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3'I. III

 

 

     

I,

 

      
   

  

[JRDADV

Michigan State
University ‘

This is to certify that the

thesis entitled

A CALORIMETRIC STUDY OF MIXTURES OF

COBALT AND NICKEL CHLORIDE HEXAHYDRATES
presented by

Nguyen Truong Lam

has been accepted towards fulfillment
of the requirements for

Ph.D. degree“, Physics

 

 

4.... for—«T

Major professor

 

Date 9/7/1977

 

0-7639

 

 

 

 

A CALORIMETRIC STUDY OF MIXTURES OF
COBALT AND NICKEL CHLORIDE HEXAHYDRATES

By

Nguyen Truong Lam

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

DOCTOR OF PHILOSOPHY

Department of Physics

1977

ABSTRACT

A Calorimetric Study of Mixtures of
Cobalt and Nickel Chloride Hexahydrates

by

Nguyen Truong Lam

Using a standard He“

adiabatic calorimeter, we have
measured the specific heats of a series of ten mixed crystals
with chemical formula CoxNil_xClz-6H20. All the samples
studied show fairly sharp transitions. The critical tempera-
tures thus determined vary continuously from 2.u3°K to

4.H7°K for x ranging from 0.89 to 0.13. The Neel tempera-
tures for x - l and x 8 0 have been determined earlier as
2.29°K and 5.3“°K. We found that the critical behaviors of '
the mixed crystals can be explained by assuming that the Ni
and Co spins behave like Ising spins randomly distributed

on a square lattice, and interact through exchange interac-
tions which do not change with concentrations. We have
applied the annealed Ising, the annealed Bethe-Peierls-Weiss
(BPW), the quenched BPW models to this system. All three
theoretical models can be solved exactly, and have varying
degrees of physical validity. The quenched models assume

the distribution function of the exchange interactions to

be fixed, while the annealed models allow this distribution

Nguyen Truong Lam

to come into equilibrium with the spins at each temperature.
We found that the TN vs. concentration diagram can be fairly
well explained by both the annealed Ising and BPW models.
However, the annealed BPW does not fit the experimental
specific heat data as well as the quenched BPW model, while
the latter predicts critical temperatures which are too high.
In all cases, the specific heat data near the critical region
agree well with the annealed Ising model. We present semi-
quantitative arguments to support the view that the quantita-
tive differences between the quenched and annealed BPW models
are due to some peculiarity in the BPW approximation itself,
and that it is unlikely that an exact solution of the
quenched Ising model will exhibit significant differences
from the annealed Ising model.

The low temperature specific heats are also found to
follow a T3 dependence between 0.5T/TN and 0.75T/TN. This
is presumably due to spin wave behavior. We also present
some preliminary magnetic phase diagram data. We discuss
these features qualitatively and suggest directions in which
further theoretical and experimental investigations on

CoxN11_x012-6H20 can be carried out.

ACKNOWLEDGMENTS

I would like to express my sincere thanks to my
thesis advisor, Dr. H. Forstat, for having suggested the
topic of this research, and for his advice and encourage-
ment, without which this project might never have been com-
pleted. His understanding and patience in attempting to
teach me experimental physics are greatly appreciated. I
would also like to thank Dr. M. F. Thorpe for many fruitful
discussions about the theory of the annealed Ising model.

To Dr. T. A. Kaplan I owe many valuable comments which help
clarify some of my ideas about the physics of random magne-
tic systems. The financial supports of the Air Force Office
of Scientific Research and of the Physics Department, during
the early and later stages of this work, are gratefully
acknowledged.

Finally, to my parents who have always given me so
much, and asked for so little in return, I dedicate this

thesis.

11

TABLE OF CONTENTS

List or Tables 0 O O O O O O O O I O O O O O O O O O 0

List of Figures . . . . . . . . . . . . . . . . . . .

IntrOduction O O O O O O O I O O O O O O O O O O O O O

I.

II.

Theory
A.
l.
2.
B.
l.
2.
C.
l.
2.
3.
A.
5.
D.
1.
2.
30
I".

General considerations . . . . . . . . . . . .

Quenched model . . . . . . . . . . . . . .
Annealed model . . . . . . . . . . . . . .

Mean field models . . . . . . . . . . . . . .

Virtual crystal approximation . . . . . .
Quenched mean field model . . . . . . . .

Annealed Ising model . . . . . . . . . . . . .

Relation between the annealed and regular
Ising models . . . . . . . .
Specific heat of the annealed Ising
model . . . . . . . . . . . . . .

Annealed Ising chain . . . . .
Annealed Ising square lattice

Correlations between bonds . .

BPW mOdel O O O O O I O O O O O O O O O I O 0

Regular Ising lattice . . . . . . . . . .
Annealed Bethe lattice . . . . . . . . . .
Quenched Bethe lattice . . . . . .
Quenched Classical Heisenberg Bethe

lattice . . . . . . . . . . . . . . . . .

Experimental Techniques . . . . . . . . . . . . .

A.

l
2
3
1;

Experimental Apparatus . . . . . . . . . . . .

Dewar and calorimeter
Nylon sample holder. .
Vacuum pumps . . . . .
Pressure Gauges . . .

111

Page
. v
. vi
. l
. M
. h
. A
. 5
. 6
. 6
. 10
. 1h
. 1H
. 18
. 20
. 21
. 22
. 2A
. 2h
. 27
. 28
. 31
. 32
. 32
. 32
. 37
. 37
. 37

B.

Page

5. Thermometer current supplies . . . . . . . . 39
6. Measuring electronics . . . . . . . . . . . 39
7. Magnet and Gaussmeter . . . . . . . . . . . “1

Experimental Procedures . . . . . . . . . . . . “3
1. Sample preparation . . . . . . . . . . . . . “3

2. Helium transfer and temperature
calibration . . . . . . . . . 45

3. Specific heat measurements . . . . . . . . . “8
A. Adiabatic Field Rotations . . . . . . . . . 51
5. Adiabatic magnetizations . . . . . . . . . . 51
6. Antiferromagnetic-paramagnetic

boundary . . . . . . . . . . . . . . . . . . 52
7. Shutdown . . . . . . . . . . . . . . . . . . 52

C. Data Reduction . . . . . . . . . . . . . . . . . 52

1. Converting Pressure into Temperature . . . . 52
2. Thermometer calibration equations . . . . . 53
3. Determination of thermometer resistances . . 5"
A. Calculations for specific heat, field

rotations and field sweep data . . . . . . . 55

III. Experimental results and discussion . . . . . . . . 57

A.

D.
E.
Referenc

Appendix

Review of calorimetric and magnetic properties of
COC12’6H20 and N1C12'6H20 o o o o o o o o o o o 57

1. Crystallographic and magnetic structures . . 57
2. Specific heat data . . . . . . . . . . . . . 51

The concentration phase diagram . . . . . . . . 69

1. Mean field fits . . . . . . 71
2. The annealed Ising and Bethe lattices . . . 7“
3. Annealed and Quenched Bethe lattices . . . . 78
u. Choice of JN1_CO . . . . . . . . . . . . . . 83

Specific heat results . . . . . . . . . . . . . 87

1. Comparisons between theoretical and experi-

mental specific heats . . . . . . . . . . . 87
2. Entropy calculation . . . . . . . . . 95
3. Validity of the annealed models . . . . . . 98
u. Low temperature behavior . . . . . . . . . .102

Magnetic phase diagram . . . . . . . . . . . . .105
Conclusions . . . . . . . . . . . . . . . . . .111

es 0 O O O O O O O I O O O O 0 O O O O O O O O .11“

. . . . . . . . . . . . . . . . . . . . . . .117

Table

I.

II.

III.

IV.

LIST OF TABLES

Page

Lattice parameters of CoClZ-6H20 and
N1012'6H20 o o o o o o o o o o o o o o o o o o o 57

Chemical compositions and Neel temperatures of

the samples for which calorimetric measurements
have been made. The values quoted for each
element are percentages by weight. . . . . . . . 70

Values of JN -Co obtained when the annealed
2-dimensiona concentration phase diagram is
required to pass through a particular data

point . . . . . . . . . . . . . . . . . . . . . 8A

Values of exchange constants in the annealed
planar Ising and BPW models . . . . . . . . . . 86

Experimental and theoretical total magnetic
entropy changes . . . . . . . . . . . . . . . . 96

LIST OF FIGURES

Figure Page
1. Pyrex helium dewar . . . . . . . . . . . . . . . . 33
2. Cross section of body of calorimeter
(Not to scale) . . . . . . . . . . . . . . . . . . 3‘l
3. Side view of upper part of calorimeter
(Not to scale) . . . . . . . . . . . . . . . . . . 35
A. Schematic diagram of pumping system . . . . . . . 38
5. Sample thermometer current supply. Similar circuit
for bath thermometer with all resistances divided
by 10 O C I O O O O O O O I O O O O O I O O C O O
6. Diagram of electrical measuring circuits . . . . . “2
7. Example of recorder pen path . . . . . . . . . . . 50
a.) Specific heat data
b.) Field rotation data
8. Magnetic structure of CoCl '6H20. J, J1, J2 are
possible exchange interact ons . . . . . . . . . . 59
9. Experimental specific heats of NiC12-6H20 and
CoClg-6H20 compared with the BPW and Ising
mOdels O I O O O I O O O O O O O O O O 0 O O O O O 6“
10. TN vs. concentration diagram for mean field
mOdels O O O O O O O O O O O O O O O O O I I I O O 73
11. TN vs. concentration diagram for the annealed
and BPW madels O O O O O O O O C O O O O O O C O O 77
12. TN vs. concentration diagram for the annealed
square lattice and simple cubic Ising models . . . 80
13. TN vs. concentration diagram for the annealed and
quenched BPW models . . . . . . . . . . . . . . . 82
1h. Experimental and theoretical specific heats of 89
N10.11 000.89 C12'I6H20 o o o o o o o o o o o o o o
15. Experimental and theoretical specific heats of

N1 012.6H20 o o o o o o o o o o o o o o 91

0.50 C°o.so

vi

Figure Page

16.

17.

l8.

19.

20.
21.

Experimental and theoretical specific heats of
N10.87C00.13C12.6H20 o o o o o o o o o o o o o o o o 93

Normalized magnetic entropy for three Co concentra-
tions. The Neel temperatures are indicated by
arrows. O O O I O O I O O O O 0 O O O O O O O O O O 98

Fractional mean square fluctuations of the distri-
bution functions for the three possible bonds in

the annealed Ising and BPW models. Note that the

right hand scale is ten times larger than the left

hand scale . . . . . . . . . . . . . . . . . . . . .101

C/T3 vs. T/TN for three different Co concentra-
tions 0 O O O O O I O O O O O O O O O O O O O O O .10“

Spin wave coefficient a vs. Co concentration . . . .107

Tentative magnetic phase diagram for two different
Co concentrations . . . . . . . . . . . . . . . . .110

vii

Introduction

 

In recent years, there has been an upsurge of interest
in random magnetic systems. As can be expected, such systems
present interesting experimental and theoretical problems.
The present work is a calorimetric study of random mixtures
of two antiferromagnetic insulators.

NiClz-6H20 and CoC12-6H20 have similar molecular
weights, formation energies and crystallographic structures.
Furthermore, the lattice constants are nearly equal. There-
fore, one can expect the two salts to form a complete series
of solid solutions and indeed, the compositional phase
diagram has been studied as early as 1928 by Osaka et al.1
The magnetic behaviors of both salts have been extensively
investigated. Calorimetric studies indicate Neel tempera-
tures in the liquid He“ region (2.29°K for CoCla'6H20 with
effective spin S - 1/2 and 5.3M°K for NiC12'6H20 with effec-
tive spin S‘- l). Magneto caloric and susceptibility studies
establish that both salts can be described as anisotropic
antiferromagnets although the source of the anisotropy may
differ in each salt.

Using a standard He“

calorimeter, we have established
the Neel temperature vs. concentration diagram for a wide
range of compositions. To explain this phase diagram, we

have made the following assumptions:

a) the Ni++ and Co++ ions are randomly distributed
on a square lattice.
b) the exchange constants between like ions retain

their "pure" values in the mixed lattice.

Thus, the only adjustable parameter is the exchange
constant between unlike ions. We have taken this exchange
constant to be antiferromagnetic. Further, it has been
possible to carry out a detailed comparison between experi—
mental and theoretical specific heats. Among the theoretical
models considered, the one which appears to explain the phase
diagram and fit the experimental specific heats the best is
the "annealed Ising" model. However, these apparent agree-
ments raise almost as many questions as they answer. We
will attempt to address those questions in the course of
this work.

As far as their magnetic behaviors are concerned,
NiClz'6H20 and CoC12'6H20 exhibit qualitatively similar
magnetic phase diagrams. However, the anisotropy energies
in NiClz'6H20 are significantly higher than those of CoClZ'6H20.
Also, the easy axes are oriented differently. Therefore,
one might expect an extensive study of the magnetic phase
diagrams of mixed crystals to reveal some interesting aspects.
Unfortunately, we have not been able to grow single mixed
crystals of sufficient quality to warrant such an investiga-
tion. We will present preliminary magnetic phase diagrams

for two concentrations and attempt to discuss them qualitatively.

This work will be divided into three sections:
-Section I sets forth the theoretical models used.
-Section II describes the experimental apparatus.
-Section III contains a discussion of the results

and offers suggestions for further work.

I. Theory
A. General considerations

Consider a lattice of N magnetic ions with spins
31: whose magnetic behaviors can be described by a
Heisenberg Hamiltonian,

Ha.- ZJ s (1.1)
{1.3} iJ 185

Following Brout,2

we distinguish between two kinds of ther-
modynamic behaviors if the Jij's are allowed to vary randomly.

l.) Quenched model

 

The positions of the ions are "frozen in", i.e. they
are not allowed to change with temperature. For a definite
ionic configuration specified by ($1, 23, ...), one writes

the Hamiltonian explicitly as

I}i(§',§ ,...) - - :2: J' (I ,§ ...)§ °§ (1.2)
i J {1,3} 11 i J i J
and the partition function as
Z(§1,§J,...)= g e"‘3H (1.3)
{81}
with
B 3 l/kT

k - Boltzmann's constant
T 8 temperature
In (1.3). the summation is carried out over all possi-

ble spin states. For the particular ionic configuration

A

under consideration, the average energy can be claculated

in the usual way as

U(;1,;J,ooo) . "' 1%:le- (1.”)

The macroscopic energy U cannot depend on a particu-
lar ionic configuration, but only on such parameters as the
concentrations of the ions. In other words, the observable
‘U must be an average of the U(;i,xj,...) over all spatial

configurations, i.e.

fi- 8 <<U(;1,;Jooo)>> (105)
or
E 8 <<.. 3%)): -aiB <<1nZ(;1,-J;J,..) >> (1.6)
38

Here, the double bracket denotes configurational
averaging. From (1.6) one sees that all thermodynamic func-

tions can be obtained from <<an(;i,xh...)*>.

As far as mixtures of magnetic insulators are con-

cerned, the quenched model can certainly be realized in practice.

2. Annealed model

In this model, one allows the probability distribution
of the JiJ's to come to equilibrium with the spins at each
temperature. In much the same way one lets the number of
particles vary in the grand canonical ensemble. One is thus
led to introduce a grand partition function

E :- Z 2: e'BHe'Bgnfn (1.7)
{81} {fn}

where

5n - chemical potential associated with a possible

value Jn of J13

fn =- thermodynamic variable conjugate to {n

The average energy U is now given by

fi - - Jig—21E. (1.8)
With the {n's determined so as to produce the average dis-
tribution P(J), which is assumed independent of temperature.

In (1.7) the summation over the spins and the fn's
are carried out simultaneously. Looking at (1.7), one can
say that, roughly speaking, the annealed model amounts to
averaging the partition function Z over the probability
distribution of the Jij's, while in the quenched model, this
averaging is done for an. As one may imagine, the former
averaging is mathematically much more tractable.

The nature of "fn" may seem obscure at the moment.
It can be related to the probability for a particular exchange
interaction Jn to occur. More details will be provided in

part C.
B. Mean field models

In almost all dissertations on magnetic insulators,
one usually begins with some sort of mean field approximation.
Let us follow this tradition by considering two random mean

field models.

1. Virtual crystal approximation
In the mean field approximation one replaces the
Heisenberg Hamiltonian (1.1) by a sum of single spin

Hamiltonians, i.e.

)imf - 22’11 (1.9)

where

)1 . -§1 . :8 J (1.10)

1 13‘s?

The bracket denotes thermal average.

For definitiveness, let us assume all J13 0, two kinds
of magnetic ions A and B (with spins SA and SB), and nearest
neighbor interactions only. Thus, we introduce three kinds
of exchange interactions: 3AA: JAB: JBB' Further, assume
the presences of small anisotropy energies which orient all
spins in the same direction, in the ordered state. In the
following, we can then drop all vector notations.

.Consider an A ion surrounded by z nearest neighbors.
From (1.10), one sees that the effective field acting on ion

A is given by

Similarly
HB - [JAB nA ”A + JBB nBch] / gBu (1.12)

Here
gA(gB) a gyromagnetic ratio of ion A(B) along the
magnetization direction
u - Bohr magneton

nA(nB) - number of nearest neighbors of type A(B)

”A a <SA> a“B 8 <SB>

Let p and q be the concentrations of ions A and B

respectively.

We have
“A + “B - z (1.13)
p + q - 1 (1.1u)

The virtual crystal approximation consists in setting
nA - zp (1.15)
nB - zq (1-15)

The mean field consistency conditions become

 

0h 8 SA BSA (xA) (1.17)
03 = 33 BSB (xB) (1.18)
where
BS (x) - the Brillouin function
- 25+1 goth géil x - l. coth §_ (1.19)
ZS ZS 28 23
S g vH
xA,= _é_£_11 (1,20)
kT
S g uHB
B B
I — 1021
xB RT ( )

The system of equations (1.17) and (1.18) possesses
non zero solutions only below a critical temperature Tc.
To find To, expand the right hand sides of (1.17) and (1.18)

around 0A a 0B a 0. Using the well known Taylor series

BS(x) . §il + 0(x3) (1.22)
38

these expansions read

 

 

S(S +1)
agAA
A 3kT [2(pJAA 01 + qJAB<TB>] (1.23)
S (s + 1)
05" B 3:? [2(pJAB “A + qJBB‘TB)] (1.2a)

Using the mean field expressions for the critical

temperatures of the pure systems

2 sA(sA + l)JAA
- z S (S + l)J

3k

 

 

(1.25)

We can rewrite (1.23) and (1.2A) as a homogeneous

linear system

J
(p — 1.)”1 + q .6203 = 0 (1.26)
TA JAA

JAB 0'
P ___ + - 2.
JBB A (q )0a = 0 (1.27)

To get a non-zero solution the determinant should

vanish. This condition leads to a quadratic equation in T.

2
JAB

2 —
T (p'I‘A 0TB)T TA TB pq (1 JAAJBB) - 0 (1.28)

The positive solution Tc is given by

 

I'-'

Tc = 5 [( TA + qTB) +:v/(pTA - qTB)2 + "Pq TATE JIB
JAAJBB

(1.29)

10

(1.29) gives the critical temperature vs. concentra-
2
tion in this approximation. Note that when JAB = JAAJBB:

(1.29) reduces to

TC ' pTA + QTB (1.30)
and the Tc vs. p curve becomes a straight line.

2. Quenched mean field model

 

The virtual crystal assumption may appear somewhat
arbitrary. It certainly ignores fluctuations in the local
surroundings of an ion. In the hope of producing a more
consistent (if not better) approximation, let us follow the
procedure outlined in part A for quenched random models.

In this case, the configurational averaging of an can be
carried out in a simple manner.

Assume that the ions are distributed completely ran-
domly on the lattice. The probability for a lattice site to
be accupied by an A ion is simply p = concentration of A ions.
The probability that this ion is surrounded by “A and n

B

nearest neighbors of types A and B is then 2! pnAan.
nAlnB!

In other words, if we let
PA(nA,nB) = probability of occurrence of a cluster with an
A ion at the center surrounded by nA A ions and “B
B ions
Then

PA(nA,nB) = p w(nA,nB) (1.31)

11

with
z!
w(nA,nB) = —;—-!- pnAan (1.32)
"A H3

The partition function associated with this particu—
lar cluster will be denoted by ZA
Similarly
PB(nA,nB) = QWUIA’HB) (1033)
with associated partition function ZB

The configurationally averaged partition function Z

 

is given by
z z
an = Z PA(nA,nB) anA + Z PB(nA,nB) anB
nA=0 nA=0
(1.31:)
z z
s p Z w(nA,nB) anA + q 2 w(nA,nB) anB
nA=O nA=0
The explicit expressions for ZA and ZB are
28 +1
z = sinh( A x >/sinh( x11) (1.35)
A ng A 28A
+
ZB = Sinh (283 1 xB)///51“h (.22.) (1 36)
233 '

283

with xA and xB defined earlier [equations (1.20) and (1.21)].

Just as for the pure lattices, the average spins 0

A
and OB are calculated as
U=Sfl=SZw(nn)alnzA Z
A A axA p A A’ B -———— = DSA W(nAsnB)BSA(XA)

3
x“ (1.37)

12

 

 

 

 

dan danB
‘7 I S I S 28w n n I S 23w n n B x
B 36 KB 9 s < A: B)dx Xa q B ( A’ 3) SB( 3)
(1.38)
Expand the Brillouin functions around.0A I 0B I 0
to get
pS (S +1)
. A A 2.0.an) [nAJAAUA + nBJAB‘TB] (1.39)
3kT
qS (S +1)
I B B Zw(nA,nB) [nAJABUA + “BJBBGBJ (1.“0)
3kT
Now 2
z: 21 nA nB
Z:w(nA,nB)nA I n . DAIHB! p q "A I zp (1.A1)
A
z
- 21 nA nB g
Z:w(nA,nB)nB Z: “AlnB! p q “B zq (l.A2)

Expressing JAA’ JBB in terms of TA and TB, we obtain

2 T) JAB¢7
p - 4- pg B ' 0 (1.143)
( TA A JAA

J
_ABO'+(2-T)0'-o 1m:
Comparing (l.AA), (1.A3) with (1.27) and (1.26) one

can immediately write down the equation for the Tc-p diagram

as
2
J

Tc ' %{}PZTA + qZTB) +«J{;2TA-q2TB )2 1 "PZQZTATB J AB ]

 

AAJBB
(1.“5)

Thus, as far as the Tc-p diagram is concerned, the quenched

13

mean field model simply amounts to substituting p2 and q2
for p and q in the virtual crystal approximation. From
(1.Al) and (l.fl2), one also sees that it is possible to
obtain the virtual crystal approximation by letting

PA(nA,nB) I PB(nA’nB) I w(nA,nB) (1.46)
which is certainly a rough way of doing configurational
averaging.

So far, we have considered only the ferromagnetic
case. In general, the zero field thermodynamic behavior
V of an antiferromagnet is identical to that of the correspond-
ing ferromagnet, provided the antiferromagnetic lattice can
be subdivided into two sublattices [called "up" and "down"
sublattices], such that the nearest neighbors of an "up"
spin belong to the "down" sublattice, and vice versa. One
can verify this equivalence directly by noting that the
effective field acting on an A spin which is "up" should
bewritten as

HA I [JAAnA0h(down) + JABnEUB(down)]/8Av (l.A7)
With all exchange constants negative, one readily verifies
that by imposing new consistency conditions

”A(up) I -0h(down) (1.”8)

03(up) I -Ob(down) (1.A9)

the same Tc-p diagram is obtained as before. For zero exter-
nal magnetic field, (1.A8) and (l.h9) certainly appear rea-
sonable. As is also well known,3 a two sublattice division
is possible for simple crystallographic structures, such as

the simple cubic and square lattices.

In
C. Annealed Ising model

So far we have considered only isotropic Heisenberg
interactions. Since S I 1/2 for the Co ions and both pure
crystals exhibit magnetic anisotropy, it seems natural to
turn to a random Ising model as the next approximation. For
mixtures of magnetic insulators one would like to try a
quenched random Ising model. However, except for the one
dimensional case, the quenched model is beset with consider-
able mathematical difficulties. The annealed Ising model
turns out to have a simpler mathematical structure. Basi-
cally, it can be related to the regular Ising model, for which
exact solutions are available in one and two dimensions.

The following development closely follows Thorpe and Beeman.
The basic ideas behind the annealed Ising model have been
discussed earlier in the literaturefj’6 But ref. A has the
merit of clearly setting forth a general approach to the

problem.

1. Relation between the annealed and regular Ising

models

The most general Ising Hamiltonian can be written as

H I - E: J (r a' (1.50)
<mn> mn m n
with spin variables
crm, (In I 11

Suppose the Jmn's can take on a set of discrete values
J1, J2, ...J1, "'Jn' With each J1 associate a chemical poten-
tial £1.

15

The extended Hamiltonian )1 can then be written as
a sum of bond Hamiltonians
)‘l. 2 Hum (1.51)
<mn>
where, for example
)4 n
12 ' ' X Jifi ‘71 c’2 ' Z Eifi (1-52)
i=1 1
With f1 I 1 if J12 I Ji; 0 otherwise.
Note that f1 can be considered as a thermodynamical variable

conjugate to 51. To have a single exchange interaction

associated with each bond, the fi should satisfy the condi-

tion

n

2 r1 . 1

i=1 (1.53)
This makes

21¢? a 1 (1.51))

The bracket < > denotes thermal average. ‘fi’ can be inter-
preted as a probability for a bond to have the interaction
J1. The requirement that <f1> be temperature independent
will determine :1. I

The grand partition function 5 involves a summation
over both the f1 and the spins 03

a. Z X e-BH (1.55)
{0J1 {f1}

16

Do the sum over the fi first. Since, the }imn'in
(1.55) commute with one another, this sum reduces to a pro-
duct of terms of the form.

{Ef-BHM . 2; exp(BJ10‘m0'n + 851) (1.56)
1

The next step consists in noting that the product¢7m0h = :1.

One can then rewrite (1.55) as

Z exp(BJ1CTm<Tn + 52:1) = AeK‘m‘Tn (1.57)
i
Explicitly
for Oman = l Eexpin + 861) = AeK (1.58)
Oman = '1 Z exp(-BJ1 + 851) = Ae-K (1.59)
i

Combining (1.59) and (1.58) we find expressions for A and K

e2K g [z eB(€1+J1)]/[Zefi(51'Ji)] (1.60)
1 J

A2 3 [: ea(ci+11)] [hem-JD] (1.61)
i J

Notice that K and A depend only on temperature T and the J1,
and not on the pair of spins Um, 0h under consideration.

Therefore, performing all the summations in (1.55), we get

E = ANB Z eKaman = ANB Z(K) (1.62)
{m,n}

17

where

NB I total number of bonds in the lattice

Z(K) I partition function of the regular Ising lattice
(1.61) tells us that the annealed Ising model at temperature
T can be effectively related to a regular Ising model at a
different temperature T', inversely proportional to K.

From now on, assume nearest neighbors interaction

only. Then for a lattice of N spins, each one of which is

11,; .

surrounded by z nearest neighbors, we have NB I 2

As mentioned earlier, <f1> can be considered as the
average probability for a bond to have the interaction J1.

It is calculated as follows

 

 

2 2)in??? 31nA 2 a1
< > a - = ____ + __ nZ 8K
f1 NZ 3 851) 351 N2 8K {FI' (1°63)

or

_ alnA 3K
<f1> _. 861 + can —351 (1.61))
where 2 aan

e(K) I N; 3K = <210é>>= nearest neighbor correlation

function in the regular lattice
Using (1.61) and (1.60), one finds that (1.6A) is equivalent
to
e B<€1+J1> a 2 <fi> (1.65)
geek: ”3) [1+e(K)] + [l-e(K)]e2(K'BJi)

 

 

18

Sum over all i to obtain
2 (fl)

3 = l .66
1 [l+e(K)] + [1-e(K)]e2(K-BJ1) 2 (1 )

Using Exfi> I 1, one shows that (1.66) reduces to a more
1

symmetrical form

<f1>
i coth(K-BJ1)-e(K) a O (1.67)

 

Going over to a continuous probability distribution

function P(J), (1.67) becomes

2(1) dJ _ .
]{coth(K-BJ)-6(K) 0 (1°68)

In summary, (1.68) allows us to calculate K as a

 

function of temperature T, once P(J) has been specified.
Incidentally, suppose we change all the signs of the J's
without altering P(IJI). From (1.59) we see that K Simply
becomes -K. Then (1.68) remains satisfied, provided e(K) I
-e(-K). In this case, there is complete equivalence between
the ferromagnetic annealed Ising model and the antiferromag-

netic one with the same P(J).
2. Specific heat of the annealed Ising model

One can proceed further and obtain general expressions

for the average energy U and the magnetic specific heat C.

19

 

N2 1.
U a " 7 2J1<f10102> = 2J1 am: (1.69)
1 1 3(8Ji)
C 3 6U
5T (1.70)
Combining (1.61), (1.60), (1.68) we derive
8 NE 1 - ecoth (K-BJ)
u 2 f coth (K_ SJ) _6 }J P(J) dJ (1.71)
Further

-1
C " 'N‘k' “2 {13 “"2 (“>1 + I22 [11"(1'82 (x)-%E> 1'1}

2
(1.72)
With
11 = ./[coth (K- BJ)- e(K)J"2 P(J) dJ (1.73a)
12 a ‘[[coth (K- BJ)- e:(1<)]"2 csch2(K- BJ)P(J) dJ
(1.73b)
I3 = JfEcoth (K- BJ)- s:(K)]"2 csch2(K- BJ)J2 P(J) dJ
(1.730)

Consider some special cases.

3. Annealed Ising chain

The one dimensional regular Ising model has first

been solved by E. Ising:7

For the nearest neighbor correla-
tion function we have

e(K) a tanh K (1.7“)

20

Substituting (1.7“) into (1.71) and (1.72), we find

C:

. .. 2112 [J tanh( BJ) P(J) dJ (1.75)

2'0
75‘

I €32 “[JZ sech2( BJ) P(J) dJ (1.76)

If we have two kinds of ions A and B randomly distri-
buted on the chain with concentration p and q, and with
interaction constants JAA3 JBB; JAB: a natural choice of

P(J) might be
P(J) = 1328(J-JAA) + 2pq8(J-JAB) + q28(J—JBB) (1.77)

where 8 denotes a Dirac delta function.

To elaborate slightly, suppose we have NA A ions and
NB B ions (NA + NB = N) which are randomly distributed on an
infinite lattice. (1.77) is the bond distribution function
which results in the limit N+I , such that Efi I p and
Na
“'77 ' q°

Substituting (1.77) into (1.76), the specific heat
becomes
NE 3.; 82 [sziAsech2 BJAA+2qu§Bsech23 JAB+q2J§BsecthBBJ

(1.78)

With 2I2, the above agrees exactly with the heat capacity of
the quenche? Ising linear chain as calculated by Katsura and
Matsubara.

So, in the one dimensional case, the annealed and

quenched models for Ising spins are completely equivalent.

21
A. Annealed Ising square lattice

As is well known, the regular two-dimensional square
Ising model has first been solved by Onsager.9 He obtained

the following expression for e(K)

e(K) I coth 2K[% + ”-1 (2 tank22K-l) F1(k1)] (1.79)
where

k1 = 2 Sinh 2K (1.80a)
(cosh 2K)2

 

F1(k) I complete elliptic integral of the first kind

W/2 2
F1(k) = ]{ de(1—k2sin2e)'1/ (1.80b)

O

(1.79) looks more formidable than it really is, since it
turns out that Fl(k) can be readily evaluated by computer.
de
A second order phase transition occurs when 3E diverges.

This happens for Kc =.% 1n (l+.¢2) and 5c I :%§.

Inserting those values of Kc and ‘0 into (1.68) and
assuming P(J) given by (1.77), we obtain the critical temper-
ature Tc vs. concentration diagram.

Further, using the full expression for e(K) and (1.68),
K has been numerically calculated as a function of temperature
T, by the standard Newton-Raphson method. (1.72) and (1.73)
then give specific heat as a function of T.

- These theoretical results will be compared with exper-

imental data in chapter III.

22

Note that e(K) is an odd function of K. This ensures
complete equivalence between the antiferromagnetic and ferro-
magnetic annealed square Ising lattices under zero magnetic

field.

5. Correlations between bonds

 

At this point, it may be worthwhile to ask the ques-
tion: to what extent does an annealed model approximate a
quenched model which one expects to be much more physically
realistic for mixtures of magnetic insulators? Within the
framework of the annealed Ising model, it is possible to
provide a partial answer to this question.

In the quenched model, the interaction constants are
independently distributed on the lattice according to some
probability function P(J) which does not change with tempera-
ture. The annealed model allows the interaction constants
distribution to come into equilibrium with the spins at each
temperature. This makes the interactions distribution fluc-
tuate around some average distribution (which we have also
called P(J)). One can also look at the situation from a
different point of view.

Consider two pairs of spins: ( 01*72) and ( ar,ar+i).
Let f1 and £3 refer to the first and second pairs of spins
respectively. The thermal average <f1fj> can be interpreted
as P(J1,JJ) I joint probability for ( 01:72) and (<Tr,0f+1)
to have interaction constants_J1 and JJ respec-

tively.

23

u

Thorpe has shown that

<f1fJ> I <f1><fj> (1 +.c) (1.81)

where
1.82
<‘7102°r°r+l> - 52(K) ( )

 

[00th (K-BJi) - €(K)][00th (K-BJJ) - €(K)]

In the quenched model, P(J1,J2) I P(Jl):P(J2). So
from (1.81) one can say that the annealing process has de-
stroyed the "true" randomness or statistical independence of
the bonds. For the Ising chain, <Ula2°rar+1>' <7102><Urar+l>

2
6 so that c I 0. This helps explain why the annealed

and quenched Ising chains are completely equivalent.

Now, let f1 I f and 01 = 0r; a? I 0}+1 in (1.81)

J
and (1.82). We obtain

 

 

2 2
c = ‘fi ’ ‘ ‘fi’ = 1 - e2(K) (1.83)
<f1>2 [coth(K-BJ1) - e(K)]2
since 012 I 022 I l

c in (1.83) can be considered as the fractional mean
square deviation of the bond occupation probability for J1
from its average value <f1>. The magnitude of c is then a
rough measure of the extent to which the annealed and quenched
models differ from one another. One should not take this
too literally, however. For example, in the linear chain,
(1.83) becomes small but does not vanish, although the annealed
linear chain is equivalent to the quenched one.

It turns out that there is one non trivial theoretical

model where one can make a direct comparison between the

2A

annealed and quenched approaches. This is the Bethe-Peierls-

Weiss model, which we will consider in the next section.

D. BPW model

The BPW approximation was first applied to the Ising

11 Later, Weiss12 extended it

model by Bethe10 and Peierls.
to the Heisenberg model. Let us first review the BPW treat-

ment of the regular Ising lattice.
1. Regular IsingAlattice

Consider a cluster consisting of a central spin with
its 2 nearest neighbors. The BPW approximation treats the
interactions within the cluster exactly, and replaces the
effects of the outside spins by an effective field acting
on the nearest neighbors. Explicitly, the cluster Hamiltonian

can be written as

Z ' Z
H01 - -J Z (roan - g pH 00 - g ”(Hi-H) Z on (1.814)

nIl n=1
where
06 I central spin I I1
an I nearest neighbor spins I I1

I external magnetic field
H' I effective field due to outside spins.
g I gyromagnetic ratio along the magnetization direc-
tion.

J I interaction constant

25

We have assumed H and H. to be direbted along the
magnetization direction. We can then drop all vector nota-
tion.

The partition function of the cluster 201 I Tre’B)U'
can be conveniently divided into two parts

ch I 2+ + Z_ (1.85)
2+ corresponds to a I l

o
Explicitly

z
2 I Z Z exp[x + (x+L+K) Z on]

+ (713:1 02:11 {1:1
I ex[2 cosh (x+L+K)]z (1.86)

Similarly Z_ correponds to UOI-l and is given by

Z_ I e'x [2 cosh(x+L-K)]z (1.87)
Here
x I BguH
L I BgvH'
K I BJ
The average spins (do) and <ah> can be calculated in terms
of ch as alnz
‘ “0’ ‘ "E—x'c‘l' . (1.88)
31nZ
‘ “n’ "2 'TITC'I' (1.89)

We restrict ourselves to the case H I 0. (1.88) and (1.89)
then simplify to
21-2.,

‘ ”o" T (1.90)

26

[Z tanh(L+K + Z_tanh(L-K)]
<an>I + ) (1°91)

ch

 

Consider first the ferromagnetic case. The effective
field is then found by imposing the consistency condition

< ab> I < an> . This yields the following equation for

 

H' (or L)
2L [cosh(L + K) z-l (1.92)
e ' cosh(L - K)
or, in a more symmetrical form
tanh K tanh L I tanh [L/(z-l)] (1.93)

By expanding (1.93) as a Taylor series around L I 0
one finds that a non-zero solution exists for L below a cri-

tical temperature Tc given by

1:1
tanh Kc = tanh ET; 2-1 (1.9u)

One can also calculate the nearest neighbor correlation

e(K) = <<TO Uh> = E ‘3‘?" (1.95)

Combining (1.86), (1.87), (1.92), we find

2
(K = 1 -
e ) eZKcosh2L + 1 (1.96)

 

For N spins, the total average energy U is then given by

u = - 3223— cm (1.97)

27

and the specific heat

2K2 de

2 dK (1.98)

C)

II
Dali):
*3 C

II

2

N

For T2 Tc’ L 0 and (1.96) simplified to

e(K) = tanhK (1.99)

which is identical to the 8(K) of the Ising chain.

For the two sublattice antiferromagnet, the consistency
condition should now be < 00> I - <¢7n>. We then find

£(K) I - e(IK|) and the specific heat is exactly the same
as in the ferromagnetic case.

Just as the mean field approximation, the BPW appro-
ximation gives a finite specific heat discontinuity at Tc-
The main improvement over mean field is that BPW predicts a
non-zero specific heat above Tc-

Finally, note that the BPW approximation becomes
exact if the Ising spins are distributed on a pseudo lattice
called the "Bethe" lattice. For details, see Katsura and

13

Takizawa.

2. Annealed Bethe lattice

 

We are now in a position to apply the annealed Ising
formalism to the Bethe lattice. Recall that the transition

temperature Tc is given by

28

 

P(J)dJ _
J/‘EOth (Kc'BcJ) - €(Kc) - O (l 100)

As 6(Kc) I tanh Kc I 2%I’ (1.100) reduces to

‘/r(J) tanh (BcJ)dJ a 2%I (1.101)

This result has been derived earlier by Matsubara.1'4

3. Quenched Bethe Lattice

Matsubara,lu Katsura and Matsubara,8 Eggarter15 have
investigated the quenched Bethe lattice using three fairly
indirect approaches. We present here a simpler (but less
rigorous) derivation, based on a straightforward configura-
tional averaging of 1n ch.

Just as for the quenched mean field model (section

B.2), we can write

2 z
<<1anl>> = p :0 w(nA,nB) anA + q :0 w(nA,nB) anB(l.102)
nA= nA=

where all the symbols have been defined in 8.2.
Just as for the regular lattice, we find it convenient

to split ZA and 23 into two parts ZA+; ZA_; 25+; 28- such that

X Z 11 1'1
ZAI I e1 A0 2 cosh AExAn+LAiKAAJCOSh BExBn+LBiKABJ (1.103)

ix 2 nA i n +
Z i g 6 BO 2 cosh [xAn +LA KABJCosh B[xBn + LB-KBB]

(1.10u)

29

where
X10 ' BgiuH 3 Kin (1.105)
KIJ ’ 3313

Bgiufli

t‘
p.
u

1 I A,B
Note that we have introduced two effective fields
HA, Hg acting on nearest neighbor ions of type A and B
respectively.
For the ferromagnetic case, HA and Hg are determined

by the consistency condition

<"‘Ao> = “’An’ (1.106)
<"136’ = <0Bn’ (1.107)

where, for example

(1.108)

 

 

a<<1nzcl>> .l a<<1nzcl>>
Z

(GAO) = 3on "OAn> : aLA

The full expressions for (1.106) and (1.107) appear
somewhat complicated. To first order in LA’ LB they reduce
to
PtAB[(Z-1){PCAA + thB}-Z]LA

+ [l-zthB + (z-1)q2t§B + pq(z-1)t§BJLB = 0 (1.109)

2
[l-zptAA + (z-l)p2tAA + pq(z-l)tfiBJLA
+ thB[(z-1){ptAA + thB}-Z]LB 3 0 (1.110)

where we have abbreviated

tAA 3 tanh KAA tBB g tanh KBB tAB 3 tanh KAB

30

For non-trivial solutions to exist, the determinant

of the above system should vanish, or

2 2 , 2
[l-zptAA+(z-l)p2tAA+DQ(z-l)tg8][l—zthB+(z-l)q2tBB+pq(z—l)tAB]

'pqt§B[(Z-1) ptAA+thB “212 3 0 (1.111)

(1.111) can be factorized into

2 2 2

[{1-(2-1)ptAA) {l-(z-l)thB) - (z-1)2p q tAB][(1-ptAA)(1-thB)

-pthBJ I 0 (1.112)

This is equivalent to

{1-(z-l)ptAA} {1-(z-1)thB} -(z-l)2pqtiB I 0 (1.113)

since the other term in brackets corresponds to z I 2 (linear
chain) and has no non-trivial solution.

To sum up, (1.113) gives the Tc vs. concentration
diagram for the quenched ferromagnetic Bethe lattice. As
usual, for the two sublattice antiferromagnet, the consistency
conditions should now be < 0A0) I -< Okn> and < 9Eo> I “aBn’ .
This then yields the same Tc vs. p equation as in (1.113),

except that now t and tBB should enter with their absolute

AA
values.

In chapter III, we will compare (1.113) with the
annealed Bethe result (eq. 1.101). Right now, we can easily
write down expressions for the specific heat above Tc'

Above Tc’ since e(K) I tanh K the annealed Bethe

lattice specific heat CA reduces to the linear chain form

[eq. 1.78], assuming P(J) given by (1.77).

31

To find the quenched Bethe lattice specific heat C

Q9
one first finds the energy per bond -% 3 <<:: ch>> . Since
there are 33. bonds the total energy U is then -§

2 38

 

 

Finally, CQ 8 %% . Above Tc, H; I Hg I 0 and one finds that

CQ reduces to (1.78). Therefore, above a certain temperature,
we have CA I CQ and the annealed and quenched Bethe lattices
become equivalent.

Below Tc’ CA and CQ must be evaluated numerically.

The results will be presented in chapter III.

A. Quenched classical Heisenberg Bethe lattice

 

Finally, as an aside, one may mention the quenched
classical Heisenberg BPW model, since the Tc vs. p diagram
looks similar to (1.113). Assuming that the spins couple
through a Heisenberg interaction and behave like classical
vectors, Boubel et al.16 [also Katsural7] have derived the
following Tc vs. p diagram, within a quenched BPW approxima-

tion.
(1.111))

[l-(Z-l)pL(BCJAA)][l-(z-1)qL(BCJBBH - (z-1)2pqL2(eCJAB) = o

where

L(x) I cothx - % I the Langevin function. (1.115)

N 3<<1n Zol>> .

II. Experimental Techniques

A. Experimental Apparatus

 

l. Dewar and calorimeter

 

The experimental apparatus consists of a pyrex dewar
shown in Fig. l and a triple can calorimeter shown in Figs.
2 and 3. The dewar was made to specifications by H. S.
Martin and Son (Evanston, Illinois) and has been described
in details earlier.18

The fundamental problem in adiabatic calorimetry
is to ensure good thermal isolation of the sample. In this
case, the surroundings consist of a) the large Hen bath at
A.2°K in which the calorimeter is immersed, b) a smaller

A contained in a Helium can which is connected

volume of He
to the inner can where the sample is placed. Good thermal
isolation can be achieved by ensuring that the sample is not
in contact with the inner can, and by pumping on the inner
and outer cans. The purpose of the Helium can is to allow
temperatures lower than A.2°K to be reached by pumping on a
small volume of He“.
To reduce heat leaks to the inner can, all pumping

lines (including the leads line) are made of german silver.
The Helium can and inner can are made of cOpper. The inner

can pumping lines and leads line are partially surrounded

u
by the He in the Helium can. When they emerge from the end
32

33

 

 

 

v

Figure l.

 

'.- LI
:' :‘l
. r
€ '5‘
..
I.

SR
SYXXV.

“SICKXS

  
 
 

,xm

‘SSCSS-

\t
UT\

I
:ERSX.

,éxasxs

 

V.

\_

   

 

m
:SSL.‘ XXX

/

Pyrex helium dewar

822?

 

M-3064O 2mm
BORE STOPCOCK
(90" FROM FILL

PORTS)

32 mm 0.0- FILL
l80°

\Poars, ..
2 LONG

APART ,

C ROSS HATCHING

REPRESENTS
VACUUM JACKETS

BOTH DEWARSARE
SILVERED, WITH A

TAPERED SLIT, IO mm
WIDE AT THE BOTTOM

HELIUM CAN—:1: ‘—
PUMPING LINE '—

 

\4

INNER CAN

/PUMP|NG LINE

 

 

HELIUM CAN

 

EIGHT 2'56 k

 

 

TAPPED HOLES\%

 

 

”7

It-

 

BATH /
THERMOMETER

BAKELITE ——--“""

TERMINAL BOARD

ELECTRICAL/I

LEADS

NYLON SPACER \>

 

 

 

 

 

L EADS LINE

LEAD O‘RING

...———-—"BATH HEATER

(—-—‘ OUTER CAN

BRASS RADIATION
S HIELD

 

 

 

 

 

 

INNER CAN

GERMAN SILVER
STRIP

CERROLOW ||7

j/SOLDER JOINT

Figure 2. Cross section of body of calorimeter

(Not to scale)

35
r1 INNER CAN
r/fi / Q/ PUMPING
/
.. LINE

/

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HECIAIIIJM J L. NEEDLE
PUMPING VALVE
LINE f //
' L / J
/
’/’//)///W’F‘/ J
JUNCTION
___—.4.
BOX tag
'1 F-
LEADS
KJ‘” LINE

 

 

+
OUTER CAN
PUMPING LINES

[ll

 

 

 

 

 

I-..

 

 

 

 

L_.-
E

 

 

Ir~

Figure 3. Side view of upper part of calorimeter
(Not to scale)

36

of the leads line, the electrical leads are glued to the
bottom of the Helium can, thus ensuring good contact to a
low temperature bath. The bottom of the outer can is made
removable, to allow insertion of nylon spacers which further
prevent thermal contact between inner and outer cans.

Heat radiation from parts of the calorimeter at room
temperature constitutes another source of thermal leak.
The standard remedy for this is to put right angle bends in
pumping lines, wherever practical. As shown in Fig. 3, the
lower part of the Helium can evacuation line is connected by
a tee joint to the upper part, so that pumping on the
Helium can is done through one side of the tee. The upper
part of the outer can pumping line ends in a junction box,
from which emerge two smaller pumping lines to the outer can.
A brass radiation shield is attached to the bottom of the
Helium can. Note however that the inner can pumping lines
go straight down. Apparently, this produces no adverse
effects during specific heat measurements. The brass shield
has two clearance holes drilled through it, to allow inser-
tion of a stainless steel shield enclosing the smaple. This
limits the sizes of the samples, and does not seem to have
any effects, in one way or another. This shield was removed
in the last run.

This calorimeter is basically a modification of a
rotator calorimeter designed by J. R. Ricks. Some of the
early measurements were done with a calorimeter designed by

18

N. D. Love and described in his thesis. In Love's calori-

meter, the inner can is simply soft soldered (with Cerrolow 117)

37

to the bottom of the Helium can. But, due to repeated fail-
ures of the soldered joints, we have resorted to a "flange"
system, whereby the sealing is ensured by a lead 0 ring be-
tween flanges On the inner and Helium cans. Details of the

arrangement are depicted in Fig. 2.

2. Nylon sample holder

For experiments in a magnetic field, we use a nylon
a C-clamp attached to a nylon support to prevent the sample
from rotating under the influence of the field. This has
19

been more fully described elsewhere.

3. Vacuum Pumps

 

The pumping system is depicted schematically in Fig. A.
A Veeco EP 2AI 350W air cooled diffusion pump, which can

attain a pressure of 10'6

mm Hg, is used for pumping on the
inner and outer cans. A Welch Duo Seal pump serves as the
forepump. Another such pump maintains vacuum on one arm of
the U tube manometer, and is used to evacuate the McLeod
gauge after a reading. A high capacity Stokes mechanical

pump evacuates the dewar or pumps on the bath in the Helium

can.

A. Pressure Gauges

 

Helium can pressures above 2.50m Hg are measured with
a mercury U tube manometer. Below that pressure, an oil-
filled manometer is used to observe qualitatively the change
in pressure with temperature during thermometer calibration,
while the McLeod gauge measures the calibration pressure

accurately.

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39

A NRC 831 vacuum ionization gauge allows monitoring
of the pressures in the inner and outer cans. Both the
Helium can and the dewar have a U.S.G. gauge which gives

rough readings (30 in vacuum to 15 P.S.I.)

5. Thermometer current supplies

Both sample and bath thermometers are l/lO W, Allen
Bradley carbon resistors, with room temperature resistances
between 56 and 60 ohms. Their resistances are measured
potentiometrically, using a standard four leads technique.

The sample and bath thermometer current supplies are
diagrammed in Fig. 5. One can prove that the current I sup-

plied to the load is given by

I .. yz (2.1)

RT

Vz I reverse break down voltage of the Zener Diode
RT I total resistance between the ground and the inverting
input of the op amp.

With Vz nearly equal to 6.2V, Rt has been adjusted to
provide 1 microampere for the sample thermometer and 10 micro-
amperes for the bath thermometer. Such a low current is needed
to avoid self heating of the sample thermometer, while this

problem is less critical for the bath thermometer.

6. Measuring Electronics

To measure the voltage across the sample thermometer,
we use a Leeds and Northrup K-3 potentiometer. The off balance
voltage from the potentiometer is amplified by a Leeds and
Northrup 9835 B microvolt DC amplifier, and then "fed" to a

A0

 

 

 

 

 

 

THERMOMETER
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Figure 5. Sample thermometer current supply.
Similar circuit for bath thermometer with all
resistances divided by 10.

A1

Leeds and Northrup dual pen Speedomax G recorder with a 5mv
range card. The amplifier can be adjusted to give the amount
of sensitivity desired. A similar system using the other
pen of the recorder (with a 10 mv range card) monitors the
bath thermometer voltage.

A filtered Lambda LM263 power supply (0 to 32v),
in series with a set of variable resistors totaling 10 meg-
ohms provides the desired current through the sample heater.
The latter can be turned on and off by a relay connected to
an electronic timer (Veeder-Root Econoflex 71 220A) which is
preset to run for a selected time interval. The relay con-
nects an external resistor (of resistance A00 ohms, nearly
equal to heater resistance) in place of the heater in the
circuit, when the timer is off. This minimizes possible
pulses from the measuring circuit.

A simple switch arrangement allows a Data Technology
323 digital voltmeter to measure either the heater voltage,
or the voltage across a precision 10 ohm resistor in series
with it. The latter voltage gives the current through the
heater when it is on.

The above circuits are diagrammed in Fig. 6.

7. Magnet and Gauss-meter

 

The magnetic field is provided by a water cooled
Harvey Wells 22KG magnet, with a Harvey Wells DC power supply
providing up to 200 amperes at 80 volts. An electric motor
with two reduction boxes can rotate the magnet at 2.8, 1A, or

70 degrees per minute. Angles are measured using the 360 gear

A2

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A3

teeth on the base of the magnet. A "home built" sweep cir-
cuit allows us to increase or decrease the field linearly
at any desired rate up to 3000 gauss per minute. The value
of the field is measured to within 10 gauss, with a Bell
660 digital gaussmeter connected to a Hall probe taped to
one pole face of the magnet. Further details are provided

in P. T. Bailey's thesis.19

B. Experimental Procedures

 

1. Sample preparation

 

The samples were grown from saturated aqueous solutions
containing appropriate stoichiometric ratios of N1C12'6H20
and CoClz-6H20. The solutions were placed in a crystal grow-
ing room which maintained constant temperature and humidity
conditions.

Most of the samples appeared polycrystalline. Their
colors ranged from dark for high concentrations of Co to
brownish for high concentrations of Ni. Fairly large samples
(weighing from 1 to 1.5 g) were readily obtained from high
Co concentration solutions. But growing sufficiently large
samples from high Ni concentration solutions proved to be a
slow process. In this case, the weights of the samples used
ranged from 600 to 800 milligrams.

With the evaporation method, it appeared difficult to
grow large single crystals, especially for high concentrations
of Ni. From some high Co concentration solutions (ranging

from l:A to 1:1 Ni:Co ratio), we have obtained a few reasonably

AA

large crystals which might qualify as single, in the sense
that crystal faces can be distinguished and the interfacial
angles corresponded to those in pure COC12-6H20. Unfor-
tunately, we also saw irregularities on the crystal faces and
they were not used in any experiments.

For magnetic phase diagram measurements, we used two
crystals previously grown by P. T. Bailey from 1:1 ratio
solutions. They appeared single.

When a sample had reached a sufficient size, it was
removed from the solution and pieces were cut from it. Both
samples and pieces were then refrigerated to prevent dehydra-
tion. After an experiment on the sample, the pieces were
sent for chemical analysis. This procedure became necessary
after it turned out that the chemical composition of the
sample could differ by as much as 10% from the stoichiome-
tric ratio of the solution from which it was grown.

The day before a run, the sample was taken out of the
refrigerator and weighed. Afterwards, it was lightly coated
with GE 1202 glyptal, and a 12% inch length of l.A mil
Evanohm’ wire (with resistance nearly equal to A00 ohms)
was wound as non-inductively as possible around it. A 1/10
watt Allen-Bradley carbOn resistor, with nominal resistance
between 56 and 60 ohms was then glued to the sample. This

thermometer was further secured with two cotton threads wound

 

”Trademark of wire supplied by Wilbur B. Driver Co.,
Newark, N. J.

A5

around the sample in opposite directions. The first thread
sewed to attach the sample to a hook below the radiation
shield, while the other thread was tied down to the base

of a german silver strip connected to the bottom of the
Helium can.

Both heater and thermometer carried a set of two leads
at each end (making a total of eight formex coated manganin
wires, each one 6 inches long, 3.1 mil in diameter, with a
resistance of 15 ohms). Once the sample was initially secured
with one cotton thread, the leads were soldered to the ter-
minal board and checked for continuity. The sample was fur-
ther tied down and a stainless steel shield slipped around
it. A lead 0 ring was placed on the inner can flange which
was then tightly bolted to the Helium can flange. To pre-
vent damage to the O ring, care was taken to tighten the
flanges uniformly. A similar procedure was followed to bolt
the outer can into place. As a precautionary step, we next
closed the inner can pumping line and leak detected the outer
can with a helium mass spectrometer leak detector Veeco
MSABl7. Past experience has indicated that when the outer
can was leak tight, the same condition would also likely
prevail for both the inner can and the leads line.

After leak detection proved satisfactory, the calori-
meter was lowered into the dewar and tightly bolted to the
upper flange of the dewar. The needle valve was then opened
and both dewar and Helium can were evacuated with the Stokes

mechanical pump, while the inner can and outer can pumping

A6

lines were closed. Next, the outer dewar was filled with
liquid nitrogen. The dewar was then closed and up to 3 P.S.I.
Helium gas introduced into the inner dewar. This allowed the
calorimeter to cool down to liquid nitrogen temperature in

a few hours. The liquid nitrogen needed periodic replenish-
ing. This was automatically done by an electronic circuit

designed around an IC 555 timer.

2. Helium transfer and temperature calibration

 

Early the next morning, the sample resistance was
noted. This gave the first calibration point at liquid
nitrogen temperature. Next, the inner and outer cans were
evacuated. If no leaks existed, the pressures in both cans
reached 2 x 10'" mm Hg rather quickly. Approximatively 2mm
Hg of Helium exchange gas was then introduced into both cans.

Before liquid Helium transfer, the needle valve and
the pumping lines to the Helium can were closed and the
inner dewar was vented. We could then proceed with Helium
transfer, via a flexible transfer tube. ApprOximatively
3 P.S.I. Helium gas was introduced into the Helium container
through a pressurizing port to force liquid Helium into the
dewar. Once liquid was.tranferred, both Helium can and sample
started cooling. The liquid level could be checked through
the dewar viewing slit with a flashlight.

After transferring, the needle valve was opened, to
allow liquid into the Helium can. The Helium can pumping
lines were also Opened. For magnetic measurements, the

magnet was then rolled into place and a suitable magnetic

I7

field applied. Otherwise, we proceeded directly to tempera-
ture calibration. A

The first calibration point was taken near A.2°K.
To proceed to the next point, the needle valve was closed and
the outer can pumped on to isolate the Helium and inner can
from the A.2°K bath. The Helium can temperature was lowered
by regulating the pumping speed through a small valve. When
the desired temperature range was reached, the pumping was
stopped. The Helium bath would immediately reach thermal
equilibrium, and at sufficiently high temperatures the sample
should do likewise not too long afterwards. The values of
the bath and sample thermometer resistances, and the vapor
pressure in the Helium can as read from the U—tube manometer
or the McLeod gauge were noted down. The pumping was then
resumed and the whole process repeated, until we had Obtained
between 10-15 calibration points in the temperature region
of interest. Below the Lambda point, it might prove diffi-
cult for the bath to reach equilibrium when the pumping
stopped. Also, due to the decreased efficiency of heat trans-
fer through Helium gas, it might take quite a while for the
sample to reach thermal equilibrium. In this case, the bath
temperature fluctuations could be observed on a moving coil
galvanometer and the small valve "fine tuned", to allow both
bath and sample to stay in thermal equilibrium long enough

for accurate readings to be taken.

A8

3. Specific heat measurements

 

After the last calibration point, the small valve was
left open, the outer can closed, and the inner can evacuated
to isolate the sample. When the inner can pressure reached
the 10"5 mm Hg range, the outer can was also evacuated. We
would now be ready to take specific data.

In principle, a specific heat measurement consists
in supplying a known quantity of heat to the sample and
determining the initial and final temperatures. The sample
temperature should remain stable before and after the heat-
ing period. In the present calorimeter, this could not be
achieved since heat leaks due to radiation and thermal con-
ducion down electrical leads could not be completely elimi-
nated. To substract this background heat, we applied the
standard procedure of extrapolating the "before" and "after"
recorder traces back to the middle of the heating period,
as depicted in Fig. 7. The extrapolated points C, D were
then considered as corresponding to the true initial and final
temperatures. For this method to give good results, the
background heat leak should remain fairly constant in the
time scale of interest, i.e. the "before" and "after" recorder
traces should remain parallel. Below A.2°K, this could be
readily achieved by keeping the bath slightly hotter than the
sample. Initially, we controlled the bath heating rate by
progressively closing the small valve. When the latter was
completely closed, we switched on the bath heater and increased

the current as required.

“9

Figure 7. Example of recorder pen path
a.) Specific heat data

b.) Field rotation data

50

(a) I
NULL ’

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I
I

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
I
l
l 90‘ RoIRoR
I
I
I

Figure 7

51

For best results, we usually chose the sample heater
current to produce a deviation of between 10-15 divisions
across the chart during the heating period (which usually
lasts for 20 seconds). Near the critical temperature, this

was usually not done to better visualize the transition.

A. Adiabatic field rotations

 

In magnetic phase diagram measurements, the first
task after temperature calibration consisted in finding the
direction Of the easy axis. With the field maintained con-
stant, the magnet was rotated until the sample showed a mini-
mum in temperature. According to a simple mean field theory,5
this corresponded to the direction of magnetization in the

antiferromagnetic state.

5. Adiabatic magnetizations

 

TO determine the first order anti ferranagnetic -spin
flop (AF-SF) boundary, the field, aligned along the easy
axis, was swept up at a suitable rate. If initially the
sample was in the antiferromagnetic state, it should start
cooling until it reached the boundary, where the temperature
should either stay constant or increase. The value of the
field at the critical point was taken as the critical field.
TO ascertain that an AF-SF transition did take place, the
field was swept down. The sample should start cooling until
it reached the SF-AF boundary, when it should start warming
again. To trace out the complete boundary, the sample was
heated to near the desired new temperature, and the new criti-

cal field determined as before.

52

6. Antiferromagnetic-Paramagnet1c boundary

This boundary was determined by specific heat measure-
ments in a magnetic field. The boundary "crossing" was fairly
easily seen by a sudden increase in temperature change with
a given heat input. The field was then set to a new value,
and the diffusion pump disconnected from the inner and outer
cans. Some Helium gas was admitted into the inner can. The
sample would cool back to the temperature of the Helium bath.
Once the sample had cooled back to the Ordered state, the
inner can was pumped on again, and a new set of specific

heat data taken.

7. Shutdown

At the end of an experiment, we closed the inner and
outer cans. The remaining liquid Helium was pumped out of
the dewar and the Helium can. The diffusion pump was turned
off, but the forepump left on until the next morning, when
it was turned off and the Stokes pump disconnected from the
dewar. Next, the Helium dewar was flushed with nitrogen gas,
and air was admitted into the inner can, the outer can and
the forepump. The sample would be removed from the calori-

meter, not too long afterwards.

C. Data Reduction

 

1. Converting pressure into temperature
We converted pressures into temperatures in the stan-
dard way. Above the Lambda point, the U-tube manometer read-

ing was first changed to pressure at 0°C. Next, the height of

53

liquid helium above the sample was estimated and a hydrosta-
tic pressure correction added to the previous value (the
exact helium height is not very critical). This hydrostatic
correction was not made below the Lambda point, since then no
temperature gradient existed in liquid helium. Also, no 0°C
correction was made to readings below 2.5cm Hg, because
those were read off the McLeod gauge which is relatively
insensitive to temperature change.

Finally, the corrected pressures were converted to
temperatures, using a computer generated interpolation table

based on the 1958 He“ scale of temperature.20

2. Thermometer calibration equation
First, we fitted the resistance-temperature relation

to the two-parameters Clement-Quinnell equation.21

3
(L33) I a + b lnR (2.2)

The relative deviations were next fitted to a fourth

degree polynomial in lnR.

 

k I A
[11%;]meas - [1%8- calc a Z Cn(1nfi)n (2,3)
[lnR]k n80
_T_'meas

This resulted in the following explicit expression for T.

5A

 

a + b lnR ’2

T I lnRL A :| (2.1;)
- Z 0,,(1nn)n
nIO

This type of fit usually resulted in relative deviations

of no more than 0.2%.

Occassionally6the following equation was also used.

I. 2 Cn(lrIR)n'1 (2.5)
T nIl

Whichever equation the better fit.

3. Determination of thermometer resistances

When the chart recorder pen was at the null position,
the thermometer resistance RT simply equalled the potentio-
meter setting Ro. However, most experimental data were taken
with the potentiometer offset from the null position. We

used the following empirical equation.

2 3

 

a a
T l + [02 + 2C3RT +3cung] D
Where
D I deviation from the null position, considered positive
(negative) when the pen was at the right (left) of the null
position.
An iteration process was employed to determine RT.
First, R0 was substituted into (2.6) to get a new value for
'RT, which was in turn substituted into (2.6), etc... . This
process was considered convergent, when two successive values

of RT differed by 0.1%.

55

The coefficients Cl to CA were obtained from a least

squares fit to the chart calibration equation

ROL - ROR

 

I 01 + 02R + c 32 + CuR3 (2.7)

3
DL - DR

Chart calibration points were taken when the pen was
near the extreme right Of the chart (with deviation DR from
the null position) for a given potentiometer setting ROR.
The latter was then changed to a new setting ROL, such that
the pen moved to near the extreme left of the chart, with
deviation DL from the null position. In (2.7), R was appro-
ximated by RQL + ROR .

 

2
The relative deviations of this fit usually did not

exceed 2%.

A. Calculations for specific heat, field rotations

 

and field sweep data

 

The heat capacity HC was computed as

80 s 123. .
AT (2 8)

where V voltage across heater

I I current through heater

t I duration of heat pulse
AT I temperature difference

Knowledge of the sample mass and its molecular weight

would then allow conversion into specific heat values. The.

chart positions corresponding to the "true" initial and final

56

temperatures were determined graphically, as described earlier.
They were converted to resistance and temperature values. The
difference between those temperatures was taken as AT, while
their average was considered as the temperature corresponding
to the computed specific heat.

The largest source of error came from reading distances
on the chart. As mentioned earlier, this error was minimized
by choosing suitable heat pulses so as to get sufficiently
large deviations. Another source of error was the heat capa-
cities of the addenda,thermometer, leads, varnish. According

to N. D. Love18

, this error was largest at h°K and amounted
to nearly 2.8% of the total heat capacity at that temperature.

For field rotations and field sweep experiments, the
recorder trace was marked whenever a certain angle or field
was reached. The chart positions corresponding to those

marks were converted to temperatures, as for specific heat

data.

III. Experimental Results and Discussion

A. Review of calorimetric and magnetic properties of

 

COC12°6H20 and N1C12°6H20

 

Before discussing the results we have obtained, it
will be pertinent to review the magnetic properties of

Co012-6H20 and N1012-6H20.

1. Crystallographic and magnetic structures

Both N1C12'6H20 and CoClg-6H20 belong to the monoclinic

base centered type, with space group C 2/m. Their crystal

structures have been determined by Mizun022’23, and the appro-

priate lattice parameters are summarized in Table I.

Table I

Lattice parameters of CoClz-6H20 and NiClg'6H20

 

 

 

a b c 3 no. of spins

O O O

(A) (A) (A) per unit cell
CoC12o6H2O 10.3u 7.06 6.67 122°2o' 2
NiClg-6H20 10.23 7.05 6.57 122°10' 2

 

 

 

 

 

 

57

58

Despite their similarities in crystallographic struc-
ture and lattice parameters, the magnetic structures of the
salts turn out to be different. Fig. 8 depicts the magnetic

structure of CoClg°6H20 in the antiferromagnetic state, as

determined by Kleinberg2" using neutron diffraction. The
easy magnetization direction lies along the c-axis. The
nearest neighbor exchange interaction J occurs along the
diagonal in the a-b plane. Appreciable exchange interactions
J1, J2 may also exist along the b and c axes.

As also determined by Kleinberg25, the spins in

NiClz-6H20 are arranged similarly, except that the easy axis

has moved in the a-c plane to a direction roughly 10° away
from the a-axis.

Part of the reason for the difference in the magnetic
behavior of NiC12-6H20 and CoClz-6H20 lies in the fact that
the Ni++ and Co++ have different ground states, and therefore
respond differently to the crystal field. Each Ni++ or Co++
ion is surrounded by an elongated octahedron consisting of
four water molecules and two chlorine ions. The four water
molecules form a distorted square, inclined by about 9°
from the b-c plane while the chlorine ions lie along an axis
perpendicular to the square. The resulting cubic and tetra-
gonal fields split the ground state uFQ/2 of the free Co++
ion into two triplets and a singlet, with a triplet lying
lowest.26 Since S = 3/2, this triplet has a fourfold spin
degeneracy. The other two water molecules introduce a small

orthorhombic component which, along with the spin-orbit

59

\

\
\\

 

 

 

 

 

 

 

 

Figure 8. Magnetic structure of CoClz-6H20
J, J1, J2 are possible exchange interactions.

60

coupling, split the lowest triplet into six Kramers doublets.
It turns out that the lowest doublet lies about lO°K below
the next lowest doublet, compared to a Neel temperature of
only 2.29°K. Therefore, at low temperatures, one may de-
scribe the magnetic properties of CoClZ-6H20 by a spin
Hamiltonian with effective spin 8 8 1/2. One may also
expect the exchange interactions and the g tensor to be
anisotropic.27

On the other hand, a cubic field lifts the orbital
degeneracy of the Ni++ ion in its 3F“ ground state, leaving
an orbital singlet lowest.28 The orthorhombic component
lifts the remaining spin degeneracy, so that the effective
spin S a l, the same as the true spin of the free Ni++ ion.
Susceptibility data and low temperature specific heat have
been successfully analysed by Hamburger and Friedberg29,
and Iwashita and Uryu30, using the following Hamiltonian.

H=-JX E's’ -J X go‘s: 4.1)[232 + 32]
(1,3,1 J 14,101 k 1 12 3:32

2 2 2 2
+ E [§ (Six - Siy) + 32: (SJX - SJy)] (3.1)

In (3.1), J and J1 denote nearest and next-nearest
neighbors exchange interactions. It is also assumed that
+ +
the nearest neighbor spins Si, 83 belong to different sub—

lattices.

61

According to ref. 30
D 8 -(l.00 + 0.02)°K E - (0.05 + 0.02)°K

Thus, susceptibility data suggest that NiC12-6H20 is

a predominantly planar Heisenberg antiferromagnet, with a

relatively strong uniaxial single ion anisotropy.

As for CoC12'6H20, analysis of the paramagnetic sus-
ceptibility data led Kimura31 to suggest that the exchange
interactions are anisotropic. Further, the ratio Jl/J
between next nearest and nearest neighbor interactions
nearly equals 3/A. This simplies that the magnetic behavior
of CoClg-6H20 cannot be two-dimensional. More recently,
Cieplak32 added additional single ion anisotropy terms to
the Kimura Hamiltonian, to account for the low temperature
dependence of the paramagnetic-spin flop boundary.

Thus far, theoretical analyses of the susceptibility
data seem to imply that CoC12-6H20 and NiC12-6H20 have dif-
ferent magnetic dimensionalities. To gain additional insight

into the matter, let us next turn to specific heat data.

2. Specific Heat Data

 

The specific heats of CoCl2-6H20 and NiC12-6H20 were
first measured by Robinson and Friedberg32, in the tempera-
ture interval l.H°K to 20°K. They determined TN 8 2.29°K
and 5.3M°K for CoC12o6H20 and NiC12-6H20 respectively. The
total magnetic entropy changes correspond to the expected

effective spins. Further, the amount of entropy gained up

62

to TN is only about 50% of the total entrOpy. This suggests
two dimensional magnetic behavior.33

In Fig. 9, we have plotted the Robinson-Friedberg
data (corrected for lattice specific heat) vs. T/TN. Fig. 9
also shows the theoretical specific heats given by the
Onsager solution, and by the planar S = 1/2 and S = l BPW
models. Note that the CoC12'6H2O data follows the exact
quadratic Ising curve rather well, from 0.8 T/TN up to 0.98
T/TN. Above TN, the fit between the CoC12-6H20 data and
the Onsager curve seems rather poor, while the NiC12-6H20
data appear to fit the latter somewhat better up to 1.2
T/TN. Below TN, the NiC12-6H20 data deviate systematically
from the quadratic Ising curve, as befits an antiferromagnet
with S - 1. What seems rather surprising is the good agree-
ment between the NiClz-6H20 data and the planar, S 8 1/2
BPW model. In the absence of physical arguments to the con-
trary, this apparent fit should be merely considered as a
mathematical accident. A priori, one might expect a S = l
Ising model to do better. We have computed the specific
heat of a planar, S = 1 Ising model in the generalized BPW
approximation of Obokata and Oguchi.3u As one can see, this
approximation does not fit the data at all. Above TN, both
BPW approximations fail badly, because they underestimate
the exact short range order.

Thus, one can say that the S = l/2, planar BPW and the
quadratic Ising models fit the NiClz'6H2O and CoClz-6H20

specific heats respectively, in some temperature range below

63

Figure 9. Experimental specific heats of NiC12-6H20
and CoC12-6H20 compared with the BPW and

Ising models

614

O.N m._ 0.. .v._ N._ 0..

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65

TN (at least between 0.7 T/TN and 0.9 T/TN). Above TN
the quadratic Ising model fits the data better. A priori,
one should not expect effective field type theories, such
as the BPW models, to account for the critical behaviors
of real systems. In view of our experimental results, it
is pertinent to ask the question: to what extent does the
quadratic Ising model fit the CoCl2°6H20 and NiC12°6H20
specific heats near TN?

High resolution calorimetric studies of CoC12-6H20
were first carried out by Skalyo and Friedberg.35 They
found that in the reduced temperature range 5 x lO'3<t<

5 x 10‘2

= -O.239 - 0.271 1nt + A (3.2)

:IJIO

with A

0 if T > TN, and A = 0.574 1: T < TN.
T - TN
TN

Here t with T = 2.2890°K'

N

0n the other hand, the critical behavior of the
NiC12'6H20 specific heat has been determined by Johnson
and Reese as36

C

fi = -0.55 1nt + 0.27 T < TN (3.3)
g = 0.61t"°-17 -0.10 T > TN (3.u)

with TN = 5.3"8°K

.66

As noted by Wielinga et al,37 for t < 0.1 the
quadratic Ising model specific heat agrees to within a few

percent with

g... -0.u9 1m: - 0.29 (3.5)

for both T < TN and T > TN.

A comparison of (3.5) with (3.2), (3.3), (3.“) shows
that the quadratic Ising model does not describe the criti-
cal behaviors of NiC12'6H20 and CoC12'6H20 quantitatively,
although the logarithmic nature of the transition may be
given correctly. The critical behavior of CoC12°6H20 has
some similarities with those of RbgCoFu and K2CoFu, in which
the Co++ ions are also subjected to crystal fields with octa-
hedral cubic symmetry. For those compounds however, magne-
tization data show a much more striking agreement with the
quadratic Ising models.38 NiC12-6H20 is more similar to
K2N1Fu, a two dimensional Heisenberg antiferromagnet whose
critical behavior is dominated by a single ion anisotropy
term. According to the recent measurements of Birgeneau
et al.,39 the magnetization curve of K2N1Fu does not follow
the prediction of the quadratic Ising model quantitatively
near TN. But the critical exponent B = 0.14 still comes
out close to the theoretical 8 = 0.125.

As far as the magnetic dimensionality of CoC12'6H20
is concerned, one should also mention an attempt to fit the

specific heat of this compound to the so called XY model.

67

The Hamiltonian for this model is

H XY 3 " 2 [stxist + JySyisyJ] (3.6)
<103>

Assuming Jx - J and the Co spins confined in the

y
be plane (so that x,y refer to the c and b directions re-
spectively), DeJongh et al.“0 found good agreement between
the CoC12°6H20 specific heat and the quadratic XY model, in
the temperature range H°K to 10°K. Since the quadratic XY'
model does not allow long range order, the Ising-like ano-
maly in CoC12°6H20 is presumably due to an in-plane aniso-
tropy (Jx can differ from Jy by about 4%).

The low temperature specific heats of CoC12'6H20 and
N1C12-6H20 also present some interesting features. The
calorimetric measurements of Donaldson and Edmonds'41 indi-
cated that the specific heat of COC12'6H20 follows a T3 law
between 0.7°K and 1.6°K (corresponding to 0.3 T/TN and 0.7
T/TN). The NiC12'6H20 specific heat does likewise between
2.0°K and 4.0°K (or from 0.9 T/TN to 0.75 T/TN). Below
2.0°K the data decrease more rapidly with temperature.

This behavior can be reasonably well explained by the
Eisele-Kefferu2 spin wave theory of the two sublattice anti-

ferromagnet with uniaxial anisotropy. According to this

theory, for T<<TN, the magnetic specific heat Cm is given by

cm . AT3 f(T/TAE) (3.7)

68

where
A . constant which depends on the lattice structure
’f(x) - function which approaches 1 for x>>l and drOps
exponentially for x < l
kTAE I a measure of the energy gap introduced by aniso-

tropy in the spin wave spectrum.

Donaldson and Edmonds found that the choice TAE - N.O°K
leads to a reasonable agreement between (3.7) and the N1012'
6H20 data. Later, Hamburger and Friedberg29 found that the
same choice of TAE produced good agreement between the low
temperature theoretical and experimental parallel suscepti-
bilities. The situation is less clear for CoC12'6H20. The
specific heat shows no sign of decreasing down to 0.7°K.

To account for this, one should take TAE < 0.6°K. However,

"3

the antiferromagnetic resonance data of Date gave TAE -
1.66°K. There has not yet been any satisfactory resolution
of the discrepancy between these two estimations of TAE'
Kimura“3 attempted to explain the spin wave behavior of
CoC12-6H20 from a three dimensional anisotropic exchange
Hamiltonian. It may be significant to note that his theory
failed to account for the specific heat T3 dependence.

To sum up, one can say that anisotropy plays an impor-

tant role in the magnetic behaviors of NiClZ'6H20 and CoClz-

6H20. The calorimetric and susceptibility data for NiClZ'6H20

can be at least qualitatively understood in terms of a planar

Heisenberg Hamiltonian, with single ion anisotropy. The

exact nature of the anisotropy in CoC12°6H20 is still a

69

matter of some controversy. On the one hand, it is possible
to fit the magnetic susceptibility data from l.2°K to A.2°K
using the Kimura three-dimensional anisotropic Hamiltonian.“3
On the other hand, zero field calorimetric data suggest a
two dimensional magnetic behavior, with nearest neighbor
interactions only. One should also add that a consideration
of the possible superexchange paths between the Co++ ions
also tends to favor two dimensional behavior.M At any
rate, the quadratic Ising model can account for the criti-
cal behaviors of NiC12-6H2O and CoC12'6H20 much better

than the BPW models.

B. The Concentration Phase Diagram

We have measured the specific heats of ten mixed
crystals, grown from saturated aqueous solutions containing
various stoichiometric ratios of NiC12-6H20 and CoC12-6H20.
Before each run, small pieces were cut from the sample and
later sent out for chemical analysis. All the samples studied
showed fairly sharp transitions, and the temperatures cor-
responding to specific heat maxima were taken as the Neel
temperatures. The chemical analysis results indicate that
all samples have chemical compositions which can be well
represented by the formula CoxNil_xC12-6H20. Table II lists
the various chemical compositions, along with the correspond-
ing Neel temperatures.

To explain the variation of TN with chemical composi-

tion, we proceed as follows:

Chemical compositions

70

Table II

a

and Neel temperatures of the

samples for which calorimetric measurements have been made.

The values quoted for each element are percentages by weight.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Chemical Formula Co Ni Cl H20 TN(°K)
0oc12-6H20b 20.77 0.00 29.80 05.03 2.29c
CoO 89N10.ll C12-6H20 21.07 2.56 29.97 00.03 2.03
Coo.8uNiO.l6 C12°6H20 20.85 3.96 30.15 05.32 2.07
000.83Nio 17 Cl2°6H20 20.31 0.20 29.92 05.30 2.09
000.57Nio.u3 C12'6H2O 9.26 7.11 29.50 00.68 2.77
0o0.52Nio.u8 C12'6H20 13.31 12.18 29.99 00.76 2.87
Coo.50N10.50 C12'6H2O 12.33 12.15 30.00 00.00 2.96
Coo_30Nio.70 Cl2°6H2O 7.09 17.86 29.69 00.96 3.07
Coo.24Nio 76 C12'6H20 5.97 19.08 30.78 00.78 3.77
CoO 16N1O.89 C12'6H20 0.18 21.01 29.99 00.30 0.26
0o0.13Nio.87 C12-6H20 3.25 21.92 29.51 05.25 0.07
NiC12°6H20b 0.00 20.70 29.83 05.07 5.300

 

 

 

 

 

 

 

aAll chemical analyses carried out by

analytical Laboratory, Woodside, N. Y.

Schwarzkopf Micro-

bValues for CoC12-6H2O and N1012°6H20 are calculated values.

cDetermined by w. K. Robinson and s. A. Friedberg, Phys. Rev.,

111, 002 (1960).

71

a.) All theoretical curves are required to give

the experimental Neel temperatures of pure NiC12'6H20
and CoC12-6H20. This condition fixes JNi-Ni and
JCo—Co in the particular theoretical model under con-
sideration.

b.) To determine JNi-Co’ we also require all theore-
tical curves to pass through a third fixed point

(TN = 2.96°K, x = 0.5). The choice of this fixed
point may appear somewhat arbitrary, and will be Jus-

tified later on.

1. Mean Field Fits

 

Fig. 10 shows the concentration phase diagram of the
mean field models, as given by the above fitting procedure
and equations (1.29), (1.05) of chapter I. As one can see,
the virtual crystal approximation fails rather badly. The
quenched model improves the situation significantly, espe-
cially at high Ni concentrations. However, both concentration
phase diagrams start out by dropping below 2.29°K, and con-
tinues to predict Neel temperatures below 2.29°K, even up to
20% Ni. This is certainly not the case experimentally. In
retrospect, one should not expect the virtual crystal appro-
ximation to work well for mixtures of magnetic ions which
interact mainly through nearest neighbors, since this appro—
ximation is valid only when the fluctuations in local concen-
trations are unimportant.“5 This happens when the interac-

tions are of long range, as in rare earth alloys for example.

72

Figure 10. TN vs. concentration diagram for mean

field models

73

OH madman

36V :23:ch :00 oo

 

 

00. he on mm o .
\\ IIH/ _ _ O N
‘\\\ 0 .I111
00 I
///
. . 2 ..l m.N
Ax ov ._.
/
/’
III
/ . l o.»
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I/ 0 .V .V
/
l/
l/
20: cooE tor—0.330 4N6
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0.0

7H

The quenched mean field model takes a more realistic con-
figurational average, and fits the data better. The failure
at high concentrations of Co is probably due to the low
effective spins and the two dimensional magnetic behavior

of the Co ions. Usually, a combination of low spin and low
dimensionality is fatal to mean field theories. We have

not tried this, but one way to improve the mean field models
may consist in explicitly including single ion anisotrOpy

or anisotropic exchange terms in the Hamiltonian.
2. The Annealed Ising and Bethe Lattices

To determine the concentration phase diagram of the
annealed Ising model, we use equation (1.68) in conjunction
with the bond distribution function (1.77). This yields the

following equation

02 200 92 g 0

+ +

 

(3.8)
The concentration phase diagram of the annealed BPW model is

more simply given by

pztanh(BcJAA) + 2pqtanh(8cJAB) + q2tanh(8cJBB) = 2&1 (3.9)

In (3.8) and (3.9). p and q denote the Co and Ni frac-
tions respectively (p+q = 1), while A and B refer to Co and
Ni. 2 is the number of nearest neighbors. For the 2-dimen-

sional case, Kc a=%1n(1 +.J5); :c 8‘J% and z = 0.

75

For each value of p, the critical temperature Tc is
determined by solving (3.8) or (3.9) numerically, using the
Newton-Raphson algorithm. The latter is considered conver-
gent when the relative difference between two successive
approximations becomes less than 10'6. All computer pro-
grams used in concentration phase diagram and specific heat
calculations are listed in the appendix.

Fig. 11 shows the concentration phase diagrams of the
planar annealed Ising and BPW models. The agreement between
both models and experimental data appears reasonable. Over-
all, the Ising model tends to do better than the BPW model.

In Fig. 11, we have also plotted the previous results
obtained by Robinson and Simmons,“6 and Takeda et al.’47
Robinson and Simmons determined TN by the disappearance of
the 0135 NQR signal in four NixCol-x012-6H20 crystals with
x - 0, 0.1, 0.25, 0.5. (Incidentally, they also stated that
"X ray powder patterns show these crystals to be isomorphic").
Takeda et al. performed calorimetric and susceptibility mea-
surements on powder samples. Robinson's data agree with ours,
except for the last TN at 50% Co. On the other hand, Takeda's
results are systematically higher than ours. The origin of
this discrepancy is not presently understood. We tend to
think that the uncertainties in concentrations are higher
than claimed in Takeda's paper.

With regards to the possibility of 3-dimensional mag-
netic behavior in CoClg°6H2O, we have also considered the
annealed simple cubic Ising model. From high temperature

series expansions, Sykes et a1.“8 estimate Kc = 0.2217 and

76

Figure 11. TN vs. concentration diagram for the

annealed Ising and BPW models

 

 

 

77

HH otswfim

Ao\ov 8:8 25250 00

 

00. m-» Om m N CON
fl _ _ d
. 0 OOIXIHI
x .+ m.N
. . Q a
I/ O
x / m.»
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..o 3 .3353”. +++ //// o
«:03 m_£._u o oo ///. V3?
x
«.0

3mm 3.35.4. null
052 .888 3.35.4

 

 

 

 

78

5c I 0.3307 for the simple cubic Ising model. Inserting
these values into (3.8), we obtain the concentration phase
diagram of the annealed simple cubic Ising model. Fig. 12
shows both square lattice and simple cubic phase diagrams.

The latter gives lower critical temperatures for high concen-
trations of Co (there is even a shallow minimum near 100% Co),
and higher critical temperatures for high Ni concentrations.
On the scale of the graph, the differences between the two

are slight but sufficiently significant to allow one to

claim that the annealed square lattice Ising model fits the

data better, at least for high Co concentrations.
3. Annealed and Quenched Bethe Lattices

The concentration phase diagram for the quenched BPW
model is given by equation (1.113). Figure 13 shows the
concentration phase diagrams of the quenched and annealed
BPW models. The same JNi-Co has been chosen for both phase
diagrams, to allow a direct comparison between the two up to
80% Co, both models give practically the same TN. But as the
Ni concentration increases, the quenched model give systemati-
cally higher critical temperatures. Fig. 13 also shows the
phase diagram of the quenched classical Heisenberg BPW model,
given by equation (1.110). Note that near 100% Co, the latter
phase diagram starts out by dropping below 2.29°K. However,
this initial minimum is much shallower than in the case of
the virtual crystal mean field model (refer to fig. 10).

For high Ni concentrations, both classical BPW and virtual

79

Figure 12. TN vs. concentration diagram for the

annealed square lattice and simple cubic

Ising models

 

 

80

NH ossmem

oo.

 

15; cozotcoocoo 00
he. on mm o
_ _ _
0 Z all
gov ._.
lo
/ O I
/
0.030 mqaifiwolll

m0....h<.._ wmdnomlla

 

 

 

0N

0N

0.m

¢.¢

N6

0.0

81

Figure 13. TN vs. concentration diagram for the

annealed and quenched BPW models

 

82

ma onswam

Ac); 5:82.85 0 o 0

 

 

3mm 62320 lllll

3am nozocoso 11111
3 mm 3.354

 

     
  

 

 

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83

crystal mean field models give similar critical temperatures.
If the phase diagram of the quenched BPW model has been re-
quired to pass through the third fixed point (TN 8 2.96°K,

x - 0.5), it would also give a shallow minimum near 100% 00,
and would practically coincide with the classical Heisenberg
BPW phase diagram for high Ni concentrations. Thus, making
the spins go from % to a does not produce any significant
change in the shape of the concentration phase diagram. In
our opinion, the poor agreement between quenched BPW models
and experimental critical temperatures is symptomatic of the
fact that the BPW model does not describe the critical be-
haviors of NiC12-6H20 and Co012°6H2O correctly. From this
point of view, the apparent agreement between the annealed
BPW model and experiment becomes more like a mathematical
accident. Evidently, there are many ways in which so and Kc
can be chosen so as to make (3.8) "agree" with experiment.
The simple cubic Ising model is a good example of this.
Therefore, to gain more insight into the physics, one should
also consider the ways in which the different theories fit
the experimental specific heats. Before doing this, we will
attempt to justify our fitting procedure for the concentra-

tion phase diagrams.

0. Choice of JNi-Co

 

Table III lists all the data points, along with the
JNi-Co obtained by requiring the annealed square lattice
Ising concentration phase diagram to pass through a particular

point.

80

Table III

Values of JNi-Co obtained when the annealed 2-

dimensional Ising concentration phase diagram is required

to pass through a particular data point.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10o '%Ni TN(°K) JN1_CO(6K)
100 0 2.29 ___
89 11 2.03 1.280
80 16 2.07 1.223
83 17 2.09 1.230
57 03 2.77 1.058
52 08 2.87 1.003
50 50 2.96 1.070
30 70 3.07 0.921
20 76 3.77 0.923
16 80 0.26 0.963
13 87 0.07 1.001
0 100 5.30 -——-

 

 

 

 

85

The average of all the JNi-Co is jNi-Co = l.072 i
0.002°K. A least squares fit of (3.8) through all the data
points (except the 100% Co and 100% Ni points) give JNi-Co =
l.032°K, which is within the uncertainty range of jNi-C0°

It seems then reasonable to choose (2.96°K, 50% Co)
as the third fixed point since 2.96°K is itself an average

over two runs, and also because the corresponding JNi-Co

is very close to jNi-Co' This will allow direct compari-
sons between the theoretical models without introducing
unnecessary complications.

So far, we have taken JNi-Ni’ JCo-Co and JNi-Co to
have the same sign. For the pure crystals, JNi-Ni and
JCo-Co are certainly both negative, and there seem to be
no compelling physical reasons to assume JNi-Co positive.
In this connection, it may be significant to note that
with JNi-Ni and JCo-Co assumed to have the same sign, and
with the fixed points on the concentration phase diagram
as chosen, the annealed Ising and BPW phase diagram equa-
tions (equations (3.8) and (3.9)) then require JNi-Co to

have the same sign as JNi-Ni and JCo-Co‘ On the other hand,

the quenched BPW and mean field phase diagram equations
allow JNi-Co to have both signs.

It is also instructive to consider the choice of
JNi-Co from another point of view. Write the different

++ ++ +4-

exchange energies as JAASA'SA; JABSA'SB and JBBSB'SB' If
the J's arise from superexchange, then a simple argument
given by Bacon et a1.”0 leads one to expect that, as a first

approximation

86

2 a
JAB JAA JBB (3.10)

Table IV lists the different J's obtained from the
annealed Ising and BPW models, assuming SCO = l/2 and

8N1 8 10
Table IV

Values of exchange constants in the Ising and BPW models

 

 

 

 

 

 

JCo-Co JNi-Ni JNi-Co JNi-Co

(°K) (°K) (OK) -JUNi-Ni JCo-Co
Ising 0.036 2.353 2.100 0.69
BPW 3.176 1.851 1.560 0.65

 

 

 

 

 

 

As one can see, the agreement with (3.10) is not
fantastic. Note however that (3.10) has been derived in a
simple superexchange case [two magnetic ions separated by
a diamagnetic ion]. It may not be valid when the superex-
change goes through many diamagnetic groups as in NiClg-6H20
and CoClg-6H20. Also, strictly speaking, the J's listed in
table IV correspond to the effective spins, not the "true"
spins. As is well known, the effective J is related to the
true J by the square of a g-tensor component.38 By calculat-
ing the true J's correctly from the effective J's, it may be
possible to find that the former will satisfy (3.10) more

closely. One should add that even for "simpler" systems

87

such as FeO - CoO oxides and rare earth alloys, other
authors [Boubel et al.16, Lindggrdus] had found it neces-

sary to impose significant deviations from (3.10).
C. Specific Heat Results

1. Comparisons between theoretical and experimental

specific heats

 

Figures 10, 15, 16 show the theoretical and experimen-
tal specific heats for samples containing 89%, 50%, 13% Co
respectively. They are the samples for which the most
extensive sets of data have been taken. The data plotted
have also been corrected for lattice specific heats, assum-
ing a T3 dependence for the latter (in the temperature
range of interest, this correction amounts at most to a
few percent). In Fig. 10, one sees that the annealed 2-
dimensional Ising model agrees reasonably well with experi-
ment below TN, while, as expected, the annealed BPW model
does not fit the data. In this case, the quenched BPW model

with the same JNi-Co gives practically the same specific heat

as the annealed BPW model, and is not shown on the graph for
clarity.

In Fig. 15, the data start out by deviating systemati-
cally from the annealed Ising curve, up to about 0.8T/Tfi.
Presumably, this is due to spin wave behavior. Beyond
0.8T/TN, critical behavior sets in, and the data follow
the annealed Ising curve reasonably well, up to about 1.2

T/TN. Note that although JNi-Co has been adjusted so that

88

Figure 10. Experimental and theoretical specific

heats Of N10.11COO.89C12'6H2O

89

:H ohswfim

C. o. H
Ne w.» an Qm em

N.N

0..

0..

 

1 . . . _

 

0° . ....o°°°

N00.

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N

N
'0

 

 

0.¢

90

Figure 15. Experimental and theoretical specific

heats Of N10.50COO.50C12'6H20

91

ma mhdwfim

3.6....

 

ofomuosoc
0qan73llll
afiom 3.35.4.1}:
Em. “02005.4.

 

 

 

 

 

(Mo/9°
|01/|00)‘“Q

92

Figure 16. Experimental and theoretical specific

heats of Nio.87CoO.13ClZ-6H2O

we shaman

 

 

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90

the annealed BPW model gives the experimental critical tem-
perature 2.96°K, it does not fit the data as well as the
quenched BPW model with the same JNi-Co- However, the
quenched BPW specific heat curve exhibits a peculiar shoulder
at around 2.5°K. This is probably an artifact of the quenched
BPW model at 50% concentration.

One can now predict beforehand the behavior of the
data with respect to theory in fig. 16. As expected, below
TN, the data agree with the annealed Ising curve only very
near TN. Above TN, the agreement "area" covers a wider tem-
perature range. Below TN, the quenched BPW model fits the
data fairly well, but the critical temperature it predicts
is too high.

In summary, among the theoretical models considered,
the annealed two dimensional Ising fits experimental data
the best in the critical region. Although it predicts the
TN vs. concentration fairly well, the annealed BPW model
does not fit the data as well as its quenched counterpart.
The overall behavior of the data with respect to the random
Ising and BPW models is what one might qualitatively predict,
by looking at fig. 9 and assuming that the Ni and Co ions
are randomly distributed on the lattice. Also, it appears
that the exchange constants JCo—Co: JNi-Co’ JNi-Ni do not
change appreciably with concentration. Since superexchange
can depend sensitively on atomic distances and bond angles,
this may mean that the latter do not change significantly
with concentration. Detailed crystallographic work will be

needed to confirm this.

95

2. Entropy calculation

 

We have also carried out total magnetic entropy change
calculations for the three samples considered. The paramag-
netic specific heats in the random Ising and BPW models are
dominated by l/T2 terms. A Green's function calculation
shows that this is also the case for the pure Heisenberg
paramagnet.50 It seems therefore reasonable to assume that
sufficiently above TN, the total specific heats of the mixed

crystals will be well approximated by

CToT = %g + BT3 (3.11)

The second term is the lattice specific heat which we
assume to have a T3 dependence.
We can then calculate the total magnetic entropy
change AS in the standard way.32 Theoretically, AS is simply

given by
AS = R[p 1n (2sCo + 1) + (l-p)1n(2sN1 + 1)] (3.12)

p = fraction of Co
SCo and 8N1 denote the effective spins of Co++ and Ni++.

Table V lists the experimental and expected entropies.

96

Table V

Experimental and theoretical total magnetic entrOpy changes

 

 

 

 

 

%Co ASth ASexp
(cal/mole/°K) (cal/mole/°K)

89 1.02 1.07

50 1.79 1.79

13 1.91 2.09

 

 

 

 

In table V, Asth has been calculated by assuming
5C0 = % and 5N1 = l. The agreement between the theoretical
and experimental AS may be considered satisfactory given
the uncertainties in entropy calculations. (for 13% Co,
the main uncertainty comes in the estimation of the A/T2
term)

Thus, entropy calculation suggest that the Ni++

and
Co++ ions retain their "pure" effective spins in the mixed
crystals. Fig. 17 shows the normalized entropy curves for
the three mixed crystals. Note that only about 50% of the
total entropy 8(a) = AS is gained up to TN, just as in the

pure crystals.
3. Validity of the annealed models

An annealed bond Ising model can, at best, provide a
good approximation to the thermodynamical behavior of mix-

tures of magnetic insulators. Recently, Harris51 has shown

97

Figure 1?. Normalized magnetic entropy for three
Co concentrations. The Neel temperatures

are indicated by arrows.

98

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C. o. ._.
os 08 on cc 0.» o.~

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99

that, at least as far as the Tc vs. concentration diagram

is concerned, a CPA (Coherent Potential Approximation) cal-
culation on the quenched site Ising system gives the same
answer as the annealed bond Ising model. For Ising systems,
many other theoretical comparisons between the annealed and
quenched models have also been carried out.“52 The results
of these studies suggest that there are probably no signi-
ficant differences between the two, when all the interactions
have the same sign. The greatest differences occur in the
critical region. When antiferromagnetic and ferromagnetic
bonds are allowed to mix, substantial differences between the
two approaches do arise. As a notable example of this, the
annealed Ising model does not allow a spin glass state to
occur."

These general conclusions appear to be qualitatively
borne out by the detailed comparisons we have made between
the annealed and quenched BPW models. At sufficiently high
temperatures, both models give the same specific heat. But
below TN, the quantitative agreement is rather poor, except
when the two models happen to give the same critical tempera-
ture. The concentration phase diagram of the quenched BPW
model lies systematically higher than that of the annealed
BPW model.

In chapter I (section 5), we have given a general ex-
pression for a quantity "c", which may be interpreted as a
measure of the fluctuation of a bond occupation probability

around its average value. In Fig. 18, we have plotted "c"

Figure 18.

100

Fractional mean square fluctuations of the
distribution functions for the three
possible bonds in the annealed Ising and
BPW models. Note that the right hand scale
is ten times larger than the left hand

scale.

101

06
00.

¢0.

00.

ma opswam

 

N00.

coo.

000.

 

 

oo .
3am}..- aoo

02.0.IIII.

 

 

0 .0.

102

for all three possible bonds, as given by the annealed Ising
and BPW models in the case of 50% Co concentration. As
expected, the magnitudes of the c's remain small (5% at most)
and reach maximas near the critical temperature. Note that
below TN, the BPW curves lie systematically higher than the
corresponding Ising curves. Similar conclusions can be
drawn for 13% and 89% Co concentrations. This tends to
suggest that the relatively substantial quantitative disa-
greements between the annealed and quenched BPW models below
TN are due to some peculiarity in the BPW approximation it-
self. A more qualitative, but perhaps more convincing argu-
ment lies in the fact that the annealed BPW model does not
fit the specific heat data as well as the quenched BPW
model, while the annealed Ising model already fits the same

data in the "expected" manner.

0. Low temperature behavior

 

In Fig. 19, we have plotted C/T3 vs. T/TN for three
different Co concentrations. There is some scatter, espe-
cially at high Co concentration, but it appears that the

specific heats of all three samples follow a T3 law, i.e.
c = 8T3 (3.13)

between 0.5 T/TN and 0.75 T/TN. For high Ni concentration,
the T3 dependence apparently persists even up to 0.9 T/TN.
Note that pure 00012-6H20 and NiC12°6H20 also follow a T3

law in roughly the same range of T/TN.u1 This suggests

103

Figure 19. C/T3 vs. T/TN for three different Co

concentrations

100

ma ohswam

ze\e

 

+ + ... +
u|+l .... ,++ I» ++nr++ +H++++ + H++Hl +c++++++¥ I.

+++++.? III.
00 o\om.

I
x
x x x x x
x x x x xx x x xx x
x x x xxx x xxx

00. o\o00

l

o 0% 0000 o

o
o o o
o o 000 00090 o .l
coo o 00

A... . 8 {.mm . .I

O
O

 

 

(,>|./9I0W/ID°) .1/0

 

105

some sort of "scaling" of the spin wave spectrum.

For pure crystals, the spin wave coefficient a in
(3.13) depends mainly on the crystal structure, and also
on the exchange constant.“2 In Fig. 20, we have plotted
0 vs. concentration. To within the accuracies of the cal-
culations, a linear variation of a with concentration is
thereby suggested. A full interpretation of this awaits
the development of a spin wave theory of mixed anisotropic

Heisenberg antiferromagnets.

D. Magnetic Phase Diagram

 

Our interpretation of the magnetic phase diagram is

based upon the magnetocaloric equation53

932-2121! 1
dH ‘ CH <3T) H (3' 0)
where

H = magnetic.field

CH = specific heat at constant field

x = susceptibility
Suppose the field is aligned along the easy axis and
increased adiabatically. In the antiferromagnetic state, the

appropriate x is x11, the parallel susceptibility. Since

a
-§%l > o, the temperature T decreases. When the spins
have flopped, the perpendicular susceptibility xi should be

used in (3.13). According to the molecular field theory x1

is a constant. Thus, the temperature should stay constant.

106

Figure 20. Spin wave coefficient a vs. Co concentration

107

00.

om ouswfim

.o\o. cozotcmocoo 00
2.

00

 

 

.

.

 

I10..

 

 

0N.

(,xyanow/Im) 70

108

This allows a determination of the antiferromagnetic spin
flop boundary using the procedures described in chapter II
(section 5). The data points taken in this way are shown on
the low temperature side of Fig. 21. (The data points on
the high temperature side represent the antiferromagnetic-
paramagnetic boundary for 57% Co).

HAF-SF (0), the antiferromagnetic spin flop critical
field at 0°K has been determined by Rives and Bhatiasu as
6.6KG for CoC12°6H2O. The corresponding field in NiC12°
6H20 is considerably higher: 39.6KG.55 The range between
those two numbers may help to understand fig. 21 qualitatively.
We note that for 57% Co, the critical fields are lower or
comparable to the corresponding fields in CoClz°6H20. This

++ ions

suggests that the anisotropies of the Ni++ and Co
have not changed, even up to a fairly high degree of mixing.
For 50% Co, the AF-SF boundary begins to rise substantially
above the CoCl2'6H20 boundary. Qualitatively, one may say
that some Ni++ spins have begun to flop. It may also be
significant to note that the boundary curvature for both
concentrations appears more pronounced than in CoC12'6H20.
Among the many theories which can be brought to bear
on this problem, the simplest is probably a recent treatment
of the diluted anisotropic antiferromagnet by Moreira et al.56
Basically, theirs is a generalization of the standard mole-
cular field theory of the pure anisotropic antiferromagnet.

They assume virtual crystal approximation and single ion uni-

axial anisotropies which do not change with concentration.

109

Figure 21. Tentative magnetic phase diagram for two

different 00 concentrations

110

am opswfim

 

 

.v. .. ._.
, can Nu m._ S o._
a. _ . _ _ a _

O 0 O O 00 I,
O K K 0 X +XT+X I

+ +
O. L. .II
o I
o I:

oo oxoom++++
o oo o\ohm 0000 ll

00 o\o00. .333.

 

 

0.

0.

ON

0N

(9)4)H

111

We have not attempted this, but it seems reasonably straight
forward to extend the approach of ref. 56 to the case of
mixed crystals, assuming the spins to be distributed on

two distinct sublattices and oriented along the same easy
axis. Although the virtual crystal approximation does not
work well in the case of short range interactions, this
approach may provide a first step towards a quantitative
understanding of the phase diagram.

Finally, we should mention that in one respect, there
is some difficulty in the interpretation of our data as
critical AF-SF fields. According to a simple mean field
theory developed in ref. 53, the temperature should exhibit
a minimum at the easy axis, when the field is rotated around
the easy axis in the antiferromagnetic state. The minumum
should become a maximum in the spin flop state. This effect
is quite evident in CoCl2-6H20. However, for 57% Co we
observe no changeover from minimum to maximum above the cri-
tical field. This may mean that the rotation may not have
been done in the plane where the spins have flopped, or that
the theory of ref. 53 needs to be revised for the case of
mixed crystals. At any rate, the magnetic phase diagrams
of fig. 21 can only be considered as tentative. Further
speculations should await experimental work on single cry-

stals over a wider range of compositions.

E. Conclusions

 

We have performed a fairly extensive zero field calori-

metric study of CoxNi1-x012°6H20 over a wide range of

112

compositions. The critical behaviors of the mixed crystals
have been found to be reasonably well described by the annealed
two dimensional Ising model. This is probably due to the

fact the that specific heats of both NiC12-6H2O and CoClg-

6H20 exhibit Ising-like behaviors in the critical region.

The fact that the annealed BPW model happens to fit
the concentration fairly well has prompted us to carry out
a detailed comparison between the quenched and annealed BPW
models. The results of this comparison show that relatively
substantial quantitative differences exist between the two
models below TN. From a consideration of the fluctuations
in the bond occupation probabilities introduced by the
annealing process, it is argued that these quantitative
differences are probably due to some intrinsic feature of
the BPW approximation. It is likely that the exact quenched
Ising model will not provide an improvement over the annealed
Ising model.

Apart from the assumption of Ising spins, we have also
made the following implicit assumptions in our analysis of
the data:

a.) the exchange constants are antiferromagnetic and

do not change with concentration

b.) 7the magnetic structure is a simple interpene-

trating two sublattice structures with all spins

directed along the same easy axis (which may vary with

concentration).

113

The Ising spins assumption will break down at low tempera—
tures. But it seems reasonable to assume that a) and b)
will continue to hold. Neutron diffraction, or NMR studies
are needed to confirm b).

Assuming a) and b), it should be relatively straight-
forward to extend spin wave and molecular field theories, as
a first step in accounting for the low temperature behavior
of the specific heat and the magnetic phase diagram.

Good quality single crystals of different compositions
needed to be grown to allow detailed measurements under
magnetic fields. High resolution calorimetric studies may
provide a good testing ground for current theoretical ideas

about the critical indices of random systems.

REFERENCES

10.
11.
12.

135

10.
15.
16.

17.
18.

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APPENDIX

117

C----PHASE DIAGRAM OF THE PLANAR ANNEALED [SING MDDEL‘-"'

3

S
2

27

17

ll

10

IS

WRITE (4:3)

FGRMATC‘XarA AND TB 7'5)

READ (4:5) TAgTB

PERMAT (2F10o3)

CGNTINUE

B 8 1o/SQRTC2.)

A 3 lo0+8

TEM 8 ALDG(I.O+SQRF(2.))/2o

VJA TA*TEM

VJB TBiTEM

WRITE (4927)

FDRMATCIX0'VALUE 0F JAB ?'3)

READ (4:17) VJAB

FORMAT(F10.3)

IF (VJABoNEoOo) GD T0 6

WRITE (4:7)

PERMAT (1X3'TEMP0 AND CDNCo FDR FIT ?'$)
READ (4:5) TC9XC

Q 3 lo-XC

U 8 OoS-(XC*XC)/(A*B*EXP(-20*VJA/TC))-(Q*Q)/(A+B*EXP(-2o*VJB/TC))
TEM = B/(2o*XC*Q/d-A)

VJAB 3 [CtALQGCTEPU/Zo

CGNTINUE

X0 3 00007

WRITE (4:11) VJAJVJ31VJAB
FORMATCIXp'JAA =':F10-3:5X:'JBB ='3F100395X3'JAB 3'sFlOo3)
DO 20 I = 1:21

P = (I")/200

0 3 IOO'P

P2 8 P*P

02 : 9*0

P92 = 2o0*P*Q

ITER = 0

CDNTINUE

DENA 8 A+B*EXP(-X0*VJA)

DENB = A+B*EXP('XO*VJB)

DENAB 3 A+B*EXP('XO*VJAB)

PCT = P2/DENA + PQZ’DENAB + QZIDENB -0.S
DPCT 3 VJAtP2*EXP('XO*VJA)/(DENA*DENA)
DFCT 8 DFCT+VJAB*P02*EXPC-X0*VJAB)/(DENAB*DENAB)
DPCT 8 8*(DFCT+VJB*92*EXP('XO*VJB)ICDENB*DENB))
XINC FCT/DFCT

IF (ABS(XINC/X0)oLEoloE-6) 60 TD ‘4

IrER = ITER+1

X0 3 XO‘XINC

IP‘ITERoLEoAO) GD T0 10

URITE(4:IS)

F0RMAT¢IX0'T00 MANY ITERATIDNS 1')

69 T0 40

14

2!
2O
40

118

TEMP = 2o/(XO‘XINC)
WRITE (4:21) P.Q,TEMP
F0RMATC3(5XDFIO¢3))
CENTINUE

PAUSE

GO T0 2

END

119

C------PHASE DIAGRAM IN THE ANNEALED BPW MoDEL- --------

3

S
2

I!

IS

17

2!

I9

10

WRITE (403)

FDRMATCIXD'TA AND TB ?'5)

READ (4:5) TAaTB

FURNATCZFIOo3)

CONTINUE

NRITEC41II)

FORMAT(IX:'N0¢ 0F NEAREST NBKS 7'3)
READ (4,15) ZN

FDRMATCF1003)

ZNI = Io/( ZN‘Io)

HRITEC49I7)

PERMATCIXn'TEMPo AND CONCo FDR FIT 7'3)
READ (4'5) TC3P

Q = IO‘P
VJAA TAtTEM

VJBB = T3*TEM

IFCTCoEQ-Oo) GD T0 6

THAA 3 TANH(VJAA/TC)

THBB = TANH(VJBB/TD)

THAB (ZNI'PtPfiTHAA'Q*Q*THBB)/(20*P*Q)
VJAB =TC* ALDG((I.§THAB)/(Io'THAB))/20
GD T0 8

WRITE (4:35)

FDRMAT(IX:'ENTER JAB 3'5)

READ (AOIS) VJAB

IF (VJABoEQoDo) VJAB
CUNTINUE

WRITE (492I) VJAA3VJ831VJAB

FDRMAT(IX:'JAA 3'1F100395X:.JBB =.DFIOO3DSXI.JAB 3.9FI003)
WRITE (41I9)

FDRMAT(IX:'INITIAL GUESS ?'$)

READ (A:I5)X0

DD 20 I =1121

P = (1-1)/200

Q 3 ‘0'?

ITER 3 0

CGNTINUE

THAA 8 TANH(XD*VJAA)

THBB 3 TANH(X0*VJBB)

THAB 3 TANH‘XO*VJAB)

FCT' P*P*THAA+Q*Q*THBB§20*P*Q*THAB'ZNI
DFCT=P*P*VJAA*(Io‘THAA*THAA)+Q*Q*VJBB*(I0-THBB*THBB)
DFCT= DFCT+2.*P*Q*VJAB*(Io’THAB*THAB)

XINC 8 FCT/DFCT

IF (ABS(XINC/XO)0LEoIoE'6) GD TD 14

ITER = ITER+I

x0 8 XO‘XINC

I? (ITERoLEoAD) 69 TD IO

SQRT€VJAA¥VJBB)

25
14
3!

20
40

120

WRITE (4:25)

FDRMAT(IX:'TDD MANY ITERATIDNS I.)
GD TD 40

TEMP = Io/(XO'XINC)

WRITE (4:31) PoQ,TEMP
FDRMATCD<5X9PIO¢3))

CDNTINUE

PAUSE

GD TD 2

END

121

C-----PHASE DIAGRAM IN THE QUENCHED apw MeoEL ---------

3

S
2

11

IS

I?

I9

21

IO

WRITE (4:3)

FDRMAT(IX:'TA AND TB ?'$)

READ (4:5) TA:TB

FDRMAT(2F10.3)

CDNTINUE

WRITE(4:II)

FORMAT(IX:'ND. DP NEAREST NBRS ?'$)
READ (4:15) ZN

FDRMAT(FIO.3)

ZNI 8 ZN-l.

WRITE(4:I7)

FDRMAT(IX:'TEMP. AND CDNC. FDR FIT ?'$)
READ (4:5) TC:P

WRITE (4:19)

FORMAT(IX:'INITIAL GUESS ?'$)

READ (4:15)XO

Q ' IO-P
TEM = ALDG(ZN/(ZN-2o))/2o
VJAA TAtTEM

VJBB = TBtTEM

IF(TC:EQ:0:) GD TD 6

THAA = TANH(VJAA/TC)

THBR = TANH‘VJBB/TC)

THAB I (Io‘P*ZNI*THAA)*(I:‘Q*ZNI*THBB)/(ZNI‘ZNI‘P‘Q)

THAB 3 SQRT‘THAB)

VJAB =TC¥ ALDG((I:+THAB)/(Io-THAB))/2:

GD TD 8

WRITE (4:35)

FDRMAT(IX:'ENTER JAB ='$)

READ (4:15) VJAB

IF (VJAB.EQ.0.) VJAB 3 SQRT‘VJAA*VJBB)

CDNTINUE

WRITE (4:2I) VJAA:VJBB:VJAB

FDRMATCIX:'JAA =':FI003:5X:'JBB 3':FI003:5X:'JAB =':FIO:3)
DD 20 I 31:21

P 8 (I’IIIZDo

Q 3 I0-P

ITER 3 0

CDNTINUE

THAA = TANH‘XD*VJAA)

THBB 3 TANH‘XO*VJBB)

THAB 3 TANH‘XO*VJAB)
FCT’ZNI*(PtTHAA+Q*THBB)'ZNI*ZNI*P*Q*(THAA*THBB'THAB’THAB)
EDT 3 Io'FCT
DFCTBVJAA*THBB.(IO'THAA*THAA)*VJBB*THAA*(I.-IHBB*rHBB)
DFCT=ZNI*P*Q*(DFCT'2:*THAB*VJAB*(I:‘THAB*THAB))
DFCT‘ZNI*(DFCT-P*VJAA*(I:-THAA*THAA)’Q*VJBB*(I:'THBB*THBB)’
KING 8 FCT/DFCT

IF (ABSCXINC/XD):LE:I¢E’6) GD TD I4

25
I4
3!

2O
40

122

ITER 3 ITER+I

X0 8 XD'XINC

IF (ITERoLE040) GD TD IO
WRITE (4:25)
FDRMATCIX:'TDD MANY ITERATIDNS 9')
GD TD 40

TEMP 8 I:/(X0'XINC)
WRITE (4:31) P:Q:TEMP
FORMAT(3(SX:F10.3))
CDNTINUE

PAUSE

GD TD 2

END

123

C°----$PECIFIC HEAT DF THE ANNEALED PLANAR ISING MDDEL """
PI 8 3:1415926536

15 FDRMAT(F10:3)
WRITE (4:17)

17 FDRMAT(1X:'NUMBER DF NEAREST NEIGHBDRS ?'S)
READ (4:15) ZN
WRITE (4:19)

19 FDRMAT (IX:°TA : TB : TC : X0 ?'5)
READ (4:25) TA:TB:TC:XC

25 FDRMAT(4F10:3)

21 FDRMAT(3F10:3)

2 CDNTINUE
WRITE (4:11)

11 FDRMAT(1X:'CDBALT CDNCENTRATIDN ?'$)
READ (4:15) P
WRITE (4:31)

31 FDRMAT(1X:'TINIT:TFINAL:TSTEP ?'£)
READ (4:21)TD:TF:TSTEP
B = 1o/50RT(2:)
A 3 IOO+B
TEN ALDG‘IOO§SQRT(20))/2o
VJA TA*TEM
VJB TB*TEM
Q 3 I:’XC
U=0.5-(XCtXC)l(A+B*EXP(-2:*VJA/TC))-(Q*Q)/(A+B*EXP(-2o*VJB/TC))
TEM = B/(2.tXC*Q/U-A)
VJAB = TC*ALDG(TEM)/2o
P2 8 PtP
Q 3 IOO'P
02 = QtQ
P02 : 20*P*Q
VJA2 = VJA*VJA
VJABZ = VJABtVJAB
VJB2 = VJB*VJ8

c ---------- START IIEHArIaN -------------------------------
ITER a 1
WRITE (4:41) VJA:VJB:VJA3

41 FDRMAT(1X:'JAA =':F10-3:5X:'JBB =':F10-3:5X:'JAB =':F10o3)
5K0 3 1:2

10 CONTINUE
AA 8 SKD-VJA/TO
AB = SKO-VJAB/TD
88 = SKO-VJB/TO
CSHAA 2 CDSH(AA)
CSHAB = CDSH(AB)
CSHBB 8 CDSH(BB)

SHAA = SINH(AA)
SHAB = SINH(AB)
SHBB = SINH(BB)

SHK = SINH(2.*SKO)
CHK 8 CDSH(2.*SKO)
THK = SHK/CHK

61

45

20

51

40

12“

AK 3 2:*SHK/(CHK*CHK)

AKC 8 2:*THK*THK‘I:0

CALL ELLIP‘AK:CELI:CEL2:DER)

TERM 8 (D:5+AKC*CEL1/PI)

CGRR 8 TERM/THK

DCDRR 3 TERM/$HK+2:*(CEL2’CEL1)/AK/PI
DCDRR 8 ~20¢DCDRRISHK

DAAI 8 (CSHAA/SHAA)-CDRR
DAB1 8 (CSHAB/SHAB)-CDRR
0881 8 (CSHBB/SHBB)-CDRR
DAA2 8 CSHAA-CDRRtSHAA
DABZ 8 CSHAB-CDRRtSHAB
DBB2 ' CSHBB-CDRRtSHBB

FX 8 (P2/DAA1)+(PQ2/DA81)+(Q2/DBBI)

SUMI 8 P2/(DAA1*DAA1)+PQ2/DA81/DABI +Q2/0881/0881
SUM2 8(P2/DAA2/DAA2)+(P02/DA82/DA82)+Q2/0882/DBB2
DEX 8 DCDRRflSUM1+SUM2

SKINC 8 FX/DFX

IF (ABS(SKINC/SKO):LE:1:E-6) GD TD 20

ITER 8 ITER+1

5K0 8 SKO-SKINC

FDRMAT(5X:I3:5X:F10:3)

IF (ITER-LEo4D) GD TD 10

WRITE (4:45)

FDRMAT(1X:'TDD MANY ITERATIDNS 1')

GD TD 40

SKI 3 Io/SKO

DAA22 8 DAA2*DAA2

DAB22 DA82*DAB2

D8822 088280882

SUM2 8 P2*VJA/DAA22 +PQ2¥VJA8IDA822 +02¢VJ8/08822
SUM3 8 P28VJA2/DAA22 +P92tVJAB2/DAB22 +02*VJ82/08322
CDRRM 8 1:-CDRR*CDRR

TERM] 8 SUM2*SUM2/(SUM1-1o/(CDRRM-DCDRR))

CP 8(SUM38CDRRM+TERM1)*ZN/(T0*TD)

WRITE (4:51)T0:5KI:CDRR:DCDRR:CP
FDRMAT(F703:5X:F10o3:2(5X:E10:3):5X:F10:3)

T0 8 T0+TSTEP

IF (TD-GToTF) GD TD 40

ITER 8 1

GD TD 10

PAUSE

GD TD 2

END

125

SUBRDUTINE ELLIP(AK:CEL1:CEL2:DER)
C-~---THIS SUBRDUTINE CDMPUTES CDMPLETE ELLIPTIC---"'-'
C-----INTEGRALS DP THE FIRST AND SECDND KIND$------------

PI 8 3:1415926536

PI2 8 PI/2.

IFCT 8 2

SUM 8 AKtAK

AKC2 8 1o-SUM

AKC s SQRT(AKC2)

CELI 8 1.0E+20

CELQ 3 I00

DER 8 1oOE+20

IF (ABS(AKC):LE.1oE-6) RETURN

AARI 3 1:0

GED 8 AKC
10 CDNTINUE

ARI 8 (AARI‘GED)/2.

DIFF 8 AARI-ARI

SUM 8 SUM+IFCT*DIFF*DIFF

IFCT I 2tIFCT

GED 8 SQRT(AARI*GED)

AARI 8 ARI

IF (GED/ARI-Oo999999) 10:20:20

20 CEL1 8 PIZ/ARI

SUM 3 SUM/2:

CEL2 8 CEL1*(1.-SUM)

DER 8 CEL1*(AK-SUMIAK)IAKC2

RETURN

END

126

C“"'SPECIFIC HEAT DP ANNEALEQ BETHE LATTICE"""°'°
ATH(X) 3 0:53ALDG((10+X)/(1:-X))
WRITE (4:3)
3 PDRMAT(1X:'TA AND TB 7'3)
READ (4:5) TA:TB
5 PDRMAT(2P10:3)
WRITE(4:11)
11 FDRMAT(1X:'ND: DP NEAREST NBRS ?'5)
READ (4:15) ZN
15 FDRMAT(PID:3)
ZNI 3 1:/( ZN'Io)
WRITE(4:17)
17 PDRMAT(1X:'TEMP¢ AND CDNC: FDR FIT 7'3)
READ (4:5) TC:P
Q 3 IO’P
TEM 8 ATH(ZN1)
VJAA 3 TA3TEM
VJ88 8 T88TEM
IP(TC:EQ:0:) GD TD 6

THAA = TANH(VJAA/TC)
THBR 8 TANH(VJ88/TC)
THAR 8 (ZNI-P*PtTHAA-Q*QtTH89)/(2.*P*D)

VJAB 3TC* ALDG((1o+THAB)/(108THA3))/2o
GD TD 8
6 WRITE (4:35)
35 PDRMAT(1X:'ENTER JAB 8'5)
READ (4:15) VJAB
1P (VJAB.EQ.O.) VJAB 3 SQRT(VJAA*VJBB)
8 CDNTINUE
WRITE (4:21) VJAA:VJBB:VJAB
21 FDRMAT(1X:'JAA 8':F10o3:5X:'J33 8':P10:3:5X:'JAB 3':P10:3)
2 CDNTINUE
WRITE (4:19)
19 PDRMAT(1X:'ENTER CDNC: 8 ?'$)
READ (4:15) P

Q 3 IO'P
P2 = P8?
02 c 0*0

P9 3 2.3P30
WRITE (4:23)

23 PDRMAT(1X:'TINIT:TINC:TMAX 7'3)
READ (4:9) TD:TINC:TMAX

9 PDRMAT(3P10:3)
WRITE (4:27)

27 PDRMAT(1X:'INITIAL GUESSES FDR K AND H0 7'5)
READ (4:5) AKO:HD

C"--'PIND EFFECTIVE FIELD CDRRESPDNDING TD K""""

12 ITER2 3 D

14 CDNTINUE
APK 3 TANH(AKO)
I? (APK:LE:ZN1) H0 8 0:0

10

37

16

127

IF(HO.EQ.0.0) GD TD 16

ITER1 8 0

CDNTINUE

THO 8 TANH(HO)

THL 8 TANH(H0*ZN1)

FCTI 8 AFKtTHO-THL

IF (ABS(FCT1).LE.1:E-6) GD TD 16
DFCT1 8 APK*(108TH03TH0)’ZNI*(IO‘THL*THL)
HINC 8 FCT1/DFCT1

IF (ABS(HINC/H0).LE.1:E-6) GD TD 16
ITER1 8 ITER1+1

H0 8 HO -HINC

IF (ITERloLEo40) GD TD 10

WRITE (4:37)

FDRMAT(1X:'T00 MANY ITERATIDNS IN FINDING H0 1')
GD TD 40

CDNTINUE '

D1LK 8 ZN1*(1.-AFK*AFK*THO*THO)-AFK*(1o-TH08THO)
DILK 8 THO*(1o-AFK*AFK)/01LK

SH2L 8 SINH(2.8H0)

CH2L 8 CDSH(2.*H0)

EX2K 8 EXP(2.*AKO)

DEN 8 EX2K80H2L+1o

CDRR 8 1.-(2./DEN)

DCDRR 8 4.8EX2K8(CH2L+SH2L801LK)/(DEN*DEN)
C-----FIND AKO CDRRESPDNDING TD T0 --------------------

CHAA CDSH(AKO-VJAA/T0)

41

CHAD 8 CDSH(AKO-VJAB/TO)
CHBB 8 CDSH(AKO-VJBB/TO)
SHAA 8 SINH(AKO-VJAA/TO)
SHAB 8 SINH(AKO-VJAB/T0)
SHBB 8 SINH(AKO-VJBB/T0)

DENAA 8 CHAA-CDRRtSHAA

DENAB 8 CHAB-CDRRtSHAB

DENBB 8 CHBB-CDRRtSHBB

FCT2 8 (P285HAA/DENAA)+(PQ*SHAB/DENA3)+(92*SHBB/DENBB)
DAA2 8 DENAAtDENAA

DAB2 8 DENABtDENAB

DBB2 DENBBtDENBB

IF (ABS(FCT2):LT.1oE-6) GD TD 20
DFCT28P2*(1.+DCDRR*SHAA8$HAA)IDAA2+PQ*(1o+DCDRR8$HA885HAB)/DAB2
DFCT2 892*(1:+DCDRR*SHBB*SHBB)/DBBZ + DFCT2

AKINC 8 FCT2/DFCT2

IF (ABS(AKINC/AKO)-LE.1.E-6) GD TD 20

ITER2 8 ITER2+1

AKO 8 AKO-AKINC

IF(ITER2.LE.40) GD TD 14

WRITE (4:41)

FDRMAT(1X:'T00 MANY ITERATIDNS IN FINDING K 1‘)

GD TD 40

20 CDNTINUE

51

53

40

128

SUM! = P2*SHAA*SHAAIDAA2 + P085HA885HABIDAB2

SUM! 8 02*SHBB8SHBBIDBBZ +SUMI

SUMZ 8 VJAA*P2/DAA2 + VJAB$POIDA82 + VJBB*92/0882
SUNS 8 VJAAtVJAA*P2/DAA2 +VJAB*VJA9*PQ/DA82

SUNS s VJBBtVJBB*92/DBRZ + SUM3

TEM 3 Io'CDRR3CDRR'DCDRR

DKB 8 SUM2/(Io‘SUM13TEM)

GP 3 ZN*(SUM3*(10°CDRR3CDRR)‘SUM23TEM*DKB)/(T03T0)
WRITE (4:51) T0:AKD:HD:CDRR:DCDRR
PDRMAT(P503:4(5X:E1104)$)

WRITE (4:53) GP
PDRMAT(IH+:5X:PII:4)

TO 3 TOOTINC

IF (TO.GT.TMAX) GD TD 40

GD TD 12

PAUSE

GD TD 2

END

129

C-----SPECIFIC HEAT DF QUENCHED PLANAR BETHE LATTICE-----

17

21

19

23

33

ATH(X)30053ALDG((10+X)’(10'X))
LDGICAL VARI

DIMENSIDN W(10):ENER(3)

VNULL 3 IoE-6

WRITE(4:3)

FDRMAT(1X:'TA I TB 7'5)

READ (4:5) TA:T8
FDRMAT(2F10:3)

FDRMAT(F10:3)

ZN! 3 1’3:

2 3 4o

WRITE(4:17)

FDRMAT(1X:'TEMP: AND CDNC: FDR FIT 7'5)
READ (4:5) TC:P

Q 3 IO-P

TEM 8 ATH(ZNI)

VJAA 8 TAfiTEM

VJBB 3 TB‘TEM

IF (TC:EQ:O:)GD TD 6

THAA =TANH(VJAA/TC)

TH88 TANH(VJBB/TC)

THAB 3 (ZNI’P‘THAA)*(ZNI'Q*THBB)/(P*D)
THAB - SQRT(THA8)

VJAB 8 TCtATH(THAB)

GD TD 8

WRITE(4:35)

FDRMAT(1X:'ENTER JAB 8'5)

READ (4:15) VJAB

'IFIVJAB.EQ.0.I VJAB8SQRT(VJAA*VJBB)

CDNTINUE

WRITE (4:21) VJAA:VJBB:VJAB

FDRMAT(1X:'JAA 3':F10:3:5X:'J88 8':F10:3:5X:.JA8 8':F10:3)
CDNTINUE

WRITE (4:19)

FDRMAT(1X:'ENTER CDNC: 8'5)

READ (4:15) P

Q 310‘?

W(1) 8 0884

W(2) 8 4:8P*9*Q*0
W(3) 8 6.8P8P8980
W(4) 8 4:8P8P8P80
W(S) 8 P884

WRITE (4:23)
FDRMAT(1X:'TINIT:TINC:TMAX 7'5)
READ (4:9)TD:TINC:TMAX
FDRMAT(3F10:3)

WRITE (4:33)

FDRMAT(1X:'RELO INC: FDR DER: 7'5)
READ(4:15) HINC

WRITE(4:27)

130

27 FDRMAT(1X:'GUESSES FDR HA & H8 7'5)
READ(4:5) X0:YD
C°"°‘START ITERATIDN‘-°--“'--'--- 888888888 ------------
10 CDNTINUE
DD 80 K31:3
GD TD (62:64:66) :K
62 TI 8 Io/(T0*(10§HINC))
GD TD 50
64 TI 8 1:/(T0*(1:'HINC))
GD TD 50
66 TI 8 1:/TO
50 CDNTINUE

ITER 8 0

BAA 8 VJAAtTI
BBB 8 VJBBtTI
BAB 8 VJABtTI
TAA 8 TANHCBAA)
TBB a TANH(BBB)
TAB 8 TANHCBAB)

12 CDNTINUE
THX 8 TANH(X0)
THY 8 TANH(Y0)
A 8 (1.0THX8TAA)/(1.-THX*TAA)

B 8 (1o+THY8TAB)/(1:-THY*TAB)
C 8 (1:+THX*TAB)/(1:-THX*TAB)
D 8 (1.4THY8T88)/(1o-THY*T88)
TAAS 8 TANH<XO+BAA)

TAAD 8 TANH(X0-BAA)

TABS 8 TANH(X0+BA8)

TABD 8 TANH(XO-BA8)

T885 8 TANH(Y0+BBB)

T880 8 TANH(Y0-BBB)

TBAS 8 TANH(Y0+8A8)

TBAD 8 TANH(Y0-BAB)

3A1 3 0.

531 3 00

SNAA1 8 0:

SNABI 3 0:

SNBB1 8 0:

SNBAI 3 0 o

SNAZA 3 0:

SNA2B 8 0.

SNAIA 3 00

SNABA 3 0.

SNABB 8 0o

SNBZB 3 00 '
SNBZA 8 0o

SNBIB 3 O.

SNBIA 3 0:

SNAIB 3 00

DD 22 131:5

22

131

NA 8 1-!

NB 8 4-NA

ZAPM 8 (A:*NA)*(B**NB)
ZBPM a (Ct*NA)t(D**NB)
SA! 8 SA!+N(I)/(!.+ZAPM)
SB! 8 SB!+W(I)/(!.+ZBPM)

SNAAI 8 SNAA!+NA*W(I)/(!.+ZAPM)

SNAB! a SNAB!+NA*H(I)/(!.+ZBPM)

SNBB! 8 SNBB!+NB*W(I)I(1.+ZBPM)

SNBA! 8 SNBA!+NB*W(I)/(!.¢ZAPM)

SNAZA s SNA2A+NA*NA*W(I)tZAPM/((!.+ZAPM)tt2)
SNAZB 8 SNA28+NA$NA*k(I)*ZBPM/((!.+ZBPM)#*2)
SNAIA = SNA1A+NA8W(I)*ZAPM/((1:+ZAPM)**2)
SNABA 8 SNABA+NA¢NB*W(I)*ZAPMI((!.oZAPM)**2)
SNABB 8 SNABB+NAtN8*H(l)*ZBPM/((!.+ZBPM)**2)
SNBZB = SNBZB+NB¥NB*w(I)*ZBPM/((I.+ZBPM)t*2)
SNBZA 8 SNBZA+NB*NB*W(I)8ZAPMI((I.+ZAPM)*#2)
SNB!B = SNBIB+NB*W(I)*ZBPM/((1.+ZBPM)**2)
SNBIA = SNBIA+NB*W(I)*ZAPM/((1.+ZAPM)**2)
SNAIB = SNAIB+NAtu(I)*ZBPM/((!.+ZBPM)**2)
CGNIINUE

FXY 8 P*(TAAS*(Z*P-SNAA!)+TAAD*SNAA!)-ZtPt(!.-2.*SA!)
FXY 8 FXY+08(TABS*(Z*P-SNABI)+fA80*SNABI)

GXY 8 Qt<{885*(Zte-SNBB!)+TBBD*SNBB!)-Z*Q¢(|.-2.*SB!)
GXY = GXY+P*(fBASttZtQ-SNBA!)+TBAU*5N3A!)

VAR! 8 (ABS(FXY).LE.VNULL).AND.(ABS(GXY).LE-VNULL)

IF (VAR!) 60 TD 30

DFX 8 PttC!.-TAAS#TAAS)#(Z*P-SNAA!)+(!.-TAAD*#2)*SNAA!)
DFX 8 DFX+Q*((!.-TABS*#2)*(Z*P-SNABI)+(!.-TABD*¢2)#SNAB!)
DFX 8 DFX+PtSNA2Am<TAAS-TAAD):#2+Q*SNA28*(TABS~TABDJ¢$2
DFX 8 DFX-2.tZtP*(TAAS-TAAD)*SNA!A
YT=P*(TAAS-TAAD)*(TBAS-TBAD)85NABA

YT 8 YT+Q*(TABS-TBAD)*(TBBS-TBBD)*SNABB

DFY = YT-z.*Z*Pt(TBAs-TBAD)*SNBIA

DGX 8 YT-2.*Z*Q*(TABS-TABD)tSNA!B

DGY - 08((1.-TBBS**2)#(Z#Q-SN831)+(1o-T880882)8SN381)

DGY 8 DGY+P¢((!.-TBAS**2)*(Z*Q-SNBA!)+(!.-fBAD**2)*SNBA!)
DGY = DGY+QtSN823*(TBBS~TBBo)**2+P*SN32A*(TBAS~r8A0)*t2
DGY 8 DGY-2.*Z*Qt(TBBS-TBBD)tSNB!B

VAR1 8 (ABS(XO)-LE.VNULL).AND.(ABS(Y0).LE:VNULL)
IF (VARI) GD TD 20

DET 8 DFX*DGY-DFY*DGX

KING 8 (FXYtDGY-GXY80FY)/DET

YINC 8 (GXY8DFX-FXYtDGX)/DET

VAR1 8 (ABS(XINC/X0):LEoVNULL)oAND:(ABS(YINC/YO):LE:VNULL)
IF (VARI) GD TD 30

X0 8 XO-XINC

Y0 8 YO-YINC

ITER 8 ITER+1

IF(ITER:LE.40) GD TD 12

WRITE (4:25)

25

20

30

80

3!

40

132

FDRMATCIX1'TDD MANY ITERATIDNS I')

GD TD 40

x0300

Y0300

CDNTINUE

IF (K-EQoS) GD TD 80

TERM! VJAA*(Z*P8TAAS-SNAA18(TAAS+TAAD))

TERM! 8 TERMI+VJAB¢(Z*Q*IBAS-SNBAItCTBAS+TBAD))
TERMZ VJBB*(Z*Q*[BBS-SNBBI*(TBBS+TBBD))

TERMZ 8 TERM2+VJAB*(Z*P*TABS‘SNAB1*(TABS+TABD))
ENERCK) = PfiTERMI+Q¥TERM2

CDNTINUE

CP 3 (ENERC2)-ENER(I))/(2¢*T0*HINC)

WRITE (433I)T0:X09Y0:CP
FDRMAT€FIO¢3:3(5X:FI1.4))

T0 8 I0+TINC

IF(TO.LE.TMAX) GD T0 10

PAUSE

GD TD 2

END