SPECIFIC HEAT STUDIES OF
MN(C2H3OZ)24HZO AND FEBIZ 4H20

IN THE LIQUID HELIUM
TEMPERATURE REGION

The“: for Hm Degree of M. S.
MECHIGAN STATE UNEVERSETY
Sister M. Charles
Joseph Zinn. O. P.
1962

m IIIIII II IIII IIIIIIIII III '

312993 01743 0152

 

M

LIBRARY LI
Michigan State "

Univcrfiity r

 

mayor:-
in I 3122.19

Innahv

i5 :97 5’3 s‘c. 3'“; tr

 

 

 

 

 

ABSTRACT

SPECIFIC HEAT STUDIES
OF Mn(C2H302)2-HH20 AND FeBr2~HH20

IN THE LIQUID HELIUM TEMPERATURE REGION

by Sister M. Charles Joseph Zinn, O.P.

In this investigation the heat capacity of single
crystals of ferrous bromide and manganous acetate were
studied in the helium temperature range. Basically the
method used was to cool the sample to the desired temperature
region in a conventional double Dewar system, isolate it, and
then put measured amounts of heat into the crystal while
observing the corresponding temperature rise. A small carbon
resistor embedded in the crystal served as a thermometer and
a fine wire coil wrapped around it provided a mechanism for
heating. At these low temperatures (l°—H.5° K) the lattice
heat capacity, which varies as T3, is negligibly small and
therefore anomalies due to magnetic ordering processes and
other effects are observable.

A-type anomalies in the heat capacity curves were
observed at 1.28 L 0.02°K in the case of ferrous bromide and
at 3.17 ; 0.02°K for manganous acetate. The latter is be-
lieved to be the Curie temperature for a paramagnetic-

ferromagnetic transition. The magnetic entropies associated

Sister M. Charles Joseph Zinn, O.P.

with these transitions are 2.6m cal/moleodeg and 1.61 cal/
mole-deg respectively. The first is in fair agreement with
the theoretical value, 3.19 cal/moleodeg, if the Spin of the
magnetic ion (Fett) is taken as 2 in R 1n(28+l). In the case
of manganous acetate, however, one finds disagreement by more
than a factor of 2 between the entrOpy calculated from
measured values and that obtained from R 1n(28+l). These

are 1.61 and 3.58 cal/moleodeg respectively.

SPECIFIC HEAT STUDIES

‘HH 0

2 2 2
IN THE LIQUID HELIUM TEMPERATURE REGION

OF Mn<C2H302)2-uH 0 AND FeBr

By

Sister M. Charles Joseph Zinn, O.P.

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

1962

Approved flu“: kal-

ACKNOWLEDGMENTS

The author wishes to eXpress her grati-
tude to D. H. Forstat for his kindness in
directing this research. His guidance has
been sincerely appreciated. The author also
wishes to thank Sister Paul Christine, O.P.,
whose help in typing this thesis was so
graciously given. For financial assistance
the author thanks the Arthur J. Schmitt
Foundation, Lemont, Illinois, for supporting
this work. This work was also performed
under a contract with the Office of Naval

Research.

ii

TABLE OF CONTENTS

Chapter Page

INTRODUCTION 0 o o O O O 0 O O O 0 O O O O O O O l

I THEORY o O D 0 O O O O 0 0 0 O O O O O O O O O O 2

Specific Heat . . . . . . . . . . . . . . . . 2
Magnetic Contributions to the Specific Heat . 8
Nuclear Specific Heat . . . . . . . . . . . . 15
EntrOpy Consideration . . . . . . . . . . . . 16

II EXPERIMENTAL APPARATUS O O O O O O O O O O O O 0 l8

Dewar System and Calorimeter . . . . . . . . 18
Preparation of the Sample . . . . . . . . . 21
Circuits . . . . . . . . . . . . . . . . . . 21
Measurement and Control of th Liquid

Helium Bath Temperature . . . . . . . . 23
Vacuum System for Calorimeter . . . . . . . . 24

III EXPERIMENTAL PROCEDURE . . . . . . . . . . o . . 28
IV DATA PROCESSING AND CALCULATIONS . . . . . . . . 31
V RESULTS AND CONCLUSIONS . . . . . . . . . . . . 35
Ferrous Bromide Tetrahydrate . . . . . . . . 35
Manganous Acetate Tetrahydrate . . . . . . . ”0

LIST OF REFERENCES . . . . . . . . . . . . . . . H8

APPENDICES o O O O O O O O O I O O O O O O O O 0 1+9

iii

11.
12.
13.

1”.

LIST OF FIGURES

Dewar System . . . . . . . . . . . . . . . .
Adiabatic Calorimeter . . . . . . . . . . .
Circuit for Crystal Thermometer . . . . . .
Measurement of Heater Voltage and Current .
Circuit for Temperature Control below 2.2°K
Calorimeter Vacuum System . . . . . . . . .

Typical Recorder Trace for Specific Heat
Measurement 0 O O 0 O I C C O O 9 O 9 O O 0

Heat Capacity of Ferrous Bromide . . . . . .
CT2 versus T5 for Ferrous Bromide . . . . .

EntrOpy versus Temperature for Ferrous
Bromide O I O 0 O O O O 0 O O O I 0 O 0 O 0

Heat Capacity of Manganous Acetate . . . . .
CT2 versus T5 for Manganous Acetate . . . .
A Plot of Cmag for Manganous Acetate . . . .

Entropy versus Temperature for Manganous
Acetate 0 O O O 0 O O O O O O I O O O O O 0

iv

Page
19

20
22
22
25

26

32
36

38

39
H1
H2

nu

H6

LIST OF APPENDICES

APPENDIX Page

I. Heat Capacity Data of Ferrous Bromide . . . . 50

II. Heat Capacity Data of Manganous Acetate. . . . 55

INTRODUCTION

This paper is a study of the heat capacity of ferrous
bromide and manganous acetate in the temperature range 1° -
u.5° K in view of the .A-type anomalies which were located
in the heat capacity curves of both salts. An anomaly of
this type corresponds to a second-order phase transition and
is characterized by a large absorption of heat for very
little rise in temperature. The heat capacity curve there-
fore exhibits a well-defined peak at the transition tempera-
ture. The purpose of this paper is to present the data ob-
tained'and to discuss the possible causes of these anomalies.

The paper includes five chapters: a brief deve10p-
ment of the theories of specific heat and magnetic ordering,
descriptions of the experimental apparatus and the techniques
used in taking data, an explanation of the method of render—
ing the data, and a discussion of the results obtained and
the conclusions reached.

Since this energy study is so closely related to the
order-disorder consideration, an estimation of the magnetic
entropy was made for each transition and the result was com-
pared with the theoretical value. Also, an indication of
the degree of order retained above the transition point was
obtained by comparing the entrOpy gains above and below this

pointo

CHAPTER I

THEORY

Specific Heat. When a system, such as a crystal

 

lattice, absorbs heat, a temperature change may or may not
be observed. If a change does occur, for example Tl _. T2
as a small amount of heat AQ is transferred, then we may de—

fine the average heat capacity by the ratio

. A
average heat capaCity = TI%%qfi; (1)

In the limit of very small temperature changes and infinite-

simal amounts of heat, this ratio becomes the heat capacity
- <30
C - 3T (2)

If this quantity is divided by the mass of the sample, the
heat capacity per unit mass or specific heat is obtained.
When determined on the basis of one mole, it is called the
molar heat capacity.

Since dQ is an inexact differential the value of C
will vary depending upon the conditions in which heat is put
into the system. For example, Cv is the heat capacity of a
substance which is being maintained at a cdnstant volume
while Cp is its heat capacity at constant pressure. Ordi-
narily one is more interested in Cv than Cp, since CV =

(§%)v where u, the molar internal energy, may be obtained

2

3
theoretically using statistical mechanics, and hence a check
may be made against theory.

In 1819 Dulong and Petit noted that the product of
the Specific heat of an element and its atomic weight is
almost independent of the element for solid materials. This
implies that the heat capacity per atom is very close to a
constant, from one element to another.

In 1893, Richarz suggested a theoretical explanation
for this fact. [1] Cv for one mole of an ideal gas having
three degrees of freedom was known from Maxwell's Bquiparti-

tion of Energy Principle to be 2R where R is the universal

2
gas constant. Richarz argued that in a solid only half the
energy is associated with the thermal action of the particles,
the other half existing as the potential energy of the par-
ticles considered as bound oscillators. Thus, if %R expressed
the part of the heat capacity associated with the kinetic
energy, then the total heat capacity may be expressed as
Cv = 3R. This equation gives the theoretical eXpression for
the Dulong-Petit Law. Difficulties with this law arose, how-
ever, as experiments showed Cv to decrease with decreasing
temperature. One of the first attempts to give this obser—
vation a basis in theory was made by Einstein in his appli-
cation of the quantum hypothesis of Planck to the problem
of Specific heats. Einstein adOpted as a model a system of
harmonic oscillators having three degrees of freedom and all

oscillating at the same frequency. Planck's expression for

the mean energy of an isotrOpic, three-dimensional oscillator

is,
— 3hw
U = (3)
e—IIQ/kT _ l

 

If we multiply by N1 the number of oscillators in one mole
of a monatomic solid and then differentiate with reSpect to

T1, we obtain the heat capacity,

 

 

c z 9.2 = 3R (1&2 8W“ (I)
V M kT (ehu/kT _ 1,2
Letting :32 = 9 , this becomes,
6 2 ee/T
C = 3R(-) (5)
V T (ee/T - 1)2

9 in this eXpression is known as the characteristic tempera-
ture. For large T, i.e., T>>e, CV -—~ 3R. This is the
classical value which agrees with eXperiment. However, as

T —+ 0, Cv -+ O exponentially, which is not observed
experimentally.

Nernst modified Einstein's expression by postulating
two frequencies, u>and % in order to obtain better agreement
with observations. But he was not able to find any theoreti-
cal basis for his choice of the two frequencies.

Debye extended Einstein's theory by replacing the
single frequency with a frequency spectrum. [2] In place of
many independently vibrating atoms, Debye adOpted for his
model a continuous medium capable of supporting a set of
elastic standing waves. The frequencies of these waves range

up to a maximumcgm corresponding to a wave length of the

5
order of the interatomic distances. We will first apply
this simplification to the case of a one-dimensional crystal
and then extend the method to three dimensions.
The internal energy of the one-dimensional crystal

may be expressed from quantum theory as,

-2 ““k
u - km (s)

e kT - 1

where the summation is over all normal modes of vibration.
A quantized lattice vibration is commonly referred to as a
phonon. For a large number of oscillators we replace this

sum by an integral,

 

I103
-

where w(k)dk is the number of modes between k and k + dk.

For a one-dimensional line of oscillators there is one mode

for each interval Ak = E where L is the length of the line.

L

Therefore,

 

w

Debye used the continuum approximation, us: vok, which is

valid for a homogeneous line. In this approximation 31:; is

simply %—, the reciprocal for wave velocity. Therefore,

0

L c0m
_ m k
U ‘ "TVOJ‘O ehm/kT _ l d“ (9)

6

The fact that the line is actually composed of discrete
oscillators is brought into the theory in order to limit the
number of modes and thereby the range of frequencies. In
actual substances the order of magnitude of the limit on k
islkm|='§:3108 cm‘l where a is the lattice constant. [3]

In order to determine the w(k) for the three-
dimensional case we apply the method of periodic boundary
conditions, requiring periodicity of the vibrational wave

at the boundaries of a cube of side L. Then ka’ k L, and

Y
kzL must be multiples of 2fl‘so that a cubic lattice may be

constructed of lattice constant fiflzin which the points re-

present the allowed values of k. The number of states with

wave number less than Ikl is represented by the volume of a

3
sphere of radius Ikl, measured in units of %E i.e.,(ufljfl: .
2 L

3k2
2112

The appearance of the factor 3 is due to the exis-

Therefore, for one unit volume, w(k) =

tence of two transverse and one longitudinal modes of vibra-
tion in an isotrOpic medium. Therefore, we may write the

internal energy per unit volume as,

km 2
: -———AL— «gunk—I—
U f0 ehw/kT .. 1 2W2 dk (10)

or using the Debye or continuum approximation, a): vok,

O
u : ____3h m_______w3dw (11)
2W2V03 0 ehQ/kT _ 1

Since the total number of modes must equal 3N, where N is

the number of atoms per unit volume, we may determine the

upper limits of km andcum,

3
m = 3N (12
211

7?

|\.)

Rewriting Equation (11) and letting x =‘%%’
318%” med
z MI “Ji— <13)

---- - -- (611’2N)l/3= $- (11+)

This equation defines the Debye characteristic temperature 9.
Differentiating with respect to temperature we obtain the

heat capacity,

XIII
c = 9Nk(I)3 955145-—- (15)
V

9 0 (ex- 1)2

At T>>9 this eXpression approaches the classical Dulong and
Petit value quite well. At very low temperatures, the upper

limit in the above integral approaches infinity and we have,

x3ch 1TH
- - l -
f - 6r ( ) 5 T 5 (l )

where5°(H) is the Riemann zeta function. Thus for T-<< 6

u
U = 3";ng1‘ (17)
and
- Q! - 12 I. 3 - I. 3
cv - dT - 5n“Nk(o ) - 23uNk(e ) (18)

This is the well-known Debye T3 approximation for low temp-

8
eratures. It may be seen from this equation that the slope
of the graph of cv versus T should approach zero as T —+.0.
Both this prediction and the T3 variation of cv agree well
with eXperiment in the temperature range 0° tog%o.

Magnetic Contributions to the Specific Heat. The

 

above consideration of lattice vibrations represents only
one of several ways in which the crystal system can absorb
energy. Other sources of absorption are due to (1) free
electrons, (2) anomalies which occur with ferromagnetics and
anti-ferromagnetics at the Curie and Neel points, (3) anoma-
lies with superconductors at their transition temperatures,
and (4) nuclear interaction with the electronic field. The
first of these, the free electron contribution, is assumed
negligible in the cases of the two salts considered in this
investigation because the salts are dielectrics. The super-
conductivity consideration is also non-applicable here. How-
ever, both salts exhibit anomalies in the specific heat curve
which seem indicative of magnetic order-disorder transitions
and one of the salts, the tetrahydrate of manganous acetate,
might possible indicate a nuclear contribution to the speci-
fic heat in the region T<?2.6°K. These effects are observa-
ble because they occur in the region where the T3 lattice
contribution to the Specific heat is quite small.

A substance which has no net magnetic moment until
an external field is applied is said to be paramagnetic.
Among the various systems exhibiting paramagnetic properties

are all free atoms and ions having an unfilled inner shell,

9
such as those of the transition and rare earth elements. In
the elements of the iron group the third shell which is re-
sponsible for paramagnetism is on the outside of the atom in
the ionic state. This shell is strongly affected, therefore,
by the powerful electric fields produced by neighboring ions
and the dipole moments of water of hydration in the crystal.
There are two effects of this interaction [U]: a) The de-
struction of the L°S coupling, so that the states are no
longer specified by their J values and, b) the Splitting of
the 2L + l sublevels belonging to a particular L. When an
electric field is directed toward a fixed center, such as the
nucleus, the plane of the orbit of the electrons remains
fixed in space and thus the components of the orbital angular
momentum are constants. In quantum theory, only one of these
components, usually L2, and the square of the total orbital
angular momentum L2 are constants in a central field. How-
ever, when an inhomogeneous field is introduced, the plane
of orbit will no longer remain fixed and the angular momentum
components may average to zero. If Lz averages to zero, it
is said to be quenched. AS a starting point for a theory of
paramagnetic substances, it is usual to regard the orbital
angular momentum of the electrons to be quenched by the
crystalline electric field. Therefore, the magnetic ions are
in S-states, with L = 0 and S = total Spin quantum number.
If the ions of the lattice are sufficiently separated so that

their mutual interaction is negligible, then in the presence

of a field H the induced moment is [5]

10

M = NSg(pP)BS(2-Ifig-%§-) (19)

is the Brillouin function,

where P9 is the Bohr magneton, BS

and N is the total number of magnetic ions in the sample.
If Lfifi%i<<<l, then this expression reduces to a simpler form

yielding

 

X = g : mfl SQS + 1) (20)

3kT

This is the Curie law for paramagnetism.

A substance is said to be ferromagnetic if it
possesses a net magnetic moment even in a zero external
field. It is useful to consider it a paramagnetic in which
some internal interaction causes partial alignment of the
ionic moments. Weiss (1907) first postulated such an inter—
action and thus it is called the Weiss field or the molecular
field. The Curie point, Tc, is the temperature at which the
Spontaneous magnetic moment disappears. Therefore, one may
conclude that at this temperature the energy of the Weiss
interaction with an atomic spin is of the same order of mag-
nitude as the thermal energy of the atom.

The Weiss field, HE, is assumed to be proportional

to the magnetization,

Above the transition temperature the Curie law holds if the

magnetic field be taken as the sum of the applied field and

11

the Weiss field,
M - Q
- T (22)

This is the Curie-Weiss law for paramagnetics above the
Curie point where.A.is called the Weiss field constant. The

susceptibility above the Curie point may then be expressed,

- __£___
X-T_TC (23)

Heisenberg showed in 1928 that the Weiss field
arises from the quantum-mechanical exchange integral. Under
certain conditions it can be shown [5] that the energy of
interaction of atoms i, j of the lattice having spins Si, Sj

contains a term,

wex = 'ZJgi’gj (2”)

where J is the temperature dependent exchange integral which
arises from the overlap of the charge distribution. It is
the difference in Coulomb interaction energy when the spins
are parallel or antiparallel.

We would like to relate the exchange integral and
the Curie point. If an atom has 2 nearest neighbors all
interacting with it, with exchange integral J and if J = 0
for all other neighbors, then we may express the interaction

energy,

wex '=‘ .4st2 = —g§_hHE = -g§|..leA(g§_|1°fl'l) (25)

where g = Spectroscopic Splitting factor ( = 2.00), Fe ::

l2

 

—22, the Bohr magneton, and.a.= the atomic volume. Then,
2mc
since N = {l1 and

1 _ g_ , NgZS (S + 1) 2

.A.- TC 3kTC *#T_' (26)
we have

 

2 2 3kT. (27)
MAE”: -
2 11 228 (S + l)

The paramagnetic-ferromagnetic and the paramagnetic-
antiferromagnetic transitions are characterized by the above
exchange integral, J. The atoms,originally aligned with
their spins parallel or antiparallel, absorb energy and
attain a random orientation. Within a very Short temperature
range, a very large number undergo this transition and thus
a sharp peak occurs in the specific heat curve.

When the exchange integral of Equation (2”) is
negative, anti—ferromagnetism results, the spins of neigh-
boring atoms locking into antiparallel orientations as the
temperature drops below the transition temperature known as
the Neel point.

Rewriting Equation (2%) in Operator notation,

vop = 2|fi25§i0p.§j0p (28)
10

where summation is over distinct pairs of nearest neighbors.
The simplest example of antiferromagnetism arises when the
crystal lattice may be divided into two interpenetrating sub-

lattices A,B so that all the nearest neighbors of an ion on

l3
lattice A lie on lattice B. VanVleck carried out the calcu-
lations for the special case in which interactions are be-
tween nearest neighbors only. [6]

The operator notation may be written in the form

VOP = IJIZ Siopo§<§flop> (29)
i n=l

where‘<§nofi>is the yet undetermined expectation value of the
Operator SnOP of one of the z nearest neighbors to atom i.
Each atom is now effectively in a molecular field,

S o

-2|J|:§:-n p

h=l gym

Taking the direction of this field to be the z direction

(30)

 

(shop) = ($13,013) = 0 (31)

And SiZOP has expectation values S, 8-1, ... -S+l, -S.
VanVleck restricts himself here to crystal lattices
for which none of the nearest neighbors of an atom are near-
est neighbors of each other. For a system in thermodynamic
equilibrium, the mean value of S2 is the same for all atoms
on sub-lattice B, and the average moments of the atoms on
the two sub-lattices are equal and opposite. VanVleck Shows

that the molecular field on an atom of sub-lattice A is

l 2|J|

 

(32)

To determine the variation of S with temperature we must
substitute this eXpreSSion in place of H in Equation (19).

This yields,

1H

2|J|zSS
-__ET_-

S = SBSE J (33)

The solution S of this equation is S as T -+ 0 but falls to

0 as T -+ TC. The physical meaning here is that at the abso-
lute zero all the Spins on one sub-lattice will be perfectly
aligned pointing in one direction and all those on the other
sub-lattice will be oppositely oriented. AS the system ab-
sorbs energy its disorder increases and when the critical
temperature is reached all long-ordering completely vanishes.
Above TC, i.e., in the paramagnetic state, S = 0. From

Equation (33) we have

- -NIJIZSZ (3a)

m
I

Therefore,

a
In.

= -2NIJIz§ (35)

min.
H ["1

CV = d

H

This may be shown [7] to reduce to

1611‘2 '62 a
c = Nk[-—--] = —— (36)
V 271“2 T2

in the paramagnetic state, where t is the characteristic

temperature defined by

 

2
’t = 2.4921118 <3 + 1) <37)

Coupling the expressions of Debye and VanVleck we may write

for the heat capacity above TC

Cv = I7 + bT3 (38)

15
where the first term gives the magnetic contribution and the
second refers to the lattice heat capacity. In the region
of interest in this paper, the second term remains at all
times negligible, within the limits of experimental error,
since b is quite small for each of the two salts being
considered.

In the case of ferromagnetic materials, Spin wave
theory predicts a contribution to the specific heat propor-
tional to the temperature to the three-halves power. [8]

The values obtained for the heat capacity of manganous
acetate in the present study seem to give good agreement with
this "T3/2 law."

Nuclear Specific Heat. In their work on the Specific

 

heat of ferrites Pollack and Atkins [8] report a nuclear con-
tribution to the Specific heat which is proportional to %7,
in the case of cobalt ferrite. This nuclear heat capacity
comes about because the Co59 nucleus has an intrinsic magnetic
moment, and therefore interacts with the hyperfine field at
the nucleus due to orbital electrons. Pollack and Atkins
suggest that it may be possible to observe this effect in
manganese ferrites. Although manganous acetate is not a
ferrite, an attempt has been made to relate the rise in the
Specific heat below 1.H°K to a nuclear Specific heat con-
tribution.

Another possible reason for this rise in the heat
capacity observed at very low temperatures for manganous

acetate is that there may be another transition below 1.2°K.

16
The work of Dzialoshinskii [9] has demonstrated the possibi-
lity of an intermediate state existing between the true anti-
ferromagnetic state and the paramagnetic state. This inter-
mediate state comes about when, instead Of attaining an
ordered state in one step as the temperature approaches abso-
lute zero, certain substances seem to have a first transition
in which the spins of the atoms on one sub-lattice shift into
canted positions with respect to those of the other sub-
lattice and then a second transition occurs in which com-
plete order ensues. Thus it is apparent that this inter-
mediate state will give the appearance of being a weak type
of ferromagnetism, since because Of the canting, atoms Of
one sub-lattice will always have some components of spin
that cannot be cancelled out by the oppositely directed spins
of atoms on the other sublattice. Moriya eXplainS the
physical interactions responsible for canting by suggesting
two distinct mechanisms which may give rise to this effect.
[10, 11] The first mechanism Operates because of the dif-
ferences in the single spin magnetocrystalline anistropy for
differing Sites. The second mechanism is through an anti-
symmetric exchange coupling.

Entropy Consideration. From statistical mechanics,

 

one Obtains the following expression for the change in
entrOpy of a paramagnetic system at low temperatures and

zero magnetic fields, [12]

A4 = R 1n (28 + 1) (39)

17

where R is the gas constant, and S is the ground state spin
of the paramagnetic ion. From this equation we obtain a
theoretical value for the entrOpy which is to be compared with
our calculated values.

We are also interested in the amount of entrOpy
change occurring above the transition temperature as com-
pared with that taking place below TC. This gives us a
measure of the degree of order still remaining in the

crystal above the transition point.

CHAPTER II

EXPERIMENTAL APPARATUS

Dewar System and Calorimeter. The specific heat

 

measurements are made in an adiabatic calorimeter, which is
mounted in a conventional double Dewar system. This system
consists of one Dewar flask, 90.7 cm long, (for liquid
helium)mounted inside a second Dewar flask, 92 cm long, (for
liquid air or nitrogen). Both Dewars are strip Silvered,
for convenience in viewing the liquid levels. Figure 1
shows this Dewar arrangement. Attached to the top of the
helium Dewar is a two-inch COpper tee. The side arm of this
tee is connected to the liquid helium pumping system, while
the top arm is used for mounting Of the calorimeter.

The adiabatic calorimeter is shown in Figure 2. The
calorimeter can is made of COpper, one mm in thickness, (h =
15.9 cm; d = u.0 cm). The top plate, labelled A ianigure 2,
is held to this can by eight 6/32" screws. To ensure a
vacuum seal (at helium temperatures), a fuse wire "0" ring
is used. The leads from the thermometer and heater on the
sample are brought out of the calorimeter through four
tungsten glass seals which are mounted on the tOp plate Of
the calorimeter. A 1/2" stainless steel tube is also
soldered to the tOp plate for evacuating the calorimeter.

A small radiation shield, D, is incorporated into the

18

19

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L
to I
pump
5.” cm
I
ll
glass 1 ( I
flange L_A J
'I
6.Hcm
4 I
N
\
x
\
\.
\
\
\
\.
\
\
vacuum -————4 \
liquid air \
vacuum
\
liquid He \ 92 cm
s \
K N
N x
\ x
N \
\ \
Figure l. Dewar System .1,

 

2O

 

 

 

to
ionization fl
gauge I
‘\ f for admission of
-:j I liquid helium
to pump <———
to manometer H——\\ I

 

B ———+L I

I

 

 

 

 

7H cm

 

 

 

 

=—\_/ /

 

 

 

[£“1JLILr-—1[ lower view

15.9 cm

 

 

 

e‘u.1 ‘I
cm

Figure 2. Adiabatic Calorimeter

21
pumping line to reduce the heat leak into the calorimeter.
The pumping tube is connected at the top of the helium Dewar
to a brass flange, labelled B in the diagram, which in turn
is sealed to the top of the c0pper tee by means of a neoprene
"O" ring. An ionization gauge (NRC - 507) and a vacuum valve
(Veeco: 1/2") are connected to the pumping tube above the
brass flange. Several other Openings in this flange are used
for introducing the liquid helium into the Dewar, for taking
out the electrical leads, and for connections to the pressure
manometers.

Preparation of the Sample. The sample, a single

 

crystal, is grown from an aqueous solution at room tempera-
ture; usually it weighs between 0.5 gram and 2 grams. A hole
is drilled through the center of the crystal and a small car-
bon resistor is inserted to serve as a thermometer, (56 ohms,
Type TR, l/10 w, Allen-Bradley). Wrapped around the sample
is approximately one foot of fine manganin wire which serves
as a heater wire, (H00!1/ft., enamel insulation). The leads
to both resistor and heater are of copper, teflon insulated,
(A.W.G. 3A). The sample is coated with glyptal to prevent
its dehydration and also to fix the heater coil in position.
It is mounted inside the calorimeter as Shown in Figure 2.

A nylon thread, passed through a small hole in the radiation
shield, provides for suspension of the sample.

Circuits. The temperature of the crystal is deter-

 

mined by measuring the voltage drOp across the carbon

resistor. (See Figure 3.) This provides a direct indication

22

 

 

10011

 

 

l K-2

J Potentiometer

 

 

K-3
Potentiometer

 

@

RT

 

§__l

___l

 

 

Figure 3. Circuit for Crystal Thermometer

 

 

 

: I

 

1-1 {HI

"\

Potentiometer

 

 

 

\ W—

 

 

W

Heater

 

Figure u.

 

Std: 10011

 

 

Measurement of Heater Voltage and Current

23
of resistance since a lOPa current is maintained in this cir-
cuit at all times. Voltage is read on a Leeds and Northrup
Type K-3 Potentiometer, used in conjunction with a L 8 N
Speedomax Recorder.

The bridge circuit for providing a steady lOPa
current is shown in Figure 3. With the L 8 N potentiometer
K-2 set at 10-3 volts, the variable resistor Rv is adjusted
until the galvanometer (a high sensitivity wall galvanometer:
L 8 N No. 2285) in the output of the potentiometer circuit
shows zero deflection. This adjustment is made before each
voltage reading.

In order to calculate the heat input to the sample
in a given interval of time, it is necessary to know the
voltage drOp across the heater coil, the current in it, and
the time during which the sample is heated. The circuit for
this measurement is shown in Figure H. When the DPDT switch
is in position 1, the potentiometer (L 8 N Type K-l using a
No. 2M20-B Galv.) reads the drop across the heater. In
position 2, it reads the drOp across a lOOIlstandard resistor
in the heater circuit. Since the current is the same for
both the resistor and the heater, this voltage reading is
readily converted to current. The time of heating is deter-
mined by means of a stOp watch which can be read to 0.1
second.

Measurement and Control of the Liquid Helium Bath

Temperature. TO lower the temperature of the liquid helium

 

bath below u.2°K, it is necessary to reduce the vapor

21+

pressure of the liquid. This is done in the present experi-
ment by using a large capacity vacuum pump, (Kinney KDH-130).
The vapor pressure is read by a conventional Hg manometer for
temperatures above the Arpoint (2.2°K), and an octoil
manometer for temperatures below the A-point (this increases
the reading scale by a factor of approximately 1”). These
vapor pressure readings are converted to temperatures using
the 1958 He” temperature scale. [13]

In order to maintain a constant temperature Of the
bath for purposes of calibration, two methods are used.
Above the.1rpoint, control is achieved by adjusting the
pumping speed of the vacuum pump. A set of coarse and fine
adjustment vacuum valves is used. Below the.2:point, because
of the large thermal conductivity of the liquid, it is pos-
sible to use a small wire wound heater (a noun) in the bath
together with a carbon thermometer (220flw 1/2 w, Allen-
Bradley). Figure 5 shows the circuit used for this method
of control. By first adjusting Rv SO that the galvanometer
reads null, the current through the HOOIIheater is continually
controlled in order to maintain a galvanometer null. Tempera-
tures can be controlled in this manner to at least 0.001°K.

Vacuum System for Calorimeter. The two branches of

 

the calorimeter vacuum system, one for the high vacuum side
and the other for roughing purposes, are diagrammed in
Figure 6. The pumps employed in this system are a Welch
Duo-seal pump and a conventional oil-diffusion pump. The

latter is protected by a liquid air trap. Pressure readings

25

 

 

  
 

Resistor in Bath
( Thermometer )

 

 

 

 

 

 

Figure 5. Circuit for Temperature Control Below 2.2° K

26

T-C
Gauge To
Calorimeter._——»

 

 

 

 

 

 

 

 

 

Liq.
Air
Trap

 

 

 

 

 

 

'+L.Oil Diffusion

Pump

 

 

 

 

 

 

 

 

T-C

Gauge I2:

 

For

Admission of
Exchange Gas

 

 

 

 

 

 

 

 

 

 

 

To Fore-pump

 

I
Ballast I

Figure 6. Calorimeter Vacuum System

27
are taken in the two branches by means Of two thermocouple
gauges, type 501. The vacuum system is connected to the
COpper tee through a brass bellows allowing a degree of
flexibility at this joint. The positions of the five valves
in this system may be seen in the diagram.

The samples used in these experiments were Single
crystals weighing between 0.5 — 2.5 grams and measuring
approximately 1.0 - 1.5 cm on a side. The chemicals used
to prepare the aqueous solutions from which they were grown
were obtained from the J. T. Baker Chemical Company and were
of analytical reagent grade. The ferrous bromide crystals
were of hexagonal type and were dark brown in color. The
manganous acetate crystals were monoclinic and were pink in
color. An analysis was made of one of the ferrous bromide
crystals by the Schwarzkopf Microanalytical Laboratory,

56-19 37th Avenue, Woodside, New York.

CHAPTER III

EXPERIMENTAL PROCEDURE

When the sample has been suspended from the top of
the calorimeter as explained in Chapter I, the calorimeter
can is screwed securely into place. The "0" ring, as shown
in Figure 2, seals the can against liquid helium. The
entire apparatus is then sealed inside the Helium Dewar Of
Figure l and the valve to the calorimeter is closed to pre-
vent dehydration of the sample while the rest of the system
is being pumped down. After removing the air from the
Helium Dewar, gaseous helium is admitted and then pumped out.
This is repeated several times in order to remove the last
traces of air, and then the helium is finally left at atmos-
pheric pressure. The outer flask is filled with liquid air
in order to pre-cool the calorimeter system. It usually
takes several hours for the calorimeter to reach liquid air
temperature. It is the usual practice in the present eXperi-
ment to allow the pre-cooling process to continue overnight.

The calorimeter is evacuated after liquid air tem-
peratures have been reached (this is evident by the resis-
tance reading of the carbon thermometer on the sample which
has increased to approximately 60 ohms) and a good vacuum
obtained. Helium gas, at a pressure of 1 mm Hg is then ad—
mitted to the calorimeter to serve as an exchange gas during

28

29
the calibration process.

The transfer of the liquid helium is carried out in
the following way: a large (25 l.) flask containing the
helium is placed on a hoist next to the experimental appe-
ratus. A long transfer tube is simultaneously lowered into
both flasks while the hoist is raised. Then helium gas is
pumped into the portable flask in order to force the liquid
helium over. When the inner flask is full, the helium gas
flow is cut off and the transfer tube extracted.

It will be recalled that the carbon resistor was
embedded in the sample to serve as a thermometer. To carry
out the calibration of this resistor, it is necessary to know
its resistance at fixed temperatures. Usually about twelve
calibration points are taken, starting from H.2°K and going
down to approximately l.2°K. It is assumed that the sample
is at the same temperature as the bath during this calibra-
tion, due to the presence of the exchange gas inside the
calorimeter. In practice, the manometers are connected into
the system just after helium transfer and a reading taken at
atmospheric pressure. Then the cooling is started by opening
the valve to the Kinney pump to increase the rate of evapora-
tion of the bath. By controlling the pumping speed, tem-
perature readings are taken at about 10 cm (Hg) pressure
intervals. This consists of reading the manometers, and
voltage drop across the carbon thermometer. The voltage
drOp is a measure of resistance Since the 10p; current in

the thermometer is held constant at all times. The vapor

30
pressure readings of the manometer are converted to tempera-
ture by using the helium vapor pressure tables. [13]

The lowest reading taken on the mercury manometer is
usually about 5 cm. Then the oil manometer is connected to
the system and is used down to the lowest temperatures. When
the calibration is complete the He exchange gas is pumped
out of the calorimeter.

Specific Heat readings are taken in this manner: the
recording potentiometer is set in the zero position and the
voltage is noted. A short period elapses during which the
temperature is slowly rising due to the heat leak. When the
trace is sufficiently long to establish the lepe of this
heat leak curve, power is turned on in the heater of the
sample. A stOp watch is used to determine the time of power
input to the nearest 0.1 second. Readings of voltage,
current, and time are made and another interval of heat
leak recorded. The method for subtraction of the tempera-
ture rise due to heat leakage during the period of power
input will be eXplained in the next chapter. This process
is repeated for as many points as possible in the range
under investigation. During an average run, between sixty

and seventy points are obtained.

CHAPTER IV

DATA PROCESSING AND CALCULATIONS

Calibration. All the pressure readings are eXpressed

 

in terms of cm Hg, those taken on the oil manometer being
multiplied by the ratio of the density of the Oil to that of
mercury. These are converted to temperature readings by ref-
erence to the Helium Vapor Pressure tables mentioned in
Chapter II.

Since the carbon thermometer acts like a semiconduc-
tor in the liquid helium region, one would expect that its
resistance would vary with the temperature as follows:

R = Aea/T where A and a are constants. By plotting log R as
a function of l/T, a straight line is Obtained. This curve
may now be used to obtain temperatures and temperature dif-
ferences caused by power input. Since the carbon thermome-
ter's resistance is not reproducible, it is necessary to take
a separate calibration curve for each run. Figure 7 shows

a typical record of a fore, during, and after heating cycle.
The steeper regions, between (1) and (2) and between (3) and
(H) called the fore and after heating periods respectively
indicate the rate of heat leak, while the mid-section of the
curve (between 2 and 3) shows a more rapid temperature rise
which is due to the heating of the sample by the power
input. The voltage readings cannot be taken at points (2)

31

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time ?

 

 

 

 

 

 

 

 

 

(\Ik
VI

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ Temp. (-V)

Figure 7. Typical Recorder Trace for
Specific Heat Measurement

 

 

 

 

 

33
and (3) directly, however, since a part of this temperature
rise is due to the heat that leaked into the sample while
power was being supplied. Therefore, to subtract this con-
tribution, we extend both heat leak regions as shown in the
figure. Then lines a and b parallel to the voltage axis are
drawn through points (2) and (3) and their median, line c, is
located with a divider. The points of intersection of this
line c and the extensions of the heat leak curves are taken
as indications of the actual temperature rise due to power
input. This is equivalent to extrapolating back to the mid-
point of the heating period.

To obtain the voltages corresponding to these points,
we must know what the voltage was at point (1) and the cali-
bration of the chart in volts/div., characteristic of this
temperature. The former is already known since the voltage
on the K-3 potentiometer is read at the start of each new
point and the latter may be found from the calibration curve
of the potentiometer. Thus, by counting the scale divisions
from the left time axis to the points of intersection, multi-
plying by the proper volts/div. value,and subtracting from
the voltage at point 1, the voltageS--and therefore the
resistances of the thermometer at the points of intersection--
are obtained. The lOgarithms of these resistances are then
plotted on the calibration curve and the corresponding l/T
values recorded. Temperatures are thus determined, and by
subtraction we know the AT for each power input. The average

of each pair of temperatures is also obtained as we assume

3..
the value of specific heat will be approximately constant
over this small interval. The heat capacity of the sample
at this average temperature may then be calculated by divi-
ding the total heat input by'AIB The heat input is Obtained
by multiplying the voltage and current readings of the

heater with the time of heating.

CHAPTER V

RESULTS AND CONCLUSIONS

Ferrous Bromide Tetrahydrate (FeBPZ'uH2O)- The cal-

 

culated values of T,t>T, and C for five experimental runs on
the ferrous bromide crystals may be found in Appendix I.
The values of heat capacity have been corrected for energy
dissipated across the leads and for the heat capacities Of
the thermometer and the glyptal. These losses amounted to
less than one per cent of the total energy input. A plot of
the heat capacity as a function of temperature in the range
1.16°K - H.20°K is shown in Figure 8. The sharp peak occur-
ring at l.28°K may well be indicative of a magnetic transi-
tion although no confirmation of this has been obtained by
other methods. However, some indirect evidence exists from
the magnetic susceptibility work on the isomorphous crystal
of FeClzoquO. [1%] It will be noted, however, that the
curve does not drOp to a constant value above the transition
temperature immediately but has a small tail extending over
about 0.2°K. This seems to indicate a slow diminution of
the short-range ordering above the transition temperature.
It is of interest to calculate the entropy change
taking place during this transition. Using the method de-
velOped by Friedberg [15], and Kapadnis and Hartmans [16],
we may evaluate the changes in the entrOpy above and below

35

ovfleonm muoHAOh.Ho huaowmwo poem .w onumfih

 

oh
...—... “ OWN m _ _ b 00PM _ _ L _ OWN . . _ _ _ 00H
1 . q 1 a 4 q q _ q _ —
o
x x ...-n... .x x.+x+x+ .u. . o o x X.“ 0.. .0 «4:3 *6. 0 In
x I If i ...“... I... I an o ‘00. o W. s.m0.° .
+ +++ ++ + .... +1.. «.... «aura... +x+£. 9. MM!” 00+! +W+$ J... shaggy-flea
dMMw .. N
....
x 0 1..
o %
..m ..
O
a .. .
000 1|
0 II
o o
O
%
. SS .2 Esme -- m
x «boa .oH mundane
e
. News .0 sumsnas
mo .Mnoa
. Homa .s nonaooon A Mao
o

33 .H 93503

 

37
the critical temperature separately, in order to observe the
total magnetic entrOpy change. We assume that above TO the
magnetic heat capacity varies inversely with the square of
the temperature while the lattice heat capacity varies
directly with the cube of T in accord with Debye and VanVleck
as given in Equation (38) where a and b are the spin and
lattice heat capacity constants. These are seen from the
CT2 versus T5 curve of Figure 9 to have the values,

a = 3.05 911L933; b = 0.078 cal ——. Above To, there-
mOle mole odegu

 

fore, we may obtain the magnetic contribution to the entropy

change by evaluating the integral,

no N
A} = (5111‘ = M—gndi‘ (00)
To i.u T

The value of T0 was determined from the plot of Figure 9,

by noting the temperature at which the curve deviates from
linearity. This integral yields a value of 0.79 cal/mole-deg
for the magnetic contribution to the entropy above To.

Below TO a plot of C/T versus T was made and the
entrOpy contribution from 0°K.to l.u°K.waS obtained graphi-
cally. We assume here that the lattice contribution to the
specific heat in this region is negligible compared to the
magnetic contribution. This plot is shown in Figure 10 and
measurement of the entropy contribution yielded the value
1.85 cal/moleodeg. Combining the magnetic contributions
above and below the TO therefore, we obtain the total entrOpy
change for this transition 1.85 + 0.79 = 2.69 cal/moleodeg.

This is seen to be in fair agreement with the value obtained

38

X

0.0.28.5 36.30% you

9 SEE

on

. m newsman

“O

O

 

 

SS .3 E835
nos” .3 Sam...
32 .m snags

_

a
11
l
m
m

 

Bo

39

31$ B

0330.8 mayhem no.“ onspmhomSoa admnob .39“an .oa oBmHm
m2” 0..” m.
_ _ 1 _ A lq _ u 4 I a

 

 

 

00
from the theoretical expression: AJ = R ln(2S + l) in which
the ferrous ion (Fe++) is taken to have Spin (S) = 2:A.J=
R ln5 = 3.19 cal/mole-deg. It should be pointed out that
about thirty per cent of the entropy change occurs above To,
indicating the persistence of short—range ordering above To.

Manganous Acetate Tetrahydrate [Mn(C2H302)2oHH20],
The corrected values of T}.AI'and C for the five experimental
runs taken on manganous acetate samples are shown in Appendix
II, and a plot of these data may be seen in Figure 11. A
gradual rise in the heat capacity, from about one cal/mole-
deg at 2°K.to about three cal/mole‘deg at TC 3.17°K, indi-
cates a second—orderuA-type transition. TC is believed to
be the Curie temperature for manganous acetate, the salt
being weakly ferromagnetic below this point, as reported by ;
Flippen and Friedberg. [17]

As in the case Of the ferrous bromide, the behavior
of the heat capacity above TC may be described by the equa-
tion C = $7 + bT3o This is verified by the CT2 versus T5
curve shown in Figure 12. The constants were found from
this curve by the method of least squares: a = 15.31;

b = 0.00363. (The solid line appearing above TC in the C
versus T plot of Figure 11 is a plot Of the above equation.
Below TC the heat capacity does not approach zero as T —+-0
but drops smoothly to a value of approximately one cal/mole-
deg at 2°K, remains constant there for about 0.6°K, and then
begins to rise again as T —+ 0. Pollack and Atkins found a

similar rise below 3°K with cobalt ferrite [8] and following

093004 Handmade Mo .mfiodmdo poem ...H 093m

 

 

 

 

. $2 ... ens
. $3 ...N genes . 33 .8 82.

x 32 .3 s3. . 3% .mm 23h.

 

 

 

3334 Sandman: mom me 3693 mac .3” Penman
o
83 83 08 .m.
_ 4 .I . _ d a a _ A q _ A.
.
.
.
.
.
.
_
_
x x x X
XX (Xx «KthJNX-XX(
( K I I} .. X
”KN Kx x x I “x l\ k X“ H X x “I!
p. X i i I K K K
I14 ‘XX x ‘ X XX X X

1+2

 

on

90

H3

the suggestion of Bleaney [18] and Marshall [19] have shown
this heat capacity to be inversely prOportional to the
square of the temperatureo It is believed that this con-
tribution is the nuclear heat capacity arising from the inter-
action between the nuclear magnetic moment and the magnetic
field set Up at the nucleus by the orbital electrons.
Pollack and Atkins predicted that a similar effect should be
observed in the case of manganese salts. The present experi-
ment seems to lend some support to this theory.

When the quantity E§97 is plotted against T3/2 as in

Figure 13, a linear graph is not achieved. However, if suc-

 

cessive values of lg-are subtracted from the experimentally
T
. . cv - F/TQ. .
determined Cv's and the quantity T3/2 is plotted against

T3/2 a well-defined straight line results. The smallest de-
viation from the linear plot was obtained by choosing for
the constant x*the value l.u6.

Reading the constants from the graph, we may write

for the linear equation,

= [.1uT3/2 + .029T3] ———99i-— (ul)

Cmag mole-deg

When this eXpression for the magnetic heat capacity is plotted

as a function of temperature and the equation,

1.U6 cal

Cnuclear : T2 l:mole-deg] (”2)

is plotted on the same graph, the sum of the two curves

yields the observed heat capacity versus temperature curveo

uu

opmpoo¢ msonwmnds now mdao no poam ¢ .MH on:mam

was i

‘r
i

1.
"x?
*r

-L-

X.
X
K

. m
grids Ame $0.0 + Qmaflov u use

. hdoaoon
AMKHMO v NB\©¢3H N o

cur-N
-b

db-

 

HS

Prior to the calculation taking into account the nuclear
contribution, several analytical functions were tried in an
attempt to fit the eXperimental curve. Below TC the heat
capacity was found to vary with T according to the equation
1DCT2=Ae‘B/T where A = 8.935 and B = 3.6Q°K. Further,
the entropy curve obtained from the experimental data is
shown in Figure 1”, together with the theoretical magnetic
and nuclear contributions. Again, to a good approximation,
the entropy change calculated from the measurements is equal
to the sum of the other two, except for the slight disagree-
ment in the range l.2°K.» l.3°K.

A calculation of the entrOpy change in this tran-
sition was carried out, using Equation (38) for the change
above TO and Equation (kl) for the magnetic part of the

entropy change below To. For the change above To, we have

3° 00

AJ=J %=J§3dt (1+3)
T 3 3T
0 ° _ ‘

The values of a and T0 were obtained from the CT2 versus
TS graph of Figure 12, and the lattice term is again ignored
in the integration as in the case of ferrous bromide, due to
the small value of b (0.0036). This integral yields an
entrOpy change of 0.70 cal/molecdeg.

Below To, we may eXpress the entropy change by the
integral,

3.3
AJ =f 3°3.1uT1/2dT +f .029T2dT (an)
0 0

3334 323mg: no.“ 0932359 mumnob hmonvnm .3 Pun—mam

 

O o .
A mov a a o.m 9m 04 . o
ILI.I._I+I_..I!._FI... _ _ _ _ _ _ a _ _ _ a _ .
- ' XI... '1 ' 'K’
_ ’ x,
. I. 1
a . . I
+ b ‘ — ... + ‘1
§ 8 £10 I t+ x 1. x. c. l I.
O O p. 0 £3.64 “vfix 0...“ 010%”?! “a... /
oioo~.343o of...... v
40! $ 0...... c: to... Cg I
C. O . ‘ C 0+3. K 0000‘ o I I
0.... 6 0 +5. s o o r/
. nae. /
++ ..... I 1
Duo 9 v...;/
.L
. . .. oA
. z
w ..
ma ,
34 u o M I
,
I
X I.
,
+ 33 K has ,
. 32 Km gamma . $2 .3 was
,_ so

 

. 33 .3 33. x 3% .mm 25s

u?
yielding for the change in entropy the value 0.91 cal/mole
deg.

Combining the entropy gains above and below To, we
obtain 1.61 cal/moleodeg. This differs from the theoretical
value, calculated by means of Equation (39), by more than a
factor of 224.1: R ln6 = 3.58, taking 5/2 for the ground state
spin of the Mn++ ion.

This poor agreement in the entrOpy calculation may
lend support to the possibility that another transition
exists below l.2°K as was discussed in Chapter I. In other
words, the entrOpy change for the lower transitions must be
added to the above, which may possibly improve the agreement.
This argument is made in spite of the fact that the curve
fitting suggests a nuclear contribution.

In summary we have, in these experiments, shown that
a definite transition, possibly magnetic in origin, occurs
very close to l.28°K in the case of FeBrzouHZO. We have
calculated the entropy change above and below the transition
temperature and have seen that at least thirty per cent of
the total change is due to persistence of short-range order-
ing, not taken into account in the theory of VanVleck.

As for Mn(C2H302)2°uH20, we have identified a second
order transition at about 3.17°K. The latter may be due to
a nuclear contribution of the form %% which seems in very
good agreement with our measurements. It may also be due to
another transition occurring at a temperature lower than our
methods were able to achieve. The resolution of this

question must await further investigation.

1)
2)

3)

u)

5)
6)
7)

8)

9)
10)
11)

12)

13)

1”)

15)
16)

17)

18)

19)

LIST OF REFERENCES

F. Richarz, Ann. Physik i8, 708 (1893).
P. J. Debye, Ann. Physik 39, 789 (1912).

C. Kittel, Introduction to Solid State Physics, (John
Wiley and Sons, Inc., 1953), p. 63..

C. Kittel, Introduction to Solid State Physics, (John
Wiley and Sons, Inc., 1953), p. 1H7.

A. B. Lidiard, Rep. Progr. Phys. 11, 201 (195%).
J. H. VanVleck, J. Chem. Phys. 2, 85 (19M1).

J. H. VanVleck, J. Chem. Phys. 5, 320 (1937).

S. R. Pollack and K. R. Atkins, Phys. Rev. 125, 12H8
(1962). "‘—

J. Dzialoshinskii, Soviet Phys. JETP 6, 1130 (1958).

T. Moriya, Phys. Rev. 117, 635 (1960).

T. Moriya, Phys. Rev. 20, 91 (1960).

A. W. Wilson, Thermodynamics and Statistical Mechanics,

(Cambridge University Press, 1957), p. 303.

F. G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement,
and J. K. Logan, J. Research Nat'l Bur. Standards 6HA,
l (1960).

R. D. Pierce and S. A. Friedberg, J. App. Phys. 33, 665

.(1961).

S. A. Friedberg, Physica 18, 714 (1952).
D. G. Kapadnis and R. Hartmans, Physica 32, 181 (1956).

R. B. Flippen and S. A. Priedberg, Phys. Rev. 121, 1591
(1961).

B. Bleaney, Phys. Rev. 18, 21% (1950).
w. Marshall, Phys. Rev. 110, 1280 (1958).

H8

APPENDICES

T(°K.)
1.4286
1.4354
1.4473
1.4575
1.4670
1.4802
1.5158
1.6428
1.5572
1.5662
1.5916
1.6036
1.6144
1.6270

1.6394

1.6507

1.6640
1.6771
1.6913
1.7053
1.7203
1.7355
1.7508

1.7735

Heat Capacity Data of Ferrous

T(°K.)

.0031
.0027
.0028
.0030
.0032
.0031
.0023
.0031
.0032
.0025
.0040
.0046
.0029
.0037
.0035
.0032
.0030
.0037
.0037
.0041
.0042
.0042
.0038

.0041

APPENDIX I

December 1, 1961

1

C(mole. eg)

 

1.64

T(°K.)

1.7868
1.8045
1.8233
1.8470
1.8667
1.8887
1.9064
1.9318
1.9552
1.9820
2.0080
2.0411
2.0701
2.0977
2.1243
2.1454
2.1699
2.1906
2.5390
2.6316
2.7552
2.7770
2.9621

3.0047

Bromide

T(°K.)

.0032
.0043
.0037
.0046
.0035
.0061
.0054
.0048
.0042
°0043
.0041
.0056
.0047
.0035
.0050
.0037
.0052
.0048
.0058
.0056
.0084
.0062
.0158

.0190

C(

cal

mole-deg

)

 

1.27
1.30

T(°K.)

 

3.3536
3.4560
3.5329

3.9502

T(°K.)
.0191
.0203
.0237

.0171

December 7,

1.8477
1.8629
1.8816
1.8997
1.9166
1.9387
1.9585
1.9791
1.9986
2.0180
2.0436
2.0666
2.0863
2.1174
2.1363
2.1594
2.1818
2.2037
2.2381
2.2727

2.3116

.0061
.0042
.0031
.0036
.0048
.0030
.0062
.0047
.0044
.0045
.0039
.0035
.0043
'".0041
.0046
.0047
.0053
.0039
.0050
.0062

.0054

C(

1961

cal

51

\

mole-deg)

1.08

1.10

T(°K.)

2.3521
2.3886
2.4334
2.4706
2.6656
2.7148
2.7578

2.8059

January 9, 1962

1.8426
1.8703
1.8910
1.9164
1.9411
1.9638
1.9914
2.0200
2.0487
2.0775
2.1083
2.1367
2.1917
2.2219
2.2504
2.2787

2.3097

T(°K.)

.0061
.0051
.0065
.0055
.0064
.0066
.0061

.0081

.0096
.0095
.0093
.0102
.0095
.0108
.0100
.0102
.0101
.0099
.0098
.0119
.0121
.0084
.0096
.0089

.0090

C

I

car

1.39
1.61
1.44
1.60

1.27

1.58

\

‘mole-deg’

52

 

 

 

T(°K.) T(°K.) c<m0129§eg> T(°K.) T(°K.) c< 1:?fieg>
2.3551 .0122 1.40 3.6798 .0122 1.19
2.3980 .0103 1.46 3.7341 .0112 1.39
2.4381 .0078 1.58 3.7893 .0144 1.35
2.4746 .0086 1.69 3.8402 .0088 1.59
2.5141 .0108 1.55 3.8925 .0091 1.51
2.5536 .0078 1.76 3.9447 .0093 1.81
2.5997 .0101 1.65 3.9721 .0111 1.58
2.6441 .0098 1.43 4.0241 .0098 1.61
2.6841 .0093 1.61 4.1008 .0117 1.23
2.7236 .0081 1.58 4.1194 .0119 1.64
2.7708 .0108 1.38 January 16, 1962'.
2.8205 .0087 1.45 1.2855 .0039 3.94
2.8591 .0107 1.40 1.2956 .0069 3.01
2.9018 .0101 1.50 1.3067 .0056 2.72
2.9537 .0096 1.59 1.3170 .0065 2.51
2.9931 .0108 1.39 1.3281 .0067 2.36
3.0371 .0101 1.51 1.3404 .0089 2.21
3.0902 .0114 1.50 1.3566 .0079 2.15
3.1387 .0098 1.43 1.3738 .0076 2.14
3.1806 .0121 1.41 1.3909 .0077 2.00
3.2294 .0114 1.39 1.4063 .0081 2.02
3.2760 .0097 1.43 1.4252 .0061 1.88
3.4358 .0106 1.43 1.4462 .0086 1.78
3.4928 .0098 1.50 1.4672 .0076 1.77
3.5523 .0101 1.58 1.4968 .0099 1.69
3.6172 .0117 1.42 1.5191 .0085 1.69

T(°K.)

1.5513
1.8521
1.6087
1.6449
1.6743
1.7089
1.7325
1.7613
1.7959
1.8238
1.9013
1.9372
1.9920
2.0439
2.0822
2.1141
2.1563
2.1908
2.2451
2.2872
2.3215
2.3778
2.4245
2.4804
2.5246

2.5749

,53

 

 

 

T(°K.) c<molgféegj
.0101 1.55
.0108 1.59
.0083 1.55
.0099 1.61
.0092 1.59
.0096 1.52
.0102 1.51
.0096 1.48
.0135 1.40
.0086 1.62
.0134 1.33
.0120 1.42
.0223 1.21
.0188 1.26
.0169 1.19
.0143 1.25
.0181 1.24
.0130 1.20
.0121 1.58
.0095 1.70
.0102 1.62
.0096 1.66
.0112 1.49
.0092 1.51
.0102 1.50
.0113 1.48

T(°K.)

2.6168
2.6830
2.7285
2.7570
2.8042
2.8506
2.9078
2.9611
3.0174
3.0637
3.1328
3.1842
3.2330
3.2959
3.3467
3.4375
3.4995
3.5631
3.6140
3.7043
3.8476
3.9131
3.9960
4.0625
4.1692

4.2291

T(°K.)

.0117
.0101
.0104
.0091
.0094
.0114
.0102
.0105
.0109
.0112
.0118
.0132
.0125
.0109
.0134
.0095
.0110
.0114
.0104
.0151
.0148
.0138
.0144
.0116
.0157

.0125

C(

__c.a.l_
mole-degi

1.36

1.60

1.24
1.02
1.21
1.01

1.11

)

54

 

 

 

T(°K.) T(°K.) c<molgi§eg5 T(°K.) T(°K.) C(mo 27:8 )
February 13,1962 1.3225 .0030 2.61
1.1723 .0033 3.25 1.3309 .0039 2.07
1.1797 .0027 2.90 1.3394 .0040 1.93
1.1862 .0023 3.28 1.3496 .0046 1.92
1.1983 .0023 3.41 1.9678 .0045 1.81
1.2040 .0022 3.62 1.3806 .0045 1.74
1.2102 .0020 4.10 1.3914 .0046 1.78
1.2158 .0026 3.85 1.4027 .0045 1.81
1.2217 .0022 3.70 1.4129 .0042 1.56
1.2265 .0021 3.77 1.4272 .0049 1.68
1.2316 .0016 4.40 1.4414 .0052 1.49
1.2366 .0017 4.65 1.4586 .0047 1.52
1.2413 .0016 4.15 1.4780 .0057 1.47
1.2457 .0015 5.16 1.4963 .0047 1.45
1.2502 .0017 4.70 1.5168 .0060 1.41
1.2546 .0014 5.02 1.5353 .0054 1.42
1.2621 .0014 6.08 1.5540 .0054 1.44
1.2663 .0011 6.77 1.5728 .0060 1.40
1.2723 .0010 7.51 1.5945 .0054 1.32
1.2758 .0013 6.36 1.6176 .0062 1.32
1.2809 .0010 7.65 1.6512 .0077 1.32
1.2846 .0012 4.98 1.6837 .0059 1.38
1.2919 .0022 3.36 1.7181 .0062 1.31
1.2986 .0028 2.76 1.7517 .0049 1.28
1.3062 .0034 2.48 1.8049 .0062 1.28
1.3143 °0030 2.44 1.8445 .0058 1.34

T(°K.)

1.5039

1.5280
1.5368
1.5510
1.5686
1.5936
1.6126
1.6277
1.6405
1.6539
1.6742
1.6959
1.7178
1.7350
1.7577
1.7816
1.8039
1.8220
1.8438
1.8725
1.9047
1.9389

1.9745

T(°K.)

.0067

.0048
.0048
.0098
.0064
.0087
.0063
.0087
.0061
.0055
.0072
.0083
.0068
.0045
.0043
.0070
.0107
.0084
.0078
.0080
.0073
.0064

.0074

APPENDIX II

June 22, 1961

moleodeg

 

1.12

55

T(°K.)

2.0082

2.0464
2.0786
2.1034
2.1289
2.1746
2.2185
2.2980
2.3485
2.4007
2.4873
2.5523
2.6680
2.7322
2.8405
2.9180
2.9873
3.0764
3.1250
3.1661
3.2113
3.2605

3.3134

Heat Capacity Data of Manganous Acetate

T(°K.)

.0069

.0071
.0060
.0053
.0063
.0241
.0133
.0121
.0155
.0144
.0130
.0130
.0100
.0105
.0056
.0085
.0187
.0123
.0117
.0090
.0124
.0149

.0175

C(

cal

3

moleodeg'

 

0.94

0.91

T(°K.)
3.3846
3.4590
3.5323
3.6068
3.6778
3.7495
3.8088
3.8752
3.9387
4.0185
4.0891
4.1314
4.1823

4.2445

T(°K.)

.0149

.0215
.0150
.0195
.0189
.0182
.0188
.0225
.0202
.0242
.0217
.0222
.0245

.0252

C!

June.27, 1961

1.6586
1.6797
1.6971
1.7143
1.7331
1.7529
1.7725
1.7912
1.8107
1.8308

1.8484

.0231
.0099
.0115
.0105
.0120
.0095
.0085
.0094
.0101
.0140

.0102

cal

1.55

\

‘mole-deé7

56

1.8698

1.8945
1.9227
1.9538
1.9831
2.0153
2.0449
2.0737

2.1004

2.1301.,

2.2007
2.2338
2.2735
2.3177
2.3642
2.4168
2.4801
2.5549
2.6075
2.6684
2.7340
2.7979
2.8785
2.9612
3.0138

3.0548

T(°K.)

.0122
.0112
.0096
.0115
.0090
.0122
.0109
.0118
.0106
.0096
.0164
.0105
.0088
.0113
.0128
.0111
.0111
.0104
.0095
.0106
.0127
.0157
.0166
.0140
.0145

.0185

C(

cal

\

mole‘deg’

1.05
1.13
1.04
1.06

1.14

57

 

 

 

T(°K.) T(°K.) 0653T§%%5§9 T(°K.) T(°K.) C molgféeg
3.0925 .0105 2.21 July 7, 1961
3.1318 .0098 2.41 1.5094 .0023. 0.98
3.1630 .0070 3.00 1.5177 .0028 1.02
3.2004 .0113 2.02 1.5259 .0020 1.16
3.2388 .0094 1.99 1.5408 .0052 1.29
3.2754 .0124 1.44 1.5533 .0058 1.13
3.3211 .0132 1.69 1.5670 .0049 1.34
3.3721 .0148 1.45 1.5837 .0055 1.02
3.4205 .0129 1.58 1.6017 .0059 1.14
3.4710 .0144 1.40 1.6189 .0058 1.02
3.5186 .0123 1.36 1.6344 .0072 1.05
3.5669 .0140 1.38 1.6485 .0054 1.06
3.6160 .0196 1.10 1.6671 .0070 0.97
3.6690 .0148 1.36 1.6865 .0060 0.96
3.7216 .0166 1.19 1.7063 .0061 1.02
3.7735 .0143 1.26 1.7256 .0066 1.01
3.8328 .0147 1.36 1.7472 .0064 1.03
3.8812 .0196 1.16 1.7654 .0053 1.15
3.9331 .0170 1.07 1.7878 .0068 0.99
3.9880 .0175 1.14 1.8137 .0063 0.98
4.0453 .0196 1.19 1.8402 .0067 1.00
4.0992 .0184 1.09 1.8654 .0070 0.87
4.2043 .0230 1.11 1.8891 .0146 1.07
4.2544 .0164 1.14 1.9191 .0134 1.10

T(°K.)

1.9549
1.9922
2.0317
2.0721
2.1103
2.1561
2.1976
2.2349
2.2691
2.3097
2.3518
2.3966
2.4435
2.4965
2.5439
2.6448
2.7027
2.7563
2.8035
2.8939
2.9381
2.9832
3.0170
3.0501
3.0878

3.1192

58

 

 

 

T(°K.) Ctmolgféeg) T(°K.)
.0164 0.91 3.1431
.0155 0.94 3.1706
.0132 1.03 3.1979
.0137 1.03 3.2268
.0067 0.82 3.2781
.0158 0.97 3.3167
.0130 1.06 3.3658
.0114 1.19 3.4141
.0103 1.25 3.4704
.0133 1.14 3.5187
.0143 1.17 3.5625
.0143 1.24 3.5849
.0126 1.22 3.6304
.0131 1.23 3.6731
.0130 1.23 3.7224
.0112 1.40 3.7681
.0088 1.35 3.8024
.0107 1.39 3.8543
.0078 1.64 3.8767
.0075 1.58 3.9246
.0095 1.64 3.9706
.0071 1.89 4.0136
.0081 2.08 4.0659
.0055 2.09 4.1144
.0067 2.32 4.1374
.0059 2.29 4.1855

 

 

T(°K.) C(molgfé§3_
.0049 2.58
.0060 2.41
.0081 1.75
.0083 1.53
.0097 1.54
.0110 1.44
.0091 1.60
.0093 1.47
.0084 1.39
.0099 1.56
.0102 1.43
.0116 1.34
.0145 1.14
.0122 1.37
.0125 1.27
.0156 1.09
.0117 1.25
.0104 1.19
.0105 1.28
.0123 1.42
.0110 1.04
.0113 1.32
.0281 1.16
.0254 1.10
.0240 1.22
.0228 1.14

59

 

 

 

 

 

 

T(°K.) T(°K.) ccmolgféega T(°K.) T(°K.) C(m01233eg)
July 11, 1961 2.8563 .0098 1.40
1.8286 .0144 1.15 2.8943 .0101 1.59
1.8553 .0158 1.04 2.9330 .0094 1.52
1.8833 .0142 1.10 2.9700 .0088 1.70
1.9129 .0143 1.00 3.0048 .0090 1.72
1.9399 .0150 1.04 3.0326 .0082 1.70
1.9689 .0194 1.01 3.0675 .0076 2.21
2.0032 .0152 1.06 3.1032 .0068 2.06
2.0354 .0157 1.05 3.1334 .0069 2.25
2.0695 .0145 1.04 3.1882 .0091 1.54
2.1104 .0218 1.01 3.2268 .0083 1.64
2.1487 .0157 1.06 3.2594 .0085 1.84
2.1889 .0148 1.10 3.2959 .0087 1.88
2.2234 .0143 1.18 3.3361 .0100 1.53
2.2540 .0137 1.24 3.3744 .0102 1.59
2.2870 .0121 1.27 3.4376 .0260 1.46
2.3213 .0118 1.24 3.5051 .0221 1.34
2.3579 .0123 1.24 3.6517 .0306 1.28
2.3963 .0115 1.20 3.7085 .0234 1.24
2.4384 .0119 1.32 3.7474 .0267 1.34
2.5850 .0140 1.15 3.7972 .0187 1.36
2.6375 .0104 1.35 3.8499 .0252 1.10
2.6838 .0115 1.36 3.9078 .0305 1.23
2.7166 .0103 1.44 3.9401 .0311 1.29
2.7411 .0113 1.35 3.9852 .0319 1.13
2.7866 .0101 1.36 4.0519 .0296 1.09
2.8208 .0111 1.57 4.1085 .0270 1.11

T(°K.)

T)°K.)

cal

 

‘mole~desm

February 27, 1962

1.1785
1.1912
1.2011
1.2088
1.2168
1.2240
1.2320
1.2398
1.2493
1.2567
1.2662
1.2742
1.2839
1.2949
1.3056
1.3134
1.3278
1.3397
1.3512
1.3630
1.3740
1.3843
1.3933

1.4046

.0035
.0032
.0042
.0034
.0032
.0027
.0039
.0026
.0039
.0028
.0034
.0029
.0024
.0089
.0049
.0031
.0036
.0040
.0033
.0028
.0031
.0035
.0031

.0037

1.4437
1.4574
1.4722
1.4917
1.5042
1.5184
1.5373
1.5632
1.5796
1.5933
1.6138
1.6615
1.6850
1.7121
1.7379
1.7713
1.8022
1.8306
1.8649
1.9060
1.9303
1.9594
1.9870

2.0216

cal

 

T(°K.) C(

 

mole-deg}
.0031 1.05
.0045 0.88
.0033 0.99
.0038 0.90
.0041 0.92
.0041 0.93
.0047 0.97
.0034 0.93
.0042 0.93
.0036 0.99
.0044 0.88
.0070 0.87
.0054 0.92
.0050 0.93
.0054 0.86
.0134 0.88
.0113 0.86
.0111 0.89
.0139 0.87
.0112 0.85
.0108 0.89
.0088 0.89
.0107 0.92
.0110 0.92

PuvsncsuATw um

BRRRIES

nI11100101