ABSTRACT

HEAT CAPACITY OF CuSiO3-HZO

FROM 20K TO 400K
by Warren Rice Eisenberg

The heat capacity of CuSiO3- H20 was measured in an attempt to
find evidence for a magnetic transition in the temperature range of 20K
to 400K. Such a transition was observed and was characterized by an
anomaly at 210K in the specific heat vs. temperature curve.

The anomalous behavior may be attributed to the fact that spins
of the magnetic ions go from random orientations in the paramagnetic
state to an antiparallel alignment in the antiferromagnetic state. The
magnetic contribution to the specific heat was found to be superimposed

on the normal lattice curve as predicted in the theories of Van Vleck

and Debye.

HEAT CAPACITY OF CuSiO3'HZO

FROM 2°K TO 400K

By

Warren Rice Eisenberg

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Physics and Astronomy

1963

AC KNOW LEDG MENTS

The author wishes to express his gratitude to Dr. H. Forstat
whose guidance and suggestions made this research possible. His
many hours of assistance are truly appreciated.

The author also wishes to thank Mr. N. Love for his help in

assisting with the experimental operations and the United States Air

Force Office of Scientific Research for financially supporting this
project.

**>'.<**

ii

TABLE OF CONTENTS

Page
THEORY ......................................... . . . . . 1
The Basic Equation for Heat Capacity Studies 1
Classical Theory of Specific Heats 3
Quantum Mechanical Theory of Specific Heats 4
Paramagnetic-Antiferromagnetic Transitions 8
Entropy Considerations 15
EXPERIMENT ............................... . ......... 16
Crystal Preparation 16
Calorimeter Design 18
Dewar System 21
External Measuring Apparatus 23
Procedure for Making a Run 25
RESULTS ............................................ 28
The Dioptase Specific Heat Curve 28
Analysis 28
REFERENCES ........................................ 35
APPENDIX ............................................ 37

iii

Figure

10.
11.
12.
13.

14.

15.

16.

17.

18.

LIST OF FIGURES

The specific heat of aluminum .....................

Specific heat of NiBr2

Magnetization vs. external field ....................
Specific heat vs. temperature .....................
Spontaneous magnetization vs. temperature .........
Antiferromagnetic spin moments ..................
<S>/s vs. T/T ................................
N
c /Nk vs. T/T ................................
v N
Dioptase and copper container .....................
Three part calorimeter ..........................
Double dewar system ............................
Heater circuit ...................................

Thermometer circuit .............................

Heat capacity (mj/OK) vs. temperature (OK) of

diopta se - c ontaine r c om binati on ...................

Heat capacity (mj/OK) vs. temperature (0K) of all

runs on dioptase and container ....... . ............

Heat capacity (mj/OK) and specific heat capacity
(cal/mole OK) vs. temperature (0K) of dioptase

crystal ..........................................

Plot of chZ (cal C)K/mole) vs. T5 (OK)5 for

dioptase above 210K .............................

Specific heat (cal/mole oK) vs. temperature (OK) for
experimental dioptase data and calculated lattice

contribution .....................................

iv

° 6H20 .....................

Page

10
IO
10
12
14
14
17
19
22
24

24

29

3O

30

32

34

THEORY

The Basic Equation for Heat Capacity Studies.

 

. . . 1 . .
The mean heat capac1ty IS defined as the ratio of a given amount
of heat put into a substance divided by the corresponding change in

temperature of the substance, or

--eg
C-AT. (1)

Frequently (especially at low temperatures) C exhibits a strong tem-

perature dependence, and thus the true heat capacity is given as

__ Iim AQ _ dQ
C -AT—)O AT ‘ dT' (2)

Experimentally one must hold the pressure or volume of the substance
constant while making the heat capacity measurement; furthermore,
the heat capacity per unit mass or mole (specific heat) is of importance.

Thus the definition is further refined so that

d
c (ca—:2“)
__"_= C =___V , (3)
m V m

where the small "c” indicates specific heat capacity, and the subscript
"v" indicates that the volume is to be held constant; m denotes mass or
moles.

Many of the theoretical discussions on specific heat capacities

are based on energy relations. To see why one needs only to consider

N

energies use is made of the First Law of Thermodynamics:

d0 = dU + pdV. (4)

Assume that U = f(T, V). Then

09

U
Cv=(?) - (5)
V

09

Equation (5) indicates a simple relation showing that if there eixists an
expression giving the internal energy of a substance as a function of
temperature, one differentiates with respect to the temperature,
keeping the volume constant, to find the heat capacity. Experimentally,
however, it is quite difficult to hold a substance at a constant volume
while performing measurements at different temperatures. It is much
easier to keep the pressure constant while making heat capacity meas-
urements. If one assumes U = f(T, P), the result obtained using Equa-
tion (4) is

c =(3—U) +p<3—") . (6)

P

8 3T
p P

Equation (6) obviously would be much more difficult to work with
theoretically. Fortunately, however, as the temperature approaches low

values CI) and cV are practically equal. This fact is expressed in the

:fiszZ
v K '

sion, and 7C is the isothermal compressibility. Thus Equation (5) is

equation cp - c where 6 is the coefficient of volume expan-
used for theoretical discussions while experimentally the pressure is
held constant. Equation (5), then, is a basic equation for heat capacity

studies.

Classical Theory of Specific Heats.

 

The question now arises as to what exactly contributes to the
internal energy of a solid; more precisely, if one can find an expres-
sion for U as a function of T, then one needs merely to insert this into
Equation (5) to find an expression for the specific heat of a solid.

Consider first a non-metallic substance. Here it can be assumed
that the electrons act in unison with the atoms. 3 The important con-
tribution to the internal energy comes from the vibrational energy of
each atom in the solid. Classical mechanics predicts that the vibra-
tional energy of each atom can be represented by the energy of three
harmonic oscillators. Each oscillator will have a potential energy
equal to l/ZkT and a kinetic energy equal to l/2kT, where k is Boltz-
man's constant. Hence each atom contributes 3kT to the solid's energy.
If exactly one mole of the substance is considered, then there will be
No (Avogadro's number) atoms, and the molar specific heat will be
(using Equation (5))

c = 3R, (7)

V

where R = Nok is the gas constant. Equation (7) is referred to as the
Law of Dulong and Petit. It is found that many solids do have a specific
heat of 3R, but only at high temperatures. The above relation is not
valid at the lower temperature region.

In the case of metallic substances one can assume that the electrons

behave as an "electron gas. " Drude4 postulated that a metal consists
of positive metal ions whose valence electrons are free to move be-
tween the ions as if they constituted an electron gas, and the electrons
are free to move throughout the entire crystal subject only to the laws
of classical mechanics. Classical theory states that the average kinetic
energy in a monatomic gas is 3/2 kT per atom. The average energy
per mole then becomes 3/2 NOkT = 3/2 RT. The specific heat contri-

bution of electrons is therefore given by
c = 3/2 R. (8)
v

Thus in the case of metals the total specific heat capacity is the sum of
Equations (7) and (8). 5 This gives cv = 9/2 R. It is observed that the

specific heats of metals tend to be more in agreement with Equation (7)
at high temperatures rather than 9R/2 and, furthermore, that Equation

(8) is invalid for low temperature electronic contributions.

Quantum Mechanical Theory of Specific Heats.

 

A typical graph of cV vs. T for most substances appears as shown
in Figure 1, here shown for aluminum. 6 At the higher temperatures
there is an asymptotic approach to the classically predicted value of 3R,
as given in Equation (7). At the lower temperatures, however, cvotT .
In an attempt to explain experimental results similar to Figure 1,
Einstein7 postulated that the atoms of a crystal oscillate independently

and with the same frequency v: A crystal containing N atoms can be

||l(|[1l|" lIIlI-[fx'lll'llrl‘l‘llfi’ll‘lIlll i |I||I I."

represented by 3N harmonic oscillators all vibrating with fre-
quency 1/. If one mole of the substance in question is considered
(N =NO) and use is made of quantum mechanics, the energy becomes
U = 3NOhV/[exp(hV/kT) - l], where h is Planck's constant. The spe-
cific heat is again obtained by substitution of U into Equation (5); the

result is

exp(h1// kT)
[exp(hV/kT) - 112

 

Z
c = 3R (5‘35) (9)

v kT

A plot of Equation (9) shows a graph similar to Figure 1 but does not

coincide with it. Einstein's equation shows fairly good agreement with

experiment only in the range cV > 3R/2. 8

F-—----—---—--_-------------------------—---—----

cp (cal/mole OK)->
O 0 HH N 60va- rhU'le

OO‘NCDI-hOO‘NCDv-ho
I

 

O 100 200 300 400

T(OK)—-—)

FIGURE 1. The specific heat of aluminum.

Consequently Debye used a model of a crystal in which he related
the specific heat to the elastic properties of the solid. Using Boltzman's
Theorem and quantum mechanics it is found that the energy of an as-
sembly of oscillators having a continuous frequency distribution is

given by V
m

hV/kT
exp(hU/kT) -

 

U = kT (hV/ZkT + 1 )q(u) dv, (10)

o
where q(v)dv is the number of oscillators in the frequency range from
V to V + (11/, and um is the maximum frequency of any oscillator. One
major difficulty in using Equation (10) is the determination of q(V). In
an attempt to find this distribution Debye assumed that, contrary to the
theory of Einstein, the atoms in a solid do not vibrate independently.
Debye's solution to the theory of specific heats involves the deter-
mination of the normal modes of vibration of a continuous solid made up

of individual atoms. The frequencies of vibration are quantized accord-

n 2
V X.
=— —— +
X

where v is the velocity of propagation and n , n , nz are integers in
x Y

ing to

 

n

2 nz 2
:2) war) »
y z

 

the direction of the three axes of length Lx’ Ly, L2. The quantity under

the root can be thought of as a radius vector R:

V
=—R. 12
V 2 ( )

[[l~'.['|ll‘"li“."‘l(llllll|[.rl‘l'").Its-II"I‘I‘RI

The number of allowed frequencies dN lying in a shell between R and

(R + dR) is 41rR2 dR. Only positive integers are allowed; thus one
considers only points in the positive octant of the shell. This gives

1/8 (41rRZdR), or from Equation (12), 4TTV2dV/V3. Each lattice point

has a unit volume associated with it that contains V allowed frequencies.

Therefore, the total number of allowed frequencies in the range d]! is

_ 41w 2 dz/

3
v

dN V. (13)

If three mutually orthogonal directions of vibration are considered, one

longitudinal and two transverse, Equation (13) becomes

dN = 41w2d1/ V -1—3 + -% (14)
v v
I t

dN
d_v is q(V) in Equation (10). Debye assumed that there existed a certain

cutoff frequency um and showed that Equation (10) takes on the form

 

V
m 3
9Nh V d]!
U = —- .
V 3 exp(hV/kT) - 1 (15)
m
In his final result Debye defined a characteristic temperature 9D =th/k

and showed by numerical integration that for T < < 9 the specific heat is

D

given by

c =aT , _ (16)

where a = IZTI'4R/59D3. Equation (16) is the universally famous Third
Power Law of Debye. Debye's results are in good agreement with
experimental work at both high and low temperatures.

One other important quantum mechanical result is the specific
heat contribution of free electrons at low temperatures. Making use
of the Pauli Exclusion Principle it is found that the electronic con-
tribution is given by

c = 8T, (17)

where 8 is a constant. Thus in the most general discussion of specific
heats at low temperatures the sum of Equations (16) and (17) expresses

c as a function of T:
v

3
c:V = aT + 8T. (18)

Paramagnetic -Antiferromagnetic Transitions .

Instead of indicating a c vs. T curve exactly as shown in Figure 1,
v

many substances exhibit a curve like that in Figure 2, here shown for

 

6..
15-
0M4-
%3#
£2-
31-
:80 .......

 

2 4 6 8 10 1214 T(°K)——>

FIGURE 2. Specific heat of NiBr2° 6HZO.

the specific case of NiBrZ' 6HZO.10 The striking feature here is the
appearance of an anomalous peak. Notice that the temperature range
plotted here is small in comparison to Figure l and that actually the
peak in Figure 2 would appear as a well defined kink superimposed on
Figure l at the indicated temperature. k-shaped curves like the above
are characterized by a maximum value at the temperature indicated

by T the Neél temperature.

N'
What is happening to the internal energy of certain solids to give
rise to a curve such as Figure 2? In answering this question reference
will be made to two important papers by Van Vleck. 11' 12
Basically the situation is most easily explained by stating that
when T>TN the substance is in a paramagnetic state and when T<TN
the substance is in an antiferromagnetic state. A brief discussion will
point out what is meant by paramagnetism and antiferromagnetism.
When matter is placed in a magnetic field, magnetic forces on the
moving electrons cause some type of reorientation of electron orbits in
the atoms so that the substance itself will set up a macroscopic field.
A paramagnetic substance is characterized by an extremely small field
proportional to and in the direction of the external field. Antiferro-
magnetic substances are characterized by antiparallel spin moments
(ordered spins) in adjacent atoms, thus producing no field. Furthermore,

paramagnetic substances are characterized by random spin moments

(disordered spins) of the atoms.

10

Figures 3, 4, and 5 show some experimental work done in Leiden,

13

Holland on the antiferromagnetic properties of CuClz- ZHZO and will

help to explain the theoretical results of Van Vleck more clearly.

Figure 3 shows the dependence of the magnetic field in the crystal on

Magnetization
Specific Heat
Spontaneous

Magnetization

//

T

T
N

N

 

 

 

 

 

 

External Field Temperature Temperature

FIGURE 3 FIGURE 4 FIGURE 5

the applied field parallel and normal to the preferred direction. While
the magnetization at right angles shows no anomaly, the parallel one
remains small at low fields and then rises abruptly to become even-
tually proportional to the applied field. This suggests the behavior of a
critical (threshold) field in addition to critical temperature. The latter
is clearly marked in the specific heat (Figure 4) indicating the disap-
pearance with rising temperature of long range order. That the specific
heat decreases only gradually to its normal value above the critical

temperature means that some short range order evidently persists

11

above T . The spontaneous magnetization (Figure 5) vanishes at T =T

N N’

the same place at which the specific heat shows an anomaly. Further-
more, magnetic susceptibility curves with kinks in them are frequently
found. 14 Any good theory on anomalous specific heats, then, should
simultaneously satisfy all of these observations.

In deriving results for the paramagnetic part of curves like Fig-
ure 2, Van Vleck considered the influence of dipole-dipole coupling.
Here the partition function Z of an ensemble of atoms is determined
assuming each atom has a magnetic moment and is subject to the ex-
change coupling with the moments of other atoms. Use is made of the

statistical mechanics relation showing c as a function of T:
v

_a 231nZ
Cv‘a'r (kT 8T

 

)- (19)

To find Z the Hamiltonian is expressed as

3(i1.- Emil.- L.)
H=Z 1 hfij- 1‘141. (m)

j>i r 3 r 2
ij ij

 

 

where Eli is the magnetic moment of atom i and ii, the radius vector
connecting atoms i and j. Use of Equations (19) and (20) in conjunction
with the fact that Z = i: exp(-W)\/kT), where the sum is over all the
energy states of the crystal and W). are the characteristic values of the

Hamiltonian, eventually leads to the result

c =l (21)

12

where y is a characteristic constant. Thus the total heat capacity in

the paramagnetic state is the sum of Equations (18) and (21) or

c =aT3+5T+V—. (22)
v T2

In the antiferromagnetic state there is, as yet, no simple rela-
tion between cV and T. Consider the atoms of a crystal to be arranged
on two interpenetrating sublattices so that the spin moments of atoms
in one sublattice array A have an opposite sense to the other sublattice
array B; furthermore, assume the nearest neighbors in A are contained
entirely in B as shown in Figure 6. Thus when one considers exchange
interactions of an atom in A, only effects produced by the atoms in B
need to be taken into account. The effective potential to which an atom i
is subjected is given by

V. =-2JS.'Z.:<S.>. (23)
1 1 J j

where Si is the spin angular momentum vector of atom i, Sj is the spin

 

FIGURE 6. Antiferromagnetic spin moments.

13

angular momentum vector of atom j, and J is the exchange integral.

By considering the effect of various external field orientations on the
spin vectors, the problem reduces to one of scalar functions. If <S>
denotes the mean value of all the spins in sublattice A, then the mean

value of all the spins in sublattice B is also given by <S>; let s denote

 

 

the spin quantum number (i. e. é, %, g, . . . ). An important result
is
<S> = sBs(yO), (24)
where
133m = 232:1 coth (553%!) - 31; coth ('2Ls) (25)
is the Brillouin function and
Y0 :ZlJllf; <S>, (26)

2 being the number of neighbors possessed by a given atom. The

solution <S> of Equation (24) is s as T—90; at T = T <S> = 0, showing

N

the complete antiparallel arrangement of the spins. The Neél tempera-

ture TN is given by

-2 a
TN—3lJlks(s+l). (27)

A plot of Equation (24) for the case s =% is shown in Figure 7. 15 Note

the agreement with Figure 5. If N denotes the number of atoms per unit

volume then

 

d <S>

=-2N J <s>
CV | lz dT

(28)

14

A
ICD—>
V
|<n—> '

o
'(0
U1
'
3‘2
077'

 

 

 

 

 

0. 0 ' - 0. 0 ~ A . ,
0.5 1. 0 0.5 1.0

T/TN—e T/TN —>

FIGURE 7. FIGURE 8.

1
A plot of Equation (28) is given in Figure 8. 6 Here note the resemblance

to Figures 2 and 4 for T<TN.
Suffice it to say that Van Vleck's theory does satisfy all of the
aforementioned diagrams and physical anomalies; thus it can be said

that in the case of a paramagnetic-antiferromagnetic transition magnetic

contributions strongly affect the internal energy.

Entropy Considerations.

It is possible to obtain some idea of the amount of disorder-order
involved in a paramagnetic-antiferromagnetic transition by considering

entropy changes. 17’ 18 Above the Nee’l temperature Equation (22) yields

2
c T = aT5 + y, (29)
V

for the case of a non-metal where the electronic contribution is negli-

2 5
gible. A plot of cVT vs. T will give a straight line, the slope of

15

which is a and the intercept of which is y; frequently 0 turns out to be

quite small and y comparatively large. For T > TN Equation (21) can

now be used to calculate the entropy S due to magnetic contributions:

T2 c 00 dT
s = —V dT = y —— (30)
T 3
T
T1 TN

Below the Neél temperature one must resort to graphical methods to
find S due to magnetic contributions. First the low temperature end

of the cv vs. T curve is extrapolated to T = 0. Then the T3 lattice
contribution to the specific heat is calculated from Equation (16). a is
found by measuring the slope of the graph obtained by plotting Equa-
tion (29). The lattice contribution is subtracted from the total specific
heat capacity curve. The resulting values are the magnetic contribu-
tion to the specific heat, Cmag’ A graph of Cmag/T vs. T is now
plotted and the area of the resulting curve measured. As is indicated
by the first part of Equation (30) the area measurement will provide the
entropy of the antiferromagnetic state. The total magnetic contribution

to the entropy is given by the expression
S =R1n(2s +1), (31)

and thus the experimental results may be compared with theory.

EXPERIMENT

Crystal Preparation.

 

In order to experimentally measure specific heats it was neces-
sary to use a modified form of Equation (3), or AQ = mchT. A known
amount of heat AQ was put into a crystal of mass m and the resulting
temperature change AT measured.

The crystal to be examined was wrapped with a manganin heater
wire enabling known amounts of heat to enter the crystal. A carbon
resistor was inserted in the crystal for temperature measurements. 19
A small amount of Glyptal was used to fasten the wire and resistor in
place. Connections to the calorimeter leads were made by small
lengths (6inches) of teflon wire. The carbon resistance thermometer
(thermistor) had a room temperature resistance of 56 ohms. The
heater wire resistance was about 150 ohms.

The particular crystal used in this experiment was dioptase, or
CuSiO 3’ H20, a mineral mined in S. W. Africa. Difficulty had pre-
viously been encountered with this crystal20 in that it showed an ap-
parent inability to retain heat put into it with the experimental setup
used. One cause of this was believed to be a possible outgassing of the
crystal creating an undesirable exchange gas. This would make thermal

isolation of the crystal, a necessary procedure for measuring heat

capacities, impossible. Thus a very small can composed of copper

16

l7

and low melting solder was constructed around the crystal (Figure 9)
to prevent the crystal from outgassing. Originally the experiment was

carried through without the special container; difficulty was again

fl— cold weld

 

copper container
heate r wir e

helium exchange ga s

dioptase crystal

 

 

thermistor

FIGURE 9. Dioptase and copper container.

encountered. Thus a slight modification was made on the wiring: The
heater wire was wrapped around the container containing the crystal
and the thermistor placed into contact with the container. Glyptal was
again used as a fastener. The can was evacuated and helium exchange
gas at 1 mm Hg pressure inserted to insure good thermal contact be-
tween the crystal and copper-solder container. A cold welding pliers
was used to seal off the low pressure helium from the atmosphere. A
small amount of solder was then placed over the weld to be certain that
no leakage would occur when in the lower temperature region. The
heat capacity of the entire setup was thus measured. Later the can was
disassembled and the crystal removed; the can was then bathed in

acetone so that no possible contamination resulted from fragments of

1 l I ‘1

18

the dioptase. The can was reassembled and a heat capacity measure-
ment made on it. From these data (given in the Appendix) the specific

heat as a function of temperature was obtained.

Calorimete r De sign.

 

Nuclear magnetic measurementsm indicated a possible transition
point (i. e. Nee’l temperature) to be around 210 K. Thus a calorimeter
was designed which would permit temperature measurements from liquid
helium temperatures (1.20 to 4. 20 K) to about 700K. The calorimeter
consisted of three brass cans, generally described as small, medium,
and large; it is shown diagramatically in a one to one scale in Figure 10.

In the small calorimeter container the crystal and copper-solder
can combination was supported by a nylon thread attached to a hole in
the pipe used for evacuation of the can. The evacuation pipes were used
to connect the calorimeter containers to the external pumping system.
Leads from the heater and thermistor were connected to a small array
of eight prongs, which in turn were connected to the external measuring
apparatus by manganin wire wrapped in teflon tubing extending through
the length of the evacuation tube. Note that in Figure 10 the ends of all
the evacuation pipes were closed off and that only two small holes on
either side of each tube were used for evacuation of the calorimeter
chambers. This served the purpose of providing radiation shields to

the calorimeter proper.

19

V evacuation pipes

\\

 

 

l

 

, radiation shields
1"] /// /1"1
A/ //
,/ \

L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘)
constant ¢ - flanges
volume
thermometer )1 fl
0
heater wire a I
u
, prongs for leads
RNH‘N

 

, ~ nylon thread

 

large can _____,

medium can

 

_ _ *mw“ dioptase-
container

 

 

 

small can

 

 

 

 

 

 

 

FIGURE 10. Three part calorimeter.

20

For measurement in the liquid helium temperature range the
small can alone would be sufficient; it would act as an adiabatic wall
between the crystal and helium bath. In going above 4.20K, however,
it was necessary to use some device to make it "appear to the crystal"
that the helium bath in which the entire calorimeter rested (as dis-
cussed under "Dewar System") was capable of going to temperatures
as high as 700K. This was accomplished by introduction ofa central
can cut with fine grooves into which was wound 1500 ohms of manganin
wire. By introducing an alternating current of varying effective
voltages into the central can it was possible to put in large amounts
of heat. From the heat produced by this method in conjunction with
the liquid helium background, it was possible to produce temperatures,
as seen by the crystal, from 1.20K to 700K.

The small and medium cans were then enclosed in a large can.
The large can served as an adiabatic wall insulating the heater of the
medium can from the helium bath.

All three cans were sealed to their respective flanges by three
ampere fuse wire "O"-rings. The leads throughout the calorimeter
were brought up through the evacuation pipes to connect to small brass
flanges containing glass seals with metal contacts. Connections to
the external measuring apparatus were then made from the flanges.
An ionization gauge was placed at the top of the evacuation pipes to

indicate pressures in the calorimeter system. A constant volume gas

21

thermometer placed in the calorimeter was to be used as a means of
accurately determining temperatures; experimental difficulties were

encountered with the gas thermometer, and it was therefore not used.

Dewar SLstem.

 

The calorimeter was placed in the inner part of a two container
dewar arrangement as shown in Figure 11. The outer dewar was used
to pre-cool the inner dewar before introduction of the liquid helium.
The jacket of the outer dewar was constructed with a permanent vacuum.
This prevented heat from entering the liquid air. The jacket of the
inner dewar was constructed with a glass stopcock. This permitted
frequent pumping on the jacket to eliminate helium gas which diffused
into it. The dewars were also silvered to prevent heat loss. A small
slit along the side of each dewar allowed observation of the liquid helium
and liquid air levels; illumination of the liquids was obtained by a
fluorescent light aligned along the slits. The inner dewar was con-
structed to be leak tight when the calorimeter was placed in it thus
allowing the temperature of the liquid helium to be lowered by pumping. _
A manometer was connected into the pumping line so as to be able to
calibrate the thermistor of the crystal-can against the vapor pressure
of the liquid helium. The liquid helium was produced in a Collins Helium

Cryostat in the laboratory.

22

vents to exte rnal

/ {equipment

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l
flange r : :1 F
.JLA;__
Dr—V— stopcock
n C {inner dewar
jacket
- outer dewar
liquid jacket
helium \g
\\L\_
"‘ . 1i uid air
\ \ 1., Mm ‘1
\ 1:;AC'GAIPP/f (T //7
”’ \r-H‘M \N‘NL
\ \\
_. 1.. )_ _
calorimeter

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE 11. Double dewar system.

23

External Measuring Apparatus.

 

Three basic circuits were set up for measuring the heat capaci-
ties. The first was used for putting heat of known amounts into the
crystal. This consisted of a potentiometer, two precision resistors,
a galvanometer, a helipot, an electric clock, an ammeter, and other
associated equipment. The basic circuit is shown in Figure 12. The
potentiometer was connected so as to allow measurement of the voltage
V across the heater of the crystal-can suspended in the small
colorimeter container. The potentiometer was used also to read the
voltage across the precision resistor connected within the heater cir-
cuit. The latter reading allowed the heater current I to be deter-
mined. A 10 ohm precision resistor was used to determine current
values at higher temperatures (above 150K) while a 100 ohm resistor
was used at lower temperatures. This was done so as to allow the
best sensitivity of the potentiometer in the current range being meas-
ured. A timer was automatically turned on and off at the same instant
a switch controlling the heater current was turned on and off. This
gave the time t that the power was put into the heater wire. The heat
AQ was computed from the relation AC2 = VIt.

The second circuit was that in which a continuous record of the
resistance values of the thermistor was kept. The circuit consisted
of two potentiometers, a D. C. Microvolt Amplifier, a Speedomax

recorder, a D'Arsonval galvanometer, a microammeter, a helipot,

/

10K 9 helipot

Edison cells

.._./

9h

24

100 S2 precision

 

 

 

 

O

milliammeter

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 S2
_ pot precision
° 0
‘1' o/ /
galv. 4 9 9
\
heater.L
FIGURE 12. Heater circuit.
microammeter
-——1 I @ WWW—#—
30K 9 100K 9 helipot 1009 precision
Edison cells 1/
WV
thermistor s-C-I P°t°

 

 

 

I
S.C. 'l'_.1

pot. 1

11

d.c. amp

I l

Speedomax

 

 

 

 

 

 

 

FIGURE 13.

 

 

 

D ' Ar sonval galv.

 

 

 

Thermometer circuit.

25

a precision resistor, and other related accessories. By appropriately
calibrating the chart of the Speedomax recorder, it was possible to
determine the temperature change of the crystal-can when power was
put into the heater. The results thus obtained in a single run gave a
value of the specific heat at about 80 points at different temperatures.
It should be pointed out here that what was actually being measured was
the mean specific heat and that the temperatures at which the various
heat capacities were found were really average temperatures computed
over a small temperature interval. This was carefully taken into
account when evaluating the data. A diagram of the thermistor circuit
is shown in Figure 13.

To put varying amounts of heat through the heater of the middle
can entailed the use of a third circuit. This consisted of an a. c. source,
a Variac for course control, and a helipot for fine control. The range
of the Variac was from 0 to 110 volts; the helipot had a maximum resist-

ance of 100, 000 ohms.

Procedure for Making a Run.

 

Six basic steps were followed in making a run on the specific heats
of the dioptase-can and the can by itself.
1. Assembly: The crystal-can and calorimeter were wired
and sealed respectively.

2. Pre-cooling: The inner dewar was evacuated and then filled

26

with helium exchange gas; liquid air was added to the outer
dewar. The system was kept filled with liquid air for at
least two to three days.

3. Transferring: The calorimeter was evacuated and then filled
with helium exchange gas at 1 mm Hg pressure. Liquid
helium was transferred from storage tanks into the inner
dewar.

4. Calibration: The thermistor was calibrated against a vapor
pressure curve of helium.

5. Evacuation and Measurement: The calorimeter was totally
evacuated to about 3 x 10.5 mm Hg and heat capacity meas-
urements made.

6. Warmup: Having gotten the necessary data, the remaining
helium was pumped out of the dewar and the liquid air was
allowed to boil off. The calorimeter was then removed for
disassembly.

The data obtained in Step 4 was used to calibrate the thermistor

by use of the equation

log10 R 1/2

T = a + b log10 R. (32)

That is, the vapor pressure values at various known resistances were
. 22
converted to temperature values at these re31stances. The values so

obtained were inserted into Equation (32). Thus "a" and I'b, " two

27

unknowns characteristic of the particular carbon resistance being used,
were evaluated. "a" was a function of temperature while ”b" was con-
stant. The helium vapor pressure method could only be used up to the
boiling point of liquid helium (4. 20K). To obtain values of "a" above this
it was necessary to extrapolate the calibration curve to a resistance
value corresponding to some known temperature. This was done by
measuring the resistance while Step 2 was in progress. This gave the
thermistor resistance at liquid nitrogen temperatures, taken to be 77.4OK.
Solving for Equation (32) for T gives

log R

2 (33)
(a + b log R)

 

T:

In Step 5 it was merely necessary to compute resistance values on the
Speedomax recorder before and after putting heat into the crystal can.
Insertion into Equation (33) for the two instances yielded the temperature

change AT.

RESULTS

The Dioptase Specific Heat Curve.

 

Three runs were made on the dioptase-container combination
and one run was made on the container by itself. Figure 14 shows the
results obtained in each dioptase-container run. Figure 15 shows the
three dioptase-container runs superimposed. In Figure 15 the data
obtained from Run 3 was plotted 1. 60K higher than is shown in Figure 14.
This was necessary as the entire curve obtained in Run 3 appeared to
be shifted to the left of Runs 1 and 2 by 1.6OK (as explained under
”Analysis"). Also plotted in Figure 15 is the data obtained on the copper
container run. Graphical subtraction of the container heat capacity
values from the dioptase-container values gives the heat capacity of
the dioptase crystal; the result is shown in Figure 16. The heat
capacity values are given on the ordinate at the left and the converted
values to specific heat capacities are shown at the ordinate on the right
side of the graph. The actual data obtained from all the runs is given

in the Appendix.

Analysis.

Evidence appears in Figure 16 for an anomaly in the heat capac-
ity of the dioptase crystal. This anomaly can also be observed in the
plot of the separate runs in Figure 15. A magnetic transition is evident

0 . . 0 . .
at about 21 K With an uncertainty of i2 K. The uncertainty in the

28

C-—)

C—>

29

 

 

 

 

5000 L
4000 - Q o
O
3000 ~ 0
O
2000 - oo
o
1000 r 0
0066
M I I l I l I I I I L I L I I I
0 4 8 12 16 20 24 28 32 36 40
T—> Runl
5000 -
O
4000 0 o
3000 — Q o 0
O
2000 -
0 o
1000 — o 9
000
o
WIIILLIIIIIILIII
0 4 8 12 16 20 24 28 32 36 4O
T-—> Run2
5000...
0
G
3000 -
0 O
o
2000 -
G o
o
1000 "’ 00
as?”
,M9‘DelolllelllLIJ I LII

 

 

0 4 8 12 16 20 24 28 32 36 40

T—> Run 3

FIGURE 14. Heat capacity (mj/bK) vs. temperature (OK)

of dioptase -container combination.

C—-)

4000

3000

2000

1000

2000

30

 

 

 

 

 

_ Legend ' g 0
a 0 GI A3
A Run 1 (diop. and can) A
0 Run 2 (diop. and can) B
_ G
a Run 3 (diop. and can) G
E) q, “PA
x Can alone x
x
"" A B X X i
a 1‘
63%
g) X
_ £49,. "
00‘,
0 a 1‘
G I?”
3%» i
W‘? I I I L 1 I I I I I I I I I I
0 4 8 12 16 20 24 28 32 36 40
T—)
FIGURE 15. Heat capacity (mj/OK) vs. temperature
(0K) of all runs on dioptase and container.
)— .1
g. I I I I I I I I L I
0 4 24 28 32 36 40

 

T—->

FIGURE 16. Heat capacity (mj/OK) and specific heat capacity
(cal/mole 0K) vs. temperature (0K) of dioptase crystal.

26.8

13.4

31

temperature is due to the method of calibrating the thermistor. Tem-
peratures were known to an accuracy of i0. 0010K from 20K to 4. 20K
and of i0. 10K at liquid nitrogen temperatures (77. 40 K). Between

4. 20K and 77. 40 K, however, the thermistor temperature values were
obtained only by interpolation. This uncertainty in temperature meas-
urement probably accounts for the apparent shifting of the data of Run 3
from the data of Runs 1 and 2.

The curves in general follow a Debye T3 form excepting the region
at which the magnetic transition occurs. Above 320K the dioptase curve
(Figure 16) appears to flatten out and then begin to dip down. This may
be due to a Schottky effect, but no generalizations should be made at this
part of the curve, however, as there is not sufficient data at the higher
temperatures to provide more concrete information. In this region, as
well, long equilibrium times were observed which might suggest insen-
sitivity of the thermistor.

An analysis of Figure 16 was made to find the entropy contributions
above and below the transition temperature as discussed previously under
”Entropy Considerations. “ In Figure 17 a plot of Equation (29) is shown
for the dioptase crystal above 210K, the data being obtained from Figure
16. In the neighborhood of 210K there is a sharp break from linearity.
This indicates the points at which Equation (29) is no longer valid. The
intercept in Figure 17 gives the value of Y and is found to be about 20

cal- OK/mole. Since TN = 210K and y = 20 cal- OK/mole, Equation (30)

2 -
cT x103

16

14

12

10

on

V
0‘

32

 

 

I J l I l l l l l I I J

2 4 6 8 10 12 14 16 18 20 22 24
T5x10'6——)

o 5

FIGURE 1?. Plot of csz2 (cal OK/mole) vs. T5( K)

for dioptase above 210K.

33

gives the magnetic contribution to the entropy in the paramagnetic
state. The value is found to be 0. 02 cal/mole - c)K.

Using a graphical method to find the entropy contribution from
the antiferromagnetic state, it is first necessary, as previously ex-
plained, to subtract the lattice T3 contribution from the curve of Fig-
ure 16. The lattice contribution can be gotten by use of Equation (16).
Here the value of a is found from the slope of the line in Figure 17.
Figure 18 shows the dioptase specific heat curve and the lattice
contribution. The difference of the curves represents the magnetic
contribution to the specific heat (Cmag). Plotting Cmag/T vs. T and
measuring the area, the entropy due to the antiferromagnetic state of
the dioptase is found. The value so obtained is l. 17 cal/mole oK.

Using Equation (31) and noting that s = 1/2 for the Cu++ ion in
CuSiO3 - H20, the theoretical value for the total magnetic contribution
to the entropy is stot = l. 39 cal/mole - oK. Experimentally

tot = Spara + Santifer = 0.02 + 1.17 = 1.19 cal/mole - 0K, a result in
good agreement with theory. The antiferromagnetic entropy contribu-

tion is about 98 %, while the paramagnetic contribution is about 2%, due

probably to short range ordering.

34

20—

18

 

   

Expe rim ental dioptase
specific heat

 

Calculated lattice
specific heat

 

lillJllllll
4 8 1216 20 24 28 32 36 40

0
FIGURE 18. Specific heat (cal/mole' K) vs. temperature
(OK) for experimental dioptase data and
calculated lattice contribution.

10.

11.

12.

13.

14.

15.

REFERENCES

F. W. Sears, Thermodynamics (Addison-Wesley Pub. Co.,

 

Cambridge, Mass., 1956).
D. H. Parkinson, Rep. Progr. Phys., _2_l, 226 (1958).

L. V. Azaroff, Introduction to Solids (McGraw-Hill Book Co. ,

 

Inc., New York, 1960).
P. Drude, Ann. Physik, l, 566 (1900).

C. Kittel, Introduction to Solid State Physics (John Wiley and

 

Sons, Inc., New York: 2nd edition, 1956).

C. G. Maier and C. T. Anderson, J. Chem. Phys., 2, 513
(1934).

A. Einstein, Ann. Physik, _22, 180 (1906).

F. Seitz, Modern Theorj of Solids (McGraw-Hill Book Co. , Inc. ,

 

New York, 1940).

P. Debye, Ann. Physik, 3_9, 789 (1912).
R. D. Spence, H. Forstat, G. A. Khan, and G. Taylor, J. Chem.
Phys., 11, 555(1959).

J. H. Van Vleck, J. Chem. Phys., _5_, 320 (1937).
J. H. Van Vleck, J. Chem. Phys., _9_, 85 (1941).

C. J. Gorter, Revs. Modern Phys., _2_5, 332(1953).

Bizette, Squire, and Tsai, Compt. Rend., 207, 449 (1938).

A. B. Lidiard, Rep. Progr. Phys., E, 201 (1954).

35

l6.

17.

18.

19.

20.

21.

22.

36

Ibid.

 

s. A. Friedberg, Physica, _1_§, 714(1952).

D. G. Kapadnis and R. Hartman, Physica, 22, 181 (1956).

J. R. Clement and E. H. Quinnel, Rev. Sci. Instr., _2_3_, 213
(1952).

G. 0. Taylor, Jr. , "Low Temperature Heat Capacities of Single
Crystals" (unpublished Master's thesis, Dept. of Physics and
Astronomy, Michigan State University, 1960).

R. D. Spence and J. H. Muller, J. Chem. Phys., _22, 961 (1958).
F. G. Brickwedde, J. Research of National Bureau of Standards,

fl, 1 (1960).

APPENDD(

Experimental Data Obtained from All Runs

First Run on Dioptase-Container, August 2, 1963.

 

 

 

'1'“ (°K> AT (°K) c (mj/°K) "I" <°K> AT <°K> c (mi/OK)
2.19 0.0010 37 3.99 0.013 44
2.21 0.0007 55 4.10 0.015 45
2.23 0.0011 32 4.29 0.022 31
2.25 0.0011 32 4.47 0.024 37
2.27 0.0010 34 4.68 0.060 16
2.32 0.0014 26 5.02 0.042 20
2.34 0.0011 33 5.27 0.024 37
2.38 0.0011 36 5.46 0.017 56
2.41 0.0008 45 5.67 0.033 30
2.44 0.0008 44 5.97 0.020 45
2.47 0.0010 36 6.26 0.021 53
2.50 0.0019 37 6.54 0.033 61
2.54 0.0016 36 6.78 0.023 83
2.57 0.0016 31 6.96 0.023 80
2.61 0.0018 30 7.16 0.035 75
2.65 0.0016 36 7.68 0.030 90
2.69 0.0012 44 8.15 0.024 100
2.74 0.0024 42 8.60 0.027 140
2.83 0.0027 26 9.08 0.024 200
2.86 0.0022 35 9.49 0.040 250
2.90 0.0043 36 9.85 0.039 260
2.95 0.0025 38 10.4 0.095 210
3.00 0.0020 47 11.9 0.086 270
3.05 0.0035 26 13.2 0.050 480
3.09 0.0042 38 14.5 0.050 790
3.15 0.0043 42 15.5 0.10 1100
3.20 0.0037 45 17.1 0.48 1200
3.26 0.0068 29 19.0 0.30 1600
3.33 0.0050 39 20.8 0.26 1900
3.40 0.011 22 23.6 0.26 2700
3.49 0.0054 36 26.2 0.42 2500
3.58 0.010 27 29.4 0.53 3400
3.67 0.0075 34 33.8 0.71 3800
3.76 0.0090 47 38.8 1.0 4000
3.85 0.010 38

37

Second Run on Dioptase-Container, August 6, 1963.

 

 

T zsr c i 131' c
2.28 0.0024 28 5.22 0.040 30
2.30 0.0027 26 5.43 0.028 46
2.33 0 0020 29 5.66 0.025 34
2.36 0.0015 35 5.98 0.024 50
2.39 0 0022 33 6.22 0.033 50
2.42 0.0021 39 6.45 0.020 59
2.46 0.0024 32 6.77 0.026 58
2.49 0.0021 34 7.07 0.025 77
2.53 0.0017 41 7.40 0.029 92
2.56 0.0016 47 7.65 0.025 110
2.60 0.0023 32 7.87 0.032 120
2.64 0 0027 36 8.17 0.038 130
2.67 0.0025 34 8.51 0.037 140
2.71 0.0021 43 8.87 0.041 190
2.96 0.0043 44 9.24 0.052 160
3.01 0 0021 72 9.60 0.050 250
3.05 0.0049 47 9.98 0.060 270
3.10 0.0033 53 10.5 0.081 370
3.14 0.0042 52 11.3 0.12 470
3.19 0.0047 53 12.0 0.064 700
3.25 0 0025 73 12.7 0.25 190
3.30 0.0038 70 13.3 0.11 840
3.36 0.0041 59 14.1 0.095 900
3.40 0.0043 51 15.1 0.22 990
3.45 0.0040 55 16.6 0.22 1100
3.51 0.0034 58 18.1 0.21 1400
3.58 0.0057 65 19.7 0.25 1500
3.64 0.0032 90 21.4 0.21 2800
4.21 0.017 42 23.6 0.34 2500
4.31 0.012 52 25.8 0.46 2700
4.47 0.020 30 28.3 0.66 3100
4.59 0 012 47 31.3 0.75 3800
4.69 0.014 41 35.3 1.1 3700
4.81 0 017 32 40.3 1.7 4000

39

Third Run on Dioptase-Container, Augist 7, 1963.

 

 

 

T .017 c T .61: c
2.27 0 0023 23 4.26 0 0068 56
2.29 0.0023 42 4.36 0.016 31
2.31 0 0033 33 4.45 0 015 32
2.34 0 0022 33 4.55 0.019 36
2.37 0 0018 41 4.65 0 012 46
2.40 0.0019 38 4.74 0 022 24
2.42 0 0020 39 4.86 0.015 35
2.45 0 0022 33 5.25 0.020 39
2.48 0.0016 46 5.42 0.021 36
2.50 0 0012 61 5.58 0.031 30
2.54 0.0021 36 5.79 0.026 36
2.57 0 0023 38 6.01 0 032 31
2.64 0 0025 33 6.21 0.022 44
2.68 0 0023 36 6.43 0.024 48
2.71 0.0029 29 6.86 0.025 99
2.75 0.0029 30 7.05 0.033 83
2.80 0.0027 31 7.21 0.023 89
2.87 0.0037 32 7.39 0 026 100
2.91 0.0024 41 7.55 0.031 80
2.95 0.0034 50 7.85 0 0090 310
2.99 0.0040 41 8.20 0 013 260
3.03 0.0029 44 8.56 0.034 130
3.06 0.0031 43 8.93 0 037 160
3.10 0.0041 45 9.27 0.054 200
3.14 0.0030 58 9.61 0.073 120
3.26 0.0043 40 9.91 0.030 390
3.32 0.0088 32 10.2 0 064 290
3.38 0 0047 46 10.6 0.048 330
3.44 0.0079 28 11.3 0.055 510
3.49 0.0046 47 11.9 0.10 470
3.55 0.0095 23 12.5 0.096 650
3.62 0.0085 40 13.1 0.16 780
3.68 0 020 13 13.8 0.20 1000
3.75 0.0080 33 14.7 0.20 1000
3.82 0 013 30 15.8 0.25 1300
3.95 0.0054 65 17.0 0.26 1300
4.01 0.014 31 18.3 0.22 1700

(continued)

40

 

 

 

 

T 131' c: T 131' c
4.10 0.0064 64 19.4 0.17 2500
4.18 0.012 32 20.7 0.28 1900
22.2 -0 33 2400 30.4 0.58 3800
23.9 0.35 2600 33.6 0.79 3800
25.8 0.27 3300 27.1 0.96 4100
27.8 0.43 3900
Run on Container Alone, August 13, 1963.

T .61‘ c: T st c
2.29 0.004 51 3.87 0.0061 39
2.31 0.0014 34 3.92 0.0072 32
2.34 0.0013 33 3.97 0.0056 43
2.36 0.0013 33 4.03 0.0044 52
2.38 0.0020 34 4.08 0.0057 47
2.41 0.0026 41 4.13 0.0048 51
2.47 0.0019 33 4.19 0.0065 52
2.49 0.0018 35 4.32 0.0064 43
2.53 0.0018 36 4.46 0.015 24
2.56 0.0017 37 4.60 0.013 31
2.59 0.0013 50 4.70 0 0098 33
2.63 0.0013 49 4.79 0.015 22
2.66 0.0017 37 4.97 0.014 30
2.69 0.0024 34 5.12 0.012 33
2.73 0.0028 26 5.21 0.013 30
2.76 0.0024 25 5.36 0.014 28
2.80 0.0018 30 5.48 0.014 28
2.83 0.0019 39 5.60 0.019 27
2.87 0.0027 34 5.76 0.022 22
2.90 0.0022 40 5.96 0.024 22
2.94 0.0018 49 6.33 0.022 34
2.97 0.0021 42 6.49 0.020 58
3.01 0.0023 38 6.72 0 021 41
3.04 0.0024 43 6.92 0.024 56
3.08 0.0020 52 7.12 0.031 51
3.11 0.0021 46 7.32 0.029 59
3.15 0.0021 46 7.50 0.037 65
3.22 0.0058 20 7.72 0.036 68

(continued)

41

 

 

T .A1? (3 T 131' C
3.58 0.0030 57 7.99 0.048 69
3.67 0.0046 43 8.28 0.034 97
3.72 0.0043 52 8.56 0.048 97
3.76 0.0034 53. 8.95 0.11 120
3.82 0.0058 52 9.38 0.052 150
9.77 0.058 150 18.9 0.47 1100
10.7 0.054 190 21.1 0.24 1200
11.2 0.048 320 23.3 0.19 1900
11.8 0.070 320 25.5 0.27 1900
12.5 0.14 260 28.3 0.45 1800
13.4 0.19 400 31.7 0.57 2600
14.5 0.14 440 35.7 1.4 2100
15.7 0.092 680 46.5 1.2 2700
17.0 0.11 1000

llIllHllllWill11111llll1111111111111111111

 

016836268