Title: Thermal expansion and isothermal compressibility of solid nitrogen and methane
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Title: Thermal expansion and isothermal compressibility of solid nitrogen and methane
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Creator: Heberlein, David Craig, 1942-
Copyright Date: 1969
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THERMAL EXPANSION AND ISOTHERMAL

COMPRESSIBILITY OF SOLID NITROGEN AND METHANE













By
DAVID CRAIG HEBERLEIN


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY














UNIVERSITY OF FLORIDA
1969



























To Martha














ACKNOWLEDGEMENTS


I would like to express my appreciation to the follow-

ing persons for their contributions towards the completion

S of this work:

Dr. E. D. Adams, the Chairman of my Supervisory

Committee, who assisted in the design and construction of

the apparatus and guided the course of this work;

Dr. T. A. Scott for his helpful suggestions in the

study of solid nitrogen;

Dr. J. S. Rosenshein for his helpful suggestions in

the study of solid methane;

Dr. J. W. Philp for his help in using the computer to

analyze the data taken in these experiments;

Doctors J. R. Gonano, P. N. Henriksen, M. F. Panczyck,

R. A. Scribner, and G. C. Straty, who all have rendered

valuable assistance to me; and,

Mr. B. McDowell, who provided valuable technical

assistance and produced the liquid helium necessary to

run the experiments.

I would like to thank my parents, Mr. and Mrs. F. A.

Heberlein, for their continued help and encouragement

during my undergraduate and graduate study. I also would

like to thank my aunt and uncle, Dr. and Mrs. R. L. Fairing,

for their help and encouragement during my graduate study.


iii






I am grateful to my wife, Martha, for her understanding

and encouragement during a long, and,at times, frustrating,

graduate study.
















TABLE OF CONTENTS


ACKNOWLEDGEMENTS.


LIST OF FIGURES .


Chapter


Page


. . . . . iii


. . . . . vi


I. INTRODUCTION . . . . .


Nitrogen . . . . . .


Methane . . . . . .


II. THEORY . . . . . . .


Nitrogen . . . . . .


Methane . . . . . .


III. APPARATUS AND PROCEDURE . .


Cryostat . . . . . .


Gas Handling and Pressure System


Sample Chamber and Pressure Bomb


Temperature Measurements, Calibration
and Regulation . . . . . .


Procedure and Sample Measurements. .


IV. EXPERIMENTAL RESULTS AND DISCUSSION.


Nitrogen . . . . . . .


Methane . . . . . . .


REFERENCES. . . . . . . . .


BIOGRAPHICAL SKETCH . . . . . .


S. .


. 36


S. 39


. 42


. 46


. 59


. 72


S. 74


* *


* .


* *


* .














LIST OF FIGURES


Figure Page

1. Phase diagram of solid nitrogen. . . . 2

2. Phase diagram of solid methane . . . 6

3. Schematic drawing of apparatus . . .. 20

4. Pressure system. .. . . . . ... 25

5. Sample chamber and pressure bomb . . .. .31

6. Schematic drawing of three-terminal
resistance bridge. . . . . . .. 37

7. Specific heat versus temperature
for solid nitrogen . . . . . . 47

8. Relative length changes versus
temperature for solid nitrogen . . .. .48

9. Semilogarithmic plot of relative length
changes versus inverse temperature for
alpha nitrogen . . . . . . . 50

10. Linear expansion coefficient versus
temperature for solid nitrogen . . .. 52

11. Measured and calculated compressibilities
for solid nitrogen . . . . . ... 56

12. Gruneisen parameter versus temperature
for alpha nitrogen . . . . . .. .58

13. Specific heat versus temperature
for solid methane. . . . . ... 60

14. Relative length changes versus
temperature for solid methane. . . .. 61

15. Linear expansion coefficient versus
temperature for solid methane. . . . 63

16. Time versus temperature for a typical
warming curve of solid methane . . .. .65







Figure


17. Specific heat (C') versus temperature
p
for solid methane . . . . . . . 66

18. Specific heat versus temperature for
solid methane at low temperatures . . .. .68

19. Relative length changes versus
temperature for solid methane at
low temperatures. . . . . . . . 70


vii


Page












I. INTRODUCTION


Nitrogen


Solid nitrogen exists in three distinct phases. The

phase diagram as determined by Swenson1 is shown in Fig. 1.

The gamma phase found by Swenson in 1959 exists at low

temperatures and high pressures. Very little about the

gamma phase is known.

From its triple point to 35.6 K, nitrogen in its beta

phase consists of diatomic molecules arranged in a hexagonal-

close-packed lattice. The space group P63/mmc was determined

from x-ray studies by Boltz et al.2 in 1959. Molecular

rotation in the beta phase was confirmed from x-ray studies

by Jordan et al.3 and Streib et al. and from studies of the

infrared spectrum by Smith et al.5 The quenching of the

quadrupole coupling in beta nitrogen can only be explained

by either spherical rotation as found in liquid nitrogen or

by hindered rotation with the molecular axis inclined at an

angle of 54.7. Because the specific heat is found to have

a value 50% greater than that needed for free rotation and

the molecular volume is only 75% of that expected for freely

rotating molecules, the rotations are not free. Thus, there

is a hindered rotation in which the molecular axis makes an

angle of 54.70 with the crystal axis.












9




'8





7




6




5



0
o
4 4
p4



3






2





1




0
0


Fig. 1.


Phase diagram of solid nitrogen.
Pressure versus temperature.


10 20. 30 40 50 60
TEMPERATURE (K)


S70







At 35.6 K, solid nitrogen at its vapor pressure under-

goes a first-order phase transition with a latent heat of

54.7 cal/gm-mole.67 The lower temperature, alpha phase

has a face-centered-cubic lattice in which the centers of

each molecule are displaced slightly from the lattice

points. X-ray measurements by Jordan et al.3 showed the

space group to be P213. Because alpha nitrogen has a rather

open structure, it is reasonable to assume that the nuclear

quadrupole coupling constant, e2qQ/h, differs very little

from its value in the free molecule. Using pure quadrupole.

resonance continuous wave techniques, Scott8 determined the

temperature dependence of e2qQ/h. The result of this study

and subsequent studies by DeReggi found the resonance

frequency to be extremely temperature dependent near the

alpha-beta phase transition.

Although most theories of solids are given for constant

volume processes, it is easier experimentally in most cases

to make measurements at constant pressure. In order to

distinguish the temperature dependence of a quantity such

as the quadrupole resonance frequency, which is measured at

constant pressure, from that caused by the expansion of the

lattice, the equation of state of the solid must be known.

To determine the equation of state, the temperature dependence

of the expansion coefficient and the isothermal compressi-

bility must be known.

Until recently, little experimental information other

than the specific heat was available on the thermal properties







of solid nitrogen. In 1966, Manzhelii, Tolkachev, and

Voitovichl0 (MTV) measured the expansion coefficient of

solid nitrogen from 22 to 44 K. In an attempt to extend

these measurements to 4.2K, the expansion coefficient was

measured from 4.2 to 38 K in this work. Recently, Bezuglyi,

Tarasenko, and Ivanov (BTI) determined the adiabatic

compressibility from 16 to 44 K from their measurements of

the velocity of sound in solid nitrogen. The isothermal

compressibility, KT, is simply related to the adiabatic

compressibility, KS, by

K = K + (2TvC1 (1)
T S P

where c is the expansion coefficient, T the temperature,

v the molar volume, and Cp the specific heat at constant

pressure. Thus, the isothermal compressibility of solid

nitrogen was measured in this work from 8 to 40 K as a

consistency check not only on the measurements of BTI,

but also as a check on the temperature dependence of the

expansion coefficient.


Methane


The existence of a anomaly in the specific heat of

solid methane at 20.4 K was first observed in 1929 by

Clusius.6 Because the maximum in the specific heat appeared

to be finite, the transition was assumed to be of second-

order. X-ray diffraction studies by Schallamach12 showed

no change in the face-centered-cubic structure of solid

methane at 20.4 K, with the lattice constants differing

by approximately 1% on opposite sides of the transition.




5


This small change in the molecular lattice indicates that

the thermal anomaly involves a change in the orientational

ordering of the molecules of the lattice.
13
Pauling originally described the transition as

involving a sudden change from molecular oscillations in the

low temperature phase to free molecular rotation in the

higher temperature phase. This was consistent with the

approximate value of 3 cal/mole for the specific heat above

the transition. Proton spin resonance experiments by
14
Thomas et al. determined that this picture of the transi-

tion was not correct. The resonance line width observed by

Thomas et al. indicated no perceptible change at 20.4 K, and

only above 65 K did the line width and characteristic time

for spin-lattice relaxation both drop toward the values

observed in liquid methane. Studies of the Raman bands at

79 K by Crawfordl5 showed rotational wings of the same size

as those found in liquid methane. Thus, while the hindrance

to molecular rotation is small above 65 K, the phase change in

solid methane at 20.4 K takes place without the establishment

of free molecular rotation. Although the passage to free

rotation does occur, it occurs gradually at higher tempera-

tures, and without the appearance of a thermal anomaly.

The phase diagram of solid methane is shown in Fig. 2.

There are four solid phases in methane. The alpha phase has

been extensively studied and the alpha-beta and alpha-gamma

phase boundaries have been determined from the specific heat

measurements of methane under hydrostatic pressure by













8



7



6 S

SII

5








m /
3













0. 10 20. 30 40 50 60
TEMPERATURE (K)

Fig. 2. Phase diagram of solid methane.
Pressure versus temperature.
r//
w
s /
v! /

vl .




" /
0 -- I -- u -- L- ---------
0 02.304 06


70.







Rosenshein6 and from the p-v isotherms determined by

Stevensonl7 and Stewart.18 The delta phase was first

reported by Stevenson in 1957 and to date nothing more is

known about this phase. The beta-gamma phase boundary has

been observed to pressures as low as 200 atm. by Rosenshein,

or about 800 atm. lower than that observed by either

Stevenson or Stewart. Recent specific heat measurements

by Colwell, Gill, and Morrison19 (CGM) show a broad excess

heat capacity in solid methane at zero pressure centered

about 8 K. An extrapolation of the beta-gamma phase

boundary determined by Rosenshein to zero pressure gives

a transition temperature of 9.5 K. While the agreement

seems reasonable in view of the broadness of the lower

temperature transition, there exists no other experimental

evidence to extend the beta-gamma phase boundary to 8 K

at zero pressure.

With the exception of the early molar volume studies

of solid methane between 20 and 21 K reported by Heuse20

in 1936, virtually no information exists on the thermal

expansion of methane at or below 21 K. In this work, the

thermal expansion of methane was measured from 4.2 to

26 K not only to determine the lattice contribution to the

thermal expansion, but also to study the nature of the

thermal anomaly at 20.4 K and to determine if possible the

existence of a lower temperature phase at zero pressure.















II. THEORY


In 1936 Pauling13 suggested that the transitions in

solid CH4, N2, 02, and CO2 were caused by a change from

oscillation to free rotation of the molecules. He approx-

imated the effect of the crystal field on the molecule

with the potential

V(9) = V (l cos29) (2)

in which 9 is the angle between a molecular axis and its

equilibrium position. This form is, however, a poor repre-

sentation of the dipole interaction which depends not only

on the relative orientation of the two dipoles, but also on

the direction of the vector separating their centers.

Furthermore, it has been shown by Kirkwood21 that a potential

of this form always leads to a prediction of a single

second-order transition.

Kreiger and James22 showed that the interaction potential

V = Acos ij + Bcos29ij (3)

could lead to a single second-order transition, a single

first-order transition, two first-order transitions or a

second-order transition followed by a first-order transition

for increasing temperature. Both of these models are much

oversimplified in that they assume axial symmetry of the

molecules and they treat the interaction of the molecules

as dependent only on the orientations of the molecules in







space irrespective of the position of each molecule in

the lattice.


Nitrogen


Kohin23 found the directional interaction potential for

solid nitrogen to be

V(9) = CIP2(cos9) (4)

where I is the average of P2(cosQi) for the neighbors of a

central molecule, 9 the angle that each molecule makes

between the molecular axis and the symmetry axis at its

lattice site, P2(cosO) = -(1/2) + (3/2)cos28, and C a func-

tion of the lattice constants and molecular parameters as

determined from nearest and next-nearest neighbor inter-

actions. Eq. (4) is a combination of the following terms:

a term due to the interaction of the quadrupole moments of

the molecules; a directional correction to the attractive

dispersion forces due to the anisotropic polarizabilities of

the molecules; and, a directional correction to the repulsive

forces.

If an assembly of classical rigid rotors with fixed

centers is taken as a model for a crystal of diatomic mole-

cules, the normalized rotational distribution function is

f(G,0) = exp(-V(9,0)/kT (5)
/exp(-V(9,0)/kTdw

where f(89,)dw is the probability that the axis of the mole-

cule lies in the solid angle dw about (9,0). V(9,0) is the

orientational potential due to the crystal field. If one now

assumes that each molecule orients itself independently in








the average field of its neighbors and that the probability

distribution for molecular orientation is the same for each

lattice site, we can then require the rotational distribution

to be consistent with the crystal field to obtain a consistency

equation relating the temperature and the order parameter.

Since I has been defined as the assembly average of P2(cosS),

the classical consistency relation for I(T) becomes

I(T) = P2(9)exp(-CIP2(9)/kT)dw (6)
Sexp(-CIP2(9)/kT)dw

There are two solutions to Eq. (6). One solution corresponds

to rotational disorder, i.e. V(9,0) = 0 for all temperatures.

The other solution, corresponding to an ordered array, has an

extremum corresponding to a maximum in the temperature. The

calculated Tmax then corresponds to a theoretical upper bound

on the alpha-beta transition. Using a classical model similar

to this, Jansen and de Wette24 obtained a value for Tmax very

close to 35.6 K. Kohin pointed out, however, that when the

directional anisotropy of the intermolecular repulsive forces

are included as in Eq. (6), the calculated transition tempera-

ture becomes much higher.

By treating the diatomic nitrogen molecule as a rigid

rotor, Kohin solved the Schrtdinger equation to find the

eigenvalues for the directional potential given in Eq. (4).

By minimizing the directional energy of the crystal field,

the acceptable orientations of the molecules agree with

the observed structure for alpha nitrogen. Since the method

used by Kohin assumes that the molecules are stationary

and consequently near absolute zero, no attempt was made to








determine theoretically the structure for beta nitrogen.

Furthermore, it would be erroneous to attempt to extend this

theory to temperatures higher than 20 K without including

such effects as the lattice vibrations and direct correla-

tions between neighbors. Since there exists no microscopic

theory to explain the behavior of alpha nitrogen above 20 K,

we must turn to some phenomenological approaches based on

experimental observations.
25
DeReggi, Canepa, and Scott found that the pure

quadrupole resonance frequency, Q', obeys an order parameter

relation of the form

= A[(Tc T)I1d, (7)


such that
-I
r d) = -l(T T) (8)
-- C
dT is

By plotting -Q (- (Q/T)-1 versus temperature, these authors

found that their experimental points gave a straight line

from approximately 20 K to the transition temperature.

Extrapolating this line to its intersection with the tempera-

ture axis gave a critical temperature Tc = 37.70 0.07 K.

From this behavior it appears that there exists a higher

order process in alpha nitrogen.- The process is interrupted

before completion by the first-order transition. A similar
26
behavior is found in solid hydrogen26 in which a change in

crystal structure interrupts the progress of a higher order

process.







A phenomenological approach to explain the large rise

in heat capacity and the thermal expansion from 20 K to the

transition at 35.6 K has been proffered by MTV.10 A number

of solids behave in a similar manner as the melting point

is neared. The changes in the thermal properties other

than that which can be ascribed to the nominal expansion

of the lattice are explained by a rapid growth in the

number of vacancies. In analogy to the rapid growth in

the number of vacancies, MTV suggested that as the tempera-

ture rises there is a rapid growth in the number of

disoriented molecules in alpha nitrogen until the lattice

becomes unstable. At this temperature a phase transition

occurs, and a different crystal structure appears.

To explain the formation of these "orientational de-

fects", the free energy of the crystal would have to decrease.

In order for the free energy to decrease, the gain in

potential energy due to the new orientation about which the

molecule oscillates has to be exceeded by a gain in entropy

from the decrease in orientational order.

If it is assumed that the production of disoriented

molecules is accompanied by an increase in volume, then the

change in volume may be expressed as

Av = Kc (9)

where K is a constant of proportionality and c is the con-

centration of orientational defects. The concentration of

these orientational defects should obey an exponential law

of the form


c = Aexp(-U/RT) ,


(10)






where A is a degeneracy factor equal to the number of posi-

tions of disoriented molecules having the same energy, U

the activation energy to create the orientational defect, R

the natural gas constant, and T the temperature. Combining

Eq. (9) and Eq. (10), we find the relative change in length,

dl/1, to be given by

AL = Av = KAexp(-U/RT) (11)
1 3v 3v

or ln(Al/l) = ln(KA/3v) U/RT. (12)

By assuming that at some temperature, T1, the influence of

the orientational defects is negligibly small and that the

coefficient of thermal expansion connected with other

mechanisms remains well-behaved as the temperature is

raised above TI, then any major portion of the plot of

ln(l1/1) versus 1/T that gives a straight line can be said

to obey the predicted exponential behavior. From their

thermal expansion data, MTV found this behavior in the

interval 29.4-34.5 K using a T1 = 23 K. From the slope

of this line, MTV calculated an activation energy of

450 cal/mole. The difficulty with this approach is that

both the slope of ln(Al/l) versus 1/T and the interval

for which a straight line pertains depends on the choice of

T1. Ideally, Al would approach zero as T approaches 0 K,

but in reality the exponential behavior is only observed

for temperatures above 20 K. This approach is useful

because it is possible to determine in approximately which

temperature interval a mechanism with an exponential

temperature dependence begins to dominate the thermal

expansion.







A similar approach by Bagatskii, Kucheryavy, Manzhelii,

and Popov27 (BKMP) to explain the excess heat capacity above

20 K gave an activation energy of 460 cal/mole. In view of

the approximations made by BKMP for the lattice contribution

to the specific heat and the inherent error in the deter-

mination of the activation energy by MTV, the agreement

appears fortuitous. The phenomenological approach of MTV

is of interest, however, because there appears to be

nothing in the potential found by Kohin to explain the

strong temperature dependence of the expansion coefficient,

the isothermal compressibility, the excess heat capacity,

or the pure quadrupole resonance frequency at temperatures

above 20 K.


Methane


Fortunately, methane with its high molecular symmetry

and relatively large intermolecular spacing is a favorable

case for studying orientational coupling. In methane the

ratio of the intramolecular distance to the molecular

separation is so small that one can expect the octopole-

octopole interaction between next neighbors to be dominant.

Keenan and James28 have shown for methane that the electro-

static interaction for non-overlapping molecules falls off

as rapidly as R-7 and by neglecting all interactions except

those between nearest neighbors in a crystal, they were able

to formulate a theory for solid methane in terms of a single

adjustable parameter, the effective octopole moment of the

molecule.






In the model developed by Keenan and James, molecular

and lattice vibrations are neglected and the methane crystal

was treated as a face-centered-cubic array of spherical

rotors carrying a charge distribution with tetrahedral

symmetry. The statistical calculation is a classical ver-

sion of the self-consistent field approach in which the

conditions for self-consistency appear as a family of

integro-functional equations, one for each molecule in the

crystal. Neglect of quantum effects makes the results

applicable only to CD4. Three solutions of these equations

minimize the free energy in a particular temperature range.

At the lowest temperatures the stable phase has a tetragonal

symmetry with the molecules oscillating about equivalent

equilibrium orientations. As the temperature rises, the

crystal undergoes a first-order transition into a phase

with octahedral symmetry in which one molecule in four

rotates freely. As the temperature rises even higher the

crystal undergoes a second-order transformation to an

orientationally disordered phase. By assigning a value to

the molecular octopole moment to make the higher transition

temperature agree with the 27.4 K value observed in CD4, the

predicted lower transition temperature becomes 24.4 K as

opposed to the observed temperature of 22.2 K. The predictions

of this theory are in agreement with integrated heats of

transition, zero-point entropy and the optical properties

of all three phases.






An extension of this model to include solid CH4 has

recently been completed by Yamamoto and Kataoka.29 The

high and low temperature ordered phases are assumed to have

the same sub-lattice structure as that proposed by James

and Keenan, but differ in that all calculations were made

on the basis of quantum statistical mechanics in the sub-

space J = 4, where J is the rotational quantum number.

Since the various spin combinations of the hydrogens pro-

duce three separate spin species, A (meta), E (para), and

T (ortho), the nuclear spin species are treated separately.

Inasmuch as the high temperature equilibrium proportions

of A:E:T become frozen in at low temperatures, the results

for individual spin species can not be compared in any

other than a qualitative way with experimental measurements

on samples with "mixed" spin species. The most interesting

feature of this quantum mechanical treatment of solid

methane is the prediction that for a sample of spin species

A, two phase transitions, one at 20.7 K and the other at

16 K, are to be expected. This is to be compared with the

specific heat measurements on "mixed" crystals by CGM,

whose results showed transitions at 20.4 K and 8 K.

Thermodynamically the large specific heat anomaly in

methane has been described as a cooperative transition. To

describe what is meant by a cooperative transition, let us

assume for the moment that methane obeys a van der Waals

equation of state


(P + av2) (v b) = RT ,


(13)








in which the term av2 is a correction to the pressure to

allow for the attractions of the molecules, and b is a term

to allow for the fact that the molecules are of a finite

size. The term av-2 increases in importance as the volume

decreases, such that if the temperature is steadily

reduced it becomes thermodynamically advantageous for the

entire lattice to change abruptly in volume. The loss in

entropy from this change would be offset by the gain in

energy from the work done by the attractive forces. Thus

the presence of a term such as av-2 becomes increasingly

important with the progress of the change which it is

causing, and it is in this sense that the transition was

called cooperative by Fowler.30'31 Furthermore, the

presence of av-2 also points out that a transition of

this type can only be understood if some account of the

molecular interactions is included in the theory of

solids displaying this type of behavior.

The dominant interaction in solid methane has been shown

to be the octopole-octopole interaction which lead to a type

of orientational ordering of the methane molecules below

20.4 K. Before ordering can occur, the energy in excess of

that allowed for the new molecular orientation must first

be dissipated in the lattice. Consequently, there appears

a large change in the specific heat. The term cooperative

is applied to the specific heat anomaly in methane because

the transition is very broad.














III. APPARATUS AND PROCEDURE


In this section the design and construction of the

apparatus necessary to measure thermal expansion and

isothermal compressibility of heavy solidified gases will

be presented. A capacitance technique will be described

for measuring changes in the length of the sample corres-

ponding to changes in temperature or pressure.

The cryogenic considerations including appropriate

plumbing and electrical connections will be considered in

a section devoted to the cryostat. The attainment of

temperatures ranging from 4.2 to 90 K is discussed in

this section with particular emphasis on the control and

measurement of temperatures in the interval 52-77 K.

The sample gas handling and helium pressure systems

are discussed in the second section of this chapter.

Particular emphasis is placed on the purification of the

samples used and on the means of controlling the pressure

transmitted to the samples.

The third section of this chapter describes the sample

chamber and the pressure bomb. The design of the sample

chamber is dictated by the thermal properties of the samples

which were studied. The concomitant problems of the design

necessary for filling the sample chamber at temperatures

from 63 to 90 K and of making sample measurements from







4.2 to 40 K are discussed. The construction of a pressure

bomb which contained the fluid helium used to transmit

hydrostatic pressure to the sample is also presented in

this section.

A germanium resistance-thermometer was used to monitor

the temperature of the sample. The associated electronics

and plumbing used in measuring, calibrating, and regulating

temperatures in the interval 4.2-40 K are discussed in the

fourth section of this chapter.

The final section in this chapter describes the

experimental procedure used to form the sample and to

make measurements of the thermal properties of each sample.


Cryostat


The cryostat used to perform these experiments is

similar to that described by Walsh32 and is shown schemat-

ically in Fig. 3. The sample chamber and pressure bomb

were surrounded by two exchange chambers which in turn were

enclosed by two large stainless steel dewars. The outer

dewar was usually filled with liquid nitrogen. The required

sample temperatures determined the refrigerant to be used

in the inner dewar. Thus, liquid nitrogen was used as a

refrigerant to obtain temperatures from 52 to 90 K and

liquid helium was used as a refrigerant to attain temperatures

from 4.2 to 52 K. The refrigerant contained in the inner

dewar will be referred to hereafter as the main bath.


















H 3


M C 0


>1

H4 .C
Sr-4 -4



MUe 04



c H
* a) 04 >
H 1.4 14

*d Mu > >
4-4 U) M 0
0),Q 0
0) 14 0)

04 0

ftaQ)
40j a)ea
vi H a 1a


0

.Q

54





) 0) 04
k C C






u) 4 C

e e r


C 3 z


H w 0) 0





E )d 0)

(d 3
m O H


0)



44



--I
e u


Cn
c e








The outer exchange chamber consisted of a vacuum jacket

connected to the stem of the cryostat with a flange-type

seal using an indium gasket. The outer exchange chamber

was used to isolate the sample chamber from the main bath.

A cylindrical container was enclosed by the outer

vacuum jacket. This container had a volume of approximately

250 cm3 and could be filled with liquid from the main bath

by means of a modified "Hoke" valve. The liquid in this

container will be referred to hereafter as the inner bath.

A smaller vacuum jacket connected to the base of the

inner bath container with another flange-type seal formed

the inner exchange chamber. The inner chamber was used to

either isolate the sample from the inner bath, or, when

filled with exchange gas, to provide thermal contact

between the sample and the inner bath.

To form the solid samples,the entire sample system

was first cooled to 77 K by filling the main bath with

liquid nitrogen. The inner bath container was then filled

with liquid nitrogen and the temperature of the sample was

controlled by reducing the vapor pressure of the nitrogen.

Temperatures from 52 to 77 K could be maintained by pumping

the nitrogen with a model KC-46 Kinney pump. The sample

chamber and the inner bath had to be maintained at a

temperature below the triple point of the sample in order

to form a crystal. Therefore, to fill the sample chamber,

the sample within the filling capillary had to be heated

above its melting point. A 0.0225" O.D. stainless steel









capillary was run through a 0.25" O.D. stainless steel tube

which extended to the top of the cryostat. The filling

capillary was electrically grounded only at the top of the

sample chamber and was thermally isolated from the main

and inner baths, permitting the temperature of the entire

length of capillary to be raised by connecting a 6-volt

storage battery between the capillary and the cryostat

ground. The capillary to the pressure bomb was connected

in a similar manner such that, if it became blocked by

frozen impurities, moderate heating would establish

pressure transmission over the entire length of the

capillary.

After forming the sample, the liquid nitrogen was

removed from the inner dewar. The sample was then cooled

to 4.2 K by transferring liquid helium into the inner

dewar. Two separate heating coils were wound on the

outside of the inner vacuum jacket. Using these heaters,

the entire inner chamber could be maintained at temperatures

much above that of the main bath. The larger heating

coil had a total resistance of 5000-ohms, and, when used

with a Heathkit Model PS-4 Regulated Power Supply, pro-

vided a course temperature control from 4.2 to 52 K. The

smaller coil had a total resistance of 1000-ohms and was

used as a fine control on the temperature.

Individual 0.059" O.D. stainless steel capillaries

run through separate 0.25 O.D. stainless steel tubes







which extended from the outer vacuum jacket flange to the

top of the cryostat formed coaxial lines which were used

for capacitance leads. The leads to the heaters and

auxiliary thermometers were introduced through the pumping

tube into the outer exchange chamber. These leads were

made from #36 Advance wire and were thermally anchored to

the inner bath container. Vacuum-tight glass-to-metal

seals were used to bring electrical leads through the inner

bath container into the inner vacuum chamber. The leads

into the inner exchange chamber for the germanium

resistance-thermometer consisted of two separate 0.059" O.D.

stainless steel capillaries introduced through the pumping

tube. A 4" section of #36 Advance wire was attached to both

of these stainless steel leads before thermally anchoring

them to the sample chamber.


Gas Handling and Pressure System


Purification of gas samples was necessary before

introduction into the sample and pressure systems. Because

impurities such as oxygen and nitrogen in methane could

pass through a cold trap held at a temperature near the

triple point of methane, a special process was used to

purify the methane samples.

C.P. Grade methane, 99.0% pure, was transferred into

a one liter cylinder and then immersed in liquid nitrogen.

This quick freezing of the methane produced a shattered

solid with a large surface area. Since the major








contaminants had a vapor pressure much higher than that of

methane at 77 K, the residual vapors could be pumped away

until the pressure over the solid was that appropriate to

the pure substance. This process was repeated until the

newly formed solid had the correct vapor pressure.

C.P. Grade nitrogen, 99.7% pure, was purified by

passing the gas sample through a liquid nitrogen cold

trap before it entered the U-tube. Similarly, the helium

gas used to pressurize the bomb was passed through a

helium cold trap before reaching the U-tube.

The gas handling system, shown schematically in

Fig. 4, consists of a sample line and a pressure line,

both interconnected to a mercury U-tube pressure system.

The pressure system was of standard U-tube design and

has been adequately described by Straty.33

In operation the sample was admitted into the U-tube

and subsequently into the filling capillary. The pressure

of the gas in the U-tube was then raised, creating a

pressure difference between the sample cell and the U-tube.

This pressure head produced rapid condensation of the gas

into the sample chamber. After forming the sample, the

valve between the filling capillary and the U-tube was

closed and the gas remaining in the U-tube was pumped away.

In order to measure the isothermal compressibilities

of samples, helium gas from a helium cylinder with a high

pressure regulator was then introduced into the U-tube and

the pressure capillary. Since the volume of the U-tube was




























































0)
1O
0) 0 -1




0 0
S4 l 0U

cn 4a



EO En
M C.) a)
H 01V


C 4 U)

SCfl -H (
1 01% B3


a) 0)
0 0
(fl Di

Cn
4>

0 n
01 O-
0) 0



U) 0
asa
r (0
Sc/I
k4d








approximately twenty times the dead volume of the pressure

bomb, the height of the mercury on the gas side of the

U-tube was used to vary the pressure of the gas trapped

in the bomb. To prevent mercury from entering the

hydraulic lines, a high-level alarm was installed on the

oil side of the U-tube. Because the entire U-tube assembly

was electrically insulated from its supporting structure,

the mercury itself was used as a switching device to

actuate the alarm.

A Texas Instruments pressure gauge was connected

through appropriate valving into the gas side of the U-tube

and was calibrated against a dead weight gauge connected

into the oil side of the U-tube. Special care was taken

to insure that the mercury levels in the U-tube were

equal when calibrating the Texas Instruments pressure

gauge reading against the dead weight gauge pressure.


Sample Chamber and Pressure Bomb


Two different approaches were available for measuring

thermal expansion and compressibility in solid nitrogen and

methane. One approach, which has been used successfully in

studies of solid helium,34 was to measure P = P(T) for

fixed molar volumes. The second approach was to measure

V = V(T) for fixed pressures.

When solid is formed in a sample chamber, the solid

also plugs the filling tube, isolating the chamber from

external pressure sensing devices. As the solid is cooled







below its melting point, the internal pressure also de-

creases. Assuming a Debye solid, this change in pressure

can be expressed as

ap = SCv-ldT =9DV-lfCvd(T/GD) (14)

where d is the Gruneisen parameter, v the molar volume,

and 8D the Debye temperature. Between 90 K, the melting

point of methane, and 4.2 K, the change in the internal

pressure of methane would be approximately 103 atm. It

therefore seemed unreasonable to attempt to design a

chamber capable of withstanding 103 atm. at 90 K and

yet retaining sufficient sensitivity to measure changes

of less than 1 atm. at 4.2 K.

Fortunately, the heavier solidified gases have a

coefficient of thermal expansion on the order of 10-4K-1

over most of the temperature interval below their melting

points. It was more reasonable to design a chamber whose

volume would be determined by the solid sample which it

surrounded.

Since the crystalline fields in both solid methane

and nitrogen are isotropic,29 it was possible to consider

geometries for the sample chamber in which only length

changes would be observed. By choosing a chamber with

cylindrical geometry, one end of the chamber could be

fixed, and the active end would then be used as the moving

plate of a three-terminal capacitor. Relative changes in

length would be directly related to changes in capacitance

as monitored by a General Radio type 1620-A capacitance








measuring assembly. A brass bellows with a small Hooke's

law constant was chosen as the sample chamber, both for

its radial strength and for its ability to distend elas-

tically over 20% of its equilibrium length.

In making measurements of changes in length, it was

necessary to have the bellows distended from its equilibrium

position by the solid which it surrounded. Since the solid

contracted uniformly as it was cooled, any solid caught in

the convolutions of the bellows would contract at the

same rate as the total length of the sample.

The change in volume upon melting in both methane and

nitrogen is approximately 10% of the total liquid volume.

To maximize the capacitive sensitivity, the distance be-

tween the moving and the fixed plate, referred to as the

gap length (1g), must be minimal. To minimize the gap

length in spite of the large volume change upon melting,

two further considerations were incorporated in the design

of the sample chamber.

The easiest means of filling the bellows with solid

was to sublimate the gas coming through the filling capil-

lary directly into the sample chamber. For sublimation to

occur, a thermal gradient along the length of the sample

chamber had to be maintained. Since the filling capillary

would necessarily have to be at or above the melting

temperature of the sample gas, the heated capillary was

introduced through the less massive, copper plug at the

active end of the bellows. The more massive, supporting







structure for the fixed end of the bellows was thermally

anchored to the inner bath. The sample formed in this

manner was not homogeneous.

Once the bellows was filled with solid, it was

desirable to form as homogeneous a sample as possible. To

form a good crystal meant that the material trapped in

the bellows would have to be melted and frozen again

very slowly. To allow for this large excursion in length,

the active end of the bellows was allowed to push a spring

loaded, "fixed" capacitor plate away from the stops in

its supporting structure. As the liquid began to solidify,

the active end of the bellows returned slowly to a position

such that the fixed plate was again pushed against the

stops in its supporting structure, and the two plates

were no longer shorted to one another.

To apply hydrostatic pressure directly to the sample,

the area both inside and outside the bellows had to be

filled with fluid helium. To surround the sample with

fluid helium, a pressure capillary was introduced through

the copper support assembly into the sample area. To

contain the pressurizing fluid outside the bellows, the

entire sample chamber was enclosed inside a pressure

bomb.

The sample chamber, surrounded by the pressure bomb,

is shown in Fig. 5. Copper was used extensively through-

out the sample chamber for two reasons. Copper has a high

thermal conductivity, insuring reasonable thermal























Sample chamber and pressure bomb
Pressure bomb
Spring
Fixed capacitor plate
Stops
Active capacitor plate
Filling capillary
Bellows sample chamber
Large hollow copper cylinder
Detachable copper piece
Copper support assembly
Copper gasket
Pressure plug
Stainless steel thermal standoff
Thermometer mounting assembly
Thermometers
Copper cage assembly
Small hollow copper cylinder
Pressure capillaries


Fig. 5.
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
L.
M.
N.
0.
P.
Q.
R.












A .. :* * * ... ,* ,* ** :
.. --'"-, t' ;."" """'" ::"'.':" a:"" '"""
s* q.* **.. *P. *: .*. .*s.. .* : *
*. -... ... ; -, ,j ,*,













.. ::






*.' .






_ '.
.'..."* .."
D .*













M' II W [' ; -
r



G 1 __S :I -
o









equilibrium times. Also the thermal expansion of copper

is not only small below 77 K, but is also well known.

The sample chamber consisted of a brass bellows,

approximately 0.50" in length, with an I.D. of 0.350"

and an O.D. of 0.592". The active end of the bellows was

soldered to a copper plug. A cylindrical capacitor plate

with a diameter of 0.500" was fastened with epoxy to the

opposite end of the copper plug. The epoxy electrically

insulated the capacitor plate from ground.

The fixed end of the bellows was soldered into a

detachable copper piece. At the opposite end of this

piece a hub was tapped to receive a plug extending from

the copper supporting structure. The plug, when covered

with 0.001" Teflon tape, seated on a 450 surface in the

bottom of the hub, thus served as the pressure seal to

the chamber. To insure that the solid was exposed only to

an active portion of the bellows, a hollow copper cylinder,

closed only at its top, was mounted on top of the threaded

copper plug. The diameter of the cylinder was 0.350" to

insure a mechanically tight, but not gas tight, seal to

the inside of the bellows. The length of the cylinder was

0.200", implying that the closed end of the cylinder pro-

truded 0.072" into the active portion of the bellows.

Inside the previously described hollow cylinder was

inserted a similar, but smaller, hollow cylinder with a

closed top. This cylinder covered the open end of the

pressure capillary which extended through the copper







support system into the sample chamber. A pressed indium

O-ring seal was made between the base of this hollow

cylinder and the top of the threaded plug. This seal was

vacuum tight and prevented the sample gas from entering

the pressure line when the chamber was being filled. Be-

cause this seal was easily broken when the pressure in the

pressure capillary exceeded two atmospheres, it was possible

to pressurize the inside of the bellows after it had been

filled with the solid sample.

The copper supporting structure not only held the

detachable copper piece, but also served as a support for

a cylindrical copper "cage". The base of the supporting

structure was threaded to fit similarly tapped threads in

the base of the cage. The top of the cage contained the

spring loaded, fixed plate assembly. To give this plate

a fixed position, stops were machined a distance 0.317"

from the top of the cage. The top of the cage was threaded

to receive a 0.875" O.D. copper plate. Between this copper

plate and the fixed plate was inserted a small spring. The

spring was pliant enough so that a small pressure inside

the bellows could lift the fixed plate above the stops in

the cage, but was strong enough to insure that the plate

returned to the stops when it was no longer in contact with

the bellows.

The fixed plate consisted of a guard ring surrounded

by a Mylar-insulated capacitor plate. The capacitor plate

was made from a 0.486" O.D. copper piece with a 0.067" taper








per foot. This piece was wound with two layers of 0.001"

Mylar and then pressed into a similarly tapered hole in a

copper plate. The outside plate formed a guard ring for

the capacitor plate.

The entire sample system was mounted on top of a

steel pressure plug. The copper supporting structure was

silver soldered to the steel plug at the point where the

supporting structure passed through the steel plug. The

outside of the pressure plug was threaded to fit the female

threads of the pressure bomb. A 0.012" copper gasket was

used to seal the pressure plug to a knife-edge surface

machined into the bomb.

Steel was chosen as the material for the pressure

bomb for two reasons. Firstly, the strength of steel per-

mitted a minimal wall thickness of the pressure bomb.

Secondly, the thermal expansion of steel is very nearly

that for a ductile material such as copper which was to

be used as the sealing gasket. If the gasket contracted

more rapidly than the steel, then the bearing pressure on

the sealing gasket would be appreciably reduced at low

temperatures.

To permit easy access to the sample chamber, all

electrical and capillary connections into the bomb were

made through the pressure plug. The capacitor leads were

attached to glass-to-metal seals which were soldered to

the inside surface of the pressure plug.








As mentioned previously, the entire length of capil-

lary to the top of the sample chamber had to be heated in

order to fill the sample chamber. A stainless steel ther-

mal standoff extended 2.5" from the outside surface of the

pressure plug. The filling capillary was soldered to a

feed-through glass-to-metal seal at the end of the thermal

standoff. The capillary which was used to pressurize the

pressure bomb was silver soldered into the pressure plug.

A large copper cylinder was soldered to that part of

the sample support system which extended outside the

pressure plug. Holes were drilled into this cylinder to

form wells into which the thermometers were inserted.

Apiezon type N grease was forced around the thermometers

and into the wells to insure good thermal contact between

the cylinder and the thermometers. All electrical leads

were thermally anchored to this cylinder by soldering them

to glass-to-metal seals set in the large copper cylinder.

Capacitor leads were run on opposite sides of the sample

support system, and,wherever unshielded, were separated as

far as possible to reduce distributed capacitance. By

disconnecting the leads at the capacitor plates, the dis-

tributed line capacitance was measured to be 0.004 pf. The

distributed line capacitance affected the total capacitance

by less than 0.4%. As mentioned previously, the capacitance

leads were made from stainless steel and had a low temperature

coefficient such that the change in the distributed line

capacitance with temperature was essentially zero.












Temperature Measurements, Calibration, and Regulation


A "three-terminal" resistance bridge was used to measure

the resistance of the germanium resistance-thermometer. A

schematic drawing of the three-terminal resistance bridge

is shown in Fig. 6. The term three-terminal means that the

ground is important as well as the current leads to the

unknown and standard resistor. By placing the current leads

to the resistors across the ratio transformer and grounding

the center tap of the ratio transformer, ground acts as a

"guard point". At balance the detector signal is zero and

no current is drawn. Therefore, any resistive or capacitive

leakage to ground in the leads has no effect on the balance

of the bridge. At balance, the unknown resistance of the

germanium resistance-thermometer plus the lead resistance

is given by

R = S(l x) (15)
x

where R is the unknown resistance, S the resistance of the

standard resistor, and x the reading of the ratio trans-

former. The use of stainless steel thermometer leads in

the cryostat minimized both the temperature coefficient of

the lead resistance and the total line resistance, while

thermally isolating the sample system from room temperature.

The bridge was composed of a Model 120 PAR lockin

amplifier with a fixed oscillator frequency of 400 Hz, a









E

E -

-~ *-


L
0)
E

'-
iC
0
L.
-


S'i
i----vw\_ *- 1








PAR CR4 differential preamplifier, a Gersch ST100-B iso-

lation transformer, a Cryocal S/N 87 germanium resistance-

thermometer, a wire-wound standard resistor (2000-ohms)

with a low temperature coefficient, and a General Radio

model 1493 ratio transformer.

The germanium resistance-thermometer was calibrated

against both the vapor pressure of hydrogen and against a

platinum resistance-thermometer. Between 12 and 20.4 K

temperatures were calibrated against the vapor pressure of

normal liquid hydrogen. All vapor pressure measurements

were taken within three to four hours after condensation

of the hydrogen gas into the vapor pressure bulb, and thus

represented values appropriate for normal liquid hydrogen.

The hydrogen was condensed into a copper bulb with a total

volume of 2.5 cm3. A strip of copper foil 0.010" thick,

0.50" wide and 10" long, was spiral wound to fit the inside

of the chamber and was silver soldered to the bottom surface

of the bulb. The large surface area provided good thermal

contact between the liquid and the chamber.

In the temperature interval 10-77 K, the germanium

resistance-thermometer was calibrated against a platinum

resistance-thermometer, No. 1634579. This platinum ther-

mometer had been calibrated in May, 1964 by the National

Bureau of Standards.

For the germanium resistance-thermometer, if the log R

is plotted against log T, the result is very nearly a

straight line. To obtain temperatures between 4.2 and 10 K







this straight line was used to find the temperature corres-

ponding to a measured value of the resistance. By using

the average value, -.0591, for the slope of this line in

the temperature interval 10-77 K, the temperature can be

expressed as
n
T = ZAiR-0.591(i 1) (16)
i=l

where R is the resistance, Ai the constants to be deter-

mined, and n the number of terms in the expansion. Since

the resistance has already been expressed in terms of the

bridge reading, the temperature is then given by

n -0.591(i 1)
T = 2A[ [1 x 2000 (17)
i=l x

A program developed by Philp35 permits the temperature to

be calculated by computer directly from the bridge reading.

This program gave a smooth temperature fit with a rms devia-

tion of 0.02 K when five terms were used in the expansion.

To regulate the temperature, the out-of-balance signal

from the lockin amplifier was used to heat the 1000-ohm coil

wound on the inner vacuum jacket against the heat leak to

the main bath. To maintain temperatures above 15 K, the

output of the lockin amplifier was supplemented by a con-

stant heating current supplied to the larger 5000-ohm coil

also wound on the outside of the inner vacuum jacket.


Procedure and Sample Measurements


The sample gas was condensed into the sample chamber

until the capacitance indicated that pressure was being








transmitted to the bellows. The liquid in the bellows

was then cooled into the solid. A thermal gradient was

established along the length of the chamber and more solid

was sublimated from the filling capillary into the sample

chamber. The heat needed to warm the capillary as well as

the warm fluid entering the bellows from the capillary

gradually raised the temperature of the entire sample

system. The heat supplied to the capillary would then

have to be reduced and the sample system allowed to cool

much below the melting temperature of the sample. This

process would be repeated until the capacitance and

temperature indicated that the bellows was distended from

its equilibrium position by the solid which it surrounded.

To insure that a good sample was formed, the sample

was warmed above its melting point and allowed to cool

slowly into the solid and then anneal for a period of from

eight to ten hours. With the sample thus formed, the

sample was cooled to 4.2 K over a two to three hour period.

To measure the expansion coefficient, heating and

cooling curves of the capacitance as a function of temperature

were taken. Near solid phase transitions, capacitance was

measured at three to four millidegree intervals. Away from

the solid phase transitions, capacitance was measured at

0.1 to 0.2 K intervals. Thermal equilibrium was indicated

when the capacitance no longer changed at a fixed temperature.

With the solid nitrogen samples, the temperature was main-

tained at 35.6 K for time periods ranging from one to three








hours to insure that the total length change associated with

the first-order alpha-beta transition was measured accurately.

To measure the isothermal compressibility, the pressure

bomb was pressurized with fluid helium. As mentioned pre-

viously, the fluid helium surrounded the sample as well as

the sample chamber. The compressibility was measured by

maintaining a constant temperature and noting changes in

capacitance corresponding to changes in pressure. The

pressure was varied by changing the height of the mercury

in the U-tube. The pressure was read to 0.01 psi on a

Texas Instruments pressure gauge.













IV. EXPERIMENTAL RESULTS AND DISCUSSION


The raw data obtained from these experiments consist

of a series of values of capacitance corresponding to

various temperatures at zero pressure, and a series of

values of capacitance corresponding to various pressures

at a fixed temperature. To relate the changes in capaci-

tance to changes in length, we must first look at the

expression for a three-terminal capacitor using a guard
36
ring

C 1f72 + ; Erw +l + Fw (18)
1g Ig + 0.22w [L 2rJJ

where C is the capacitance, E the permittivity, r the

radius of the plate with the guard ring, 1g the length

between the capacitor plates, and w the half width of the

distance between the inside radius of the guard ring and

the outside radius of the capacitor plate surrounded by

the guard ring. For w = 0.001", 1 = 0.020", and r = 0.486",

the second term in the above expression affected the total

capacitance by less than 1% and the ratio 1 dClby less
C [dT
than 0.4%.

To calculate the expansion coefficient, we find that

a change in the gap length, dlg, corresponding to a tempera-

ture change, dT, is given by

dl = ITr2 dC] + 2Irr [dr] (19)
dc LdJ
dT C2 dT C dTJ







Because the radial expansion of the copper capacitor plates

is small, rIdCj > 104 d, the second term on the right-hand
C dTJ r [daT

side of Eq. (19) shall be neglected. By noting that the

change in the gap length is equal and opposite to the change

in the sample length and that the coefficient of linear

expansion is given by 1 fdls, we find that
IdTJ

1 = 7-r2 dCl. (20)
isC2 dTI

A correction to this term to be considered is the change in

capacitance contributed by the expansion of the construction

material. Because the entire sample system with the ex-

ception of the bellows was made from copper, only the ex-

pansion of copper over a length equal to the plate spacing

plus the length of the sample had to be considered. A

calculation of this correction showed that it affected the

results by less than 0.5% and was well inside the scatter

of the data.

A third correction to be considered was a correction

for any tilt of one capacitor plate relative to the other

capacitor plate. A relation developed by Philp35 gives the

correction to the expansion coefficient as

C= C[ 1 r292l
sample measured [2--i J (21)


where r is the radius of the capacitor plate surrounded by

the guard ring, Q the angle of tilt, and 1 the gap length

at which the measurements.of capacitance were taken. A

measure of the effect of non-parallel plates was afforded





44


when the bellows, then filled with solid, became disengaged

from the spring loaded capacitor plate. As the sample

continued to cool, it was reasonable to assume that the

shape of the sample would remain intact. A typical value

of 48 pf was measured when the moving plate no longer

touched the ground of the guard ring. This value of

capacitance implied a plate spacing of less than 0.001"

and therefore a 9 = (0.001"/r). For a plate spacing of

0.020", this effect amounted to a correction of less than

0.07%.

To compute the expansion coefficient, the constant

1T-r211 = Co was calculated and inserted into a computer

program designed to determine the quantity C dC As

has been mentioned, the temperature had been fitted by

computer to a function of the bridge reading. Thus, with

the exception of computing Co, the computer calculated

the expansion coefficient as a function of temperature

from the raw data.

Because the fluid helium used to pressurize the

sample also filled the space between the capacitor plates,

the measured differences in capacitance caused by changes

in pressure had to be corrected for the accompanying

change in the dielectric constant of the fluid helium.

To calculate the isothermal compressibility, we find

that a change in the sample length, dls, corresponding to

a pressure change, dP, is given by

[dli = rgr2[dCl WTr2 d] (22)
dP T C2 LdPJ C TdpJ






Since (dC/dP)T is the quantity that is actually measured,

we must calculate (dc/dP)T from published data.

The dielectric constant, #(= E/eo, can be calculated

for a given pressure and temperature by using the Clausius-

Mosotti equation,

0= 3M ( 1- 1) (23)
4T ( K + 2)

in which C( is the molar polarizability, M the molecular

weight, and S the density. This equation has been veri-
37
field from above 300 K (gas) down to 1.62 K (liquid He II).

Because the dielectric constant is equal to unity to within

1% over the pressure and temperature studied in this work,

this equation may be rewritten as

= [4i-a + 1 = 0.39 f+ 1 (24)

with f expressed in gm/cm3. A change in permittivity

corresponding to a change in pressure is then simply

[d = o =d 0.39Eo d[ ] (25)
LdPj LdPJ [dPJ
T dP T dPT
Because the relative change in volume is three times the

relative change in length, the expression for the iso-

thermal compressibility becomes

S= s = 3KErr2 31dCs 0.39 1dfg (26)
s dPJ T is CIdP C TdPJT

Because the density changed most rapidly with pressure at

low temperatures, and the compressibility itself was a slowly

varying function of temperature, the error in determining

what fraction of the total capacitive change was due to a

change in length of the sample was largest for the lowest

temperatures.











Nitrogen


The Debye model for solids has been chosen for

convenience in comparing the results of the present deter-

mination of the expansion coefficient and the isothermal

compressibility with related quantities such as the specific

heat and the velocity of sound in solid nitrogen. For a

Debye solid, C the expansion coefficient, and KT, the

isothermal compressibility, are related to C the specific

heat at constant volume, by

Cv = 0v (27)
KTi

where T is the Gruneisen parameter and v is the molar

volume. The degree to which the ratio (v/K Cv varies from

a constant is a measure of the specific applicability of

this model to nitrogen and will be discussed later in terms

of the temperature dependence of the Gruneisen parameter.

The temperature dependence of the specific heat at

zero pressure for solid nitrogen is depicted in Fig. 7.

Comparison of the specific heat measurements given in

Fig. 7 to the temperature dependence of the relative changes

in length of nitrogen shown in Fig. 8 suggests that not

only is there an abrupt change in the lattice size corres-

ponding to a change in structure, but also that the lattice

must expand rapidly in the alpha phase near the transition




47

1.5







































0 10 20. 30 40 50
/

/


B/



/ /















nitrogen. Dashed curve represents behavior
w I
/
0




0.5









0 I



0 10 20. 30 40 50
TEMPERATURE (K)
Fig. 7. Specific heat versus temperature for solid
nitrogen. Dashed curve represents behavior
expected from a Debye solid with 8D = 80K.















0-




O -4 4

H R
4 J

,C *u-I






) 0
C,




4J, 4 0)

-4 > -.4





Eo 44
0





m .0






E4-
H OC




















a 0E
C 4
0 C









> E)
(a 0





0) 4C

rq







o










*r4
(B H *
a 10

















l
Qkd'


* ^


I/.TV OT
1/1 0







in order to accommodate the large amount of energy given to

the solid as it is warmed.

Recent thermal expansion measurements by MTV10 give

the relative change in volume at 35.6 K as 0.5%, which is

lower than the 0.8% value obtained earlier by Swenson.1

The magnitude of this change as determined by this work

gave a 1.4% change on warming and a 0.9% change on cooling.

Several efforts were made to reconcile the difference in

volume change when warming and cooling through the transition.

After cooling to initiate the transition, the transition

temperature was maintained for two to three hours. The

length of the sample, as detected by the capacitance bridge,

would change for the first hour, after which time no change

in length would occur as long as the transition temperature

was maintained. Cooling below the transition produced a

thermal contraction of the same magnitude as the thermal

expansion observed when warming through the same temperature

interval. Thus, there was no evidence of supercooling. An

analogous approach to determine if superheating had occurred

also produced a negative result. MTV also noticed the

presence of hysteresis in the vicinity of the phase transition.

The ratio l1/1 versus 1/T is shown in the semilogarithmic

plot given in Fig. 9. By choosing T1 as absolute zero, the

slope of ln(l/l) versus 1/T remains constant from 24 K to

the transition temperature. From the slope of this line

an activation energy of 300 cal/mole was calculated from





50





90

80 O- T1 = 0 K

70
S- T1 = 15 K
60
S- T = 23 K
50


40



30





S20
0








10
9
8

7

6

5


4



3
1 2 3 4 5

102/T (K-1)

Fig. 9. Semilogarithmic plot of relative
length changes versus inverse
temperature for alpha nitrogen.







Eq. (2) in Chapter II. By choosing T, = 15 K, the slope

remains constant from 20 K to the transition temperature

and gives an activation energy of 310 cal/mole. Finally,

by choosing the same T1 as MTV, T1 = 23 K, an activation

every of 370 cal/mole was found. This is to be compared

with the 450 cal/mole activation energy found by MTV and

the 460 cal/mole activation energy found by BKMP.

The temperature interval for which the slope remains

constant found in this work is much larger than that found

by MTV and compares favorably with the 24-35.6 K interval

in which the excess specific heat has been attributed to

orientational defects. This is also the same temperature

interval in which the pure quadrupole resonance frequency

has been found to obey the parameter relation given in

Eq. (8) in Chapter II. While the data do seem to obey

the postulated behavior for orientational defects, it is

apparent from the differences found in calculating activa-

tion energies corresponding to different choices of T1

that the experimental evidence of such behavior is far from

conclusive.

A point-by-point differentiation of length changes

corresponding to temperature changes gave the expansion

coefficient shown in Fig. 10. The scatter in the data

is a reflection of the limit of capacitive and temperature

sensitivity. As can be seen from the reproducibility of

the heating and cooling curves in both Fig. 8 and Fig. 10,

thermal equilibrium was established for all measurements.





























0D 30 6
-. 4 *p4 >



00

0 0U 0
0>O rJ C
mC O



CU 0 Ci
4 4OC



4*H C (
OUCC



X r
-4 oo -- 4 *




C U '
w 0 A







\ $0 U)41

0 CU)
-AC 40 0H








Coc








C'J CD 0 -T C 0









,-q -I
( N Oa l
(TSX- sCox I I o O H
^ 0 0 -)
;!r i- ^ Q<


~ I ac
y 9








The expansion coefficient is discontinuous at 35.6 K, as

is the specific heat, and the temperature dependence of

the expansion coefficient is similar to that of the

specific heat.

Also shown in Fig. 10 are the results of MTV. As

mentioned before, the measurements of MTV near the phase

transition exhibited hysteresis and no thermal expansion

data were reported from 34.5 to 36.5 K. Experimentally

the results quoted by MTV represent length changes

measured over 1 and 2 K intervals, except near the phase

transition, where measurements were taken over 0.25 K

intervals. Since no mention is made by MTV about their

limit of resolution in detecting changes in length, it is

fair to assume from the relatively small expansion

coefficient of alpha nitrogen that the large temperature

intervals were required to detect length changes.

Because length changes were measured over smaller

temperature intervals in this work, it is nct surprising to

find that this determination of the expansion coefficient

gave smaller values for I at lower temperatures and larger

values for o( at higher temperatures. What is surprising

is the degree to which the two results differ. The lowest

data point given by MTV at 21 K is 30% higher than that

measured in this work. The highest data point given by

MTV in the alpha phase of solid nitrogen at 34.5 K is 26%

lower than that found in this work. The differences in these

results become more acute when we consider the Gruneisen








parameter, but to compute the Gruneisen parameter we must

first determine the isothermal compressibility of solid

nitrogen.

The isothermal compressibility, as measured in this

work, and the adiabatic compressibility as measured by

BTIll are shown as a function of temperature in Fig. 11.

The isothermal compressibility, KT, is related to the

adiabatic compressibility, KS, as given by Eq. (1) in

Chapter I. To compare the measured values of the adia-

batic compressibility with measured values of the

isothermal compressibility, the following three curves

are also plotted in Fig. 11. The first curve is the

adiabatic compressibility and was calculated from the

isothermal compressibility and thermal expansion measured

in this work. Agreement with the measured values of

BTI at temperatures near the transition is good, but the

agreement becomes worse as the temperature decreases. The

second curve is the isothermal compressibility as calculated

from the adiabatic compressibility measured by BTI and the

expansion coefficient measured by MTV. Agreement with the

isothermal compressibility measured in this work is good

at low temperatures, but becomes poorer as the alpha-beta

transition is approached. The third curve is the isothermal

compressibility as calculated from the measurements of the

expansion coefficient given in this work and from the

measurements of the adiabatic compressibility given by

BTI. Agreement with the measured isothermal compressibility























Fig. 11. Measured and calculated compressibilities
for solid nitrogen.
O Isothermal compressibility measured in
this work.
0 Adiabatic compressibility measured by
BTI, Ref. 11.
O Isothermal compressibility as calculated
from the adiabatic compressibility
measured by BTI and the expansion
coefficient measured in this work.
Z Adiabatic compressibility as calculated
from the isothermal compressibility and
expansion coefficient as measured in this
work.
O Isothermal compressibility as calculated
from the adiabatic compressibility as
measured by BTI and the expansion coeffi-
cient as measured by MTV, Ref. 10.




















































































( -UMp OT) A1InIgISSaHdwoo
1- 9


'56


0






m





(N








to



-1








is excellent from 28 K to the transition temperature, but

at temperatures less than 28 K the agreement becomes

poorer. In any event, the worst disagreement between the

measured and calculated compressibilities is less than

2%.

Having determined the isothermal and adiabatic

compressibility of solid nitrogen in its alpha phase, the

specific heat at constant volume, C can be simply related

to the specific heat at constant pressure, C as

Cv =[KS Cp (28)
KT]

Having found Cv, the Gruneisen parameter can now be

calculated.

The temperature dependence of the Gruneisen parameter

was calculated from Eq. (27) using the isothermal compressi-

bility and the expansion coefficient measured in this work.

The temperature dependence of the Gruneisen parameter was

also calculated using the adiabatic compressibility measured

by BTI and the expansion coefficient measured by MTV.

These results are shown in Fig. 12. As can be seen in

Fig. 7, above 20 K the behavior of the specific heat for

alpha nitrogen is markedly different from that expected

for a Debye solid. It is not clear how the Gruneisen

parameter can remain constant over a temperature interval

where the specific heat differs appreciably from that of a

Debye solid. Yet, the Gruneisen parameter as calculated

from the measurements of BTI and MTV would seem to indicate

that alpha nitrogen is nearly a perfect Debye solid.


























2



















TOP
I




8 12. 16 20. 24. '28. 32. 36
TEMPERATURE (K)

Fig. 12. Gruneisen parameter versus temperature
for alpha nitrogen. 0's represent
values calculated from measurements of
MTV and BTI, Refs. 10 and 11.








The Gruneisen parameter as calculated solely from the

results found in this work is more easily understood.

Clearly, the Debye model predicts no first-order phase

change at any temperature below the Debye temperature, 9D.

The degree to which the Gruneisen parameter varies can be

considered a measure of how a particular solid deviates

from the behavior of an ideal Debye solid. As can be

seen from Fig. 12, the Gruneisen parameter has a strong

temperature dependence as the transition temperature is

approached.

From the nearly constant behavior of f below 18 K

found in this work, alpha nitrogen can be said to behave

like a Debye solid in the so-called "T3. region. There

is some question as to the absolute magnitude of I as

found in this work, because one would nominally expect

to have a value between 1 and 2. It is not clear, however,

what shape the expansion coefficient would have to assume

from 0 to 20 K to maintain a higher value for the Gruneisen

parameter.


Methane


The specific heat of solid methane as measured by

Clusius6 is shown in Fig. 13. The solid undergoes a change

of phase at 20.4 K as is evidenced by the large I anomaly

in the specific heat.

Relative length changes in methane measured in this

work are shown in Fig. 14. The reproducibility of the




60











15


















































16 18 1020 22 24

TEMPERATURE (K)
Fig. 13. Specific heat versus temperature for solid
methane. O's represent data points given
by. Clusius, Ref. 6.
0
'l-
0



























16 18 20 22 24.
TEMPERATURE (K)
Fig. 13. Specific heat versus temperature for solid
methane. 0's represent data points given
by Clusius, Ref. 6.
































































































%0 0
Cl Cl


TN

.C >9


CCOCV


4) cQ) >
1) CU -4 '4-









V0 >U CC












0 H
N U) > 0
,C -c

o C


















01 C 0



Q) 9) 4a
4 -4 )-4
CUa)


















(0 04

4-> 4J 4J *4



O M
10 U
















CC1H
Sr )l C)
r0 CU 4. C) ^










C C C




















CM 0'
H~c



N m
H r 1l i


T/TV OT








warming and cooling curves establishes that all points

were taken in thermal equilibrium. Fast heating rates

displaced the warming curves to higher temperatures. Also

shown in Fig. 14 are the warming and cooling curves found

by Heuse.20 It is important to note the similarities in

the results of Heuse and the results presented in this work.

The transition is spread out below 20.4 K with the

consequence that neither result gives a maximum slope at

20.4 K. Furthermore, while nl/1 remains continuous, dl/dT

is discontinuous at 20.4 K.

The expansion coefficient of methane as measured in

this work is shown in Fig. 15. The circles and triangles

represent two samples of different length and formed from

different gas. The maximum in the expansion coefficient

is found to be at 19.8 K. It is also apparent that at

20.4 K the solid is entirely in the higher temperature

phase. Also shown in Fig. 15 are the thermal expansion
10
measurements of MTV. Unfortunately, MTV reported no

measurements below 22 K.

If for some reason the temperature calibration was

grossly in error, a determination of the specific heat would

show a maximum at a temperature different from that given

by the calibration. A crude determination of the specific

heat was effected by making simultaneous measurements of the

temperature and length changes at equal time intervals when

heating at a slow constant rate. The temperature dependence

of the expansion coefficient remained the same, but the

maximum slope in the time versus temperature plot, shown in




























2


E-4
Z
W
U













O 4
0














0














14 16


Fig. 15.


18 20.
TEMPERATURE (K)


22..


24


Linear expansion coefficient versus temperature
for-solid methane. O's and A's. represent values
for two different samples. O's represent data
points given by MTV, Ref. 10.








Fig. 16, occurred at 20.4 K. The specific heat at constant

pressure is proportional to dt/dT, which we shall call C'.
p

A plot of C' versus temperature is shown in Fig. 17. Since
p
points were taken at five minute intervals, the detailed

structure of the specific heat near 20.4 K could not be

determined. It is now obvious that the temperature cali-

bration was not in error, and that the temperatures at which

the specific heat and the expansion coefficient are maxima

are indeed different.

To explain the difference in the temperature at which

the specific heat is a maximum and the temperature at which

the expansion coefficient is a maximum, let us consider the

general thermodynamic relation

C = Cv + c2TvK 1 (29)

Eq. (29) is a different way of expressing Eq. (1) given

in Chapter I. If it is assumed that Cv is a smoothly-

varying function of temperature and that for small values of

the expansion coefficient Cv is approximately equal to C ,

then any "bump" in Cp must be attributed to the term (2TvKT.

At 19.8 K, where the expansion coefficient is a maximum, the

magnitude of this term is 1.37 cal/mole-K. The specific

heat, as measured by Clusius6 and given in Fig. 13, exhibits

a small bump at 19.8 K with a height of approximately 1.6

cal/mole-K above that assumed to be contributed by Cv. These

two quantities are in reasonable agreement. Rosenshein16

reported double peaks in the specific heat in all the upper

transitions which he measured. The smaller peak usually











































































0 0 0 0
O (O O M
(u\O ) 3






















6.0-





5.4






84.8









3.6









3.0
c i
























2.4 PII1


Fig. 17. Specific heat (C') versus temperature for
solid methane.
solid methane.







occurred 0.25 K below the main peak in the specific heat.

While the specific heat measurements reported in this work

were much too crude to resolve and confirm this structure

in the specific heat at 19.8 K, it does appear that the

expansion coefficient measured in this work is not at

variance with existing measurements of the specific heat.

It is also helpful to look at the general properties

of the transition. The transition is of the coQperative

type, in which the modes of motion of each molecule are

directly affected by those of its neighbors. There is

no noticeable change in the crystal structure at 20.4 K,

but as the transition temperature is approached, each

molecule is reoriented in the lattice. In order for this

reorientation to occur, the lattice must expand. From the

results of this work, it appears that the greater part of

the lattice expansion occurs before 20.4 K and that most

of the energy assumed by the lattice goes into the new

rotational modes allowed for the new orientation of each

molecule.

Recent specific heat measurements by CGM,19 which are

shown in Fig. 18, exhibit a broad excess heat capacity

centered about 8 K. Because equilibriation times become

longer at lower temperatures, varying from several hours

at 5 K to approximately 20 minutes at 11 K, only a few

experimental points were reported by CGM. Specific

heat measurements by Rosenshein16 have shown the presence

of a second, lower temperature transition to pressures






68





















ci
0w


OU

OO
\ \ <"


0 04


0 mi
4 rO














01
$4
> oV





0

0 \




o -E

















(-om/ ) IVHH DI
0 0) )
(x o o IS OI3D


\ n+J 3
\ i 4-1cT^l
fc ttKSi-
^ ^ s
xo a) .
V u a -i
Y -^ e v






















()I-3-oni/-[Bo) ivatH 3I3I39aS < 0








as low as 200 atm. An extrapolation of the lower temperature

phase boundary determined by Rosenshein to zero pressure

gives a transition temperature of 9.5 K. Because the transi-

tion becomes broader at low temperatures, the agreement is

reasonable. It must be pointed out, however, that there

is no other experimental evidence to verify that the high

pressure phase observed by Rosenshein is the same as the

excess heat capacity found by CGM.19

Relative changes in length as a function of temperature

as determined in this work are shown in Fig. 19. It is

readily apparent that negative expansion occurs below 8.75 K.

Calculation of the expansion coefficient does not clarify

the issue. In fact, the expansion coefficient as calculated

from Fig. 19 has a negative minimum at 7.5 K. It is

difficult to compare this behavior to existing specific

heat measurements in any other than a qualitative manner.

Attempts to explain the long equilibriation times and

the peculiar thermal behavior below 12 K in terms of hydrogen

spin conversion are inconclusive. The sample apparently

does reach thermal equilibrium, as was evidenced in this

work. The length of the sample, as measured by the capaci-

tance, would change rapidly when warmed or cooled. The

length would continue to change slowly when the temperature

was held at the new value. Thermal equilibrium was estab-

lished in a time related to the temperature. As was

noticed by CGM, the lower the temperature, the longer the

equilibriation time. Attempts to ascribe a rate constant






















1-)



'0
*4



0
S a



I,






01 A
o 0



p,-I
















40
4
(d
-I



M 0n






f*U,

^ n


Ca
CC





c



i-I

-(U


0 .


I/TV ol
1/'Y0







to the equilibriation time similar to that found for ortho-

para conversion in liquid and solid hydrogen were unsuccess-

ful. If conversion were taking place, it would have been

an irreversible process, as in liquid and solid hydrogen

where the para hydrogen becomes frozen in the lattice. If

the negative expansion observed below 8.75 K were caused

by spin species conversion, then by warming the sample to

a higher temperature, the conversion would continue to

occur and the sample would continue to expand. The results

of Fig. 19 do not bear this out. It would appear that the

best approach to determine the nature of the low temperature

properties of solid methane would be to make more sensitive

specific heat or thermal expansion measurements at pressures

up to the lower limit of the phase boundary determined by

Rosenshein.














REFERENCES


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Acta Cryst. 12, 247 (1959).

3. T. H. Jordan, H. W. Smith, W. E. Streib, and W. N.
Lipscomb, J. Chem Phys. 41, 756 (1964).

4. W. E. Streib, T. H. Jordan, and W. N. Lipscomb, J. Chem.
Phys. 37, 2962 (1962).

5. A. L. Smith, W. E. Keller, and H. L. Johnston, Phys.
Rev. 79, 728 (1950),

6. K. Clusius, A. physik.Chem. B3, 41 (1929).

7. W. F. Giaque and J. O. Clayton, J. Am. Chem. Soc. 55,
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8. T. A. Scott, J. Chem. Phys. 36, 1459 (1963).

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10. V. G. Manzhelii, A. M. Tolkachev, and E. I. Voitovich,
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11. P. A. Bezuglyi, L. M. Tarasenko, and Yu. S. Ivanov,
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16. J. S. Rosenshein, Ph.D. dissertation, Massachusetts
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17. R. Stevenson, J. Chem. Phys. 27, 656 (1957).

18. J. W. Stewart, J. Phys. Chem. Solids 12, 122 (1959).








19. J. H. Colwell, E. K. Gill, and J. A. Morrison, J. Chem.
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20. W. Heuse, Z. physik. Chem. A147, 266 (1936).

21. J. G. Kirkwood, J. Chem. Phys. 8, 205 (1940).

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BIOGRAPHICAL SKETCH


David Craig Heberlein was born in San Antonio, Texas,

on January 8, 1942. He attended high school in Arlington,

Virginia, graduating from Washington-Lee High School in

1959. After four years at the University of Virginia,

he received a B. S. in Physics degree. Since entering

the University of Florida in 1963, he has worked as a

graduate teaching assistant and as a graduate research

assistant.

Mr. Heberlein is married to the former Martha Lois

Walkden of Lake Worth, Florida. He is a member of

Sigma Pi Sigma.












This dissertation was prepared under the direction of

the chairman of the candidate's supervisory committee and

has been approved by all members of that committee. It was

submitted to the Dean of the College of Arts and Sciences

and to the Graduate Council, and was approved as partial

fulfillment of the requirements for the degree of Doctor

of Philosophy.


August 1969



Dean, Col /e/ of/Arts and Sciences



Dean, Graduate School




Supervisory Committee:



Chairman






/(" l74'-'\ /f^




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