.. € -5 A
IL K 0490!. it ..I ...»r\‘b‘0»
»\r ‘3‘; .3....oV"Ym mIK.v¢ofll.l3vle s I'D?
. Ovvz UN \4.|.4.v00$n00. . I... . .“l‘txl.ih‘lnl.lv
.9 l n x. I .. .'|&. v
. . Hum . with... My? !.fcI..J km). I. h 6‘ in!“
c . .A :4 "13;..- .u. t.“ 0
It 'I l D 6an VIII 1!: (Irv
0?. a“: U .x u.
u ..

     

 

. v1 a’!!!i|ll.l[n|.£{dl
. .IIAIIIJ‘}. .4».
it!“ .II‘. I.’
(It. ‘ . .r... '9‘.
v . X >1....JI;. Kl . .4
. ‘8’. .A.»O.v1t
. . nu, . 1".VD..’II4'A-uu
.‘ V V gilttfil
$11.; t £4. ».H
:I 1"
\cn _!A {Info-la It If.
...... . \u. y ([3 I4 . .’.|.|.llnu
\n 111‘"... .s i . ~ ~ . h vfltlu‘lnyll .
o 2. {p . . . ‘ 4
94:43,... use b.” , a .s t 12), its). 3|...

 

sl:¥.‘2§§.§~th§l ,
3151. - v|1l 31V .\‘\ If,
I)! 1.:

   

     

3:31:03

31% was
A .l

g?
ah

  

             

  

v . v ‘ . n y c.
I"! 4 .lulhllvot‘. 3
Lanii‘xivftutiollt. .1. I) .0 afioll . -.l I
‘l)~ilv?1-qull\'1uti.lllr 1 .v a
\)v(1vl.1\\)..1\s \\.A‘:IV‘:II\(0I£|£V" §.|.|11l1 .8
. {VS 3.; :3. $.lgll‘} 1:1 .
Pt“is\|\(.v‘»uug 2’1 , .
valavaviavsl. I: ~ ..

 

 

          

   

‘ . . .1 .
.y .7333... ;
11.1.3.1. . , .11.! Pianr‘
. ‘32]; . \‘ulliix‘huvuruézfi . .zx.3..xzfif..:I-aif ill \. gk
. . 113.313.1115}: .5 titan} 3. .313. 1!. .. 1 ‘ é?
{$.13,§§1§u113§£ 1...... it‘l} «
‘ I. I
it: . ,

 

.11.).

     

f iii?! A

      

(ails t aliquifzdizlii. ‘

u I .1.le
étif if; \KIIHI

D“ £‘.‘fr‘(l§.t fulfil-7... Dix .gf‘tlr’gff I‘EI‘.‘

' { Ill‘ ££»£§v‘§~§ u

u... l .) lillr villi;

 

 

THESlS

 

This is to certify that the

thesis entitled

Thermal Conductivity of Solid Argon

presented by

Juan Javier Bautista

has been accepted towards fulfillment
of the requirements for

Ph.D. _ Physics
degree 1n

 

 

g Major professor

A ril 4, 1980
Date p

 

0-7 639

THERMAL CONDUCTIVITY OF SOLID ARGON
BY

Juan Javier Bautista

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

 

Department of Physics

1980

 

 

 

ABSTRACT
THERMAL CONDUCTIVITY OF SOLID ARGON
BY

Juan Javier Bautista

An apparatus was designed and constructed to measure the
thermal conductivity of rapidly cooled samples of solid argon
grown from the melt. A linear heat flow method was used to
obtain the in situ measurements while the samples were under
their own equilibrium vapor pressure in the temperature range
of 2.2 K to 83 K. The measurements obtained indicated the
presence of a thermal contact resistance which was quantita-
tively taken into account.

The corrected data are in reasonable agreement with the
constant and equilibrium volume data of previous experimen-
ters. In addition, the present data are of sufficient

density to compare to previous theoretical calculations.

 

 

To Simona, Carlye and Geraldo

 

ii

   

ACKNOWLEDGMENTS

I would like to express my extreme gratitude and
appreciation to Professor G. L. Pollack for his unyielding
support and encouragement throughout the course of this
research and my graduate career at Michigan State University.
I would also like to thank Professors J. Kovacs, S. D. Mahanti
and T. Kaplan for their support and especially Professor
J. Cowen who provided suggestions and guidance at crucial
moments.

I am also greatly indebted to my wife, Simona, for her
patience, love and support throughout our stay at Michigan

State University. I would also like to thank her for her

 

dedication and persistence toward the typing and completion
of this manuscript. I would also like to thank Rob Godbey
for his assistance in preparing some of the figures.
Finally, I would like to thank the Michigan State
University Physics Department, the Department of Energy
(formerly E. R. D. A.) and the Ford Foundation for con—

tributing to the financial support of this work.

 

TABLE OF CONTENTS

Chapter Page
I. INTRODUCTION....... .......... ................ l

A. Discovery and General Properties of the
Rare—Gas Solids...... ........... ........ l
B. Thermal Conductivity of an Insulator.... 2

C. Thermal Conductivity of Solid Argon-

Experimental Background ................. 11

D. Purpose. ................................ 25

II. THE EXPERIMENT ............................... 27
A. Cryogenic Apparatus ..................... 27

1. Description of Apparatus............ 27

2. Temperature Control ................. 33

3. Temperature Measurement ............. 40

B. Sample Preparation ...................... 45

1. Sample Growth ....................... 45

2. Sample Manipulation ................. 52

3. Thermal Conductance Measurements.... 57

4. Thermal Conductivity Measurements... 60

III. EXPERIMENTAL RESULTS ......................... 63
A. Results of the Force Experiments ........ 64

1. Measured Thermal Conductance versus
Force ............................... 64

iv

 

 

 

Chapter Page

2. Remarks on the Observed Effects of

the Applied Force on the Measured

Thermal Conductance ....... .......... 65
3. Force Experiment Errors..... ..... ... 73

B. Results of the Thermal Conductance Meas-
urements........... ..... ..... ......... .. 74

1. Thermal Resistance versus Sample

Length............. ................ . 75
2. Calculation of the Thermal Conduc-
tivity ....... .. ...... ... ........... . 75
3. Remarks on the Thermal Conductivity
Measurements ...... ...... ........... . 90
4. Thermal Conductivity Errors......... 94
IV. DISCUSSION AND CONCLUSION ................... . 99

 

 

LIST OF APPENDICES

 

Appendix Page

A. Effective Thermal Conductance at Constant
Force .............. .. ....... ... ......... .... 109
B. Effective Thermal Conductance Data .......... 111
C. Defect Scattering — Classical Analogs ....... 120
D. Error Associated with the Magnitude of AT... 122
LIST OF REFERENCES .......................... 124

vi

 

LIST OF TABLES

Table Page
1. Summary of Sample Preparation............ ..... 64

2. Effective Thermal Conductance at Constant
Temperature.... ....... ................ ........ 66

3. Effective Thermal Conductance at Constant

Force............... ..................... ..... 109
4. Effective Thermal Conductance Data. ...... ..... 111
5. Smoothed KEff Values..... ..................... 79

6. Calculated Thermal Conductivity and Thermal
Contact Conductance ....... ........ ............ 81

7. Experimental High Temperature Thermal Conduc—
tivity Data.... ................. ....... ....... 88

8. Thermal Conductivity Data of Short and Long
Samples ........ . ........... . ............. . ... 92

 

 

Figure

l.

2.

10.

LIST OF FIGURES

A normal process and an Umklapp process in a
one dimensional crystal of lattice constant a

Typical behavior of thermal conductivity for
a finite crystal with no defects and a crys-
tal with dislocations ........ . ...............

A cross section of the thermal conductivity
apparatus .............. .... ...... . ...........

A block diagram of the Artronix temperature
controller ...... . ........................... .

A schematic diagram of thermometer and lower
Al block heater circuits... ........ ... .......

A plot of the effective thermal resistance of
sample 6 as a function of the reciprocal of
the applied force ............................

A plot of the effective thermal resistance of
sample 7 as a function of the reciprocal of
the applied force ................... . ........

A plot of the effective thermal conductance
for different methods of making mechanical
contact..... ....... ........ ....... . ..... .....

A plot of the effective thermal conductance
versus sample length for several isotherms
between 2.25 and 5.00 K ......................

A plot of the effective thermal conductance

versus sample length for several isotherms
between 8 and 26 K.... .......................

viii

Page

29

35

43

68

69

72

76

77

 

 

Figure Page

11. A plot of the thermal conductivity versus

temperature. Included for comparison are
the data of Clayton and Batchelder and
Krupskii and Manzhelii ........... . .......... 82

12. A plot of the effective thermal conductance
versus temperature for samples of different
1engths........ ........................ . ..... 84

13. A plot of the thermal contact conductance
versus temperature. .......................... 85

14. A plot of the effective thermal conductivity
versus temperature for samples of different
lengths.. .............. . ........ ... .......... 86

15. A plot of the high temperature thermal con—
ductivity data and the three—phonon thermal
conductivity calculated by Christen and
Pollack... ................................... 87

16. A plot of the calculated thermal conductivity
versus temperature for short and long samples 93

17. A plot of the effective thermal conductance
for two samples of approximately the same
length displaying different values of RC0nt"

18. A plot of the low temperature (solid curves)
and high temperature (dashed curve) theoreti-
cal calculations of Christen and Pollack.
Included for comparison are the present data. 101

96

 

19. A plot of the thermal conductivity data of
several workers for Ar under its own
equilibrium vapor pressure....... ...... . ..... 104

20. A plot of the thermal conductivity data of
Christen and Pollack from runs 8 and 10. In—
cluded for comparison is a plot of the
"effective thermal conductivity" of the pre-
sent data, sample 12 run 1 ................... 105

ix

 

Figure Page

21. A plot of the thermal conductivity versus
temperature of the present data. Included
for comparison are the data of Christen and
Pollack, runs 7,8 and 10 ......... ............ 107

 

 

I. INTRODUCTION

A. Discovery and General Properties
of the Rare—Gas Solids

Historically, the first of the rare-gases to be identi-
fied was argon (Ar). In 1785 Henry Cavendish was the first
to observe argon while attempting to identify the constitu-
ents of atmospheric air.1 However, it wasn't until approxi-
mately a century later that the "lazy one", argon, was
identified as a new element by Lord Rayleigh and Sir William
Ramsay. Subsequently, Ramsay within a short span of four
years went on to discover the rest of the elements (He, Ne,
Xe and Kr) comprising the eighth column of the periodic
table with the exception of radon (Rn) which was discovered
two years later by Ernest Rutherford.2

Since the introduction of quantum physics the scientific
community has devoted considerable attention to the problem
of understanding the properties of the rare-gas solids.
Although the bulk of the work has been devoted to lattice
dynamics,3 work has also been carried out over a wider range.
For example, there have been recent studies of the inter-

’5 and

action of gaseous xenon with biological materials,4
also recently, solid xenon has been converted from an

insulator to a conductor by the application of very high

 

2

pressures.6 Of the solid state properties of rare-gas solids
(RGS) investigated to date the relatively least understood is
thermal conductivity.7

Although the rare-gas solids are easily exploited as
theoretical models, they are less amenable to experimental
handling. The rare-gas solids are well suited for theoretical
thermal conductivity studies since they form simple close
packed structures containing one atom per unit cell and are,
ordinarily, electrical insulators due to their closed shell
electronic structure.8 Hence, when thermal conductivity is
to be investigated it is only necessary to consider the heat
transported by lattice vibrational waves. The complicating
effects due to the heat transport of conduction electrons
present in metals and semiconductors or due to optic modes
occurring in solids with more than one atom per unit cell can
safely be ignored. Furthermore, the intermolecular forces
are essentially central in nature and the lattice dynamics
are relatively well understood.7

B. Thermal Conductivity of an Insulator
The specific question that currently arises is the

following: What can one learn by investigating the thermal

 

conductivity of an insulator? The answer is that since heat
is transported by atomic vibrations in insulators one can
gain valuable information about their nature. The thermal
conductivity for these materials is entirely determined by
the way in which lattice vibrational waves (phonons) interact

with one another and how they interact with defects,

3

impurities and the sample boundaries.9

For the purposes of calculation the total lattice poten—
tial may be expanded in a Taylor series in terms of the
displacements of the atoms from their equilibrium sites.
For small oscillations the harmonic term is taken to repre-
sent the ideal crystal, while the higher order terms are
then treated as perturbations.lO The energy of these lattice
vibrations or phonons is quantized with one phonon having
energy E = hm and crystal momentum E = ha (where w is the
phonon frequency and q is the phonon wave vector). The
average number of phonons having this energy is given by the
Planck distribution

mm) = l/(exp(hw/kBT)-l) (l)

where kB = Boltzmannconstant.

Hence one can View a finite insulating solid as a rigid-
walled box containing a phonon gas. By heating one end of

the box we locally increase the number of phonons having

 

energy hm. These will then tend to diffuse toward the cooler
end and a net flow of energy will result. A useful expression
for the thermal conductivity of an insulator is given by

K = (l/3)CVU£ (2)
where Cv is the specific heat per unit volume at constant
volume, v the velocity of sound and 2 the phonon mean free
path.ll It should be pointed out that although Equation (2)
also gives the thermal conductivity for an ideal classical

gaS, the number of phonons (or particles) does not remain

constant as in the case for a classical gas.

—

4

Although Equation (2) is not accurate in general, it can
be used to gain valid physical insights. Since the specific
heat CV is a well known function of temperature and the
velocity of sound U is not strongly temperature dependent,
it is then only necessary to define the mean free path 2.

In general R is a very complicated function of temperature
and frequency. However, in order to gain a good qualitative
understanding it is sufficient to consider the mean free path
characteristic of the dominant scattering mechanism.

For temperatures above the Debye temperature (0), Cv

is constant and the mean free path is proportional to the

inverse of the temperature (T_l). Thus K is proportional

to T_l. Since phonons cannot interact (scatter from one
another) in the harmonic approximation this behavior is
entirely explained by the anharmonic terms in the Taylor
series which couples the phonons. That is, a phonon may now
combine with a second to give a third or it may break up to

give two phonons.

Although it may be obvious that £«T_l (since lal/n

 

and n+kBT/hm as T+W), not all of the phonon—phonon inter—
actions contribute to the thermal resistivity. Only the so
called Umklapp processes contribute directly to the thermal
resistivity (l/K) while the normal processes only redistri—
bute the phonon energies. Although normal processes do not
contribute directly to the thermal resistivity, they do
contribute indirectly by keeping those states occupied which

scatter by Umklapp processes.

 

5

This can be understood by considering the conservation
rules for combining two phonons to produce a third inside a
crystal. These are the energy conservation

fiwl+hw2 = hw3 (3)
and wave vector conservation given by

+ —) —>

ql+q2 _ q3ifi . (4)
Here R is a reciprocal lattice vector of the crystal. When

R = 0 in Equation (4) the scattering processes are called

normal processes and when R # 0 Umklapp processes.12

To simplify the argument we will consider the inter-
action of two phonons in a one dimensional solid of lattice
spacing a. Figure 1(a) illustrates two phonons combining to
yield a third that is within the first Brillouin zone.
Normal processes do not change the direction of energy flow,
hence cannot directly contribute to the thermal resistivity.
In Figure 1(b) the phonons combine to yield a third that
lies outside the first Brillouin zone. This phonon is
equivalent to one lying within the first zone but pointing
in the opposite direction. For this case there has been a
reversal of energy flow, hence only Umklapp processes
contribute directly to the thermal resistance.

As the temperature is lowered toward the Debye
temperature the number of phonons present that can partici—
pate in Umklapp scattering decreases exponentially, i.e.
naexp—(O/bT). Thus the thermal conductivity will increase
exponentially as the temperature is lowered further.7

For a perfect infinite solid the mean free path would

 

 

 

 

 

 

 

(a) (b)
I ' ' t
1 C -K ‘
I I I < I 4'
‘ a I q ,q
l q]. qz I l l 2
————+»———r——+
I | I
0 q3 q3 '
I ‘ ’ | | <.-. '
l I I
l | 1 J
- n/a 0 n/a —n/a 0 n/a
93 = ql + q2 q3 = q1 + q2 - K, K = 2n/a
Figure l. (a) A normal process and (b) an Umklapp process

in a one dimensional crystal of lattice constant a.

 

an(T)

 

 

 

 

Figure 2. Typical behavior of thermal conductivity for (a)
a finite crystal with no defects and (b) a crystal
with dislocations.

 

 

7
continue to rise exponentially without bound as the tempera-

ture is lowered significantly past the Debye temperature.
However for a finite but otherwise perfect solid the mean
free path will become constant and of the order of the
dimensions of the solid. That is, phonons propagate without
scattering to the surface of the solid where they are
absorbed and reemitted. Thus at very low temperatures the
thermal conductivity will be proportional to the specific
heat, i.e. KmT3 for a perfect but finite solid.

For defected solids the mean free path is determined
by the type of defects and their concentration. The kinds
of defects that concern us are point defects (vacancies,
impurities, isotopic impurities and interstitials) and line
defects (dislocations).

At low temperatures the mean free path due to phonon
scattering from imperfections is not intrinsically temperature
dependent. It is strongly dependent on the phonon frequency
w. Hence in order to establish the temperature dependence

of the thermal conductivity associated with a particular

 

type of defect a slightly more complex expression than
Equation (2) is needed.

It can be shown that in the Debye approximation (m=Uq)
for a cubic and isotropic crystal the thermal conductivity
is given by

2 339/114 -2
K(T) = kBu/Zn (kB/hu) T g x expx(expx—l) £(q(x))dx (5)

where O = Debye temperature and x = hm/kBT, a dimensionless

8
variable. In the special case where only one type of defect
is present in the crystal £(q(x)) can be adequately repre-
sented by a power law of the form
£(q(x)) = Aq'n = A(KBT/hu)‘nx'“ . (6)
If we now take the low temperature limit (T<<O) we note that
equation (5) yields that

_ 3-n
K(T) — Bn T (7)

where Bn = kBu/2n2(kB/‘hu)3-nAfnx4expx(expx—l)_2dx.l3
0

Although quantum mechanical perturbations methods are
required to obtain the exact expressions for £(q), we will
for the present rely on useful classical analogs that are
reasonable approximations for phonons of long wavelengths.
As we shall see this is a valid assumption at low tempera—
tures.

Defects in a solid tend to distort (or strain) the
lattice within their vicinity. This defective region will
in general have slightly different physical characteristics
(e.g. density, compressibility, etc.) than the surrounding
region and hence scatter incoming phonons.l4

A phonon scattering from a point defect is analogous
to the scattering of a plane sound wave from a fixed solid
sphere. Rayleigh showed that in the long wavelength the
limit, i.e. >\>>Vl/3 where V = volume of the sphere, the
intensity of the scattered sound wave is proportional to
q4 (see Appendix C).15 Phonons of wavelengths greater than

the dimensions of the defect (a few lattice spacings) will

 

9
then display a mean free path that is inversely proportional

to q4 (since £(q)¢l/O(q)). Thus at low temperatures a solid
with only point defects present will have KmT-l.

The scattering of phonons from static dislocations is
characterized by two main features. These are the core of
the dislocation consisting of a narrow region along its
axis and the surrounding strain field (whose radial exten-
sion is of the order of several wavelengths). The strain
of the crystal about the dislocation varies as b/r where
b = Burgers vector of the dislocation and r = radial
distance from the core.

To incident phonons the core can be regarded as a long
and narrow cylindrical obstruction with cross sectional
area A = flaz. Rayleigh also showed that in the long wave-

1/2

length limit (A>>A ) the scattering cross—section per

unit length of dislocation varies as a(a/)\)3 (see Appendix
C).15 3

Thus 2core is proportional to q .

For the surrounding strain field the Rayleigh theory is
not applicable, since lgAl/Z. The effect of the surrounding
strain field can be estimated by considering an analogy with
geometrical optics. As a phonon passes through the strained
region the phonon's velocity will be altered due to the
anharmonicity in real crystals. The phonon velocity in this
region is given by

u = u0(ltyb/r) (8)
where U0 = velocity in the unstrained region, and Y is the

Grueneisen constant (a measure of the anharmonicity). To

 

10

first order for small scattering angles the incident phonon
will be deviated from its original direction by an angle
¢~yb/p where p is approximately the closest distance that
the phonon will approach the dislocation core. For a small
scattering angle ¢ the scattering cross-section per unit
length of dislocation will be proportional to YZbZ/pO where
pO is the smallest allowed value for p. Our geometrical
optics analogy breaks down unless p031. Hence, the

scattering cross—section is proportional to q and thus

2 a —1 16,17
strain q ‘
The ratio of the mean free paths is thus Q . /2
strain core
. . 3 2 2 .
which varies as a(a/A) /(Y b /l). Since Y_l'£strain/2core

thus varies as (a4/y2b2)/12=(a2/b)2/A2. In the low tempera-
ture region the wavelength of a typical phonon is much
greater than both the dislocation core radius a and the
<2

Burgers vector b, hence £ Thus a defective

<
strain core’

solid dominated by dislocations will display a thermal
conductivity that is proportional to T2.14

Therefore K(T) is seen to be a sensitive indicator of
crystalline quality at low temperatures, while at the higher
temperatures it is a sensitive indicator of the interatomic
potential anharmonicity. Figure 2 contains a graph of

the thermal conductivity of an insulator displaying the

main features we have just discussed.

 

11

C. Thermal Conductivity of Solid Argon-
Experimental Background

Since the intermolecular forces are weak and short
ranged the RGS are characterized by low melting temperatures,
high vapor and sublimation pressures and a relatively large
ratio of heat of fusion to heat of vaporization, it is
necessary to carry out experiments at cryogenic temperatures

and/or high pressures.8’54

In addition, since the proba-
bility of stray nucleation is relatively high, large grained
single crystals are difficult to obtain. The difficulty of
obtaining a defect free sample is compounded by the small
activation energies necessary for inducing various types
of crystal defects and an unusually large thermal expansivity
which may well be 100 times larger than that of the container
it is grown in. Hence, experimenters must be content to
work with plastic and easily deformed solids at low tem-
peratures.8’18

The thermal conductivity for an isotropic solid is
defined by the relation

3 = —K§T (9)

where h is the heat flux, ET is the temperature gradient
and the constant of proportionality K is the thermal
conductivity. Hence in order to measure the thermal
conductivity in principle it is only necessary to know the
heat flux and the temperature gradient associated with it.

To date six groups have measured the thermal conducti-

7,19,20,21,22,23,24,25

Vity of solid Ar. These experiments

 

 

12

can be classified as either constant volume or equilibrium
volume measurements. There are of course inherent trade—
offs when one choses to perform one type of experiment over
another.

The major disadvantage of performing constant volume
measurements is that high pressures must be applied. Hence
chambers containing the samples are necessarily opaque and
usually have thermal conductivities comparable to the sample
itself. This latter problem may be partially obviated in
one of two ways depending upon the geometry employed for
the measurements.

First, the standard, direct linear heat flow method
may be employed for constant volume measurements. In this
method heat is conducted parallel to the sample walls. Now
the thermal conductivity of solid Ar at high temperatures
is comparable to that of glass. This is usually much
smaller than the thermal conductivity of a typical metal
sample chamber. Thus, this method has the disadvantage of
tending to make the high temperature region very difficult
to measure.

A second method that may be chosen is the radial heat
flow method. In this method heat is conducted radially
outward toward and perpendicular to the sample chamber
walls. Since the sample chamber can be chosen to have a
large heat capacity and thermal conductivity, it can be
used as a heat sink that defines a thermal equipotential

surface.23 This method has the advantage over the first

 

13
that most of the heat is carried by the sample itself, hence

measurements can be carried out over the entire temperature
range without difficulty.7

Once the difficulties associated with the application
of high pressures to RGS are overcome, the thermal conducti-
vity as a function of T at constant volume can be studied
as well as the thermal conductivity as a function of density
at constant T.7 Furthermore, since the volume is kept con-
stant, the sample is usually in good thermal contact with
the heat sinks and the temperature probes throughout.

The major drawback associated with equilibrium volume
experiments is the unusually large thermal expansivity of
solid Ar. In cooling down from its triple point (83.8 K)
to liquid helium temperatures the volume of the sample of
Ar will contract approximately 9%, hence the temperature
probes as well as the thermal heat sinks will tend to pull
away from the sample. The thermal contact problem is also
aggravated by the vapor phase transport of material from
the heat sinks and probes in the presence of a thermal
gradient. This causes the formation of voids about the
heat sinks and probes.8 Furthermore, a small yield strength
near the triple point and relatively high brittleness at low
temperatures make the introduction of defects particularly
easy whenever large thermal gradients are introduced.26

Hence in spite of the difficulties mentioned so far,
the major advantage of the equilibrium volume method is

that the application of high pressures are not required so

 

14

that transparent, thin-walled and low thermal conductivity
sample tubes can be employed. The advantages of using this
type of sample tube are that most of the heat will be con-
ducted by the sample throughout the entire temperature range
of interest; and one can visually inspect the quality of the
sample at any time during an experiment. Another advantage
is that it has been demonstrated that growing Ar crystals
from the melt at equilibrium volume is the best method so
far for growing single crystals.27

To date all of the thermal conductivity experiments
performed at equilibrium volume have employed a linear
steady—state heat flow method. Of the samples examined most
were grown from the melt using a Bridgman technique. That
is, the samples were directionally grown solidifying from
bottom to tOp as heat was extracted from the bottom of the
sample tube. Although the sizes of the samples varied all
were cylindrical. As was mentioned earlier, since these
samples are allowed to contract the thermal boundary re-
sistance at the heatLSinksq;probes, and other interfaces may
not have been properly taken into account.

White and Woods19 were the first to measure the thermal
conductivity of solid neon, argon and xenon. Their samples
were grown from the melt in thin-walled Inconel tubes of
1.3 cm diameter by 7.6 cm length. The temperature difference
was measured by two He gas thermometers connected to two
copper wires approximately 5 cm apart.28 These wires were

in turn stuck through the tube perpendicular to the axis

 

 

 

15
of the sample tube and soldered in place.

Although White and Woods were not able to visually
examine their samples, they were able to estimate the quality
of their samples from trial experiments performed using a
glass sample tube. From these they estimated that the
growth rates varied from 1 to 2 mm/min. For these growth
rates one might expect grain sizes between 0.1 and 1 mm.26
The trial samples were initially transparent and were found
to become opaque and cloudy when rapidly cooled from 77 K
to 4 K.

The results of their thermal conductivity measurements
supported their preliminary observations. The six samples
they studied displayed very low thermal conductivity values
at low temperatures, indicating that they were highly de-
fective due to the constraints imposed by the sample tube
during cooling. It was further noted that the fifth sample
yielded the lowest values even though greater care was
exercised to minimize the thermal strains.19 This result
may have been caused by the loss of good thermal contact.
Since this sample was cooled at a slower rate than previous
samples a significant amount of mass migration through the
vapor phase may have occurred at the thermal probes.

Berne, Boato and DePaz22 cognizant of the difficulties
encountered by White and Woods19 succeeded in eliminating
some of the earlier experimental difficulties. Berne at 31.

grew over 50 samples of solid argon from the melt. Only 12

of these yielded data. These specimens were grown in pyrex

l6
tubes (0.55 cm inner diameter by 6 cm long) at rates of l

to 3 mm/hr (1/60 of the rates used by White and Woods).

The average grain sizes were observed to be between 1 and

4 mm. In addition, the thermal strains induced by the con—
tainer were completely removed.

After the sample was grown, near the triple point
temperature, it was cooled down to a uniform temperature
of about 75 K. The sample was then separated from the
container walls by subliming a small amount of material at
the sample chamber walls. This was accomplished by pumping
gently on the vapor above the sample.27 Once the sample
was completely separated, it was lifted out of its container
by means of a pyrex rod which was embedded in the top of
the sample during crystal growth. The sample was then
positioned between four copper spring clamps that were
anchored to the conductivity measuring stage of the appara-
tus (a heat source, two helium gas thermometers and a heat
sink).

To prevent the sample from subliming further and to
reduce the thermal gradients during cooldown the sample was
surrounded by helium exchange gas at a pressure of 200 Torr.
The samples were then slowly cooled to 4 K. Although the
sample was slightly reduced in size (since some sublimation
nonetheless occured), the samples remained optically
transparent.

With the sample at 4 K the clamps were then closed,

gripping the sample, and the measurements were performed.

 

 

17
Spring clamps of moderate strength were used, since it was

found that clamps which were too strong would break the
sample and clamps which were too weak would achieve very
poor thermal contact.

Since the vapor pressure of argon increases quite
rapidly as the temperature is increased Berne, 3: 31.
confined their measurements to low temperatures between
3 K and 15 K. Although their values were much higher than
those obtained by White and Woods (indicating better quality
samples), the data were not reproducible from sample to
sample as well as for a single sample. This was attributed
to the different ways in which the samples were grown and
to thermal contact which may have varied from sample to
sample.

Krupskii and Manzhelii20 unlike Berne, at 31. concen-
trated their efforts on making precise measurements at high
temperatures. Three specimens were grown from the vapor at
70 K in glass sample tubes (1.9 cm inner diameter by 5 cm
length) at rates less than 5 mm/hr and the expected grain
sizes are of l to 4 mm.27 To measure the thermal gradient
the sample was conveniently grown about a differential
copper-constantan thermocouple.

Since their work covered the temperature range of 24 K
to 73 K, where the thermal conductivity is defect indepen—
dent, Krupskii and Manzhelii made no attempt to remove the

20

strains induced by the sample chamber. By cooling their

samples slowly at rates of 10 K/hr to the desired temperature,

18
Krupskii and Manzhelii were able to keep all but one of the

samples free of any visible defects. That is, the first two
samples remained transparent while the third became translu-
cent after cooling. The data of Krupskii and Manzhelii

were nonetheless reproducible from sample to sample.

As with other workers, Krupskii and Manzhelii also
experienced trouble with loss of thermal contact at the
heat sink and source as the sample was cooled. This pro-
blem was partially overcome by the introduction of He gas
into the sample chamber to reestablish contact.

Since it was established by Peterson, 33 31.27 that
crystals grown from the vapor are much more defective than
those grown from the melt, it is mildly surprising that at
about 25 K Krupskii and Manzhelii obtained higher values
of thermal conductivity than White and Woods. Krupskii
and Manzhelii attributed this difference to having obtained
better quality crystals than White and Woods. However,

a more likely explanation is that Krupskii and Manzhelii
obtained better thermal contact than White and Woods, since
at 25 K the thermal conductivity is essentially defect
independent.

Christen and Pollack24 unlike previous workers attempted
to remove both the thermal boundary resistance (at the tem-
perature probes, heat source and heat sink) and the thermal
strains. The samples tested were grown from a "seed" in a
thin, transparent Mylar tube (1 cm diameter by 3 cm length)

at the rate of 0.7 mm/hr. The seed was prepared by locally

l9
maintaining the bottom the liquid filled sample tube slightly

below the Ar triple point temperature until a thin wafer of
solid about 0.5 mm thick appeared. It was then allowed to
anneal for a period of 12 to 24 hours before the remainder
of the sample was grown. It was observed that samples pre-
pared in this way had grains which were 5 mm to 10 mm in
size.25
The principal difference between the measurements per-
formed by Christen and Pollack and those of other workers
was the number and the location of the thermometers used to
measure the temperature gradient. Since argon tends upon
cooling to separate from the temperature probes, a single
germanium thermometer was embedded in the heat source
located at the bottom of the sample tube. The temperature
difference between the top and the bottom was measured in
the following way. First, the entire sample was allowed to
reach a uniform temperature. In this case the temperature
was that of the heat sink situated at the tOp of the sample.
This was accomplished by electronically controlling the heat
needed to keep the heat sink at a constant temperature.
This initial temperature was noted and then heat was applied
to the bottom of the sample via the heat source located
there. Once a steady—state condition was reached, the
temperature at the heat source was once again noted. Hence,
assuming a thermal contact resistance was not present at the
interfaces the temperature gradient along the sample is the

difference between the final and initial temperatures at the

20
heat source divided by the length of the sample.

To reduce thermal strains Christen and Pollack first
partly detached the sample from the walls of the sample
chamber in essentially the same manner employed by Berne,
gt 31. Once separated the samples were cooled to liquid
helium temperatures at the rate of 1 K/hr.

To prevent the sample from separating from the heat
source and sink the sample tube was lifted toward the heat
sink during cooling. This was accomplished by the use of
a metal bellows attached to the top of the sample tube. Due
to the high vapor pressure of argon near 83.8 K an over-
pressure of over half an atmosphere keeps the bellows ex—
panded. As the temperature is lowered the vapor pressure
drops quite rapidly8 and hence nearly a constant positive
stress is exerted on the sample as the temperature is lowered
further.

In order to prevent atoms from migrating away from the
interface between the heat source and the bottom of the
sample during the cooling process Christen and Pollack kept
the bottom of the sample slightly colder than the tOp. This
was accomplished by the introduction of a small amount of
helium exchange gas around the sample tube. It should be
noted that the presence of this thermal gradient tends to
encourage migration of atoms toward the bottom of the sample.
Thus possibly reducing the contact area at the interface
between the heat sink and the top of the sample.

Of the 10 samples grown by Christen and Pollack only

21
three yielded low temperature data. The data were found to

be reproducible for a given sample but not from sample to
sample. Most of the samples were found to turn cloudy and
all suffered at least some surface defects on cooling. This
was attributed to the sample bridging to and subsequently
separating from the walls of the sample chamber.

To check for a thermal boundary resistance Christen and
Pollack measured two samples of different lengths. If the

thermal contact resistance (l/K ) and the thermal resis-

Cont
tivity (l/K) is the same for both samples, then the effective

thermal resistance (l/KEff) should be a linear function of

the lengths with a positive non zero y-intercept (l/K ).

Cont
Quantitatively expressed we have

l/KEff = L/KA + l/KCont

where

KEff = Q/AT . (11)

In Equations (10) and (11) K is the effective (or meas-

Eff

ured) thermal conductance, L is the length of the sample
and A is its cross sectional area, K is the thermal conduc-

tivity of the argon crystal, K is the thermal contact

Cont

conductance, O is the power supplied to the heat source and
AT is the temperature difference between the heat source
and sink associated with the application of O. Since their
two samples yielded results that implied that 1/KCont was a
negative quantity Christen and Pollack assumed that a ther-
mal boundary resistance was either not present or negli-

gible.25

22

In view of the great care exercised in handling the
samples it is surprising the data at low temperatures in-
dicated that these samples were only slightly better than

19

those tested by White and Woods. It should be pointed

out that in Equation (10) K and KCont could have also
changed for the two samples tested by Christen and Pollack.
Ideally, the same sample of two different lengths would
have been preferred. The possibility of the existence of
a boundary effect should not have been so easily dismissed,
since it may have been, in fact probably was, present.

Clayton and Batchelder23 were the first workers to
perform constant volume thermal conductivity measurements.
Their work has been recognized as the first direct veri-
fication of one of the oldest predictions of solid state
physics. In 1914 Debye9 showed that at high temperatures
the thermal conductivity at constant volume should be
inversely proportional to the temperature. Clayton and
Batchelder's experiments showed that this prediction is
valid.

For this extremely difficult experiment,23 5 samples
of various molar volumes were grown from the melt at
essentially constant pressure. The pressures used to grow
the samples were in the 1 to 5 kilobar range. A high
thermally conducting copper tube (7.0 cm long and 1.5 cm
i.d.) contained within a high pressure cell served as the
sample chamber.

The thermal conductivity measuring part of the appa-

ratus was located within the sample chamber tube concentric

23
with the tube's axis. The heat source was constructed of

a long thin steel rod wound with heater wire which lay along
the tube axis. Two concentric copper rings suspended with
nylon string served to define two thermal equipotentials.
A difference thermocouple attached to the two rings measured
an average radial temperature difference.

These samples were grown from the melt at constant
pressure in a manner similar to that of Crawford and

30 and Leake, gt gt.3l Before a

Daniels,29 Daniels, gt gt.
sample was grown it was first determined what the melting
temperature and pressure should be such that at a given
temperature the sample experienced no external pressure.
The sample chamber is filled with liquid at the temperature
Tm and a pressure slightly less than Pm and left to equili-
brate for several hours. Next, in the presence of a slight
thermal gradient along the length of the sample chamber the
pressure was gradually increased to Pm and crystallization
was marked by a momentary rise in temperature. As solid
continued to fill the sample chamber more argon was intro-
duced to maintain a constant pressure.26 The samples were
then left to anneal for 16 to 39 hours near their respective
melting temperatures.

Although the samples could not be visually examined,
the low-temperature thermal conductivity data obtained by

Clayton and Batchelder23

indicated that the samples were of
the same quality as those grown by Berne, gt g1. Although

Clayton and Batchelder had no direct way to test for

24

a thermal contact resistance, it is unlikely that one was
present, since the external pressure on the samples is not
expected to pass through zero.

The nylon string used to support the concentric COpper
rings aided Clayton and Batchelder in determining the
strained state of their samples. That is, if the strings
were found broken after performing an experiment, it was
assumed that the samples had been severely strained and
the data were discarded.

Clayton and Batchelder found that for a given sample
the data were reproducible for the entire temperature range
investigated, except in a small region slightly below the
melting temperature. Between the melting temperature and
just 20 K below it their data fluctuated randomly. The in—
stability in this region was attributed to significant re-
crystallization of the sample during the measurements.

It has been demonstrated that samples grown at high
pressures consist of a large single crystal surrounded by
many small ones26 having grains of l to 10 mm in size. This
is a consequence of the high thermal conductivity of the
sample chamber walls.

One of the experimental difficulties encountered by
Clayton and Batchelder was that near the thermal conducti-
vity maximum the error in measuring the temperature differ-
ence was largest. Near the maximum they found that the
relatively larger amount of power required to produce a

measurable temperature difference caused an appreciable

 

25
warming of the sample. It was later suggested by Clayton

and Batchelder that this difficulty could be overcome by
employing a linear heat flow method.

Weston and Daniels7 subsequently pursued this suggestion
and performed measurements at constant volume near the con-
ductivity maximum. The temperature range covered was bet-
ween 5 K and 40 K. Weston and Daniels' sample preparation
techniques were similar to Clayton and Batchelder's. How-
ever, the results of their work indicated that their samples
were of a significantly better quality, than those studied
by previous workers.

D. Purpose

In View of the difficulties experienced by previous
experimenters with the thermal contact resistance, one of
our chief interests was to determine what role, if any, the
thermal contact resistance plays in the measurement of the
thermal conductivity of solid Ar at equilibrium pressures.
Once this was determined our next goal was to provide
reliable data so that the various theoretical models cur-
rently available can be tested.

Natural Ar was chosen because of its value as a re-
search material. Its low triple point temperature (83.8 K)
allows the use of inexpensive liquid nitrogen as the cryogen
during crystal growth. The interatomic potential is the
best known of all the rare gases. Natural argon is rela-
tively inexpensive and is available in very pure form (less

then 10 ppm total impurity content) and is nearly isotOpically

26
pure. The argon content of atmospheric air is 99.6% Ar40,

36 and 0.063% Ar38.2

0.337% Ar

For the present work we grew several crystalline sam-
ples of Ar under controlled conditions. The measurements
of the effective thermal conductance were all performed 22
gttg. We first measured the effective thermal conductance
at constant temperature as a function of the force applied
to the ends of the crystal at the heat source and sink in-
terfaces. We then measured the effective thermal conduct~
ance for various sample lengths while we varied the tem-
perature.

The data obtained from these experiments will be com-

pared to those of previous experimental and theoretical

workers.

II . THE EXPERIMENT

A. Cryogenic Apparatus
To perform the current experiments an apparatus was
designed and constructed to measure thermal conductivity
at low temperatures. The apparatus was mounted within a
double dewar system to allow the use of liquid He or liquid

N as the cryogen.

2

1. Description of Apparatus
The design of the apparatus is similar in principle to

the steady-state linear heat flow systems used by previous

19'20'22’24 In particular, it is patterned on the

system employed by Christen and Pollack.24 Although the

workers.

present system incorporates many of the same features as
theirs, the use and construction of the sample chamber is
considerably different. A sketch of the cryostat is shown
in Figure 3.

The cylindrical sample chamber was constructed from a
sheet of transparent Mylar 0.005" in thickness. A tube 3
to 4 cm long was formed by gently heating the Mylar sheet
with a heat gun while it was wrapped tightly around a Teflon
mandrel until it conformed to the mandrel. The seam was

then secured with ordinary quick-setting epoxy glue.

27

C—HICW'nmmDnCD)

I'— 7\"‘

N

Figure 3.

‘

Ar Gas Line
He Gas Line

Brass Gears

Upper Cu (Block) Heat Sink

Pt Sensor

Ge Sensor

Ar Needle Valve
Threaded Brass Rods
Large Cu Bellows
Vacuum Chamber

Short Cu Bellows

Sample Tube Cu Flange

Mylar Sample Chamber

Al Lower Block Heater

Ge Thermometer

apparatus.

28

~

-<><Z<C-—iC/);UD—UOZ
I l

Nl\l
l I

Pt Thermometer
Stainless Steel Tube
He Needle Value
Braided Cu Wire
Upper Block Heater
Exchange Gas Chamber
Cu Lifting Flange

Cu Piston

Cotton Thread
Annular Heater

Cu Radiation Shield
Al Lower Block

Glass

Glass

A cross-section of the thermal conductivity

 

 

 

 

30
Securing a leak-tight seal at the ends of the sample

tube proved to be quite troublesome. Either the copper
flange at the top or the Al block at the bottom, which were
also secured with epoxy glue to the tube, develoPed leaks
during several experimental runs. (This problem was due
largely to the differences in thermal expansivity of the
materials and the epoxy glue). This problem was later
solved by increasing the bonding surface and by using a
special low temperature epoxy glue (Stycast #2850 GT) which
is ideally suited for low temperatures because of its low
thermal expansivity and high thermal conductivity.

The sample tube is then secured by its copper flange
with low melting temperature solder (Cerro-Bend) to the Cu
lifting flange. The flange was employed to move the sample
tube relative to a stationary Cu piston protruding into the
sample tube. Furthermore, a short Cu bellows could also
be installed between the lifting flange and the sample tube
flange to enable us to measure the force exerted on the
ends of the sample.

By varying the distance between the Cu piston and the
Al lower block the sample chamber volume could be altered.
This could be accomplished in one of two ways; either by
replacing the Cu piston or by compressing (or expanding) the
large Cu bellows soldered between the upper copper heat sink
and the Cu lifting flange.

For the sample lengths of interest (0.3-3.0 cm) it was

only necessary to change the length of the large Cu bellows.

 

31

This was accomplished by turning two long threaded brass
rods screwed into two tapped holes in the Cu lifting flange.
To keep the Cu piston and the sample tube aligned a set of
brass gears soldered to the top of the rods and meshed to

a third central gear served to synchronize the rotation of
the rods.

To allow external control of the brass rods a thin-
walled stainless steel tube was passed through a demountable
O-ring seal at the top of the cryostat. An Allen wrench
attached to the bottom of the stainless steel tube was used
to turn an Allen cap screw soldered to the top of one of
the brass rods.

To initiate crystal growth gaseous Ar was condensed
directly into the sample chamber through the Ar needle valve
in the upper block. This gas was supplied directly from a
steel bottle through a stainless steel gas line and its
pressure monitored by a Wallace and Tiernan pressure gauge.

An annular heater suspended around the sample tube
aided in controlling the growth rate of the crystals. It
consisted of a Cu ring wound with heater wire and was sus—
pended from the top of the apparatus with cotton threads.
(To allow external control the threads were tied to a pair
of stainless steel tubes that exited through demountable
O—ring seals at the top of the cryostat). The heater served
to locally keep the sample slightly above the triple point
and thus the growth rate of the sample was regulated by the

rate at which the heater was raised.

 

32

To help maintain the sample at a constant temperature
and to reduce heat loss by radiation a Cu radiation shield
enclosed the sample chamber. The radiation shield was
provided with long slits covered with transparent Mylar to
allow visual examination of the sample. The introduction of
helium exchange gas into the vacuum chamber (between the
sample chamber and the exchange gas chamber) helped reduce
the thermal gradients present during crystal growth and
during the cool-down process. During thermal conductivity
measurements this vacuum region was evacuated to approxi—
mately 10-7 Torr as measured by an ionization gauge. Alumi-
nized Mylar glued onto the surface of the Cu radiation shield
helped to further reduce heat loss by radiation to the
surrounding cryogen.

The exchange gas chamber also served as a coarse tem-
perature control by acting as a variable heat leak from the
upper heat sink to the surrounding cryogen. For temperatures
above 5 K this was accomplished by introducing an appro-
priate amount of helium exchange gas through the He gas
inlet line. This pressure was monitored by two Wallace and
Tiernan gauges at pressures above 1 Torr and by a Veeco
thermo—couple gauge at pressures below 1 Torr. For tempera-
tures below 5 K the chamber was filled with liquid He by
siphoning it directly from the surrounding bath through the
liquid He needle valve. To improve the heat exchange
braided copper wire was soldered to the copper flange of the

upper heat sink. This aided the exchange of heat by

 

33

presenting a large effective area to the He exchange gas.
All of the common practices used to minimize heat conduction
in cryogenic apparatus were employed in constructing the
remainder of the apparatus. For example, all of the vacuum
and gas lines of the cryostat were constructed of low thermal
conductivity, thin-walled stainless steel tubing. In
addition all of the electrical wires leading to the sample
chamber were made of #36 gauge Cu wire and were thermally
anchored to the He bath. This was accomplished by wrapping
the wire several times around the Cu heat sinks in contact
with the bath and then varnishing the wires in p1ace.33’34
External to the cryostat the gas handling system
(except for the Ar gas line) was constructed of Cu tubing.
A pumping station was used to evacuate the system through
a pumping manifold containing the necessary valves. This

station consisted of a liquid N trap, a diffusion pump and

2
a rotary backing (and roughing) pump.
2. Temperature Control

For the duration of the experiment, especially during
the thermal conductivity measurements, it is necessary to
keep the sample chamber at a constant temperature. In prin-
ciple this can be accomplished by controlling the rate at
which heat is lost from the sample chamber to the cryogenic
bath.

Throughout the temperature range of interest precise
temperature control was obtained by employing a combination

of different methods. A coarse temperature control was

34

maintained by changing the vapor pressure of the cryogenic
bath and by carefully adjusting the exchange gas chamber
pressure. The vapor pressure of the bath was controlled by
pumping vapor of the cryogenic bath through a manostat type

pressure regulator with a mechanical vacuum pump.33

Using
this procedure it was possible to regulate the temperature
of the bath to within i0.02 K for liquid N2 and 10.002 K
for liquid He.34 However, a drOp in the bath level, the
desorption of He atoms from the exchange gas chamber walls,
etc., can alter the thermal loads at the upper heat sink
and make it difficult to maintain the sample chamber itself
within these limits.

This difficulty can easily be overcome and precise
temperature control can be had by compensating for the
change in thermal loads with an electronically controlled
heater. The temperature controller employed for this ex-
periment is a Model 5301 manufactured by Artronix Instru-
mentation Inc. (St. Louis, Missouri). Its principle of
operation is briefly summarized in the paragraphs that
follow and a block diagram is included in Figure 4.

The temperature controller monitors temperature de-
viations with an AC bridge and a phase sensitive detector
in conjunction with a differential AC amplifier. One of
the legs of the bridge consists of a temperature sensor
(either a Ge or Pt resistance thermometer) secured in place

with vacuum grease in a drilled cavity located in the upper

Cu block. The other leg located on the front panel of the

 

35

T, S. - Temperature Sensor (Ge or Pt)

POT - 10 Turn Potentiometer (Temperature Set Point)
AMP1 - Narrow Band Amplifier

AMP2 — Differentiating Amplifier (Rate)

AMP3 - Proportional Amplifier (Size)

AMP4 - Integrating Amplifier (Duration)

DET - Phase Sensitive Detector

OSC - Oscillator

O, P, - Output Circuit

H — 120 0 Heater Wire

8 - Servo

O, POT, - Output 10 Turn Potentiometer (Power Output)

Figure 4. A block diagram of the Artronix temperature
controller

 

 

 

 

AMP3

 

 

 

 

 

 

 

 

 

 

DET

 

 

 

 

 

37

temperature controller is a variable ten-turn potentiometer
that determines the temperature set point. A generator (or
oscillator) provides a stable AC signal to both the bridge
circuit and the phase sensitive detector and operation of
the temperature controller is the result of comparison of
the phase relationship of the input and output of the AC
bridge.

During the bridge balance condition the potential drop
across the temperature sensor and across the set point re-
sistance are equal. Whenever the temperature of the upper
block changes, the sensor resistance changes and the bridge
is no longer in balance. The output signal from the bridge
is then in or out of phase with the generator signal, de-
pending on whether the temperature deviation is above or
below the temperature set point. That is, the input vol-
tages to the phase detector from the bridge circuit and the
generator are either both of the same sign tending to be
in phase or of the opposite sign tending to be out of phase.

The out-of—balance signal is then amplified by the AC
differential amplifier and phase analyzed relative to the
oscillator signal. The difference in phase is then used to
produce a DC signal which is further amplified and fed into
the output control section containing the heater wire which
is wound about and varnished to the upper Cu block. The
power delivered to the heater is adjusted continuously
according to the size (proportional amp), duration (inte—

grating amp) and rate (differentiating amp) of the

38

temperature deviation. Furthermore, by minimizing the
power output requirements of the temperature controller it
is possible to maximize the overall sensitivity of the
system and thus maximize the temperature controllability
of the sample chamber.

During the sample growth process the sample chamber
temperature must be kept nearly constant for several days.
This is accomplished by admitting an atmosphere of He
exchange gas into the liquid helium dewar (inner dewar),
admitting a few Torr of He into the exchange gas chamber,
admitting 1 Torr in the vacuum chamber and maintaining the
level of liquid N2 in the outer dewar constant. The liquid
N2 level was maintained essentially constant by an automatic
liquid N2 filler system of 25 liter capacity that required
refilling approximately every three days. In this way
enough heat exchange between the sample chamber and the
liquid N2 bath was present to enable the temperature con-
troller to maintain the sample chamber temperature close to
the triple point of Ar. It should be noted that although
the temperature control occurs at the upper Cu block the
presence of the Cu radiation shield which is soldered to
the upper block defines a thermal equipotential and the He
exchange gas in the vacuum chamber maintains the sample
chamber at a constant temperature.

To carry out thermal conductivity measurements in the

high temperature region, liquid N was transferred directly

2
into the liquid He dewar (inner dewar). The liquid N2 bath

39

temperature was kept constant by regulating its vapor pres-
sure through a manostat and monitored with a Hg manometer
while the sample chamber temperature was kept constant by
the temperature controller. In order to extend this tem-
perature range below 63 K (the triple point temperature of

liquid N2) the liquid N was solidified by reducing its

2
vapor pressure below its triple point pressure of 94 Torr
and then regulating the sublimation pressure of the solid
N2. Enough cooling capacity was available to enable us to
extend the temperature range down to 48 K by these means.
For obtaining temperatures below 48 K liquid He was
transferred directly into the He dewar surrounding the
thermal conductivity apparatus. The nature of the cooling
process required that we first carry out measurements at
the lowest temperatures from 5 K to 2 K. After cooling to
4.2 K liquid He was admitted via a He needle valve directly
into the exchange gas chamber from the surrounding bath.
Temperature control was accomplished by employing the same

technique used for liquid N Next, for temperatures above

2.
5 K and below 48 K there is no convenient cryogenic liquid
and it was necessary to continue using liquid He in this
range. Between 5 K and 10 K it was necessary to pump out
the liquid He from the exchange gas chamber and to reduce
its pressure from 1 Torr to 10 mTorr as we slowly increased
the temperature. Above 10 K it then became necessary to

completely evacuate the exchange gas chamber, since more

than sufficient thermal contact was present to require the

40

use of the temperature controller to raise the temperature
of the sample chamber above 10 K.

Throughout the experiment the bath vapor pressure and
the exchange gas pressure were judiciously adjusted to en-
able us to use the temperature controller at its maximum
sensitivity. Use of this technique allowed us to achieve a
constant temperature control to better than $0.001 K at the
lowest temperatures (2 K-10 K), i0.005 K at the interme-
diate temperatures (10 K-40K) and i0.0l K at the highest
temperatures (40 K-83 K).

3. Temperature Measurement

The temperature and temperature gradient must be known
precisely during the thermal conductivity measurements and
during the sample growth process. To accomplish this cali—
brated Ge and Pt resistance thermometers (manufactured by
Scientific Instruments, Lake Worth, Florida) were installed
in the Al lower block. In the upper heat sink were in-
stalled initially uncalibrated Ge and Pt thermometers to
serve as sensors. These Ge and Pt sensors are then cali—
brated against the thermometers located in the Al lower
block. To insure that the thermometers and sensors are in
good thermal contact all were imbedded in drilled cavities
and secured in place with vacuum grease. To avoid falsely
elevating the temperature of the thermometers due to thermal
heat leaks from room temperature through the electrical
leads all of the leads were made of #36 gauge copper wire

and were thermally anchored to the upper heat sink.

41

To measure the resistance of the thermometers a four-
leaded DC potentiometeric method was used. Two of the leads
carried the excitation current which was 10 uamps below
34 K and 100 uamps above 34 K, while the remaining two were
used to measure the potential drop across the thermometer.
A Leeds and Northrup K—5 potentiometer and null detector
system was used to standardize the excitation current and
a Keithley 174 digital multimeter was used to measure the
voltage drop across the thermometer and thus its resistance.
The potential drop was actually measured twice-once with
the excitation current flowing in the forward direction and
again in the reverse direction. The two results were then
averaged to cancel out the thermal emf's present. Figure
5 contains a schematic of the circuit used to measure the
thermometer resistances and also included is the circuit
for the lower Al block heater.

The calibration for the Ge resistance thermometer was
provided in tabular form by the manufacturer, Scientific
Instruments. This calibration was performed against a
secondary N.B.S. Ge standard thermometer. Calibration
points were provided for every 0.25 K for the lowest tem-
peratures (1.5 K to 5.0 K), every 0.5 K to 2.0 K for the
intermediate temperatures (5.0 K to 20 K), and every 5.0 K
for the highest temperatures (40 K to 100 K). In order to
interpolate for temperatures between these points the cali-
bration data were fitted (for overlapping temperature in-

tervals) to a quadratic equation of the form33

42

R1 = O‘lOKQ V1 = 6 Volts

R2 = O'lOOKQ V2 = 6 Volts

R3 = O-lOOKQ RGe = Germanium Thermometer
R4 = 680-780KQ RPt = Platinum Thermometer
R5 = lOOQ Precision RH = Lower Block Heater

R6 = 100 Precision

Figure 5.

A schematic diagram of the thermometer and
lower Al block heater circuits.

43

m 20322—0 _A-m

m7:H

 

44
l/T = a + a lnR + a2(lnR)2 (12)

0 1
using the method of least squares. The set of constants
a0, a1 and a2 for each line segment were determined by using
at least 5 calibration data points. Furthermore, the set
of constants chosen were those that yielded line segments
that were smoothly connected. The temperatures between 1.5
K and 50 K were calculated by using the above formula after
substituting the apprOpriate constants and the measured
value of the Ge thermometer resistance.

Since above 50 K Pt thermometers are more sensitive
than Ge thermometers, all of the temperature measurements
above 50 K were taken with the Pt thermometer. We cali-
brated this particular thermometer against another Pt
thermometer (which had been calibrated by N.B.S.) by com-
paring the values of their resistances every single degree
from 45 K to 90 K. The calibration data were then fitted
for overlapping temperature intervals to a straight line
of the form33

T = mR + b (13)
by the method of least squares. The set of constants m and
b for each temperature interval were determined by using at
least 10 points with one overlapping point at each end. By
using the above formula and substituting the measured values
of R along with the appropriate constants (m and b) the
temperatures above 50 K could be determined.

The repeatability of the thermometers after thermal

cycling between 4 K and room temperature is reputed by the

45

manufacturer, based on tests of the thermometers supplied,

to be within $0.5 mK. Our estimated accuracy

of the tem-

perature measurements, accounting for possible calibration

errors, is no larger than $2.5 mK in the 2-30

K temperature

range and $5.0 mK in the 30-90 K temperature range.

B. Sample Preparation

The technique chosen to grow the samples
similar to the technique used by Christen and
previous workers.27’35’36 Basically, it is a
the Bridgman and the zone melting techniques.
first a "seed" is grown from the melt and the
crystal is directionally grown from bottom to
heater locally maintains the liquid above the
at a temperature slightly above 83.8 K. This

trol the rate of growth.

of Ar is

Pollack24

and
synthesis of
That is,
rest of the
top. A zone
growing sample

serves to con-

It has been found that employing this technique and

growing the samples in a slow and even manner

grained crystals of Ar. In fact,

found to be inversely proportional to the growth rate.

1. Sample Growth

Before cooling the apparatus to liquid N

yield large-

the grain size has been

37,38

temperatures

to begin the sample growth process, all of the gas lines and

vacuum regions were evacuated and flushed at room temperature.

In particular,

it is imperative that the Ar gas line and the

sample chamber be charged with fresh Ar and evacuated sev-

eral times to remove any contaminants or impurities present.

After flushing the Ar line and the sample chamber a fresh

46

charge of Ar gas at approximately an atmosphere was admitted
into this region.
To initiate the cool down process the outermost dewar

was filled with liquid N To speed up the cooling rate a

2.
"thumbfull" of air was admitted into the vacuum jacket of
the liquid He dewar through a stopcock normally used to
pump out this vacuum region. In addition He exchange gas
was admitted into the other vacuum regions-l atmosphere into
the He dewar, 14 Torr into the exchange gas chamber and 1
Torr into the vacuum chamber.34
While the apparatus cooled we adjusted the position of
the sample chamber so that the tip of the Cu piston protruded
slightly into the sample chamber. When the sample chamber
reached 79 K the pressure inside the (closed off) Ar gas
line drOpped below 300 Torr. We then slowly increased the
Ar pressure to approximately 100 Torr above the triple point
pressure of 516 Torr and liquid Ar was observed to readily
condense into the sample chamber. Once the sample tube is
filled with liquid (in approximately 30 minutes) the Ar in—
let needle valve was closed. At the end of this time the
sample chamber temperature had risen above 83.8 K. The
sample was then allowed to cool down gradually to a few
tenths of a degree above its triple point temperature of
83.8 K. The sample was then held at this temperature for
about an hour to allow the liquid Ar to reach a uniform

temperature.

Although care had been taken to decouple the apparatus

47

from mechanical vibrations transmitted through the floor,
all unneeded vacuum pumps were turned off during the fol-
lowing phase of the sample growth process. This was done to
minimize the possibility of inducing defects during this
early phase.

To begin the seed growing phase the annular heater was
lowered into place. It was positioned near the bottom of
the sample chamber so that a 1-2 mm gap was visible between
the top of the Al block and the bottom of the annular heater.
To locally maintain the liquid inside the region of the
annular heater above 83.8 K,10 mwatts of power were de-
livered to the heater as the temperature set point, which
controls the upper block temperature, was slowly reduced at
the rate of 0.25 K/hr. The temperature was continuously
lowered until a thin wafer of solid Ar of about 0.5 mm in
thickness appeared at the Al lower block. The apparatus
was then left in this steady state condition for a period
of 12-24 hours.

During this annealing period it is expected that the
larger crystal grains will absorb the smaller ones until a
lower energy configuration is reached. This increases the
likelihood that the wafer or seed will present itself as a
single crystal to the melt (liquid Ar).

The remainder of the crystal was then grown by slowly
raising the annular heater at the rate of 0.5 mm/hr. A
driving mechanism situated outside and mounted above the

apparatus served to raise and lower the heater. The driving

48

mechanism was essentially a winch powered through a reducing
gear box coupled by a pair of demountable gears to a l and
2/3 r.p.m. synchronous motor. By replacing the driving
gears with a pair having a different ratio or by replacing
the synchronous motor with one of a different r.p.m., the
rate of raising the annular heater could be altered.

Besides determining the rate of growth of the sample,
the annular heater also served to inhibit the nucleation of
additional crystals at the Mylar walls. Since the thermal
conductivity of the solid Ar is considerably less than the
thermal conductivity of the Mylar walls the heat of fusion
would be easily conducted away through the Mylar walls if
the annular heater was not present.

As the sample continued to grow a vapor space was ob-
served to very slowly increase in size in the region between
the solid Ar growing downward from the upper Cu piston and
the upper surface of the Ar melt. The appearance of this
gap is attributable to two of the many unusual properties
of solid Ar. First, the density of solid Ar is 15% higher
than the density of its liquid, so that approximately 13%
of the sample chamber volume will be left vacant when all
of the liquid within the sample chamber freezes. Second,
the sublimation pressure of solid Ar near its triple point
temperature is very high (516 Torr), so that the presence
of a thermal gradient induces vapor phase mass migration
from the slightly warmer sample tube to the cooler region

near the Cu piston. Thus solid will tend to condense in

 

49
the free volume between the Cu piston and the large Cu

bellows.

The presence of the vapor gap, unfortunately, provided
ideal conditions for the appearance of vapor snakes,39 an
interesting phenomenon in itself. The growth process was
momentarily halted on a few occasions when the sample chamber
became filled with vapor snakes.

As the upper surface of the melt drops below 83.8 K a
crust will freeze on top of the melt. The melt in the closed
volume between the crust above and the solid below is now
effectively sealed off. As more of the melt below the crust
continues to solidify a vapor filled bubble appears just
below the crust. As the size of the bubble continues to
increase atoms from the surrounding liquid evaporate to fill
the available volume. Due to argon's large heat of vapori-
zation compared to its heat of fusion,8 a solid shell will
freeze around the bubble as Ar atoms on the liquid sur-
rounding the bubble evaporate to the available vapor space.
Further solidification of the liquid leads to more free
volume and the bubble propagates into the melt as a vapor
filled tube with a transparent thin-walled solid sheath.8
If the vapor snake prOpagates beyond the annular heater and
reaches the crystal's upper surface, it then becomes neces-
sary to halt the growth process in order to melt the vapor
snake.

This was accomplished by momentarily stopping the

annular heater's ascent and increasing the power delivered

50
to it from 10 mwatt to 40 mwatt until the vapor snakes re-

ceded or melted. The power was then slowly decreased to a
value slightly above 10 mwatt and we then resumed raising
the annular heater. This procedure was necessary to prevent
the nucleation of new crystallites.

For the longer samples studied (2.0-3.0 cm), after
approximately two-thirds or more of the sample had grown,
the amount of liquid present above the sample was insuffi-
cient to continue growing the sample from the melt. It
thus became necessary to stop the growth process in order to
condense more liquid Ar. This was accomplished by first
gradually raising the temperature of the upper Cu block to
83.8 K and then admitting more Ar gas through the Ar needle
valve until the vapor gap was filled with liquid. The tem-
perature of the upper Cu block was then slowly reduced back
down below 83.8 K and the growth process resumed.

When the sample had reached the approximate desired
length the growth process was stopped and the power deliv—
ered to the annular heater was turned off. To allow visual
examination of the entire sample the annular heater was
lowered to the bottom of the sample chamber.

At the end of the growth process solid Ar usually froze
in the space between the Cu piston and the Mylar tube; this
bound the two together. In order to free the Mylar tube
from the Cu piston (to allow relative vertical displacements
of the sample tube), it was necessary first to raise the

temperature of the upper Cu block slightly above 83.8 K.

51
The Mylar sample tube was then raised until the top of the

sample was in intimate contact with the bottom surface of
the Cu piston. The Ar was then frozen in contact with the
Cu piston by quickly reducing the temperature of the upper
Cu block slightly below 83.8 K.

To reduce the likelihood of the Mylar and/or the Cu
bellows from binding to the Cu piston during the subsequent
experiment, it was necessary to remove the excess solid Ar
within this region. This was effected by gently pumping on
the sample through the Ar inlet needle valve. The pumping
rate was carefully adjusted so that the sublimation pressure
was reduced no more than 10 Torr below the equilibrium subli-
mation pressure. After pumping for approximately an hour the
solid Ar was observed to sublime quite readily from the re-
gion between the Mylar tube and the Cu piston. This subli-
mation process was stopped when the vapor gap between the
top of the sample and the Cu piston had increased to approx-
imately 0.1 mm.

To achieve thermal contact once again the sample was
raised toward and gently pressed against the Cu piston.
Since solid Ar is quite plastic at these temperatures, the
force required to deform the top surface of the sample so
that it conformed to the surface of the Cu piston was rela-
tively small, less than 15 nt.

Although it would ordinarily be difficult to determine
whether the entire surface area of the Cu piston was in con-

tact with the sample, the transparency of solid Ar made this

52
determination quite simple. By viewing the top surface of
the sample from below it was easy to observe when the two
surfaces were in uniform contact. That is, when the sur-
faces are separated light is almost totally reflected from
the Ar surface and thus the Cu is only slightly visible.
As the Cu surface begins to make contact, light can be
observed to be reflected from the Cu surface where the Cu
and Ar are in good thermal contact.

The same procedure which we have just described was
used to prepare the shorter samples (less than 2.0 cm),
except that enough melt was present initially to grow the
samples for their entire length without interruption.
Furthermore, in spite of the interruptions the average
growth rate for all of our samples was approximately 0.2-
0.4 mm/hr. All of the samples were then gradually cooled
down to 79 K at about 0.5 K/hr and allowed to anneal for
24 hours before measurements were begun.

2. Sample Manipulation

It has been demonstrated that the study of gross crys-
talline characteristics may be carried out without the use
of X-ray techniques. In particular, it has been shown that
solid Ar is ideally suited for thermal etching techniques to

8'22r24'37 The grain boundaries of a poly-

study grain size.
crystalline sample of Ar can be easily revealed by gently
pumping on the solid while the sample temperature is near

83.8 K.

53

The mechanism for thermal etching can be understood by
noting that atoms between grain boundaries are on highly
defective sites. These are more loosely bound than those on
a true crystal face and are more easily evaporated. In
addition, atoms forming a surface at these highly defective
sites have a higher surface free energy than those on true
crystal surfaces and thus tend to migrate away from the
grain boundaries as the surface energy approaches a minimum.
Thus, the resulting preferential evaporation and surface
migration results in the appearance of etched lines along
the grain boundary.

Since the manner in which we prepared the samples for
thermal conductivity measurements at low temperatures in-
evitably results in etched samples, it was not necessary to
perform a separate experiment to determine the grain size
of our samples.

In order to carry out thermal conductivity measurements
in the temperature range of 2 to 48 K, the samples were
first separated from the sample tube and then rapidly cooled
from 79 K to 4 K in less than an hour. Data were then ob-
tained as we first cooled down to 2 K and then as we warmed
up from 2 to 50 K.

It was hoped that, by separating the sample from the
sample chamber walls (or boundaries) prior to the cool-down
process, the thermal strains induced by the constraints of
the sample chamber could be reduced. With the sample at

79 K the vapor pressure over the sample was reduced from an

54

initial value of 260 Torr to a final value of 250 Torr by
pumping on the Ar sample through the Ar needle valve. To
speed up this process the annular heater was used to locally
warm up the lateral sides of the sample to induce preferen-
tial evaporation of Ar from the sample tube walls. That

is, as we continued to pump on the sample the annular heater
was supplied with 6 mwatts of power as it traversed the
length of the sample (down and up once) at the rate of 2.4
cm/hr. As soon as the annular heater had completed its
sweep the Ar inlet needle valve was closed and the power
supplied to the annular heater was turned off. At this point
we noticed that the sample was nearly completely separated
from the sample chamber boundaries, including the thermal
heat source and sink boundaries, and the lateral surface had
the appearance of etched glass. We further noticed the
absence of etch lines associated with grain boundaries, in-
dicating that the samples studied were possibly single
crystals.

As we mentioned earlier, all of the samples that we
studied were cooled at a very fast rate. This was accom—
plished by first admitting 250 Torr of He exchange gas into
the vacuum and exchange gas chambers in order to minimize
the size of thermal gradients present during cool—down.
Next, liquid He was transferred directly into the He dewar
and as the temperature of the sample neared 4 K the vacuum

chamber was pumped down to as low a pressure as possible.

55

The temperature at which we attempted to establish
mechanical contact with the sample ends was determined by
the type of study we wished to carry out. First, to study
the effect on the effective (or measured) thermal conduct-
ance caused by varying the force exerted on the sample ends
at constant temperature, we attempted to establish mechan—
ical contact only after the sample had been cooled down
to 4 K.

The results of this study indicated that in order to
achieve good thermal contact, mechanical contact would have
to be maintained throughout the cool—down procedure for the
subsequent experiments devoted to measuring the thermal
conductivity of Ar.

Due to the high vapor pressure of Ar the sample has a
tendency to reattach itself to the Mylar walls during cool—
down. It was also noticed that solid Ar tended to condense
onto the Cu piston. For the first set of experiments these
problems were obviated by continuing to pump on the sample
while it cooled down to 4 K. Mechanical contact was then
achieved at the desired temperature. Unfortunately, for
these samples the combined effects of rapid pumping and
cooling produced samples that were quite opaque.

For the second set of experiments we attempted to
maximize the contact surface area at the sample ends by
continuously pressing on them by raising the sample tube
up to the Cu piston during the cool-down procedure. This

procedure also prevented Ar from condensing from the vapor

56

state onto the Cu piston surface at the Cu-Ar interface.
Since Ar is extremely plastic between 79 K and 54 K, maxi-
mum contact area is easily achieved. At 79 K a force of
approximately 15.2 mt was required to achieve contact at
the Cu-Ar interface. However, below 54 K Ar starts to be—
come quite hard and brittle and hence will only be slightly
deformed with the application of the same force as the
temperature approaches 4 K. This means that as the sample
volume decreases and atoms evaporate away from the Cu-Ar
interface to cooler regions, it becomes necessary to contin—
uously apply a larger force on the ends of the sample as it
cools below 54 K up to a maximum of 30 nt at 10 K.

We should note at this point that the positive stress
exerted on the Ar sample between 50 and 79 K causes the
ordinarily cylindrical sample to bulge slightly and flow
out towards the Mylar walls, reattaching the two once more.
This is not unusual since carefully prepared samples of Ar

40 This

have been observed to sag under their own weight.
effect coupled with the atomic migration which causes re-
condensation between the Ar and Mylar tube walls firmly
binds the sample to the sample tube once again. As we
pressed and cooled these samples we heard a single "click"
and noticed that the samples suddenly turned cloudy (but not
opaque) at approximately 45 K. This phenomenon was probably
due to the sample suddenly contracting away from the Mylar

walls. Although some gross defects appeared near the ends

and the middle of the largest samples, all of the samples

57

possessed a uniform cloudy appearance. Closer visual in-
spection of the samples revealed that the cloudiness
appeared to be confined to the sample surface. We also
observed that the smaller samples appeared less defective
than the larger ones .

Since the thermal conductivity at temperatures above
the thermal conductivity maximum is less dependent on
crystalline quality, the precautions exercised in the low
temperature experiments need not be observed for measure-
ments between 48 K and 83 K. It was only necessary to
separate the Mylar tube from the Cu piston prior to the
measurements to allow relative motion of the Cu piston.
Since the data were taken from 83 K down to 48 K, the
sample on cooling tends to separate from the A1 lower block.
To obviate this difficulty we would press on the sample ends
(in the same manner as was previously described) until ther-
mal contact was reestablished.

3. Thermal Conductance Measurements

All of the thermal conductance measurements were per-
formed in the same manner. With the vacuum chamber pumped
out to the lowest vacuum possible and the upper Cu block
maintained at a predetermined temperature Tu by the tempera-
ture controller, we first waited for the sample to reach a
uniform temperature. At the lowest temperatures this time
was of the order of seconds, while at the highest tempera-
tures it was of the order of hours. The criterion for this

initial steady state condition was that the thermometer at

58

the A1 lower block had reached a constant temperature. This
temperature was then recorded as the initial temperature Ti'

Next, joule heating was applied at a known rate to the
lower Al block heater (See Figure 3). This heater was wired
in a four leaded configuration to allow convenient potentio-
metric measurements of the voltage drOp across it with the
Keithley D.M.M. A 100 precision resistor wired in series
with the heater was used to measure the current I delivered
to the heater. The power éApp delivered to the heater was
then computed according to the relation QAPP = IV.

After the sample had reached a steady state condition
with a constant éApp being delivered to the bottom of the
sample this higher final temperature Tf of the thermometer
at the Al lower block was also recorded. The temperature
difference AT between the final and initial temperature was
then computed, and the thermal conductance was calculated
from the relation K /AT. The temperature asso-
ciated with KEff was taken to be the mean of the initial
and final temperature.

That only one calibrated thermometer at the Al lower
block is necessary to measure the effective thermal con-
ductance can be easily demonstrated. Suppose that in the
initial steady state condition (éApp = 0) that heat loss
nonetheless occurs at the bottom of the sample. Then the
rate at which heat is being supplied to the lower block is

given by Oi = KEff(Tu-Ti) where Tu>Ti. Now after the Al

lower block heater is turned on in the final steady state

59
the net rate of heat conducted from the lower block to the

upper block is given by

Q

QNet = QApp- i = KEff(Tf_Tu)'

that is,

QApp = KEff(Tf_Tu)+Qi

Substituting for Oi in the above expression we see that

QApp = KEff(Tf-Ti)'
that is

K = K =

Eff meas QApp/(Tf-Ti)'

It can also be easily seen that the above result is also
true for T ST..
u i

To test for a possible dependence of KEff on AT we
measured K for several values of AT. We observed no sys-
tematic dependence on AT. In general the AT's used were
0.02 K at the lowest temperatures near 2 K, 0.05 K near
the thermal conductance maxima, 4-10 K, and 0.25 K near
83 K. These values were large enough to provide a small
error in the measurement of AT, yet small enough so that
the fractional difference between KEff(T) and K(T), the
true thermaJ.conductance,.is much less than 1% (See Appen-
dix D). The latter criterion is fulfilled when ATEO.7 at
the lowest temperatures and AT528 K at the highest.l4’25

To investigate what effect pressing on the sample ends
would have on the values measured for the thermal conduct-
ance, the short bellows was soldered in place between the

sample tube flange and the Cu lifting flange (See Figure 3).

To perform these measurements we started by first making

60
very light contact at the Cu-Ar interface. The thermal con-

ductance was then measured as the applied force was in-
creased while the temperature was kept constant. The
applied force was determined by measuring the relative ex-
pansion of the bellows With a Wild cathetometer. The pre-
viously measured force versus bellows expansion was then
used to determine the applied force.

A variation of the above experiments was conducted to
determine the effect on the temperature dependence of the
thermal conductance for a constant applied force at two
different values. For this study and the subsequent ones
the short copper bellows was removed. The first set of
measurements was carried out after thermal contact was
established at approximately 5 K. We then maximized the
thermal conductance at this temperature by squeezing on the
ends of the sample until we reached the maximum force that
could be applied. Measurements of KEff were carried out
for temperatures between 3-20K.

A The temperature was then lowered back down to 5 K
where the thermal contact was first broken and then re-
established. However, in this case the force applied was

such that K was approximately half the thermal conduct-

Eff
ance of the previous experiment at 5 K and measurements
were then conducted over the same temperature range.

4. Thermal Conductivity Measurements

The results of the previous experiments indicated the

presence of a thermal contact resistance that would have to

61
be determined in order to determine the thermal conductivty

of the sample itself with the present apparatus. It was
thus necessary to measure the thermal conductance as a
function of temperature for samples of various lengths at
the same thermal contact resistance.

Of course the ideal experiment would have been to
measure K for various lengths of the same sample. We did
attempt to follow this route. However, vacuum leaks in
the sample tube limited the number of different lengths we
could examine for the same sample. In spite of the diffi-
culties we were able to measure KEff for two different
lengths of the same sample for at least three different
samples.

The procedure was to first measure K as a function

Eff
of temperature for the longest sample. Next, the sample's
temperature was raised to slightly above 83.8 K and the top
of the sample was melted until the sample was the desired
length. The excess Ar was then pumped out in the usual
manner and the sample was allowed to anneal at 82 K until
it regained its optical clarity. We then repeated the ex-
periment for this next length.

The lengths of the samples were measured with the Wild
cathetometer and their cross-sectional area was taken to be
the same as the inner cross—sectional area of the Mylar
sample tubes corrected for thermal expansion.

The thermal conductivity as well as the thermal bound-

ary conductance was determined by fitting data to a straight

62
line of the form given by Equation (10) using the method of

least squares.

III . EXPERIMENTAL RESULTS

A total of sixteen samples were grown to carry out our
investigations. Table 1 contains a summary of the growth
conditions for the samples that gave meaningful data. The
average growth rate in Table l is defined as the length of
the sample divided by the total time required to grow that
length. For each of the first runs the annealing period
is the time interval between the end of the growth period
to the actual beginning of the first measurement of thermal
conductance. For subsequent runs it is the time between
the end of the previous run to the beginning of the next
measurement of thermal conductance.

Samples 6, 7, and 10 were used to study the dependence
of the thermal conductance on the force applied to the sam-
ple ends. Specifically, samples 6 and 7 were used to in—
vestigate the thermal conductance as a function of the
applied force from 0.0 to 18.0 nt, while sample 10 was used
to study the thermal conductance as a function of tempera-
ture for the two forces 15 nt and 30 nt. Samples 10, ll,
12, 13, 14, 15, and 16 were used to determine the thermal
conductivity of solid Ar as a function of temperature from

2.2 to 83.0 K. We should further note that sample 15 was

63

64

used to carry out extensive thermal conductivity measure-

ments in the high temperature range of 48 to 83 K.

Table 1

Summary of Sample Preparation

Sample Run No. Length Ave. Growth Annealing
No. (cm) Rate (mm/hr) Time (Days)
6 2 0.86 0.20 5
6 3 0.67 0.20 2
7 2 1.72 0.34 2
7 3 0.83 0.34 l
10 l 1.57 0.34 1/2
10 2 1.17 0.34 2
11 3 1.97 0.26 1/2
11 5 1.34 0.26 6
12 1 2.84 0.31 4
13 l 0.33 0.36 0
13 3 0.95 0.28 2
l4 1 3.00 0.25 4
l4 2 2.98 0.25 4
15 l 1.28 0.19 1
l6 1 0.63 0.25 2
l6 2 2.41 0.37 1

A. Results of the Force Experiments

Measured Thermal Conductance versus Force

Table 2 contains the results of samples 6 and 7. Sam-

ple 6 run 4 consists of two sets of data for different but

fixed temperatures one for 5.2 K and the other for 34 K.

Sample 7 also consists of two sets ofcknxh that is,

and 4.

runs 3

These are both for two different lengths approxi-

mately two to one in ratio taken at 9 K (See Table l for

sample lengths).

The results of sample 10 run 1 are contained in Table

3 (See Appendix A).

Included in this table are two sets of

65

data for the measured thermal conductance as a function of
temperature for two different applied forces. Although the
short Cu bellows (that allowed us to measure the applied
force) had been removed for these and subsequent experiments,
we can nonetheless estimate the two forces.

The applied force for the first set of data of Table 3
was approximately the maximum force we could apply 30 nt.
The force for the second set of data of Table 3 was adjusted
so that the thermal conductance at 5.0 K was approximately
equal to one-half the thermal conductance at 5.0 K for the
maximum applied force. To a first approximation the rela-
tionship between the thermal conductance and the applied
force can be assumed to be linear, as we shall later show,
so that the second force is approximately 15 nt.

2. Remarks on the Observed Effects of the Applied
Force on the Measured Thermal Conductance

The data in Table 2 indicate that the measured thermal
conductance increases as the applied force increases. This
is expected since the area in contact at the Cu—Ar and
Al-Ar interfaces increases, since the Ar will yield plas-
tically at these interfaces as the force is increased. Since
the yield stress of Cu and A1 is 102-103 times greater than
the yield stress of Ar throughout the temperature range
studied, it can be safely assumed that Ar is the only mate-
rial to plastically deform during these experiments. Ber-

man41 measured the yield stress of Cu at 4.2 K and found it

to be 3.8 kbar, while Leonteva, gt gt.42 found the yield

66

Table 2

Effective Thermal Conductance at Constant Temperature

Sample No. 6 Run No. 4
T = 5.2 K T = 34.0 K
Force(nt) KEff(mw/K) Force(nt) KEff(mw/K)
0.0 0.12 i 4% 0.0 1.62 $ 6%
3.6 $ 6% 0.75 $ 4% 3.6 $ 6% 4.60 $ 6%
7.9 i 6% 1.70 i 4% 7.9 $ 6% 5.18 i 6%
17.5 i 3% 2.29 $ 4% 13.3 $ 4% 5.85 $ 6%
Sample No. T = 9.0 K
Run No. 3 Run No. 4
Force(nt) KEff(mw/K) Force(nt) KEff(mw/K)
0.0 0.12 $ 4% 0.0 0.43 $ 4%
0.8 i 12% 0.33 $ 4% 0.6 t 17% 3.75 $ %
1.7 $ 12% 7.56 $ 4% 1.4 t 14% 5.13 $ %
3.6 i 6% 9.89 $ 4% 2.1 $ 10% 6.79 $ 4%
5.6 $ 9% 13.07 i 4% 2.9 $ 7% 7.37 $ 4%
7.9 i 6% 13.84 $ 4% 6.2 $ 8% 11.12 $ 4%
13.1 i 4% 14.92 i 4% 7.2 $ 7% 12.10 $ 4%
7.2 $ 7% 11.25 $ 4%

 

67
stress of polycrystalline samples of Ar to be 14.0 bar at

4.2 K.

That the Ar does indeed deform plastically at the sur-
faces can be made clearer by plotting the effective thermal
resistance (REff = 1/KEff) as a function of the reciprocal
of the applied force (F_l) for reasons which will be de-
scribed below (See Equation (15)). The plots of these
variables calculated from Table 2 are displayed on Figures
6 and 7. Although the data are sparse for sample 6 run 4,
the 4 sets of data on Figures 6 and 7 clearly fit a straight
line.

This behavior can be understood by noting that REff
consists of the thermal resistance of the crystal (R = L/KA)

in series with the thermal contact resistance

(RCont = l/KCont) at the two interfaces (Al-Ar and Cu-Ar).
That is,

REff = R + RCOnt (14)
but

REff = R + CF_1 (15)

where C is just some proportionality constant.

Since on a microscopic level the ends of the sample are
quite rough, we expect that the contact area will increase
since the surface asperities deform readily while the force
is increased. We can thus conclude that the thermal contact
conductance at the interfaces is proportional to the applied

force, that is,K F.

0:
Cont

 

68

 

1.4-

c5 c3 t3 FL
01 00 C: no

O
.2

EFFECTIVE THERMAL RESISTANCE R5,,(K/Mw)

0.2

 

Figure 6.

SAMPLE N0. 6 '
RUN NO. 4

IS.2 K

A3“) K

 

1 1 1
0.1 0.2 0.3
INVERSE FORCE F'1(NT)'1

A plot of the effective thermal resistance of
sample 6 as a function of the reciprocal of the
applied force.

 

EFFECTIVE THERMAL RESISTANCE R5,,(K/Mw)

69

 

.20_

    
    

SAMPLE N0. 7
A RUN N0. 3
O RUN N0. 4

T=9.0 K
.15

 

0,0 1 1 1 1 1 1 1
0.0 0.2 0.4 0.6
INVERSE FORCE F'1(N'I’)'1

Figure 7. A plot of the effective thermal resistance of sample 7
as a function of the reciprocal of the applied force.

 

70
This result is in agreement with the experiments of
41 43 44
Berman, Berman and Mate, and Mate who measured the

thermal contact conductance between pressed surfaces of
various materials (Cu—Cu, Cu-diamond, Au-Au, sapphire-
sapphire) as a function of the applied force at liquid He
temperatures and observed an almost linear dependence.

An approximate relationship between K and F can be

Cont

obtained by considering the following simplified picture of

 

the surfaces. Assume that the average sized asperity is
contained within a cube of side d, then the total boundary
conductance of n asperities in contact (which can be con-
sidered to be in parallel) is given by

KCont = an (16)
where K is the thermal conductivity of the material of the
individual asperities. The total surface area A in contact
is determined by yield stress PC of the material and the
force applied F. That is, as force is applied the asperi-

ties will continue to deform plastically until the ratio

F/A = Pc the yield stress. That is A = F/PC = nd2 so that
d = (F/nPC)% . (17)
Substituting this result in Equation (16) yields that
KCont = (nF/Pc)%K. (18)
Tabor has shown that for two steel surfaces in contact
nng, hence KContaF3/4’ which is in approximate agreement
with our results. Tabor has also shown that K FO‘6

CI
Cont

for deformations that take place elastically.45

71
In Figure 8 we show the data plotted from Table 3 (open

circles, filled circles and filled triangles) including the
data from sample 10 run 2 (filled squares). The two sets of
data from Table 3 (open and filled circles) diSplay essen-
tially the same qualitative behavior, except that the maxi-
mum of KEff(T) has been shifted. The shift of the peak
toward a higher temperature for the smaller applied force
(filled circles) is probably due to the increased dominance
of RCont over R.

Curiously, below the peak the two lower sets of data
(Open and filled circles) in Figure 8 have a definite T2
dependence, while the data of sample 10 run 2 (filled
squares) have a temperature dependence somewhat less than
T2. It should be recalled that for sample 10 run 2 we
attempted to maintain thermal contact with the crystal
ends during the cooldown process. Although better thermal
contact is achieved as is evidenced by the almost two-fold
increase in the effective thermal conductance near the peak
for run 2 (filled squares) over run 1 (open circles), it
is unfortunately accomplished at the expense of the quality
of the sample. Although some deformation of the entire
sample is expected, severe deformation is probably confined
to the sample ends. However this may be an irrelevant
point since the samples are thermally strained 1% or more
during cooldown. Even a strain this small is large enough

to induce plastic deformation above 65 K.42

TOO

 

tn
c:
I

I I II

Ln
I

EFFECTIVE THERMAL CONDUCTANCE K5,,(Mw/K)
E3

 

II
.I I
I I
I I
I I
000 o
I o 0 0C I
o
f 0
Q 93
04 o -
O ‘ o a... o
0 o
. O I
A
o o '0 o
n . O

ICONTACT ACHIEVED AT 80 K
‘INCREASING FORCE
CMAXIMUM FORCE AT 5 K
OHALF-MAXIMUM FORCE AT 5 K

 

 

1 I I I I I | I I I I I 1 Jill I I I

l l 50
TEMPERATURE T(K)

Figure 8. A plot of the effective thermal conductance for different

methods of making mechanical contact.

100

 

 

73

3. Force Experiment Errors

The error in measuring the applied force F is due
principally to the hysteresis associated with the non-
elastic behavior of the small Cu bellows when large forces
are applied. In a separate experiment this bellows was
calibrated at 77 K by pressurizing its interior with He
gas. The applied force F was determined from the applied
over pressure and the cross-sectional area of the bellows
(1.13 cm2). Its extension was then measured as the force
(pressure) was increased from zero to a maximum of 21 nt
(1390 Torr) and then decreased to zero. Since these meas-
urements yielded two sets of curves, the calibration curve
was taken to be the mean of these two measured curves.
Thus the maximum error in measuring the applied force F due
to the inelastic behavior of the bellows was determined to
be $0.1 to $0.2 nt for applied forces between 0.0 and 5.0
mt and $0.5 nt for forces between 5.0 nt and above. The
percentage error varied from 17% for the smallest applied
force to 3% for the largest applied force. The error in
measuring the absolute temperature T is due principally
to the limits on our ability to keep the sample temperature
constant (See Chapter II, Section 2). We estimate that the
maximum error in measuring T (ST) is then $0.001 K for
2 KST<10 K, $0.005 K for 10 KST<4O K, and $0.010 K for
40 KETE83 K.

Since we were able to measure O to better than 0.1%

the largest source of error introduced in determining KEff

74

is in measuring AT. The maximum error that AT will in-
troduce is twice the maximum error in T (26T). The AT's
used varied from 0.05 K near 4 K to 0.25K near 80 K. We
estimate from these values that the maximum percent error
in KEff varies from 4% at 4 K to 8% at 80 K.
B. Results of the Thermal Conductance Measurements
Table 4 in Appendix B contains all the measured thermal
conductance values and their corresponding temperatures for
samples 10-16. These samples were used to determine K.
The data presented in this table satisfy two experimental
criteria.

First, the sample tube remained reasonably leak tight
during the experimental run. That is, the sample did not
gradually leak out of the sample tube into the surrounding
vacuum region over a 24 hour period. A gross leak of this
size could be easily verified by the inability of the vacuum
pumping system to reduce the vacuum chamber pressure below
10 mTorr.

Second, mechanical contact with the sample was main—
tained throughout the cooldown process. That is, it should
have been possible to continuously remove the slack on the
lifting screw due to the contraction and sublimation of
the sample. If the sample tube and the Cu piston remained
frozen together for more than 15 minutes, we assumed that
the mechanical contact had been lost.

Any data that did not meet these two criteria were

discarded.

75

1. Thermal Resistance versus Sample Length
As suggested earlier, in the Introduction, the simplest
model for the effective thermal resistance is two thermal

resistances in series. One of these is due to the imperfect

1

ont' The

contact achieved at the sample ends RCont = K;
other thermal resistance is due to the presence of the Ar
sample which may be written, by definition, R = L/KA (L =
length of the sample and A = cross—sectional area of the
sample). Recalling Equation (10), this is expressed as

1/KEff = L/KA + l/K (10)

Cont ‘

In order to determine whether the above relation is
reasonably represented by our data, we must know KEff for
each L at exactly the same temperature for the temperature
range investigated. Since it is not feasible to measure
KEff at exactly the same temperature for each L, we drew

the smoothest curve through the raw K (T) data for each

Eff
length to interpolate and extrapolate for KEff at conven-
ient temperatures. The values of KEff(L) for these temper-

atures were determined graphically and are contained in
Table 5 along with the corresponding temperature.
2. Calculation of the Thermal Conductivity

Figures 9 and 10 are plots of REff as a function of L
(length) at constant temperature for various isotherms
from 2.25 K to 26 K. It is apparent from Figure 9 that be—
low 3.5 K the longer samples show a greater scatter than
the shorter samples. However as the temperature increases

above 3.5 K the scatter diminishes and the data approach

EFFECTIVE THERMAL RESISTANCE Raff(Mw/K)

76

 

 

 

 

.20
— o 2.25K O
_ 0 2.50K _
0 3.00K o
— I 3.50K
_ 4A 5.00K D
o
D
.15 __
D
I c
o
_ o
D
3 o
.10 __ O C I
o o D I
— C] D O I
D D
.
I ;
— o o
o . ,
I A i
I I ‘ |
I “ I
.05- ' I
A I
— A |
A A A A i
” |
.00 l I I I L I
0.0 1.0 2.0 3.0

LENGTH L(CM)

Figure 9. A plot of the effective thermal conductance versus
sample length for several isotherms between 2.25
and 5.00 K.

EFFECTIVE THERMAL RESISTANCE REff(K/Mw)

77

 

 

 

.20 __ A
A1 26K
P II 22K I
0 18K
L C1 12K I
r CD 8K
I- A .
.15 __
L o
|
o
I- I
I o
I— A
.1oI_
I A A .
L I
_ D
I ' D
_ O D
_. O
.A. . D
.05.__
I I
D
P D
; A. ° 0
1 . a
1. “
I
I.
,OOI I I I I 1 I
0.0 1.0 2.0 3.0
LENGTH L(CM)
Figure 10. A plot of the effective thermal conductance versus sample

length for several isotherms between 8 and 26 K.

 

78
very nice straight lines.

Since the samples were grown and manipulated in essen-
tially the same manner, we assume that K (thermal conducti-

vity) and K (thermal contact conductance) are the same

Cont

for each sample at the same force. Thus, in order to deter-
mine these two quantities we fit the data of Table 5 using

the linear regression method to a straight line of the form

REff = mL + RCont (19)
where K - (mA) and KCont — RCont .

calculations along with the correlation coefficients are

The results of these

contained in Table 6. A plot of the thermal conductivity
K(T) is contained in Figure 11 along with the data of
Clayton and Batchelder23 and Krupskii and Manzheliizo’21
for comparison. Above 65 K our plotted thermal conductivity
values in Figure 11 are those obtained from Table 5 for
sample 15 run 1.

Before we present the high temperature data in row
form, we mention three interesting effects displayed by the
raw data contained in Table 4. The three effects we are
considering are as follows: (a) the dominance of RCont(T)
for the shorter samples (L S 1.34 cm) at low temperatures
(T E 6 K), (b) the broadening and shifting of the maximum
(T) and (c) the

of (T) due again to presence of K

KEff Cont
dominance of K at the higher temperature (T Z 38 K).

It is clear by comparing columns 2-5 in Table 5 with
column 3 in Table 6, for temperatures below 6 K, that the

KEff is very nearly equal to the thermal contact conductance.

79

on.m om.m ov.w 00.x N.HH m.mH m.ma v.mm N.mm oo.mm
om.m ma.m v.5 H.m h.NH o.¢a o.mH m.vm o.mm oo.om
om.h om.h n.m o.oa m.¢a m.ma m.na v.hm o.mm oo.ma
oa.m om.m H.OH m.mH m.ma m.mH 0.0m m.om N.H¢ oo.oa
m.oa ~.oa o.mH o.mH N.om o.mm e.mm o.vm v.66 oo.va
m.ma m.ma o.va m.na m.vm «.mm m.>m H.mm m.ov oo.ma
b.¢a m.¢a m.mH 0.0m m.mm m.>m n.mm m.mm m.>v oo.HH
n.mH m.oH m.ha N.mm m.mm m.mm o.mm w.ov 0.56 oo.oa
5.5H m.ha m.ma e.mm 0.0m v.om «.mm 6.Hv o.>v om.m
m.ma m.ma m.ma v.vm m.Hm o.Hm m.vm o.m¢ m.ov oo.m
o.om m.ma m.om m.mm m.mm m.mm m.mm N.m¢ m.wv om.m
o.HN N.HN >.HN m.mm m.mm o.mm o.om o.mv o.mv oo.m
v.am o.mm m.mm H.nm m.vm ~.mm m.mm o.H¢ m.mv om.n
m.Hm v.mm v.mm 6.5N m.vm o.mm m.mm «.mm S.H¢ oo.w
m.om m.mm m.HN 0.5m m.mm m.mm o.vm m.>m w.mm om.o
m.ma h.am m.am m.mm o.mm o.Hm m.mm o.mm o.mm oo.m
m.ma m.om 0.0m N.mm 0.0m m.mm m.om «.mm m.mm om.m
o.wa «.ma m.nH m.mm m.hm e.mm w.nm o.mm 0.0m oo.m
h.ma 0.0H m.mH h.ma o.vm H.mm o.vm v.mm N.mN om.v
>.HH w.ma w.mH m.ma b.om H.ma h.om m.am m.mm oo.v
om.m v.HH v.oa o.vH m.ha o.mH m.>a v.mH m.ma om.m
oo.m ov.m om.m v.HH m.¢H +m.ma m.va o.mH «.ma oo.m
om.h cv.m om.> +H.0H o.ma +m.HH o.ma m.ma n.ma mn.m
om.m ov.h om.m +m.m +m.HH +m.oa m.HH m.HH H.ma om.m
on.m ov.m +mm.m +m.h +0.0H +N.m o.oH +v.oa +m.oa mN.N
I s ~ ~ I s s s s
m mmmm m mmmm n mmmx m mwmm m mwmx v wwmx m mmmm m mmmm H wmmm Amve
mmsam> AM\3EvmmmM ConuOOEm

m mHQmB

80

mm.H
0H

mh.a
vm.a
NH.N
0¢.N
Hm.m
¢H.m
mm.m
mm.v

u
0H mmmm

wm.m
m

00.N
Hbm.m
+0N.m
+00.m
+Nv.v

00.m

m wmmm

«m.m
m

mo.m
mN.N
Hm.m
mh.m
00.m
mm.m
mm.m
ma.v
mm.v

\
m mwmx

HV.N
h

mm.m
mm.m
mh.m
00.m
mm.m
mw.m
00.v
0m.v
00.m
00.m

0 mmmx

>0.H
0

0N.m
H00.m
+0m.0
00.0

s
w wmmm

wm.a
m

mm.m
ow.m
NN.m
wo.m
ma.v
00.0
m0.m
mm.m
om.m
0m.0
00.0
0m.h
0N.m
00.0
00.0

m mmmm

A©.pcoov

0H.H
v

00.0
+0m.0

v.0H
m.HH

w mmmm

m magma

mm.0
m

vu.m
HmH.0
+mm.0
+0H.n
00.0
0m.m
00.0
00.0

0.0H
0.HH

m mmmm

m0.0
N

m0.n
mm.m
N0.m
mv.0H
I0.HH
mm.0a
m.aa
m.NH
0.0H
m.ma
m.0H
N.mH
0.0a

N mmmx

mm.o
H

ba.a
Ho.HH
mm.NH
ma.ma
m.mH
1m.mH
Mm.aa
ma.ma
0.0m
m.am
6.mm
m.am
m.mm
o.mm
6.am

H mmmm

mmsHm> pmumHommnuxm +

15004

“C

s
C mwmx

00.00
00.mm
00.00
00.m0
00.00
00.mm
00.0m
00.0w
00.Nv
00.00
00.mm
00.0m
00.vm
00.Nm
00.0m
00.0w
00.0N
00.0w

Axes

81

Table 6

Calculated Thermal Conductivity and
Thermal Contact Conductance

T(K) K(mw/ch) KCOnt(mw/K) Correlation KCont/K(Cm)
CoeffiCient
2.25 33.4 $ 22% 13.9 $ 20% 0.900 0.42
2.50 38.9 $ 23% 15.9 $ 21% 0.912 0.41
2.75 44.4 $ 23% 18.0 $ 20% 0.922 0.41
3.00 49.7 $ 21% 20.2 $ 18% 0.931 0.41
3.50 61.9 $ 20% 24.4 $ 17% 0.941 0.39
4.00 75.6 $ 17% 28.8 $ 14% 0.948 0.38
4.50 90.8 $ 17% 33.1 $ 14% 0.951 0.36
5.00 107.9 $ 17% 37.5 $ 13% 0.954 0.35
5.50 122.1 $ 16% 41.3 $ 11% 0.966 0.34
6.00 129.4 $ 12% 44.8 $ 9% 0.977 0.35
6.50 128.9 $ 12% 48.1 $ 10% 0.977 0.37
7.00 126.3 $ 12% 51.0 $ 10% 0.984 0.40
7.50 119.8 $ 12% 53.7 $ 11% 0.988 0.45
8.00 110.6 $ 11% 56.3 $ 12% 0.989 0.51
8.50 99.2 $ 10% 59.1 $ 12% 0.992 0.60
9.00 89.0 $ 9% 61.5 $ 13% 0.993 0.69
9.50 80.8 $ 8% 63.1 $ 13% 0.994 0.78
10.00 73.1 $ 7% 64.8 $ 13% 0.993 0.89
11.00 61.0 $ 6% 70.3 $ 14% 0.995 1.15
12.00 52.3 $ 5% 73.8 $ 15% 0.995 1.41
14.00 40.6 i 4% 78.7 $ 16% 0.994 1.94
16.00 33.2 $ 4% 76.3 $ 16% 0.994 2.30
18.00 27.4 $ 4% 79.1 $ 22% 0.995 2.88
20.00 23.0 $ 4% 82.1 $ 22% 0.995 3.58
22.00 19.7 $ 3% 84.8 $ 23% 0.994 4.31
24.00 17.1 $ 3% 81.0 $ 22% 0.994 4.73
26.00 15.1 $ 3% 96.0 $ 26% 0.994 6.40
28.00 12.7 $ 3% 211.3 $ 57% 0.997 16.69
30.00 11.3 $ 3% 353.3 $ 96% 0.997 31.21
32.00 10.2 $ 2% 221.6 $ 48% 0.999 21.68
34.00 9.27 $ 2% 356.9 f 77% 0.999 38.50
36.00 8.39 $ 2% 1504.7 >$ 100% 0.998 179.30
38.00 7.58 $ 2% 666.6 >$ 100% 0.998 87.942
40.00 6.86 f 2% -1l4.0 >$ 100% 0.997 >102
42.00 6.65 $ 2% -236.5 >$ 100% 0.996 >102
46.00 6.38 $ 2% ~469.7 >$ 100% 0.996 >102
50.00 5.65 $ 2% -275.3 >$ 100% 0.997 >102
55.00 4.67 $ 2% +276.8 >$ 100% 0.992 >102
60.00 4.33 $ 2% 274.2 >$ 100% 0.997 >102
65.00 3.75 $ 2% 1213.9 >$ 100% 0.992 >10

82

 

 

 

 

200
A
A A a°°°a
A n o
100.. A A o
_. £5 0 o
C A ° :0
501 A a A 0
AN 0 1A
_. 0 0
g A
a P D "‘ D
\ 0
£1
C; D
X __ £5 0
E: A‘ D
>
s in
§10__ '0
c.) _ m
:s; -— “3
E — Chev
FE _' II ‘7
St A CLAYTON a BATCHELDER DATA g V
_ v SMOOTHED CLAYTON a BATCHELDER DATA '3 vV
_ . SMOOTHED KRUPSKII a MANZHELII DATA ‘8 v
o PRESENT DATA in
O
l I 11L11111 I 11111111
1 5 10 50 100

TEMPERATURE T(K)

Figure 11. A plot of the thermal conductivity versus temperature.
Included for comparison are the data of Clayton and
Batchelder and Krupskii and Manzhelii.

83
For example, at 2.25 K,(RCOnt/REff)(100%) varies from 77%

to 42% for the shortest to the longest sample. To illus-
trate this effect more clearly we show in Figure 12 plots

(T)

of the effective (or measured) thermal conductance KEff

for samples of various lengths. Below 6 K the data converge

to KCont as the samples become smaller in length.

Figure 12 also illustrates the second feature, that

ff lS

broadened and shifted toward higher temperatures. We expect

is, as the samples become shorter the maximum of KE

that as L+O, K That this is approximately correct

Eff+KCont'

can be seen by comparing Figure 12 with Figure 13 which con-

tains a plot of K (T) taken from the data of Table 6.

Cont
Between 2 and 20 K it can be seen that KEff(T) does indeed
approach KCont(T) as L becomes shorter. (Above 20 K the
behavior of KCont(T) is slightly more complicated. We shall

discuss this later.)

The third feature is that above 38 K the conductance of
the sample is now dominant. For example at 38 K (RCont/REff)
(100%) varies from 2.8% to 0.3% from the smallest to the
largest sample. To further illustrate this effect in Figure
14 we plot KEff(L/A), the effective thermal conductivity,
as a function of temperature for several sample lengths.

In view of this last effect we can safely assume that
above 38 K the measurements we have carried out have yielded
K(T) directly. In Figure 15 we plot the raw thermal con-
ductivity data (corrected for KSp the spurious heat conduc—

tance due to the mylar tube, wire leads, etc.) of Table 7.

 

84

 

 

 

 

100
L.
I—
50..
I.‘. .
_. moo
.0
:2 EB. '0. U
\\ _. are .
g .000000
V C
: Ido AA‘ C
01 D I “ A o
:4 P— l. o ‘ .
g ..00. A -
E I C. O
L) o.
:3
Q . OA
€3.10L_ o
L) __ 4 . o
__J A
< )—
E A
m )—
LLJ A
g; .—
LU 5..
E: I 0.33 CM
*—
tfl 7' D 0.63 CM
ft
uJ __ 0 0.95 CM
0 1.97 CM
1. ‘ 2.84 CM
1L 1 1 1 11111 1 11
1 S 10 50

TEMPERATURE T(K)

Figure 12. A plot of the effective thermal conductance versus
temperature for samples of different lengths.

85

 

1000

1..
F
r—
500%
/\ __
.\_4
E
r F—
V
w
c
O
U __
:2
uJ
L)
2:
<
p.—
L)
Fa”
z 100_
C)
U )-
.._ I——
L)
< r—
P—
2: __
8
,
__, :0.
<2
E
33 F—
I—-

 

 

IIIIII I I IIIIII

l 5 10 50 100
TEMPERATURE T(K)

Figure 13. A plot of the thermal contact conductance versus temperature.

......
C)
I—

 

100

/CM K)

KMW

I

K:

,, x LENGTH/AREA

U1

h

50

p._1
CD

 

86

 

 

 

 

 

L.
_. - II
1... I
I
I- . _\41CO(‘_ I
-c *
__ J L I
. C A“A CC I
. Q ‘A‘ AA
e— cC A‘ 0 ‘ '
. C A C] DOD.
o ‘ D
C) C] .
I C ‘ D A
C D D -
I— A
I ‘ 00 ..° 00/
A 0 ,°'° . . I
A D 0
. 0 ° ° DC
’ U ... o .0
A D o
._ ‘ [P . ' 9:-
y—_ C] O I
I— ° '7
o ‘0
F— 0 °
+— 0 $7.
I- . I
+— .
| I 2.98 CM I
I. 0 1.34 CM '
I . 0.95 CM I
L— l
1 D 0.63 CM .
i - 0.33 CM 1
I
1 I I I I I II 1 I 1 : I
l 5 10 SO
TEMPERATURE T(K)
Figure 14. A plot of the effective thermal conductivity versus

temperature for samples of different lengths.

THERMAL CONDUCTIVITY K(MN/CM K)

87

 

15.0%.

10.0L

8.0u

 

   
 

 

 

6.0 _
A.0_ A 0.33 CM
0 0.63 CM
. 1.28 cm
_ A 1.34 CM
0 SMOOTHED DATA CORRECTED
FOR THERMAL BOUNDARY °
CONDUCTANCE
2.0—- ——— THREE PHONON THEORETICAL
THERMAL CONDUCTIVITY (CHRISTEN a POLLACK)
1.5 l 1 l l L a a L
15 20 A0 60 80 100
TEMPERATURE T(K)
Figure 15. A plot of the high temperature thermal conductivity

data and the three-phonon thermal conductivity
calculated by Christen and Pollack.

88

Table 7

Experimental High Temperature Thermal Conductivity Data

T(K) KEff(mw/K) Ksp(mw/K) K(mw/cm K)

A. Sample No. 11 Run No. 5

37.83 5.51 0.05 8.38
45.40 4.28 0.07 6.47
63.45 2.63 0.10 3.89
B. Sample No. 13 Run No. l
39.51 19.26 0.05 7.31
49.53 16.43 0.07 6.23
60.87 10.49 0.09 3.96
C. Sample No. 15 Run No. 1
83.08 1.82 0.12 2.51
81.70 1.77 0.12 2.44
79.90 1.90 0.12 2.63
78.27 2.00 0.12 2.78
76.00 1.97 0.11 2.74
72.85 2.14 0.11 3.00
68.72 2.37 0.10 3.35
65.75 2.55 0.10 3.62
62.92 2.86 0.10 4.08
59.65 3.04 0.09 4.35
55.93 3.29 0.09 4.73
52.85 3.83 0.08 5.54
51.34 3.92 0.08 5.68
48.29 4.36 0.07 6.33
45.99 4.61 0.07 6.71
52.50 3.91 0.08 5.66

89

Table 7 (cont'd)

T(K) KEff(mw/K) Ksp(mw/K) K(mw/cm K)

D. Sample No. 17 Run No. 1

41.38 11.11 0.06 8.03
42.25 10.07 0.06 7.28
44.61 9.34 0.06 6.74
47.61 7.87 0.07 5.60
50.39 8.25 0.07 5.94
53.81 6.72 0.08 4.82
E. Sample No. Run No.
39.31 2.64 0.05 7.19

9O
Along with these data we also present for comparison the

theoretical three phonon K(T) calculation of Christen and
Pollack.24 We should note that between 30 and 80 K the
experimental data and the theory are in good agreement. We
shall discuss this later.

3. Remarks on the Thermal Conductivity Measurements

The three main features we have just discussed can be
understood by recalling some of the unusual properties of
solid Ar: (i) the large coefficient of thermal expansion
and (ii) the low yield stress. These two properties are

clearly manifested by the behavior of K (T) (Figure 13).

Cont
Between 4 K and 10 K the relative change in the lattice

constant of Ar is negligible (approximately 0.01%)46 so

that for a constant applied force (as is the case here) d

and n in Equation (16) should be essentially constant in

this temperature range. In the last column of Table 6 we

show K = nd which does remain essentially constant

Cont/K
in this temperature range. Thus KCont(T) is proportional
to K. This temperature behavior (Figure 13) below 10 K is
in quantitative agreement with the results of M05547 who
measured K(T) of plastically deformed crystals of CaF2 and
found that K varied somewhat less than T2.

Above 10 K, n and d are expected to increase with an
increase in temperature. This is because the sample as
well as the asperities will begin to experience a signifi-

cant increase in size. In going from 4 K to 20 K the rela-

tive increase in the lattice constant is 0.13%, while in

91

going from 4 K to 40 K it is 0.69%.46

As the sample in-
creases in length relative to the stationary sample tube
ends, the applied force to the sample ends is effectively
increased. The increasing force coupled with a diminishing
yield stress results in a rapid increase in the contact

area above 20 K. Hence, we expect that above 20 K,K (T)

Cont
and K /K will both rapidly rise. A quick glance at

Cont
Figure 13 and at the last column of Table 6 show convinc-
ingly that our results support this conclusion.
The first and second effects, (a) and (b), mentioned
in the previous section are due mainly to a reduction in
the contribution of R (thermal resistance of the sample)
because of a reduction in L and also possibly an increase
in the thermal conductivity K for the shorter samples. This
latter possibility can be investigated by calculating K(T)
separately for the short samples (0.33 cm 5 L S 1.34 cm)
and long samples(l.97cm1$ L S 2.98 cm). In Figure 16 we
have plotted only the meaningful data contained in Table 8.
The plots in Figure 16 indicate that the shorter sam-
ples are apparently less damaged (defected) than the larger
ones. That is, the thermal conductivity peak of the shorter
samples is higher (in fact a factor of 2 higher) and occurs
at a lower temperature than for the longer samples. This
is consistent with both the experimental observations and
the fact that the shorter samples will experience smaller

thermal gradients during the cooldown process. This result

supports our belief that the thermally induced strains cause

92

hmm.o om.bm ma.mn Hmm.o mH.Hm H.m> oo.oa
wvm.o m~.mm Hm.mm mmm.o mm.mm m.mm om.m
mvm.o wo.vm ma.vm .mmm.o mo.>m m.HoH oo.m
mmm.o mm.om >.moa nnm.o nm.vm m.mHH om.m
Nam.o om.>v m.mma mom.o mm.am m.mma oo.m
ham.o ~¢.m¢ m.mma mmm.o am.mv v.nma om.>
oam.o mm.om H.omH mmm.o mm.mv m.mma 00.5
nbm.o No.mv m.¢ma mmm.o oa.mw H.mam om.o
mom.o mv.mm H.NHH vvm.o Ho.mm m.¢mm oo.w
vom.o mm.am mm.am Hom.o mm.mm m.m¢m om.m
5mm.o va.mm mm.mn mmn.o mm.Hm m.oom oo.m
vmm.o mw.vm Ho.om mom.o mm.>~ m.mam om.v
vvm.o mm.mv vm.mm mm>.o ¢¢.mm N.oma oo.v
mmh.o Hm.mm hm.mv mmh.o Ho.aa m.m>a om.m
vm>.o mm.vm mh.mv mmn.o Hm.mH m.nma oo.m
omm.o mm.na vm.mv mmm.o mm.ma o.mmm m>.m
wmm.ol Hm.¢ vw.h¢1 omm.o mm.ma h.mvm om.m
www.cl wm.m vo.mm| omm.o mm.oa c.0HN mm.m

ucwflowmmmoo ucmHOmemOU

coflumamuuoo AM\3Evucoox AM EO\BEVy cowumamuuoo AM\BEVuCOUM Ax EO\BEVy AMVB

EU mm.m w d w E0 hm.H EU vm.H w a w EU mm.o

mmHmEmm mcoq paw uuonm mo mumo wpfl>flpospcoo Hmanmce

m mHQMB

1000

93

 

500

100

50

THERMAL CONDUCTIVITY K(Mw/CM K)

T E

T

 

10

° SHORT SAMPLES 0.33 - 1.34 CM
0 LONG SAMPLES 1.97 - 2.98 CM
I ALL SAMPLES 0.33 - 2.98 CM

 

 

0 o
c. . .
. o

o

....

w.-
e 9
r '9
P— M?
__ I
__ I
_. I

I
P.
I
I
r- I
I
I
I
L, T l T 1 l 111 l l | l l
l 5 10 50
TEMPERATURE T(K)

Figure 16. A plot of the calculated thermal conductivity versus

temperature for short and long samples.

94

more damage to the entire sample than pressing on the sam-
ples ends.

The third effect (c) we mentioned in the previous sec-
tion is simply due to the rapid increase in contact area
resulting from the combination of the increasing applied
force and the reduction of the yield stress of solid Ar.
This results in the significant reduction of RCont relative
to R near 38 K for all samples.

4. Thermal Conductivity Errors

In determining K for Ar from 2.25 K to 65 K we recall
that the data were fit to Equation (10) using from 9 to 3
different lengths (See Table 5) for a given isotherm using
the method of least squares. For clarity we recall Equa-
tion (19)

REff : mL + RCont (19)

l = 1/K

where REff = KEff’ m = l/KA and R Since we

Cont Cont'
were able to measure L directly within i 0.3%, we will
assume that L is known exactly to determine the uncertainty

in K and K The quantities contributing to the uncer-

Cont'
tainty are therefore REff and A the cross sectional area
of the samples.

'In this case if the error in REff is the same for each

length Li’l°e'6REff,i = SREff,then the variance of m is
. 48
given by
v<m) = <(6m)2> = (a )2/(2 L 2 - (2 L )Z/N) <20)
REff ii ii
and the variance of R is given by

Cont

95

V( = <(5 )2>

RCont
2

RCont)

_ 2 2 _ 2
— (aREff) (i Li )/(N§ Li T; Li) ). (21>

l

Ordinarily, if R were exactly reproducible for each run

Cont
at a given length Li' then GREff would be proportional to
6(AT). Because of the way in which we smoothed the data,
we expect the errors associated with AT to average to zero.
Hence, the major contribution to the uncertainty in the
determination of REff is the uncertainty of reproducing the
contact resistance SRCont for each run.

To estimate the maximum uncertainty associated with
RCont we compared the data of sample 14 runs 1 and 2 shown
in Figure 17. These two runs we carried out on the same
sample for approximately the same length 3.00 cm and 2.98
cm, respectively. We note that for run 1 mechanical contact
was lost and then reestablished at 25 K, thus these data
were not used in determining K.

For the expressed purpose of estimating 6R we

Cont'
will assume that K remained unchanged from run 1 to run 2
and that L and A are known exactly. This is a valid as-
sumption, since the sample was manipulated in essentially
the same manner for both runs and annealing between runs

is expected to restore the sample to its original unstrained

49,50

state. If we take run 1 to be representative of the

maximum value of RCont and run 2 to be representative of

the minimum, then the error in RCont (or REff) is given by,

6Reont = 6REff = i 1/2 (REff,l ' REff,2) (22)

100

EFFECTIVE THERMAL CONDUCTANCE K5,,(Mw/K)

96

 

SO._

 

 

 

100

_ ltggoou.
.30 0'
ID -
.C] D.
ID -
10.. . o D I
— D
_ I a DI
_ I n u
I o D -
__ I D
S_ C] DI
_ D
_ E]
SAMPLE N0. 14
a RUN N0. 1 L = 3.00 CM
I RUN NO. 2 L = 2.98 CM
1 TIIIIII I IILIlll‘
l 5 10 50

Figure 17.

TEMPERATURE T(K)

A plot of the effective thermal conductance for two samples
of apprOXimately the same length displaying different values

Of RCont-

97
where REff,l and REff,2 are the effective thermal reSist-

ances for runs 1 and 2, respectively, at a given tempera-
ture. The values of SREff determined in this way were

found to vary from 0.021 K/mw at 2.25 K to 0.001 K/mw at

40 K.
The fractional error in KCont is then determined from
Equation (19) and the relation
SKCont/KCont = GRCont/RCont (23)
The value of (SKCOnt/KCont)(100%) was determined to vary

from a minimum of 9% at 6 K to a maximum of over 100% above
36 K. This large error near 36 K is just the consequence
of RCont approaching zero as T approaches 38 K. In fact

the appearance of the physically unreal, negative values

of in Table 6 in the vicinity of 40 K is just the

KCont
result of the straight line to which we fit our data,
statistically missing zero as the y-intercept.

Since K = l/mA, the fractional error for K is then
given by

(SK/K = i (<6m2>/m2 + (6A/A)2);i. (24)

Recalling that (6A/A)(100%) = 2.0%, the values of
(6K/K)(100%) were determined to vary from 22% at 2.25 K to
i 2.0% at 65 K. All of the percentage errors for KCont and
K are shown in Table 6.

For the raw thermal conductivity data, corrected for
the spurious heat conductance (Ksp), shown in Table 7 and

plotted in Figure 15, we compute the percent error in a

straightforward manner. Since for these data

98

K = (KEff - Ksp)L/A, the fractional uncertainty in K is
given by
_ _ _ 2
éK/K — : {(6(KEff Ksp)/(KEff Ksp))
L
+ (6L/L)2 + (6A/A)2}2 . (25)

In a separate experiment with the sample tube and
vacuum chamber evacuated we measured KSp directly and
found it to vary from 1.11 x 10.1 mw/K at 83 K to 1.10
x 10_2 mw/K at 10 K. The uncertainty in this measurement
was less than 0.7% throughout that temperature range. The
magnitude of KSp represents a correction of 7% to KEff at
83 K and a correction of less than 1% at 38 K for a sample
of intermediate length, say 1.3 cm. (Since KSp represents
less than 1% of the total thermal conductance below 38 K
it was unnecessary to correct KEff data for KSp at tempera-
tures below 38 K.) Thus the evaluation of Equation (21)
gives the result thatthe percenterror for the raw data

representingkcis approximately i 8% for the temperatures

between 38 K and 83 K.

IV. DISCUSSION AND CONCLUSION

In the paragraphs that follow we shall briefly discuss
how our data compare with the results of previous workers
and with the theoretical calculations of Christen25 and
Christen and Pollack.24 In our concluding remarks we will
cite the work of Kimber and RogersSI, who carried out meas-
urements of K for constant volume samples of Ne, to aid us
in suggesting possibly a better approach to measure the
thermal conductivity of Ar.

At high temperatures (T20) K is expected to be
proportional to l/T for a constant volume sample. Clayton
and Batchelder who measured K of isochoric samples of Ar
having approximately the same molar volume as ours did in-
deed find that K c l/T above 20 K. Our data, on the other
hand decrease faster than l/T above 20 K and are in good

agreement with the results of Krupskii and Manzhelii21

(see Figure 11) who found that l/K = AT + BT2 (A,B > 0).
The high temperature, theoretical, K calculation of

Christen and Pollack appears to reconcile the deviation of

our data and the data of Krupskii and Manzhelii from the

l/T behavior. In this complicated computer calculation

Christen and Pollack24 used the best known interatomic

99

100

potential (Barker and Pompesz) to calculate the contribution
of the anharmonic components of the crystal potential (three—
phonon contribution) to the thermal resistivity. In their
calculation, in order to take into account the effects of

the lattice expansion, the lattice frequencies were eval-
uated for the equilibrium density at a given temperature.

The results of their calculation are plotted in Figures 15
and 18 along with our data for comparison.

In the temperature range of 30-80 K our data are in
reasonable agreement with the three-phonon calculation of
Christen and Pollack.24 However, as the temperature falls
below 30 K the theoretical curve (dashed curve in Figure
18) falls significantly below our data. This discrepancy
as suggested by Christen and Pollack is probably due to
the model used. That is, it was assumed, for this calcu-
lation, that the collision frequency for normal processes
was much larger than that for Umklapp processes. This has
the effect of pOpulating those states which scatter strongly
by Umklapp processes, depressing the value of the theoretical
K between 6 K and 30 K.

In the low temperature region for a defective solid
dominated by dislocations the expected temperature depen-
dence of K is T2. In general below 10 K our data are in
qualitative agreement with the constant volume data of

Clayton and Batchelder23

(see Figure 11). However, unlike
the T2 behavior displayed by the data of Clayton and

Batchelder, our data show a behavior that is somewhat slower

101

 

 

 

 

 

200 \
D‘IUOD
100 L. o a
E. 0 0
i \ ”a
u. D
50 __ 0
__ 0
Q \ 0
§ ~ \ .
é \ a
X
i - \\ D 0
ES \\\o
L; o
g 10 \ C
8 : \30 I
2;; VERA
85 I‘ 0
55 SE—' /// \9 1
‘— - o PRESENT EXPERIMENTAL DATA ‘10 I
_ / -— RELAXATION TIME THEORY \) '
_/ — — THREE PHONON THEORETICAL R I
THERMAL CONDUCTIVITY “\o
lI— I
ll 1 I lllllll I ITIIJILI
l 5 10 50 ‘
TEMPERATURE T(K)
Figure 18. A plot of the low temperature (solid curves) and high
temperature (dashed curve) theoretical calculations of
Christen and Pollack. Included for comparison are the

present data.

 

102

than T2 as the temperature decreases below 5 K. This is
the same temperature dependence that was observed by
Christen and Pollack. This observed temperature dependence
may well be due to the deformed state of the samples we
studied. However, since our error in this range is so
large this question can only be properly answered after
further investigation.

In Figure 18 we have also included the relaxation
time theory calculation of Christen, although this theory
of low temperature K is still quite phenomenological.
The three solid curves shown in this figure were computed
for three different dislocation densities (0) having values
of 2.5 x 109 cm_2, 5.2 x 109 cm_2 and 1.5 x 1010 cm"2,
respectively, from the uppermost curve to the lowest. These
calculations were performed considering only the conven-
tional scattering mechanisms (dislocations, isotopic impu-
rities, sample boundaries and 3 phonon, normal and Umklapp,
scattering). The calculations also parametrically take
into account the importance that normal processes, relative
to the other scattering processes, have in redistributing
phonon states among the various scattering mechanisms.

Below 3 K the lst curve in Figure 18 (o = 2.5 x 109

cm ) is in agreement with the data of Clayton and

Batchelder, while below 5 K the 2nd curve (0 = 5.2 x 109

cm_ ) turns out to be in rough agreement (within experi-
mental uncertainty) with our data. Unfortunately, these

values are 3 orders of magnitude higher than the values

 

103

27

determined by Peterson, 32 31. from x-ray analysis of sam-

ples of Ar who set a lower limit of o = 106 cm-z.

To end this part of the discussion we will present the
results of previous workers who also carried out equilibrium
volume measurements.

In Figure 19 we show a plot of our results along with
the data of Berne, gt £1.22 and White and Woods.19 Between
3 K and 13 K our data do appear to be in reasonable agree-
ment with the data for one of the samples of Berne, gt 31.
(filled circles). In general, near 20 K,K is expected to
be independent of scattering from defects, however, our data
appear to be significantly higher than the results of White
and Woods.19 This difference can probably be attributed to
a presence of a thermal contact resistance at the tempera-
ture probes, in the data of White and Woods, that was not
properly taken into account.

To conclude this discussion we show in Figure 20 the
data of Christen and Pollack for their runs 8 and 10 along

with the "effective thermal conductivity" = K x L/A)

(KEff Eff

of our sample 12 run 1. If there were no thermal contact
resistance, from the apparent agreement of the data we would
probably, conclude that the thermal conductivity is inde-
pendent of the cooling rate. This is unlikely since our
cooling rates were approximately 70 times those of Christen
and Pollack. The more likely explanation is that the ther-

mal contact resistance in the experiments of Christen and

Pollack probably changed from run to run.

200

THERMAL CONDUCTIVITY K(Mw/CM K)

100

50

5.;
O

104

 

I I
I I l
I dime
II ' o "0
r— 0. . O
_. D
_. CL. . 0
O
__ U. I - U
r— C] C _8l3 . D
_. o A o
‘r I" o
__ .AP 0
D
I
U
.A
A.

 

 

 

”' o

_ ODD

‘—' O

P - . o DERNE, BOATO a DEPAz DATA

__ A WHITE 3 NOODS DATA DA

0 PRESENT DATA “’0

I I I l i ' i I L '

1 S 10 50

TEMPERATURE T(K)
Figure 19. A plot of the thermal conductivity data of several workers

for Ar under its own equilibrium vapor pressure.

100

THERMAL CONDUCTIVITY K(MW/CM K)

 

105

 

100
— o oeoonl0
— . 3"”‘4‘.
50 ,_ 0 .% Q?:.A
O 0? 'kA .
L. ch.' (“‘5‘
2 1.
_ (3“ 2.
10__
L: a
E- o RUN N0. 10 CHRISTEN & POLLACK DATA
SP- - RUN N0. 8 CHRISTEN a POLLACK DATA
r- . EFFECTIVE THERMAL CONDUCTIVITY
__ SAMPLE N0. 12 RUN N0. 1 PRESENT DATA

 

 

_f.___'e

1, l Llllllll 4 1111111

5 10 50 100
TEMPERATURE T(K)

Figure 20. A plot of the thermal conductivity data of Christen and Pollack
from runs 8 and 10. Included for comparison is a plot of the

"effective thermal conductivity" from the present data,
sample 12 run 1.

pa

106
Figure 21 contains a plot of our data corrected for

RCont including the data of Christen and Pollack for runs
7,8 and 10. The data of run 7 seem to be in agreement with
our data and suggest that possibly RCont for this run was
smaller than for the other two.

It is unfortunate that the presence of RCont masks the
true thermal conductivity in measurements of this type,
since the quality of the samples studied by Christen and
Pollack is probably greater than what the data would imply.

In summary, we have demonstrated that above 38 K the
linear heat flow method we have employed yields K of Ar
directly. Unfortunately, at low temperatures (2 to 15 K)
RCont becomes significant and must be reproduced reliably
in order to determine K. In addition, this method yielded
only an average value of K for all of the samples studied.
Thus, unless one is willing to work with infinitely large
samples, our method will not yield K directly.

If one is to avoid the complicating effects of a
thermal contact resistance a truly potentiometeric method
must be used. That is, the probes used to measure AT must
be in good thermal contact with the sample and must conduct
a negligible amount of heat current during the K measure-
ments.

The experiments carried out on equilibrium volume
samples of Ne by Kimber and Rogers51 employed just such a

technique. Kimber and Rogers grew cylindrical samples of

Ne in a glass tube (in a manner similar to ours) around

THERMAL CONDUCTIVITY K(MW/CM K)

107

 

 

 

 

 

 

 

150
00000
D D
100% O a
L. . 0
t 0 008.0000 .00
L— [3 (>03’ru‘.': . 0
50E_ 0 O 2‘? q o
D 3 o
_ C) 0 ° 0
D 0.! g o
I—- 00 D
O o
L. 00 o
D
°o
D
D
lOI_ C]
_ D
O
r“ o
_ Ebb
5‘ - - c CHRISTEN a POLLACK DATA 0
I RUNS 7, 88.10 ”D.
j“ o PRESENT DATA 0
IH- P -
DD
1 I IiIIIILL I IIllLll
l 5 10 50 100

TEMPERATURE T(K)

Figure 21. A plot of the thermal conductivity versus temperature of the
present data. Included for comparison are the data of
Christen and Pollack, runs 7, 8 and 10.

 

108
looped platinum wire 0.5 mm in diameter. Before measure-

ments were commenced the samples were first separated from
the glass tube and then cooled in a slow and even manner to
the temperature range of interest (from 24 K to 4 K in 3
hours). The results for isotopically pure 20Ne samples
were in excellent agreement with the results of Clemans53
who carried out measurements of K for constant volume sam-
ples of 20Ne of the same purity.

In this cleverly designed experiment Clemans had the
capability of measuring the K of the sample in segments and
was thus able to test whether the sample was of a uniform
quality throughout. The sample chamber used by Clemans
consisted of a thick-walled stainless tube with three annu-
lar heaters equally spaced one cm apart and a single ther-
mometer located at the bottom was used to measure AT. In
this way Clemans was able to measure K for the portion of
the sample between the upper heat sink and any one of the
three annular heaters during a single run.

The excellent agreement between these two methods, un—
doubtedly suggests that the successful method used by Kimber
and Rogers be employed in future experiments to obtain

direct measurements of the thermal conductivity of equilib-

rium volume samples of Ar.

 

 

APPENDICES

APPENDIX A

Table 3

Effective Thermal Conductance at Constant Force

Force Gradually Increased to Fmax = 30 nt
T(K) KEff(mw/K) T(K) KEff(mw/K)
5.01 8.72 5.12 12.17
5.07 8.74 5.12 12.77
5.06 10.57 5.13 13.16
5.05 11.65
Force = 30 nt

5.13 13.47 6.52 16.05
4.86 12.41 6.80 16.22
4.49 10.64 7.04 16.29
4.14 9.89 7.30 16.45
3.86 8.19 7.57 15.97
3.59 6.87 7.78 16.25
3.94 8.25 8.01 16.37
3.58 7.10 8.27 15.62
3.28 5.62 8.52 15.70
3.27 5.60 8.74 15.73
4.59 9.42 9.00 15.27
4.60 10.27 9.23 15.15
4.61 10.82 9.50 14.94
4.61 10.79 9.76 14.12
4.83 11.66 9.99 14.12
4.99 12.39 10.46 13.46
5.25 13.94 11.00 12.59
5.50 15.05 11.46 12.25
5.79 14.08 12.02 9.63
5.82 14.47 12.96 11.60
6.02 14.53 14.00 10.40
6.26 15.04 15.00 8.59

109

T(K)

15.01
15.00
17.79
20.04
20.02
20.03
20.03
12.54
12.98
12.01

5.00
5.00
4.51
4.05
4.05
5.52
6.01
6.48
7.01
7.50

KEff(mw/K)

9.58
9.23
8.01
8.29
6.95
6.53
6.94
11.75
11.01
11.68

Applied Force

5.88
5.88
4.72
3.66
3.66
6.77
8.09
9.13
9.68
10.36

110

Table 3 (cont'd)

T(K)

9.99
7.96
6.06
6.95
6.95
5.01
5.01
5.01
5.00

15

8.06
8.52
9.08
9.58
10.01
11.04
12.02
12.99
13.98
14.98

nt

KEff
14

10.
10.
10.
10.
.00
.39
.34
.14
.15
.39

oxxiooooxo

(mW/K)

.57
17.
17.
18.
18.
14.
14.
14.
14.

15
01
68
18
20
34
75
27

71
36
35
46

T(K)

5.05
5.05
5.15
5.21
5.50
5.99
6.02
6.26
6.51
6.78
7.00
7.23
7.42
7.44
7.43
7.42
7.41
7.74
8.02
8.20
8.51
9.03
9.52
9.96
11.08
12.00
13.07
14.03
15.30
16.18
17.24

KEff

Sample No.

29.03
27.66
28.84
28.00
26.70
29.53
31.83
32.56
32.62
33.00
33.10
33.83
33.74
34.32
33.33
33.38
34.23
32.95
32.77
32.80
32.11
31.73
29.91
28.93
26.89
25.44
23.35
21.11
19.40
18.64
18.33

(mW/K)

Table

111

APPENDIX B

4

Effective Thermal Conductance Data

10 Run No.

T(K)

19.87
21.44
23.32
24.79
26.75
28.61
28.59
28.61
28.61
26.09
24.23
23.14
21.78
20.05
17.88
16.07
14.82
13.78
11.95
11.08
9.98
9.52
9.03
8.52
8.03
7.49
7.03
6.51
6.06
5.73
5.52

KEff

14.70
14.23
11.39
11.94
10.44

9.75

8.57

8.80

9.34
10.72
10.41
10.80
13.06
14.52
17.64
20.81
19.84
21.57
23.92
27.32
29.10
30.01
31.51
32.03
32.40
31.57
33.57
32.08
31.02
28.72
28.37

(mW/K)

T(K)

5.25
5.01
4.76
4.51
4.52
4.24
4.00

4.91
5.00
5.12
5.30
5.45
5.64
4.81
4.67
4.49
4.35
4.21
2.91
2.94
2.98
3.01
3.06
3.16
3.26
3.35
3.47
3.52
3.47
3.53
3.64
3.77
3.87
4.02
4.14
4.27
4.38
4.52
4.64
4.59

KEff(mw/K)

27.58
24.71
24.37
22.09
23.05
21.07
20.49

Sample No.

23.84
23.89
24.09
24.56
24.32
24.73
21.21
20.96
19.94
18.72
18.85
11.03
11.97
10.81
11.20
11.66
11.93
12.89
13.42
13.88
13.55
13.81
14.06
14.97
15.61
16.08
16.82
17.77
18.12
18.80
19.40
20.12
20.51

Table 4 (cont'd)

T(K)

3.77
3.49
3.27
3.22
3.27
3.27

11 Run No.

4.79
4.92
5.01
5.08
4.99
5.13
5.22
5.29
5.41
5.52
5.65
5.81
5.89
5.98
6.09
6.09
6.21
6.43
6.60
6.84
7.03
7.18
7.40
7.56
7.78
8.05
8.24
8.38
8.01
7.38
8.38
8.69
8.86

3

18.00
15.95
14.64
15.02
15.15
14.16

21.58
23.24
23.10
24.06
23.94
23.65
24.30
25.41
25.95
25.10
25.34
25.90
26.13
26.25
27.56
27.16
27.04
26.90
27.20
26.27
26.37
27.34
28.72
27.19
26.33
27.09
24.80
25.76
27.06
25.90
23.54
22.98
25.12

KEff(mw/K)

T(K)

8.84
9.06
9.23
9.56
9.79
9.97
10.55
10.99

2.92
2.96
3.00
3.04
3.07
3.16
3.24
3.33
3.40
3.51
3.61
3.73
3.80
3.90
4.03
4.17
4.28
4.42
4.55
4.75
4.95
4.85
5.09
5.26
5.45
5.68
5.64
5.86
5.98
6.12
6.22

KEff)mw/K)

26.94
25.98
24.93
22.82
21.60
20.57
21.21
19.65

Sample No.

14.15
14.27
14.47
14.71
14.95
15.30
15.55
16.41
16.95
17.37
18.56
19.52
19.86
20.50
21.37
22.21
22.99
23.90
24.52
25.51
28.12
27.49
29.01
29.10
30.13
30.08
30.03
29.61
30.26
33.05
32.94

Table 4 (cont'd)

T(K)

11.94
13.08
14.17
15.89
18.36
20.78
24.66

11 Run No.

6.47
6.51
6.73
6.99
7.22
7.47
7.77
8.00
8.23
8.55
8.78
8.32
8.65
8.96
9.20
9.37
9.57
9.78
9.96
10.56
11.07
12.29
14.19
16.19
21.15
25.25
31.16
37.83
45.40
63.45

KEff(mw/K)

17.91
15.47
15.08
13.10
10.09

8.77

6.54

33.57
34.15
34.38
34.78
34.00
34.20
33.68
33.43
33.26
33.28
32.15
33.11
32.05
32.00
30.85
30.48
29.55
28.76
28.76
27.32
26.00
23.41
19.67
16.91
11.71

9.46

6.82

5.51

4.28

2.63

114

Table 4 (cont'd)

Sample No. 12 Run No. l

T(K) KEff(mw/K) T(K) KEff(mw/K)
4.67 16.79 4.23 15.00
4.67 16.97 4.35 15.37
4.74 17.22 4.48 15.92
4.89 18.08 4.59 16.63
4.99 18.37 4.69 16.72
5.10 18.83 6.02 21.82
5.20 19.54 6.19 22.03
5.21 19.34 6.47 22.14
5.33 19.83 6.62 22.31
5.47 20.24 6.87 22.14
5.58 20.96 7.00 22.31
5.66 21.18 7.27 22.13
5.76 20.96 7.46 21.93
5.85 21.38 7.66 21.88
5.94 21.73 7.85 21.75
6.02 21.77 8.01 21.24
6.08 21.96 8.25 20.42
6.17 22.09 8.48 19.95
6.27 22.44 8.71 19.42
6.37 22.20 8.90 18.86
2.30 6.70 9.04 18.62
2.35 6.74 9.24 18.06
2.40 6.99 9.49 17.35
2.47 7.27 9.70 17.03
2.54 7.60 10.04 16.07
2.61 7.87 10.63 14.93
2.68 8.19 11.09 14.17
2.77 8.52 11.59 13.29
2.85 8.80 12.07 12.47
2.93 9.10 12.69 11.52
2.99 9.25 13.11 11.15
3.08 9.71 14.25 9.91
3.13 9.88 15.10 9.14
3.21 10.05 10.60 15.30
3.29 10.34 11.04 14.28
3.38 11.03 12.57 11.91
3.50 11.43 13.71 10.67
3.63 12.05 14.73 9.63
3.75 12.52 15.65 8.75
3.86 13.07 16.41 8.32
3.97 13.58 17.65 7.53
4.10 14.21 18.71 6.63

T(K)

17.70
18.79
19.95
21.52

4.59
4.69
.87
.02
.18
.32
.97
70
.48
.23
.98
.77
.57
40
.76
.98
.55
.06

\IONU'IU'IU'INNNWWUJWbUIUIub

.42
.61
.79
.14
90
.68
.41
.18
.90
.75
.28
.39
.49
.62

NNNNNNwwwwebpp

KEff

7.38
6.99
6.73
5.54

Sample No.

27.01
27.51
29.06
30.11
31.22
24.98
22.41
20.61
18.98
17.10
15.23
13.95
12.63
33.17
35.55
36.06
38.81
41.32

Sample No.

23.92
24.96
26.07
22.00
20.38
18.89
17.28
15.56
13.61
13.00
10.14
10.86
11.45
12.33

(mW/K)

Table 4 (cont'd)

T(K)

24.19
26.93
32.00
37.87

13 Run No.

7.55
6.02
8.03
8.51
8.93
9.51
10.01
11.07
12.51
14.94
17.16
20.31
24.85
31.36
39.51
49.53
60.87

13 Run No.

5.00
5.20
5.52
5.82
5.56
5.98
6.27
6.52
6.73
7.03
7.49
7.96
8.49
9.05

KEff

4.61
4.00
3.02
2.33

44.29
36.55
45.66
45.61
46.64
47.06
46.90
49.09
46.23
43.47
39.60
37.13
29.17
25.23
19.26
16.43
10.49

27.65
28.65
29.21
32.50
30.95
32.26
33.07
33.74
34.26
37.26
35.55
36.82
35.46
34.28

(mW/K)

T(K)

9.51
10.01
11.08
12.56

4.61
4.76
4.96
5.20
4.29
4.42
4.54
3.99
4.08
4.19
2.39
2.49
2.59
2.73
2.88
2.99
3.11
3.22
3.34
3.48
3.70
3.80
3.94
5.16
5.36
5.48
5.66
5.84
6.03
6.18
6.36
6.57

KEff

116

Table 4 (cont'd)

(mW/K)

32.75
32.19
29.69
26.48

Sample No.

12.35
13.04
13.88
14.88
11.17
11.67
12.25
10.01
10.40
10.75
4.92
5.22
5.54
5.94
6.34
6.57
6.93
7.18
7.70
8.16
8.95
9.31
9.87
14.52
15.49
15.92
16.75
17.12
17.74
17.85
18.32
18.88

14 Run No.

T(K)

15.01
23.84
28.16
35.20

6.71
6.87
7.02
7.20
7.41
7.65
7.87
8.03
8.28
8.56
8.80
9.03
9.28
9.50
9.72
9.74
9.99
10.37
10.53
11.09
11.85
11.46
13.21
12.39
14.20
15.23
15.91
18.80
21.57
25.90
33.26

KEff

(mW/K)

21.56
11.85
10.22

7.32

18.85
18.84

19.05

19.13

19.13
19.22
19.15
19.05
18.55
17.94
17.50
17.00
16.53
16.22
15.61
15.79
15.42
14.58
14.25
13.75
12.39
12.82
10.43
11.51

9.81

9.13

8.31

6.75

5.44

4.30

3.06

117

Table 4 (cont'd)

Sample No. 14 Run No. 2

T(K) KEff(mw/K) T(K) KEff(mw/K)
4.60 14.34 5.92 19.79
4.60 14.35 6.05 20.05
4.70 14.79 6.23 20.41
4.85 15.41 6.40 20.61
4.96 15.73 6.61 20.89
5.07 16.46 6.77 21.02
5.16 16.83 7.02 21.25
5.28 16.44 7.23 21.20
2.28 5.76 7.54 21.43
2.37 6.08 7.75 21.43
2.50 6.55 8.03 21.01
2.62 6.90 8.30 20.27
2.75 7.30 8.74 19.36
2.90 7.73 8.97 18.67
3.01 8.01 9.35 17.81
3.09 8.24 9.67 17.32
3.22 8.78 9.90 16.85
3.29 9.03 10.30 15.86
3.38 9.34 6.03 20.01
3.50 9.81 6.50 20.79
3.55 9.98 7.02 21.27
3.63 10.29 7.45 21.23
3.76 10.69 7.97 21.16
3.90 11.38 8.99 18.64
4.03 11.82 10.08 16.58
4.14 12.39 10.58 15.69
4.23 12.63 11.01 14.90
2.36 13.26 11.95 13.24
4.55 14.09 12.56 12.42
5.40 17.81 13.62 11.14
5.52 18.27 15.01 9.90
5.64 18.95 16.69 8.39
5.73 19.18 18.53 7.46
5.83 19.79 20.92 6.36
5.93 19.83 23.88 4.98

6.06

20.10

118

Table 4 (cont'd)

Sample No. 15 Run No. 1

(T) KEff)mw/K) T(K) KEff(mw/K)
83.08 1.82 62.92 2.86
81.70 1.77 59.65 3.04
79.90 1.90 55.93 3.29
78.27 2.00 52.85 3.83
76.00 1.97 51.34 3.92
72.85 2.14 48.29 4.36
68.72 2.37 45.99 4.61
65.75 2.55 52.50 3.91

Sample No. 16 Run No.

4.66 26.86 9.00 42.23

4.88 27.60 9.46 41.47

5.12 29.74 9.98 40.77

5.44 32.91 10.48 40.66

2.59 12.50 11.04 39.18

2.65 12.89 11.97 37.48

2.87 14.20 13.16 36.20

3.00 14.93 13.99 33.65

3.27 16.83 14.94 32.80

3.50 18.43 16.14 30.16

3.77 20.39 18.02 27.46

4.02 22.25 20.08 25.44

4.26 23.91 23.40 19.86

4.42 24.98 25.11 17.77

4.54 26.02 26.87 18.09

4.75 27.45 28.20 17.30

5.00 29.03 30.21 15.32

5.48 32.41 32.18 13.94

5.69 33.94 34.42 12.56

5.99 35.43 36.23 11.84

6.28 36.44 38.14 10.97

6.66 37.89 41.38 11.11

6.99 39.31 42.25 10.07

7.39 40.54 44.61 9.34

7.81 42.54 47.61 7.87

8.01 42.00 50.23 7.37

8.50 41.96 53.81 6.72

119

Table 4 (cont'd)

Sample No. 16 Run No. 2

T(K) KEff(mw/K) T(K) KEff(mw/K)
4.69 16.39 6.67 22.36
4.82 17.82 6.82 22.31
4.99 17.98 7.01 22.21
5.23 19.07 7.22 22.39
5.40 19.86 7.41 22.30
2.47 6.09 7.72 22.26
2.62 6.76 7.94 22.12
2.79 7.48 8.21 21.56
3.00 8.24 8.46 20.87
3.25 9.17 8.71 20.14
3.46 10.42 8.97 19.41
3.76 11.81 9.14 19.23
4.01 13.18 9.39 18.78
4.23 14.20 9.70 18.29
4.50 15.48 10.00 17.73
4.78 16.77 10.35 17.53
4.32 14.51 10.78 16.39
4.62 16.15 11.06 16.02
4.82 17.09 12.02 14.12
5.12 18.47 12.82 12.92
5.29 19.06 14.16 11.69
5.51 20.09 15.11 10.60
5.66 20.70 16.09 9.71
5.75 21.28 18.42 8.69
6.00 21.66 20.66 7.18
6.15 21.55 22.63 6.16
5.99 21.15 24.27 5.36
5.83 21.04 27.15 4.82
5.65 20.43 29.65 4.09
6.00 21.13 34.43 3.23
6.32 21.69 39.31 2.64

6.47

21.88

APPENDIX C

Defect Scattering - Classical Analogs

1. Rayleigh Scattering from a Spherical Object -
Dimensional Analysis

The Rayleigh scattering law for a spherical obstruction
can be verified by energy conservation and dimensional anal-
ysis considerations.15 Consider a plane wave of unit ampli-

tude wi = exp(iqz) incident upon a spherical object of volume

V such that 1>>Vl/3. The amplitude of the scattered wave

ws is necessarily proportional to 1/r since the scattering

intensity is proportional to lwslz |2 a 1/r2. This

is because flwslzrzdfl must be equal to a constant. Now the
Q

and st

only other guantities that ws can depend on are u (velocity
of sound), A (wavelength) given by 2'rrq_l and V (volume).
Since U is the only quantity that involves time as one of
its dimensions, it is discarded. Hence, the simplest prod-
uct that we can form from 1, r and V such that 05 remains
dimensionless is V/Azr = L3/L2L = 1. Thus, $5 a V/Azr and
the scattering cross section is proportional to V2/14. That

is, o is proportional to q4

120

121
2. Scattering from a Cylindrical Object -
Dimensional Analysis
The Rayleigh scattering law foreulinfinite cylindrical

obstruction can be verified as in the preceding case.15 In

this instance since we have cylindrical symmetry, the pro-
blem can be reduced to 2 dimensions. Scattering intensity
considerations yield that the amplitude of $5 a 1/03, since
glwslzpde must be a constant. The other quantities on which
ws can depend are A and A = naz the cross sectional area of
the cylinder. Thus the simplest product we can form is
A/A3/Zpg, i.e., $3 « A/A3/20;5 = Lz/LB/ZL15 = 1. Therefore,
the core scattering cross section per unit length of disloca-
'2

tion is o a [05 ~ AZ/A3 = a(a/A)3.

core
3. Scattering from the Strain Field Surrounding
the Cylindrical Core - Geometrical "Optics"
Limit

For a phonon that is deviated from its original direc-
tion by an angle 0 4 yb/p on passing through the strain
field, the change in momentum along the original direction
is proportional to (1-cos¢).32 Since the change in heat
current will be reduced by an amount which is proportional

to (1-cos¢), the scatteringcmoss secthmnper unit length of

dislocation is thus proportional to

p 2 2 2
f¢m(l-cos¢>)d¢> = meWb/p) dp = Y b /pO
0
where p0 is the least value of p (p0 ~ 1). Thus 0

is proportional to yzbZ/X.

strain

APPENDIX D

Error Associated with the Magnitude of AT
In our measurements the expression we use to evaluate
the effective thermal conductance is given by
KEff (T) = Q/AT (01)
where T = TO + AT/2 is the average temperature. For small

AT Equation (D1) is a good approximation of the following

equation
. TO+AT
K = Q/AT = (f K(T)dT)/AT (D2)
Eff T0
. . 14,25
where K(T) is the true effective thermal conductance.
At low temperatures K(T) 2 ATZ. On substituting this

expression into Equation (D2) and evaluating the integral
we obtain for the effective thermal conductance

K = ((T

3 3
Eff + AT) - To )A/3AT . (D3)

0

The true effective thermal conductance at T is given by

K(T) = A(TO + AT/2)2. (D4)

The relative difference (K(T) - KEff)/K(T) to lowest order
in AT/T0 is given by

AK/K : (AT/T0)2/12 . (05)

Hence, if we want Equation (D1) to stay within 1% of the

122

123
true effective thermal conductance, then the largest AT

that we should not exceed at the lowest temperature (2 K)
is given by
AT : 2(o.12);5 = 0.7 K . (D6)
Similarly at the highest temperatures, where K 2 BT-1
after performing the appropriate expansions, the error will
also be given to lowest order in AT/TO by Equation (D5).
Hence, if at 80 K we wish to stay within a 1% error, then

the largest AT that we should not exceed is given by

L
T S 80(0.12)2 = 28 K.

LI ST OF REFE RENCES

10.

11.

12.

13.

14.

15.

16.
17.

18.

LIST OF REFERENCES

H. Cavendish, Phil. Trans. R. Soc., 13, 372 (1785).

G. A. Cook in Argon, Helium, and the Rare Gases I,
G. A. Cook ed., (Interscience Publishers, New York,
1961).

 

 

G K. Horton in Rare Gas Solids l, J. A. Venables and

. L. Smith eds., (Academic Press, London, 1977).

 

 

L. Pollack, Biophysical J. 33, 49 (1977).

. Unsworth and F.C. Gillespie in Diffusion Processes
, J. N. Sherwood, g; 33. eds., (Gordon & Breach,
cience Publishers, Inc., New York, 1971).

 

 

B
G
J
2
S
D. A. Nelson and A. L. Ruoff, Phys. Rev. Lett. 33,

383 (1979).

D. N. Batchelder in Rare Gas Solids 2, J. A. Venables
and B. L. Smith eds., (Academic Press, London, 1977).

 

G. L. Pollack, Rev. Mod. Phys. 33, 48 (1969).

R. E. Peierls, Quantum Theory 9£ Solids (Oxford Univer-
sity Press, New York, 1964).

 

P. Carruthers, Rev. Mod. Phys. 33, 92 (1961).

C. Kittel, Introduction 39 Solid State Physics (John
Wiley and Sons, Inc., New York, 1971).

 

 

R. E. Peierls, Ann. Phys. Leipzig 3, 1055, 1101 (1929).

P. G. Klemens in Thermal Conductivity 1, R. P. Tye ed.,
(Academic Press, New York, 1969).

 

R. Berman, Thermal Conduction ig_Solids (Oxford Univer-
sity Press, Oxford, 1976).

 

Lord Rayleigh, Theory 9f Sound 2 (Dover Publications,
New York, 1945).

F. R. N. Nabarro, Proc. Roy. Soc. A209, 278 (1951).
J. M. Ziman, Nuovo Cimento Z, 353 (1958).

B. L. Smith, Contemp. Phys. 33, 125 (1970).

124

19.

20.

21

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

125
G. K. White and S. B. Woods, Phil. Mag. 3, 785 (1958).

I. N. Krupskii, V. G. Manzhelii, and L. A. Koloskova,
Phys. Stat. Sol. 33, 263 (1968).

I. N. Krupskii and V. G. Manzhelii, Phys. Stat. Sol.
33, K53 (1967) and Soviet Phys. JETP, 33, 1097 (1969).

A. Berne, G. Boato and M. DePaz, Nuovo Cimento 33, 182
(1969).

F. Clayton and D. N. Batchelder, J. Phys. 93, 1213
(1973).

D. K. Christen and G. L. Pollack, Phys. Rev. 812, 338
(1975).

D. K. Christen, Thermal Conductivity 9: Solid Argon,
Ph.D. Thesis, Michigan State University (1974).

 

 

J. A. Venables and B. L. Smith in Rare Gas Solids 2,

J. A. Venables and B. L. Smith eds., (Academic Press,
London, (1977).

 

O. G. Peterson, D. N. Batchelder and R. 0. Simmons,
J. Appl. Phys. 33, 2682 (1965).

G. K. White and S. B. Woods, Canad. J. Phys. 33, 58
(1955) and Nature, London 177, 851 (1956).

R. K. Crawford and W. B. Daniels, J. Chem. Phys. 32,
3171—83 (1969).

W. B. Daniels, 25 33., Phys. Rev. Lett. 33, 548-50
(1967).

J. A. Leake, g3 33., Phys. Rev. 181, 1331—60 (1969).

J. M. Ziman, Electrons and Phonons (Clarendon Press,
Oxford, 1960).

 

G. K. White, Experimental Techniques 32_Low—Temperature
Physics (Clarendon Press, Oxford, England, 1968).

 

 

 

A. C. Rose-Innes, Low Temperature Techniques, (English
University Press, Ltd., London, 1964).

 

V. I. Grushko, D. N. Bol'shutkin, G. N. Scherbakov and
N. F. Shevchenko, Soviet Phys.-Crystallography 33, 536
(1970).

36.

37.

38.

39.

40.

41

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

126

M. Gsanger, H. Egger, G. Fritsch and E. Lfischer, Z.
Angew. Phys. 33, 334 (1969).

G. L. Pollack and E. N. Farabaugh, J. Appl. Phys. 33,
513 (1965).

I. Lefkowitz, K. Kramer, M. A. Shields and G. L.
Pollack, J. Appl. Phys. 8, 4867 (1967).

G. L. Pollack and H. P. Broida, J. Chem. Phys. 38, 968
(1963). _—'

C. R. Tilford and C. A. Swenson, Phys. Rev. 33, 719
(1772).

R. Berman, J. Appl. Phys., 33, 318 (1956).

A. V. Leonteva, Yu. S. Stroilov, E. E. Lakin and D. N.
Bol'shutkin, Phys. Stat. Sol. 33, 543 (1970).

R. Berman and C. F. Mate, Nature, Lond., 182, 1661
(1958). ‘—

C. F. Mate in "Heat Flow below 100 K", p. 101, Annexe.
1965-2 Suppl. Bull. Inst. Refrig. (1965).

D. Tabor, The Hardness 93 Metals (Oxford University
Press, 1951).

 

O. G. Peterson, D. N. Batchelder and R. 0. Simmons,
Phys. Rev. 150, 703 (1966).

M. Moss, J. Appl. Phys. 33, 3308 (1965).

S. L. Meyer, Data Analysis for Scientist and Engineers,
(John Wiley and Sons, Inc., New York, 1975).

 

W. R. G. Kemp, P. G. Klemens and R. J. Tanish, Phil.
Mag. 3, 845 (1959).

J. N. Lomer and H. M. Rosenberg, Phil. Mag. 3, 467
(1959).

R. M. Kimber and S. J. Rogers, J. Phys. 33, 2279 (1973).

J. A. Barker and A. Pompe, Australian J. Chem. 33, 1683
(1968).

J. E. Clemans, Phys. Rev. B 33, 1072 (1975).

G. Boato, Cryogenics 3, 65 (1964).