ABSTRACT

LOW TEMPERATURE ADIABATIC STUDIES OF MAGNETIC
PHASE BOUNDARIES IN ANTIFERROMAGNETS

by James Nathaniel McElearney

A theoretical treatment of antiferromagnetism, for
arbitrary Spin values, is given, based on previous exten-
sions of the molecular field model of antiferromagnetism.
Theoretical predictions involving the magnetic phase bound-
aries are given. A new method to predict critical fields
for spin flopping, using three magnetic susceptibility
measurements, is proposed and the theoretical basis for
adiabatic invemjgations of antiferromagnetic spin flop
states is given. Apparatus and methods for such experiments
are described.

The results of adiabatic experiments on CoC12°6H20 and
MnC12-4H20, whose spin flop states have been previously
studied, are given, indicating the validity of the new
method. Also, results are given indicating that two pre-
viously studied substances, CoBr2c6H20 and FeC12-4H20, ex-
hibit spin flopping in accessible experimental regions.
CoBr2-6H20 has a critical field in the region 1.15°K to the
triple point, 2.91 i .OIOK, which is well fitted by the
equation Hc = 7523 - 230.7 T + 292.5 T2. FeC12-4H20 ap-
parently spin flops below 0.680K above a field equal to
about 5500 gauss. Finally, results are also given of in-
vestigations done on MnBr2-4H20, CszMnBr4°2H20, and

NiC12-6H20, which indicate the lack of spin flop states in

James Nathaniel McElearney

these substances in fields below 10,000 gauss and in tem-

peratures down to 1.0°K.

LOW TEMPERATURE ADIABATIC STUDIES OF MAGNETIC

PHASE BOUNDARIES IN ANTIFERROMAGNETS

BY

James Nathaniel McElearney

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Physics

1968

ACKNOWLEDGMENTS

The author wishes to express his appreciation and thanks
to all those who have helped in this study; to Dr. H. Forstat
for suggesting this study and for invaluable aid during the
course of the work; to Dr. J. A. Cowen and Dr. R. D. Spence
for several fruitful discussions; to Mr. Peter T. Bailey
for his assistance with the experiments; to the M.S.U. Com-
puter Center for computer time made available; and especially
to the Air Force Office of Scientific Research, Office of
Aerospace Research, United States Air Force for their finan-

cial support of this work.

ii

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1

CHAPTER
I. THEORETICAL BACKGROUND . . . . . . . . . . .

2

I. Theory . . . . . . . . . . . . . . . 2
A. Antiferromagnetism . . . . . . . 2
B. Generalization of Garrett's Theory 4
C. Behavior of the Sublattices in

a Field . . . . . . . . . . . . 7
D. Equation for Magnetization and
Entropy . . . . . . . . . . . . 12
E. Introduction of Anisotropy . . . 13
F. Antiferromagnetic-Paramagnetic
Boundaries . . . . . . . . . . . 14
G. Susceptibilities . . . . . . . . 18
H. Spin Flop . . . . . . . . . . . 22
II. Applications of the Theory . . . . . 25
A. Observation of Spin Flop
Boundaries . . . . . . . . . . . 25
B. Observation of Paramagnetic
Boundaries . . . . . . . . . . . 30
II. EXPERIMENTAL METHODS . . . . . . . . . . . . 32
I. Experimental Apparatus . . . . . . . 32
A. Triple Can Calorimeter
Modifications . . . . . . . . . 32
B. Main Dewar and Calorimeter . . . 33
C. Sample Holder . . . . . . . . . 37
D. Pumps . . . . . . . . . . . . . 40
E. Pressure Gauges . . . . . . . 40
F. Thermometers and Current Supplies 42
G. Measuring Electronics. . . . . . 43
H. Magnet . . . . . . . . . . . . . 44
II. Experimental Procedures . . . . . . . 46
A. Sample Preparation and Precooling 46
B. Helium Transfer and Calibration 48
C. Specific Heat Mbasurements . . . 50
D. .Magnetic Field Experiments . . . 51
E. Shut-Down . . . . . . . . . . . 54
III. Reduction of Experimental Data . . . 55
A. Conversion of Pressures to
Temperatures . . . . . . . . . . 55

B. Temperature Calibration Equation 56

iii

TABLE OF CONTENTS (Cont.)

CHAPTER Page
C. Determination of Thermometer
Resistances . . . . . . . . . . 58
D. Specific Heat Calculations . . . 62
E. Calculation of Magnetic Field . 62
III. RESULTS AND CONCLUSIONS . . . . . . . . . . 65
I 0 Results 0 o o o o o o o o o o o o o o 65
A. Results of Experiments on Samples
with Known Spin Flop Phases . . 65
B. Materials Discovered to Exhibit
Spin Flop . . . . . . . . . . . 81
C. Materials Investigated Which Did
Not Exhibit Spin Flop . . . . . 95
II. Conclusions . . . . ..... . . . . . . 104

REFERENCES . . . . . . . . . . . . . . . . . 106
APPENDICES
I. COMPUTER PROGRAMS . . . . . . . . . . . . . 109
Part A. FORTRAN Subroutines for Theory . 110
Part B. FORTRAN Subroutines for Data
Reduction . . . . . . . . . . . 117
Part C. Typical Data Deck Input . . . . 137

II. RESISTANCE CALCULATION WITH AN UNBALANCED
POTENTIOMETER SETTING . . . . . . . . . . . 144

III. EXPERIMENTAL DATA . . . . . . . . . . . . . 148

iv

LIST OF TABLES

TABLE Page

1. CoC12.6H 0 BC plane isentropic rotations
(C = 55° . . . . . . . . . ... . . . . . . . . 149

2. CoC12'6H20 AB plane isentropic rotations
(B = 0°) . . . . . . . . . . . . . . . . . . . 150

3. CoC12-6H 0 AC plane isentropic rotations
(c = 70° . . . . . . . . . . . . . . ... . . 151

4. CoClz-GHZO C-axis isentrOpic magnetizations . . 152

5. CoC12-6H20 antiferromagnetic-Spin flop
boundary points (H // C-axis) . . . . . . . . . 152

6. CoC12-6H20 BC plane paramagnetic isentropic
rotation (C = 55°) . . . . . . . . . . . . . . 153

7. MnC12-4H 0 AC plane isentropic rotations
(c = 45° . . . . . . . . . . . . . . . . . . . 154

8. MnC12°4H 0 BC plane isentropic rotations
(c = 5503 . . . . . . . . . . . . . . . . . . . 155

9. MnClz'4H20 C-axis isentropic magnetizations . . 157
10. MnC12°4H20 specific heat maxima (H // C-axis). 160

11a. CoBr2-6H 0 BC plane isentropic rotations
(c = 60° . . . . . . . . . . . . . . . . . ... 161

11b. CoBr2-6H20 AC plane isentropic rotations (C = 22°)162
12. CoBr2°6H20 C-axis isentropic magnetizations . . 163

13. CoBr2-6H20 antiferromagnetic-Spin flop
boundary points (H // C-axis) . . . . . . . . . 165

14. Specific heat maxima (H // C-axis, CoBr2-6H20). 165
15. FeC12°4H20 isentropic rotations . . . . . . . . 166
16. FeClz-4H20 B-axis isentropic magnetizations . . 167
17. Specific heat maxima (H // B-axis, FeC12.4H20) 168
18. MnBr2-4H20 C—axis isentropic magnetizations . . 169
19. CszMnBr4-2HZO isentropic magnetizations . . . . 172

20. CszMnBr4-2H20 specific heat maxima (H //[I I I]) 173
v

LIST OF TABLES (CONT.)

TABLE

21.

22.
23.

NiClz-GHZO AC plane isentropic rotation
(A. = 1100) C O O O O O O O O O C O C O O O O

NiC12'6H20 A'—axis isentropic magnetizations.

NiClZ-GHZO Specific heat maxima (H // A') . .

vi

Page

173
174

177

LIST OF FIGURES

FIGURE Page
1. Definition of angles . . . . . . . . . . . . 8
2. Vector relations for a perpendicular field . 11

3. Parallel and perpendicular antiferromagnetic
paramagnetic boundaries . . . . . . . . . . . 17

4. Parallel, perpendicular and paramagnetic

susceptibilities . . . . . . . . . . . . . . 21
5. Normalized spin flop cirtical fields . . . . 26
6. Predicted temperature variation in an

isentrOpic rotation . . . . . . . . . . . . . 29
7. Pyrex helium dewar . . . . . . . . . . . . . 34
8. Cross section of body of calorimeter . . . . 35

9. Front and Side cross sections of top of
calorimeter . . . . . . . . . . . . . . . . . 36

10. C-clamp sample holder . . . . . . . . . . . . 38
11. Schematic diagram of pumping systems . . . . 41
12. Diagram of electrical measuring circuits . . 45
13. Example of preliminary reduction of chart data 60

14. Voltage calibration points from FeC12-4H20

(sample 4) runs . . . . . . . . . . . . . . . 61
15. Voltage calibration derived from Fig. 14. . . 61a
16. ,Magnet calibration curve . . . . . . . . . . 64
17. CoC12°6H20 BC plane isentrOpic rotations . . 67
18. CoClz'GHZO AB plane isentrOpic rotations . . 68
19. CoC12-6H20 AC plane isentrOpic rotations . . 69
20. CoC12°6H20 isentropes . . . . . . . . . . . . 71
21. CoC12°6H20 phase boundaries . . . . . . . . . 73

vii

LIST OF FIGURES (Cont.)

FIGURE

22.

23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.

37.

CoC12-6H20 BC plane paramagnetic isentropic
rotation . . . . . . . . . . . . . . . . .

MnClz-4H20 AC plane isentropic rotations .
MnC12°4H20 BC plane isentropic rotations .
MnC12°4H20 phase boundaries and isentropes
CoBr2‘6H20 BC plane isentropic rotations .
CoBr2-6HzO AC plane isentropic rotations .
CoBr2-6H20 isentropes . . . . . . . . . . .

CoBrz'GHZO phase boundaries . . . . . . . .

FeC12-4H20 AB and BC plane isentropic rotations

FeC12'4H20 phase boundary and isentropes .
MnBr2'4H20 isentropes . . . . . . . . . . .
CszMnBr4-2H30 phase boundary and isentropes
CszMnBr4-2H20 isentropes (expanded scale) .
NiC12-6H20 AC plane isentropic rotation . .
NiC12-6H20 phase boundary and isentropes .

Circuit diagram of a Simple potentiometer .

viii

Page

74
76
77
79
83
84
85
87

94
97
99
100
102
103
147

INTRODUCTION

Magnetic prOperties of matter have interested physicists
for a long time. In recent years, there has been a great
deal of study done on magnetic systems in their ordered
states. Many different experimental measurements supply in-
sight into the phenomenon of magnetism.

Although there have been many experimental investigations
of different magnetic systems in different substances, at
present there have been few investigations involving anti—
ferromagnetic systems in the spin flop state (although, some
of those done have been thorough). The basic reason for
this is that few substances exhibit the phenomenon of spin
f10p in regions of magnetic field and temperature accessible
by common methods.

The purpose of this investigation is to develop further
methods of study of Spin f10p states and to discover more
materials exhibiting spin flop properties in accessible ex-
perimental regions. The study has been divided into three
parts. The first part will develop the theoretical basis
for adiabatic observations of spin f10p states. The second
part will describe the experimental apparatus and methods
used in such a study, and finally, the third part will re—
port results indicating the validity of the new method and
'will also report on the new materials which exhibit Spin

flop prOperties.

CHAPTER I
THEORETICAL BACKGROUND

I. THEORY
A. Antiferromagnetism

The concept of antiferromagnetism, interpenetrating
lattices of Oppositely directed magnetic moments, was first
postulated by Neel1 in 1932. Since then, many antiferro-
magnetic materials have been discovered and studied. There
exist many excellent papers on the subject and reference 2
contains a general review of the subject.

To give a theoretical analysis of the phenomenon, one
needs to describe the energy of interaction of several ions
with Spins. The model presently accepted as being correct
was first suggested in 1928 by Heisenberg3. Heisenberg
type exchange interactions, in the Dirac formalism4, are

equivalent to the interatomic potential

_ _ —> .->
vij — 1/2 J(1 + 4 si sj). (1.00)

s: and S; are the quantum mechanical spins of ions 1
and j. J is the exchange integral and is positive for
ferromagnets and negative for antiferromagnets. Apart from

an additive constant, then, the potential energy of atom i

is given by

_ = _ '—> .—>
vi — i vij 2 J i (Sj Si). (1.01)

3
Thus, the Heisenberg model corresponds to magnetic isotrOpy.
Perhaps one of the most widely used approximations to
equation 1.00 is that of the Ising model5, in which it is
assumed that only the components of the spins in one direc-

tion interact. That is,

Vi = - 2J i sz

5 (1-02)

iz'
This model obviously corresponds to extreme magnetic aniso-
tropy.

The most commonly used approximation is that which Neel

used, the Weiss molecular field approximations. In this

approach, the potential energy of atom i is given as
z =;—>
V. = i V.. - — 2J z Sj Si’ (1.03)

where z j atoms interact with atom i with the same J
and S: is the statistical average of S; over the j
sublattice. Since the magnetic moment of an atom may be
written as E9=-g “B S9, where g and ”B are the usual
magnetic factors, equation 1.00 becomes

= - —> o—>
Vi Hj ui, (1.04)

where
-—> 2 ->

Hj = (- 2J z/g2 uB) uj (1.05)

is the effective magnetic field (the molecular field) due
'to the j sublattice which is acting on atom i.
There have been many other approaches and approximations

used, but for the purposes of this study, the molecular

4
field approach will supply the necessary insight for under-
standing the experimental data involved. Bitter7, Van V1eck3,
Garrett9, Andersonlo, Nagamiyall, Yosidalz, Gorter and
Haantjesl3, and many others have made extensions and refine-
ments to Neel's original theory. In particular, this study
will deal with a further extension of Garrett's approadh
which itself was an extension, to include high magnetic

fields, of Van Vledk's work on the Neel model.

B. Generalization of Garrett's Theory

 

Garret's work was for a spin 1/2 substance, and his
equations did not explicitly include anisotropy. This study
will consider his equations for Spin S and, in addition,
solutions will be considered for magnetic fields applied at
arbitrary angles to the preferred axis of magnetization,
rather than just parallel or perpendicular to it. Also,
anisotropy will be introduced to the solution through
Yosida's14 approach which was Similar to Neel's calculation
at absolute zerol5. The importance of anisotrOpy to anti—
ferromagnetic phenomena cannot be overemphasized, as many
of the writers have pointed out. Apparently the best
treatment, to date, of a molecular field theory which in-
cludes anisotropic effects is that of Gorter and Van Peski-
Tinbergenla. However, since that approach, which was for
Spin 1/2 systems, involved graphical solutions, it is not
the simplest to comprehend, and a generalization of Garrett's

'work is worthwhile.

5
In the Van Vleck model, the net magnetization per unit

volume of an antiferromagnetic crystal is given by

—>_ 1: -> —> _ -> —>

M - 2 g ”B N(si + sj) - Mi + Mj. (1.06)
where g is the usual Landé factor, ”B is the Bohr
magneton, N is the number of magnetic atoms per unit
volume, and. S: and S; represent the average Spins of
the ions of the two sublattices. The magnitudes of S:

and S: are given according to standard magnetic theory17

by

si = s BS(Hi s g “B/k T)

(1.07)
s. = s BS(Hj s g uB/k T), _
where k is Boltzman's constant, T is the absolute tem-
perature, S is the Spin quantum number of the magnetic
atom in the crystal, Hi and Hj are the magnitudes of
the effective fields due to the j and i sublattices

respectively, and B is the Brillouin function, given by

S
_ ZS + 1 28 + 1 _ ;L_ 3L.
BS(x) —--—§§——- coth (-§§-—-x) ZS coth (ZS). (1.08)

For an arbitrary applied field H? H: and H; are given by

—>

Hi ‘fi’+ (2z J/g BB)

—>

3 (1.09)

(D

H>+ (22 J/g uB) 8:,

with the additional requirement for internal consistency of

the model that H: is parallel to S: and H; is parallel

to gjo

6
Following Garrett's method of introducing "reduced"

fields, Spins and temperatures, equations 1.07 and 1.09

 

become
si = bS(hi/t)
(1.10)
'. = b h. t
S) s< 3/ >
H? = in E?
l 3 (1.11)
K? = Ki §>,
j 1
where
_ 3s
bS(x) — BS( 5 + 1 x), (1.12)
when the following definitions are made:
Ho = (-22 J 5/9 93) (1.13)
Tn = - 2J z s(s + 1)/3k (1.14)
H? = E.)/Ho
1 1 (1.15)
F? = E? H
J 3/ °
3°: fi7Ho (1.16)
t = T/Tn (1.17)
S1 = Si/S
(1.18)
. = 5.
SJ J/S
—> —>
. = M./(g N s/2)
1 1 93 (1.19)

m. = Mg/(g “B N S/2)

(Note nOW' m: = s. and m; = 3;, so the reduced magnetization

is m=fi§+fig=sg+§;.)

7

C. Behavior of the Sublattices in a Field

It is convenient now to define angles as in reference
2. Consider the z-x plane as in Fig. 1. Then let the
preferred axis of magnetization be 2 and let h be ap-
plied in the z—x plane. It is shown in reference 2 that
if

E) = (Ej/Sj) - (SE/Si), (1.20)

then Z) is also in the z-x plane. The angle between h">

and the z axis is 9 and the angle between '5'> and 'Z>
is ¢. (w is not equal to 9 due to the anisotropy, as
will be seen shortly.)

Note that the definition of A) insures that the
angles (denoted as 0 in Fig. 1) between S; or S: and
Z) will be equal. Now, for the molecular field model,
as stated in reference 2, the component of h> perpendicu-
lar to 3’ causes the Spin vectors of the two sublattices
to cant equal angles away from Z) toward 3). Further-
more, the component of 3) parallel to Z) causes the
magnitudes of S: and S; to change. It is not clear
whether the authors mean the above statements to be true in
general, regardless of the intensity of the applied field,
or whether the statements are only true for small enough
fields such that the angle of cant of the sublattices, ¢,
is small.

This description of the behavior of the sublattice

vectors certainly has a better chance of matching reality

O
F""'
.L‘ o

 

 

1.

 

Definition of angles.

(Pl

9
than that of reference 18, in which it is stated that the
components of h'> parallel and perpendicular to A) cause
the parallel and perpendicular components of the magnetiza-
tion to increase by 1/2 xuh" and 1/2 thL respectively.
Were this so, then, as shown in reference 18, the follow-

ing relations would hold:

S. =(so+(x,,h(cos ¢)/2))cos(z//-9)+(XLh sin #1 Sin(z//-9)/2)

32
ij= -(so+(xoh(cos w/2))sin(¢-9)+(XLh sin w cos(w-e)/2)

, (1.21)
Siz= -(so-(X"h(cos w/2))cos(w-e)+(XLh sin w sin(¢-6)/2)

six=(so-(X"h(cos fiV2))Sin(w-9)+(XLh Sin w cos(¢-e)/2),

where so is the zero field magnetic moment of one sub-
lattice. Then if M and M' denote the sum and differ—
ence of the magnetic moments on the i and j sublattices,

the components of these vectors are:

- th cos w cos(w-9) + XLh sin ¢ Sin(¢-9)

N3

M = XLh sin w cos(¢-9) — th cos w sin(¢-G)
x (1.22)
M; = 2 so cos(¢-6)
M: = 2 so sin(¢—6).
x
But this implies M' = 2 so always. However, this is not

possible, as is evident from consideration of the case
where the applied field is perpendicular, or nearly perpen—

dicular to the preferred axis and is large enough to almost

10
cause saturation. M' must then obviously approach zero.
The behavior of the sublattices in a high field at some
angle to the easy axis seems to need clarification. For
a field applied either parallel or perpendicular to the
easy axis, the picture of reference 2 would seem plausible
(with the necessity that the relationship between w and
e be modified for high fields).

Furthermore, Garrett has given the following argument
to determine the effect of a perpendicular field of arbi-
trary value. Consider F9 applied perpendicular to 2,
as shown in Fig. 2, which represents the vector relations
between h9, h:, E:’ SE, and S;. Then the effect of the
field must be to equally rotate S: and S; without
changing their magnitudes. For, if h"> were to change in
magnitude and if the magnitudes of S: and S; were to
change, they would have to change prOportionally to the
magnitudes of hi and hj' as can be seen from Fig. 2.
But this would imply a linear relation between si and hi
and also between 3j and hj' Therefore, since the rela-
tionships of equations 1.10 are not linear, the magnitudes
can not change and the only alternative is that they keep
the same magnitude as in zero applied field and rotate
toward the perpendicular direction when the field is ap-
plied. (This argument can not be applied in general to a
field oriented at some arbitrary angle to the easy axis be—

cause of the effect of the anisotropy field.)

11

 

 

 

 

 

Fig. 2. Vector relations for a perpendicular field.

12

D. Equation for Magnetization and EntrOpy

Thus, adopting the viewPoint of reference 2 regarding
the behavior of the sublattice vectors, with the reserva-
tion that the fields used must be small enough so as not

to introduce any saturation effects, the magnitudes of S?

1
and S; may be determined by merely considering the per—
pendicular component of the field to be zero. Then equa-
tions 1.10 become
5. = b «s. -’h cos ¢)/t)
1 S 3 (1.23)

s. = bs«h cos w + Si)/t),

where it has been assumed that the parallel components of
Sj and h point in the same direction. (Note that these
equations give positive values for Si and sj, whereas
Garrett's equations have Signed values for Si and sj
built in.)

Besides the magnetization, which may be calculated
from si and Sj’ the entropy of the system is also of
interest. Garrett has given a statistical argument to de-
termine the entropy for a spin 1/2 system in terms of the
sublattice magnetizations. The general thermodynamic
argument to determine the entropy of an antiferromagnet of

spin j will be given here. In general,
3 = bj (h/t). (1.24)

From the combined first and second laws of thermodynamics

for a magnetic system,

13

dU=TdS +HdM, (1.25)

where S now is entropy and U is the energy of the sys-
tem (including the field energy). For dU = 0, T dS =

- H dM. In reduced units,

 

 

 

 

 

h k N 3J
t dS = - 2 (J + 1) ds. (1.26)
Thus, h/t = — ZLJagfil) d(§£k) . (1.27)
_ _ 2(J+1) d(s[k)
So, 5 — BJ( 3JN ds ). (1.28)
-1 _ 2(J + 1) dis/k)
Therefore, BJ (s) — - 3JN ds . (1.29)
3 -1
So, S/k = - 2(J 1N1)f BJ (s) ds. (1.30)
Thus, for an antiferromagnet,
S/kN = - 3J (f 3‘1 (S.)ds + f 3‘1 (s ) ds ) (1 31)
2(J+1) J 1 i J j j '_ '

which reduces to Garrett's eXpression for a spin 1/2

system.
E. Introduction of Anisotropy

To introduce anisotrOpy into the solution, w is al-
lowed to be different from 9 such that the sum of the

magnetic energy,

1 .
Em = - §(XL Sin2 w +-Xu cos2 w) h2, (1.32)

and the anisotropy energy (assuming uniaxial anisotropy

14

and no saturation effects),

Ea = a sin2 (w-e) (1.33)

(where a isin reduced units and is greater than zero),
takes on its minimum value. This requirement leads to the
equation

Sin 26
cos 26 - (h/hc)2

 

tan 2w = (1.34)

 

where hC = 4' 2a7(XL - X..) . (1.35)

F. Antiferromagnetic-Paramagnetic Boundaries

 

1. Perpendicular Boundary

Garrett has given the following argument to determine
the antiferromagnetic-paramagnetic boundary in the case of
a perpendicular field. The component of magnetization in
the direction of the field is 2so sin 0 which must equal

XL h. But X1 = 1, as will be seen later, so that

2so sin ¢ = h (1.36)

describes the magnetization in the perpendicular direction
until the sublattice spins have been dragged into complete
parallelism with the field. This occurs at ¢ = 90°, or

h = 23o. But for a perpendicular field, equations 1.23 re-

duce to

so = bS(so/t). (1.37)

15
But on the boundary h = 2So, so
h/2 = bS(h/2t),
, (1.38)

or t = h/2 bgl (h/2),

where bgl refers to the inverse Brillouin function.
(Note that at t = 0, the boundary intersects the h axis

at h = 2.)
2. Parallel Boundary

For the parallel case (¢ = 0), Garrett has noted that
the antiferromagnetic-paramagnetic boundary is obtained
by letting sj approach -si as the difference of sj
and 51 remains constant. The latter condition is equi-

valent to dsi/dsj = 1. Application of these conditions

to equations 1.23 results in the equations which define the

boundary:
—Si - h
51 = bS(—-—t-——) (1.39)
and ' h + Si
t bS(-—¥f——2 ! (1°40)

where bé is the derivative of the Brillouin function with

respect to its argument. So, denoting the inverse of b;

by bé-l, 1.40 becomes

h + s.

5*‘1(t) = ——E——i , (1.41)

and therefore, using 1.41 in 1.39,

-s1 = bS(bS—1 (t)). (1.42)

16

Also, 1.39 is equivalent to

_1 h + si
bs (-Si) = —-t——- . (1.43)
Solving 1.43 for h,
-1
h — t bS (-si) - Si. (1.44)
Therefore, using 1.42 in 1.43,
_ I-1 I-1
h — t bS (t) + bs(bs (t)) (1.45)

defines the paramagnetic boundary for a field applied in
the parallel direction. For S = 1/2 this reduces to the

equation which Garrett derived:

}1==t tanh-l J1 - E -+ J1'--t . (1.46)

For S not equal to 1/2, equation 1.45 cannot be solved
in closed form and numerical solutions are necessary.
Computer programs were written to solve the boundary equa-
tions for fields applied both parallel or perpendicular

to the easy axis, (see Appendix IA) and the results for

two values of S are shown plotted in Fig. 3. Ziman19,
using the higher order Bethe-Peierls4Weiss method has found
that the critical curve, for the parallel case, drops con-
tinuously from h = 1 at t = 0 to h = 0 at t = 1,

so that the peculiar maximum in the parallel critical curve
near t = 0.2 for the molecular field model, as shown in

Fig. 3, probably does not exist in an exact solution.

17

AF“P BOUNDARIES
“"“ H//Z
“—7 Hi2

 

 

.0 0.5 1.0 L?) 2.0

Fig. 3. Parallel and perpendicular antiferromagnetic—
paramagneLic boundaries.

18

G. Susceptibilities

1. Perpendicular Case

AS will be seen shortly, the magnetic susceptibil-
ities of antiferromagnets are related to some of the phenom-
ena connected with antiferromagnetism, so it is worthWhile
to calculate theoretical expressions for them. The sus—
ceptibility for a perpendicular field is most easily
calculated by noting that the torque on a sublattice spin,

due to the field acting on it, must be zero. That is,

S? x 3?] = 0, (1.47)
1 1 .
or. IS: x (h9— S;)| = 0 (1.47')

.

(having neglected anisotropy). For E) perpendicular to

A , si = sj = so, and the angle between 'S: or S; and
A. is 0. So

si h cos 0 - si sj Sln 20 = 0 (1.48)

or, h = 2 so sin 0. (1.49)

But the right hand side of this equation is the magnetization
in the perpendicular direction which must equal XL h.
Therefore,

X1 = 1. (1.50)
2. Parallel Case

For a parallel field, the susceptibility may be calculated

from equations 1.23, since X,, = m/h: (Sj-si)/h. Following

19
Van Vleck's argument, the Brillouin function is expanded
about so/t as a Taylor's series in its argument. Con—

sidering only the first two terms,

So h " So ' so

by? = bur) + (7—) bs(T:—) . (1.51)
Since
11 hi
m = SJ - si = bS(t ) bS(E—-), (1.52)

 

 

_ = _J____i. '.__
SJ SI t bs(t )
2h + Si - si ' so
= t bS('E-—) . (1.53)
Solving 1.53 for sj - Si:
2h b'(So/t)
s. - s1 = S . (1.54)
3 t + bé(so/t)
Therefore,
2 b'(So/t)
x" = s . (1.55)

 

t + bé(so/t)

3. Paramagnetic Case

In order to calculate the paramagnetic susceptibility
(t greater than 1), it is necessary to assume t large
enough so that the argument of the Brillouin function is
small enough so that the function may be expanded. For

small y,

386) = (143—1) y. (1.56)

20

or bS(y) = y. (1.57)

Then, since in the paramagnetic state the sublattice
vectors are in the same direction as the field, equations

1.10 become

 

 

 

h - 3.
Si = ——t———1 (1.58)
h - si
and $3. = —-E—-—— - (1.59)
2h - (si + Si)
So, m = si + sj = t . (1.60)
_ 2h
Or, si + sj — 1 + t , (1.61)
and therefore X = 2 (1 62)
’ para 1 + t '

for t large. Note that the derivation of this expres-
sion does not guarantee its validity for t only slightly
greater than 1, although it may actually be valid.

X" and XL, along with X are Shown in Fig. 4

para’
for two values of S. Two points are to be noted. First
that XL is greater than X". and second that X" de-

creases to 0 at t = 0.

4. Arbitrary Orientation Case

If a field is applied at an arbitrary angle from the
easy axis, the susceptibility in that direction is calculated
as follows. For a relatively small field, the parallel

and perpendicular components of the induced magnetization

|.0

0.2

0.0

21

 

 

0.0

Fig. 4.

 

.l.
(Xpara
/X" (S=|/2)
XII (S = 5/2)
,fifldu L_J L I _I I I I I l I I ' I J I I J
0.5 ' |.0 L5 2.0

T/ TN

Parallel, perpendicular and paramagnetic suscepti—
bilities.

22

are:

m" = X" h" (1.63)
and mL = X; h‘L , (1.64)
where h,I = h cos ¢ and hL = h sin 0. The component of

the induced magnetization in the direction of the field
is given by the sum of the projections, in the direction

of the field, of the above magnetization components:

X" h,' cos w +’X1 hL sin w (1.65)
or, X" h cos2 w +'X1 h sin2 w. (1.66)
So, X = X" cos2 w +'X1 sin2 ¢- (1.67)

H. Spin Flop

1. The Phenomenon of Spin Flop

 

An immediate consequence of the difference between
X" and XL is the phenomenon of spin flop. This invol-
ves a 900 rotation of the direction of magnetization when
h'> is applied along the easy axis. That this is possible
is seen upon comparison of the free energy in the case
that h-> is parallel to the easy axis and ‘5) is also
parallel to it to the free energy in the case that h'> is
parallel to the easy axis and 2?. is perpendicular to it.
The free energy of the first case is - é-X" hz, and for
the second case it is a - %-XL hz, where a is the

anisotropy constant of equation 1.33. Thus there is a

field such that

23

.l 2 _ l. 2
-2X,,hC—a-2XLhC. (1.68)

 

_ L 2a
Or, hC — J'xL _‘ X" . (1.69)

This is the same as equation 1.35. As can be seen from
equation 1.34, for 9 = 0 and h greater than he’

w = 90°. That is to say, A's direction flops at h = he'
This phenomenon was first predicted by Neel15 in 1936

and was first observed in 1951 by Gorter et. al.2° in a

crystal of CuC12°2H20.
2. Prediction of the Critical Field

It is worth noting at this point that hc may be
predicted from three susceptibility measurements. If X
is measured along h'> at an angle 9 from the easy axis,
and if the near zero field values of X_L and X" are
obtained, then the following two equations,previously
derived, can be used to predict hc:

X = X,, cos2 ¢ + X1 sin2 8
(1.70)

tan 2¢ = Sln 29 .

cos 29 - (h/hc)2

This possibility does not seem to have been previously
considered. (Indeed, approximations are involved in the
above equations, but an order of magnitude prediction is

certainly a realistic possibility.)

24

3. Equations for the Critical Field

Upon substitution of the previously derived values of
X.I and XL in equation 1.69, the following equation may

be written:

h: = 2 (1.71)
t

bé(50/t)

 

 

To evaluate this equation, one needs to know a as a func-
tion of temperature and spin. Yosida14 has made a calcu-
lation, based on statistical mechanics, of the temperature
variation of a for S greater than %n based on the as-
sumption that the ionic energy causes the anisotropy of
the entire crystal (versus the Zeeman energy and the ex-
change energy of neighboring magnetic ions). His result
for the anisotropy constant is

a = A $2 (§—§¥l-- 3BS(x) ég- coth(§§)) (1.72)
where

x = -2J z s So/kT.

At T = 0 this becomes AS(S - é). So,

 

 

 

 

 

but) = =~=<——s g: 1 - 3bs<e3> §1§C°th(T?—'i 1 5%»
32(57 (3 --l) (1 -‘ 2 . ) (1.73)
' . H 2 . 1 + t
L‘ bé(50/t) “

This function has been calculated numerically on the computer

25
and the results are shown in Fig. 5 for several values of
S.

For the case S = %” the expression giving a in equa-
tion 1.72 identically vanishes. Yosida12 has made a calcu-
lation of the anisotropy constant for this case under the
assumption that the anisotrOpy energy arises from the
crystalline electric field at a magnetic ion site and from
an anisotropic coupling of two spins (such as dipole-dipole
interaction). In this case the anisotropy constant shows
the same temperature variation as $3. Thus,

.. ._ 1/2
hc(t) _ S0

h (0) — 1 _ 2 . (1.74)

 

 

 

1 + ' t
bS(SO/t)~

 

 

This equation is also shown in Fig. 5.

II. Applications of the Theory

A. Observation of Spin Flop Boundaries

1. Variation of Temperature in an Isentropic Mag-

 

netization

 

On the basis of the preceding theory, it is obvious
that the phenomenon of spin flop may easily be observed by
adiabatic means. Whether the system is in the normal
antiferromagnetic state, or whether it is in the spin-flop
state may be determined by observation of the quantity

dT/dH. Now.

26

 

2-0 F" SPIN FLOP CRITICAL. FIELDS
,1... ['8 .-
O" .—
o
I l.6 - -
\ ' . S= l/2
A .. S=l
\ [.4 “""' 8:5/2
(...... (note that order. of ‘ .-
V . "' curves inverts at
3:0 | 2 low temperatures) /“
LO LlfiJc-e'fiii/X I l I I I I

 

 

0.0 0.2 0.4 0.6 0.8 I.0
T/ TN

Fig. 5. Normalized Spin flop critical fields.

27

as
)

i—E. = 9%) “Ti-"1:? , (1...)
55

where S now is entropy, H is field, and T is tempera-

ture. But

T (%%)H = CH(T,H) . (1.76)

where CH is the constant field specific heat, and by a

Maxwell relation:

I§%>T (3%) (1.77)
dT _ -T 8M

SO, EFT-El; ( 8—17)}1 , (1.78)
dT _ TH a

or, afi- - - E; (5%)}.1 (1.79)

Thus if the system is in a state such that X = X", then
dT/dH is negative, since qu/dT is positive. If the
system is such that X = XL, then dT/dH is zero, since
dXL/dT is zero. So, if the field is aligned along the
easy axis and increased adiabatically, the temperature of
the sample should decrease until the spins flop. Thereafter,
there Should be no magnetocaloric effect.

On the basis of the molecular field model Garret has
computed the values of h and t along several isentropes
for a spin 1/2 system and it is seeh that the isentrOpes
indeed have negative slopes in the h-t plane. IsentrOpes

could be calculated for a spin S system using the

28

generalized Garrett equations previously derived above,
but their general features should not be expected to vary

much from the Spin 1/2 system.

2. Variation of Temperature in an Isentropic Rotation

 

In addition, equation 1.67 may be combined with equa—
tion 1.79, for the case when the field is at an arbitrary

angle 9 from the easy axis. Then,

 

91 . _ It 2 5X"
dH C cos ¢ (Sf-) . (1.80)
H H
where W is given by equation 1.34. In terms of 9,
£1 = "‘lfl'l(1 + COS 29 — (H/Hc) )(axu)
dH CH 2 (1 — 2(H/HC) cos 29 + (H/HC)2)1/2 5T ”H
(1.81)
. 8x" . .
Then, if (5E7-) 1s relatively constant between To and
TH, and if CH is also relatively constant, equation 1.81

may be integrated from (To, 0) to (T H). Then,

HI

h

Th = To exp[c f h'(1 + cos 29 - h
0

1
(1 - 2h' cos 29 + h'z) /2

 

)dh'], (1.82)

9 - 2
where C:— (-Xmg H /2C , and h = H/H .
5T H c H c

(For the type of substances used in this study, C turns
out to be of the order 1/10 to 1.)
(Th/To)1/C has been calculated from 1.82 and is

shown in Fig. 6. Th/To has

I.2 " 'ISENTRQPIC ROTATIONS (h<l)

 

 

l .
0 90 . ISO 270 360
9 (DEGREES)

 

 

 

(a)
I.2" ISENTROPIC ROTATIONS (h>I)
..8
8
7.... 0.8---
F0 f2: :\
\ 0.6-
I“;
... 0.4 -
0.2
0 ISO 60
9 (DEGREES)2
03)

Fig. 6. Predicted temperature variation in an isentropic
rotation.

30
been plotted against 9 for several values of b. As can
be seen in Fig. 6a, for h less than 1, the sample takes
on its minimum temperature at 9 = 0. This fact can be
used to align a magnetic field along the easy axis in an
isentropic Situation.

Furthermore, note should be taken of the fact that for
h greater than 1, the minimum in temperature (as a func—
tion of angle in an isentropic rotation of the field)
becomes a maximum, as seen in Fig. 6b. This may be used
as confirmation of Spin-flopping, although caution must be
used, Since the model is basically a two dimensional model,
and the same results are not to be expected if the Spins
flop out of the plane of rotation of the field. Also,
since, in fact, there also exists anisotropy in the para-
magnetic state, if there is a change of principal axes
between the antiferromagnetic and paramagnetic states, a
temperature minimum becoming a temperature maximum may only
indicate an antiferromagnetic-paramagnetic transition.

As an aside, note that although basically a two di-
mensional model has been used, Nagamiya21 has done calcu-
lations for fields in different planes of a three dimensional
system, and has found that Spin-flop can exist in one
crystal plane for values of h within a critical hyperbola

region.
B. Observation of Paramagnetic Boundaries

Finally, as noted by Garrett, the transition from the

31

antiferromagnetic state to the paramagnetic state is a
second order transition. Thus, the point of the transi-
tion may easily be determined adiabatically by finding the
temperature at which there is a discontinuity in the slepe
of the Specific heat as a function of temperature in a

constant field.

CHAPTER II

EXPERIMENTAL METHODS

I. Experimental Apparatus

 

The experimental data were taken, using two distinct
types of calorimeters. With the exception of minor modi-
fications, the main experimental apparatus has been de-
scribed elsewherezz, but for the sake of completeness it
will be briefly redescribed below. A secondary triple-can
calorimeter system, with modifications to cut down heat
leaks, was used on a set of zero field specific heat runs
and has also been previously described23. Only the modi-

fications to it will be noted here.

A. Triple—Can Calorimeter Modifications

 

To reduce the heat leaks to the inner sample can of
the triple-can system, the leads which previously came
down this can's pumping line were brought in from the liq-
uid helium bath through an epoxy seal similar to those of
Wheatley24. A small tube was also attached to the end of
the inner can's pumping line sudh that the line terminated
in a T-joint, the pumping being through the sides of the
T. Also, only the outer and middle cans were used on the
runs. Prior to the above modifications, there were appreci-
able heat leaks to the sample at 1.2°K; these were elimin-

ated for all practical purposes.

32

33

B. Main Dewar and Calorimeter

The main experimental apparatus consisted of the
narrow-tailed pyrex helium dewar shown in Fig. 7 and the
calorimeter Shown in Figs. 8 and 9. The dewar is of com-
mon style, designed for insertion between the poles of a
magnet, and was made to order by H. S. Martin & Son,
Evanston, Illinois. The calorimeter was designed so that
hopefully, temperatures below 1.2°K could be reached.

This was the lowest temperature that could be attained
with the standard medium sized pump, which was available,
pumping on a large volume of liquid helium through a large
line. By pumping on a smaller volume of liquid helium
which was surrounded by liquid helium, although still iso-
lated from it, lower temperatures could be attained.

This calorimeter was somewhat successful in this aspect,
in that the temperature of the helium can normally could
be reduced to 10K. Note also should be made of the at-
tempts to cut heat leaks to the inner can: 1) an extra
shield was placed inside the top of the outer can to cut
down radiation from the outer can evacuation line; 2) to
reduce radiation into the inner can, right angle bends were
put in its pumping line inside the tOp of the outer can
flange; 3) the inner can evacuation lines (and the elec—
trical leads in one of those lines) were partially sur-
rounded by the inside helium bath; 4) the leads were wrap—
ped around and varnished to the bottom of the helium can

for good thermal contact; 5) a brass radiation shield to

 

/\ A H

 

85mm
I 55 m m
2' Ofim‘: .... I

I

—-—-—————-—————

\\\\ \ \\ )l\\\}\WI\\\\\\\\\\\\\\\<\\D 3‘. ’..'.'~ 1/ r '. - - '- .3

 

 

<1“ 95 mm;
ID.

34'

III-30640 2mm
BORE STOPCOCK
(90° FROM FILL

PORTS)

  

432 mm OD. FILL
w\PORTS, l80°
APART, 2'I LONG

C R0 88 HATCHING

 

WIDE AT THE BOTTOM

 

’- — I75mm 7, REPRESENTS

a 0.0. a? VACUUM JACKETS

)1: - I45mm-M

; LD. j BOTH DEWARSARE

/ -Hsmm- ; SILVERED,WI'I'HA
IOOOmm ; 0.0 5 TAPERED SLIT,I0mm

/

5 a

i r

x E

 

 

If.

 

J4

\ \\\\\\\L\‘

 

475 mm

 

 

\\\\ \ \\ \\\-}‘.

 

 

 

 

 

 
    
 
  

\\\\\\l~ \\\\\\\\\\

 

€S\\\\\V\\\\\\\\\ \\\\\

\

 

 

\.

Fig. 7.

Pyrex helium dewar.

   
   

 

~---- 66 mm 0.0.
58 mm 0.0.

‘-—— 50 mm 0.0.

42 mm 0.0.

 

 

 

KR

-«-(—-

I-g--

 

Lb

'_‘ _..__._""—.:::T._‘ ‘

 

I
[ii
D
I

A

 

A

35

~L~L/”"~T\fi
WI°":U\\\HELIUM CAN
PUMPING LINE
a;
\‘\> INNER CAN
/ EVACUATION LINES

 

}\

 

ITNNH

-2!

HELIUM CAN
4—— OUTER CAN

CERROLOW II7
SOLDER J OINT

BATH THERMOMETER

Ll... EMBEDDED IN BOTTOM

TIP OF HELIUM CAN

,/ BRASS RADIATION SHIELD

 

INNER CAN

GERMAN SILVER
SUPPORT FOR

 

(

 

 

 

 

 

TERMINAL BOARD AND
SAMPLE HOLDER

NYLON SPACER
BAKELITE

// TERMINAL BOARD

CERRLOW II7

 

.o/SOLDER JOINT
h— I"-*'I

Cross section of body of calorimeter.

36

.HODOEHHOHNU mo mow mo mooIBUOm mmouu mounm cam usoum .m .mfim

TI

 

 

 

 

_I._

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I\. [\I j
mmozquLLHOm oe cum: z<o eukao IIIIIN I I A:
mammom cam: zmuBc J
ummem mmmuzqew uueoz III _
mmno: omcc<EIII . _ II
mmIm Exo:m IImwwmr.I ...wm n
...ihw IA rIp IIOIID
82:70 cam... A\mUv//I/AM w M U/é e
/ / / /1
muse: omcc<eIII I 7 M A
I// 9.1. we... WWI/fl M m)m 4.
RI)“. . / .I.
I I. ,_ / /.
mm we m mmmm w
W \/
Wm ..I/I/M on:
m m
_ 1 L.
m>u<> (I m _
uuomuz _ _

 

 

m2: 02_n:>:4n_ Z<0 EDJMI

 

“12:.292845
zqo mweno

3:

m2: ZO_._.<DO<>w 240 mmzz.

37

block any radiation that did get down the inner can evacua-
tion line was placed directly below the ends of the line;
6) copper, which has a higher thermal conductivity than
brass, was used in most of the calorimeter below the outer
can flange; 7) German silver, which has a low thermal con-
ductivity and is non-magnetic, was used in the pumping
lines which essentially support the outer can and its
flange to the upper flange, and the inner can and the helium
can to the outer can flange; 8) the bottom of the outside
can was made removable to allow insertion of a nylon
spacer which insured no direct physical thermal contact of
the inner and outer cans.

The calorimeter differs from its previous description
only in the brass radiation Shield (which replaced an
epoxy terminal ring-radiation shield) and the German silver
extension strip which now supports a bakelite terminal
board. This latter modification increased the ease of
soldering leads to the terminals and also increased the
thermal path from the lead board to the helium can. It
was important that this path be as long as possible, Since

the sample holder was supported from the terminal board.
C. Sample Holder

The C-clamp sample holder and its support are Shown
in Fig. 10. (The C-clamp section was made to size for each
crystal used.) Since a magnetic field experiment on a

single crystal depends on the relative orientation of the

38

00

/ SUPPORT

 

 

 

  

2-56 k

~2" SCREWS N \CLEARANCE HOLE
FOR 2-56 SCREW

(ALL OTHER HOLES
IL TAPPED WITH
7h

&
om.)

 

 

 

 

 

 

 

‘ 2-64 THREADS)

C)

\L 0
' J ' ALL PARTS NYLON

N

Fig. 10. C~clamp sample holder.

39
crystal and the field, it was important that the crystal
be immobile. An early experiment showed that when a single
crystal was supported by nylon threads and a field rotated
about it, there was enough torque on the crystal to break
the threads and free the crystal.

So, a more solid support was needed. However, since
adiabatic experiments were to be done, a support with low
thermal conductivity was needed. In addition, it was
highly desirable that the support have some internal degrees
of freedom so that a crystal's orientation might be
changed. other workers have used pyrex glass25 and carbon26,
but it was felt that nylon had advantages over both of
these, since it can be tapped to take a screw and since
its fixumal conductivity is lower than most other mater—
ials27. It should be noted that the threads tapped in
the holder were for the next size smaller screw than was
actually used in the holes. This procedure insured that
the holder would maintain its position, once set.

It must be noted that the holder was not completely
satisfactory, Since a certain amount of heat leak still
existed. How much these heat leaks actually affected the
measurements depended ontflmaheat capacity of the sample.
The larger the heat capacity, the less a given heat leak
affected its temperature. So, the holder provided only an
apparent adiabatic situation and was not satisfactory in

a region where the sample's specific heat was small.

40

D. Pumps

Five pumps were used in the experiments. The medium
size, high capacity pump used to pump on the liquid helium
was a Kinney KDH-130. The high vacuum system used to
evacuate the inner and outer cans consisted of a Welch
DuoSeal pump used as a forepump for a water cooled diffusion
pump which, in turn, was used as a forepump of another
water cooled diffusion pump. Pressures as low as 10-6mm
Hg were possible with this system. These pumping systems
are shown schematically in Fig. 11. Another Welch DuoSeal

pump was used to maintain a good vacuum on one Side of the

U—tube manometer system.
E. Pressure Gauges

Pressures in the high vacuum system could be read by
means of an NRC thermocouple-ion gauge meter. Pressures
in the liquid helium pumping system could be read on any
of three gauges which could be connected to the system:

a mercury filled U-tube manometer, an oil filled U-tube
manometer, or a McLeod type gauge. Above 2.5 cm Hg of
pressure, pressure was measured by measuring the difference
in height of the mercury levels in the two arms of the
mercury U-tube manometer. Below 2.5 cm Hg , the Todd Mc-
Leod type gauge was used. (It is important that no con-
densable vapors, e.g., air moisture, be allowed into a Mc-
Leod gauge system. Therefore, dry nitrogen gas was used

to pressurize the gauge for each reading, and as there could

41

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

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42

be a small amount of leakage of air into the system when
it was shut down, the gauge was completely pumped out
prior to its usage.)

The oil filled U-tube manometer was used below 4.0
cm. Hg for a qualitative observation of decrease in
pressure (temperature) during the thermometer calibration
part of a run. A Duragauge gauge (30 in. of vacuum to
15 # of pressure) connected to the pumping system at all
times could be used as a crude qualitative measure of vacu-

um also.

F. Thermometers and Current Supplies

 

To measure temperatures, 1/10th watt 56 ohm Allen
Bradley carbon resistors were used. Two Six inch 31 mil
manganin wires were soldered to each lead of a resistor,
close to its body. Soldering was done at as low a tempera-
ture as possible while the resistor was being air cooled
in order to lessen damage to the resistor. Current was
passed through the resistor through two of the leads and
the voltage across the resistor was measured potentiometri—
cally through the other two leads. A one micro—ampere
current was supplied by two 28 volt Mallory mercury bat-
teries in series with three precision wirewound resistors
totalling 56 megohms and a variable 20 megohm carbon po-
tentiometer. The variable resistor was set to give the
required current by potentiometrically checking the voltage

across a 100K ohm precision resistor which was also in

43

series with the batteries. To insure a stable temperature
environment, all the above resistors and batteries were en-
cased in a 1/2-inch thidk wooden box, as it was determined
that fluctuating room temperatures significantly affected
the current output (this probably was due to variations in
the carbon resistor or the batteries). Two of these cur-
rent supplies were used: one to supply the sample there

mometer, and another to supply the helium bath thermometer.

G. Measuring Electronics

 

A Leeds and Northrup K-3 potentiometer with a gal-
vanometer system consisting of a Leeds and Northrup 9835-B
microvolt amplifier used in conjunction with a Leeds and
Northrup two-pen Speedomax G recorder (with a 5 millivolt
range card) was used to either measure the resistance of
the sample thermometer (the potentiometer balanced), or
observe its change in time (the potentiometer unbalanced).
The amplifier was attached to the potentiometer so as to
give maximum sensitivity to the system. (The sensitivity
of the system could be changed by means of the amplifier's
range selector.)

A similar system (with only a one-pen recorder) was
used to observe fluctuations in the temperature of the bath.
This latter system was also used to measure the voltage
across a 400 ohm heater on the sample. (A heater consisted

of one foot of 14 mil Evanohm wire* with four leads similar

 

*
Available from Wilber B. Driver Co., Newark, N.J.

44

to those attached to the thermometer.) (In this case the
other pen of the two—pen recorder was used in the galvanom-
eter system.) When a heater voltage was being measured,
the current through the heater was monitored by using the
one-pen recorder to read the voltage across a precision
resistor in series with the heater. (The heater current
supply consisted of three five—cell 6.6 volt Edison batter-
ies in series with a variable resistor used to adjust the
current output.)

The heater was switched onanuioff by a relay attached
to an electronic timer which could be preset to run for a
given length of time, once started. The relay connected
an external resistor (of the same value resistance as the
heater) in place of the heater in the heater circuit when
the timer was off so as to minimize any pulses from the
current monitoring recorder.

All the measuring circuits are box-diagramed in Fig.

12.

H. Magnet

The magnet used in the experiments was constructed
from a yoke and poles made to order to accommodate 36 c0p-
per strip coils 21 inches in diameter. The poles were
made from 10-10 cold rolled steel and were 6.75 inches in
diameter. The pole gap was 2.75 inches. Current for the
magnet was obtained from two D.C. generators placed in

series with the coils. The current output of the generators

 

 

 

 

 

 

 

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46
was controlled by a variable rheostat in the field circuit
of the generators, and a 50 ampere Simpson meter was used
with a 50 mv shunt to measure the current in the coils. A
Rawson-Lush rotating coil gaussmeter was used with a type
501 Rawson-Lush indicator to measure the field between the
pole faces. The field was Observed to be uniform, within
the reading sensitivity of the indicator (about 20 gauss),
over about five cubic inches in the center of the gap.
The magnet was placed on a frame with a rotating base. A
Selsyn motor was attached to the magnet frame and the
motor's shaft was attached to the non-rotating center sec-
tion of the base. This motor was connected in parallel to
a similar motor attached to a two foot diameter dial. The
latter motor's shaft had a pointer attached to it. This
arrangement allowed easy angular adjustment of the magnet
(a lever arm was attached to the magnet frame to ease its
rotation). The magnet was air cooled by two fans placed

beneath it.

II. Experimental Procedures

 

A. Sample Preparation and Precooling

 

An experiment actually begins with the growing of the
sample to be used. Single crystals of one to two cc in size,
of the types of materials used in this study, are often dif-
ficult to grow and much patience is required. Once an ade-

quate size single crystal was available, it was refrigerated

47
to prevent its losing or gaining water. When it was to be
used in an experiment, the external angles between its
faces were measured with a protractor-goniometer, and by
comparison of these angles with those given in the litera—
ture, the crystallographic axes were determined. The
sample was then weighed, a heater was wound around the
crystal. and a carbon thermometer was tied to the crystal
with a cotton thread. Usually the sample, the heater and
the thermometer were then coated with G. E. 1202 varnish
to insure good thermal contact. (Since the varnish reacted
with FeC12-4H20, Krylon clear plastic spray was used on
the fourth sample of that substance, instead.)

The sample was then placed in the C-clamp portion of
the nylon holder, and the clamp then tightened. Care was
taken so that major edges and faces of the crystal remained
visible. The entire holder was then screwed tight to the
bakelite terminal board and the leads from the sample
heater and thermometer were then soldered to their termin—
als. Then, it having been previously decided which crystal-
lographic plane was to be parallel to the rotation plane of
the magnet, the holder was adjusted so that the apprOpriate
edges or faces of the crystal were horizontal or vertical.
This was done carefully by eyesight, and, depending on the
orientation desired and the sizes of the faces and edges
used for the orientation, the amount of misorientation was

estimated to be from two to five degrees.

48

The inside can was then slipped up under the helium
can and soldered closed with Cerrolow Alloy 117*. The out-
side can was then slipped up onto its flange (with five-
ampere Buss fuse wire O-rings in place on the O-ring
gnxwes) and tightened down. The nylon Spacer was slipped
up between the inner and outer cans from the bottom of the
outer can, after which the bottom was soldered in place
with the Cerrolow alloy. The calorimeter being completely
assembled, it was put inside the dewar and all flanges
were sealed tight and electrical connections were checked
for continuity.

The inner and outer can evacuation line valves were
closed, the helium can pumping valve closed, the needle
lvalve Opened, and the dewar evacuated with the high capacity
pump. Liquid nitrogen was then put into the dewar's
nitrogen jacket; when the jacket was filled to the top,
the pump was closed off and helium gas at atmOSpheric pres-

sure was admitted to the inside of the dewar.
B. Helium Transfer and Calibration

The above procedure was generally accomplished in two
to three hours so that there would be little damage to the
crystal from exposure to air at room temperature. Several
hours later, when it was certain the crystal was cold enough
to be pumped on without damage, the air which had been
trapped in the inner and outer cans was pumped off with the

Welch pump of the high vacuum system. When the pressure

 

*Available from Cerro Sales Corporation, New York.

49
was below 50 microns Hg, the diffusion pumps were cut into
the pumping system and were allowed to pump until the pres-
sure was about 10-4 mm Hg. The high vacuum system was
then closed off and helium gas at atmOSpheric pressure
was admitted to the inner and outer cans. This procedure
insured complete pre-cooling of the sample overnight.

Early the next morning, a thermometer calibration
point at liquid nitrogen temperatures was taken by record-
ing the thermometer's resistance, atmospheric pressure and
room temperature. The helium exchange gas pressure was
then reduced with the Welch pump to about 1000 microns Hg;
with this pressure inside the cans and with the needle
valve closed, liquid helium.was transferred from a storage
dewar into the experimental dewar. Both the bath and sample
thermometer resistances were monitored on the two-pen re-
corder during the transfer. The transfer was done Slowly
enough to insure the sample's complete cooling to liquid
helium temperatures before the end of the transfer. This
procedure usually resulted in a five to seven liter transfer.

After the transfer, liquid helium was admitted into
the helium can by opening the needle valve. The outer can
valve was opened to the high vacuum system, the helium can
pumping valve was opened, and the dewar vent valve was
opened. When both the bath and sample thermometers had come
to equilibrium, and when the pressure in the outer can was
less than about 8.10"5 mm Hg, another thermometer calibra-

tion point was taken by recording the resistance of the

50
sample thermometer, room temperature, atmospheric pressure,
and the approximate height of liquid above the sample. The
needle valve was then closed, the dewar vent valve connected
to a helium recovery system, and the high capacity pump
needle valve opened to pump on the helium in the helium
can. Approximately every 0.2-0.3°K the pumping was slowed
down to allow both bath and sample thermometers to come to
equilibrium. Equilibrium of the bath was attained by ad-
justing the needle valve to keep the resistance of the
bath thermometer (observed on the recorder) constant. When
both thermometers were stable, the sample thermometer's
resistance and atmospheric pressure were recorded, and the
pump valve was again opened until the next calibration
point was reached. Each calibration point generally took
from ten to thirty minutes, and the entire calibration pro—
cedure lasted about three hours.

When the calibration was finished, the outer can valve
was closed, and the inner can was evacuated. The sample
was considered to be isolated when the pressure in the
inner can was less than about 5°10“5 mm Hg; the outer can

valve could then be reopened.
C. Specific Heat Measurements

For a specific heat experiment, the potentiometer sys—
tem which was being used to observe the bath thermometer
was then set to observe the voltage across the sample heater,

and the variable resistance in the heater circuit was set

51
to give a heater current of about 0.1 ma. The potentiometer
observing the sample thermometer was set off balance so
that the recorder pen was near the left side of the chart,
and the potentiometer reading (converted to resistance)
was recorded. The automatic timer was started, the sample
was heated, and the effect on the sample's temperature
was observed on the chart recorder. The heat input was ad-
justed so that in 20-30 seconds there would be 10-20 divi—
sions change in the position of the pen observing the
sample. Since there are always some heat leaks present
in a calorimeter, the sample's temperature was seldom con-
stant when the heater was off. Depending on its temperature
and the temperature of the bath, the sample was normally
either warming or cooling. Heat was introduced to the
sample only after there was a relatively constant rate of
change of temperature for about a minute. When the re-
corder pen was at the right Side of the chart, the potenti-
ometer was adjusted so the pen moved back to the left side
and the above procedure was repeated. (Each time the
timer was on, the length of time it was on, the amount of
current being used, and the voltage across the heater were

recorded.)
D. Magnetic Field Experiments

Four different types of magnetic field experiments
were done in this study: 1) adiabatic rotation of the mag-

netic field about the sample, 2) adiabatic magnetization

52
of the sample, 3) discontinuous heat input to the sample
(specific heat measurements), and 4) continuous constant
power input to the sample. Generally a rotation was done
immediately after isolating the sample and turning on the
field.

By adiabatically rotating the field, the preferred
axis of magnetization could be determined by looking for
the magnet position in which the temperature of the sample
was a minimum. For this positioning to be unambiguous,
there could only be two minima 180° apart, thus insuring
the sample to be in the AF state at the position of the
minima. (If the field were higher than some critical value,
the sample would be in some other state in the easy direc-
tion, and other minima would appear.) Generally, in order
to be able to see a temperature difference upon rotation,
the field had to be several kilogauss.

Although most rotation data points were taken five or
ten degrees apart, the position of the easy axis was always
determined to within one degree by careful Observation of
the temperature change on a slow continuous rotation. For
each point taken in an adiabatic rotation, the magnet posi-
tion was read from the Selsyn dial and the sample's tempera-
ture was determined by making a null measurement with the
potentiometer of the voltage across the sample thermometer,
thus determining its resistance.

After the direction of the easy axis was determined,

isentropes could be measured along either the easy or hard

53
axis (the hard axis being 90° away from the easy direction).
Points on a given isentrope were obtained by potentiometri-
cally measuring the sample's temperature for different
values of field (the value of the field being determined
from the current in the magnet).

After the initial isentrope was obtained, higher tem-
perature isentrOpeS could be obtained by warming up the
sample through its heater and then measuring the temperature
at different fields again. Since the nylon holder permit-
ted a heat leak, any measured isentrope actually consists
of parts of many close-lying isentropes. So, although the
isentrOpes are not quantitatively correct in every detail,
their shapes should be qualitatively correct.

Isentropes along the easy axis direction were used
to observe first-order phase boundaries. In this case there
was sometimes a change in Sign of the slope of the isen-
trope at the boundary in the H-T diagram and the point of
change could easily be followed in the H-T plane by ob-
serving only a small part of the isentrOpe near the change.
Points were obtained by slowly varying the field and re-
cording resistance and field values as near as possible to
the point of change. In order to get a good average, since
the measurements were quasi-static, several points were
generally taken, both with the field being raised and with
it being lowered. (If there was no point of change, a

complete isentrope was taken with static measurements.)

54

To observe a second-order transition boundary, a con-
stant field specific heat experiment was done until the
boundary was crossed (easily seen by the sudden increase
in the amount of temperature change for a given heat input).
Then the field was set to a new value, the inner and outer
can valves Shut, the high vacuum pumping system closed off,
and about .05—.2 mm Hg of helium gas admitted to the pump-
ing line outside the inner can valve. The valve was slowly
Opened and the sample would cool back to the temperature
of the helium bath. The amount of exchange gas determined
the speed of cooling. The inside can was evacuated with
the high vacuum system as soon as it was evident that the
sample had cooled enough so that it was sure to be in the
ordered state again. Then specific heat measurements were
done again.

In some cases, for instance, when the heat leak was
very large, the second-order transition point was observed
by loOking for a change in the slope of a continuous heat-
ing track on the recorder. (The point of Slope change was
also looked for in the specific heat measurements.) In
these measurements , the potentiometer reading (converted
to resistance) was recorded whenever the recorder pen was

moved to the left side of the chart.

E. Shut-Down

 

At the end of a run, all pumps were opened wide, the

diffusion pump heaters turned off, and liquid nitrogen

55
maintained in the dewar jacket until about twelve hours
before the calorimeter was to be removed from the dewar.
The pumps were shut off and dry nitrogen gas was admitted
to the dewar about four hours before removal of the calor-
imeter. Through these procedures, the sample was still cold
when it was removed, so that damage to the crystal (from
Sitting in air at room temperature) was minimized. The
sample was returned to the refrigerator after it was removed

from the calorimeter.

III. Reduction of Experimental Data

Except for preliminary reduction of data from the re-
corder chart, all data reduction and calculations were done
on the M.S.U. CDC 3600 computer. Appendix I contains a
listing of the programs used plus a listing of a typical

input deck.,
A. Conversion of Pressures to Temperatures

Most of the data reduction is involved with determining
the temperature of the sample. The pressure readings taken
for each run, for the calibration of the thermometer, were
first changed into pressures at 0°C. Then hydrostatic cor-
rections were made for the pressure of the amount of liquid
helium above the sample, since the vapor pressure measurements
correSpond to the temperature at the surface of the liquid.

A correction was added to the 0°C corrected manometer read-

ing, equal to the height of the helium times the density of

56
helium at its estimated temperature divided by the density
of mercury at 0°C. (For the main calorimeter, the height
of the helium could not be seen after the initial point,
Since the helium was isolated inside the cans, so the height
was guessed to be six or seven cm.) Since a temperature
gradient cannot exist in liquid helium below the lambda
point (2.17°K), hydrostatic corrections were not made below
this temperature. Also, below a pressure of 2.5 cm Hg, no
pressure corrections to 0°C were made, since those read-
ings were taken on the McLeod gauge which is relatively in-
sensitive to temperature changes.

The corrected pressure readings were translated to
temperatures using "The 1958 He4 Scale of Temperatures"23.
The nitrogen temperature calibration point, taken before
the helium transfer, was translated to a temperature from

the equation29

_ 255.821
T ' 6.49594 - log10 p + 6‘6 '

 

where P is the 0°C corrected manometer reading in mm.

B. Temperature Calibration Equation

 

An equation was then fitted to these resistance—ther-
mometer calibration points for interpolation (or extrapola-
tion) of temperatures from thermometer resistances. The

standard equation that has been used3° is

 

J log R7T = a + b log R,

57

where R is resistance and T is temperature. However,
it has been observed that when the calibration points are
least-square fitted with the above equation, there are
systematic deviations of the points from this curve for
the thermometers used in this study. To take advantage of
this fact, the relative deviations are then least—square
fitted to a polynomial in log(resistance). (Normally, a
fourth degree polynomial was used, although, in order to
reduce extrapolation errors, a third degree fit could be
used if there was to be any extrapolation from the equation.)
Thus, given a thermometer resistance, the corresponding
temperature could generally be calculated from the follow-

ing equation:

T = log R a + b log R

4 n
1 - 2 C (log R)
0 n

This type of fit usually resulted in no more than
0.1-0.2% deviations of the measured points from the fitted
curve. If there was to be much extrapolation above the
helium calibration region, the unmodified two parameter
equation was used for the fit, since a separate experiment
in which a carbon resistance thermometer was calibrated
against a germanium resistance thermometer showed there to
be less extrapolation error in this case. Similarly, if
there was much extrapolation below the lowest calibration
point, the stndght line fit was assumed to induce less er-
ror. (Another possiblity of a calibration equation is

l/T = a polynomial in log R. The fits resulting from this

58

equation were never as good as those from the modified

straight line fit, so it was never used.)
C. Determination of Thermometer Resistances

If a potentiometer null reading was taken, the thermom-
eter resistance was merely determined by dividing the read—
ing by the current in the thermometer. For the cases, as
in specific heat measurements, where null measurements
were not made, but the distance of the recorder pen from
the null position was known, a "voltage calibration" had
to be made in order to relate the resistance of the thermom-
eter and the potentiometer setting when the pen was at a
given distance from the potentiometer null position. The
relationship used was (see Appendix II):

Ro - [c1 - C2R - 2C3R2 - 3C4R3]D

R = .
T 1 + [c2 f 2C3R I 3C4R2]D

 

where RT = the actual thermometer resistance, D = the
number of divisions on the chart the recorder pen was to

the right of the null position; R initially was set equal

to Ro, the potentiometer setting divided by the thermometer
current, and the C's are constants determined from the

fit to the voltage calibration points. The calculation was

reiterated then, using R = R of the previous calculation

T
until two successive iterations differed by less than 0.1%.
The C's in the above equation were the coefficients ob-

tained from fitting the voltage calibration points taken

59
off the chart to a third degree polynomial. A voltage
calibration point was automatically obtained each time the
recorder pen was made to move across the chart by changing
the potentiometer setting. RoL was the potentiometer set-
ting when the pen was at the left of the change and RoR
the setting when it was at the right (both readings were
converted to resistance). DL and DR were the number of
divisions to the right of the null position when the pen
was at the left and right sides of the change respectively.
(See Fig. 13.) The abscissa and ordinate used in the fit
were (RoL + RoR)/2 and (RoL - RoR)/(DL - DR) respectively.
(Generally, the change was made across the null position
so that (RoL + RoR)/2 approximated RT.)

During the runs on sample four of FeClz-4H20, in
which the thermometer resistance was very large, it was
discovered that there was considerable non-linearity across
the chart page. Some of the observed voltage calibration
points, plotted against the average number of divisions
from the null position at which each point was taken are
shown in Fig. JAI- A graph of voltage calibration points
plotted against thermometer resistance, for different dis-
tances from the null position, derived from the previous
plot, is shown in Fig. 15.. Using this last graph, it was

possible to estimate R for the pen sitting D divisions

TI
right of the null position and a converted potentiometer

setting of Ro, using the crude equation

Ro=RoL

 

 

 

 

 

 

j RECORDER
...... NULL POSITION

   

 

\TRACE OF I
RECORDER PEN

I ' ' '* ‘ ' ' da‘a.
Fig. 13. Exanple of preliminary reduction of chart L
I

.61

 

 

 

I2 E g
- "' VOLTAGE CALIBRATION
'0 _ (5/I AND 5/2/68)
I; + R3=550 K0.
ES " + o R;=IOG K0
(Q ' '
:2 _
I
C? 6 -
>6 . i.
V " +
C)
'0 ,4 T
0\:
'0 T
2 ... C’00 +~I~
C) C) c)
-- O O (3 Cg) 000600
O I I I I I ......l I I I
~40 ~20 0 20 40
Davg.

Fig. 14. Voltage calibration points from FeC12°4H20

(sample 4) runs.

61a

K5
1

DERIVED

- VOLTAGE
CALIBRATION

5
I

00
I
I
a

d R/dD A ( K-OHMS/DIV.) 1
I: m
I I

 

 

 

h)
..J I
II

 

C) I l I I I I

O 200 400 600
R. (Km

Fig. 15. Voltage calibration derived from Fig. 14.

where the quantity in parentheses is the appropriate num-

ber estimated from the second graph.
D. Specific Heat Calculations

The variable quantities in a specific heat measurement
were the heat input to the sample and the temperature change
of the sample produced by that heat input. The quantity of
heat was equal to the (voltage across the heater) x (the
current through the heater) x (the length of time the heater
was on). The effect of heat leaks on the sample was sub-
tracted out by the graphical method indicated in Fig. 13.
The method minimizes any effect from a change in the rate
of the heat leaks and approximates the ideal situation
where the heat input is instantaneous to the sample. (For
a small heat input, there should be little effect to the
heat leak rates.) Thermometer resistances were calculated
for the positions DS and DE with a converted potenti-
ometer setting of Ro and were then transformed into tem-
peratures. The difference of the temperatures was used to
calculate the specific heat and their average was considered

to be the temperature of the sample.
E. Calculation of Magnetic Field

In order to obtain slightly better values for the mag-

netic field as a function of the magnet current, an eighth

63
degree polynomial in the current was fitted to about 50
current-field calibration points. Any necessary inter—
polation was then made from the fitted curve. The cali-

bration curve of the magnet is Shown in Fig. 16.

0.0m

64

.0>HDU soaumnnflamo umcmmz .®H .mwm

Ame—2.3 Hzmmmao

0.0¢ . 0.0m 0.0N 0.0.
_ _ ~ _ _ . — _

ZO_._.<mm_I_<o ...mzoflz wm\om\mo ..

 

L 0.0.

(SSDVO-M) C'IEILI

CHAPTER III
RESULTS AND CONCLUS IONS
I. Results

A. Results of Experiments on Samples with Known Spin Flop

Phases

Since the adiabatic method used to observe the spin
flop state of an antiferromagnet was new, observations were
made on two materials whose Spin flop states have been pre—
viously studied. The studies on CoClz-6H2031r32 and
MnClz-4H2033'34 have indicated that both substances exhibit
spin flop properties near 1°K below 10,000 gauss. Thus,
since these regions of field and temperature were attain—
able with the present equipment, these substances offered
a good test for the adiabatic method of observing the Spin
flop state.

A nearly single crystal of CoClz-GHZO, weighing about
0.8 grams, was grown from an aqueous solution at room tem-
perature. COC12'6H20 is monoclinic with the angle 9 =
122°19'. The habit has been described by Groth35 and the
crystal was oriented on the basis of that description. AS
noted by Schmidt and Friedberg32, large Single crystals of
CoClz-6H20 are difficult to grow. Since the faces of the
crystal used in this study exhibited some irregularities,

more than likely it was not completely single.

65

66

Experimental measurements were made in three different
crystallographic planes: the BC plane, the AB plane and
the AC plane. Isentropic field rotations were made in each
of these planes at two different fields, one below and one
above the critical field for Spin-flopping. (The previous
measurements on COC12'6H20 have indicated a critical field
of 7200 gauss at 1°K which rises to a value of 7825 gauss
at 2.06°K.) The results are shown in Figs. 17, 18, and 19,
and the data are listed in Tables 1, 2, and 3 of Appendix
III. (It should be noted that in most of the isentropic
rotations in this study, there were imperfect adiabatic
conditions, so that at the end of a complete rotation, a
sample's temperature may have been different from its tem—
perature at the beginning of the rotation.)

Note should be made of the peculiar shift of the maxi-
mum in the two BC rotations, for which there is no explana-
tion at present. Also, a third BC rotation, which is not
shown, was done in a higher field of 8830 gauss. The re-
sults of this rotation reproduced, almost point for point,
the results of the low field rotation of Fig. 17a. Although
it is doubtful, there is a slight possibility there was a
misreading of the field in one of these rotations. Thus,
further investigation into the behavior of the magnetization
in the BC plane is indicated.

Since in two of the isentropic rotations, in fields
below the reported critical field (Figs. 17a and 19a), the

temperature is a minimum in the C direction; this direction

a

|.4

|.3

L2

TEMPERATURE (°K)

I.5

I.4

L2

TEMPERATURE (°K)

67

.— CoClz-GHZO (5550 gauss,BCpI0ne,C=55°I

 

|.3.

..LI . . L. . I ."14

0 90 I80 270 560

MAGNET POSITION (DEGREES)
(0)

"" CO‘Clz-GHZO (7860 gauss,BG plane, C=55°I

 

O 90 I80 270 360

MAGNET POS I T IO N (DEGREES)
(b)

Fig. 17. CoClz-GHZO BC plane isentropic rotations.

TEMPERATURE (°K -)

TEMPERATURE (°K)

 

68

COCIZ'GHZO (6 4|O gauss, AB plane, B=0°I

 

 

 

I.5 '-
l.4 -
F
|.3 ""
I.2 -— ' .
I.I — I I I n I ' I .1 I [A I I LJ
O 90 I80 270 360
MAGNET POSITION (DEGREES)
(0)
I.5 '"' CoCI2-6H20 (9700 gau35,AB pIane,B=O°I
I.4 -'
'53 L: ..‘. ..’..,,.‘.. . .. .....-
I.2 +- ' ' ' . .
I,I I I L L I I I I I I I I
O 90 ISO 270 360
MAGNET POSITION (DEGREES)
(b) '
' . 18. CoClz-CHZO AB plane isentropic rotations.

69

"" COCI2°6HzO (64I0 gauss,AOpI0ne,C=—7O°I

 

 

 

 

. A '05
X ’.
1L
_ I.4--
UJ .
9% ' t o. .' .'.°
I.3 "‘0 0 .. °
'21 _ . . , .-
33 o o . .
a. I.2 "'" . .. .. ' . . 0
2 _. " ...
E I,I I I I I I . I I I I I I I
0 90 ' I80 , 270 560
MAGNET POSITION (DEGREES)
(G)
A I.5 "" CoCl2-6H20 (9620 gauss,ACpIane,C-'=70°I
L4“
LLI _ 0.. '0
0: ° . o . o
E3 [(3 _‘ . o .
<1 _ a
a: o ..,oo... . . ..’.,‘,.
til] I.2 "" O .. Q 0..
5 . _
f— I.I I I I I I I I I I I l I
O 90 ISO 270 360
MAGNET POSITION (DEGREES)
(b)
Fig. 19. CoClz'GHZO AC plane isentropic rotations.

70

is, on the basis of the discussion of isentropic rotations
in Chapter I and the predicted behavior Shown in Fig. 6a,
the preferred axis of magnetization. This is consistent
with the susceptibility results of Haseda36, which gave the
same indication.

The AB rotations (Fig. 18) Show nothing remarkable,
although it is worth noting that it can be seen from these
rotations, that since the temperature is a minimum in the
A direction, after the C axis, the A axis (or perhaps
the A' axis) is the next most preferred axis of magneti-
zation. Thus if the Spins flop, they Should flop to the A
(or maybe A') direction, and an AC plane isentropic ro-
tation, above the critical field, would be expected to
produce results as predicted in Fig. 6b.

The results of the AC plane high field rotation, Shown
in Fig. 19b, do indeed qualitatively correspond to Fig. 6b,
and that CoC12-6H20 undergoes spin flop (the spins flopping
to the A' direction) is partially indicated by these re-
sults.

More conclusive evidence that CoC12°6H20 enters a spin
flop phase between 7200 and 7800 gauss in the temperature
region 1.2-1.7°K is given by the results of the isentropic
magnetizations along the C axis. The isentrope points
thus obtained are listed in Table 4 of Appendix III and
Fig. 20 shows smooth curves which have been drawn through
the individual data points. AS can be seen, there is cool—

ing upon magnetization initially, which disappears when the

spins flop.

71

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(SSOVO-MICTEHA

72

Note should be taken of the change in sign of the isen-
tropes' slopes near 7200 gauss. This fact, as mentioned in
Chapter II, allowed determination of the point of the
phase change without measuring a complete isentrope. The
points thus obtained are listed in Table 5 of Appendix III
and are Shown in Fig. 21, along with a smooth line repre-
senting the phase boundary results of Van der Lugt and
Poulis. (The results of Schmidt and Friedberg coincide
closely to those results and therefore are not shown.) No
points were taken above about 1.76°K, since the change in
Sign of the isentropes' lepeS disappeared above that tem-
perature. However, it is clear that the points obtained
coincide very well with the previously published results.

(To further verify that there is no magnetocaloric ef-
fect when the spins are perpendicular to the field, an
isentropic magnetization along the B axis was done when
the crystal was in the AB plane of rotation orientation.
When the field was changed from about 6400 gauss to near
10,000 gauss, there was less than 5 millidegrees fluctua-
tion in the sample's temperature, thus confirming the pre-
dictions of the theory.)

Also, an isentropic rotation was done in the paramag-
netic state in the BC plane. The data is given in Table 6
of Appendix III and is shown in Fig. 22. It is seen that
in this plane, from observation of the temperature varia-
tion in the rotation, the anisotropy is considerably smaller

in the paramagnetic state than in the ordered state. Also

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A 2.52 COCIz-GHZO (BSBOgauss, BCpIane, C=55°I

Ed ,

32 2.50

DJ . . O. .

(I 2.48 O . . . o o

I? o . ' . . .

< 2.46 . . ‘ ° , ' °

m o O .

z ' - , , . -

It! 2442 l I I I I L I I I I I I
0 90 ISO -270 360

MAGNET POSITION (DEGREES)
Fig. 22. CoClz-GHZO BC plane paramagnetic isentropic

rotation.

75

it is seen that there is no change of principal magnetic
axes between the ordered and unordered state, at least as
far as the B and C axes are concerned.

Two points that may merit further investigation are
the following. First, the magnitude of the field used in
the high field AC rotation, relative to the critical field,
Should not have been expected to produce the relatively
large change in the behavior of the sample when the field
was near the C axis, as seen in Fig. 19. Secondly, the
observed change in Sign of the isentrope Slopes was not
predicted by the Simple molecular field theory, and is not
yet understood.

For the measurements on MnC12'4H20, two different por-
tions of single crystals, borrowed from R. D. Spence, were
used in separate experiments. The first crystal weighed
about 1.0 grams and the second about 1.7 grams. The morpho-
logical description given by Groth35 was used as the basis
of the orientation of each crystal. Like CoClz-GHZO, MnC12:
4H20 is monoclinic with B = 99.74°.

The previous measurements have indicated that the pre-
ferred axis in MnC12-4H20 is the C axis and that a transi-
tion to the Spin flop phase takes place in a field near
8000 gauss near 1°K. Thus, measurements were made only in
the AC and BC planes for MnC12°4H20. The results of
two isentropic rotations (below and above the critical field)
in each of these two planes, using the first crystal, are
Shown in Figs. 23 and 24, and the data are listed in Tables

7 and 8 of Appendix III.

76

MnCIz'4H20 (64I 0 gauss, AC plane, C=45°I

 

 

 

 

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2...

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MAGNET POSITION (DEGREES)
(0)

A |,3 "' MnCIa-4H2'0 (8570 gauss,AC pIane,C=45°I
g: -

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g b. 0000 no .
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MAGNET POSITION (DEGREES)
(b)
Fig. 23. MnC12°4H20 AC plane isentrOpic rotations.

77

LI

... InClz'4H20 (6690 GGUSS.BCp|ane.C-‘55°I

 

 

x ..
2,
I.2-—
LIJ
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a I I __ "no... ......
< - ... 0.. ... 00..
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MAGNET POSITION (DEGREES)
(0)

 

 

A |.3 "' MnClz-4H20 (9970 gauss,BC planO,C=55°)
E 'T'
I .2 ...
“J ..
n: u. .00.
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|_. I I — .. O o ..
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9: I O '9'; . . 0
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I'- 009 ..f....‘l‘000.t I I I ..l.....J...'OI J I J
O 90 I80 270 ‘ 360

MAGNET POSITION (DEGREES)
(b)

Fig. 24. MnC12°4H20 BC plane isentropic rotations.

78

As in CoClz-GHZO, the results of the rotations below
the critical field indicate that the C axis is the pre—
ferred axis of magnetization. Other than the flatness near
the C axis in the high field rotation (Fig. 23b), nothing
remarkable is seen in the AC plane rotations. However,
in the BC rotation above the critical field, shown in
Fig. 24b, the behavior is again similar to that of Fig. 6b.
Thus there is partial indication that MnClz'4H20 undergoes
a spin flop transition and that the Spins flop to the B
axis. Rives34, in one of his experiments, with the field
initially aligned along the C axis, experienced a 90°
rotation of his crystal, about the A' axis, upon a quick
demagnetization. This is consistent with the conclusiOn
that the spins flop to the B axis.

Further evidence of the spin flop phase in MnC12-4H20
is given by the C axis isentropic magnetization results.
The data points are given in Table 9 of Appendix III and
smooth curves drawn through the points are shown in Fig. 25.
Also shown in Fig. 25 as a dashed line is the spin flop—
antiferromagnetic boundary obtained by Rives.34 (Rives
results are about 100-200 gauss higher than the results of
Gijsman, et. al.33, which are not shown.) From the measured
isentropes, it is seen that the initial cooling upon mag-
netization disappears in the spin f10p phase.

Also shown in Fig. 25 are several points obtained
from the maxima of specific heat measurements made in this

study (these points are listed in Table 10 of Appendix III).

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80

It is seen that there is close agreement between the speci—
fic heat and nuclear resonance paramagnetic boundary re-
sults of Gijsman et. al.33 which are shown in Fig. 25 also.
(Note, however, that Rives' paramagnetic boundary results,
which are not shown, are considerably higher in temperature
than the boundary points shown in Fig. 25.)

The lowest temperature isentrope was obtained from the
first crystal when it was in the BC plane orientation,
while the others were obtained from the second crystal.
(The second crystal was oriented in the AC plane. For this
crystal, isentropic rotations in this plane gave results
similar to those obtained for the first crystal.) Although
there is a change in sign of the slope of the isentrOpe
from the first crystal, similar to the CoClz-GHZO results,
there is none for the second crystal results and the isen-
tropes behave as predicted by the theory. Unfortunately,
the gradual decrease of the magnetization cooling effect
near the spin flop boundary makes it nearly impossible to
define a single point as a phase change point. However,
the results, at least, do show the existence of the spin
flop state, with a critical field in the vicinity of the
results of the previous studies.

Also worthy of note are the measured isentropes which
start in the antiferromagnetic state and cross over into
the paramagnetic state. It is seen that adiabatic mag—
netization in the paramagnetic state, near the antiferro-

magnetic boundary produces cooling. Although this is

81

unusual, such an effect has been observed before by Fried-
berg and Schelleng37 in MnBr2-4Héo. In contrast with

those observations, though, these antiferromagnetic-para-
magnetic isentropes apparently do not cross the phase
boundary tangentially. It is possible that this is due

to imperfect adiabatic conditions, although further measure-

ments are needed for a better determination of the true

isentropes.
B. Materials Discovered to Exhibit Spin Flop

Since the above results on CoC12°6H20 and MnC12-4H20
showed the practicality of the adiabatic method of ob-
serving the spin flOp state in antiferromagnets, a search
was begun to discover substances with previously unobserved
spin flop states. From these investigations, it has been
found that two previously studied substances, CoBr2‘6H20
and FeC12'4H20, undergo spin flopping in accessible field
and temperature regions.

Previous specific heat38 and susceptibility39 measure—
ments had been made on CoBr2'6H20, indicating it is anti-
ferromagnetic below about 3.2°K. Although it is isomorphic
to CoClz'GHzo, CoBr2°6H20 is a more desirable substance to
study than CoC12-6H20, since it easily grows in long, large
single crystals, while single crystals of CoC12-6H20 are
grown with difficulty.

About a cubic centimeter portion of a single crystal
of CoBrz-BHZO grown from an aqueous solution at room tem-

perature was used in the measurements. Since the previous

82
susceptibility results indicated they C axis to be the
preferred axis of magnetization, measurements were made in
the BC and AC crystallographic planes. Groth's mor—
phology of CoC12-6H20 was used as a basis for orienting
the crystal. The AC plane orientation was an easier
orientation to make than the BC orientation, and it is
believed, for this reason, that there was less mis-align-
ment of the crystal in the AC orientation than in the
BC orientation.

Isentropic field rotations at two different fields were
made in each of these planes. The results are listed in
Table 11 of Appendix III and are shown in Figs. 26 and 27.
As in CoC12-6H20. and MnClzo4H20, the results show that
the C axis is the preferred axis. It is interesting to
note the similarity of the shift of the maximum in the BC
rotations to that observed in CoC12-6H20. This provides
further evidence of the reality of that shift.

The results of the 8830 gauss AC plane rotation
(Fig. 27b) are similar to the higher field rotation in
C0C12-6H20 and the predictions shown in Fig. 6b. Thus
there is partial evidence, from the rotation data, that
CoBrz-GHZO undergoes spin flop below 8830 gauss near 1.3°K.
with the Spins flopping to the A' direction.

As further evidence of the existence of the spin f10p
state in CoBr2-6H20, the isentropes obtained in isentrOpic
magnetization along the C axis are shown as smooth lines

through the data points in Fig. 28 (the data points are

 

83

 

 

 

A I.5 '" COBrg'GHao (G4IO gauss,BCplane,C=60°)
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MAGNET POSITION (DEGREES)
(0)

A I.5 7‘ CoBrzveHzo (8830 gGUSS,BCpIGne,C=60°)
X - ..
2.,

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3? ”-0 ...... o .....
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F. I.I I I I J I I I I I I, I J

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MAGNET POSITION (DEGREES)
(b)

Fig. 26. CoBr2°6H20 BC plane isentropic rotations.

TEMPERATURE (°K)

TEMPERATURE (°K)

84'

 

 

 

 

I.5 '7‘ CoDrz-GHZO (64l0 gouss,ACp|Gne,C=22°)
|.4--
L;— ....o o... ...,' ....
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MAGNET POSITION (DEGREES)
(0)

I.5 "" CoBrzioBHzo (883C gouss,AC plane;c=22°)
l..4-
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MAGNET POSITION (DEGREES)
(b)

Fig. 27. CoBrz-GHZO AC plane isentropic rotations.

85

 

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86
listed in Table 12 of Appendix III). Similar to the
CoClz-Gfléo results, there is initial cooling upon magnetiza-
tion in the antiferromagnetic state which disappears when
the spin flop state is entered. The change in sign of
the slopes of the isentropes, seen in CoC12-6H20, also
exists in CoBr2-6H20.

Measurements of this point of change were made both
when the crystal was in the AC plane orientation and
when it was in the BC plane orientation. The BC plane
results gave a slightly higher critical field than the AC
plane measurements; this would correspond to a poorer align—
ment in the BC plane than in the AC plane as hypothe-
sized.

Measurements of the paramagnetic boundary, using
constant field specific heat measurements, were also made,
both above and below the critical field. These results,
along with the spin flOp boundary results, are shown in
the phase diagram shown in Fig. 29. All the data points
are listed in Tables 13 and 14 of Appendix III.

The value of the triple point is (2.91 i .010K,

9320 i 40 gauss). A second degree polynomial was used to
fit the observed spin flop boundary. The result of the fit
was

HC = 7523 - 230.7 T + 292.5 T2 , (3.00)

where HC is the critical field at temperature T. Thus

on the basis of the observed data, a critical field of

about 7250 gauss is expected at 00K. Note that the spin

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88
flop boundary results do not quantitatively agree with the
predictions of Yosida's theory for a spin 1/2 system, which
is assumed for CoBré-6Hé0. The spin flop boundary rises
more slowly than the prediction. However, note that the
antiferromagnetic-paramagnetic boundary corresponds to the
predictions of the theory quite well with an H0 of 18,000
gauss.

It is also worth noting that the antiferromagnetic-
paramagnetic boundary apparently intersects the temperature
axis perpendicularly. This is what was predicted from
the results of the simple molecular field theory shown in
Fig. 3a. (It should be noted that the zero field Neel
temperature measured in this study, 3.152°K, differs from
the result of the previous specific heat study, 3.07°K.

At present, there is no explanation for this.) Again, as
in MnC12-4H20, it is also interesting to note the behavior
of the isentrope measured in the paramagnetic state.

Previous specific heat4°I41, susceptibility42 and
NMR43 studies on FeC12-4H20 indicated it is in an ordered
state below about 1.1°K. This temperature is near the
lowest temperature attainable in the apparatus used in
this study, but since cooling takes place upon magnetiza-
tion of an antiferromagnet, and since spin flopping was
observed near 5400 gauss at 0.40K in the NMR studies44, a
search for the spin flop state in FeC12-4H20 was made.

The samples which had been used in the resonance ex—

periments were borrowed from R. D. Spence and were used in

89
some initial measurements. However, it subsequently was
discovered that the McLeod pressure gauge of the tempera—
ture measuring system was giving spurious measurements.
Since there also was some question about the orientations
of Spence's crystals (they had been ground into cylinders),
it was decided to use the initial results only as quali—
tative indications of the behavior of FeC12-4H20.

A new single crystal of FeC12-4H20 was grown from an
aqueous solution in a desiccator so that more precise
measurements could be made. It weighed about 1.3 grams
and was stored in a refrigerator until its use, since, as
noted by Schriempf and Friedberg42, FeClz°4H20 tends to
lose or gain water when exposed to air at room temperature.
Also, before its use, it was spray coated with Krylon
clear plastic spray to further protect it from contamination.

Like the crystals previously studied, FeC12°4H20 forms
monoclinic crystals with B = 112°. The growth habit de-
scribed by Groth35, was used as a basis for orientation of
the crystal. The susceptibility results indicated that the
B axis is the preferred axis of magnetization, so experi-
ments were done in the AB and BC crystallographic planes.
It is felt that, since the BC plane is prominent on the
crystal, that the BC plane orientation was more precise.

The results of high field isentropic rotations in these
planes are shown in Fig. 30, and the data is listed in
Table 15 of Appendix III. The earlier data gave the same

qualitative results. In addition, the earlier data showed

iv

TEMPERATURE (°K)

'Iu

TEMPERATURE (°K)
S

C)
GD

90

"' Fe Cl2‘4H20 (8830 gauss, AB plane, B=22°I

 

0 90 ISO . 270 360
MAGNET POSITION (DEGREES),
(0)
‘" FeCl2-4H20 (9080 gauss,BCplane,B=2°)

 

 

I I I I I I I I I I I I
0 90 I80 270 360
MAGNET POSITION (DEGREES)
(b)

Fig. 30. FeC12-4H20 AB and BC plane isentropic

rotations.

91
that the rise in temperature near the B axis for the BC
rotation disappears at lower fields and also that the dip
in temperature near the A axis in the AB rotation dis-
appears at lower fields. Also, the earlier data included
AC plane isentropic rotations which gave results similar
to the AB plane behavior discussed here. Furthermore,
the earlier results also included a high field (about 8000
gauss) BC plane rotation at higher temperatures (about
1.5-1.6°K). The results of that rotation were similar to
the low field rotations done in previous crystals. However,
in direct contrast to the FeC12-4H20 low field BC plane
rotations at lower temperature, mentioned above, the tem-
perature was a maximum with the field along the B axis
and a minimum.when the field was along the C axis. These
qualitative results seem to imply a change in the principal
magnetic axes between about 1.1°K and 1.5°K.

Although the BC plane results, to some extent, cor-
reSpond to what would be seen in a spin flopping material,
the AB plane results definitely do not correspond with the
results seen in the other crystals studied.

In addition, when the crystal was in the AB plane
orientation, with the field directed along the B axis,
isentropic magnetizations produced cooling to a point and
then warming. This also does not strictly correspond with
the spin flop results previously seen (although the initial
measurements were reported45 as being partially similar to

the MnClz-4H20 results.)

92

For the above reasons, it was deemed worthwhile to
measure the antiferromagnetic-paramagnetic boundary to de-
termine in which phase the isentropic magnetizations had
been taken. The results of these measurements indicated
the previously measured isentropes were probably anti-
ferromagnetic-paramagnetic or paramagnetic isentropes simi-
lar to the ones previously seen in other crystals. In one
experiment, the turning point of the paramagnetic isentropes
from cooling to warming was followed to near 1.30K. (Such
a large region of negatively sloped paramagnetic isentropes
is probably due, to some extent, to the large anisotropies
seen in the susceptibilities.)

After the BC plane orientation had been made, it was
discovered that a region of temperature (near 0.75°K) could
be reached upon isentropic magnetization where the isen-
tropes behaved similar to those in the Spin flop state of
MnC12'4H20. Unfortunately, the interval between 0.75°K
and the first calibration point of the thermometer, 1.0°K,
is relatively large, so there may be as much as 0.05°K
error in the absolute temperatures measured, due to extrapo-
lation of the thermometer calibration. However, the obser-
vations made are certainly at least, qualitatively correct.

Several times when temperaunxe near 0.75°K were reached
upon isentropic magnetization, the field was held constant
at different values and the heat leaking in to the sample
was allowed to warm it. In this manner, second order phase

transitions were easily seen as discontinuous Changes in

93

the rate of rise of temperature of the sample.

The measured isentropes (with smooth lines drawn
through the points obtained) and all the second order
phase transition points are shown in Fig. 31. All the data
points are listed in Tables 16 and 17 of Appendix III.

Several points may be made from these results. First,
from observation of the lowest temperature isentrope, it
seems that FeCl2°4H20 goes into a definite spin flop state
above about 7000 gauss near 0.680K. Second, the paramag-
netic boundary points indicate that, if there is a spin
flop state, the triple point is about 5500 gauss near 0.760K.
Third, since this field is near the one reported by Spence,
it is possible that the critical field is actually near
5400-5500 gauss in the entire region 0.4-0.750K. Fourth,
if that is so, it would seem, on observation of Fig. 31,
that one conclusion is that the inflection point of the
antiferromagnetic-spin flop isentropes may be the point of
spin flop. (If this method of determining the spin flop
point were then applied to the MnC12-4H20 results, the crit-
ical field observed would turn out to be slightly lower
than the previously published results.) Fifth, the nearly
constant temperature of the paramagnetic boundary above
the critical point is highly unusual. Sixth, these results.
coupled with the four-sublattice models that have been pre-
viously hypothesized‘IZI46 for FeC12°4H20 and the unusual
NMR results between 0.7 and 1.1°K, indicate that further

isentropic measurements, perhaps using a He3 refrigerator,

 

 

 

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95

in the region 0.7-1.10K from 0-7000 gauss would be inter-
esting. Seventh, and finally, it should be noted that the
zero field Néel temperature determined here (1.1160K) is
slightly higher than that reported previously (1.0970K),

and that, in either case, the antiferromagnetic-paramagnetic
phase boundary would not seem to intersect the temperature
axis quite perpendicularly, in contrast to what was seen in
the other crystals studied (although finer measurements

are needed to verify this fact).
C. Materials Investigated Which Did Not Exhibit Spin Flop

In the search for substances exhibiting spin f10p
states, experimental observations were made on three other
hydrated salts: MnBr2°4H20, CSZMnBr4°2H20, and NiC12-6H20.
None of these exhibited spin f10p in the region 0-10,000
gauss between 1°K and 40K.

The first of the above three salts, MnBr2-4H20,has
been previously studied by means of susceptibility32o33,
Specific heat37'47'43, proton resonance33, optical absorp—
tion49, and antiferromagnetic resonance5° experiments.

The latter two experiments have inferred spin fIOpping
near 9000 gauss near 1.2-1.4°K, while the others have shown
no such results.

A single crystal of MnBr2°4H20, weighing about 2 grams
was grown from an aqueous solution at room temperature.
MnBr2-4H20 crystals are monoclinic with B = 99.60. Under

the assumption that the lattice structure is similar to

96
the isomorphic crystal, MnC12°4H20, the crystal was aligned
with the AC crystallographic plane as the plane of rota-
tion of the field. (The susceptibility measurements have
indicated the C axis to be the easy axis.) The field
was precisely aligned along the C axis by observation of
the temperature variation in an isentropic rotation (no
data points were taken). Isentropic magnetizations were
then done at several temperature regions. The isentrope
results are shown in Fig. 32 with smooth curves drawn be-
tween the points measured and the data is given in Table 18
of Appendix III. As can be seen, from the continuous cool—
ing effect of magnetization, there is no evidence of spin
flop in the regions observed.

Although a previous investigation22 of the paramag-
netic phase boundary in CszMnC14-2H20, which is a triclinic
crystal, failed to exhibit any spin f10p characteristics,

a single crystal of the isomorphic compound CszMnBr4-2H20
was grown so that a search could be made for the spin flop
state in it. A solution of CsBr and MnBr2-4H20 in water
at room temperature was used to grow the crystal. A
specific heat study of this substance indicated it to be
in a magnetically ordered state below 2.82°K.

The bromide was assumed to have the same preferred
axis as the chloride ([i i i] direction) so it was oriented
with the [0 i 1] direction parallel to the axis of rotation
of the field. Isentropic magnetizations were done with

the field aligned parallel and perpendicular to the

.mmmonucomH Onmv. mums: .Nm .wflm

gov mmscfimmo—zms
9N m... m; I. N; o._ m.o

/ i
m mmaomszmmH omxehtmcz -
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I 08
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. Amy.» I 0.0.

 

 

(SSHVO-M) O'IEIj

98
apparent preferred axis, determined from an isentropic
rotation. The isentropes are shown in Fig. 33, along with
several paramagnetic boundary points measured from constant
field Specific heat measurements. (The data points are
given in Tables 19 and 20 of Appendix III.) As can be seen
from the expanded portion of the data shown in Fig. 34,
there is no indication of spin flopping from the behavior
of the parallel isentrOpes. However, it should be noted
that the perpendicular isentropes are not temperature inde—
pendent, as would be expected. Thus, it might be concluded
that the magnetization direction in CszMnBr4-2H20 (and
probably CSZMnCl4-2H20, also) is not in the plane perpen-
dicular to the [0 i 1] direction. (In fact, resonance
experiments on the chloride51 do indicate the magnetization
direction to be out of that plane.) Thus a better align-
ment might be necessary, before any conclusions are drawn
concerning spin flopping either in the bromide or the
chloride.

Previous Specific heat52 and magnetic susceptibility53
experiments indicated that the salt NiClz-GHZO, which is
isomorphic to CoC12-6H20, becomes antiferromagnetic below
5.34°K. For this reason, an investigation was undertaken
to search for spin flopping in NiC12°6H20.

A single crystal of NiC12°6H20, weighing about one
gram, was borrowed from R. D. spence. The crystal is
monoclinic with B = 122.50 and the morphology is given

in Groth35. Since the susceptibility experiments indicated

 

 

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Q L mmakdmmoémh
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(SSDVO-M) O'IEIlzl

101
the A' axis to be the preferred axis, the crystal was
oriented so that its AC plane was in the plane of rota-
tion of the field.

The field was aligned along the A' axis by observa—
tion of the temperature in an isentropic rotation and isen—
tropic magnetizations were then done at different tempera-
ture regions. Nothing remarkable was seen in the isentropic
rotations done, other than a rather small temperature
variation. The results of the only one which was recorded
are shown in Fig. 35 and are given in Table 21 of Appendix
III. The results of the isentropic magnetizations are
shown in Fig. 36 and are listed in Table 22 of Appendix III.
These results certainly are not similar to the previous
results in the other crystals studied.

The nearly constant temperature isentropes would nor-
mally indicate a perpendicular spin state. However, since
the field had been aligned along the axis of the isen-
tropic rotation temperature minimum, the spins should have
been expected to be in the parallel state. At present,
the best explanation seems to be that the sample was not
actually isolated, so that there would be a constant
temperature in a magnetization. The lack of isolation can
be explained by noting that the sample's specific heat is
very small in the temperature region in which observations
were made. Since, as noted in Chapter 2, the degree of
apparent isolation depends on the specific heat of the

sample being used, the NiC12-6H20 sample may have been

102

 

 

A I.l0"' NiCl2°6H20 (7980 gauss,ACplanG,A’=l|0°I

x I-

L l.08-

m _o.... ..ooo.... ....O O

a: IIOG— .....00. .00....

E) _

4 I004—

m -

E I02

5 '. :-

E LOO I I I I I I I I I J I I
O 90 I80 270 360

MAGNET POSITION (DEGREES)

Fig. 35. NiC12-6H20 AC plane isentropic rotation.

 

 

 

 

 

 

 

 

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Axe mmskquESMP .
0.0 0.0 04. 0.0 0.0 0..
_ _ _ _ . _ . _ . 0.0
H I, I 0.N
O + ._ I
m L .
a P e e r O .V
00 + I. . I
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. W A i
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m. i
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O M w .Ww Loo.

 

(S.SDV9->II O'IEIId

104
relatively unisolated.

In addition to the isentrOpic magnetizations and rota-
tions done, constant field Specific heat measurements were
used in an attempt to determine the paramagnetic phase
boundary in NiC12-6H20.

The results of these measurements are also shown in
Fig. 36 and are given in Table 23 or Appendix III. As can
be seen, there is a large amount of scatter in the points.
Careful observation of the raw data shows the scatter to
be independent of the calibration curves connected with
determining temperatures. Thus the scatter is real, or
more probably, there are hysterisis effects in crossing
the NiClz-GHZO paramagnetic boundary. In any case, all the
NiC12-6H20 measurements should be considered as preliminary

results and further observations on this salt are needed.

II. Conclusions

In conclusion, this study has shown that adiabatic
methods for observing the spin flop state of an antiferro-
magnet are practical, although there is yet room for their
refinement. There are some drawbacks with the present
method of mounting the crystal. Better adiabatic conditions
would be desirable. Also, so that any crystalline axis
Inight be easily aligned parallel to the field, a method to
rotate the sample about an axis in the plane of rotation
of the field, while it is in the calorimeter during the

eXperiment, would be highly desirable. It would also be

105
desirable if there were less metal in the calorimeter, to
reduce eddy current heating which exists when the field
is being rotated.

With the discovery of spin flopping properties of
CoBr2°6H20 and FeC12‘4H20, new areas of research into some
of the basic properties of these substances are possible.
Also, it would be interesting to do a complete set of mag-
netic field specific heat experiments, magnetic suscepti-
bility experiments and adiabatic magnetization experiments
so that the theoretical relationship between the measured
quantities might be tested for antiferromagnets.

Finally, from the theory, a new method to predict the
critical field for spin flopping has been proposed. It
would be interesting to do some susceptibility measurements
to test the predictions of the simple molecular field
theory regarding the critical field value. Such measure—

ments on MnBr2°4H20 would be especially interesting.

17.

18.
19.
20.

21.

22.

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(1907); Ann. Phys. _1_7, 97 1932

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K. Yosida, Prog. Theor. Phys. 1, 25 (1952).

C. J. Gorter and J. Haantjes, Physica 18, 285 (1952).
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C. J. Gorter and Tineke Van Peski-Tinbergen, Physica
22, 273 (1956).

J. H. Van Vleck, The Theory of Electric and Magnetic
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K. Yosida, Prog. Theor. Phys. 1, 425 (1952).
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106

107

23. W. R. Eisenberg, M.S. Dissertation, Michigan State
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25. W. F. Giauque, R. A. Fisher, E. W; Hornung, and G. E.
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26. J. Butterworth (private communication).

27. A. C. Anderson, W. Reese, and J. C. Wheatley, Rev.
Sci. Instr. 34, 1386 (1963).

28. F. G. Brickwedde, H. van Dyk, M. Durieux, J. R.
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29. G. T. Armstrong, J. Res. Nat. Bur. Stand. 33, 263
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30. J. R. Clement and E. H. Quinnel, Rev. Sci. Instr. 33,
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31. W. van der Lugt and N. J. Poulis, Physica 33, 917 (1960).

32. V. A. Schmidt and S. A. Friedberg, J. Appl. Phys.,
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33. H. M. Gijsman, N. J. Poulis, and J. van den Handel,
Physica 33, 954 (1959).

34. John E. Rives, Phys. Rev. 162, 491 (1967).

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36. T. Haseda and E. Kanda, J. Phys. Soc. Japan 13, 1051
(1957).

37. S. A. Friedberg and J. H. Schelleng, Proceedings of
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39. M. Garber, J. Phys. Soc. Japan 13, 734 (1960).

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Phys. Soc. Japan 11, Suppl. B-1, 515 (1962).

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42.

43.

44.

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108

J. T. Schriempf and S. A. Friedberg, Phys. Rev. 136,
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R. B. Flippen and s. A. Friedberg, J. Appl. Phys.
315, 3388 (1960).

APPENDIX I

COMPUTER PROGRAMS

109

 

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140

0000 200 00000000
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142

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0.0000
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143

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0.0000
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0.0000
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0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

APPENDIX II

RESISTANCE CALCULATION WITH.AN UNBALANCED

POTENTIOMETER SETTING

144

APPENDIX II

RESISTANCE CALCULATION WITH AN UNBALANCED
POTENTIOMETER SETTING

When Kirchoff's Circuit Law is applied to the poten-

tiometer circuit shown in Fig. 37, the result is

(IE - IP)RP + IER + (IE + IT)RT = 0. (1)
IP is the potentiometer current; RP is the potentiometer
resistance; RT is the thermometer resistance; IT is the
thermometer current; R is the lead resistance; and IE
is the unbalance current. When IE = O (1) becomes
IP RP = IT RT' If V0 is the potentiometer voltage read-
ing, then V0,: IP RP , so (1) becomes

(1 — IE/IP)V0 = IER + (1 + IE/IT)ITRT- (2)

In the actual experimental apparatus, the galvanometer
consisted of a microvolt amplifier used with a chart re-

corder. Experiments have shown that I is proportional

E
to the pen deflection from the null position, and that it
is less than, at most, 0.1 microampere. For the K-3
potentiometer IP is never less than 200 microamperes.

Thus IE/IP is negligible compared to 1. Then, if the

recorder deflection is D, and IE = aD, (2) becomes

v0 = aDR + (1 + aD/IT)ITRT. (3)

145

Fig. 37.

146

 

 

 

 

 

 

 

 

 

 

 

 

Vp
:%
IP
-—'WWW\AN\MMANww——
Rp - T ' Rp
RL LGALVANOMETER]
IE
wwvw
RT
V CT
'F——J\NVWW\——-—
1
vT R3

Circuit diagram of a simple potentiometer.

147
Since the thermometer current is constant, (3) be-

comes, setting R0 = Vo/IT.
R0 = (a/IT)RD + [1 + (a/IT)D]RT (4)

If the potentiometer setting is changed, while RT

and R are constant, then, denoting the new values of R0

and D by primes:
R3 = (a/IT)RD' + [1 + (a/IT)D']RT. (5)

If (5) is subtracted from (4), then

 

R0 - R3 = (a/IT)(R + RT)(D - D') (6)
or, ARC/AD = C1 + CZRT' (7)

where C1 = (a/IT)R and C2 = a/IT. This is the voltage
calibration curve, in theory.

To calculate R using (7), (4) is solved for R -

 

T T'
RT = [R0 - (a/IT)RD]/[1 + (a/IT)D1 (8)
or,

T 1+C2D

In practice (7) turns out to be non-linear, so a modified
form of (9) was used:
R0 - [c1 - C2R - 2c3R2 - 3C4R3]D

RT = . (10)
1 + [c2 + 2C3R + 3C4R2]D

 

This equation was arrived at empirically.

APPENDIX III

EXPERIMENTAL DATA

148

149

 

 

 

Table 1. CoC12-6H20 BC plane isentropic rotations (C = 55°)
9 T(°K) e T(°K) e T(°K)
1/5/68, 6350 Gauss
70 1.287 175 1.278 305 1.336
73 1.294 180 1.270 310 1.336
77 1.301 185 1.262 315 1.330
80 1.306 190 1.253 320 1.324
85 1.315 195 1.243 325 1.321
88 1.318 200 1.233 330 1.316
90 1.319 205 1.223 335 1.311
93 1.320 210 1.214 340 1.316
95 1.321 215 1.204 345 1.298
98 1.323 220 1.215 350 1.291
100 1.323 225 1.201 355 1.284
103 1.324 230 1.186 0 1.276
105 1.324 235 1.184 5 1.266
108 1.324 240 1.203 10 1.256
115 1.322 245 1.222 15 1.245
120 1.322 250 1.243 20 1.234
125 1.322 255 1.260 25 1.222
130 1.321 260 1.274 30 1.210
135 1.320 265 1.287 35 1.193
140 1.313 270 1.299 40 1.178
145 1.309 275 1.309 45 1.162
150 1.305 280 1.317 50 1.149
155 1.300 285 1.324 55 1.145
160 1.295 290 1.329 60 1.162
165 1.291 295 1.332 65 1.185
170 1.285 300 1.334 70 1.206
1/5/68, 7860 Gauss
35 1.303 265 1.293 135 1.367
25 1.333 255 1.262 125 1.358
15 1.357 245 1.227 115 1.345
5 1.376 235 1.202 107 1.329
355 1.389 225 1.225 95 1.302
345 1.397 215 1.261 85 1.271
335 1.400 205 1.294 75 1.237
325 1.398 195 1.320 65 1.199
315 1.392 185 1.342 60 1.185
305 1.381 175 1.358 55 1.179
295 1.368 165 1.368 50 1.189
285 1.347 155 1.371 45 1.207
275 1.322 145 1.371

 

150

 

 

 

Table 2. CoClz-GHZO AB plane isentropic rotations (B = 0°)
6 T(°K) e T(°K)' e T(°K)
1/11/68, 6410 Gauss
70 1.224 200 1.259 320 1.266
80 1.223 210 1.256 330 1.268
90 1.228 217 1.253 340 1.267
100 1.238 220 1.252 350 1.265
110 1.246 230 1.245 0 1.265
120 1.255 240 1.240 0 1.262
130 1.260 250 1.237 10 1.259
140 1.262 260 1.237 20 1.257
150 1.263 270 1.241 30 1.253
160 1.265 280 1.248 40 1.247
170 1.265 290 1.256 50 1.242
180 1.263 300 1.263 60 1.236
190 1.262 310 1.269 70 1.232
1/11/68, 9700 Gauss
270 1.197 160 1.293 40 1.286
260 1.297 150 1.289 30 1.294
250 1.218 140 1.284 20 1.301
240 1.240 130 1.276 10 1.304
230 1.257 120 1.265 0 1.306
225 1.262 110 1.250 350 1.305
220 1.272 100 1.233 340 1.303
210 1.281 90 1.215 330 1.299
200 1.287 80 1.210 320 1.292
190 1.291 70 1.236 310 1.283
180 1.294 60 1.257 300 1.272
170 1.294 50 1.273 290 1.256

 

 

 

151

Table 3. C0C12°6H20 AC plane isentropic rotations (C = 70°)

 

 

e T(°K) ' ‘6 T(°K) ' e T(°K)

 

1/9/68, 6410 Gauss

335 1.315 215 1.225 95 1.182
325 1.300 205 1.255 85 1.167
315 1.279 195 1.284 75 1.164
305 1.250 185 1.308 65 1.171
295 1.221 175 1.321 55 1.184
285 1.194 165 1.328 45 1.205
275 1.172 155 1.327 35 1.234
265 1.161 145 1.314 25 1.265
255 1.157 135 1.292 15 1.291
245 1.160 125 1.264 5 1.315
235 1.175 115 1.233 355 1.329
225 1.198 105. 1.203 345 1.334

1/9/68, 9620 Gauss

350 1.359 230 1.221 110 1.182
340 1.365 220 1.216 100 1.208
330 1.349 210 1.200 90 1.219
320 1.313 200 1.223 80 1.227
310 1.263 190 1.283 70 1.230
300 1.206 180 1.331 60 1.230
290 1.175 170 1.363 50 1.225
280 1.203 160 1.372 40 1.217
270 1.215 150 1.357 30 1.197
260 1.222 140 1.326 20 1.232
250 1.225 130 1.278 10 1.296

240 1.224 120 1.220 0 1.341

 

152

 

 

 

pa?

 

 

Table 4. CoC12-6H20 C-axis isentropic magnetizations
T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)
1/9/68
1.207 6450 1.226 8290 1.238 9480
1.195 6780 1.229 8480 1.239 9620
1.196 7090 1.231 8660 1.240 9760
1.203 7370 1.234 8830 1.241 9890
1.212 7620 1.235 9000 1.241 9970
1.219 7860 1.236 9160
1.223 8080 1.237 9320
1/6/68
1.244 8570 1.221 7230 1.335 4310
1.242 8380 1.222 6940 1.358 3360
1.240 8190 1.236 6600 1.372 2260
1.237 7970 1.256 6200 1.372 1850
1.232 7740 1.279 5700
1.225 7500 1.303 5080
Table 5. C0C12°6H20 antiferromagnetic-spin flop boundary

points (H // C-axis)

 

 

 

T(°K) H(Gauss)
1/5/68
1.219 7140
1.273 7110
1.365 7260
1.502 7420
1.679 7440
1/6/68
1.186 7060
1.218 7100
1.279 7120
1.353 7190
1.445 7310
1.513 7420
1.579 7540
1.664 7680
1/9/68
1.491 7290
1.532 7380
1.588 7450
1.674 7610

 

153

Table 6. CoC12-6H20 BC plane paramagnetic isentropic
rotation (C = 55°)

 

 

 

0 T(°K) _ 9 T(°K) 6 T(°K)
1/6/68

55 2.459 285 2.467 5 3.209
65 2.459 295 2.472 355 3.215
75 2.463 305 2.476 345 3.219
85 2.468 315 2.478 335 3.221
95 2.474 325 2.478 325 3.222
105 2.479 335 2.475 315 3.223
115 2.484 345 2.471 305 3.223
125 2.487 355 2.464 295 3.222
135 2.489 5 2.457 285 3.222
145 2.489 15 2.448 275 3.221
155 2.486 25 2.442 265 3.224
165 2.482 35 2.436 255 3.227
175 2.476 45 2.431 245 3.228
185 2.469 55 2.430 235 3.231
195 2.461 65 2.432 225 3.233
205 2.456 75 2.435 215 3.238
215 2.450 85 2.441 205 3.242
225 2.447 95 2.447 195 3.246
235 2.446 55 3.192 185 3.252
245 2.447 45 3.194 175 3.257
255 2.451 35 3.195 165 3.262
265 2.455 155 3.267

275 2.461

 

154

Table 7. MnC12°4H20 AC plane isentropic rotations (C = 45°)

 

 

 

e T(°K) e T(°K) e T(°K)
2/6/68, 6410 Gauss
10 1.079 260 1.080 110 1.122
360 1.097 250 1.063 100 1.111
355 1.107 240 1.043 90 1.096
350 1.115 230 1.037 80 1.079
345 1.122 220 1.037 70 1.060
340 1.126 210 1.040 60 1.041
335 1.130 200 1.050 50 1.036
330 1.133 190 1.070 45 1.035
320 1.135 180 1.091 40 1.035
315 1.135 170 1.110 35 1.036
310 1.134 160 1.123 30 1.039
300 1.129 150 1.131 25 1.042
290 1.122 140 1.135 20 1.049
280 1.110 130 1.134 15 1.058
270 1.096 120 1.130
2/6/68

75 1.054 200 1.018 320 1.135
80 1.066 205 1.015 325 1.133
85 1.078 210 1.013 330 1.130
90 1.088 215 1.016 335 1.124
95 1.097 220 1.016 340 1.116
100 1.107 225 1.016 345 1.107
105 1.115 230 1.017 350 1.066
110 1.121 235 1.018 355 1.081
115 1.127 240 1.022 360 1.096
120 1.131 245 1.026 355 1.081
125 1.133 250 1.042 360 1.096
130 1.135 255 1.054 5 1.051
135 1.135 260 1.066 10 1.034
140 1.134 265 1.077 25 1.016
145 1.132 270 1.088 30 1.016
150 1.129 275 1.098 35 1.014
155 1.123 280 1.107 40 1.015
165 1.104 285 1.114 45 1.015
170 1.093 290 1.120 50 1.016
175 1.078 295 1.126 55 1.018
180 1.063 300 1.130 60 1.022
185 1.048 305 1.133 65 1.026
190 1.031 310 1.135 70 1.040

195 1.017 315 1.135

 

Table 8. MnCl§°4H20 BC plane isentropic rotations (C = 55°)

155

 

 

 

e T(°K) e T(°K) e T(°K)
8/21/67, 6690 Gauss
65 0.899 185 0.987 305 1.038
70 0.909 190 0.971 310 1.045
75 0.919 195 0.954 315 1.057
80 0.933 200 0.939 320 1.052
85 0.948 205 0.924 325 1.052
90 0.963 210 0.910 330 1.050
95 0.977 215 0.897 335 1.046
100 0.991 220 0.886 340 1.041
105 1.005 225 0.878 345 1.033
110 1.017 230 0.873 350 1.022
115 1.028 235 0.872 355 1.011
120 1.038 240 0.876 360 0.999
125 1.047 245 0.882 5 0.984
130 1.053 250 0.891 10 0.968
135 1.057 255 0.904 15 0.952
140 1.058 260 0.919 20 0.936
145 1.058 265 0.934 25 0.921
150 1.056 270 0.950 30 0.905
155 1.052 275 0.966 35 0.892
160 1.045 280 0.981 40 0.882
165 1.037 285 0.994 45 0.874
170 1.027 290 1.007 50 0.869
175 1.015 295 1.020 55 0.868
180 1.001 300 1.029 60 0.870

 

‘Lj

156

 

 

 

Table 8. (Cont.)
e T(°K) 6 T(°K) e T(°K)
8/21/67, 9970 Gauss
55 0.858 180 0.973 305 1.050
65 0.858 185 0.943 310 1.063
65 0.857 190 0.912 315 1.072
70 0.857 195 0.883 320 1.076
75 0.856 200 0.865 325 1.077
80 0.854 205 0.858 330 1.075
85 0.852 210 0.857 335 1.068
90 0.864 215 0.858 340 1.058
95 0.899 220 0.859 345 1.044
100 0.933 225 0.860 350 1.025
105 0.959 230 0.860 355 1.005
110 0.986 235 0.861 360 0.981
115 1.009 240 0.861 5 0.953
120 1.029 245 0.861 10 0.926
125 1.045 250 0.860 15 0.899
130 1.058 255 0.859 20 0.878
135 1.067 260 0.857 25 0.872
140 1.072 265 0.853 30 0.871
145 1.073 270 0.870 35 0.872
150 1.070 275 0.904 40 0.873
155 1.063 280 0.937 45 0.875
160 1.052 285 0.965 50 0.878
165 1.037 290 0.992 55 0.879
170 1.019 295 1.014
175 0.997 300 1.034

 

157

Table 9. MnC12'4H20 C-axis isentropic magnetizations

 

 

 

T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)

8/21/67
0.875 6640 0.839 7690 0.847 8430
0.867 6780 0.839 7740 0.848 8480
0.862 6860 0.839 7800 0.848 8520
0.858 6940 0.839 7860 0.849 8570
0.856 7010 0.840 7920 0.849 8660
0.852 7090 0.841 7970 0.849 8740
0.850 7160 0.842 8030 0.850 8830
0.847 7230 0.843 8080 0.850 8910
0.843 7300 0.843 8130 0.851 9000
0.841 7370 0.844 8190 0.851 9080
0.841 7430 0.845 8240 0.851 9160
0.840 7500 0.845 8290 0.852 9240
0.840 7560 0.846 8340 0.854 9620
0.839 7620 0.845 8390 0.855 9970

3/1/68
1.007 6410 0.965 8080 0.963 9160
0.991 6780 0.964 8290 0.963 9400
0.980 7090 0.963 8480 0.965 9620
0.972 7370 0.962 8660 0.965 9760
0.969 7620 0.962 8830 0.967 9970
0.967 7860 0.963 9000

3/1/68
1.123 4910 1.011 7620 1.003 9000
1.101 5410 1.007 7860 1.003 9160
1.078 5970 1.005 8080 1.003 9400
1.056 6410 1.004 8290 1.003 9620
1.039 6780 1.003 8480 1.003 9760
1.025 7090 1.003 8660 1.003 9890
1.014 7370 1.003 8830

3/1/68
1.171 4980 1.049 7620 1.039 9000
1.154 5410 1.045 7860 1.039 9160
1.126 5970 1.043 8080 1.039 9400
1.106 6410 1.041 8290 1.039 9620
1.086 6780 1.040 8480 1.039 9760
1.068 7090 1.040 8660 1.038 9970
1.056 7370 1.039 8830

158

Table 9. (Cont.)

 

 

 

T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)

2/20/68
1.129 9970 1.131 8570 1.157 7230
1.129 9620 1.131 8380 1.169 6940
1.130 9240 1.132 8190 1.185 6600
1.130 9080 1.134 7970 1.201 6200
1.130 8910 1.137 7740 1.220 5700
1.130 8740 1.144 7500 1.232 5320
1.235 5080 1.113 7740 1.106 9080
1.207 5700 1.110 7970 1.106 9240
1.186 6200 1.107 8190 1.106 9480
1.167 6600 1.106 8380 1.106 9690
1.149 6940 1.106 8570 1.106 9970
1.134 7230 1.106 8740
1.121 7500 1.106 8910

2/20/68
1.218 5700 1.134 7740 1.130 8910
1.199 6200 1.132 7970 1.130 9080
1.182 6600 1.130 8190 1.130 9240
1.166 6940 1.130 8380 1.129 9620
1.152 7230 1.129 8570 1.129 9970
1.142 7500 1.130 8740

2/20/68
1.164 9400 1.165 8380 1.213 6940
1.164 9240 1.166 8190 1.228 6600
1.164 9080 1.169 7970 1.246 6200
1.165 8910 1.174 7740 1.266 5700
1.165 8740 1.184 7500 1.282 5320
1.165 8570 1.198 7230

2/20/68
1.336 5250 1.231 7500 1.203 8910
1.320 5700 1.219 7740 3 1.203 9080
1.298 6200 1.207 8190 1.202 9240
1.278 6600 1.205 8380 1.201 9620
1.262 6940 1.204 8570 1.199 9970
1.246 7230 1.204 8740

159

Table 9. (Cont.)

 

 

 

T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)

2/20/68
1.392 5700 1.285 7740 1.260 8910
1.372 6200 1.273 7970 1.261 9080
1.347 6600 1.266 8190 1.262 9240
1.328 6940 1.262 8380 1.266 9620
1.312 7230 1.261 8570 1.274 9970
1.298 7500 1.260 8740

2/20/68
1.293 9409 1.278 8380 1.335 6940
1.289 9240 1.279 8190 1.353 6600
1.286 9080 1.283 7970 1.371 6200
1.283 8910 1.293 7740 1.393 5700
1.280 8740 1.304 7500 1.417 5080
1.279 8570 1.317 7230

2/20/68
1.449 5410 1.402 7500 1.426 8740
1.435 5700 1.404 7740 1.433 8910
1.418 6200 1.407 7970 1.439 9080
1.406 6600 1.411 8190 1.445 9240
1.402 6940 1.416 8380 1.465 9620
1.402 7230 1.421 8570 1.486 9970

 

160

Table 10, MnC12‘4H20 specific heat maxima (H // C-axis)

 

 

 

T(OK) H(Gauss)
2/20/68

1.458 5410
3/1/68

1.210 9970
3/2/68

1.220 9320

1.225 8480

1.238 7980

1.303 7230

 

161

Table 11a. CoBr2°6H20 BC plane isentropic rotations (C = 60°)

 

 

6 T(°K) 6 T(°K) 6 T(°K)

 

1/22/68, 6410 Gauss

90 1.281 220 1.214 350 1.307
100 1.301 230 1.197 0 1.291
110 1.311 240 1.198 10 1.275
120 1.317 250 1.231 20 1.256
130 1.320 260 1.262 30 1.238
140 1.317 270 1.288 40 1.216
150 1.313 280 1.304 50 1.196
160 1.305 290 1.318 60 1.195
170 1.294 300 1.326 70 1.224
180 1.280 310 1.327 80 1.258
190 1.266 320 1.327 90 1.287
200 1.250 330 1.325
210 1.233 340 1.318

1/22/68, 8830 Gauss

350 1.373 230 1.230 110 1.320
340 1.375 220 1.259 100 1.300
330 1.372 210 1.294 90 1.275
320 1.365 200 1.319 80 1.251
310 1.352 190 1.340 70 1.224
300 1.337 180 1.353 60 1.223
290 1.318 170 1.362 50 1.234
280 1.295 160 1.367 40 1.266
270 1.270 150 1.366 30 1.297
260 1.246 140 1.360 20 1.321
250 1.221 130 1.351 10 1.344

240 1.219 120 1.338 0 1.358

 

162

Table 11b. CoBr2°6H20 AC plane isentropic rotations (C - 22°)

 

 

6 T(°K) 6 T(°K) 6 T(°K)

 

1/17/16, 6410 Gauss

210 1.259 330 1.304 90 1.359
220 1.274 340 1.285 100 1.365
230 1.292 350 1.266 110 1.365
240 1.310 0 1.253 120 1.358
250 1.329 10 1.246 130 1.344
260 1.344 20 1.247 140 1.324
270 1.355 30 1.254 150 1.303
280 1.362 40 1.268 160 1.281
290 1.363 50 1.285 170 1.263
300 1.357 60 1.307 180 1.250
310 1.344 70 1.327 190 1.242
320 1.325 80 1.345 200 1.242
1/17/68, 8830 Gauss

70 1.314 200 1.296 330 1.310
80 1.351 210 1.293 340 1.271

90 1.382 220 1.281 350 1.239
100 1.401 230 1.234 360 1.276
110 1.405 240 1.271 10 1.294
120 1.396 250 1.312 20 1.296
130 1.374 260 1.350 30 1.294
140 1.341 270 1.381 40 1.282
150 1.305 280 1.399 50 1.235
160 1.264 290 1.405 60 1.272
170 1.233 300 1.397 70 1.313
180 1.281 310 1.378

190 1.293 320 1.347

 

(1

163

Table 12. CoBr2-6H20 C-axis isentropic magnetizations.

 

 

 

T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)
1/22/68
1.254 6410 1.249 8080 1.288 9160
1.243 6780 1.268 8290 1.288 9320
1.238 7090 1.279 8480 1.291 9620
1.232 7370 1.285 8660 1.296 9970
1.231 7620 1.285 8830
1.233 7860 1.288 9000
1/22/68
1.258 4910 1.210 7500 1.285 9080
1.255 5080 1.209 7620 1.288 9240
1.249 5410 1.208 7860 1.289 9400
1.245 5700 1.228 8080 1.293 9550
1.236 6200 1.265 8380 1.293 9620
1.222 6780 1.273 8570 1.295 9760
1.216 7090 1.278 8740 1.297 9890
1.211 7370 1.282 8910 1.298 9970
1/17/68
1.291 6410 1.269 7860 1.327 8830
1.280 6780 1.302 8080 1.328 9000
1.273 7090 1.309 8290 1.328 9160
1.267 7370 1.324 8480 1.334 9620
1.265 7620 1.325 8660 1.333 9720
1/22/68
1.864 5080 1.793 7090 1.798 8660
1.852 5410 1.783 7370 1.802 8830
1.843 5700 1.773 7690 1.806 9240
1.832 5970 1.768 7860 1.809 9620
1.826 6200 1.765 8080 1.811 9970
1.818 6410 1.771 8290
1.804 6810 1.788 8480
1/22/68
2.549 5080 2.492 7500 2.464 8740
2.537 5700 2.486 7740 2.465 8910
2.526 6200 2.480 7970 2.470 9080
2.516 6600 2.475 8190 2.474 9240
2.506 6940 2.469 8380 2.476 9620
2.499 7230 2.466 8570 2.477 9970
1/24/68
2.267 5080 2.205 7500 2.193 8740
2.254 5700 2.199 7740 2.201 8910
2.240 6200 2.194 7970 2.204 9080
2.230 6600 2.188 8190 2.206 9240
2.221 6940 2.184 8380 2.208 9620
2.213 7230 2.183 8570 2.209 9970

164

Table 12. (Cont.)

 

 

 

T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)
1/24/68
3.107 5350 .3.055 7860 3.022 9240
3.087 5970 3.047 8190 3.012 9620
3.082 6600 3.040 8480 3.004 9970
3.071 7090 3.034 8740 3.004 9970
3.062 7500 3.027 9000
1/26/68

2.636 1680 2.559 6940 2.522 8660
2.631 2260 2.555 7090 2.520 8740
2.628 2710 2.552 7230 2.518 8830
2.623 3360 2.549 7370 2.518 8910
2.615 3860 2.548 7500 2.518 9000
2.608 4310 2.545 7620 2.522 9080
2.602 4720 2.542 7740 2.524 9160
2.595 5080 2.539 7860 2.525 9240
2.590 5410 2.538 7970 2.525 9320
2.584 5700 2.534 8080 2.526 9480
2.579 5970 2.532 8190 2.526 9620
2.574 6200 2.530 8290 2.526 9760
2.570 6410 2.527 8380 2.527 9970
2.567 6600 2.525 8480

2.563 6780 2.523 8570

 

 

 

165

 

 

 

 

 

 

 

Table 13. COBr2°6H20 antiferromagnetic-spin flop boundary
points (H // C-axis).
T(°K) H(Gauss) T(°K) H(Gauss)
1/17/68
1.180 7670 1.906 8150
1.211 7680 1.962 8200
1.270 7690 2.030 8270
1.331 7740 2.098 8320
1.417 7790 2.224 8470
1.506 7830 2.337 8590
1.564 7870 2.506 8790
1.651 7930 2.739 9070
1.755 8010
1/22/68
1.260 7860 1.850 8180
1.296 7890 1.970 8290
1.351 7900 2.062 8380
1.425 7940 2.195 8530
1.485 7970 2.383 8740
1.539 8000 2.485 8870
1.610 8040 2.607 9030
1.705 8050 2.730 9240
1.777 8100 2.771 9260
Table 14. Specific heat maxima (H // C-axis, CoBr2‘6H20)
H(Gauss) T(°K)
1/24/68
2.981 8380
3.030 7090
3.087 5350
1/26/68
2.938 9970
2.989 8290
3.109 4310
3.125 3090
3.143 1970
3.152 0
1/27/68
2.941 9100
2.923 9420
2.930 9550
2.916 9360

 

Table 15.

FeC12-4H20 isentropic rotations

 

 

 

6 T(°K) 6 T(°K) 6 T(°K)
4/25/68, AB Plane, B = 220
330 1.078 210 0.973 90 1.097
320 1.092 200 0.968 80 1.096
310 1.085 190 0.977 70 1.078
300 1.039 180 0.995 60 1.051
290 0.989 170 1.023 50 1.021
280 1.061 160 1.054 40 0.997
270 1.095 150 1.083 30 0.983
260 1.090 140 1.097 20 0.979
250 1.072 130 1.088 10 0.988
240 1.045 120 1.043 0 1.008
230 1.014 110 0.996 350 1.036
220 0.991 100 1.061 340 1.066
5/1/68, BC Plane, 20

70 1.045 190 1.022 310 0.990
80 1.096 200 1.013 320 0.991
90 1.130 210 1.002 330 1.004
100 1.109 220 0.991 340 1.020
110 1.055 230 0.991 350 1.030
120 1.006 240 1.013 360 1.036
130 0.982 250 1.056 ‘10 1.034
140 0.982 260 1.106 20 1.024
150 0.995 270 1.138 30 1.011
160 1.011 280 1.117 40 0.998
170 1.020 290 1.065 50 0.995
180 1.025 300 1.016 60 1.015

 

167

 

 

 

Table 16. FeC12-4H20 B-axis isentropic magnetizations.
T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)

5/2/68
0.693 6600 0.678 8190 0.679 9240
0.686 6940 0.678 8380 0.680 9400
0.680 7230 0.678 8570 0.684 9620
0.679 7500 0.679 8740 0.692 9820
0.679 7740 0.679 8910
0.678 7970 0.679 9080

5/1/68
0.991 1910 0.683 6940 0.677 8570
0.955 3360 0.679 7230 0.677 8740
0.919 4310 0.678 7500 0.678 8910
0.878 5080 0.677 7740 0.679 9080
0.740 5700 0.677 7970 0.681 9240
0.709 6200 0.677 8190
0.692 6600 0.677 8380

4/26/68
1.111 1560 0.912 6940 0.918 8740
1.102 1910 0.906 7230 0.925 8910
1.056 3360 0.903 7500 0.934 9080
1.014 4310 0.901 7740 0.943 9240
0.977 5080 0.902 7970 0.953 9400
0.952 5700 0.904 8190 0.970 9620
0.933 6200 0.907 8380 0.988 9820
0.920 6600 0.912 8570 1.008 10060

4/25/68
1.030 2820 0.780 7500 0.826 9080
0.956 4310 0.776 7740 0.839 9240
0.901 5080 0.779 7970 0.853 9400
0.863 5700 0.782 8190 0.876 9620
0.833 6200 0.792 8380 0.900 9820
0.812 6600 0,798 8570 0.925 10060
0.795 6940 0.806 8740
0.786 7230 0.815 8910

4/25/68
1.189 3040 1.102 7230 1.173 8910
1.169 3710 1.107 7500 1.185 9080
1.148 4310 1.113 7740 1.197 9240
1.126 5080 1.121 7970 1.208 9400
1.113 5700 1.130 8190 1.228 9620
1.104 6200 1.140 8380 1.248 9820
1.100 6600 1.150 8570 1.267 10060
1.100 6940 1.161 8740

 

168

Table 17. Specific heat maxima (H // B-axis, FeClQ-HHQO)

 

 

 

T(°K) H(Gauss)
4/26/68
1.121 0
1.056 2820
1.055 2820
1.020 3860
4/29/68
0.972 4310
0.941 4640
0.912 4830
5/1/68
0.769 8070
5/2/68
0.752 9570
0.766 10010
0.755 8760
0.748 9230
0.740 9820
0.761 7500
0.759 6620
0.759 5990
0.755 10000
0.759 9240
0.756 10060
5/6/68
0.762 9620
0.756 9000
0.765 8080
0.756 8740
0.758 7500
0.759 6600
0.760 5970
0.763 5970

0.765 5500

 

169

Table 18. MnBr2-4H20 C—axis isentropic magnetizations.

 

 

 

T(°K) H(Gauss) T(°K) H(gauss) T(°K) H(Gauss)
3/6/68
1.022 1620 0.952 6600 0.913 8570
1.018 2030 0.945 6940 0.909 8740
1.015 2540 0.938 7230 0.905 8910
1.006 3360 0.933 7500 0.902 9080
0.990 4310 0.929 7740 0.899 9240
0.978 5080 0.924 7970 0.895 9480
0.966 5700 0.920 8190 0.892 9690
0.958 6200 0.916 8380 0.888 9970
3/6/68
0.907 9810 0.952 8570 1.030 6600
0.910 9690 0.959 8380 1.045 6200
0.915 9550 0.967 8190 1.062 5700
0.920 9400 0.976 7970 1.079 5080
0.926 9240 0.985 7740 1.109 4090
0.932 9080 0.995 7500 1.149 1210
0.938 8910 1.005 7230
0.945 8740 1.017 6940
3/6/68
1.185 1850 1.025 7230 0.946 9080
1.174 2660 1.014 7500 0.939 9240
1.165 3200 1.003 7740 0.934 9400
1.135 4310 0.994 7970 0.928 9550
1.108 5080 0.984 8190 0.922 9690
1.087 5700 0.975 8380 0.918 9820
1.068 6200 0.968 8570 0.913 9970
1.052 6600 0.960 8740
1.037 6940 0.953 8910
3/6/68
0.914 9970 0.964 8740 1.044 6940
0.919 9820 0.972 8570 1.058 6600
0.924 9690 0.980 8380 1.075 6200
0.930 9550 0.990 8190 1.093 5700
0.936 9400 0.999 7970 1.111 5150
0.942 9240 1.009 7740 1.145 4180
0.949 9080 1.020 7500 1.190 1910
0.956 8910 1.031 7230

 

170

Table 18. (Cont.)

 

 

 

T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)
3/6/68
1.249 1910 1.073 7230 0.979 9080
1.236 2710 1.059 7500 0.972 9240
1.225 3260 1.048 7740 0.964 9400
1.192 4310 1.037 7970 0.957 9550
1.166 5080 1.026 8190 0.950 9690
1.141 5700 1.014 8380 0.944 9820
1.120 6200 1.006 8570 0.937 9970
1.104 6600 0.996 8740
1.088 6940 0.988 8910
0.938 9970 1.000 8740 1.091 6940
0.945 9820 1.010 8570 1.107 6600
0.952 9690 1.020 8380 1.124 6200
0.959 9550 1.030 8190 1.145 5700
0.966 9400 1.041 7970 1.165 5080
0.974 9240 1.052 7740 1.199 4220
0.982 9080 1.063 7500 1.248 1970
0.991 8910 1.077 7230 _
1.342 1850 1.151 7230 1.042 9080
1.329 2660 1.138 7500 1.033 9240
1.319 3100 1.124 7740 1.024 9400
1.282 4310 1.110 7970 1.014 9550
1.252 5080 1.098 8190 1.005 9690
1.226 5700 1.085 8380 0.996 9820
1.204 6200 1.074 8570 0.988 9970
1.185 6600 1.064 8740
1.167 6940 1.053 8910
0.990 9970 1.066 8740 1.169 6940
0.998 9820 1.078 8570 1.186 6600
1.006 9690 1.088 8380 1.205 6200
1.015 9550 1.100 8190 1.226 5700
1.026 9400 1.112 7970 1.248 5080
1.035 9240 1.125 7740 1.285 4180
1.045 9080 1.140 7500 1.334 1910
1.055 8910 1.154 7230

171

Table 18. (Cont.)

 

 

 

T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)

3/6/68
1.511 1850 1.308 7230 1.189 9080
1.499 2660 1.292 7500 1.177 9240
1.442 4310 1.280 7740 1.167 9400
1.409 5080 1.264 8190 1.155 9550
1.384 5700 1.238 8380 1.144 9690
1.363 6200 1.226 8570 1.134 9820
1.344 6600 1.212 8740 1.123 9970
1.325 6940 1.201 8910

3/6/68
1.298 9890 1.377 8740 1.485 6940
1.302 9820 1.390 8570 1.501 6600
1.312 9690 1.401 8380 1.519 6200
1.323 9550 1.414 8190 1.541 5700
1.334 9400 1.427 7970 1.565 5080
1.344 9240 1.440 7740 1.603 4040
1.355 9080 1.453 7500 1.650 1850
1.366 8910 1.468 7230

3/6/68
1.974 1850 1.795 7230 1.679 9080
1.963 2710 1.781 7500 1.668 9240
1.953 3100 1.768 7740 1.656 9400
1.921 4310 1.755 7970 1.645 9550
1.891 5080 1.740 8190 1.633 9690
1.868 5700 1.728 8380 1.622 9820
1.846 6200 1.715 8570 1.607 9970
1.829 6600 1.704 8740
1.802 7090 1.691 8910

 

 

172

Table 19. CszMnBr4-2H20 Isentropic magnetizations

 

 

 

T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)
3/20/68, H // [1 i I]
1.026 9600 1.041 8570 1.052 7230
1.030 9400 1.043 8380 1.053 6940
1.032 9240 1.045 8190 1.055 6600
1.035 9080 1.047 7970 1.055 6200
1.037 8910 1.049 7740 1.057 5700
1.039 8740 1.050 7500 1.060 5080
3/21/68, H // [i 1 I]
2.210 9120 2.208 9240 2.284 7500
2.204 9240 2.216 9080 2.293 7230
2.197 9400 2.225 8910 2.304 6940
2.185 9620 2.232 8740 2.313 6600
2.179 9760 2.241 8570 2.326 6200
2.168 9970 2.250 8380 2.338 5700
2.181 9760 2.258 8190 2.353 5080
2.189 9620 2.267 7970
2.201 9400 2.275 7740
3/20/68, H‘i [1 1'1]
1.031 9780 1.029 8570 1.025 6940
1.031 9620 1.028 8380 1.024 6600
1.030 9400 1.028 8190 1.023 6200
1.030 9240 1.027 7970 1.022 5700
1.030 9080 1.027 7740 1.020 5080
1.029 8910 1.026 7500
1.029 8740 1.026 7230
3|20|68, H.i [I I I]
1.036 4800 1.041 7740 1.040 9080
1.038 5700 1.041 7970 1.040 9240
1.039 6200 1.045 8190 1.040 9400
1.040 6600 1.041 8380 1.040 9620
1.040 6940 1.041 8570 1.039 9760
1.040 7230 1.041 8740 1.039 9970
1.040 7500 1.040 8910

 

 

 

 

173

Table 20. CSZMnBr4-2H20 specific heat maxima (H // [1 1 1])

 

 

 

 

 

 

 

T(°K) H(Gauss)
3/21/68

2.584 9200

2.542 9920

2.642 8080

2.692 7010

2.751 5420

2.795 3860

2.828 1970

2.834 0000
Table 21. NiC12-6H20 AC plane isentropic rotation (A' = 110°)
6 T(°K) 6 T(°K) 6 T(°K)

4/9/68

80 1.059 200 1.068 320 1.067
90 1.057 210 1.066 330 1.071
100 1.058 220 1.063 340 1.072
110 1.059 230 1.061 350 1.073
120 1.060 240 1.060 360 1.072
130 1.062 250 1.059 10 1.072
140 1.065 260 1.058 20 1.070
150 1.068 270 1.059 30 1.067
160 1.070 280 1.061 40 1.065
170 1.070 290 1.062 50 1.063
180 1.070 300 1.063 60 1.062

190 1.069 310 1.065 70 1.061

 

 

174

 

 

 

Table 22. NiCl2°6H20 A'—axis isentropic magnetizations.
T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)

4|9|68
1.344 1790 1.345 5410 1.345 8290
1.343 2260 1.344 6200 1.345 8660
1.344 2820 1.344 7090 1.347 9000
1.345 4310 1.344 7370

4|11|68
1.839 5080 1.826 7740 1.827 9080
1.836 5700 1.826 7970 1.827 9240
1.832 6200 1.826 8190 1.826 9400
1.823 6600 1.828 8380 1.825 9620
1.829 6940 1.827 8570 1.824 9760
1.828 7230 1.827 8740 1.824 9970
1.827 7500 1.827 8910

4|11|68
1.846 9970 1.859 8570 1.874 6410
1.848 9820 1.860 8380 1.874 6200
1.850 9620 1.862 8190 1.877 5700
1.852 9400 1.864 7970 1.880 5080
1.854 9240 1.866 7740 1.883 4310
1.854 9080 1.867 7500 1.887 3100
1.856 8910 1.869 7230 1.888 1680
1.858 8740 1.871 6940

4|11|68
1.929 1680 1.904 6940 1.894 8910
1.928 1970 1.903 7230 1.893 9080
1.928 2260 1.901 7500 1.893 9240
1.924 3360 1.899 7740 1.892 9400
1.920 4310 1.899 7970 1.890 9620
1.916 5080 1.897 8190 1.890 9760
1.912 5700 1.896 8380 1.888 9970
1.909 6200 1.895 8570
1.907 6600 1.894 8740

 

175

 

 

 

Table 22. (cont.)
T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)
4|11|68
1.972 9970 1.988 8570 2.003 6600
1.974 9820 1.989 8380 2.006 6200
1.977 9620 1.991 8190 2.009 5700
1.979 9400 1.993 7970 2.012 5080
1.981 9240 1.995 7740 2.016 4310
1.982 9080 1.997 7500 2.021 3360
1.985 8910 1.999 7230 2.021 2820
1.986 8740 2.001 6940 2.011 1680
4 |11 [68
2.156 1680 2.134 6940 2.123 8740
2.155 2260 2.132 7230 2.123 8910
2.152 3360 2.130 7500 2.122 9080
2.148 4310 2.129 7740 2.122 9240
2.144 5080 2.127 7970 2.121 9400
2.140 5700 2.126 8190 2.119 9620
2.138 6200 2.125 8380 2.118 9820
2.136 6600 2.124 8570 2.118 9970
4 |18 |68
2.264 7730 2.257 8740 2.252 9620
2.263 7970 2.256 8910 2.250 9820
2.262 8190 2.255 9080 2.249 10060
2.260 8380 2.254 9240
2.259 8570 2.253 9400
4 |18 |68
2.268 9760 2.281 8570 2.295 6940
2.270 9620 2.283 8380 2.297 6600
2 .272 9400 2 .285 8190 2 .300 6200
2.274 9240 2.287 7970 2.303 5700
2 .276 9080 2 .288 7740 2 .305 5080
2.277 8910 2.290 7500 2.307 4800
2.279 8740 2.292 7230

 

176

 

 

 

Table 22. (Cont.)
T(°K) H(Gauss) T(°K) H(Gauss) T(°K) H(Gauss)
4 |11|68
2.477 9690 2.494 8380 2.514 6200
2.479 9550 2.496 8190 2.519 5700
2.481 9400 2.498 7970 2.522 5080
2.483 9240 2.501 7740 2.528 4310
2 .485 9080 2 .503 7500 2 .533 3360
2.488 8910 2.505 7230 2.535 2820
2.489 8740 2.508 6940 2.538 1680
2.492 8570 2.511 6600
4 |11|68
3.134 1680 3.118 6600 3.114 8380
3.134 2260 3.117 6940 3.116 8570
3.133 2820 3.116 7230 3.116 8740
3.128 4310 3.116 7500 3.116 8910
3.124 5080 3.116 7740 3.115 9240
3.121 5700 3.115 7970 3.112 9620
3.119 6200 3.115 8190 3.111 9970

 

Table

23.

177

NiClz-SHZO specific heat maxima

(H //A').

 

 

 

T(°K) H(Gauss)
4/18/68
5.229 4940
5.241 6410
5.112 9840
5.087 8950
5.088 8270
5.139 7550
5.112 7230
5.149 7230
5.159 7150
5.212 6600
5.212 6600
5.151 5120
5.154 5120
5.263 2880