO 8 SINQHB=10.*W*XMIN*(1.,0.) CONTINUE A(1,1)=C2Gl*COSPH+2.*SGlZ*COSQH A(l,2)=2.*Pl*SGl*CZGl*(COSPH-COSQH)/AK1T A(1,3)=(0.,1.)*(SZGI*SINQH-AKX*CZGI*SINPHA) A(l,4)=-(0.,1.)*P1*(CZGl2*SINPHA+SZGIZ*SINQHB) A(2,1)=AKX*(COSPH—COSQH)/P1 A(2,2)=2.*SGlZ*COSPH+CZGl*COSQH A(2,3)=-(0.,1.)*(AKX2*SINPHA+AK1TZ*SINQH)/P1 A(2,4)=(0.,l.)*(S261*8INQH-AKX*CZGI*SINPHA) A(3,l)=(0.,l.)*AKX*(CZGl*SINQHB-2.*AKlZ*SINPH/AK1T2) A(3,2)=-(0.,1.)*Pl*(4.*AKlZ*SG12*SINPH/AK1T2+CZGI2*SINQHB) A(3,3)=2.*SG12*COSPH+CZGl*COSQH A(3,4)=2.*P1*SGl*CZGl*(COSPH;COSQH)/AK1T A(4,1)=-(O.,1.)*(AKIZ*SINPH+AKX2*SINQHB)/Pl A(4,2)=(0.,1.)*AKX*(CZGI*SINQHB-2.*AKIZ*SINPH/AK1T2) A(4,3)=AKX*(COSPH-COSQH)/P1 A(4,4)IC261*COSPH+2,*SGI2*COSQH IF(NV.EQ.O)GO TO 13 POVIPO v1v=v1 CALL VISCUS(POV,V1V,A) XMIN1=XMAXV CONTINUE U(1)=(0.,1.)*PS*CZGST 87 Table Dl (cont'd.) U(2)I(O.,1.)*AKX U(3)=2.*(o.,1.)*PS*AKSLz*AKX/AXST2 U(4)=(O.,l.)*AKSLZ V(1)=(0.,l.)*PS*SZGST V(2)I-(O.,1.)*AKSTZ V(3)=-(O.,l.)*PS*C2GST V(4)I(O.,1.)*AKX ANUM-(o.,0.) ADEN=(0.,O.) DO 1 I=1,4 ANUM=ANUM+A(3,I)*U(I) 1 ADEN=ADEN+A(3,I)V(I) IF(ADEN.EQ.(O.,0.))GO TO 9 ATAL=-ANUM/ADEN w1=(0.,o.) w4=(o.,o.) DO 2 I=1,4 W1=W1+A(1,I)*(U(I)+V(I)*ATAL) 2 w4=W4+A(4,I)*(U(I)+V(I)*ATAL) GO To 10 9 CONTINUE w1=(0.,o.) W4I(0.,O.) DO 3 I=l,4 W1IW1+A(1,I)*V(I) 3 W4=W4+A(4,I)*V(I) 10 IF(W4.EQ.(O.,O.))GO TO 11 2N1=w1lwa GO TO 12 11 ZNl=(l.E50,1.E50) 12 CONTINUE P1=P11 $ v1T=v1T1 $ XMIN=XMIN1 RETURN END SUBROUTINE VISCUS(Pov,v1v,A) COMMON/Y/VI,W,N COMMON/XZV/LFLAG,XMAXV,NV COMMON/VECTOR/AKSLZZ,AKSTZZ,AKX,AKX2,AKIZZ COMMON/DENSE/AX,EX,XMIN,XMAX COMMON/SAVE/PHMAX,QHMAX COMMON/VINS/PlS,VIS,XMINS COMMON/VINH/PIH TYPE COMPLEX AKX,AKX2 TYPE COMPLEX AKSLZZ,AKSTZZ,AKIZZ TYPE COMPLEX AK12,AK1T,AK1T2,AKlz,AKlTZ TYPE COMPLEX SGl,CGl,CGlZ,SZGl,SZGlZ,CZGl,C2G12 TYPE COMPLEX QH,PH,SINQH,COSQH,SINPH,COSPH,SINQHB,SINPHA TYPE COMPLEX A,B,C DIMENSION A(4,4),B(4,4),C(4,4) 100 FORMAT (115) 101 15 16 17 8 88 Table D1 (cont'd.) FORMAT(5E15.5) DO 2 L=1,Nv DO 1 I=1,4 DO 1 J=1,4 B(I,J)IA(I,J) D1I(XMAXV-XMIN)/NV D=D1*(2.*L-1.)/2. P1IPOV*(1.+AX/((D+XMIN)**BX)) IF(PIS.NE.P1H)P1IP1H V1IV1V*(P1/POV-1.) ClI-72437.+637606.*P1 IF(LFLAG.EQ.2)ClI-44173.+468509.*P1 AK12=(1.-(0.,1.)*V1)*(Cl**2) AK12=l./AK12 GO TO (3,4),LFLAG C1T=.5*Cl v1T=v1 AK1T2I(1.-(O.,1.)*V1T)*(C1T**2) AKlT2=1./AK1T2 AK1T=CSQRT(AK1T2) GO TO 5 C1T=C1*(SQRT(1.5*V1)+.0001) AK1T=(1.+(O.,1.))/C1T AK1T2=AK1T**2 CONTINUE AKIZICSQRT(AK12-AKX2) AKITZ=CSQRT- - 8 . ‘ 3m‘; 3 E - 1 .3. I ‘ um‘: : C I I- d Ir‘: : F d 5 1 I 1 1 1 1 1111 1111 ‘VLJ ms Lo LI mUlmlm (0“! 1°. .. I I I r [FIUIIIIIIS r "0.13 .1 10": I: E I- -1 a . - §m4 : Q -1 8W" 1 10‘ :1 I 1 1 1 1 1 1 1111 1111 "1m as no LI Figure E1: m 9' 1mm (rd! 10' I I I I I I I II'IIIFE .1 _. V8 0.12 If': : I. .. a I- .1 s .1 — 8w4_ j u- d 5 h- -1 I- -1 5 flw‘: : p- .1 .- .1 no": 11 p- d I- 4 P Id 1 111 1 1 11111 ”I. as no 15 “I" “In “"1 1°. .. I I I I T I IITITIHE - 10" g :1 3 '- -I 8 " " §IO"‘- I: 3.0: : 5 . g C1 am‘ i ‘ .1 .1 m‘ : d d 1‘ 1 1 1 1 l1 1 11111 ‘rE, OJ L0 L5 on: U 11!:qu (rd) TR calculated at frequencies corresponding to the tempera- tures; 1.6, 1.75, 2.0 and 3.0 K. Only longitudinal waves are included in the first He layer. 96 ‘0. r- j I W T 1 V'rr'l'I'r lv 1- ' I 1 1 I "I‘li'I‘i '- F F- b _. 11:01: +- v: 0.12 |°"E 10" 1: fl - B p §10" ; Em" - 5 , g . E F 9 ' mo" : 31¢. I E P .- b p g - I- II I- IO“ : j 10‘" ; )— II b I- '4 P 1- < I- 4 1 1 1 1 l 1L11l1111|1 1 ‘° 6.6 o. 1.0 1.: ”"110 0.6 1.0 1.6 m 0' 11mm (full “I V min (rd) '0... I T I I [FTUUIIIHP 10': I I I I [IIIIIII )- -1 I- 1 Icon: 1 _ 1:042 - 10" g; 1 10" : :1 b d - d h d h d l U l i 1 IT? j 1T1 ‘1 I 1'" 33. "i ”18310 MICE]? II 1 L I 11 “1310 MIC!” II d P 1 I II I- - I 10“ F “'4 I - 1- p- I- 1' 1 1 r l l l l l l l l llll l ”"110 0.5 1.0 1.: “’"M 0.5 1.0 1.: out ur mum 16.11 out I! 11::qu In") Figure E2: TR calculated at frequencies corresponding to the tempera— tures; 0.1, 1.0, 1.25 and 1.4 K. 97 3 :- T IIII f1” "3. III! I: I.” I 5 II" T mans": MICKEY I! .- a TIIWT 1 o. m G 1mm (ad) ,3 10. I I I I ITTFI'IHII 4 V8 0.12 d 1111 l 3 II. II" I “1310 MIC]!!! TI 1 7171 T TI” 1 j ‘o-C L L 0.0 0.! 1.0 LC on: 0’ 1mm ltd) ’1 El ”1810 MICKEY I! i 3 5 mans". summon TI 10" ”nu. I I I EIIIIIIIIII ”0.12 I: b p P E h b h 1 1 1 ”40.0 o. 10 1.6 m U [film (Pd) '0. h- I I p 1. h p p b p I. I b- p- I: r- h h : h '04 J 0.0 1.0 15 0.8 mu G 1mm (null) Figure E3: TR calculated at frequencies corresponding to the tempera- tures; 1.43, 1.44, 1.45, and 1.46 K. 98 i MISS!“ WEIGHT 1'! 10.. 1 1 1 1 1111111111 0.0 0.5 (.0 15 m0 (mlm (M1 10. I. T I I I 1 I III‘IIIIt )- 1— d )- V=0Jz .1 10" 1: l1 '- -I e I- -1 E - ~ Q1041 :1 E .4 5 -1 5.0-'1': «I g 1- 10": h P 1 1 1 00 0.5 10 1‘ on: 0 11am (rod) 10' momma MICE!" Tl ITT b I III’ I TIII I I )— )— 10“ : 1 1 1 1J111111111 ”-11.0 0.5 1.0 1.5 m 0' (Elm (rd) 10' i 3. 1 ”19!“ our (C107 1! 10" 1 1 1 111 11111111 0.5 ma1m1mu-‘1 Figure E4: TR calculated at frequencies corresponding to the tempera- tures; 1.47, 1.48, 1.49, and 1.5 K. 99 10' 10. I I I ffIfiI'IIII )— r d 10" 10": 21 e e Z 3 E164; E1114; 1 " 5 ’ I- 1— 8 " 3 b 5104: in“: .. __ .- 1. g I- I F- I- 1- 4 10‘E |0“: .1 b q 1- 1- .1 1- r- - 1 1 1 11111111111 1 1 1 11111111111“ ”In 05 L0 L: ”In 05 L0 LS MW "cull: (M1 In“? It!“ (Pd) 10' 10. I I I I 1 IIIIIIIIIP .. ”0.12 .. 'r' IT': : .- -1 e g .- .1 §Ir’: §Io+ I- 1- 5 ~ 5 ~ g " 9. 1' Im": : 1110*: 1 ” ‘ 1 ’ r- - 1- 10": :1 10": I- II p- )- II 1- 1- d 1— 1 1 1 11111111111“ 1 1 11111111111 'VEE L0 L6 '"1m 13 15 0.5 0.5 on: N 11:11:10: (roll “I G 1mm (rd) Figure E5: TR calculated at frequencies corresponding to the tempera- tures; 1.55, 1.6, 1.75, and 2.0 K. 10' (I I ‘1'. ‘1. ”(SIM WICIM I! I "i T T I T IIIIIIIIIP V8 0.12 ‘ 1111 J 1111 111111 1 100 P p- b b b 10'. L L l 0.0 0.5 1.0 I 5 m 0' (film (M1 10. I— r I I I I I IIIIIUIII p P d 1‘ V8 0.12 .. -' d In I; I e C . '8 t ‘ ~ 9.10" - = x : d 5 1' I ~ £104 1; : b ‘ - p ‘ P d 10" ; 2. b- d b- on r- d .04 1 1 1 111 1 1 111 0.0 ‘00 '0 Figure E6: 0.5 m (I' 1mm (rd) 10' I I I I IIII IIII I 1 1 1 b d ,_ V80.“ .1 10" t = P .4 a - . a - - §m4- ~ .1 P d 5 _ . II I ‘ 8W*: 3 - P E b 4 P II! 10‘: : h cl F. d .1 J 1 1 1 l 1 11111 'rio ms L0 m0 ltlm (rd) '0' I I I I IIII IIII : 1 1 II _ Ci 1- W0.” a -1 to : i h ‘1 E i- -1 a 1- -1 on 2104 :1 I- -1 - ‘ h d .5. um‘: : u P d l b a 1- -I 10‘: 2 F d h d 1- l 1 1 1 1 1 1 1 11 1111 'vOCO ‘00 '0 0.3 m 0' 11mm (ad) Iv TR calculated at frequencies corresponding to the tempera- tures; 2.25, 2.5, 2.75, and 3.0 K. 101 ‘0. .. I I I I IIIIIIIIIIE '0. 1. I I I I IIflIIIIII P P 1. v: 0.12 .4 +- v: 0.12 .. IO" : : IO‘l : : I- d 1- -4 a I- .1 a F- “ 8 ~ - 3 _ QIO" __ : QIo-I 3 g 1- -I [g —4 1- -* n 5 " d 5 .4 131°“ : : filo" : g r. —I g -4 P -' p— —4 10“ _-_ : 10" j I I 4 cl 1- - '04 1 1 1 1 11 1 1 [1111 '04 1 1 1 1 l 1 1 1 111111 0.0 0.: 1.0 L5 0.0 0.5 1.0 I. m 0’ It!” (Pd) ME W "Clm (rod) 10° I I I I IIIIIIIIIIh 10° I I I I] IT1IIIHI #— ‘1 1— ,_ 1 v: 0.12 d __ v: 0.12 _ -l ,- 4 E " E — -I - 1- § ‘ 5 P a 210°? 3 310-1 I. I: 1.0 I: .. t .. _ I 8 -1 g _ .1 U .5. i 5 r a 310" : 8H0" : I ’0‘ :1 10" : :1 q p u d F- _‘ .1 ,_ q .043 0 l l 1 1 olstl 111611115 '04 L L 1 1 l 1 1 111111 - . ~ - 0.0 0.5 LO I. NE W "£1ch ‘9“) MI W "Clm (Pd) Figure E7: TR calculated at frequencies corresponding to the tempera— tures 3.25, 3.5, 3.75, and 4.0 K. 102 10' 10. I I —Tj_[ IIIIIIIII P P - V3 0.0' III N“ Ir'; 1 I. d a g I- 4 p- -I § § 310-. 2104 I: 1.0 I :- E E ‘1. -4 h d r- r- -1 § , 5 am': an”: a .— _ ... _ :1 g b g b .1 F 1- .1 10" : It)4 1; .1 1— b q r- I- —1 P- i- 10" 1 4 1 1 11 1 1111111 .04 1 1 1 1 1111111111 0.0 0.8 1.0 1.3 0.0 0.5 1.0 1.5 m 0 11mm (rd) nut U mum (I'd) 10° I I I I IIIIIIIII 1°. 1- l T I I IITTIIHW b- i- 1— V: 030' -1 p v: 000' d - -I 10' F: :1 W I p- A —1 95 1- 4 fl -4 .— « g - 2104‘ :1 210'. 1 E -1 E 4 d d E I ‘ .9. ‘ 18de : £111" :1 .4 -< 2 p d E -1 v- I- -4 I- '0‘. L' :1 10" ;-_ 1- -4 p- 1- 1- '04 1 1 1 11 111111111 10" _L 0.0 0 S 1.0 1.5 0.0 5 1.0 1.5 mazarmumuxumfi Figure E8: TR calculated for a small attenuation, V - 0.01, at frequencies corresponding to the temperatures; 0.], 1.0, 1.25, and 1.4 K. 103 '0. .- T I T T [FUTIIIIUU 1’: U I I 1 [IIIIIIIII I . t E , hOJl . . hom1 . 10" - 3 10" I; : : c1 1- d fi .- .1 E .. .1 -‘ -1 § § .1 210":1= 1.113‘ 21 8'04 :1 E : : 5 I d a 1— d a P o- .1 on 0‘. p ‘ gw4: : £1 : : : «:4 E I- d r Cl 1- IO“ 1: :1 10‘ f; l- -1 1- r h 1111111 1 ”"0 l l L l of? 1.0 1.: ”11.0 0.5 1.0 1.: ' m It 1mm 1rd) m 0' 11:11:11: (rd) '0. .. T I T I T I l UVI‘IUE '0. p- I I I 1 [I I IIIUIII : " P ‘1 _ wom1 . . nom1 1 Ir' : Ir'; 1 a n- q E -‘ E I- -4 E « a 1 ~ - awdzhnun gw4EuLut 3 E : E . . 5 F g - . - mo" : '10“ 1'. u- _ c— I- 1- 1- W‘ I 10“ : b D I- h- r- 1 1 L P 1 1 1 l 111 11 'Vfim L0 L8 "3. ms L0 L8 Figure E9: 0.! on: 0' 1mm (ad) on: 0’ 11:11:11: (rd) TR calculated for a small attenuation, V = 0.01, at frequencies corresponding to the temperatures; 1.43, 1.44, 1.45, and 1.46 K. 104 10' >— I I I I 1 I ITTIVHIh 10' 1. I I I I IIIIIIIIIIt 1- 1- womn . t “0&1 4 10°I :1 IO" : : -1 1— -1 E -4 I I— -I a '1 § " d Em“ : am‘ : E . 5 1 B " g o- - d c- P- _‘ 1— -4 g P -n 1- -1 1- 1 10" z 2 10" I: :1 r- P P 1' )- 4 1 1 1 1 1 1 1 1111111 ‘04 1 1 4 1 11 11111111 '°om 0. L0 L5 do ms I& 15 an: N 1$1m (Pd) ”1 0' [film 1"“) W. T I I I [ITTIIIIII 10' 1. i If I I IIIIIIIII 1— r- '- -‘ p v: 000' p '= 000‘ .4 10" 10" t : E g I- -1 3 8 ~ I: 1.5 1 u Q11)" 810'. : : P- .1 g a *' .1 £5. 15. 0,, _ 1 1 '- q 5”” 5 : : 1- .1 I- -1 “ ,. 10'. 1o __ :1 1- 1 1 1 '040.0 0.5 1.0 '0‘ '0100 005 ’00 1.5 m U [film 1rd) “1 W [film (HI) Figure E10: TR calculated for a small attenuation, V - 0.01, at frequencies corresponding to the temperatures; 1.47, 1.48, 1.49, and 1.5 K. 105 10' I IIIIIIIIIII '0. L— V3 000‘ 10" 10" 5 s p. .5 § , £3.10" : 810". : E ’ 5 " 1' 1' 8 1' 5 " u- -. ’- ~ h 210 __ £111" E P F 10" '3 IO‘ : I. 1— 1- 1- 1- 1- 4 L 1 1 1 1 1 11111111 1 m 0.0 o. 1.0 1.5 ”11.0 . 1 o 1.5 m 0 [film (MI ”I W 1‘1”: (M1 10' W"- I I I IIIIIIII r- c4 ._ V3 0.01 ., 10" In" : q E 5 - .— 5 E q 2:10" E104 _ a 5 ‘ mo" 81"" : on ~ 1- ”" 10*E '04 1 1 1 1111111111 10* 1 1 1 1111111111 0.0 0.5 1.0 1.5 0.0 1.0 1.5 m: N 11mm 1M1 0.5 m 0' 1mm 1rd) Figure Ell: TR calculated for a small attenuation, V = 0.01, at frequencies corresponding to the temperatures: 1.55, 1.6, 1.75, and 2.0 K. 106 10' I I fI I III IIII 10. I I I I ITII II" : I I 1' L: ' ' 1- 4 I— -1 t v- 11.01 .. ,. v: 0.01 .1 -1 -I 10 a 10 a s I ; ~ 6 " d 5 ‘ E ‘ é ‘ Q 4 am" 1 am" 1 CI! 4 — 4 -1 1o" 1 10" : c1 -1 d d ‘ d '04 1 1 1 1 1111111 ‘04 1 1 1 1°51 1111(1’1 0.0 0.5 1.0 ‘05 0'0 O ‘ 0 NE 0' "mm (Rd) W W 1‘1“” 11*.” 1°. .. I I I I [IIII]IIII 10' .. I I I I 1 IIIIIIIII - d h p- V: 000‘ .4 - V: 000' .4 IO" :3 1 IO" 1 I- d -I a 1" J I q E P “ E 4 E10" : Em" : 1— d I- d E » « 5 .. - 6 ’- 4 E I- q u w" c : a 104 : : n r- -1 — F- .1 P d F d 10" ; :1 10“ : :1 P d b d 1- 4 1- d r- -1 1- - .04 1 1 1 1 111111111 10" 1 1 4 1111 1 11111111 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1. mt W "am If“) an: 0’ [film 11") Figure E12: TR calculated for a small attenuation, V - 0.01, at frequencies corresponding to the temperatures; 2.25, 2.5, 2.75, and 3.0 K. 107 10' 1. I I I I IIIIIIIIII 10' I I I I I IIIIIIIII : - ‘ .. v- 0.01 .. v: 0.01 .1 Ir' : 10' : q -1 l 1 fl -1 B 1 a 1 En" : Q10" : E ' E ‘ 4 d 9 ‘ g - 810-. :1 gnu-3 : E d g -4 10" : 10‘ d d A q .1 1 1 1 1 1 11111111 4 111111 1'"11.0 0.] 1.0 1.1 '° 0.0 0.5 1.0 1. m W 11:11!!! 10") 10° I I I I 1 I I IIIIIII 10' 1. I I I I I I IIIIIIII b r- i- '1 ,_ v: 0.01 .J 1. v: 0.01 .1 10" j 10" 3 d --1 E d E -1 E -< '5' a 7:310" :1 E10" 3 t -‘ E a: g ‘ 9. I 3110" :1 £10" 3 g a g 2 .- 4 d 10" j 10'. 1 d d .1 q 1- -1 d 10.. 1 111111111 10" 1 1111111 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1. m G Imlm 1"“) m U l‘lm 1M1 Figure E13: TR calculated for a small attenuation, V - 0.01, at frequencies corresponding to the temperatures; 3.25, 3.5, 3.75, and 4.0 K. APPENDIX F: TABULATION OF CALCULATED RKT3 VALUES TABLE F1 Kapitza resistance, RK, in the form RKT3, as a function of phonon at- tenuation in the capper. Loss Factor Kapitza Resistance x (Temperature)3 RKT3 V (cm2 K4 /W) 0.0 1270 0.001 540 0.002 501.9 0.003 480.6 0.004 465.2 0.005 452.1 0.006 440.7 0.007 430.4 0.008 420.6 0.009 411.6 0.01 401.8 0.02 334.3 0.03 287.2 0.04 250.8 0.05 223.4 0.06 201.4 0.07 183.2 0.08 168.3 0.09 155.8 0.10 145.0 0.11 135.5 0.12 127.2 0.13 119.9 0.14 113.4 0.15 107.6 0.16 102.4 0.17 97.6 0.18 93.3 0.19 89.4 0.20 85.8 0.21 82.5 0.22 79.4 0.23 76.6 0.24 74.0 108 109 Table F1 (cont'd.) Loss Factor Kapitza Resistance x (Temperature)3 R103 (c1112 K4 I W) < 71.5 69.2 67.1 65.1 63.2 61.4 48.1 39.6 33.8 29.7 26.5 24.0 22.0 o. omeO‘MbWNNNNN CWQNO‘UI HOOOOOOOOOOOO 110 TABLE F2 Calculated RKT3 values used in Figure 8. T(K) VDQN6M§UNH NHHHI—‘HHHHHHOOOOOOOOOOOOOOOOOO owmfia‘m-wat-‘OOmNO‘Uwar-‘OOOOOOOOO RKT3 (cm2 KAIW) 128.2 129.8 131.3 132.8 134.2 135.6 136.8 138.0 139.0 140.0 143.5 136.4 122.3 107.4 95.4 87.3 82.2 79.1 77.2 76.1 75.6 75.4 75.5 75.7 76.0 76.3 76.7 77.0 77.3 Lgpper curve1_¥glower curve) 127.6 128.0 128.4 128.8 129.2 129.5 129.7 129.9 130.0 130.1 126.0 113.2 94.8 77.3 63.6 51.7 46.9 45.5 45.3 45.7 46.5 47.7 49.0 50.2 51.5 52.7 53.8 54.8 56.0 111 TABLE F3 Calculated RKT3 values used in Figures 9-11. T(K) RKT3 (cm: K4/W) Figure 9 Figure 10 Figure 11 V-0.12 VI0.25 V-0.12 VI0.25 V-0.12 V-O.25 0.01 127.6 71.7 127.6 71.8 127.6 71.8 0.02 128.0 72.0 128.2 72.1 128.3 72.2 0.03 128.4 72.3 128.8 73.5 129.0 72.7 0.04 128.8 72.6 129.4 72.9 129.7 73.1 0.05 129.2 72.9 130.0 73.4 130.4 73.6 0.06 129.5 73.2 130.5 73.8 131.0 74.0 0.07 129.7 73.4 131.0 74.1 131.6 74.5 0.08 129.9 73.6 131.4 74.5 132.1 74.9 0.09 130.0 73.8 131.8 74.8 132.6 75.3 0.1 130.1 73.9 132.1 75.1 133.1 75.7 0.2 126.0 72.4 131.8 75.9 134.9 ‘ 77.7 0.3 113.2 65.6 123.7 72.4 130.7 76.4 0.4 94.8 55.1 108.2 64.3 120.3 71.4 0.5 77.3 44.8 89.1 53.9 104.8 63.3 0.6 63.6 37.2 71.0 43.5 86.7 53.5 0.7 51.7 30.9 57.9 35.8 70.9 44.4 0.8 46.9 28.4 49.7 30.9 59.2 37.4 0.9 45.5 27.6 44.8 27.8 51.1 32.5 1.0 45.3 27.5 41.7 25.9 45.3 29.0 1.1 45.7 27.7 40.1 24.8 42.0 26.9 1.2 46.5 28.4 39.2 24.3 39.6 25.1 1.3 47.7 29.2 38.9 24.1 37.9 24.0 1.4 49.0 30.1 38.8 24.0 36.9 23.4 1.5 50.2 31.0 39.2 24.3 36.2 22.9 1.6 51.5 31.9 40.0 24.5 35.9 22.7 1.7 52.7 33.0 41.2 25.3 35.7 22.7 1.8 53.8 33.7 41.7 26.0 35.7 22.7 1.9 54.8 34.4 42.6 26.6 36.0 23.0 2.0 56.0 35.2 43.5 27.2 36.5 23.2 112 TABLE F4 Calculated RKT3 values used in Figures 12 and 13. T(K) 11113 (cm2 x4 m) Figure 12 Figure 13 V-0.12 V-0.25 V-0.12 v-0.25 0.01 127.7 71.8 127.8 71.8 0.02 128.4 72.3 128.7 72.4 0.03 129.2 72.8 129.6 73.1 0.04 129.9 73.3 130.5 73.7 0.05 130.7 73.8 131.4 74.3 0.06 131.3 74.3 132.3 74.9 0.07 132.9 74.7 133.1 75.5 0.08 132.5 75.2 133.8 76.1 0.09 133.0 75.6 134.5 76.6 0.10 133.5 75.9 135.2 77.1 0.2 134.1 77.5 138.5 80.4 0.3 127.5 74.9 135.4 80.0 0.4 114.3 68.0 126.3 76.0 0.5 96.6 58.5 113.3 69.2 0.6 78.7 48.2 98.8 61.2 0.7 64.3 39.9 84.7 52.5 0.8 54.3 33.9 72.6 45.2 0.9 47.7 29.8 62.8 39.1 1.0 43.5 27.1 55.3 34.3 1.1 40.7 25.3 49.5 30.7 1.2 39.1 24.3 45.2 27.9 1.3 38.2 23.7 42.2 26.2 1.4 37.7 23.4 39.8 24.6 1.5 37.7 23.4 38.1 23.6 1.6 37.8 23.6 36.9 22.9 1.7 38.3 23.9 36.1 22.4 1.8 38.9 24.3 35.7 22.1 1.9 39.5 24.7 35.4 22.0 2.0 40.3 25.2 35.3 22.0 113 TABLE F5 Calculated RKT3 values for Figures 14-17. T(K) RKT3 (cm? Kfilw) gfiigyre 14 Figure 15 Figure 16 Figure 17 0.01 126.2 178.2 175.3 186.0 0.02 125.3 166.0 161.2 174.5 0.03 124.4 154.8 148.3 163.3 0.04 123.5 145.1 137.1 152.8 0.05 122.5 137.1 127.6 143.1 0.06 121.5 130.5 119.8 134.2 0.07 120.5 125.1 113.3 126.2 0.08 119.4 120.6 107.8 118.9 0.09 118.3 116.8 103.1 112.4 0.1 117.1 113.5 99.0 106.5 0.2 102.4 95.3 76.2 71.7 0.3 84.7 78.6 59.9 53.5 0.4 66.9 60.6 45.1 39.8 0.5 52.2 46.4 34.2 30.0 0.6 41.3 36.3 26.6 23.3 0.7 32.9 28.8 21.0 18.2 0.8 29.2 25.4 18.7 16.1 0.9 27.9 24.2 17.8 15.3 1.0 27.4 23.8 17.5 15.1 1.1 27.4 23.7 17.4 15.0 1.2 27.7 23.9 17.6 15.2 1.3 28.2 24.3 17.9 15.5 1.4 28.8 24.8 18.3 15.8 1.5 29.4 25.3 18.7 16.1 1.6 30.0 25.8 19.1 16.4 1.7 30.7 26.3 19.5 16.8 1.8 31.2 26.8 19.9 17.1 1.9 31.7 27.2 20.2 17.4 2.0 32.4 27.7 20.6 17.8 LIST OF REFERENCES 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. LIST OF REFERENCES C.W. Leming and G.L. Pollack, Phys. Rev. BE, 3323 (1970). C.W. 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