ABSTRACT

SUBLIMATION PRESSURES OF
SOLID ARGON, KRYPTON, AND XENON.

BY

Charles William Leming

An eXperiment was performed to measure the sublima-
tion pressures of solid argon, krypton, and xenon over
wide temperature and pressure ranges. Data are reported
from near the reSpective triple points to about (2.3 x 10-6
Torr, 25.506K) for Ar; (2.1 x 10-4Torr, 43.13OK) for Kr;
(3.8 x 10-4Torr, 70.705K) for Xe. Pressures were measured
with a mercury manometer, a McLeod gauge, and a calibrated
Bourdon gauge. The pressure measurements were corrected
for thermomolecular flow and streaming. Temperatures were
measured with a National Bureau of Standards calibrated
platinum resistance thermometer using the 1968 Internation-
al Practical Temperature Scale. The required sample temp-
eratures were achieved by means of a liquid oxygen bath
above 55K and by means of a liquid helium bath below 55K.
Electrical heating from an ac bridge temperature controller
was used to regulate the sample chamber temperature.

Samples were condensed from Matheson research grade

gases. Impurity concentrations were reduced by distilling

the samples in situ. The gas handling system and sample

 

chamber were constructed so that contamination of the

sample by adsorbed gases could be minimized.

Application of the law of correSponding states was
investigated by analyzing the reduced pressure curves.
Values for static lattice energy, geometric mean of the
lattice vibrational Spectrum, heat of sublimation, and
lattice vibrational energy are calculated using theoreti-
cal sublimation pressure curves. Corrections were applied

to these values to account for the effect of vacancies.

SUBLIMATION PRESSURES OF
SOLID ARGON, KRYPTON, AND XENON.

By

Charles William Leming

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1970

To Paula

11

ACKNOWLEDGEMENTS

This experiment was suggested by Professor G. L.
Pollack. I am very much indebted to him for his guid-
ance and encouragement. Thanks are also due to Mr. Carl
James Duthler and Mr. Garold Fritz for assistance in tak-
ing data. Also I wish to thank my wife, Paula, for assis-
tance with foreign language materials and help with typ-
ing of the thesis. Finally I would like to acknowledge

the financial support of the U.S. Atomic Energy Commission.

iii

I.

II.

III.

IV.

TABLE OF CONTENTS

INTRODUCTIONooooooooooo

THEORETICALooooooooooooooo

General Properties of Rare-Gas Solids
Vapor Pressure. . . . . . . . . . . .

EXPERIWNTALOOOCOOOOOOO

Cryostat Design . . . . .

Gas Handling and Sample Formation

Temperature Control. . . .
Temperature Measurement . .
Pressure Measurement. . . .

RESULTS 0 O O O O O O O O O O O O 0

Law of Corresponding States

Static Lattice Energy . . . I :
Heat of Sublimation . . . . . .
Lattice Vibrational Energy. . .

GENEML CONCLUSION O O O O O O O 0

LIST OF REFERENCES . . . . . . . .

APPENDIX 0 O O O O O O O O O O O 0

iv

0...

Page

.39
.47

.47
.53

.70

.75

.78

Table

10

11

12

13

LIST OF TABLES

Values of the Mie-Lennard-Jones poten-
tial parameter 6 and r0 for m212, n=6.

Values of the Buckingham potential para-
meters 6 and r0 for m = 12.

Values of the parameters e ,O' , and A*
for the Mie—-Lennard-Jones all-neighbor
potential with m = 12, n = 6.

Impurity concentrations in gas samples
used in this experiment. These tables
are the results of a mas spectrometer
analysis supplied with the gases.

Values of the parameters A*, 8*, and C*
of equation (38).

Temperature intervals used for analysis
of vapor pressure equations.

Values of the parameters a and b of equa-
tion (30) found from vapor pressure data.

Values of E0 and mg calculated from the
parameters of Table 7.

Values of the parameters a and b of equa-
tion (36) found from vapor pressure data.

Values of Eo andkalg calculated from the
parameters of Table 9.

Values of the parameters a and b of equa-
tion (23) found from vapor pressure data.

Values of heat of sublimation, L calcu-
lated from the parameters of Table 11.

Values of vibrational ener y, Evib' cal-
culated using equation (41%.

Page

23

45

58

58

S9

62

63

68

(List of Tables continued)
Table Page

A1 Measured pressure and temperature 78
points for Ar.

A2 Measured pressure and temperature 80
points for Kr.

A3 Measured pressure and temperature 82
points for Xe.

vi

Figure

10

11

LIST OF FIGURES

Sectional view of cryostat used for this
experiment.

This drawing represents the gas handling
system used in this eXperiment. The gas
storage reservoir, vacuum lines, and stOp-
cocks are shown.

The Operation of the ac bridge temperature
controller is indicated in this schematic
drawing.

This potentiometer circuit was used for
temperature measurement. Current to the
thermometer was reversed to account for
thermal emfs in the system.

Measured sublimation pressures are plotted
for Are

Measured sublimation pressures are plott-
ed for Kr.

Measured sublimation pressures are plott-
ed for Xe.

Reduced sublimation pressure curves are
plotted for Ar, Kr, and Xe.

Typical plot of lnPT5 versus 1/T for Ar.
Upper line is corrected for vacancy forma-
tion.

Typical plot of lnPTg versus l/T for Kr.
Upper line is corrected for vacancy forma-
tion.

Typical plot of in PTk versus 1/T for Xe.
Upper line is corrected for vacancy forma-
tion.

vii

Page
18

21

31

36

48

49

50

52

54

55

56

(List of Figures continued)

Figure Page
12 Typical plot of lnP versus l/T for Ar. 65
13 Typical plot of lnP versus 1/T for Kr. 66
14 Typical plot of lnP versus 1/T for Xe. 67

viii

I. INTRODUCTION

Properties of the rare-gas solids have long created
much interest because the nature of the attractive forces
between atoms is simple and rather well understood.1’2
These forces may be approximated as short range, pairwise
additive, central forces which have the same form for all
the rare gases.3 Many thermodynamic properties of these
solids have been predicted on the basis of simple models.
Deviations in the eXperimental data may be used to study
such details as anharmonicityf’5 electron exchange,6
and lattice defects.7’8

Although lending themselves well to simple theoretical
models, experimental studies of the rare-gas solids have
met with many difficulties. Because the triple point
temperatures of rare gases are relatively low, low temp-
erature techniques must be applied to study these solids.

The purpose of this experiment was to provide accurate
sublimation pressure data extending over several orders
of magnitude for each of the rare-gas solids. In order to
accomplish this, a low temperature cryostat was constructed
to operate in the temperature range from 200K to 20K.

Measurements for all gases were made using this apparatus.

These data have been analyzed on the basis of vapor

pressure curves predicted by classical thermodynamics9
and vapor pressure curves predicted by lattice dynamical
theory.7 From this analysis values were calculated for
heats of fusion, vibrational energies, static lattice
energies, and for the geometric mean of the lattice vib-
rational spectra. The law of corresponding states has
been applied to test the consistency of published poten-
tial parameters for Ar, Kr, and Xe.
This thesis describes the eXperiment and calculations.

Results of this experiment are reported in terms of para-
meters for vapor pressure curves and also tables of pri-

mary data.

II. THEORETICAL

General Properties of Rare-Gas Solids

Many calculations have been performed to predict
the thermodynamic properties of rare-gas solids. The
reason for this theoretical interest is that the forces
between the atoms may be closely approximated as simple
central forces which have the same form for all the rare
gases.3 Forces of this type are a good first approxima-
tion because the atoms consist of tightly bound, filled
electronic orbitals.

For dilute gases, central forces can be applied
almost exactly. For solids, however, the possibility
exists that the actual intermolecular forces may consist
of various nonadditive three-body forces in addition to
the expected two-body forces.

The interatomic potentials which are normally used
to calculate prOperties of the solids are central poten-
tials which have adjustable parameters. These parameters
are used to fit theoretical calculations to eXperimentally
determined thermodynamic prOperties of the solid. Thus,
the parameters are chosen as if the actual potential were
a central potential. However, it must be remembered that
these parameters are only effective parameters which re-

sult from assuming no three-body effects are present. The

4
actual potential may contain three-body effects so that
the assumed two-body potential cannot perfectly represent
the actual potential. Therefore, the two-body potential
can only be adjusted to give as good a fit as possible.
Because the actual potential cannot be calculated,
many analytical potentials have been suggested to repre-
sent the intermolecular forces. The simplest and most
commonly used form is the well known Mie-—Lennard-Jones
potential given by3 :
w e“) is?
n r
Here,-€:is the depth of the potential and r0 is the dis-
tance from the origin to the lowest point in the potential
well. The values of m and n are usually taken to be 12
and 6 respectively.
The n=6 attractive potential at large separations
can be calculated from the induced dipole - induced dipole
interaction as calculated by London using second-order

perturbation theory.10

For this calculation, ground
state wave functions are assumed and higher order attrac-
tions are neglected.

Although the attractive part of the Mie—-Lennard-Jones
potential is theoretically plausible, the repulsive part
has no such satisfactory theoretical basis. An accurate
calculation of the repulsion due to overlapping electron
wave functions would most likely yield an exponential form

for the repulsion.3 However, in order to simplify computa-

tions, the value m=12 is usually chosen for the exponent of

5

the repulsive term. In this case the Mie—-Lennard-Jones

potential becomes: &(I) = {(20 12 -2(I_‘93 .
r r .1

Other forms of the binding 1potential include the

Buckingham potential given by

@(r) = 6m€ {1 exp -m (r/ro-) fl-_ H()}.(3)
m - 6 m

Here, m, re, and ‘7 have the same meaning as in the
Mie-—Lennard-Jones potential. Although this potential
seems more acceptable physically, it is not often used
because of computational difficulties. Also this poten-
tial does not seem to give significantly superior theo-
retical predictions.

Other potentials such as the Morse potential12 and
the Munn-Smith13 potential have been considered as likely
potentials to represent rare-gas solids. Neither the
Morse potential nor the Munn-Smith potential deviate
greatly from the potentials described previously.

The parameters *5 andro are usually determined
from experimental values of the sublimation energy and

14 Typical values of§ and r0

lattice parameter at O K.
for the Mie-—Lennard-Jones potential and the Buckingham
potential are found in Table 1 and Table 2- respectively.
The results are presented both for the case when the
potentials act between all neighbors and also for the
case when the potentials act between nearest neighbors

3

only. Since the potentials considered are only estimates

of what may actually be happening in the crystal, neither

model is obviously superior.15

a TABLE 1
Values of the Mie-—Lennard-Jones potential parameters
6 and r0 for m = 12, n = 6.

(All Neighbor)

_ Argon Krypton Xenon
€ (10 16erg) 165 227 319
ro(10'8cm) 3.820 4.084 4.446

6 (Nearest Neighbor)
€.(10'1 erg) 236 325 458
ro<10'8¢m) 3.709 3.966 4.318
aRef. 3-
TABLE 2

Valuesa of the Buckingham potential parameters 6: and r0

for m = 12.
(All Neighbor)

Aroon Krypton Xenon °
36 (10'16erg) 160.9 222.8 314.3
ro(10‘86m) 3.855 4.121 4.485
6 (Nearest Neighbor)
-6 (10‘1 erg) 222.2 323.6 456.6
ro(10-8cm) 3.712 3.968 4.319
8Ref. 3.

Although the equation of state of solids cannot yet
be calculated from any known analytic potential, certain
aSpects of the equation of state can be investigated by
applying the law of corresponding states. This law shows
that the equations of state for simple substances are
identical when expressed in terms of suitable non-

dimensional reduced variables.

7
If classical statistical mechanics applies, the
equation of state for simple molecules becomes a func-
tion of reduced temperature, Tr, reduced pressure, Pr,
and reduced volume, Vr.16 Thus the equation of state
may be written
Pr = Pr (Vr,Tr) ° (4)

The reduced variables are found by dividing P, V, and
T respectively by the correSponding critical constants
P

V and TC so that:

C’ C’

Pr = P/PC, Vr = V/VC, and Tr = T/TC - (5)

This form of the law of corresponding states has been
found to apply mainly to simple gases and liquids.
Solids generally deviate from this law.

A more modern form of the law of corresponding
states has been found to be applicable to some solids.
Consider Spherically symmetric molecules whose potential
energies depend only on the intermolecular separation
and have the form &(r) = € f(r/gr), For such sub-
stances the equations of state may be eXpressed in terms
of the modern reduced variables:

p* = PU3/C , v* = V/Na", T* = ”/6 - (6)

Here (I is the depth of the intermolecular potential well
and c’ is a characteristic length for which &(r=d) = 0.
This form of the law of corresponding states is
derived from quantum statistical mechanics.17’18 The

partition function E can be calculated from the sum

over states
.3. = g exp (an/k1“) ' (7)
Here En are the steady state energy levels determined
from the eigenvalues of the Schrddinger equation of the
system. The Schrddinger equation for a system of N inter-

acting Spherically symmetric molecules is
‘fia A! a
[4 /2m) gm,» iZ-Ktfluik) aggcrrum) = o . (8)
3 >

Written in terms of non-dimensional reduced variables

* 3
'k 2- 0
En = En/Nc , f(r*) =Q/e , andVi = a“ 72, the equa-

tion becomes

*2 N 1 *2 * Y * *
[-/\ ;(§T’" V2. *;f(r§k)-NEHJ n(r1,°'°, N) = O . (9)
=1

In equation (9)/\* is the reduced de Broglie wavelength
given by
* 8
A = h/a’ (m6) ° (10)
As can be seen from the form of equation (9), the reduced
* a *
eigenvalues En depend on v and A .
The partition function can then be written in terms
of reduced variables.
:3 ~k *
Z = éexp [-En/(kT/ffl = Zexp-En/T ° (11)
W
From this it follows that
* * *
Z=Q(V:T9A) ' (12)
The equation of state can be calculated from the
partition function using the thermodynamic relationship
a
P* = 1‘ ._a_1n Q (V*. T’ZA‘U
av*

Thus the equation of state is only a function of reduced ,1

(13)

variables and may be written

8* = 9* (v*, '1'", A") ° (14)
The form of equation (14) is the same for each type of
molecule and depends only on the reduced variables P*,
V*, T*, and/\*.

The values of the parameters {f andma’used to cal-
culate the reduced variables depend on the exact form
of the potential assumed. Table 3 shows accepted val-
ues3 off , or, and A3) for the Mie—Lennard-Jones all-
neighbor potential. For this potential a’is related to

V6

ro defined earlier by r() = 2 O’.

8 TABLE 3 *
Values of the parameters , CV, and /\ for the
Mie-—Lennard-Jones all-neig bor potential with m=12, n=6.

_ Arson Krypton Xenon
€ (10 16e1c‘8) 165 227 319
«(10'8cm) 3.503 3.745 4.077
A*
0.0289 0.0158 0.00980

aCalculated from parameters presented in reference 3.

The crystal structure of solid rare gases is a pro-

perty which cannot be predicted using the two-body poten-

tials presented earlier. Both the Mie-—Lennard-Jones19

20

and Buckingham potentials predict that at T a O K rare

gases crystallize in the th phase. However, eXperiments

21 have revealed that the

at temperatures as low as 2.5K
rare-gas crystals have fcc structure with some hcp pre-

sent as stacking faults.

10

Several explanations have been given to this crystal
structure. Probably the observed fcc structure indicates
inaccuracies in the analytic potentials assumed22 or that
three-body effects are significant in these solids.23

Vapor Pressure

The condition for vapor-solid equilibrium given by
classical thermodynamics is that the specific Gibbs
functions, g, of the respective phases must be equal.24
From this condition it is possible to calculate equations
for the vapor pressure of crystals.

Surfaces between solid and vapor phases may also
be considered using this principle. It might be asked
what effect the exact nature of the processes occuring
at the surface between solid and vapor phases has on
vapor-solid equilibrium. The solution to this problem
is to consider the surface as a separate phase which is
different from the solid or vapor. The condition for
phase equilibrium may then be extended to become.

g(solid) - g(surface) a g(vapor)

From the above equation it can be seen that the
effects of surface prOperties can be ignored when con-
sidering vapor-solid equilibrium. To calculate the con-
ditions for vapor-solid equilibrium, it is only necessary
to equate the specific Gibbs functions of the bulk solid
and the vapor.

For a monatomic solid, the vapor pressure may be

accurately calculated from the Clausius-Clapeyron

ll
equation of classical thermodynamics.25 This equation

may be derived from the condition for vapor- solid equili-
brium.

The specific Gibbs function is defined as
g = u - Ts + Pv. In this equation u is the internal
energy and s is specific entrapy. The condition for
equilibrium then becomes

gc = gv . (15)
Here the subscript c refers to the solid phase and the
subscript v refers to the vapor phase.

If the equilibrium temperature is changed slightly
from T to T + dT, the vapor pressure changes from P to
P + dP. The condition for maintaining the Gibbs func-
tions equal is then dgc = dgv. In terms of T and P
this may be written

-s¢ dT + vc dP e -sv dT + vv dP 0 (16)
Taking the ratio dP/dT in equation (16) gives

dP/dT 2 3c - sv/vc - vv . (17)
Using the definition of specific enthalpy h = u + Pv,

equation (17) becomes:

dP _ (hc'hv) - (Sc-5v) -
alr- — T (Vc _ Vv) . (18)

However, using the condition for equilibrium, gc = 8v:
and the differential relation d in P = dP/P, equation

(18) becomes

dlnP- hc-hv
dT T (va) (VG/Vv'l)

. (l9)

 

12
Using the virial expansion for va gives:
d 1n P = he ' hv , (20)
dT RT2 (1 + BP/RT) (v/vV - 1)
where B is the second virial coefficient defined by
Pv = RT (1 + B/v+.-.-) . (21)
Noting that vC/vv (<1 and BP/RT<( 1, equation (20)

may be written

 

d ln P = he ' hv , (22)
d I/T R (1 - vC/vv + BP/RT)

The difference in enthalpies of the two phases, hC-hv,
is equal to the heat of sublimation, L.

Using equation (22) it is possible to calculate
the heat of sublimation of simple molecular solids.
Sublimation pressure data may be plotted in the form
ln P versus 1/T. The slepe of this plot may then be
measured and the heat of sublimation calculated.

Because the lepe of this curve varies slowly
with temperature, the data can be assumed to be a .
straight line over narrow temperature ranges. The
method of least squares may be used to fit the data
to the equation

ln P = a/T + b - (23)
The parameter a is then equal to the right side of
equation (22).

A slightly different sublimation pressure curve

for simple molecular solids may be found from lattice

7

dynamical theory. The partition function for a single

harmonic oscillator may be written as a sum over states,

13
z = Z eXp (-En/kT) in which 13,):th (n+5) - <24)
73
Performing the sum over states yields

2 = exp (- "hW/kT)
1 - exp (hUV/ZkT)

. (25)
For a system of 3N oscillators, the Helmholtz free
energy,N F = U-TS, is then
4”ng ln s: :11“ + kT ln [1 - eXp(JAWi/kT)] (25)
For an ideal single crystal whose lattice vibra-
tions are assumed to be harmonic, equation (25) gives
the thermal contribution of the Helmholtz free energy.
To find the total free energy, the static lattice en-
ergy, E0, must be added to this expression. Physically,
E0 represents the depth of the potential well binding
the solid.

If this equation is then expanded for high temp-

eratures, F can be analytically expressed as

F = BC + 3Nle-;n%8 + :01) PIS—(flé if] (27)
n n

In this eXpression'7z2n are the even positive moments

of the frequency spectrum,‘712n = WZn, an are the

Bernoulli numbers, and.\Lé is the geometric mean of

 

the lattice vibrational spectrum:
3N

We :3 (E W; N ° (28)

In statistical mechanics the Gibbs function for
a system of N particles is expressed as G =44(N where
44{ is the chemical potential. Thus the statistical

nechanical equivalent of equation (15) for equilibrium

of phases 152%: =MV. The chemical potential of the

14

vapor iszdfv

The chemical potential of the solid,/¢(C, is form-
ed from144g = F/N + Pvc. Here, F is the Helmholtz free
energy of the crystal. If,‘(c is now set equal to the
chemical potential bf the gas phase, the following ex-
pression is found for the equilibrium vapor pressure of
an ideal solid.7

lnP = -% ln T +-E O/NkT + PVC/an + 3 lnIJ/g

Q-l

_ B2
+ 3;} 1) 23162111)! hull EMH - — + Vlng[(27r)% ’18] (29)

In this equation m is the atomic mass and B is the second
virial coefficient defined in equation (21).

The validity of this equation depends on perfect
crystal structure, quasi-harmonic lattice vibrations,
and gas imperfection so small that terms higher than
the second in the virial expansion may be ignored.

For temperatures higher than one-half the Debye
temperature, the eXpansion in l/T may be ignored. Re-
Spective values of one-half the Debye temperatures,
are approximately 42K for Ar, 32K for Kr, and 28K

for Xe.2

Neglecting the crystalline atomic volume
in comparison with the vapor phase atomic volume and
ignoring gas imperfection, equation (29) can be re-
duced to the form7:

ln PT% = a/T + b, (30)

where, a = EO/Nk, b = 3..-]:n Wg + 8 ln [(m/ZTT')3 l/k] .

15
Thus the lepe of the curve, 1n PT% versus 1/T

yields E0, the static lattice energy, and the intercept
of the curve at A = 0 yieldsln/g, the geometric mean
of the lattice vibrational Spectrum. The parameters
Eo andle/g depend on volume and thus change slowly
with temperature. By using the method of least squares
to fit eXperimental vapor pressure data to equation
(30), E0 and Wg may be calculated.

In calculating the chemical potential which led
to equation (30), a perfect crystal structure was
assumed. For relatively low temperatures this assump-
tion is valid; however, for temperatures near the triple
point the effect of vacancy formation becomes signifi-
cant. Equation (30) may be corrected for vacancies
by considering the change in chemical potential due to
vacancy formation.

The change in entrOpy of a lattice of N molecules
due to the introduction of n vacancies is

S = k ln [(N+n)! / N! nil . (31)

Thus the Gibbs function for a crystal containing n

O C 7
vacanCies may be written

Gvac = GC + ngS - kT ln [(N-m)! / N! n!] . (32)
In equation (32) Gvac is the Gibbs function for the
lattice containing n vacancies, GC is the Gibbs func-

tion for a perfect crystal, and gS is the Gibbs func-

tion for vacancy formation.

16
The chemical potential of the imperfect lattice is
then
Avac = anac /¢9N =/(.(C - kT ln(1+n/N) . (33)
If the vacancy concentration, n/N, is small, n/N“<<1,
equation (33) becomes
Mme _._ /((C - kT n/N . (34)
The equilibrium vacancy concentration is given by
n/N = exp (~gS/kT). Thus the chemical potential for
a simple lattice containing vacancies is
Mvac = Me + M 9m -ss/kT) . (35)
If this chemical potential is now used to obtain
the vapor pressure equation analogous to equation (30),
one gets7
ln FTP + exp (-gS/kT) = a/T + b . (36)
The parameters a and b have the same definition as in
equation (30). Equation (36) may be used to fit vapor
pressure data for temperatures near the triple' point

where vacancy concentration becomes significant.

III. EXPERIMENTAL

Cryostat Design

In order to perform this eXperiment over the wide
pressure and temperature ranges of interest, it was
first necessary to construct a constant temperature
cryostat for use in the temperature range 25 - 170K.
The most stringent requirement for the cryostat used
to measure vapor pressure was the capability of hold-
ing the sample temperature stable for long periods of
time. This was necessary in order to assure that the
sample was in equilibrium with its vapor and that the
temperature was uniform throughout the sample. The
cryostat also was versatile enough to allow the temp-
erature to be changed easily and uniformly in order to
facilitate crystal growth in the sample chamber.

Figure 1 shows a sectional view of the basic con-
struction of our cryostat drawn to scale. The drawing
does not show the glass dewars which contain the liquid
oxygen bath and the liquid nitrogen outer jacket.

The sample chamber within the massive copper block
was connected to the gas-handling system and pressure
measuring devices (not shown) by a % in. i.d. stainless
steel inlet tube with 0.010 in. wall thickness. This

tube was heated by means of a 1000J1.manganin wire

17

18

To Pressure Gauge

L---—Stoinlsss Steel Joell

 

Monoonln Heater

 

- -----A --~u.

 

 

—-—Somple Chamber
Cu Bloch

 

-—-flonoonln Healer
~—-Cu Collar

Pl Thermometer

 

 

011 Temperature
Sensor

 

 

 

 

Figure 1: Sectional View of the cryostat used
for this experiment.

19

heater wound on the tube. The tube was coated with
cigarette paper and glyptal varnish to provide electri-
cal insulation and to aid thermal contact of the heater
and inlet tube. Power to this heater was supplied by

a variable transformer.

A stainless steel outer jacket enclosed the Cu
block. In order to allow the Cu block to be insulated
from the liquid oxygen bath, the stainless steel jacket
could be evacuated to a pressure of about 5 x 10-5 Torr.
Electrical leads inside the stainless steel jacket were
coated with Teflon to assure adequate electrical insul-
ation. The leads exited from the t0p of the outer
jacket through Kovar seals. The inlet tube was soldered
in place at the top of the outer jacket and provided
support for the Cu block containing the sample chamber.

Thermal contact between the sample chamber and
the low temperature bath was achieved with He exchange
gas. The gas was introduced into the stainless steel
jacket directly from a He gas cylinder. In order to
change the gas pressure, thus changing the thermal con-
tact of the sample chamber and the bath, He gas was
pumped away with a vacuum pump until the desired gas
pressure was attained. The pressure of the He gas was
measured with a Pirani gauge from 2 Torr to 0.01 Torr.
Below 0.01 Torr a cold cathode ionization gauge was used

to measure the He gas pressure.

20

Gas Handling and Sample Formation

The essential features of the gas handling system
are shown in Figure 2. Pyrex was used to construct most
of the system. Pyrex to Kovar seals were used to attach
tfluaglass system to the c0pper vacuum lines and to the
stainless steel sample chamber inlet tube. High vacuum
ground glass stOpcocks were used throughout the system
wherever valves were required. The stOpcocks in this
system were greased with Apiezon-L high vacuum grease.

An oil diffusion pump and nitrogen cold trap were
used to evacuate the gas handling system. These pumps
were capable of evacuating the system to about 4 x 10"6
Torr. While evacuating the system, pressures were mea-
sured on a cold cathode ionization gauge.

The ionization gauge sensor was located near the
vacuum pumps as indicated in Figure 2. Because of the
slow rate at which gases diffuse through the gas handl-
ing system at low pressures, the pressure in the remote
parts of the system might have been slightly different
than that measured by the ionization gauge. This possi-
bility was checked by independently measuring the pressure
using the McLeod gauge. The McLeod gauge was not highly
accurate at the lowest pressures measured, but was accu-
rate enough to indicate if the system had been evacuated
throughout.

Originally the gas handling system was constructed

of c0pper and brass. This system proved to be inadequate

21

hOdeOU gauge Vacuum pump

 

 

 

Cryostat ___ GB —eGas supply
Gas storage
McLeod gaude reservoir Lanometnr
Figure 2: ”his drawing represents the gas handling
xperiment. The gas

1
system used in this
storage reservoir, vacuum lines, and stop-

cocks are shown.

22

because of outgassing of adsorbed materials from the
surface of the metal. Even after heating and evacuat-
ing the system for a period of several hours, the pres-
sure in the sealed-off system could be observed to in-
crease at a rate of about 1 mTorr/hr.

After rebuilding the system, the gases adsorbed
on the glass surfaces were removed by heating the sys-
tem and evacuating the purged gases until no pressure
increase could be observed. Following this degassing
technique, the system could be sealed for several hours
before any pressure increases were detectable. The
system was always evacuated when not in use. Before
each new gas sample was introduced, the system was
heated for approximately 45 minutes to assure that
adsorbed gases from the previous eXperiment were re-
moved. The system was then sealed for several hours
and the pressure monitored to guard against vacuum leaks
or excessive outgassing.

Gas samples were transferred from metal storage
cylinders to the gas handling system through a sealed
stainless steel regulator. To estimate the number of
moles of gas transferred to the system, the pressure
of the gas in the system was measured on the mercury
manometer. Since the total volume of the system was
about 3500cm3 and assuming that the gas followed the
ideal gas law, the number of moles of gas in the system

could be calculated. Immediately after the gases were

23

admitted, the manometer and regulator were sealed off
in order to minimize contamination of the sample by
mercury evaporated from the manometer.

All gas samples used were Matheson research grade
gases. Mass Spectrometer analyses provided with the
samples listed the concentration of impurities. Table
4 shows the impurity concentration for the gases used
in this eXperiment. More impurities may have been
present because of outgassing from the walls of the
storage cylinders.

TABLE 4
Impurity concentrations in gas samples used in

this eXperiment. These tables are the results of a
mass spectrometer analysis supplied with the gases.

Argon
(impurity) (concentration)
02 less than 0.5 ppm
02 ' 3.0 ppm
H2 less than 1.0 ppm
CO less than 0.5 ppm
N2 less than 2.0 ppm
H20 3.5 ppm
CH4 less than 0.4 ppm
Kr ton
N2 __XE—__ 2.0 ppm
02 1.0 ppm
Xe 13.0 ppm
Xenon
N2 2.0 ppm
Kr 18.0 ppm
02 1.0 ppm

After the system was filled with gas, the solid
sample was carefully formed. Careful temperature

measurement and control was necessary while condensing

the sample in order to avoid condensation of gases on

24

the walls of the inlet tube.

In order to assure no condensation occurred on the
inlet tube, the entire system was first electrically
heated to a high enough temperature that gas could
not condense. The st0pcock which admitted gas from
the gas storage reservoir to the sample chamber was
then Opened. Electrical heating was maintained on the
inlet tube while the heating of the Cu block was slow-
ly decreased. This assured that the block was the cold-
est part of the system.

While lowering the temperature of the Cu block,
the pressure reading on the Bourdon gauge was monitored.
When the sample began to condense, the gas pressure be-
gan to drOp. By lowering the temperature slowly, the
gas pressure remained near the equilibrium vapor pres-
sure as the sample was formed. Because gases only con-
dense on surfaces where the gas pressure is higher than
the equilibrium vapor pressure, slow condensation assur-
ed that condensation occurred only in the coldest part
of the system.

Samples were condensed at various temperatures.

The ultimate vapor pressure data did not depend on the
initial condensation temperature. Three distinct meth-
ods were used for condensation of the samples.

In the first method, samples were condensed above
the triple point temperature so that the gas condensed

into the liquid phase. The liquid rare gas was then

25

slowly frozen and the sample was used for measurements.
The second method consisted of condensing the sam—

ple at temperatures below the triple point. In this

case, the gas was condensed directly into the solid phase.
In the third method, samples were condensed as in

the second method; however, after condensation the sam-

ples were annealed near their triple points in order

to increase the grain size of the polycrystalline sample.
According to previous studies of crystal growth of

26

the rare gases, the different growth rates of each of
these techniques produces different average grain sizes
in the solid formed. No change in vapor pressure data
was observed which depended on the technique used to
form the sample. Therefore, it was concluded that
vapor pressure is not a function of grain size.

Another effect which was investigated was the
possible change in vapor pressure due to preferential

27 It is known that

evaporation at grain boundaries.
rare-gas solids show thermal etching at grain boundaries.
However, it is not known if the etched lines result

from preferential evaporation or from surface migration
away from grain boundaries. If thermal etching is a
result of preferential evaporation, there is a possi-
bility that the vapor pressure of a newly formed sample
might be higher than the vapor pressure of a solid which
has already undergone thermal etching.

By observing the variation in pressure with time,

it was found that the pressure of the solid sample

26

reached equilibrium in a short time after crystal grow-
th stOpped. The sample was then maintained at a cons-
tant temperature for up to an hour and no further mea-
surable pressure changes were observed. Because no
measurable pressure changes occurred, it was assumed
that if preferential evaporation were responsible for
thermal etching, the expected change is too small to

be observed.

The volumes of the condensed samples were approxi-
mately 0.8cm3. In order to estimate the sample volume,
the number of moles of gas condensed was determined.
Using the Bourdon gauge, the pressure change in the
gas storage reservoir was measured as the sample was
condensed. Knowing the volume of the gas storage reser-
voir (3500cm3) and assuming the ideal gas law applies,
the number of moles condensed was calculated. By us-

2,it was then

ing accepted densities of rare-gas solids
possible to calculate the volumes of the condensed
samples.

After the samples were formed, the system was
checked for parasitic condensation of gases on the
walls of the inlet tube. This was done by maintaining
the sample temperature constant and increasing the cur-
rent to the inlet tube heater. If the pressure was
observed to increase under these conditions, it was

assumed that gases had condensed on the walls of the

inlet tube. It was found that if gases were condensed

27

on the inlet tube, the entire sample had to be evapor-
ated and replaced before taking data.

After condensation the sample was distilled Lg
gigg to lower the concentration of non-condensable
impurities. Distillation was accomplished by first
lowering the sample temperature until almost all of
the primary gas component was condensed. The gas stor-
age reservoir was then evacuated to the lowest pressure
attainable, about 6 x 10'6Torr. After evacuation, the
gas storage reservoir was first sealed off from the
vacuum pump by closing the stopcock. The gas storage
reservoir was then opened to the sample chamber and the
vapor above the sample chamber expanded into the evacuated
gas storage reservoir. After this expansion the gas
storage reservoir was again sealed off from the sample
chamber, Opened to the pump, and evacuated. This tech-
nique assured that most of the vapor phase which con-
tained the non-condensable impurities was removed and
discarded. After the distillation, pure vapor sublimed
from the solid and replaced the impure vapor which had
been removed. After successive distillAtions caused
no further change in the measured sublimation pressure,
data were taken using the purified sample.

Temperature Control

Many techniques exist for accurate temperature

regulation of cryostats. Some control may be achieved

by simply immersing an experiment in a cryogenic bath

28

and controlling the bath temperature. However, this
technique is of limited value because suitable cryogenic
baths only exist over narrow temperature ranges. Also
the problem of accurately controlling bath temperature
over wide ranges is difficult.

In order to provide more reliable control and to
expand the range of available temperatures, electrical
heating may be applied. The type of electrical temper-
ature controller used depends on the application. For
example, in adiabatic calorimeters, the heat input re-
quirements are quite stringent so that control systems
are employed which minimize the amount of heating applied
directly to the sample.

In order to attain the temperatures desired for
our experiment, a combination of methods was employed.
For temperatures from 90 - 55K the apparatus was immer-
sed in a liquid oxygen bath. The bath temperature was
lowered by controlling the vapor pressure above the
liquid with ancautomatic pressure regulator.

The pressure regulator could maintain the vapor
pressure of the liquid stable to about I 0.1 Torr.

This permitted control of the bath temperature to about
30.02K over most of the range between 90 - 55K. Tempera-
ture of the bath was determined from vapor pressure
measurements made with a mercury manometer. Accepted

28

vapor pressure tables were used to calculate the temp-

erature from these pressure measurements.

29

When temperatures above 90K were required, the Cu
block containing the sample chamber was thermally insulat-
ed from the liquid oxygen bath by removing the He exchange
gas. Electrical heating was then applied to the Cu block
by means of the manganin wire heater wound on the block.
The amount of electrical heating was regulated to pro—
duce the desired temperature.

To achieve temperatures below 55K the apparatus was
suspended above a liquid He bath while cold vapor was
evaporated from the bath and pumped around the outer
stainless steel jacket. In order to produce the desir-
ed temperature, it was necessary to carefully vary the
exchange gas pressure and the rate of evaporation from
the liquid He bath.

The techniques described were used to attain the

temperatures desired. However, none of these techni-
ques quite provides a reliable and convient method of
controlling the cryostat temperature with sufficient
accuracy. More carefully controlled electrical heat-
ing was needed to maintain the temperature stability
within the tolerances required for accurate results.
This controlled electrical power was supplied to the
manganin heater, which was on the Cu block, by means
of a Model 1053, Hallikainen Instruments, Thermotrol
ac bridge temperature controller.

In principle an ac bridge temperature controller

consists of a Wheatstone bridge circuit driven by an

30

ac voltage source. The bridge circuit is connected to
an amplifier to supply power to a heater as shown in
Figure 3. One arm of the bridge is temperature sensi-
tive and is thermally anchored to the Cu block contain-
ing the sample chamber. A change in the temperature of
the sensitive arm of the bridge produces an unbalanced
condition in the bridge. The signal from the unbalanced
bridge is then phase analyzed to determine whether the
controller should increase or decrease its power out-
put. If a power increase is required the out-of-balance
signal from the bridge is amplified and supplied to the
heater.

The temperature controller used in this eXperiment
supplied pulses of power to the heater. Power of the
pulses supplied to the Cu block matched the thermal
losses to the bath. When the bridge circuit became
unbalanced, the duration of the pulses changed to re-
turn the system tO thermal equilibrium.29

A resistive temperature sensor Of 40 gauge Cu
wire served as the temperature sensitive arm of the
bridge. The wire was wound non-inductively on a Cu
collar which was placed around the Cu block containing
the sample chamber. Room temperature resistance of this
sensor was approximately 3004/L .

COpper thermometers are not normally used for low-
temperature measurements because their resistivity is

not reproducible over several cooling cycles. However,

31

 

  
 

Cu sensor 10011

 

 

 

 

 

 

 

 

 

 

 

Temperature Phase sensitive
Controller detector and
amplifier
Output to heater
Figure 3: {no Operation of the a: bridge temperature

controller is indicated in this schematic
drawing.

32

the resistivity changes rapidly with temperature above
about 20K so that such a thermometer is quite sensitive
to temperature changes. It is this latter condition
which is important when the thermometer is used as a
sensor to detect small temperature changes. Thus
COpper makes a good sensor for a temperature controller
but cannot be reliably calibrated for absolute temper-
ature measurements.

The output of the temperature controller was sup-
plied to the 1250afl; manganin heater wound on the block
containing the sample chamber. A layer of cigarette
paper coated with glyptal varnish was placed between the
copper block and the heater in order to assure electrical
insulation and to improve thermal contact between the
copper block and the heater. Series resistors vary-
ing from 100J'b to 4700-11' were attached in series with
the manganin heater to reduce the amount of power sup-
plied to the heater.

When the copper block temperature was nearly in
equilibrium with the bath, only small amounts of electri-
cal heating were necessary to maintain temperature con-
trol. It was found that by reducing the amount of power
to the heater by means of series resistors, more stable
control was achieved. Improved control resulted because
the pulses of power directly from the controller were
large enough to cause temperature oscillations in the Cu

block due to the alternate heating and coOling as pulses

33

were applied. By reducing the amount of power supplied
to the Cu block with each pulse, the size of the temper-
ature oscillations between pulses was greatly reduced.

With this system it was possible to control the
sample temperature to $1 mK for the length of time
necessary to take measurements and to :5 mK for longer
periods of time. Some slow drifts of sample temperature
were observed due to changes in room temperature.

Temperature Measurement

The sample temperature in this eXperiment was mea-
sured using platinum resistance thermometers. Resistance
of the thermometers was measured using a potentiometer.

Two different thermometers were used to measure
temperature in this experiment. Calibration of these
thermometers was supplied by the National Bureau of
Standards. These calibrations were based on the 1968
International Temperature Scale for which the following
relations apply: triple point temperature of water =
273.16K = 0.0100, and boiling point temperature of oxy-
gen = 90.188K = ~132.962°c.25

The thermometers used were four—lead Model 8164
Leeds and Northrup capsule-type Pt resistance thermometers.
For measurements below 91K we used the thermometer with
serial number 1644176. The thermometer used for measure-
ments above 91K had serial number 1737395.

When data were taken, the thermometer in use was

imbedded in the Cu block as shown in Figure 1. If it was

34

desired to use a different thermometer, the apparatus
was brought to room temperature, the outer stainless
steel jacket removed, and the thermometer in use was
replaced.

Thermal contact between the Cu block and the resis-
tance thermometer was aided by a thin layer of Apiezon-
L vacuum grease. The thermometer leads were thermally
anchored to the surface of the Cu block.

In order to assure that the sample and the thermo-
meter were at the same temperature, the existence of
thermal gradients in the Cu block and in the sample
was investigated in the following way. The pressure of
the exchange gas in the vacuum space around the Cu
block was increased to improve the thermal contact of
the Cu block and the He bath. Electrical heating from
the temperature controller was simultaneously increased
to maintain the block temperature constant. The vapor
pressure reading was monitored to insure that no change
in sample temperature occurred when heater power was
increased. This technique is capable of detecting
changes in sample temperature Of less than 1 mK.

Small thermal gradients of about 2 mK were Observed
when the heater power input approached its maximum value,
10 watts. In order to minimize these thermal gradients,
the heater was never Operated above about 10% of maxi-

mum power when taking data.

35

Thermal gradients due to the heater on the inlet
tube were similarly investigated; however, none were
detected. In this case the temperature of the Cu block
measured with the Pt thermometer was held constant while
the power to the inlet tube heater was varied. Although
it was Observed that heat was conducted into the Cu
block from the inlet tube, no measurable thermal grad-
ients were Observed in the block.

Thermometer resistance was measured using a Leeds
and Northrup calibrated K - 5 guarded potentiometer
using a single-potentiometer technique.31 The null de-
tector used was a Leeds and Northrup Model 9834. All
parts of the measuring circuit were guarded to prevent
error due to leakage currents.

Potentiometer readings have an estimated accuracy
of I 0.24 V. The potentiometer calibration was certi-
fied by the manufacturer. It was reported that no
corrections were necessary in order to assure readings
within the desired accuracy.

A schematic representation of the temperature
measuring circuit is shown in Figure 4. The four plat-
inum leads from the thermometer were soldered to term-
inals attached to Cu leads from the potentiometer sys-
tem. Two of the leads supplied current from a 6.0V
lead-acid cell to the thermometer windings. Thermometer
current was controlled by a ten-turn potentiometer used

as a shunt across a 100/1 resistor. The voltage drOp

Thermometer

36

Current

7\

 

 

 

 

 

 

 

 

 

#21

xeversing
switch

Control

 

1.0 11 standard

 

 

 

 

 

 

Em E

 

A
Aux.
Emf

Potentiometer

 

 

Figure 4:

This potentiometer circuit was used for
the

temperature measurement.

Current to

thermometer was reversed to account for
thermal emfs in the system.

37

across a IOJI standard resistance in series with the
thermometer and lead-acid cell was measured on the
"Auxiliary Emf! scale of the pOtentiometer. This mea-
surement was used to determine the thermometer current.
For measurements below 90K the current through
the thermometer was maintained at 2.0 mA. These cur-
rents corresponded to the currents applied when the
respective thermometers were originally calibrated.

These currents were controlled and measured to within

t0.05/£(A.

The potential drop across the resistance thermo-
meter was measured on the "Emf" terminals of the pot-
entiometer. By using a potentiometer, lead resistances
may be ignored in these measurements. No current flows
through the thermometer leads when the potentiometer
is balanced.

Although lead resistances do not affect measured
voltages, the effect of thermal emfs must be eliminated
from all measurements. Thermal emfs were minimized in
the measuring system by using continuous Cu leads from
the resistance thermometer to the potentiometer termin-
als. However, even with this precaution, thermal emfs
as large as 4 AV were sometimes present in the measur-
ing circuit.

In order to compensate for the effect of these
thermal emfs, the direction of the current through the

thermometer was reversible. For each temperature

38

determination, two potential measurements were made with
the current reversed between measurements. The first
measurement was taken using the potentiometer in the
"Emf" setting. The current was then reversed and
readjusted to the prOper value. Finally, a second
potential reading was taken using the "Reverse Emf"
setting of the potentiometer.

The difference in these two potential measurements
is twice the value of the thermal emfs present. The
average value of the two readings is equal to the pot-
ential drop across the resistance thermometer.

Care was taken to eliminate random errors due to
changes in thermal emfs as the sets of potentiometer
measurements were being made. All terminal posts and lead
wire connections were thermally insulated to reduce temp-
erature fluctuations at connections where thermal emfs
might be present. For each data point, six to eight
pairs of potential measurements were made. This assured
that the size of thermal emfs was remaining constant. If
rapid changes in thermal emfs were observed, the measure-
ments were repeated when the thermal emfs reached equilib-
rium.

Temperature measurements made using the techniques
described above have an estimated sensitivity of approxi-
mately : 0.5 mK. The sensitivity is reduced at low temper-
ature due to the decreased sensitivity of the resistance

thermometer. Absolute temperature accuracy depends on

39

accuracy of the thermometer calibration and potentio-
meter calibration. Assuming negligible error in the
thermometer calibration, the absolute accuracy of volt-
age measurements permits temperature measurement to
within :2 mK.

Pressure Measurement

Vapor pressure was measured over eight orders of
magnitude in this experiment. Because of the wide range
of pressures measured, different techniques and correc-
tions were applied depending on the pressure range be-
ing considered.

Pressures above 1 Torr were measured using a Texas
Instruments Model 142 quartz spiral Bourdon gauge. Basic-
ally this gauge consists of a fused quartz Bourdon tube
and a readout device to measure the tube deflection.

Deflection of the Bourdon tube is measured Optically.
Light is reflected from a mirror attached to the end of
the Bourdon tube. The reflected light beam is located
by means of a photocell. The Bourdon tube is calibrated
to determine the relation between the angle Of deflection
and the pressure.

For this experiment two Bourdon tubes were used
which covered pressures from 0 - 250 Torr and 250 - 500
Torr, reSpectively. The photocell detector was attached
to a counter which divides the full scale deflection
into 300,000 counts.~ Thus the full scale deflection of

250 Torr was divided into 300,000 counts resulting in a

40
gauge sensitivity of better than 1 mTorr.

At low pressures oscillations of the Bourdon tube
which were driven by vibrations present in the room
were sometimes observed. The entire system was isolat-
ed from vacuum pumps and other sources of vibrations
to eliminate these oscillations. At high pressures,
vibrations presented only minor problems because damp-
ing was supplied by the gases in the Bourdon tube.

The reference Space surrounding the Bourdon tube
was evacuated to 10"5 Torr by means of an Oil diffusion
pump. It was possible to admit air into the reference
Space in order to damp vibrations which were sometimes
started by accidental shocks to the system. Small
amounts of air in the reference Space could damp vibra-
tions and could then be evacuated before pressure mea-
surements were made.

The Bourdon gauge was calibrated using a H3 mano-
meter read with a Wild cathetometer. The manometer
was constructed from 12mm i.d. glass tubing. Before
being filled with mercury, the manometer was carefully
cleaned with commercial glass cleaner and then with
dilute nitric acid. The manometer was then rinsed with
distilled water followed by methanol. The methanol was
then evaporated by means of a vacuum pump.

After being cleaned, the manometer was filled with
reagent grade mercury. ”hen not in use the manometer

was evacuated to prevent oxidation of the mercury.

41

Using the cathetometer, the mercury level of the
manometer could be determined to within 10.02mm. How-
ever, corrections were applied to these readings to ac-
count for capillary depression of the mercury.32 For
each pressure reading the height of the mercury meniscus
was recorded and the tables in reference 32 were used
to calculate capillary depression of each reading.

In order to eliminate the effect of thermal eXpan-
sion of the mercury, the manometer readings were then
corrected to correspond to 00C density of mercury. Care
was taken to keep the temperature of the manometers
stable while the calibration was being made. Room
temperature was measured and the mercury density was
corrected using accepted tables of mercury density.33

Manometer readings were also corrected to corres-
pond to readings taken under conditions for'standard
gravitational attraction. For standand gravitational
attraction, g = 980.665 cm/secz. The accepted value
for local gravitational attraction in our laboratory
was, g = 980.350 cm/secz.

The effect of the combined corrections to adjust
the manometertreadings to 0°C density of mercury and
standard gravity was to introduce a factor which made
the corrected mercury level lower than the actual
mercury level. These correcticns never exceeded 0.5%.

The absolute accuracy of the Bourdon gauge calibra-

tion using the manometers described above is f0.02 Torr.

42

Because the gauge reSponse was nearly linear over narrow
pressure ranges, the gauge can be used to measure accurate-
ly small pressure changes to within : 2m Torr.

Below 1 Torr pressures were measured using a Con-
solidated Vacuum Corporation Type GM-lOO-A McLeod gauge.
Calibration of this gauge was supplied by the manufac-
turer. The McLeod gauge readings agreed with the man-
ometer below 1 mm to within the limits of error of the
manometer readings. The McLeod gauge was welded into
the glass vacuum system connecting the gauge to the sam-
ple chamber. When not in use the gauge was evacuated
and sealed from the rest of the system by means of a
high-vacuum stopcock.

Before data were taken, the McLeod gauge was de-
gassed by heating and evacuating the gauge. Degassing
was necessary not only to prevent contaminants from
reaching the sample chamber but also to prevent stick-
ing of the mercury column in the gauge capillaries.

A cold trap separated the McLeod gauge from the
sample chamber. The cold trap prevented contamination
of the sample by mercury diffusing from the McLeod
gauge. A bath of dry ice and acetone was used to re-
frigerate the cold trap. This mixture produced a bath
temperature of 200K which was cold enough to condense
gaseous mercury but was warm enough to prevent condensa-

tion of rare gases.

43
All McLeod gauge readings were corrected for mercury

streaming.34 Because of the flow of mercury vapor from
the McLeod gauge to the cold trap, the pressure in the
McLeod gauge is reduced. This process is similar in
principle to the Operation of a diffusicn pump. The
diffusing vapors collide with gas molecules and impart
momentum in the direction of diffusion. This causes a
pressure gradient between the McLeod gauge and the cold

trap which can be calculated from the equation,34

In PERea1)/P(r~1cLeodfl = .905 r PHgmlf/Dlz) . (37)
In equation (37), r is the radius in cm of the
tube connecting the McLeod gauge with the pressure to
be measured, T is the room temperature in Kelvins, D12
is the diffusion coefficient in cmz/ sec at 1 atmOSphere
and 300K for the gas in the gauge diffusing into Hg

vapor, and P is the vapor pressure of mercury in Torr

Hg
at room temperature.

Accepted values for the vapor pressure of mercury
were used for these calculations.35 Values used for
diffusion coefficients are34: for Ar, D = 0.12 cmZ/sec;
for Kr, D = 0.093 cmZ/sec; for Xe, D = 0.079 cmZ/sec.
The radius of the tube connecting the McLeod gauge
with the pressure to be measured was, r = 0.3 cm. Using
these values the correction never exceeded 10% of the
theod gauge reading.

Near 1 Torr, the McLeod gauge readings have an

estimated accuracy of 1%. Between 10"4 - 10'5 Torr,

44

the lower usable limit of the gauge, the gauge readings
are estimated to be accurate to within 5%. Additional
error may be present due to inaccuracy in the diffusion
coefficient values used for the mercury streaming correc-
tion. This error could be as much as %.

Readings below 1 Torr were also corrected for therm-
al transpiration. This effect is observed when two
vessels connected by a narrow tube are kept at different
temperatures. When the pressure of the gas in the
vessels is low enough that the mean free path of the
gas molecules is several times the diameter of the
connecting tube, it is found that the pressure is highs

er in the warmer vessel. The ratio of the pressures for
36

this case may be simply expressed as Pl/P2 = TZ/Tl
Here, P2 and P1 are the pressures in the reSpective
vessels and T2 and T1 are their absolute temperatures.

If the mean free path of molecules in the gas is
short compared to the diameter of the connecting tube,
no pressure gradient is observed.

HoweVer,_for most actual pressure measurements,
the mean free path of the gas molecules is between the
two extremes. In this case, the ratio of pressures in

the two vessels may be calculated from the empirical

 

equation,37
£1, = 'JTl/TZ '1 .
P _W' *2 k % k ‘T + 1 , (58)
2 X + B X + C Al?“ + 1

* r
where T2)T1 and X = 2 PZd/l‘l + T2.

45

In equation (38), P2 represents the pressure measur-
ed at room temperature, T2, and P1 represents the pressure
of the sample in the cryostat at temperature, T1. The
parameter d is the diameter in mm of the tube connect-
ing the sample and the room temperature pressure gauge.
For this experiment the value of d was 6.35 mm.

*, and 0* depend on

The values of the parameters A*, B
the gas which is being used for pressure measurements.
Table 5 shows the values A*, 3*, and C* used for this

correction.37

TABLE 5

Values of the parameters A*, 3*, and 0* of
equation (38).

Gas A*(105K2/Torr2 mmz) 3*(102K/Torr mm) 0*(K3/Torrkmmk)

Ar 10.8 8.08 15.6
Kr 14.5 15.0 13.7
Xe 35 41.4 10

This correction appeared to be adequate for pres-
sures above about 0.1 Torr. At this pressure, the sam-
ple pressure was approximately 20% lower than the pres-
sure measured at room temperature.

For lower pressures, deviations appeared which
could only be attributed to the correction. The vapor
pressure parameters presented in Table 11 have anomal-
ously low values for reduced temperatures between 0.7
and 0.5. These values indicate a large overcorrection in

this range. Deviations probably occur at lower pressures

46

also but are masked by other effects. This correction
for thermal transpiration becomes unreliable at low

pressures but no adequate correction is available.

IV. RESULTS

The measured pressure and temperature points are
presented in Table A1, Table A2, and Table A3 in the
Appendix. Figures 5, 6, and 7 of the text show the
measured sublimation pressures, P, plotted as functions
of the temperature, T, for Ar, Kr, and Xe, reSpectively.

Law of Corresponding States

The sublimation pressure curves of rare-gas solids

may be related by means of the law of correSponding

38

states. As discussed earlier, the reduced equation

of state for rare-gas solids should depend only on the
reduced temperature, T*, the reduced volume, V*, and
the reduced de Broglie wavelength Ai‘. Thus, for all
the rare-gas solids the reduced sublimation pressure
curve is given by equation (14).

It is further expected that the reduced sublimation
pressure curves should increase monotonically with in-
creasing /\*.39 The parameter/\* defined by equation
(10), indicates the relative size of quantum effects

in the crystal.

3
Physically, a large value of/\{ represents a

7':

crystal with large reduced zero point energy, Ez.

Because the reduced zero point energy is larger for

47

48

 

 

 

 

 

 

IOOO , ‘
6,:lTl‘I'I‘llli
‘ .. {a
2 w :3 ,_
'00—17 0.. _—
8 d;- I. 1
4 db .0 .-
2 '4' ’.. ‘1'
A ‘0 ‘17 ." r:-
s. 6 ‘t l q
E; 4,«~ ‘3 ._
t 2 " . '0-
a- '91? ' ‘1.—
B ’ 0 r
42 1' ARGON "
it e ..
LIJ
a: 0.: , -—.;—
53 2’: ;. T
U)
2 > 3 .
m l .
m '04:;— ° -—;_--
a. 3 .. ° L
4 1:- O in
Z 21— .' ,
9 104*"- '
.4. o .4.
F' 6M+ ' a
‘1 44.. i
E 2.. 3 ,,
5'; I041:- . q
3 2,: ; d
U) 2
"figi-o '—h-
“'o. r-
24 -.
IO'L _-

7
TEMPERAT URE T (K)

Figure 5: Measured sublimation pressures are
plotted for Ar.

49

 

 

1000 '
.M‘l'V'W'V'
4 * .l
2 4» .I.’ «f
100 ___. ..' p
Slt '0', 7"
4‘? 3' ‘*
t 2‘? 0" ‘1-
(D
t;- I01... 3’ ‘1'."
6"? e. ‘1'
a. 44" '0’ KRYPTW A.
2 T. j; n'
LLJ 1.0-j:- ..' ~1—
S 6" e. '1’
m 44- :0 ""
m 24- . "P'
a: l
0- °' “IL“ - T
6v s ‘1
2: 4.” g n
4 1°.2q— . _—
Z 6*: ' 1r
_1 ‘ . ’
m 4.1,, ’ .1.-
.cra-w- ° ‘“‘
,1: . :1
4‘? .0 "l"
zj- ..
10V“ , I 1 1_ 1 1 L l 1 1 1 l 1, 1

 

 

so 6070 so 90100110
TEMPERATURE T(K)

Figure 6: Measured sublimation pressures are
plotted for Kr.

50

 

 

I000 1
gj'l‘l‘l T'T'I'I'I';
‘d— ’0’:-
z-- -" 1.
1001.“ at". ‘2?
s~r -
c 4‘1’ ’0" l—
b .'-' d!-
E 2 /’
a_ 'C12:' 0 ‘_
3:: f "’
g 2"" 0.. '7”
:3 -n— o. -*-
g; '21: .’ ::
m 411-- .. du-
E 2'“- : ‘7
' 0" e ‘-
s . -~
; 2" c T
< .
EIO'Lr' -' ‘2‘?
a :1 .c- 2:
5’; 2" a: ,_
IO':::-. '2?

 

 

BE
2 . .-
w-‘ml l 1 l 1 l L I 1 J 1 1 1 l 1 l 1

so 00 100 no 120 1:0 140 150 100
TEMPERATURE T (K)

Figure 7: Measured sublimation pressures are
plotted for Xe.

51

solids with large values of /\*, the molecules are

more easily removed from the lattice. Hence, the larger
the value of /\*, the higher the expected reduced vapor
pressure.

Values for /\* found using the all-neighbor
Mie-—Lennard-Jones potential are presented in Table
3. Because /\* is larger for Ar than for Kr and Xe,
the reduced vapor pressure curve for Ar should lie above
the curves for Kr and Xe, respectively. Using the pot-
ential parameters of Table 3 the reduced pressure curves
were plotted using data from this eXperiment. These
curves deviate from the expected order as can be seen
in Figure 8.

The reduced pressure curves lie close together
as eXpected. However, the Kr and Xe curves are inter-
changed from the predicted order. This effect has been
observed in other properties and by other investigators?”39

The reasons for this deviation are not clear. In
the pressure range considered, the deviation is larger
than the expected error in the vapor pressure data. Be-
cause all the data were taken using the same apparatus,
the effects of systematic error should not change the
relative position of the curves.

Another possible reason for the inversion of order
is the inaccuracy in the determination of the potential
parameters 5 and 0". A change of only 2-3% in the val-
ues of 6 and a'would reverse order of the Kr and Xe

curves. The stated error of the values of E and 0"

52

 

 

 

 

'5 T I I
_ ,gx‘
vs
39”
x. ,-
‘7V" 4§€" '7
1461’.
so?
.5515
k.
-8— kg.. —-1
x C
i? :1? °
9: 5" '
I— h 0 —
c 0
_1 .235.
'9”- .‘g‘; x- ARGON —
o , °= xsuou
£3: - KRYPTON .1
o 0
£0
.22. -'"
'10 —- £0. —-4
0 O
’8;
-— O. ._,
x O
6’...
wig; l 1 l
. 55 so .65 .70

REDUCED TEMPERATURE 7*

Figure 8:

Reduced sublimation pressure curves
are plotted for Ar, Kr, and Xe.

53

is less than the amount required to change the order
of the Kr and Xe curves.3 However, it is likely that
the actual potential deviates from the analytic hie-
Lennard-Jones potential for which E and 6' were deter-
mined. Thus, the values of e and 0' for the hie-—
Lennard-Jones may differ from the parameters for the
actual potential. Although the hie-Lennard-Jones
potential has the necessary form, QD(r) = e-f (r/o-),
the accepted values of E and 0' only represent the
values which make the shape of the calculated poten-
tial as much as possible like the shape of the actual
potential.
Static Lattice Energ
By applying equation (30) to the data of this

experiment, the static lattice energy, E and the

O,

geometric mean of the lattice vibrational spectrum,

was

ed earlier, the quantities Bo and uug may be determined

, could be determined for Ar, Kr, and Xe. As describ-

from a plot of ”In PTL2 versus l/T. P is the measured
sublimation pressure and T is the sample temperature.
Typical plots of the data for Ar, Kr, and Xe are shown
in Figures 9, 10, and 11, respectively. The slope of
each plot yields E0 and the intercept yields 00%;.

In practice however, it was found that more accu-
rate results could be obtained by fitting the data to
equation (30) by using the method of least squares.

Equation (30) is first linearized into y = a + bx

 

 

 

 

8.3 «- q-
8.2" -_
Bet-“- ‘-
8.0“" .1...
7.9%” _—

“7.8"" L.

a 1

:7 7

£3 ' 7

V

42.7.6+ -..

‘3 \

d
705-!- ARGON \—1-0
7.4T'
7e31- '—
7.2""' __
7.1“ -—
1.. 1 1 1 1 1 1 1 1 1

12.0 12.5 13.0

RECIPROCAL TEMPERATURE T'l(10‘3K-1)

Figure 9: Typicdl plot of lnPTk versus l/T for Ar.
Upper line is corrected for vacancy
formation.

8.2

8.1

8.0

7.9

7.8

m3? (131:: 1:3)
'4: in b.

N
e
U

7.2

7.1

7.0

6.9

6.8

 

 

 

 

RECIPROCAL TEMPERATURE 'r' 1 ( 10'3x'1)
Figure 10: Typical plot of lnPT9 versus l/T for Kr.

Upper line is corrected for vacancy

formation.

55

1 I I l l l l J l

I I I I I l I I l
-1-- d}-
-— \\ .1.-
-- \ 1mm --
11-. _—
-I- —11-
‘11" ""

l 1 L l L l l l 1

I I I I I» I I I I
9.0 9.5 10.0

56

 

 

 

 

 

8.4 I 1 I
8.3 --
8.2 -_
8.1, 4~
8.0 ....
7.9 a-
.{;7.8 q;
”€7.11 -1-
7.5 --
7.4 -_.
7.3 J -_.
7.2 --
7.1 1? +-
1
7'0 6.5 I] ' I I 7:.0 j] ' i I 7.5

RECIPROCAL TEMPERATURE T'l(10'3K'1)

Figure 11: Typical plot of lnPTa versus l/T for Xe.
Upper line is corrected for vacancy
formation.

57
where y = 1n PT% and x = l/T. Then using the standard

method for linear regression, a and b can be determined

40,

from the equations

a = ( éyi " bZXi) /n,

“gXiyi‘ 2}":ng

b:
n gxf- (gypz

Computing was done using a Hewlett-Packard model

9100A programmable calculator. Computation of ln PT%

andl/T was performed and equation (39) was applied to
the data.

Because E0 and Lfilg change slowly with temperature,
data from narrow temperature intervals may be separately
fit to equation (30). Parameters a and b may then be
calculated for each temperature interval. The values
of E0 and Uplg calculated then are referred to the
temperature corresponding to the center of each interval.

In order to permit comparisons of the variations of
go and mg for different gases, the temperature intervals
chosen correspond to the same reduced temperature inter-
vals for each gas. The temperature intervals used for

this analysis are shown in Table 6. Reduced temperatures

were calculated using parameters presented in Table 3.

58

TABLE 6

Temperature intervals used for analysis of vapor
pressure equations.

 

 

Columns 3 and

Reduced Actual
temperatures temperatures
(nondimensional) (K)

Ar Kr Xe

0.70 - 0.65 84.5 - 78.0 115 - 107 162 - 150

0.65 - 0.60 78.0 - 72.0 107 - 98.4 150 - 139
0.60 - 0.55 72.0 — 66.0 98.4 - 90.2 139 — 127
0.55 - 0.45 66.0 - 54.0 90.2 - 73.8 127 - 104
0.45 - 0.35 54.0 - 40.0 73.8 - 54.7 104 - 76.2

4 of Table 7 show values of the para-

meters a and b of equation (30).

These parameters

were calculated using the temperature intervals shown

in Table 6.

sures eXpressed in dynes/cm

The values of a and b are found for pres-

in order to simplify cal-

culation of E0 andll) in proper units.
8

Gas

Ar
Ar
Ar

Ar

Kr
Kr
Kr

Kr

TABLE 7

Values of the parameters a and b of equation (30)
found from vapor pressure data.

Temperature
Range (K)
84.5 - 78.0
78.0 - 72.0
72.0 - 66.0
66.0 - 54.0
54.0 - 40.0
115 - 107
107 - 98.4
98.4 - 90.2
90.2 - 73.8

-a(K)

1988.
999.

74
32

995.01

993.
1061.
1387.

95
28
78

1386.79

1393.48

1368.86

27
27
27
27

27.
27.
27.
27.

.4565
.5921
.5327
.5150
28.

6928
8683
8608
9299
6426

Gas

Kr

(Table 7 continued)

Temperature
Range (K)
73.8 - 54.7
1162 - 150
150 - 139
139 - 127
127 - 104
104 - 76.2

59

-a(K)

1441.08
1933.79
1929.86
1931.63
1896.91
2028.45

28.5093
28.1399
28.1137
28.1250
27.8466
29.2002

Values of E0 andli g correSponding to these val-
ues of a and b are shown in the third and fourth col-
umns of Table 8. These values of anndLIJg are referr-

ed to the temperatures in the second column.

TABLE 8
Values of E0 andu)p calculated from the parameters
of Table 7. 0
Gas Temperature (K) -Eo(cal/mole) hug(10123ec'1)
Ar 81.2 1964 6.60
Ar 75.0 1985 6.91
Ar 69.0 1976 6.77
Ar 60.0 1974 6.73
Ar 47.0 2108 9.97
Kr 111 2756 5.23
Kr 103 2754 5.22
Kr 94.3 2768 5.34
Kr 82.0 2719 4.85
Kr 63.2 2862 6.48

60

(Table 8 continued)

Gas Temperature (K) -Eo(cal/mole) hug(10123ec'1)
Xe 156 3841 4.57
Xe 145 3833 4.53
Xe 133 3837 4.55
Xe 115 3768 4.15
Xe 90.1 4029 6.51

A similar analysis was performed using equation
(36). This equation is corrected to account for vac-
ancy concentrations. The values of E0 and MU g may be
calculated from the parameters a and b of equation (36).
These values will reflect the effect of vacancy forma-
tion and will differ slightly from those presented in
Table 8.

Estimates of the effect of vacancy formation on
specific heat measurements have yielded values for gs,
the Gibbs function for vacancy formation, for Ar and_

Kr.41 These values may be expressed as:

30 exp [-644.2/T(K)] and

for Ar, exp [-gs/kT]

for Kr. eXp [-sS/k'r] 30 exp [-890.8/T(K)] .

No vacancy concentration measurements were avail-
able for Xe so the value of gS was estimated using the
law of corresponding states. Using the definition,

g = u-Ts + Pv, we have exp(-gs/kT) = expE(u-Ts+Pv)/kT].

This may be rewritten as eXp (-gS/kT) =

(exp s/k) exp [(-u-Pv)/kTI The parameter s/k in the

61

first term is dimensionless so the reduced value is the
same for Ar, Kr, and Xe. However, u + Pv has units of
energy so that the apprOpriate reduced variable is

u + Pv/E .

The reduced value of gs for Xe was assumed to be
equal to the average of the reduced values of gs for
Ar and Kr. Using the values of € found in Table 3, the
value of gS for Xe was found to be

exp (-gS/kT) = 30 exp [-1248/T(K)] .

Using the values reported above for exp (-gs/kT),
the xperimental data for Ar, Kr, and Xe were fit to
equation (36) and were plotted in Figures 9,10, and 11,
respectively. Data were also fit to equation (36) by
the method of least squares. Data from the temperature
intervals shown in Table 6 were used for the fit. The
values of a and b for each temperature interval are
presented in Table 9. Values of a and b are again found

for pressures expressed in dynes/cm2

in order to simpli-
fy calculation of EO andia/g. Values of E0 and Lng
corresponding to these values of a and b are present-

ed in Table 10.

62
TABLE 9

Values of the parameters a and b of equation (36)
found from vapor pressure data.

-a(K) b

Gas ngpgzafޤe
Ar 84.5 - 78.0 995.89 27.5553
Ar 78.0 - 72.0 1004.12 27.6609
Ar 72.0 - 66.0 996.42 27.5557
Ar 66.0 - 54.0 994.62 27.5269
Ar 54.0 - 40.0 1061.30 28.6932
Kr 115 - 107 1396.12 27.9533
Kr 107 - 98.4 1390.84 27.9052
Kr 98.4 - 90.2 1395.47 27.9533
Kr 90.2 - 73.8 1378.55 27.7718
Kr 73.8 - 54.7 1439.73 27.7718
Xe 162 - 150 1945.65 28.2259
Xe 150 - 139 1932.43 28.1377
Xe 139 - 127 1934.22 28.1469
x. 127 - 104 1855.76 27.5149
x. 104 - 76.2 2007.64 28.9352

63

TABLE 10

Values of E andial calculated from the parameters

of Table 9.° 8
Gas Temperature (K) -Eo(callmole) hug(lolzsec'l)
Ar 81.2 1978 6.82
Ar 75.0 1994 7.07
Ar 69.0 1979 6.82
Ar 60.0 1976 6.76
At 47.0 2108 9.97
Kr 111 2773 5.38
Kr 103 2762 5.29
Kr 94.3 2772 5.38
Kr 82.0 2733 5.05
Kr 63.2 2860 6.42
Xe 156 3864 4.71
Xe 145 3838 4.57
Xe 133 3842 4.58
Xe 115 3686 3,71
Xe 90-1 3988 5.96

64

Heat of Sublimation

The data were also analyzed using equation (23) to
calculate values for the heat of sublimation,L. The
data are plotted in the form in P(Torr) versuser in
Figures 12, 13, and 14. The techniques described in
the previous Section were used to fit the data to equa-
tion (23) by the method of least squares. As in the
previous Section, the parameter a was expected to vary
slowly with temperature. Because of the expected varia-
tion, data from the temperature intervals of Table 6
were used for the fit. As before, the data from each
interval were fit to equation (23) separately. Thus,
the values of the parameter a are referred again to the
temperatures at the centers of the intervals.

The heat of sublimation, L, is related to the

parameter a of equation (23) by:

a g L . (40)
R (1 - vslvg + BP/RT)

 

The parameter a was determined from the least square
fit described earlier. The specific volumes of the
solids, v8, were obtained from density curves.2 Val-
ues of the second virial coefficient, 8, were obtained
from an extrapolation of reduced curves."2
Values found for the parameters a and b of equa-
tion (23) are presented in Table 11. Calculated val-

ues of heats of sublimation are presented in Table 12.

lnP (Torr)

d1—
“r
.—
_-
.—
.—
~1—
1...

 

H—

6.6

6e5"- ——

2"
‘1’

5.7-1

5.6“
5.51

5.4-1-

5.3-

 

 

 

12.0 12.5 13.0

11110111110011. TEMPERATURE 1'100'316'1)
, Figure 12: Typical plat of 111? versus l/T for. Ar.

-
—-11—
-

5.8

5.7%

5.6-

66

 

 

 

 

 

J
I

Figure 13:

l 1 1 1 1 l
I I I T I I
9.5 10.0

RECIPROCAL TEMPERATURE T‘l(10‘3x'1)

1
I

Typical plot of In? versus l/T for Kr.

67

 

5.7‘
5.6‘

5.5d

4.9‘7
4.8"
4.7%“

4.6"

 

 

 

l
I

 

IIJ 11 114
“'56.5' ‘ 7 ‘ 7.0 I ' I 7.5

RE01PR00AL TEMPERATURE T‘ 1(10'3R- 1)
Figure 14: Typical plot of In? versus l/T for Xe.

68

TABLE 11

Values of the parameters a and b of equation (23)
found from vapor pressure data.

Gas ngpgzagޤe -a(K) b
Ar 84.5 - 78.0 946.35 17.5408
Ar 78.0 - 72.0 962.22 17.7428
Ar 72.0 - 66.0 960.59 17.7214
Ar 66.0 - 54.0 963.39 17.7623
Ar 54.0 - 40.0 1038.03 19.0767
Kr 115 - 107 1332.30 17.8184
Kr 107 - 98.4 1334.63 17.8413
Kr 98.4 - 90.2 1346.76 17.9656
Kr 90.2 - 73.8 1330.73 17.7765
Kr 73.8 - 54.7 1399.08 18.5731
Xe 162 - 150 1856.41 17.9236
Xe 150 - 139 1857.02 17.9276
Xe 139 - 127 1860.70 17.9513
Xe 127 - 104 1836.37 17.7526
Xe 104 - 76.2 1960.37 18.9607

69

TABLE 12

Values of heat of sublimation, L, calculated from
the parameters of Table 11.

Gas Temperature (K) L (cal/mole)
Ar 81.2 1852
Ar 75.0 1900
Ar 69.0 1904
Ar 60.0 1914
Ar 47.0 2062
Kr 111 2612
Kr 103 2636
Kr 94.3 2670
Kr 82.0 2644
Kr 63.2 2780
Xe 156 3480
Xe 145 3596
Xe 133 3661
Xe 115 3632
Xe 90.1 3895

70
Lattice Vibrational Energy

The lattice vibrational energy, Evib’ may now be
calculated using the thermodynamic relation,

Evib a ’Eo - L + P(vg-v8) , (41)
Physically, this equation is similar to the equation
82(OK) . -EO(OK) - L(0K). This latter equation defines
the zero-point vibrational energy at 0 K. Equation (41)
applies for temperatures above 0 K; thus, the work done
by expanding gases must be subtracted from the heat of
sublimation. This adds the term P(vg-v8) which repre-
sents the amount of work done by expanding gases remov-
ed from the solid.

The specific volumeq'v8 andxvg, of equation (41)
are the same as those used to calculate the heats of
sublimation.2 The calculated values of Evib are pre-
sented in Table 13.

71

TABLE 13

Values of vibrational energy, E ib’ calculated
using equation (41). v

Gas Temperature (K) Evib (cal/mole)
Ar 81.2 285
Ar 75.0 243
Ar 69.0 213
Ar 60.0 181
Ar 47.0 139
Kr 111 381
Kr 103 331
Kr 94.3 288
Kr 82.0 258
Kr 63.2 205
Xe 156 693
Xe 145 529
Xe 133 445
Xe 115 281

Xe 90.1 272

V. GENERAL CONCLUSION

As can be seen from Figure 5, Figure 6, and Figure
7, the sublimation pressures of solid Ar, Kr, and Xe
have been measured over several orders of magnitude.
However, for each gas the pressure curves began to level
off at low pressures.

The cause of this deviation was the presence of
non-condensable impurities such as He in our gas samples.
As the sample temperature was lowered, most of the pri-
mary gas component was condensed. The concentration
of nonpcondensable impurities then became large in the
vapor phase. The vapor pressure curve then began to
level off because lowering the sample temperature fur-
ther had little effect on the vapor phase.

These curves were reproducible for different gas
samples and different runs. This indicates that the
impurities were present in our gas samples and not
evolved from the walls of the gas handling system. If
the data had not been repeatable or had changed with the
length of time the samples had been contained in the sys-
tem, this would have indicated that outgassing of our

system was contaminating the sample.

72

73

Assuming the impurities present were non-condensable
gases such as Be, it was possible to estimate the concen-
tration of the impurities present in the gas samples.
This was done by assuming that the original gas pressure
before the sample was condensed was the pressure due to
the primary gas component plus the partial pressure of
the impurities. The lowest vapor pressure measured then
was equal to the partial pressure of the nonscondensable
impurity gases.

Assuming that the ratio of the partial pressure of
impurities to the total gas pressure equals the impurity
concentration, the impurity concentration may be calcu-
lated. The estimated impurity concentrations found from
this analysis are: 0.2 ppm for Ar, 13 ppm for Kr, and
10 ppm for Xe. These values may be compared with the
impurity levels reported in Table 4.

The estimated values are probably accurate for Ar
but may not be so reliable for Kr and Xe. For these
latter two gases, the partial pressure of the lowest
temperatures measured of condensed impurities such as
N2 and 02 at the lowest temperatures measured, may be
higher because the sample temperature is higher. A
further source of inaccuracy at low temperatures is
the correction for thermal transpiration. Even though
the gas inlet tube used was of relatively large diameter
(kin. i.d.) this correction became large at low temp-

eratures. The anomalously low values of static lattice

74

energies and heats of sublimation for reduced tempera-
tures, T*£:50.5, indicate a large overcorrection for
thermal transpiration.

Thus it is expected that actual vapor pressures
below 0.1 Torr should be somewhat higher than observ-
ed. This effect is masked at still lower temperatures
at which the measured pressure is increased due to im-
purities. For better low temperature measurements, an
improved correction for thermal transpiration and im-
proved techniques for gas purification are required.

At high temperatures our data may be compared with
that of previous workers. Most previously available
data is reported in terms of the parameters a and b in
equation (23). For comparison with our results some
of the values for a and b found by other workers follow.

For the temperature interval 83.6 - 66.1K Flubacher

£5.31-43 found a a -953.897K, b a 17.62836 for Ar.

For the temperature interval 115.8 - 83.3K, Beaumont
££H£lo41 found a a ~1127.58K, b s 16.04625 for Kr. For
the temperature interval 161 - 110K, Freeman and Halsey
found a - -1840.0 and b a 16.972 for Xe. Our parameters
calculated for Kr differ significantly from those found
by Beaumont,g§'gl.but are closer to the parameters

a - -l3461 and b a 17.833 found earlier by Freeman

and Halsey.“4 for the temperature interval 115 - 80 K.

LIST OF REFERENCES

6.
7.
8.

9.

10.

11.

12.

13.

14.

15.

16.

LIST OF REFERENCES

E.R. Dobbs and 6.0. Jones, Repts. Progr. Phys 20,
516 (1957).

G.L. Pollack, Rev. Mod. Phys. 36, 748 (1964).
G.K. Horton, Am. J. Phys. 36, 93 (1968).
M.L. Klein, J. Chem. Phys. 41, 749 (1964).

Melee Klein and JeAe RBiS‘land, Je Chemo Physe 41’
2773 (1964).

L. Jansen, Phil. Mag. 8, 1305 (1965).
L.S. Salter, Trans. Faraday Soc. 59, 657 (1963).

.D.L. Losee and R.0. Simmons, Phys. Rev. Letters 18,

451 (1967).

R. Becker Th e g3; warm; (Springer-Verlag,
Berlin, 195 p. 8.

L.l. Schiff a m ngchanicg (McGraweHill New
York, waffle—813". 2 . ’

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APP ENDI X

APPENDIX

TABLE A1

Measured pressure and temperature points for Ar.

Ar Ar
Pressure Temperature Pressure Temperature
(Torr) (K) (Torr) (K)
561.099 84.495 53.339 69.886
538.557 84.128 53.106 69.867
513.970 83.730 49.988 69.560
492.247 83.412 46.716 69.219
464.519 82.994 41,593 68.643
448.290 82.742 37.246 68.106
443.067 82.661 36.375 67.992
420.063 82.266 34.477 67.735
385.901 81.666 32.091 67.397
358.409 81.155 28.104 66.777
340.153 80.797 26.967 66.586
313.180 80.238 24.661 66.178
299.849 79.947 22.703 65.802
284.865 79.647 19.304 65.080
271.425 79.287 17.436 64.637
261.737 79.038 16.172 64.312
254.242 78.858 14.199 63.754
236.758 78.399 13.547 63.556
218.673 77.892 10.636 62.552
204.450 77.463 8.132 61.521
189.298 76.984 4.703 59.443
173.501 76.444 3.983 58.808
167.508 76.226 3.243 58.095
159.117 75.916 2.212 56.608
145.576 75.383 0.601 52.488
128.462 74.613 0.204 50.018
119.122 74.286 0.0675 47.845
114.188 74.000 0.0660 47.739
98.050 73.138 0.0262 45.929
90.947 72.719 0.0218 45.492
.79i972 72.013 0.0158 44.807
73.900 71.589 0.0108 44.015
68.564 71.191 7.5x10-g 43.160
62.115 70.673 6.0x18'3 42.769
57.647 70.286 3.8x10' 41.871

78

 

I iilll‘l' [[111 If" I
l 1‘ Il’ll‘ll ‘I J '11 |.1‘ 111

Pressure
(Torr)

Ar

79

(Table A1 continued)

Temperature

(K)

38.707
36.803
34.977
33.930
32.146
30.400

Pressure
(Torr)

5.0X10'2
3.5x10'

Ar

Temperature

(K)

29.099
27.601
27.525
26.065
25.506

80

TABLE A2

Measured pressure and temperature points for Kr.

Kr Kr

Pressure Temperature Pressure Temperature

(Torr) (K) (Torr) (K)
505.261 114.914 92.135 100.213
502.293 114.855 82.361 99.378
482.453 114.457 '73.122 98.506
460.901 114.010 62.813 97.418
441.980 113.608 56.989 96.734
421.227 113.141 51.082 95.974
417.699 113.072 43.539 94.889
400.137 112.649 39.466 94.238
393.148 112.494 21.663 90.450
380.794 112.179 20.965 90.250
370.278 111.939 20.202 90.029
359.921 111.650 19.003 89.660
351.269 111.433 18.953 89.644
341.909 111.169 17.324 89.114
327.079 110.772 14.773 88.189
318.285 110.512 13.995 87.877
310.807 110.305 12.781 87.366
300.348 109.982 12.709 87.328
287.134 109.600 11.733 86.882
283.954 109.476 10.923 86.468
263.805 108.828 10.044 86.027
261.753 108.746 8.820 85.316
249.681 108.341 7.754 84.635
241.268 108.029 7.065 84.138
238.618 107.926 6.733 83.879
209.386 107.799 5.893 83.215
222.243 107.317 5.387 82.761
221.013 107.279 4.549 81.896
209.386 106.799 3.604 80.746
193.818 106.138 3.181 80.156
181.424 105.588 2.525 79.054
175.569 105.304 2.110 78.199
168.667 104.978 1.962 77.853
160.055 104.504 1.554 76.773
156.598 104.370 1.133 75.278
147.423 103.886 0.841 74.027
128.521 102.776 0.523 72.279
123.549 102.463 0.389 71.057
119.813 102.226 0.295 70.342
113.961 101.834 0.186 68.663
105.260 101.224 0.140 68.070
98.954 100.749 0.131 67.589

Pressure
(Torr)

0.092

0.0879
0.0619
0.0608
0.0467
0.0337
0.0231
0.0172
0.0136
0.0119
8.0X10

Kr

81

(Table A2 continued)

Temperature

(K)

66.822
66.590
65.675
65.620
64.909
64.084
62.152
61.244
61.128
60.122

Kr

Pressure

(Torr)

7.5x10:§
4.6x10 3
3.5x10-3
2.1x10-3
1.7x10:3
1.0X10 4
8.61410-4
5.2x10'

3.0x10'4
2.1x10‘4
2.1x10‘4

Temperature

(K)

59.727
58.304
57.565
56.297
55.415
54.380
53.467
51.961
49.259
46.017
45.130

..|,1 [fifplc I ‘l ‘1' l ‘ (1 l i i III! ..‘.I ' {Ill ‘ l ‘1‘ l 1 I... ll 11

82

TABLE A3

Measured pressure and temperature points for Xe.

Xe Xe
Pressure Temperature Pressure Temperature
(Torr) (K) (Torr) (K)
505.969 158.710 81.521 137.310
496.632 158.458 74.512 136.409
486.381 158.170 74.491 136.400
465.899 157.588 67.318 135.403
444.649 156.972 65.980 135.198
424.359 156.354 64.131 134.931
403.954 155.706 58.299 134.006
383.428 155.028 57.887 133.928
373.930 154.709 52.315 132.972
363.577 154.342 50.895 132.699
355.436 154.058 47.631 132.088
342.665 153.582 44.742 131.498
335.974 153.341 42.894 131.117
322.624 152.820 39.719 130.405
316.179 152.579 38.325 130.088
302.091 151.999 35.224 129.326
295.350 151.727 34.664 129.176
284.928 151.282 31.178 128.240
281.254 151.114 30.294 127.978
273.829 150.798 28.910 127.577
270.013 150.607 28.844 127.543
268.649 150.561 26.533 126.822
252.950 149.831 26.367 126.764
250.344 149.692 25.702 126.538
249.147 149.645 24.894 126.279
241.695 149.296 23.177 125.686
235.097 148.949 22.829 125.544
231.717 148.781 20.236 124.530
230.188 148.684 20.168 124.520
213.153 147.800 18.011 123.585
205.518 147.350 17.567 123.358
196.376 146.840 15.988 122.613
187.374 146.271 15.575 122.375
181.283 145.921 14.222 121.671
170.785 145.211 13.940 121.485
166.702 144.965 12.475 120.640
154.186 144.088 12.227 120.448
153.757 144.058 10.963 119.599
139.989 142.018 9.661 118.619
138.631 142.831 3.594 111.004
128.865 142.115 2.859 109.519
118.179 141.183 2.323 108.183
100.106 139.419 1.809 106.962
96.094 138.992 1.383 105.018
90.905 138.418 1.034 103.316
84.552 137.680 0.767 101.672

Pressure
(Torr)

0.367
0.246
0.162
0.132
0.106
0.0843
0.0522
30.0321
0.0205
0.0179
0.0136
0.0108

Xe

9.0x10"3

83

(Table A3 continued)

Temperature

(K)

97.826
95.935
94.136
93.106
92.117
91.759
89.851
88.013
86.253
85.963
84.830
83.646
83.02

Pressure
(Torr)

7.2x1033
5.6x10_3
4.8X10_3
4.3x10 3
3.4x10-3
2.9x10'

2.3x10'3
1.6x10"3
1.4x10'Z
8.1x10:4
6.0x10

4.7X10-4
328210"

Xe

Temperature

(K)

81.923
81.528
80.982
80.806
79.587
78.865
78.082
76.474
76.034
74.054
72.522
71.372
70.075

 

.9. [IE