THE ANISOTROPIC ANTIFERROMAGNET-a f :
THEORY AND EXPERIMENT

Thesis for the Degree of Ph. D. R
MICHIGAN STATE UNIVERSITY 1
CHRISTOPHER WARREN FAIRALLV'

rHcfi‘h

 

This is to certify that the
thesis entitled
THE AN IS OTHOPIC. fiNTIFERROIvaNhTa
THEORY AND EXPERIMENT

presented by
Christopher Warren Fairall

has been accepted towards fulfillment
of the requirements for

_.Eh...ll._ degree in 211181.08—

2 Major professor

0-7639

 

 

 

 

 

 

 

ABSTRACT

THE ANISOTROPIC ANTIFERROMAGNET-
THEORY AND EXPERIMENT

BY

Christopher Warren Fairall

The phase transitions of the two sublattice antiferro-
magnet with general second order anisotropy have been studied
in the molecular field approximation. The interactions
included were the isotrOpic and anisotrOpic exchange, uni-
axial crystal field anisotropy and Dzyaloshinski-Moriya
antisymmetric exchange anisotrOpy of the form B'§1X§2' The
3 vector was chosen perpendicular to the antiferromagnetic
axis and the phase transitions induced by applied magnetic
field were calculated numerically from the equations of
equilibrium and stability. The antiferromagnetic to spin
flop transition remains first order while the second order
paramagnetic transition is destroyed by the D-M interaction
unless the field is applied parallel to B.

The principal axis susceptibilities were calculated,
revealing an inflection point corresponding to a quasipara—
magnetic transition and an infinite anamoly at the Spin flop

critical field. Included is a calculation of the angle

 

Christopher Warren Fairall

dependence of the susceptibility of the uniaxial antiferro-
magnet in applied field.

Magnetic susceptibility measurements were made on
CszMnCl4'2H20, szMnCl4-2H20 and CuClz-ZHZO. The suscepti-
bility was measured as a function of temperature and
magnitude and orientation of applied field in liquid He4.
2MnCl4-2H20 and
RbZMnCl4-2H20 were measured and no spin flop boundaries

The magnetic H-T phase diagrams of Cs

were found to exist above 1.2 Kelvins.

The molecular field theory developed in the text was
used to interpret the data including data taken from the
literature on MnC12-4H20. The results are as following:

1. CsZMnCl4-2H20 and szMnCl4-2H20 eXhlblt an

unusually large anisotrOpy that is significantly larger in
the ordered state than in the paramagnetic state. Molecular

field calculations indicate CszMnCl4-2H20 will spin flop

below Ttp=o'55 K at a critical field Htp=l7 kOe and

szMnCl4-2H20 will spin flop below Ttp=o'83 K at a critical

field Htp=21 k0e.

2. Excellent agreement between theory and experiment
was found for the susceptibility as a function of magnitude
and orientation of applied field for CuClz-ZHZO.

3. Molecular field theory gave consistent results

for phase diagram and susceptibility data on MnC12-4H20.

 

THE ANISOTROPIC ANTIFERROMAGNET-

THEORY AND EXPERIMENT

BY

Christopher Warren Fairall

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

 

ACKNOWLEDGMENTS

The author wishes to express thanks to his mentor Dr. J. A.
Cowen and to Drs. R. D. Spence and E. H. Carlson for many helpful
discussions and assistance; to Dr. G. L. Pollack for much appre-
ciated encouragement; to John Ricks and Edward Grabowski for help
in the laboratory; and to the machine shop for construction of
apparatus. The author is especially grateful for financial assist-
ance from the National Science Foundation and an N.D.E.A. Title

IV Fellowship.

ii

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF APPENDICES

INTRODUCTION

I.

II.

III.

THE HEISENBERG HAMILTONIAN

A.

C.

D.

Exchange

1. The two electron system

2. Many electron exchange

3. Superexchange

Anisotropy

1. Dipolar anisotropy

2. Crystal field anisotropy

3. Spin orbit anisotropic effects

The General Second Order Interaction

The Question of Generality

MAGNETIC ORDERING

A.

B.

The Simple Paramagnet
The Ordered State
1. Molecular field theory

2. Green function theory
3. Other methods

MAGNETIC PHASE DIAGRAM

A.

B.

The Classical Energy
The Zero Temperature Phases

1. The zero field equilibrium configuration

2. The parallel configuration

3. The perpendicular configuration

4. The D configuration'

5. Discussion of the effects of hD on the
zero temperature phases

iii

Page

vi

viii

13
15
16
16
17
18
21
23
26
27
31
31
32
41
42

43

 

Chapter

C. The Temperature Dependence of the Critical
Fields and the H-T Phase Diagram

1. The paramagnetic boundaries
2. The spin flop boundary

IV. THE THEORETICAL DEPENDENCE OF THE SUSCEPTIBILITY ON
APPLIED FIELD AND TEMPERATURE - THE ANTIFERROMAGNET
A. The Zero Field Susceptibility
l. The susceptibility asea function of temp-
erature for H=0
2. The susceptibility as a function of angle
for H=0
B. The Susceptibility in Applied Field
1. The susceptibility as a function of H for
the canted antiferromagnet at T=O
2. The susceptibility as a function of angle
in applied field for the uniaxial anti-
ferromagnet at T=0
V. EXPERIMENTAL APPARATUS - THE MEASUREMENT OF
ANISOTROPIC SUSCEPTIBILITY
A. The Great Unit Debate
B. The Measurement of Susceptibility
C. The Parametric Apparatus
l. Susceptibility vs. temperature
2. Susceptibility vs. orientation
3. Susceptibility in applied field
D. Low Susceptibility Materials
VI. EXPERIMENTAL RESULTS AND DISCUSSION
A. CszMnCl4-2820 and RbZMnCl4-ZHZO
B. CuClZ-ZHZO
D. Conclusions
REFERENCES
APPENDICES

iv

Page

43
44
45
47
47

47
53

53

54

64

68
68
7O
71
72
72
77
81
88
88
108
121
127
130

134

Table

II.

III.

LIST OF TABLES

Physical parameters of the susceptibility coils

Experimental data and molecular field calcula-
tions for CszMnCl4-2HZO and szMnCl4-2H20

The zero temperature phase boundaries and
molecular fields for MnC12-4H20

Page
76

109

126

Figure

l.

10.
11.

12.

13.

14.

15.

16.

LIST OF FIGURES

Definition of the angles describing the spin
array

The superheated critical angle ash(hD) vs.
hD with hK=O and hL=0.1

The spin f10p critical fields vs. hD with
hK=O and hL=0.l

A comparison of numerically calculated h
with the approximation of equation 3.27

Susceptibility vs. temperature in the
molecular field approximation

Susceptibility vs. applied field with hD=0,
hK=O and hL=0.l

Susceptibility vs. applied field with hD=O.05,
hK=0 and hL=O.1 ‘

Numerical calculation of dx/dh vs. h with
hD=0.05, hK=O and hL=0.l

Numerical calculation of susceptibility vs.
angle in applied field

The crystal rotation device
Coil #5 with rotator

Calibration of resistance vs. H for the
magnetoresistor

A cross section of the apparatus for measuring
susceptibility in applied field

Susceptibility vs. angle for szMnCl4-2H20
in the paramagnetic and antiferromagnetic

'states with zero applied field

Zero field susceptibility vs. temperature for
CszMnCl4-2H20 in the parallel and perpendicular
orientations

Zero field susceptibility vs. temperature for

RbZMnC14-2HZO in the parallel and perpendicular
orientations

vi

Page

30

35

37

4O

52

58

60

62

67
74
79

83

85

91

93

95

Figure

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

Magnetic phase diagram of CszMnCl4-2H20. The
predicted Spin flOp boundary is represented
by a dotted line.

Magnetic phase diagram of Rb2MnC14-2H20. The
predicted spin flop boundary is represented by
a dotted line

Susceptibility vs. temperature in applied
field for CszMnCl4-2H20, typical data used
for obtaining phase boundaries

Extrapolation of the phase boundaries to
T=0 for Cs MnCl -2H 0

2 4 2
Extrapolation of the phase boundaries to
T=0 for Rb MnCl ~2H O

2 4 2
Susceptibility vs. temperature in applied
field for CuC12-2HZO

Extrapolation of the amplitude of the spin
f10p transition vs. temperature for CuC12-2H20

Magnetic phase diagram of CuClZ-ZHZO (Ref. 58)
Susceptibility vs. angle for various applied
fields for CuClz-ZHZO; eXperimental and
theoretical results normalized to the zero
field parallel and perpendicular values

Susceptibility vs. applied field at constant
temperature for CuClz-ZHZO

Magnetic phase diagram of MnC12-4H20
(Refs. 66 and 67)

Susceptibility vs. applied field at constant
temperature for MnC12-4H20 (Ref. 66)

vii

Page

98

100

103

105

107

111

113
115

118

120

123

125

LIST OF APPENDICES

Appendix Page
A. Newton's Method 134
B. Crystallography 136
C. Computer Programs 138

viii

 

 

 

\-

 

 

INTRODUCTION

Magnetic ordering, though known at least experimentally
for centuries, was first understood with the development of
quantum mechanics in the 1920's. It is now known that the
magnetic moments associated with magnetic ions in crystals can
take on many possible arrangements including ferromagnetic, anti-
ferromagnetic, canted antiferromagnetic, ferrimagnetic and spiral
configurations. The arrangement a given lattice of spins will
acquire depends upon the interactions of the system. The inter-
action that has been most successful in describing the arrange-
ment and pr0perties of magnetic insulators is the general
Heisenberg interaction, second order in the spin operators.

Magnetic materials can be studied by a number of experi-
ments including resonance, specific heat, neutron diffraction
and susceptibility. Antiferromagnets are particularly inter-
esting because in applied magnetic fields they can undergo var-
ious phase transitions corresponding to realignment or satura-
tion of the spin system. The basis of these transitions is
the interplay between the isotrOpic and anisotropic parts of
the interaction.

The first successful theory describing these magnetic sys-
tems was the molecular or effective field approximation where
one considers the interaction of a given spin with its neigh-
bors to be an average effect that can be replaced by an effect-
ive magnetic field. Although more saphisticated theories now
exist, our primary concern in this work is to explore the abil-

ity of molecular field theory to explain the properties of the

1

.
5"

I...

 

lr‘

 

anisotropic magnetic insulator. Specifically, we wish to accomp-
lish the following:

1. Unify the existing theoretical base of the Heisenberg
interaction and the magnetic ordering it produces.

2. Extend the theory of the magnetic phase diagram to a
general second order interaction for the canted antiferromagnet.

3. Write a theoretical description of the magnetic sus-
ceptibility as a function of orientation, applied field and temp-
erature. Particular emphasis is placed on the anisotropic prOp-
erties of canted or uniaxial antiferromagnets in applied fields
and the behavior of the susceptibility near critical points.

4. Describe the construction and use of apparatus capable
of measuring suceptibility as a function of orientation, applied
field and temperature.

5. Present the results of susceptibility measurements
made with this apparatus and their theoretical interpretation.
we also apply the theory to previously existing data on a mater-
ial that is well understood. The purpose iS-tO evaluate the

consistency and accuracy of the theory.

I. THE HEISENBERG HAMILTONIAN

The interactions that govern the behavior of a system of
electron spins in an insulating solid are multitudinous, but
one can describe many of the magnetic properties by considering
two separate but not wholly orthogonal interactions: exchange

and anisotropy.

A. Exchange
1. The two electron system.

Exchange is a purely quantum mechanical effect and, having
no classical analogue, it is difficult to employ physical intui-
tion when discussing it. However, the next best thing to phys-
ical intuition is history, so I will describe a simple example
demonstrated by Heisenberg1 in 1926.

Suppose we have two electrons in similar potentials. The

Hamiltonian for the pair is

where l and 2 refer to coordinates of the respective electrons

+ +

and r12 = Ir1 - er-

Let
H = H + e2/r (1.2)
o 12
and assume e2/r12 is a perturbation on ”0'

H W9. = 29.??. (1.3)
o 13 1] 1]

where

(1.4)

o _ o o
E0 =E‘?+E
1] 1 3

We can make this separation because HO contains no interaction

between 1 and 2.

First order perturbation theory gives a correction to the

energy
E.. = ER. +
13 l]
= *0 O 2

where Bij is the average repulsive
electrons in states i and j.

The Pauli exclusion principle
function to be antisymmetric under

and 2.

by the Slater determinant

w‘i’umm

I-‘

W..(l,2)= -

61

 

w‘j’um (1)

where alpha is the "spin up" wave function.

2

more possible wave functions W , W , W4

ent Spin configurations.

(1.5)
o o
Wi(l) Wj(2)dr

coulomb interaction of the

requires the total wave

"exchange" of particles 1

The apprOpriate wave functions including spin are given

w‘i’(2)a(2)

(1.6)

 

9"]?(2) (1(2)

There are three

corresponding to differ-

Diagonalization of the matrix elements

ch'with these wave functions results in a singlet of energy

(1.7)

‘oi

‘ I

 

and a triplet of energy
E = E.. + B . - J.. (1.8)
where

_ *o *o 2 o o
Jij — f wi(1) wj(2)e /r12 wj(1) wi(2)dr (1.9)

is called the direct exchange integral.

2. Many electron exchange
Unfortunately, one cannot describe the behavior of a crys-
tal in terms of two electron wave functions. If we consider N
electrons (designated :1) on N reasonably localized lattice

+
sites (R ), the Hamiltonian is

alpha
2
P. 2 2
_ Z 1 _ Z Ze Z ‘1 e
H — i 2m i.a Iri-Ral + ij 2 rij- (1'10)

. . . . 2
and a convenient set of wave functions are Wannier functions

th

¢nl(; - Ra) which resemble the n atomic orbital with spin 1

on lattice site a but drop off through the lattice in such a

‘way as to be orthogonal. If one rewrites the Hamiltonian in

nd 3,4

, one can show that the

2 quantized field Operator form

exchange term can be expressed

.I’ '\
_.2 ++. ;l__ +.++|'
Hex — aa' Jnn'(Rc'Ra )‘34 + S(Ra) 8(Ra ){ (1.11)
nn' “
where
J ,(R ,R ') = (an;a'n'|VIo'n';an)

(1.12)

Equation 1.11 is referred to as the Heisenberg exchange

 

rt. N u. a.

 
 
 

 

Hamiltonian, originally derived for two electrons by Diracs.

3. Superexchange
In most insulating crystals the magnetic ions are separated
by large distances with intervening ligands; it follows that the
direct exchange of equation 1.12 is very small and cannot ac-
count for the much larger interactions that are usually observed.
Let us consider a system of two Mn++ ions joined by an

0 ion. The ground state is illustrated by

++ -- ++
(A)

where p and d refer to atomic orbital states. The excited
state corresponds to promotion of one 0-— p electron into the

d shell of Mn++.

Mn+ 0’ Mn++
' (B)
dldl p 62

The wave functions for the configurations are writtens'7

W

A wdl(1)wp(2)wp.(3)wd2(4)

(1.13)

‘l’ =‘I’ (l)‘l’ 1(2)!I (3)‘lt (4)
B d1 (11 p d2

The perturbing Hamiltonian is assumed to consist of a spin inde-

pendent part, V , connecting states A and B, and a spin depend-

 

t
ent part, Ve’ diagonal with respect to orbital states. Spin
dependence shows up first in the 3rd order energy correction as
I I I I
E _ 2 (AtIVtIBt )(3t [2213“ )(B“ IvtIA“) (1.14,
3 t,u (Et' _ EtHEu' _ Eu)
B A B A

.i‘

t, u denoting spin configurations.
One rewrites the wave functions in states of definite
parity in terms of the spin coupling of the Mn-to one 0 elec-
tron. For example WA[(pdl)3(p'd2)l] is a ground state wave
function with the d1 electron of Mnl and the p electron of
0-- in a triplet state and the d electron of Mn

2 2
electron of 0-- in a singlet state. The energy of this system

and the p'

can be written

— .__l_ - 1 . 2 .+
E - [AE(t)2 m2] 2b de(§l 52)

(1.15)
1 3 1 2
" If [AEZtIZ + AE(S)2] 21’ de
where the transfer integral is
b = fwdl(1)wp(1)vtdr (1.16)
and
*
de = ] wp(1)§d(2)vewp(2)wd(1)dr (1.17)

is the exchange of the Mn++ 0 system. The energy of the Mn+
in the singlet and triplet states is given by AE(S) and AE(t)
respectively.

Let us suppose that Mnl has Spin up. Now assume that

J >0, therefore, the p electron associated with Mn1 will have

Pd
spin up. The p' electron will have spin down and therefore an
will have spin down - the interaction is antiferromagnetic.
Another way of describing this process is to assume de>0 implies

AE(S)+00 and AE(t)+U, the average coulomb interaction of an

electron with another electron on the same ion

 

.hv

 

8
E — 393 J (S E 1 18
spin _ U2 pd l. 2) ( ' a)
Since de>0, the coupling is antiferromagnetic.
Now we assume de<0, the p electron associated with Mnl

will have spin down, p' will have spin up and Mn will have spin

2
down - the interaction is antiferromagnetic if de<0, let
AE(t)+00 and AE(s)+U. Consequently
E = -§—33- (J )(S -S ) (1 18b)
spin U2 pd 1 2 '
but now J <0 so
pd
E - sz IJ |(§ § ) (1 1e )
spin ‘ U2 pd 1 2 ' c

which is an antiferromagnetic coupling. We therefore express
the superexchange in the form of equation 1.18 and note that it
is always antiferromagnetic.

It is customary8 to define direct exchange and superexchange

as

o _ * 2
Jij — f§i(1)wj(2)e /r12wj(1)wi(2)dr (1.19a)

Jij>0, always ferromagnetic and

2
s _ _2b
J1] — ——2 ldel (1.19b)

U

Jij<0 always antiferromagnetic.
The Hamiltonian is written

H.. = (IJ?.|-J?.)§.-§. * (1.20)
l] (13' l] 1 3

Of course there exist many other exchange interactionsa,
of either sign, but these rarely dominate the magnetic prOperties.

Theoretical calculations of J are very difficult so one usually

 

-
v1

’9

a

’I

 

considers J to be an experimentally determined quantity.

B. AnisotrOpy
It is a fact of life that magnetically ordered Single crys-
tals are anisotropic. Although many of their properties can be
described purely on the basis of exchange, we are particularily
interested in their anisotrOpy. I will consider three sources
of anisotrOpy: dipolar, crystal field, and spin-orbit inter-

actions.

1. Dipolar anisotropy
The interaction between two magnetic dipoles a distance

r apart is

_ 3 +.+ - +.A +.A
Hij — 1/rij[ui uj 3(ui rij)(uj rij)] (1.21)

The magnetic moment of an electron is

H _ HV V
ui - uBgi Si (1.22)

Let us assume an isotrOpic g tensor and define
uv _ 2 2 -3 uv _ u v _
where ng is the direction cosine of Eij with the u axis. The

9,10

total Hamiltonian can now be written to lowest order

H . = E. 2 CRYSRSY (1.24)
dipolar 1] u,v 13 1 3

If one knows the structure of a crystal with magnetic ions,
one assumes a magnetic arrangement and performs the sums of

equation 1.24 on the ubiquitous computer. A typical calculation

might be over 10,000 neighbors with 75 per cent of the interaction

 

\

.u»

 

10
being due to nearest neighbors.

2. Crystal field anisotrOpy

CryStal field anisotrOpy is intimately connected with the
spin-orbit interaction, but I feel it warrants individual dis-
cussion. The coulomb interaction between each unpaired electron
and the charge distribution of the crystal can be described by
an electrostatic potential V(?). The charge distributions sur-
rounding the magnetic ion may overlap the electron and analysis
of this rather hairy problem is called ligand field theoryll.
It is much easier, and still very educational, to consider the
neighbors as point charges, a consideration I shall designate

as crystal field theory. This has the advantage that V(;) can

be expanded in spherical harmonics Y£m(6,¢).

V(r.e.¢) = iiafimrzyzm(e.¢) (1.25)

The symmetry Of the crystal field resides in the Alm' Because
the electrons will be in nearly atomic orbital 2' states, the
£>£' terms in the expansion will be negligible. The spherical
harmonic y2m(6,¢) is a spherical tensor of rank R, order 22+l.
The spin Operator 5: is a spherical tensor Operator Of rank one
and order three, the elements being 32, 8+, and 8-. The Wigner-
Eckart theorem justifies an expansion Of Y2m(6,¢) in a linear
combination of the products Of 2 spin Operators. In the absence
Of applied magnetic field, time reversal invariance allows us

tO eliminate Odd product terms in our expansion. The crystal
field can be written as a bilinear expansion of spin operators,

the second order part can be written

a. = S.- ..-s. (1.26)
l 1 ll 1

 

tn:

ll

3. Spin-orbit anisotropic effects
Suppose we express the interaction of an electron spin with

the field produced by its orbital motion in a potential V(;) aslz

HS = hZ/szCZE-(TV(r) x E) = AI-S (1.27a)

In a magnetic field we add a Zeeman term and the total Hamil-

tonian is
H. = Ali-E. + U (Ei+2§.)-fi (1.27b)

Assuming that the states Of the unperturbed system can be
reasonably described by orbitally nondegenerate wave functions,

a second order perturbation Calculation gives

Heffective = 2 -u guvHuSv _ A2Auvsusv
1 u,v B

- UgAuvHqu-. (1.28)

where
gUV = 2(6W - 1A“V)
represents the admixture Of orbital angular momentum into a

Spin only ground state.

Auv _ 2 (OIRUIn)(nI£VIO) (1.29)

n#o En-Eo

n represents excited states and 2AA1N is called the g-Shift, Ag.

The second term Of equation 1.28 is the single ion aniso-

tropy. Note that, as mentioned in section 2, the Auv reflects

the crystal field symmetry. The wave functions we used can be

written13

In) = IFIY) ISIMS)

12

where F is the irreducible point group symmetry representation
and is orbitally nondegenerate.

The most interesting effect of spin orbit interactions is
their ability to couple with superexchange.to produce anisotrOpic
exchange effects. Moriyal4 has done a rather general calculation
Ofthis effect but I prefer to examine a simplification employed
by Nagamiya et a16. Assume that the unperturbed states for two
magnetic ions are crystal field split orbitally nondegenerate

states. Let our perturbation be

v = A(Ii.si) + 1(Ij-sj) - J(Si-Sj) (1.30)

The second and third order energy corrections are, respectively,

3 +

 

 

 

+ + + +
H.. = D..-(S.XS.) + S.-K..-S. (1.31)
13 1] 1 j 1 1] j
where
r J . o I. m. Jm. o I. m. \
B = 11(2 ml( I ll 1) - Z ‘3( I 3' 1)I (1.32)
ij Lmi EOi-Emi mj EOj-Emj I
HV )2 (l uvZ nn nn uv uv
= '- f- . + . - . + o o
Kij 2 136 n(Fl P3 ) (P1 P3 )} (1 33)
Jm. (0|!LPIm.) (m. IQYIO)
rpv = i 1 1 1 g 1 (1 34)
1 (EOi - Emi)

and Jm is the superexchange between the i th ion in its m th
1 1
excited state and the j th ion. Comparing equations 1.34, 1.32

and 1.29 we may roughly estimate D and K as
D = g9 J K 2 ($3)2 J (1.35)

The Dij-(SiXSj) anisotrOpy is called Dzyaloshinsky-Moriya

antisymmetric exchange anisotropy because it is antisymmetric

 

13

under exchange Of coordinates. The Kij term is called anisotropic
exchange and is symmetric.

The D-M interaction is particularly interesting because it
exists only when the crystal symmetry is low. Moriya14 has de-
vised criteria for determining the direction of 5 based on
crystal symmetry. Consider atoms 1 and 2 located at lattice
sites A and B, the line connecting them is AB.

(i) When a center Of inversion is located at the point
halfway between A and B,

B=o
(ii) When a mirror plane perpendicular to AB bisects AB,
D is parallel to the mirror plane
(iii) When there is a mirror plane including A and B
D is perpendicular to the mirror plane
(iv) When a two-fold rotation axis perpendicular to AB
passes through the midpoint Of AB
D is perpendicular to the two-fold axis
(v) When there is an n-fold axis along AB
+

D is parallel to AB

The total Hamiltonian for our system is written

H = Hex + Hdd + HDM + HCF + HAK + Hz (1.36)

The terms are respectively isotropic exchange, dipolar, D-M
antisymmetric exchange, crystal field anisotropy) anisotropic

exchange and Zeeman interactions.

C. The general second order interaction

The general second order interaction between a spin on a

lattice site i with a spin on lattice site j is given by6'9'lo

14

X uv u v
o. = - .0803. O
1] Uvall 1 J (1 37)

Let us decompose the tensor Ag; into symmetric and antisymmetric

partsls'16
uv _ 1 uv vu
Aij(s) — 2(Aij + Aij) (1.38a)
uv =‘1 uv _ vu
Aij(a) 2(Aij Aij) (1.38b)
where

AP? = APY(s) + APY(a)
1] 13 13

We define the isotrOpic exchange by the trace of A:;(s)

- uv
Jij — 1/3 tr(Aij(s)) (1.39)

The anisotropic part Of the symmetric exchange is

uv _ _ uv

The antisymmetric coefficient is defined by

bi. = APY(a)e (1.41)

13 13 uvA
where Euvl is the completely antisymmetric tensor Of rank 3.
The Hamiltonian is written

1
3.. = -J..§.-§. + E..-(§.x§.) + S.-K..-S. (1.42)
13 13 1 J 13 1 J 1 13 3

which represents the decomposition of a second order Heisenberg

Hamiltonian. Note that when i = j

H. = §.-x .-s. (1.43)

which is a general second order crystal field interaction.

Finally,

 

 

'5

w (1.44)

D. The question Of generality
The Heisenberg Hamiltonian has been successful to the point
of exceeding one's best expectations in describing, at least
qualitatively, a host Of different magnetic effects. There
exists only a handful Of materials that require more than second
order terms to explain their gross features. The crystals
and CoF which have orthorhombic coordination and d

2 2
orbitals permitting fourth order anisotropy are very well under—

14an , FeF

stOOd on a basis Of second order terms only6.

The most important failure of the spin Hamiltonian occurs
when the crystal field interaction is too weak to raise the
excited states significantly above the ground state, a situation
common in rare-earths. Even more disastrous is the non-Kramers,
or even number Of electrons, case in which the ground state

may be orbitally degenerate and all bets are off.

 

II. MAGNETIC ORDERING

The theory Of magnetic ordering is fraught with difficulty,
in fact the most complicated problem yet tO be solved exactly
is the two dimensional Ising latticel7. The fact that one's
beginning Hamiltonian is an approximation makes one reluctant to
worry about exact solutions. However, before discussing the

various magnetic ordering theories, I wish to examine a very simple

magnetic system.

A. The simple paramagnet
Consider a system Of N identical noninteracting spins with
total angular momentum S and magnetic moment us. The Hamil-

tonian for one spin in an applied magnetic field is

+

+
H = -US'H = ”uSzH (2.1)
If we let usz = guBm, then
Hm = -guBmH (2.2)
The partition function is
if ’ ‘~
ZS = m=—s exp(guBmH/kBT (2-3)
which we can write
. 28+1 ' . x
23(8) = S1nh -§§— x) /51nh (EEI (2.4)
where
x = guBSH/kT (2.5)
The magnetization is
M = NguB<m> (2.6)

1‘

l7
. . 18
wh1ch can be wr1tten

M = NguBSBS(x) (2.7)

BS(x) is called the Brillouin function.

Bs(x) = 13§§11 coth(x(2$+1)/2$) — %§ coth(x/ZS) (2.3)

The susceptibility Of this system is

2 2
dM N(gUB) S

x = 55 = kT

BS'(X)

 

(2.9)

The zero field susceptibility we Obtain by letting H+0

and expanding BS(x) for x<<l.

BS(x) = (S+1)x/3S (2.10)
Bé(x) 2 (s+1)/3s (2.11)
Consequently
x0 = Ngzu§S(s+1)/3kBT (2.12)
'Written in the form
x0 = C/T (2.13)

it is known as Curie's law for paramagnetic susceptibility.
Since <m> = <sz>

<sz> = SBS(x)
(2.14)

or <Sz> = 0 for H=0

B. The Ordered State
A.more realistic appraisal Of N spins in a crystal will

suggest that one must include interactions between spins. In

18

chapter I we discussed the Heisenberg Hamiltonian as a general
second order spin-spin interaction. For simplicity, the present
consideration Of magnetic ordering may use only the isotrOpic,
exchange part Of the Hamiltonian. Given a system Of magnetic
ions we ask the Obvious questions: at what temperature does

the system order and what is the nature of the ordered state?

1. Molecular field theory
The MFT was employed in the earliest solutions to magnetic
ordering problems. The interactions of all the ions in the crys-
tal with a given spin are replaced by an effective field. This
. . . + + + .
18 equivalent to lett1ng <Si¢Sj> = <§i>o<sj> and 1gnores all
correlation effects.

Consider the interaction of the i th spin

1

H. = -§.-§J..-§. (2.15)

1 1 J 13 3
or

+ +

Hi = -guBSi°Hi (2.16)
+

E = '1_’ 23 .go (2017)

i guB j ii 3
where H1 is the molecular field. The average Of S is

+ _2 + )3 39°13
01 - 1 Si exp{Si-HiguB/kBT}/i exp{§i HiEET} (2.18)

The molecular field approximation is made by letting

I
E. = —l— E J..-O. (2.19)
1 guB J 13 3

Equation 2.18 is written

+ + ++
oi — iBS(guBni-oi/kB'I-) (2.20)

I v
v

I

19

I a + O
The disordered state is represented by 0:0. If we are in the
ordered state, near the ordering temperature Tc' 0 will be small

so we eXpand the exponentials and keep the first order termslg.

+ +
j ij oj — 3kBTC/S(S+l)oi 3 (2.21)
This coupled set of equations is solved by letting 31(R) repre-
sent the i th sublattice ion in the primitive cell at R. We

now must sum over primitive cells

.(§,§')-3j(§') = 131(8) (2.22)

where
A = 3kBTc/S(S+l) (2.23)

Suppose we let the u th component of the spin be written

,->->
a 1k R (2.24)

1»:

(fi) = 02(0)e

where the k's are apprOpriate prOpagation vectors consistent
with periodic boundary conditions. Equation 2.23 is now writ-

ten in component form

j,§. J““(o, R' )0; (0)e ik’fi' = 102(0) (2.25)
let

€§;(i) = g. J§;(o,§')eii'§' (2.26)

j :ug:;(k)ag (0) = 102(0) (2.27)

j 2” {§"V(k) - 6““61j1(ii} 03(0) 5 o (2.28)

Solutions to 2.27 exist if the determinant is zero

uv + _ uv + =
detlgij(k) 5 eij)(k)| 0 (2.29)

20

The transition temperatures are prOportional to the eigenvalues

of the Fourier transform Of the exchange Operator.
1(E) = 3kBTc/S(S+l) (2.23)

Let us consider the following example, suppose we have a
simple cubic lattice with lattice parameter a. Let J be iso-

trOpic and nearest neighbor only. Considering two sublattices

0 J[cos(kxa)+cos(kya)+cos(kza)]

HE)

J[cos(kxa)+cos(kya)+cos(kza)] 0
The eigenvalues of this matrix are
1(k) = iJ[cos(kxa)+cos(kya)+cos(kza)] (2.30)
Consequently, the ordering temperatures are given by

Tc = S(S+1)J[cos(kxa)+cos(kya)+cos(kza)]/3kB (2.31)

Since the system will order in the mode with the highest transi-
tion temperature, we maximize the eigenvalues with respect to I.
If J>0 then i=0, corresponding to ferromagnetism. If J<0 then
+

k = (l,l,l)n/a, corresponding to antiferromagnetism. The ordering

temperature is
To = 3(s+1)|J|/3kB (2.32)

The general molecular field theory that we have discussed
here is particularly poor for large crystal field anisotrOpy
because it assumes <SzSz> = <Sz><Sz>. If one suspects crystal

field terms tO be large, a better approximation Should be usedzo.

21

2. Green function theory
The main fault Of molecular field theory is that it neg-
lects correlation and short range order effects- Green func-
tions are statistical mechanical generalizations Of the concept
Of correlation. For a general review of Green functions I recom-

22

mend a paper by Zubarev21 Or books by Kadanoff and Baym and

Abrikosov et a123.
The double-time temperature dependent retarded Green func-

tion involving two Heisenberg Operators21 A(t) and B(tl) is

<<A(t);B(t')>> = “19(t-t')<[A(t))B(t')l> (2.33)

where 6(t-t') is the step function, <> denotes the thermal aver-
age and [] denotes a commutator. We can Fourier transform this

quantity into a function Of E = hm, whose equation Of motion is

 

_ 1 .
The correlation function Of A and B is
. <<A;B>> . - <<A;B>>
, _ Lim m w+ie w-
<B(t )A(t)> - 6+0 I-.. w7kBT
e - l
(2.35)
-- - u
x e lwIt t )dw

For simplicity we choose the isotrOpic Heisenberg Hamiltonian

1 z ->
H = --2- ijJijSi sj (2.36)

Choosing A and B as spin Operators, equation 2.34 becomes

- 2S 2:
E<<S ;S >> = 5? dgh - J {

+
9 ffg
(2.37)

+ z - z +. -
<<sfsg,s >>E <<sfsg,sh>>E}

22

where g, h denote lattice Sites and S is the average value of
82. The equation will be solved in the Tyablikov or random

phase decoupling approximation where

+ z - _ z +. -
<<sfsg,s >>E _ (sg><<sf,sh>>E (2.38)

Using this decoupling scheme the equation Of motion isz4

IN

S->>

+
(S f; E

><<S

(Q
KIN

- 2
E<<S ;s >>E = dgh - f Jf

k)

+
g N
(2.39)

<Sz><<S

+ fg f

J ;S >>

E

med
Q +

We now assume we have n sublattices and define Fourier transforms

of the Green functions
.+

+ o
for Egand Rh on sublattice l and

Gi(R,E) = 2 <<S

+ iE-(fig—fih)
9.h 9

;s'>>e (2.41)

for R9 on sublattice i and Rh on sublattice 1.

We define the Fourier transform of the exchange Operator as
+ _ z ii~(§ 1 )
gij(k) ‘ fig-fih Jghe .9 §h (2.42)

for R9 on the i sublattice and Rh on the j sublattice.
we define the parameter 6i = <S;>/S for R9 on sublattice i,

and

I
U M

E = . gij(0)010j (2.43)

Equation 2.39 can be written in matrix form25

E {gij(i) - (E -ejE/2n)eij}cj(i,s) = +513.”Tr (2.44)

co

 

 

 

23

In order to find the ordering temperature we need only 61(E,E)

evaluated at E=0. If 11(E) is the i th eigenvalue of Eij(i),

 

then
-11‘“ 1
G (k,0) = -—1— .§ . _ (2.45)
1 2n n 1-1 A1(E)'§
and
Tc = Séfi+1I i (2.46)
B <Gl(k,0)>k

There are two points worthy Of comment. The average ex-
change interaction Of equation 2.43 is defined assuming that
all magnetic sublattices are crystalOgraphically equivalent
(called a Bravais system). The molecular field result for this

system can be Obtained by setting Ai(i)=0.

3. Other methods
Molecular field theory gives a good qualitative descrip—
tion of many magnetic systems but has two serious shortcomings:
quantitatively, MFT is only an order of magnitude theory and it
contains no discussion Of effects based on correlation between
spins. The MFT takes no account Of short range order and fails

tO predict the correct behavior of the magnetization near T=0 and

T=Tc26. GFT is much better quantitatively but suffers from a

complexity requiring elaborate approximation schemes such as

27 28

Callen decoupling and moment conserving ..7 These are essent-

ially all temperature theories, Green function theory becoming

29

equivalent tO spin wave theory at low temperatures. Some

other methods are:

(a) The Lyons-Kaplan3o or generalized method Of Luttinger

31,15

and Tisza is a method for finding the classical ground

24

state configuration by minimizing the Hamiltonian subject to a

”weak constraint"

2+,+ _):2
n,v nv Snv) — stnv (2'47)

or
z 2 I E - z 62 52 (2.48)

a - —
n,v nv nv nv n,v nv nv

The ground state spin configuration is found by solving the
eigenvalue problem for the Fourier transform Of the exchange Oper-
ator subject to conditions on the anv'

(b) Probably the best available method for calculation
transition temperatures involves high temperature expansions32.
At temperatures above the ordering temperature the thermodynam-
ical quantities are expanded in powers of J/kBT. Given the

Heisenberg Hamiltonian with Zeeman interaction

Z = tr[exp(-H/kBT)] (2.49)
32(1) ~ ()

x = -—— k T nz 2.50
3H2 B

Defining reduced susceptibility and temperature we find

on

- _ 2 n
x — 1/3 S(S+1)n=oan/6 (2.51)
where
Q = -%—-§-x (2.52)
N9 “8
6 = kBT/J (2.53)

The real work is involved in finding the an's. For a simple

cubic ferromagnet with nearest neighbor interactions only,

32

Rushbrooke and WOOd quote the relation

25
kBTc/J = (5/96)(z-1)(11$(S+1) -l) (2.54)

for z nearest neighbors.

I

The high temperature expansion is exact in the limit Of
including enough coefficients (for the interaction assumed).
Ordering temperatures by HTE are 30 per cent to 50 per cent

lower than the MFT values.

 

III. THE MAGNETIC PHASE DIAGRAM

Antiferromagnets have very interesting behavior in applied
magnetic fields. Suppose we have an.antiferromagnet with iso-
trOpic exchange and uniaxial anisotrOpy L. The energy Of the
system with applied field H in the easy, or parallel, direction

is, using MFT33

2

e" = -%x"(H")2-NLS (3.1)

where N is the total number Of spins. The energy for H perpend-

icular to the antiferromagnetic axis is
J. 11. 1’2
5 = -5x (H (3.2)

Since antiferromagnets in the ordered state are characterized
by xt>x", when H = H" becomes large enough, the perpendicular
configuration will be energetically favorable and the spins will
flOp perpendicular to the field - called spin-flOp. The crit-

ical field is found by equating the energies
6": e" (3.3)

which yields

 

HSF =VIZNLSZ/(X‘L-x")' (3.4)

If one continues to increase the field, the spins will saturate
and the system is said to be in a paramagnetic state. We are
interested in investigating in detail the behavior of antifer-

romagnets with a more general anisotropic interaction.

26

..'

 

Let us

27

A. The Classical Energy

consider the prOperties Of an antiferromagnet at

T=0. we will rewrite the Hamiltonian, in the molecular field

approximation, as a free energy and solve the problem classic-

ally. We assume a simple Hamiltonian but one which includes

each type Of second order anisotrOpy.

where
Hex
HaL
HaK
HD
Hz
where Kii =

tonian as a

H = Hex + HaL + HaK + an + Hz (3.5)
= -l §.J,,§.-§. isotropic (3.6a)
2 13 13 1 3 exchange
= -%L' i SIS: uniaxial (3-6b)
anisotrOpy
= %.i. 1.8:8? anisotrOpic (3.6c)
3 3 3 exchange
= I'ED Di (S. sz_ stz) Dzyaloshinsky- (3.66)
j j 3 Moriya anti-
symmetric exchange
= -9uBfi°i§i Zeeman (3.66)
interaction
J.. = 0 and D.. = -D. We now express the Hamil-
ii Ii 31

free energy, considering the spins tO be classical
34

vectors interacting with molecular fields .

' — =
e — E/NS S 2{J cos(a1a2)
-lL(coszo +cosza ) + K cosa cosa
2 l 2 l 2
+ D cos 6 Sin(a1fo2)} -guBS{ (3.7)
Hx(sinolf Sinaz)cos 0 + Hy(sinaif sindz)sin 0

+ Hz(cosa + cosa2)}

1

28
where N3 is the number Of Spins per sublattice, 01 denotes sub-
lattice A, aeA, and c2 denotes sublattice B, beB (Fig. l).

X Z

J = -aeAJab = -b€BJab (3.8a)
L = L'(1-1/2s) (3.8b)
K = z Kab = Z Kab (3 8c)
aeA beB '
D = 2 Dab = Z Dab (3 8 )
aeA beB ' c
We define the molecular fields as
HE = JS/guB
HL = LS/guB
(3.9)
HK = KS/guB
HD = DS/guB

The angles are expressed in a more convenient set by the trans-

formation

a = a + 6

(3.10)

a n + a - ¢

2

The energy is expressed as a dimensionless quantity by dividing

out the exchange field

e = E/NJS2 = -cos(2¢) - hL[cos(2¢)coszo

+ sin20] - hK[cosza - sinzel
(3.11)

- hDsin(2¢)cOS 0 — thsin ¢ cos 0 cos 0

- Zhysin ¢ cos a sin 6 + thsin 0 sin a

29

Figure 1. Definition of the angles describing the spin array

30

 

 

Figure l

31

where hL = HL/HE, etc.
Our problem is now thermodynamical - we must solve the equa-

tions of equilibrium and stability. The equilibrium equations are

 

 

 

-§%— = 0 (3.12)
i
The stability equations are
2
2 2 2
3 e 3 e 3 e
‘ "rr?—— = (n..n.) > 0 (3-13)
2 2 an.an. S i 3
ani anj i j
where
a = (¢Iale)

B. The Zero Temperature Phases
The zero temperature phases of the uniaxial antiferromagnet
(hD=0) have been studied in the molecular field approximation6’34,
in the spin wave approximation35, and.in the RPA and Callen

36 Mere recently37'38'39,

decoupled Green function approximations.
the solutions to equation 3.11 have been considered in the limit
Of hD<<l. However, three subjects need to be investigated in

detail: the nature Of the spin flop discontinuity, the effects

Of large hD and the paramagnetic boundaries.

1. The zero field equilibrium configuration
‘If we let h=0 and evaluate equation 3-12 for a, 6 and o

we find the equilibrium conditions are

and

_ 1 —1
90 - Etan [{ZhD/(Z + bx + hL)} (3.14b)

32

The Spins lie in the xz plane, perpendicular to the D vector,
"antiferromagnetically" aligned on the Z axis but tipped an angle

¢o toward the X axis. This produces a net moment

M: = sin do

in the X direction

2. The parallel configuration
If we let E = (0,0,h) the applied field is parallel to the
easy axis,1rom symmetry we see 6:0. The spin flOp transition
described in the beginning of Chapter III is actually a first
order transition. The first order phase transition is charac-

36 by discontinuities in the energy or magnetization and

terized
a magnetic hysteresis. There is no unique spin flOp field but
a region in which both the AF state and the SF state are stable.
The upper boundary of this region is hSh and the lower is hsc'
If one increases the applied field from zero, the AF state re-
mains stable up to hsh' analogous to the superheated liquid-gas
transition. If one then reverses the process, reducing the field,
the SF state remains stable down to hsc' analogous to the super-
cooled gas-liquid transition. The energies Of the AF and SF
states are equal at the thermodynamical transition hth implying

that if h <h<hsh, the AF state is metastable, and if hsc<h<hth,

th
the SF state is metastable. Let us examine the equations Of
the parallel configuration. The equilibrium equations are

ae/3¢ = [2+hK+thos(2a)]sin(2¢)—2thOS(2¢)+2h cososina = 0 (3.15)

ae/aa = [hK+thOS(2¢)]sin(2a)+2h sinocosa = 0 (3.16)

33
The stability function is
S(¢,a) = {[2 + hK + thos(2o)]cos(2¢) + 2hDsin(2¢)

-h sin ¢ sin a}{[hK + thos(2¢)]cos(2a)

(3.17)
-h Sin ¢ sin aI-{thin(2¢)sin(2a)
2
-h cos 0 cos a} = 0
If cos (d)#0, o and a are related by equation 3.16
sin a = -h sin ¢/(hK+thos(2¢)) (3.18)

The transition from AF to SF states, called the super-
heated transition, is defined by S(a,¢)=0. The solution for

hD=0 is well known and is given by

 

hsh(0) = 4R2+hx+hL)(hK+hL) (3.19)

The hD=0 case is characterized by a=¢=0 for h<hsh(0) and
a=-%-for h>hsh(0). When th0, the net moment is pulled from the
x axis, increasing in magnitude as it approaches the z axis.
Therefore, when hnfio, h<hsh(hD), a and o are not zero. As h
approaches hsh' Ial becomes larger and approaches a critical
angle ashIhD) (Fig. 2). When h exceeds hsh' then a=-%-with an
accompanying discontinuity. The hnio equations were solved by
computer (Fig. 3) using Newton's method for three unknowns
(Appendix A).

The spin flOp state is defined by letting a=-%u The bound-
aries of this phase occur at S(-%,¢)=0. When hD=0, there are
two phase boundaries, a first order spin flop tO antiferromagnetic,

called the supercooled boundary, and a second order spin flap to

34

Figure 2. The superheated critical angle ash(hD) vs. hD with

hK=0 and hL=0.1

ash (DEGREES)

80

 

 

 

l l 1 1 l

l

l

 

l

 

.02 .04 .06 .08 .l0

.l2

“0

Figure 2

.l4

.l6

.l8 .20

 

36

Figure 3. The Spin flOp critical fields vs. hD with hK=0 and
hL=0.l

 

 

.45

.40

.35

.30

 

 

I l I I I— l l I l T
hsh
"" hm
\\
- \ _ "" hsc
\ \
.. ‘\ \\
.\ \
_ .\ \
.\ \
I" .\\
- '\\
o\$
b ‘\.»
\o
.. \.
L
l 1 n 1 I I 1 1 1 I
.02 .O4 .06 .08 .IO .l2 .l4 .l6 .I8 .20
h0

Figure 3

 

38

paramagnetic boundary at higher field. The hD=0 solutions are

 

hsc(0) = (2+hK—hL) /(hx+hL)772+hx+hn) (3.20)
hp(0) = 2 + hK - hL (3.21)

Consequently, the Spin flOp state is stable for hsc<h<h;.

The tho solutions were found numerically (Fig. 3). The
paramagnetic transition, which corresponds to ¢=%, does not
occur at finite fields, since the D-M interaction will lower the
energy Of the Spins if they cant slightly. TO see this, examine
the energy when h is large, but finite, and ¢=%,-5. From

equation 3.15 we see that 5:52.80 the energy) to order hg/h, is

e = 1 + hK - hL - 2h - hDG

No matter how large h is, 6#0 will give a lower energy.
Experimentally, one almost always Observes the thermody-

namic transition defined by e = s This is because for

AF SF‘
fields in the metastable regions, a Slight disturbance foments
the transition. One usually argues that nucleation centers in
the crystal prevent metastable states from being long lived4o.
The procedure for finding the thermodynamic critical field hth

is tO define

and solve for A=0. When hD=0, the solution is

 

hth(0) = {(2+hK-hL)(hK+hL) (3.22)

which is equivalent to equation 3.4. The hD#0 equations were

solved numerically (Fig. 3) and as one would expect ath<ash'

39

Figure 4. A comparison Of numerically calculated hSC with the

approximation Of equation 3.27

40

Or. on on. mm. om. mm. 00. mt Ct mm. on.

a;

mu. ON. 9.

0..

mo.

 

 

 

-

u 4 q q — 4 a -

zo_._.<_2_xomn_d<
2072.50.30 (SEEM—>52

J -

amp.— III

on: I

1

4

 

 

0..

0m.

OSLI

Figure 4

41

The critical fields Obey hsc<hth<hsh but

hth = Vfischsh

only when hD=0.

Suppose we let 6:0 and a=-%q then if hD<<1 and hL<<l, we

can solve equations 3.15 and 3.17 to Obtain an approximation

for hsc(hD).
(2+hK-hL)sin(2¢)- ZthOS(2¢)-2h cos ¢ = 0 (3.23)
hK + thos(2¢) — h sin ¢ = 0 (3.24)

Assuming ¢ is small, equation 3.23 gives
Sin(¢) = (h+hD)/(2+hK-hL) (3.25)

It is important to notice that the term (h)sin(¢) is second

order in 0, SO we write equation 3.24
. 2 . __
hK + hL 2hL31n ¢ - h Sin ¢ - 0 (3.26)

Substituting sin(¢) and neglecting h: terms we get, assuming

hD<hL,

 

hsc(hD) = (-hD + «h: + 4h:c(0) )/2 (3.27)

In Figure 4 we have a comparison Of the numerically calculated

solutions and the approximate solution Of equation 3.27.

3. The perpendicular configuration
The magnetic field applied perpendicular to both D and the
easy axis, E=(h,0,0), results in 6=o=0. When hD=0, a second
order paramagnetic transition occurs at

h;(0) = 2+hK+hL (3.28)

42

As in the previous case, hD#0 destroys the second order paramag-

netic transition

4. The D configuration
When E=(0,h,0) is parallel to B and perpendicular to the
plane Of the spins (for zero field), we can solve the equations
exactly. For hD=0 the problem is equivalent to the perpendicular
configuration (section 3). For hnfio, the equilibrium equations

are, for c=0

ee/e¢ = (2+hK+hL)Sin(2¢) —2thos(2¢)cose

(3.29)
-2h cos ¢ cos 6 = 0
and
35/86 = 2hDsin¢ coso sine - 2h sin¢ cose = 0 (3.30)
The stability function is
S(¢,6) = {(2+hK+hL)cos(2¢) + 2hDsin(2¢) cose
+h sine sine} {h sin¢ sine + thOS¢ sin¢ cose (3.31)
-{hDsin9 cos(2¢) - h coso cose} 2 Z 0
Equation 3.30 yields the immediate result
cose = (hD/h) cosd sine (3.32)

We see that the magnetic field pulls the net moment an angle 0
from the x axis. The solution to equations 3.29 and 3.31 is

6=¢=§ at a critical field

 

2 I

D )/2 (3.33)

.L = .1 ‘L 2
hp (hp(0) + Jmpwn +4h

43

5. Discussion Of the effect Of hD on the zero
temperature phases

The antisymmetric canting interaction has had an inter—
esting effect on our system. The net moment produced by the
canting is rotated toward a magnetic field-. For h applied par-
allel to the easy axis, the Spin flOp transition is still
first order, but as hD increases the transition occurs at great-
er angles and the hysteresis is reduced until the metastable re-
gion becomes extremely narrow. The effect of hD is tO reduce the

spin flOp fields; we can see this by examining equation 3.25

sin ¢ z(h+hD)/(2+hK - h (3.25)

L)
Imagine that hD=0, hL<<1 and h=hsc(0)+dh, where 6h<<1, so that
the spins are in the Spin flOp state. we can simultaneously

decrease h and increase hD without changing ¢p consequently,

for small hD

h (h

sc 0’ ' hSC(0) m-hD

The paramagnetic transition occurs at h=w unless E is
parallel to 3.1 If E is parallel to D the.D—M-interaction is
weakened as 6+; and the hD term in the energy Of equation 3.22
becomes second order in (hD/h) and isoverpowered by lower order

terms in h.

C. The Temperature Dependence Of the Critical Fields
and the H-T Magnetic Phase Diagram
One can calculate with little difficulty the temperature
dependence Of the critical fields for a uniaxial antiferromagnet

in the molecular field approximation. The quantitative results

44

(All be inaccurate but we can get a gOOd picture Of the qualita-

tive behavior .

We assume a Hamiltonian of the form

_ 1 Z + .+ _ 1 ,2 z 2_ .2+
H — 2 IJIijSi sj 2 L i(si) guBfi isi (3.34)

l. The paramagnetic boundaries
Consider an antiferromagnet in the ordered state in an ap-
{flied field H. The effective fields acting on the respective
sublattices are, where M0=nguBS

+effective _ + _ + +

(3.35)

+effective _ + _ + +

The equilibrium condition for H perpendicular to the easy axis

 

gives 41
+effective _ +effective _ +
H1 — H2 — (HE + HL)M1/Mo (3.36)
+ + . . . 41
where M1 = -M2. The AF to paramagnetic tran31tion occurs at
L —
HP(H) - (2HE + HL)Ml(0,T)/M.0 (3.37)
If T=TN(0), then41
2 § _1_
Ml(0,T) = MO«(10(S;1) . (l-T/TN)2 (3.38)
L3(2$ +zs+1),

We can express the field dependence of the critical temperature as

TN(0)-TN(H)
TN (0)

 

= B[H/H;(0)]2 (3.39)
where

B = (3/10)(2s2 + 25 + 1)/(S + 1)2 (3.40)

45

A similar treatment for H parallel to the easy axis gives

TN(0)-TN(H)

 

 

_ 1 2
TN(0) — 38[H/Hp(0)l (3.41)
for H<HSF and
T (0)-T"(H)
N N _ " 2
TN(0) — BBIH/Hp(0)] (3.42)

for H>HSF. We can also express this as a temperature depend-

ence Of the critical field, for T=T

N
e
HP(T) = Hp(0)[(l-T/TN)/BI l (3.43)
. _ , _ 2
Hp(T) - Hp(0)[(l T/TN)/3B] (3.44)
For T20, we write, assuming H>HSF
Hp' (T) = Hp' (0)Ml(0,T)/Mo (3.45)

2. The Spin flop boundaries
If we neglect the hysteresis effects, we can write the Spin

flop critical field
1

HSF = [ZNSLSZ/(XJ'anz (3.46)

If we examine the equations we used to derive this result, we
see that the anisotrOpy term is actually the difference of the
L'<(Sz)2> term in the Hamiltonian for the spin flOp and anti-

ferromagnetic states.

L52 = L'[<(sz 2>

AF (3.47)

- <(sz)2>éF]

Since the spin flop corresponds tO a spin axis reorientation, let

<(sz)2>S =<(sY)2> = §IS(s+1)—<(sz)2>AF1 (3.48)

F AF

46

Therefore
2 . z 2 .. 1
LS = (L /2)[3<(S ) >AF-S(S+l)] = L S(S-§)F(T) (3.49)
Yosida42 has shown that F(T) can be written

F(T) = (S/(S-%))[(S+l)/S—BBS(x)coth(x/2$)/28] (3.50)
x = JS<Sz>/kBT

and <Sz>=SBs(x). The susceptibility in the AF state in the
moleculariield approximation is given by (see section IV)41

xL -x" = x‘10)(1—T/TN) '(3.51)

so we can express the temperature dependence of the spin flOp

field as
. .1.
2
HSF(T) = HSF(0)[F(T)/(1-T/TN)] (3.52)

Where F(O)=l and F(TN)=0. At T=0, equations 3.8b and 3.49 are
equivalent.
The temperature dependence of F(T) is such that, in the

present approximation, H increases slightly with temperature,

SF

increasing by 12 per cent at the Néel point. In practice, HSF

can increase or decrease with increasing temperature.

IV. THE THEORETICAL DEPENDENCE OF SUSCEPTIBILITY
ON APPLIED FIELD AND TEMPERATURE -

THE ANTIFERROMAGNET

There are a number of techniques for studying magnetic in-

sulators: nuclear and electron spin resonance43, specific heat44,

45, and susceptibility to name a few. The

neutron diffraction
primary advantage of susceptibility measurements is their sheer
simplicity and minimal equipment requirements.

Our goal is to theoretically describe the behavior of sus-
ceptibility as a function of fi and T so that we can determine
the interactions dominating a given crystal by interpreting sus-
ceptibility data. The three parameters of interest are T, H,
and the orientation of H with respect to the magnetic axes.

We will limit our treatment to the two sublattice antifer-
romagnet with second order anisotrOpy. In order to lay the ground-

work, we will first consider the zero field susceptibility and

then the field dependence.
A. Zero Field Susceptibility

l. Susceptibility as a function of temperature for H=0
we will be considering only the antiferromagnet, so we

write the exchange as a positive quantity in the Hamiltonian

1 Z

_ l_£ _ ,__ z 2 _ +.2
— 2 ijJijsi 53. L 2 i {(51) } guBH i§i (4.1)
The partition function Z is defined as
Z = tr[exp(-H/kBT)] (4.2)

47

48

The susceptibility for small H is

x = (kT/H)(aZ/3H)/z (4.3)

a. The paramagnetic susceptibility is found by expand-

ing Z in terms of H/kBT<<l, which is the condition that TN<<T.

In order to make this expansion converge more rapidly, we

write the Hamiltonian as a traceless quantity.

_ + .+ _ , z 2
H. — J si s. (L /2)[(Si )

1 3 -S(S+l)/3] - guBfi.§i (4.4)

The partition function is expanded
z = tr(I) - (l/kBT)tr(H) + (l/kBT)2tr(H2/2) . . . . (4.5)

where tr(I) = S(S+l) and tr(H) = 0. Neglecting terms in

(l/kBT)3 and H2, we find42

NgzuBS(S+l)/3kB

 

 

x =
P T + J 5(s+1)/3kB + (l-3cosze)L'(ZS+3)(ZS-l)/60kB (4.6)

which we write

x = C 2 (4.7)
P T + e + y(l-3cos e)

 

where cos(6)=l for xp and cos(6)=0 for x;. The two constants

are given by

e JS(S+l)/3kB = guBHE(s+1)/3kB (4.8)

.<
ll

_ v =
(28+3)(2$ l)L /60kB guBHL(ZS+3)/30kB (4.9)
Using equation 4.5, we can show that

l/xp - l/xp = (28+3)guBHL/10kBC (4.10)

49

Moriya46 has calculated the paramagnetic susceptibility for

a canted antiferromagnet with Dzyaloshinsky-Moriya interaction
included in H. Although his calculation is for D parallel to
the easy axis, whereas our previous work assumed D perpendi-
cular to the easy axis, one expects the existence of a weak
moment to be the most important factor. The paramagnetic

susceptibility is given by

x; = C"/(T+To) (4.11)
xp = c [1 + (TN-To)/(T-TN)J/(T+TN) (4.12)
N

xp = C Ta/[(T+TN)(T-TN)] (4.13)

where denotes parallel to the easy axis, N denotes parallel
to the net moment and J-denotes the perpendicular to these

directions. The temperatures are given by

To = JS(s+1)/3kB (4.14)

Ta = DS(S+l)/3kB (4.15)
2 2 4

TN — (To + Ta) (4.16)

Note that in the canted antiferromagnet the paramagnetic
susceptibility measured perpendicular to D increases much

faster than l/T as one approaches the critical temperature.

b. The susceptibility in the antiferromagnetic state
is calculated by writing the effective field acting on the i th
spin and expanding the Brillouin function about a small applied

field. The effective field acting on the i th spin is

fi?ffeCtlve = E - H <§.>/s + H.<s?>/s z (4.17)
1 E j 1.. l

50

If we let H = H2, Si and Sj will remain parallel to the z axis,

with thermal average values

<si> = SBS{[guBS/kBT][H-H <sj>/s + HL<Si>/S]} , (4.18)

E

we now expand the Brillouin function6 about H=0, letting

681 = GSj and <Si>o = -<Sj>o ,

<Si> = SBS(xO) + 55x Bs(xo) (4.19)
Combining equations 4.18 and 4.19 we can write
_ 2 _ .
58 - (guBS /kBT)[H + ( HE+HL)5S/S]Bs(xo) (4.20)

Therefore, if we define

em = NguBGS (4.21)
and

x = 6M/6H (4.22)

then the parallel susceptibility is

2 2

n _ 2 . _ .
x - [Ng uB S Bs(xo)]/[kBT+guBS(HE HL)BS(XO)] (4.23)

where x0 = guB(HE+HL)<S>o/kBT. A Similar treatment shows the
perpendicular susceptibility to be approximately temperature

independent
I _
x — NguBS/(ZHE+HL) (4.24)

Since the derivative of the Brillouin function goes to zero
exponentially as T goes to zero, the parallel susceptibility
is zero at T=0. The results for the zero field susceptibility
are summed up in Figure 5 for the antiferromagnet with negli—

gible paramagnetic anisotrOpy.

51

Figure 5. Susceptibility vs. temperature in the molecular

field approximation

52

 

 

2h P

 

 

Q. 0.. t. m._ N... Z 0.. m. 0.. N. w. n. t. m. N. ..
a 1 q 1 JW _ a, q a q q . - q q
4
..x a
1
L
.x
b h b n . n h h . - p — n . b .

 

 

«(0.6!

0020.".

(a. In.
("l)X/x

Figure 5

53

2. Susceptibility as a function of angle for H=0.
Consider a system whose susceptibility is uniaxial.
If a magnetic field is applied at an angle.6 with respect to
the easy axis, then the magnetizations produced will be,

assuming H is very small,

M" = x"Hcos(6) (4-25)

.L

M

The magnetization in the direction of the applied field is

the sum of the components of M" and M; in this direction.

MH

or

Ma

The susceptibility measured in the direction given by e is

XLHsin(6) (4.26)

M"cos(6) + Misin(e) (4.27)

x"Hcosz(6) + XLHsin2(6) (4.28)

x(6) = dMa/dH = X"cos2(e) + x‘sin2(e) (4.29)

B. The Susceptibility in Applied Field

We will first consider the susceptibility in applied field

along the principal axes for the canted antiferromagnet. we
are particularly interested in the directions perpendicular
to D in which we discovered that no second order paramagnetic

critical field exists at T=0. Since the behavior for T>TN is

that of a simple paramagnet, that is a gradual saturation with

increasing H, we will limit our discussion to the ordered
state at T=0. The behavior of the susceptibility with the
applied field off the principal axes is more difficult so we

will consider it only for the uniaxial antiferromagnet.

54

l. Susceptibility vs. H for the canted antiferromagnet
at T=0.
The canted antiferromagnet has easy, medium and hard

axes. The components of the magnetization written in reduced

form.are
mx = sin(¢)cos(a)cos(6) (4.30)
my = sin(¢)cos(a)sin(6) p (4.31)
m2 = —sin(¢)sin(a) (4.32)

where 0 designates the location of the spin plane in the xy
plane,a designates the orientation of the antiferromagnetic

axis relative to the z axis and ¢ is the angle of cant (Fig. l).
The zero field equilibrium of this system is

do = 0

6o 3 0 (4.33)
tan(2¢o)= ZhD/(2+hK+hL)

The zero field reduced susceptibilities are found by evaluating
dml/dhl = x1 (4.34)

in the limit that h + 0.

x: = cosz¢o/[h;(0)cos(2¢0)+2hDsin(2¢o)] (4.35)
y 1 .1

X0 = 2/[hp(0) + hp(hD)] (4-36)
x: = sin240/[hx+thos(2¢o)l (4.37)

The most interesting point is that the parallel susceptibility
x3 is not zero at T=0.
The parallel susceptibility xz is written from equations

4.32 and 4.34 as

55

x" = -sin¢cosa da/dh - cos¢sina d¢/dh (4.38)

If we let hD=0, we can find the exact solutions for three
possible states
i) Antiferromagnetic state

a=0 ¢=0 and

(4.39)
Xir=°
ii) Spin flOp state
a=-n/2 Sin¢ = h/(2+hK-hL)
(4.40)
ng = l/(2+hK-hL)
iii) Paramagnetic state
a = -fl/2 ¢ = w/2
(4.41)
"=0
xp

The susceptibility is zero until the field reaches hSF' it is
a delta function at hSF' then it is a step function out to
h;(0) (Fig.6).

The hD#0 solutions were calculated numerically by computer.
we see that the step function has become rounded (Fig.7) and
x” remains finite until h+¢, indicative of the fact that the
paramagnetic transition has been destroyed by the D-M term.
However, we note that a quasi-paramagnetic transition can
be defined as an inflection point in the susceptibility.
we have calculated dx"/dh numerically and.we see broad maximum
(Fig.8) that becomes narrower as we decrease hD.
The perpendicular susceptibility xx is calculated from

equations 4.30 and 4.34 as
xL’= cos(¢) cos(a) d¢/dh - sin¢ sin(a) da/dh (4.42)

6._

'4

5‘:

 

.1

'(

56

If we let hD=0, the exact solutions for the two possible
states are
i) Antiferromagnetic state

a = 0 Sin¢ = h/(2+hK+hL)

xAF = l/(2+hK+hL) , (4.43)

ii) Paramagnetic state
a = 0 ¢ = n/2

= 0 (4.44)

The hD=0 perpendicular susceptibility is a step function out

to hp(0)(Fig.6). The hD#0 solutions were calculated numerically
by computer. The perpendicular susceptibility is also rounded
(Fig.7) and we can calculate de/dh (Fig.8) for the inflection
point.

The susceptibility xy for hD=0 is the same as the perpendi-
cular susceptibility. Since the system undergoes a phase
transition even when hD#0, the susceptibility in the y direction
goes to zero at h;(hD) of equation 3.33.

The quasiparamagnetic transition which manifests itself
as an inflection point in the susceptibility can be defined
by the criteria

dzx/dh2 | h = o (4.45)
qp

The equilibrium equation for h>hSh can.be written from
equations 3.15 and 3.29 as
hp(0) sin(2¢) - 2thos(2¢) - 2h cos¢ = 0 (4.46)

where hp(0)
hp(0)

hp(0) = 2+hK-hL for h in the z direction

h;(0) = 2+hK+h for h in the x direction

L

57

Figure 6. Susceptibility vs. applied field with hD=O, hK=O
and hL=0.l

58

s

0..

t.

 

N.»

0.n QN 0N tN N.N
d i 4 1 d

0.N 0..

A

1

d

N..

1
m

J 0..

llx

 

Figure 6

 

 

 

L 0..

1 0.N

 

 

Figure 7.

59

Susceptibility vs. applied field with hD=0.05, hK=0
and hL=0.l

 

60

...

 

 

 

 

 

 

 

 

NM on QN 0N tm NM 0N m. 0.. t. N. 0.. m. w. v.
. _ .1. .fl. . 4 a . . .
in.
10..
..m..
(ON
JON
. _ . .4/_. . . . . . . .
Ln
10..
-0.
(ON
...p...p.tpr... .md

 

 

T

Figure 7

61

Figure 8. Numerical calculation of dx/dh vs. h with hD=0.05,

hK=0 and hL=O.l

62

m mnomam

...

 

 

 

 

Nfi O.m 0N 0N QN NN ON m. 0.. t. N. 0.. m. m. .V. N.
. — q q q u q u u . 7 u u u q d
A d A . a q _ . q . 4 u E a J. .r
p — - n p p — . p P p — p b n b

 

 

 

0. 0. “2 "I: N
UP
pr

63

We divide the equilibrium equation by cos(¢)

hp(0) sin¢ — h - thos(2¢)/cos(¢) = 0 (4.47)

and write the magnetization as

m = sin(¢) (4.48)
The susceptibility is then written

x = cos¢ d¢/dh (4.49)
Differentiating the equilibrium equation (4.47) we obtain
an expression for d¢/dh

-1 + {hp(0) - hDsin¢[4-cos(2¢)/cos3¢)] cos¢ d¢/dh=0 (4.50)

From equation 4.49 we express the susceptibility as

x = l/[hp(0)+hDsin¢(4-cos(2¢)/cos3¢)] (4.51)

If we consider hD<<l, we can solve equations 4.51 and 4.45 to
calculate the quasiparamagnetic critical field. This transition

occurs at a field hqp and an angle of cant ¢qp given by

3 _
cos (4q ) - (4/5)hD/hp(0) (4.52)

P

hqp = hp(0) + (3/5)hD/cos(¢qp) (4.53)

For the case hD=0.05, hK=O' and hL=0.l the numerically

calculated quasiparamagnetic points are

hz = 1.990 42 = 74.44°
qp QP

h“ = 2.195 4“ = 74.97°
qp qp

The approximate equations (4.52 and 4.53) give

hz = 2.009 42 = 74.0°
qp qp
hx = 2.212 x = 74.5°
qp ¢qp

64

2. The susceptibility vs. angle in applied field for
the uniaxial antiferromagnet at T=0.

The angle dependence of susceptibility in applied
fields is more difficult to treat than the zero field case of
part A, primarily because the field is strongly affecting the
system. Fortunately, a simple calculation for the uniaxial
antiferromagnet will shed considerable light on the subject.
Let us consider an antiferromagnet with uniaxial crystal field
anisotrOpy, with a magnetic field applied at an angle 6 with
respect to the 2 (easy) axis. The energy at T=0 can be

written from equation 3.11 as

e = -cos(2¢) -hL(cos(2¢)cosza+sin2¢) - 2h sin¢cosasin0

+ 2h sin¢sinacose (4.54)
The equilibrium equations are

[2+thos(20)]sin(2¢) + 2h cos¢[sinacose-cosasin0] = 0

(4.55)
thos(2¢)sin(2a) + 2h sin¢lcosacos6+sinosin81 = 0
The susceptibility is written
The components of the magnetization are
x .
m = cosa31n¢ (4.57)
m2 = -sinasin¢ (4.58)

The magnetization in the direction of the applied field is

m = mxsin(e) + mzcos(0) (4.59)

65

The reduced susceptibility is
x(e,n) = d/dh [sin¢sin(6-a)] ‘ (4.60)

The susceptibility was calculated for various applied fields

using Newton's method on the computer47. The behavior of

the susceptibility can be described in four regions (Fig.9).

Region I 0 < h < hsh(0) AF
Region II hsh(0) < h < hp(0) AF-SF
Region III h§(0) < h < H;(0) SF-P
Region IV h;(0) < h P

The points in the figure are identified by equations 4.39

through 4.44 with hK=0

XA = XXF = O (4.61)
XB = Xgr a l/(2+hL) (4.62)
xC = xgF = l/(2-hL) (4.63)

The reduced susceptibility we have been using can be converted
to cgs units by multiplying by the total saturation magnetiza-
tion

x = NguBSx lHE

cgs reduced

where N is the total number of spins.

66

Figure 9. Numerical calcualtion of susceptibility vs. angle in

applied field

67

 

REG'ON I O<H<Hc|
REGIONH Hc.<H<Hc2(|l)

 

 

 

 

REGION III Hc2(||)<H<ch(-L)
.11
xc HI
1 X3
XAllllllilJllJl
0 30 60 90

9° 9(degrees)

Figure 9

.4-

2.

V. EXPERIMENTAL APPARATUS - THE MEASUREMENT

OF ANISOTROPIC SUSCEPTIBILITY

Susceptibility measurements are well suited to observing
magnetic anisotrOpy because they are inherently directional,
as Opposed to, for instance, specific heat measurements. In
this section, our primary aim is to discuss the apparatus
required to measure susceptibility and vary the three
parameters mentioned in chapter IV: temperature, applied field
and orientation. First, we will attempt to clear up one of the
great mysteries of physics - what are the units of suscepti-

bility?
A. The Great Unit Debate

If one considers the magnetic permeability of a substance
and wishes to express it in terms of the susceptibility, then
(using c.g.s. units throughout)

u=1+x (5.1)

where x is a dimensionless quantity measured in "electromagnetic

) .

units per cubic centimeter" or e.m.u./cm3. In the literature

one can find susceptibility expressed in the following units:
cm3/mole, cm3/gm, c.g.s. units/mole, c.g.s. units/gm, e.m.u./mole,
e.m.u./gm, dimensionless, and arbitrary units. One can obtain

a consistent relation among these systems with the following

three definitions. The total magnetic energy of a system is

E = affi-Hdv or erg = gauss-Oersted cm3 (5.2)

The magnetization is written

M = NguB<S> . (5'3)

68

a.

.1!

'7-

 

69

and the Bohr magneton is defined as

20

= 0.927 x 10- erg/gauss (5.4)

“13
Since 9 and S are dimensionless and

_ N(erg/gauss)
X — dM’dH Oersted

we see that the units of susceptibility are N(erg)/(gaussOersted)
which can be converted using equation 5.2 to the equivalent
unit: N(cm3) where the units of N have yet to be chosen. Since
we have previously stated that e.m.u./cm3 is-a dimensionless
unit, we let

1 e.m.u. = l c.g.s. unit = 1 cm3 (5.5)
The susceptibility as expressed in equation 5.1 is dimensionless
so we must choose N as the number of spins per unit volume.

The following table shows possible other choices for N and

the resulting susceptibility units, assuming one spin S per

molecule.

ChOice of No No/Mw Nop/M.w
N

Units of #/mole #/gm. #lcm3
N
. 3 3

units of cm /mole cm./gm. l
x e.m.u./mole e.m.u./gm e.m.u./cm3

Where No is Avagadro's number, M.W is the gram molecular

weight, and p is the density in gm/cm3.

70

B. The Measurement of Susceptibility

There are at least four methods of measuring magnetic
susceptibility: Faraday or Gouy balance48, vibrating magneto-
meter49, nuclear magnetic resonance4 and ac mutual inductanceso.
Since we have used only the mutual inductance technique for the
measurements to be discussed, the description will be limited
to this method.

The ac mutual inductance method utilizes the fact that
the mutual inductance of two concentric solenoids is proportional
to the magnetic susceptibility of a material within the solenoids.
If we let AM be the change in mutual inductance of the coils
upon introduction of a sample of mass m, then

AM 51—”. x (5.6)
where w is a constant that is a characteristic of the coil.

The susceptibility coils used consist of a primary and two
Oppositely wound secondaries. The number of turns on the
opposing secondaries is adjusted to make the mutual inductance
approximately zero. The change of mutual inductance produced
by the introduction of a sample into one secondary is measured
with a Cryotronics51 Model 17B electronic mutual inductance
bridge, Operating at 17 Hertz. With the range switch (R) on
0.1, the coarse dial (c) and the fine dial (f) balanced, the
mutual inductance is given by

M = Rcf(50x10-5) microHenries (5.7)
The susceptibility of a sample is given by combining equations

5.6 and 5.7

x = Ron/m(50x10'5)cm3/gm (5.3)

71

which we write
x = Rcf K/m cm3/gm (5.9)
The constant K is a property of the coils being used.
Since the coils are never exactly balanced to zero without a
sample, and since their balance changes with magnetic field
and temperature, it is necessary to take readings with the sample
in (fi, sometimes called "full") and with the sample out (f0,
sometimes called "empty") of the coils. Consequently

x = Rc(fi-fo)K/m cm3/gm (5.10)

The constant K is determined by measurement of a known
paramagnet, in this case Ferric Ammonium.A1um,.whose suscepti-

bility is given by52

_ -3 3
XFAA — 9.02 x 10 /T cm /gm (5.11)

The constant K can be determined to at least 1% accuracy. If
we assume that the practical limit of measurement with a typical
set of coils is 10 fine units with c=10 and R=0.l, then in terms
of susceptibility of a 0.1gm sample this practical limit (see

Table I for K) is

7 3

x . = 2 x 10_7 cm3/gm or about 10-

e.m.u./cm
min

C. The Parametric Apparatus

We have discussed the theoretical behavior of susceptibility
as a function of the parameters H, T, and 0. we now wish to
describe the design, construction, and use of the apparatus

necessary to vary these parameters.

*v

v:

 

 

 

 

 

72

1. Susceptibility vs. temperature.
All experiments were done in conventional Helium four

cryostatssz'53

between 4.8 and 1.2 Kelvins.. The temperature
was determined from vapor pressure measurements using the
T 58 temperature table.

Measurements of susceptibility vs. temperature are
generally done with the hope of attaining accurate values for
chi. It is essential that each data point shall consist of
a balance taken with and without the sample in the coil because
the empty value of the coils is a function of temperature,
applied field, and time. The zero field measurements are made
with quartz sample holders eliminating the need for background
corrections (see part C). The zero field data is taken with
coil #2, which has a small inside diameter and enough turns of
wire to make it very sensitive. The susceptibility vs. T
measurements done in applied field are, naturally, made in the
superconducting solenoid apparatus (see section 3) where one is

usually interested in the transition temperature or field rather

than absolute susceptibility.

2. Susceptibility vs. orientation.

In order to vary the orientation of a crystal while
measuring the susceptibility one needs apparatus capable of
rotating a crystal inside the susceptibility coils while immersed
in liquid helium. The crystal rotation device (Fig. 10) shall
be referred to as the rotator and the small axle on which the
crystal is mounted shall be referred to as the Egtgg. The

rotator essentially consists of two perpendicular threaded

73

Figure 10. The crystal rotation device

74

 

 

=6 .

 

Figure 10

75

shafts, a vertical shaft that is passed through an O-ring vacuum
seal and a horizontal shaft (the rotor) to which the crystal is
attached. The vertical shaft has a simple screw thread that mates
a thread cut around the circumference of the horizontal rotor with
a tap set perpendicular to the rotor axis. When the vertical
shaft is turned by hand at room temperature, a corresponding
rotation is produced in the rotor. The turns ratio for the
rotator in coil #5 is 29:1, the original rotator for coil #6 was
19:1, the present rotator is 29:1. The rotors are made of nylon
with fiberglass-epoxy bearings to allow for thermal contraction.
The body of the rotator is made of a low susceptibility material
(see part C) to reduce the background. The vertical shaft is
indexed at the dewar head, the 29:1 ratio allows one to easily
change the orientation of the crystal by less than one degree.
Measurement of susceptibility vs. angle for various tempera-
tures is an excellent method of determining the principal magnetic
axes of the crystal and the type of ordering it undergoes.54
Many of these experiments in zero field can be understood from

equation 4.29 for the uniaxial antiferromagnet
x(0) = x"c0520 + x’sinze

where the easy or hard axes correspond4to extrema in the sus—
ceptibility. Using equations 4.10 and 4.23 we see that the
parallel axis susceptibility is a maximum in the paramagnetic
state and a minimum in the antiferromagnetic state. Two
independent rotations allow one to determine the easy direction
with an accuracy that depends on the amount of anisotrOpy and

the actual magnitude of the susceptibility.

Table I. Physical parameters

76

of the susceptibility coils.

 

 

 

 

 

 

Coil Number 2 5 6
Function x vs. T x vs. x vs. H
Sensitivity, x 108 1.35 2.20 2.73
Type Secondary double triple triple
Primary Turns 2093 3285 ' 1100
Gauge #30 #34 #36
Length 7" l4" 4"
I.D. 5/8" 15/16" 5/8"
R,300Kelvins 35 220 100
Secondary Turns 10,946 34,208 8,255
Gauge #36 #36 #38
Length 3" 8" 2"
0.D. l" 2" 29/32"
R,300Kelvins 1100 4400 1100
Rotator Ratio none 29:1 29:1

 

77

Since most of the rotator is actually in the susceptibility
coils and since it has some susceptibility, measurements of chi
are not extremely accurate. The best x vs. 0 data is taken
with coil #5 (Fig. 11) which is very long and has uniform sensi-
tivity over an approximately one centimeter length. This is
necessary becuase the rotors produce some up and down motion

of the sample.

3. Susceptibility in applied magnetic fields.

The magnetic fields for our experiments are generated
by a superconducting solenoid wound on a coil form of Synthane55
type G-ll fiberglass epoxy. The solenoid itself has an i.d.
of 2.61 cm, and o.d. of 5.02 cm and a length of 10.0 cm. The
wire is 0.0178 cm diameter Niobium - Zirconium with copper
coating and nylon insulation. The solenoid is wound with two
pieces of wire with a total length of 1700 m giving 14,712

turns. The field at the center of the solenoid is given by56

‘.a + «a: + b2 )

H = [41rIAb/(10aw)] 2n n 2 2 2 ,~ (5.12)
a + «a + b i
. l l =
where A is the filling factor
2
A — N(1r/4)(dw)/[L(a2 al)] (5.13)
and
dw = 0.0178 cm the diameter of the wire
aw = 2.48 x 10_4cm the cross sectional area of the wire

a1 = 1.31 cm the inside radius of the solenoid
a = 2.51 cm the outside radius of the solenoid
b = 5.00 cm the half length of the solenoid

I is the current

Figure 11.

78

Coil #5 with rotator

 

 

 

—-d

,79

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 11

'.

«e

80

The result of this calculation is

H = 1720 Oersted/ampere
The magnetic field is measured with a Bismuthmagnetoresistor57
that is calibrated in place (Fig. 12) against the spin flop
critical field58 of copper chloride mounted on the rotor.
This calibration gives

Hmr = 1680 t 10 Oersted/ampere

The homogeneity is estimated from an RCA Magnet Design
Aid59 as 0.3% on a 1 cm diameter sphere.) The magnet is
usually capable of reaching 12.0 amperes before quenching
producing a field of 20.0 kiloOersted.

The spin flOp transition can be quite narrow in angle,
for instance in copper chloride the spin flOp transition can
only be observed with magnetic fields applied within one degree
of the easy axis, so measurements of parallel and perpendicular
susceptibility require very careful orientation. Consequently,
the apparatus for measuring susceptibility in applied field
includes a crystal rotator similar to that described in section
2 (Fig. 13).

The present rotators have only one axis of rotation, so
one must mount the crystal with the easy axis in the plane
perpendicular to the axis of rotation. Assuming that this is
done prOperly, the crystal is rotated in some constant magnetic
field Ho' If we examine the results of the theoretical calcula-
tion of chi vs. 9 in applied field (chapter IV, part BZ) we see

that if no is slightly greater than H the susceptibility will

SF

be strongly peaked around the easy axis with a precipitous dip

at the easy axis (see Fig. 9). As one approaches HSF from

81

above, the peak becomes nearer to the axis and the dip becomes
sharper, providing an excellent guide to orient the crystal.

We have used this apparatus primarily for plotting the
H-T phase diagrams described in chapter III. In general, the
boundaries are easier to see at lower temperatures while spin
flOp boundaries, representing greater changes in the magnet-
ization, are much easier to see than paramagnetic boundaries.
Spin flOp boundaries are usually found by orienting the crystal
as described above and measuring susceptibility-as a function
of applied field at constant temperature. The very strong
peak in the susceptibility (the theoretical results are shown
in Fig.'s 6 and 7) occurs at the spin flOp field. The para-
magnetic boundaries, especially at temperatures near the Néel
point, can only be observed by measuring the susceptibility
as a function of temperature at fixed field, the transition
occurring at the point of maximum dx/dTGo.

The apparatus has two characteristics that make absolute
measurement of susceptibility impractical: (the empty value of
the coils is a strong function of field, and the noise induced

by current instabilities and flux jumps greatly reduces the

accuracy.
D. Low Susceptibility Materials

Accurate measurement of susceptibility is dependent upon
the ability to measure the susceptibility of the sample of
interest and only the sample of interest. Our attempts to
subtract background due to extraneous materials such as

rotators or sample holders degraded the results by an

82

Figure 12. Calibration of resistance vs. H for the magnetore-

sistor

RESISTANCE (OHMS)

83

 

 

    

1 l

1J4

14141.;

I

8
o

14

1

NM
G
00

l

8

§§§§§

8

7

§§§§§

0|

IOO

O
4

 

 

 

11111
2 35“ 5 6

 

7
H

 

 

9 9 )0): )2 (314mm ITIBISZC'
(KILO-OERSTED)

Figure 12

i???

84

Figure 13. A cross section of the apparatus for measuring

susceptibility in applied field

85

ROTOR
SHAFT

 

L—n :1

 

     
      
   
   

   

    
   
 
   
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SUSCEPTIBILITY 1' II SUPERCWDUCTING
COILS I" , ‘ ll 3‘ SOLENOID
I. -” E II E
'. - .' E I I :
y ' 'f'? II E ; '(
_T 3 ll : 9' “'.
I; '-'. . II : ~ I
l 5 ll : «pl
' : E 5 " " s I
I "1 = : . : ~ .. .:_ , ~‘ '
'1 ‘ ' ‘.-".-.I
E f 271
: : ,f7
: g .{j
SECONDARY .,::...;' SAMPLE
2 '4i HCWJMER
.._-..1

.""_'.‘. 0A 0".
g 7., s.‘." .5. e"
O. ~‘n 4.. .0
. a. . I
‘0 w" ”I
. '.‘ . .,
. ffipw a]
: 1"‘?.--.. 4: 3 .‘.
. :03‘."."{.‘.‘?.‘ ~‘\‘
‘..".( ~£‘._~o :. - - ~.'
. ‘0‘" ‘e, . ‘v «’
e '. , ' .> ' . .
-..:>‘ t) ”I... l

PRIMARY

4 . - .51.
4 "

 

 

 

 

 

I. """'vv " I'D OOCOCOO'O'IO'VIIIIUICOOUUUI‘0.

Figure 13

86

objectionable amount. Consequently, material in the coils

must have very low susceptibility. In general, metals should

be avoided because of eddy current effects—andunonemetals should
be easily machinable (eliminating quartz from.most.app1ications).
The following is a list of the approximate susceptibilities

of various materials at four Kelvins, those available from
Synthane Corporation55 are denoted with an R. The.materials

are divided into four groups according to desirability.

SUSCEPTIBILITY OF MISCELLANEOUS MATERIALS AT

FOUR KELVINS

I. 0<x<l x 10.6 cm3/gm

l. Quartz

2. xxXP-IR* . paper phenolic
3. C-M* cotton melamin
4. EP-22* paper epoxy

5. Black nylon

6

II. l<x<5 x 10- cm3/gm

1. White nylon

2. XP* paper phenolic
3. XXP* paper phenolic
4. C* cotton phenolic
5..L* cotton phenolic

6. Hardwood(Birch)
7. Pure epoxy

8. Epibond 100A

87

III. 5<x<10 x 10’6 cm3/gm
l. G-ll* glass epoxy
2. Glastic

3. Pyrex

IV. x>10 x 10.6 cm3/gm

1. xx* paper phenolic

2. L—XB* cotton phenolic and graphite
3. G-3* glass epoxy

4. G-5* glass epoxy

5. N-l* nylon phenolic

6. Lavite

The best material by far is EP-22 epoxy and paper which
has almost unmeasureable susceptibility and is an excellently
machining material, second best is C-M. The superconducting
solenoid and all susceptibility coils were wound on forms of

6-11 glass epoxy which is an unusually strong material.

VI. EXPERIMENTAL RESULTS AND DISCUSSION

Susceptibility measurements have been made on RbZMnCl4'2H20,

CsZMnCl4-2H20 and CuClzflZHZO. We have measured zero field

susceptibility as a function of temperature and angle in

CszMnCl4-2H20 and szMnC14-2H20. We have also measured suscepti-

bility in applied field for all three crystals and in the cases

of Cs and szMnCl4-2H20 we have used this data to plot the H-T

magnetic phase diagram. Our goal is to compare these results

with the theory and to evaluate the molecular field constants.

In addition, we have taken from the literature data on MnC12°4H20,
a very thoroughly investigated material, and applied the molecular
field theory to these results. For crystallographic details

concerning these crystals, see Appendix B.

'2H 0

A. Cs MnCl '2H 0 and szMnCl4 2

2 4 2

Cs MnCl °2H O and Rb MnCl '2H

2 4 2 2 4 2
crystals with one chemical formula per unit cell. Nuclear

O are isostructural triclinic

magnetic resonance61 and specific heat62 measurements indicate
that they order antiferromagnetically with TN = 1.84 K for the

cesium salt and TN = 2.24 K for the rubidium salt.. They are

two sublattice antiferromagnets, which our susceptibility
measurements indicate are essentially uniaxial. The manganese
ion resides in the center of a distorted square of chlorines

with oxygen atoms above and below. The easy direction was
determined by measuring chi vs. 0, (Fig. 14) to be approximately
10° from the O-Mn-O direction, in agreement with earlier measure-

61

ments by Spence et al. The angle dependence of the zero field

susceptibility is well represented by equation 4.29.

88

89

The temperature dependence of the zero field suscepti-
bility is typical of antiferromagnetic ordering (Fig.'s 15,16).
We can compare this data with the molecular field result
shown in Figure 5. Note that the molecular field calculation
for the paramagnetic susceptibility included the assumption

that T>>T and does not apply near the transition.

N
The paramagnetic susceptibility has been fit to the

I

Curie-Weiss law (equation 4.7)

x = C/(T+F) cm3/mole (6.1)
where the theoretical value of C is
c = Ng20325(s+1)/(3k3) = 4.39 cm3K/mole (6.2)

The experimental results (Table II) show a large deviation from
this value, indicating that the measurements were made sufficient-

ly near T to be affected by both short range order and neglected

N
terms in the expansion42.

The paramagnetic data can be used with equation 4.7 to
obtain a value for the crystal field anisotrOpy

_ u _ i:
Q — l/xp l/xp (28+3)guBHL/(10kBC) (6.3)
Solving for the anisotropy field

HL = lOkBCQ/((28+3)guB) (6.4)

and using the experimental values of C and Q we find the
anisotropy to be approximately 8 kOe (Table II) for both
materials. The exchange field can be calculated from the
experimental value of the perpendicular susceptibility at
the transition temperature. we solve equation 4.24 for the

exchange field (Table II)

90

Figure 14. Susceptibility vs. angle for szMnC14-2H20 in the
paramagnetic and antiferromagnetic states with zero

applied field

91

 

l'l'l'l'l'l'l'l'I'
00r- 00 00000;
'8 ”000000 0 0 000000 0—
I600—°—- ° °oo°°° °°.°oo°—
o
+-o o -'
3|4OO: O ‘:
:2 o
‘=I200*--'o ‘—
§:I(D()CI"" o o ""
Zé _ o o 'I
3 800— O o O —
x I— 000 000 q
600— —
4001— ——1
200— ' —'
OEI l I l I l I 1 I LI 1 I l I l I [—

 

 

 

0 20 60 IOO I40 (80 220 260 300 340 380
9 (degrees)

Figure 14

Figure 15.

92

Zero field susceptibility Vs. temperature for
CsZMnCl4.2H20 in the parallel and perpendicular

orientations

)

|()lc.-

0.70

x (co/mole)
.0
.h
0

0.30

O.|O

 

_'l l l I II I l l I l l l l I I

 

 

 

O I.O 2.0 3.0

T (deg. K)

Figure 15

94

Figure 16. Zero field susceptibility vs. temperature for
szMnCl4-2H20 in the parallel and perpendicular

orientations

95

 

 

 

 

I I I I I I I I I I I I I I I I I I I I I T I I
0.7 ' — _
)— -(
r—- -(
0060 — 630°C” 0000 —'
0° 00
—- O O 0 w o 00 -
0 o
._ : 00 .4
— °o H
o °o
-— 0 q
A
Q) _' o 1
6 - ° 7
E 0.40 — o -
\ - .
O
(D _ -
3 L. o .1
x — o c-I
0.30 _ —
-- -+
- <9 ..
0.20 — .—
I— -1
0.I0 -— _
—- —(
)— d
I4 I I I I I I I I I I LA I I J I I I I I I I
0 LC 2.0 3.0 4.0

T (deg..K)

Figure 16

96

HE = NoguBS/(Zx‘(TN)) - HL/Z (6.5)

Since these materials have essentially spin only 9 values of
2.0, we have assumed that HK and HD are zer063.. Neglecting
the anisotropic term in equation 6.5, Smith andeFriedberg63
have calculated HE = 20 kOe from susceptibility data for

CsZMnCl4-2H20. Excellent agreement with our valueHE = 16.7 kOe
can be obtained by subtracting HL/Z = 3.8 kOe from Friedberg's
value.

We have measured the H—T magnetic phase diagrams for both
materials (Fig.'s 17, 18) using the temperature dependence of
the susceptibility in applied field. In this temperature
range (1.2 - 4.2 K) we have found only paramagnetic boundaries.
An example of the chi vs. T data in applied field for
CszMnCl4'2H20 is shown in Figure 14. A log-log plot of the

phase boundaries (Fig.'s 20, 21) shows excellent agreement

with the T35

dependence of the critical fields predicted in
equations 3.43 and 3.44.

we have extrapolated the phase boundaries to zero tempera-
ture and used equations 3.21 and 3.28 (assuming HK=0) to

evaluate the molecular fields.

HE = (116(0) + H;(0))/4 (6.6)

1" ll
HL = (Hp(0) - Hp(0))/2 (6.7)

The exchange fields are in good agreement with the paramagnetic
values but the anisotropy fields are almost a factor of two
larger (Table II). From the phase boundary results we find

the reduced anisotropy field

97

Figure 17. Magnetic phase diagram of Cs MnCl4-2H O. The pre-

2 2
dicted spin flop boundary is represented by a dotted

line

35

04
O

N _
U"

H (KlLO-OERSTED)
a 8

5

98

 

 

 

Figure 17 T (K)

H; "
.. .. -— — H3
\' 3'
I I J I I
0.5 LO |.5 2.0 2.5

 

99

Figure 18. Magnetic phase diagram of szMnCl4-2H20. The pre-
dicted spin flop boundary is represented by a dotted

line

100

 

 

 

 

 

l.5 2.0 3 2.5

TIK)

Figure 18

IO

0.5

 

 

35"

30—

2 2

_
.b

5m...mmw0..0)..v.. I

 

l0-

101

hL = HL/HE

is approximately 0.8 for both materials. A calculation of the

zero temperature spin flop field from equation 3.22

_ _ k
ch — ((ZHE HL)HL) (6.8)
gives
ch = 16.7 kOe CsZMnCl4-H20
ch = 20.0 kOe RbZMnCl4-H20

If we assume the temperature dependence of equation 3.52, we

find the theoretical value of the triple point to be

H

tp 17.4 kOe Ttp 0.55 K CSZMnCl4'H20

H

tp 21.0 kOe Ttp 0.83 K RbZMnCl4'H20

The predicted spin flop boundaries are shown as dotted lines
in Figures 17 and 18.
The anisotrOpy of these two salts has several unusual
features:
1. The easy axis is approximately 10° from the local
coordination axis formed by O-Mn-O.
2. The anisotrOpy appears to be much larger in the
ordered state than in the paramagnetic state.
Friedberg63 has shown that the dipolar anisotropy

is negligible in the paramagnetic state but this

calculation has not been done in the ordered state.

102

Figure 19. Susceptibility vs. temperature in applied field for
CszMnCl4-2H20, typical data used for obtaining

phase boundaries

 

 

 

00.. 00.. 0k... 00.. 00..
q a q q a a 4 - a a q a q q 1 u a
4 :00m
.. I
. .
m .. 000
1 .. .
.. mo. 00 o
.. 00x on e
.. mo. 0 . . 00..
p p — P b p . — p b p p — n P p -

 

 

(5.1.an AHVBIIQHV) X

Figure 19

104

Figure 20. Extrapolation of the phase boundaries to T=0 for

~2H O

CszMnCl4 2

IIUOLO-OERSTED)

105

 

lTllllll] IIIIIIIT[11 111111

I IIII

I

I

     

HDC)

I I lIlll]

I

 

ES

1 1.111111

I

 

 

1 IIIIIHI I 111111d 1,1 1g111u

(DI JC) II) IC)
' TN’TIK)

 

Figure 20

106

Figure 21. Extrapolation of the phase boundaries to T=0 for

RbZMnCl4'2H20

107

 

I IIIIII

I

5
o
I IIIITTTI

'0'

_

H (KILO- OERSTED)

I

 

I IIIIIII IT I IIIIIII

6/0/2-

 

lJIIllll llIlIlLII

[111)”:

I

_

 
  
 

I

III]

I ILIIIII

 

 

.0)

ID I
TN -T (K)

Figure 21

.0

I I IIIII

IO

108

3. The anisotropy is an order of magnitude greater64

than any obtained by paramagnetic resonance of Mn++
ions as impurities in diamagnetic lattices. Unfortun-

ately, no diamagnetic host has been found with the

same local coordination.

B. CuClz'ZHZO

COpper chloride has been studied extensively and is believed
to be a four sublattice antiferromagnetss. We have chosen this
crystal in order to study the behavior of the susceptibility
in applied field particularly near the spin flop critical field.
COpper chloride is a good choice to test our zero temperature
theory because at 1 Kelvin T/TN = k.

We have measured chi vs. T at constant field from the spin
flOp region at low temperatures to the antiferromagnetic region
at high temperatures. The prominent peaks in the susceptibility
(Fig. 22) occur at the spin flop-antiferromagnetic boundary
and are in sharp contrast to the small effects seen in the
paramagnetic transition of CsZMnCl4°H20 (Fig. 19). we can
extrapolate the transition to the limit of zero amplitude (Fig.
23) to estimate the triple point temperature as T = 4.31 K

tP

in excellent agreement with the values Ttp = 4.31 K and Htp =

8.50 kOe quoted by Butterworth and Zide1158.

C0pper chloride has orthorhombic symmetry and may have a
D-M interaction, so we take great liberties in applying our
uniaxial theory to it. We have measured the zero field sus-
ceptibility in the perpendicular direction and find

2

x*(TN) = 3.3 x 10‘ cm3/mo1e

109

Table II. Experimental data and molecular field calculations

 

 

 

 

 

for CszMnCl4-2H20 and szMnCl4'2H20.
CSZMnCl4'2H20 szMnC14'2H20

TN K 1.84 2.24
x*(TN) cm3/mole 0.68 0.58
0 mole/cm3 0.16 0.16
c" KcmB/mole 5.2 5.7
C"Kcm3/mole 5.1 5.8
F" K 4.0 6.5
F‘ K 4.9 7.8
H;(0) kOe 20.0 25.0
Hé(0) kOe 48.0 56.0
HE paramagnetic 16.7 20.0
kOe phase diagram 17.0 20.3
HL paramagnetic 7.7 8.5
kOe phase diagram 14.0 15.5
Theoretical

HSF(0) kOe 16.7 20.0
Htp kOe 17.5 22.5
T K 0.53 0.77

1113

110

Figure 22. Susceptibility vs. temperature in applied field for

CuC12-2H20

111

3.. ...

 

I

T

T

 

ones

00¢
E

0N0

0N.»

. o.d.

 

m.03. 0&0 .

e05. 0mm 4
00x .140 o

P

034
-

00¢ 00¢

L

l

l

l

00.
on.
com
08 x
co» m
on... e
co.» w.
8.. w
IA
08 n
on... m
com .4...
one
ooe

 

 

09.

Figure 22

112

Figure 23. Extrapolation of the amplitude of the spin flop

transition vs. temperature for CuClZ-ZHZO

113

 

 

 

 

 

600 -

O O O O O
O O O O O
In <1- 0 N '-

(SIINO AHVHIISHV) XV

Figure 23

405 4J0 415 4.20 4.25 4.30

4.00

TIK)

114

Figure 24. Magnetic phase diagram of CuC12-2H20 (Ref. 58)

115

 

 

 

 

I I . I I
paramagnenc
_. .. o.
w
1
I. .. O.
.2 . '8
.‘.
0
£5. '#
g” o
s 2 N
8 3':
e; 1
S
P -1<:2
I I 1 1 I
8 9 0 IO 0°

(90)) 91;!) OIIaubow

Figure.24

5.0 .

temperature (°K)

116

From Butterworth's phase diagram (Fig. 24) we estimate the
zero temperature spin flOp field as

ch = HSF = 6.0 kOe

Solving for the exchange and anisotrOpic exchange fields using
equations 3.22 and 4.43 (HL=0 for spin 8) we find

H 85 kOe

E

HK = 0.21 kOe

CuC12°H20

We have measured chi vs. 8 for various fields and fit the
data to the theoretical curves of chapter IV by normalizing
the experimental parallel and perpendicular susceptibility at
zero field to the theoretical zero field values. If the
anisotrOpy is much less than the exchange, crystal field
anisotropy or anisotrOpic exchange give the same angle depend-
ence of the susceptibility so we apply the calculations of
chapter IV by substituting hK for hL' The theoretical values
were calculated assuming
hK = HK/HE = 2 x 10"3
There is surprisingly good agreement between the theoretical
and experimental behavior of the susceptibility in applied
field (Fig. 25).

we have measured x vs. H at constant temperature (Fig. 26)
for this crystal and again the agreement with the theory of
chapter IV is very good. Notice that the susceptibility
remains very small above the spin flOp transition in agreement
with equation 4.40
xgp = Mo/(ZHE+HK)

Here HE is very large.

117

Figure 25. Susceptibility vs. angle for various applied fields
for CuClZ-ZHZO; experimental and theoretical results
normalized to the zero field parallel and perpendicu-

lar values

X ARBITRARY UNITS

N
' I

 

 

00

 

 

 

.A
o H=0
x H=5.5 kOe
A H=6.6 kOe
o H=7.7 kOe
A
A
A
° 8
A
\.
x I N_} A
. . ' 3‘. e
_- ‘ l I 1 1 I L L
l0° 20° 30° 40° 50° 60° 70° 80° 90°
@DEGREES

Figure 25

119

Figure 26. Susceptibility vs. applied field at constant temp-

erature for CuClZ-ZHZO

X (ARBITRARY UNITS)

.5000

120

 

6000

5500

4500
4000
3500

3000

h)
(A
C)
C)

N
C)
C)
(3

I500
I000
500

I

I

I

I

I

I

 

° 4.08I K I)
A 2497K

- |.220K

 

 

 

 

 

I.0 2.0 3.0 4.0 5.0 6.0 TO 8.0 90 IQO
H (KILO-OERSTED)

Figure 26

 

I

121

C. MnC12°4H20

Manganese chloride is a particularly good candidate to
which to apply the molecular field theory. A great deal of
data is available and the phase diagram is quite complete.

The crystal symmetry is monoclinic with rather odd local
coordination. Manganese chloride orders antiferromagnetically
66

at TN = 1.62 K. Rives

diagram (Fig. 27) by measuring x vs. H at constant temperature.

has plotted the parallel phase

An example of his data (Fig. 28) can be compared with the
theoretical results of chapter IV (Fig.'s 6, 7). Gisjman

et al.67 have used proton and electron spin resonance to
determine the perpendicular paramagnetic boundary and Giauque

et al.68

have measured the entire phase diagram employing
isentropic and specific heat techniques. A summary of their
results extrapolated to zero temperature is given in Table III.

Assuming that ch = HSF' equations 3.21, 3.22, and 3.28

can be solved for the molecular fields69
.. -" __ 2 u
HE - slnp<0) Hsp‘°’/Hp‘°’] (6.9)
BL = lSIREN) - HEIOH (6.10)
_ 2 n _ L _ u
“x — HSF(O)/Hp(o) lep(0) Hpco>1 (6.11)

The results of these calculations are given in Table III.

The zero field perpendicular susceptibility and the spin

flop parallel susceptibility measured by Lasheen et al.70 and

66

Rives , respectively, are compared with the values calculated

from equations 4.40 and 4.44 using the molecular fields

122

Figure 27. Magnetic phase diagram of MnC12-4H20 (Refs. 66
and 67)

H (KILO - OERSTED)

I2

DIV-505(1)

123

 

 

 

 

I l l .1

M~CI2°4H20

 

 

 

 

II
2.4

.6 .8 l0 I2 I
T(K)

Figure 27

l l
.4 LG |.8 2.0

124

Figure 28. Susceptibility vs. applied field at constant

temperature for MnC12-4H20 (Ref. 66)

 

 

 

 

 

.0x
.58 .3}
xommofl
oNIv was).

 

 

 

 

00.-

O
O
N

00¢

000

v-OIX (WWW) OX-HX

Figure 28

126

 

 

 

Table III. The zero temperature phase boundaries and molecular

fields for MnClz°4H20

Rives-Gisjman Giauque et a1.

HSF(O) kOe 7.55 7.30 ]
Hg(0) kOe 20.6 19.0
Hé(0) kOe 25.5 . 25.9
HE kOe 11.6 11.5
HL kOe 2.50 3.45
H kOe -0.24 -O.26

127

determined from the Rives-Gisjman phase diagram.

EXPERIMENTAL $HEORETICAL
XX? 1.09 cmB/mole 1.09 c103/mole
ng 1.2 cm3/mole 1.35 cm3/mole

One should note that fix could well be zero to within

experimental accuracy.
D. Conclusions

The molecular field theory developed in chapter III and

IV has been applied to Cs MnCl ‘2H 0, Rb MnCl °2H 0,

2 4 2 2 4 2

20 and MnC12°4H20. The data used.for the first three

was taken in apparatus described in chapter V. The analysis

CuC12-2H

of MnC12-4H20 was based on data taken from the literature.

The results are very encouraging but not conclusive.

Molecular field theory gave consistent results for MnC12°4H20

within the accuracy of the data. In the case of CuC12'2H20

we correctly described the behavior of the susceptibility in
applied field. The theory gave different results for the
4'2H20 from data taken
in the paramagnetic or ordered state. It is possible this

anisotrOpy of CsZMnCl4°2H20 and szMnCl

difficulty can be resolved by a dipolar anisotrOpy calculation
in the ordered state.

It is of particular interest to compare the theoretical
and experimental values for the ordering temperature. Using
the experimentally determined values for H in the molecular

E
field equation (4.8)

TN = guBHE(S+l)/3kB

128

and Rushbrooke and Wood's equation for the two sublattice

32

cubic antiferromagnet with z nearest neighbors (assumed to

be 6) derived by high temperature expansion

TN = (J/ZkB)(5/96)(z-l)(llS(S+l)-l)(l+2/(3zS(S+l))

we calculate the transition temperatures for isotrOpic anti-
ferromagnets. Using Lines20 calculations of TN(L')/TN(0) 2
employing molecular field and Green function theory we obtain

transition temperatures for the anisotropic antiferromagnets

m4
TN(MFT) K TN(HTE-GF) K TN(Exp.) K
MnC12°4H20 1.85 1.60 1.62
CszMnCl4-2H20 2.85 2.53 1.84
RbZMnCl4°2H20 3.45 3.05 2.24
CuClZ-ZHZO 5.70 4.10 4.33

These transition temperatures are calculated using exchange
and anisotropy constants determined by molecular field inter-
pretation of experimental susceptibility and phase diagram
data. The discrepancies for the cesium and rubidium suggest
two things: applying a simple cubic theory to a triclinic
lattice may be overly Optimistic (z=4 greatly improves the
results) and the difference in the anisotropy above and below
the transition implies that our understanding of these crystals
is very incomplete.

We have used molecular field theory expressions for the
phase diagram and susceptibility of the.antiferromagnet to

determine the exchange and anisotrOpy of four magnetic salts.

129

We have tested the consistency of the molecular field by using
these results to accurately predict the behavior of the sus-
ceptibility in applied for CuC12'2H20 and MnC12’4H20. Similarly,
we have predicted the existence of a spin flop transition in
CszMnC14°2H20 and szMnC14'2H20. An experimental test of this
prediction cannot be made with our present apparatus. The
transition temperatures were calculated and, as expected, the
molecular field theory overestimated TN by 30%-50%. .A combina-

tion of high temperature expansion and Green function theory was

also used and the results of TN for CuCl2 20 and MnC12-4H20

were very good. In conclusion, molecular field theory has given

'2H

a reasonable description of the behavior of the susceptibility
of the antiferromagnet as a function of temperature, magnitude

and orientation of applied field.

REFERENCE S

9.
10.

ll.

12.

13.

14.

15.

16.

17.

18.

19.

20.

REFERENCES

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R. M. White, Quantum Theory of Magnetism, (McGraw-Hill,
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D. C. Mattis, The Theory ofiMaqnetism, (Harper and Row,
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P. A. M. Dirac, Proc. Roy. Soc. A123, 714 (1929).

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C. J. Ballhausen, Introduction to Ligand Field Theory,
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J. Kanamori, "Anisotropy and magnetostriction of ferromag-
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M. Tinkham, Group Theory and Quantum Mechanics, (McGraw-
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E. F. Bertaut, "Spin Configuration of Ionic Structures
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L. Onsager, Phys. Rev. 65, 117 (1944).

J. S. Smart, Effective Field Theories of Magnetism, (W. B.
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L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics,
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A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski,
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M. E. Lines, Phys. Rev. 11;, A1336 (1964).

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P. Heller and G. Benedek, Phys. Rev. Lett. 11, 71 (1965).
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D. H. Lyons and T. A. Kaplan, Phys. Rev. 110, 1580 (1960).
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A. H. Morrish, The Physical Principles of Magnetism, (Wiley,
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H. Rohrer and H. Thomas, J. Appl. Phys. 19, 1025 (1969).
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T. Moriya, "Weak Ferromagnetism" in Magnetism edited by
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D. Flood, unpublished.
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G. J. Butterworth and V. S. Zidell, J. Appl. Phys. 29, 1033
(1969).

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4.:

APPENDICES

 

APPENDIX A

NEWTON'S METHOD

Newton's method is a numerical process for finding the
zeroes of sets of functions. If we consider the equations
f(x) = 0 (A1)
Newton's method consists of evaluating f(x) and f'(x) at an
arbitrary point x0 and approximating the solution to A1 by find—
ing the intersection with the x axis of a striaght line con-
structed tangent to the curve f(x) at x0. The line intersects

the x axis at a point given by
l _ _ l
x - xO f(xo)/f (x0) (A2)

Newton's method is usually used in an iterative process where
each successive x' is employed in equation A2 as an x0.

If we have N equations in N unknowns

2

fn(x°,x ,...xN) = 0 (A3)

and we define
i _ i
fn - dfn/dx (A4)
The approximation to a simultaneous root of N equations is
.i _ i_ i
x — x0 x (A5)

where x:L is the solution to the set of N linear equations

Exifi(xo) = fn(xo) (A6)

134

135

The parameter x1 of equation A5 can be represented by

 

 

 

fi fi...fi-l fi fi+l f?
xi = f5 f2 I f2
5; . . . fN . . . f:

det (F)

where F is the matrix formed by the coefficients of f;(xo).
Newton's method is very rapidly convergent but one can have
difficulty inducing convergence to a particular solution of

interest.

APPENDIX B

CRYSTALLOGRAPHY

l. CszMnCl4 2H20 and szMnCl4 2H20

These crystals are grown by evaporation from aqueous solu-

tions of XC1 and MnCl2 4H20. They are whitish pink in color

and fairly stable in air. The structure is triclinic, P1 with

one chemical formula per unit ce117l. The lattice parameters

are
Cs2Mnc14 2H20 szMnC14 2H20

a A 5.74 5.66

b " ' 6.66 6.48

c " 7.27 7.01

deg. 67.0 66.7

" 87.8 87.7

" 84.3 84.8

2. MnCl2 4H20

Manganese chloride tetrahydrate is grown by evaporation of
aqueous solutions of manganous chloride and HCl. The crystals
are reddish pink in color and fairly stable in air. The struc-
ture is monoclinic, P2/m with four chemical formulae per unit

ce1172. The lattice parameters are

MnCl2 4H20
a A 11.19
b " 9.51
c " 6.19
deg. 99.74

136

137

3. CuCl2 2H20

Copper chloride dyhydrate is grown by evaporation of aqueous
solutions of CuClz. The crystals are blue-green in color and
fairly stable in air. The c axis defines.an excellent cleavage
plane. The structure is orthorhombic, Vz(Pbmn) with two chem-

ical formulae per unit ce1173.

CuCl2 2H20
a A 7.38
b " 8.04

c " 3.72

”‘33,

APPENDIX C

COMPUTER PROGRAMS

RUN($)

LGO.

oooooooooooooooooooooo

. PROGRAM ANIs(INPUT.OUTPUT.TAPE61=OUTPUTI

c TMIs PROGRAM CALCULATEs TME EQUILIBRIUM ANGLES FOR A Two SPIN

c SYSTEM MITM GENERAL ANISOTROPY. IT ALSO CALCULATES THE

c MAGNETIZATION. ENERGY. AND STABILITY OF EACH STATE MITR AN

c APPLIED FIELD.

c

c

c

c

c DEFINE FUNCTIONS

c

c AF STATE
EBAF(A989H0HK9HL9HD)=(20HK0HL*COSI2*A))‘SIN(2‘B)-2*HD'COS(2*I)
102'H*SIN(A)'COS(B)
EAAFIAoBOHoHKIHLoHO)=I(HKOHL'COS(2*BI)'SIN(A)0 H'SINIBII'COSIA)
EBAFB(AoBoHoHKcHLoHD)=Z*((ZOHKOHL*COS(2'A)I'COS(2'B)¢2*HD'SIN(2'B)
l-H*SIN(A)*SINIB))
EAAFB(AoBvHoHKoHLoHD)=COS(A)“(H’COSIB)-2*HL*SIN(2*B)”SINIA)I
EBAFA(AoBoHoHKoHLoHD)=2'IH'COSIA)*COS(B)-HL'SINI2*BI'SIN(2*A)I
EAAFA(AoBoHoHKoHLoHD)=COS(A)‘ (HKOHL'COS(2*B))‘COS(A)-SIN(A)“((
1HK0HL'COSIZ‘B))‘SINIAIOH'SIN(B)I
SAF(A980H9HK9HL9HDI = ((2 . HK . HL‘COSIZ'A))’COSI2*BI . 2*MO-SIN
1(2’8) - H’SIN(A)*SIN(BI)’I(HK ¢HL*COS(2*B))‘COSIZ'AI-H‘SINIA)
2*SIN(B)) - (HL'SIN(2*A)’SIN(2'B) - H'COS(A)*COS(B))**2
UAFIAIBIHoHKoHLoHD)I-COS(2*B)-HL9(COS(2*8)*COSIAI*COSIA)OSINI8I*
lSINIB))-HK*(COS(A)’COS(A)-SIN(8)'SIN(B))-HD'SIN(2*B)
2o2»M951N(A)9SIN(8)

c sr STATE
E5E(8.M.MK.ML.MO) = (2 . HK - HL)’SIN(2*B) - Z'HD'COSIZ'B) - 2*
1H'COSIBI
USFIBIHOHKIHLIHDI=-IOSIN(B)‘((ZOHK-HL)*SINIB)-2’H-2*HD'COS(8II
SSFIBoHoHKcHLoHD) = -((2 . HK -HL)*COS(2*B) . 2*MO-SINI208) . H”
ISINIBII'IHK #HL’COSIZ'BI-H851NIBII
E5F8I8.M.MK.ML.MOI s 29(2 . HK'- HL)*COS(2*8) + 4'HD'SINI2‘B)
Io 2*H'SINIB)

c

c PERPENOICULAR STATE

EPIBQHQHKOHLOHDI 3 I2 0 HK 0 HL)'SINI2*B) - Z'HD'COSIZ'BI - 2'H'
1005(8)

SPIBOHOHKOHL’HD) 3 (2 O‘HK 0 HL)’COSI2*B) 0 ZPHD'SINIZPB) O H'SIN
1(8)

138

139

UPIBOH’HK’HL’HD) 31-(20HKOHLI’ICOSIB)IPPZ-HD'SINIZPB) 'ZPH'SINIBI
PARAMAGNEIIC SIAIE

UPMRLIBOHOHK'HLOHDI 3 1 3 "K - HL '2’H

UPHDRIBOHOHKQHLIHDI 3 1 O MK 0 HL '2.H

L 3 0

CHIP 3 1/IZOHKOHLI
CHIAF 3 0

SHZAF 3 0

SHYP 3 0

PI 3 301415926536
HK30.1

HL 3 001

HD30.1

SHD 3 H0

SHK 3 HK

SHL 3 HL

A3000
B3HD/IZOHK0HLI

C 3 8

”3.00005

00 100 J 3 19500
H3H300005

SH 8 H

K = ‘J '

10 ASA'IEAAFIA.BOH9HK0HLIHDI*EBAFB(AOBOHQHKOHLOHDI-EBAF(AOBOHQHKOHLQ
1H0).EAAFBIAOBQHOHKOHLVHD))/IEAAFA(AOBOHOHKOHLQHDI.EBAFB(AOBOHOHKO
ZHLQHDI’EAAFBIAQBIHOHKIHLQHDI’EBAFA(AOBOHOHKOHLOHDI)

B3B-IEBAF(AOBOHOHKOHLOHDI“EAAFAIAOBQHOHKOHLOHDI’EAAF(AOBOHQHKOHLO
‘IMOIPEBAFAIA.B.M.MK.ML.HD)II(EAAFA(A.8.M.HK.ML.HOI°EBAFB(A.8.M.MK.
2HL9HDI“EAAFBIA.BOH9HK0HL9HDI”EBAFA(AQBOHOHKIHLOHDII

K = K01

EAAFX= ABSIEAAFIAOBQHOHK’HLOHD)I

EBAFX 3 ABSIEBAFIAOBOHOHKOHLOHD)I

S 3 EAAFXOEBAFX '

IFIK.GT.SOIGO TO 35

1FI50050000000001)60 T0 10

3S SEAF I S

SSAF 3 SAF‘AQBOHOHKOHLOHD)
SUAF 3UAFIA¢80H9HK9HL0HDI
XSSMZAF

SMZAF 3 -SINIAI'SINIBI
CHIAF 3ISMZAF-X)/0.005

KSKAF 3 K
SA 3 A'18000/PI
SBA? 3 B’lBO-O/PI
K 8 0
BIC
50 B 3 B‘ESFIBOHOHKOHLOHDIIESFB(BOHOHKOHLOHD)
K s K 01

ESFX 3 ESFIBOHOHKOHLOHDI

.15,

“’1! I"‘ l

000

140

S 3 ABSIESFX)
IFIK.GT.SO)GO T0 60
IFIS.GE.0.00001)GO T0 50

60 SESF 3 ESFIBOHOHKOHLOHD)
SSSF 3 SSFIBOHOHKQHLOHDI
SUSF 3 USFIBOHOHKOHLOHDI
SMZSF 3 SINIBI
SBSF 3 8.180.0/PI
KSKSF = K
C38
K80
70 B 3 B-EP(BOHOHKQHLOHDIlIZ’SPIBoHoHKoHLoHDII
K=K91

EPX 3 EPIBOHGHKOHLOHD)

S 3 ABSIEPXI
IFIK.GT.50)GO T0 80
IFIS.GE.0.00001IGO T0 70

80 SEP 3 EPIBOHQHKOHLOHD)
SSP 3 SPIBOHOHK’HLOHD)
SUP 3 UPIBOHOHKOHLOHD)
Y=SMYP

SMYP = SIN(8I
CHIP=ISMYP-Y)/0.005
SBP = 8'180.0/PI

SUPMRL 3 UPMRLIB'HOHKQHL0Hn)
SUPMDR = UPMDRIBOHOHKQHLOHDI
KSKP = K

HRITE OPERATIONS

L 3 L'l

IFIL.LE.0)2000300
200 HRITEI619210)

210 PORMAT(IHI . HK HL HD H EAF BAF
IEP 8P SAP SSF SPY MzAP MZSP
2HIP 0)

L = 60

A K ESF 85F
MVP CHIAF C

300 HRIIE‘619310)(SHKOSHL’SHDOSHQSEAFQSBAF!SACKSKAFOSESFOSBSFOSEPQSBPO

lSSAFoSSSFoSSPoSMZAFoSMZSFOSMYPoCHIAFoCHIP)

310 FORMATIX9F4.292E5.2076039F4.192F7.Zo139F4.19F7.29F4.10F7.206F7.49

IZF9.4)
100 CONTINUE
120 CONTINUE
130 CONTINUE

END

0000000000000000000000

141

FAIRALLQTJO 9 CM441009 P2. FAIRALL
MAP(0FFI
RUN(S)

L00.

0000000000000000000000

10

35

N0 LIST

PROGRAM HSC(INPUTOOUTPUTOTAPE613OUTPUTI
USE(COHIHKOHLOHD)3’IOSIN(CI'((ZOHK-HL)*SIN(CI'2'H-Z.HD.COS(C)I
ESFICOHOHKQHLOHD) 3 (Z O HK - HL)‘SIN(2'CI ' 2'HD'COS(2*CI ' 2’
IHPCOSICI

ESFH(C9H9HK9HL9HDI '2'C05(CI
ESEC(C9H0HK9HL9HD) 3 23(2 9 HK - HL)‘COS(2.C) O “PHD'SINIZPCI
lo Z'H'SINIC)

SSFC(C9H9HK9HLOHDI 3 '2’HL'SIN(2'C)'H'COS(CI

SSFH(C9H¢HK0HL9HDI 3 “SIN(CI

SSF(COHOHK9HL9HDI 3 HKOHL'COS(2'CI-H'SIN(CI

SAF(AOBOHQHKOHLOHDI 3 ((2 9 HK o HL’COS(Z*AII'COS(2’B) 9 2'HD'SIN
1(2‘31 ’ H’SINIAI’SIN(B)I’((HK OHL’COSI2“BII“C0$(2'A)'H'SIN(AI
2*SINIBI) - (HL’SIN(2'AI*SIN(Z‘BI - H’COS(A)'COS(B)I§'Z

L=0

P13361415926536

HK 3 0

HD=0.01

HL3005

SHL3HL

SHD3HD

H = ((Z‘HK-HL)”(HK‘HL)I'°O.S

H=H300001

C '-' ((HK9HLI/(20HK-HL) I.*005

HK=-0001

DO 110 J319150

HK=HK30e01

SHK=HK

K=0
C3C'IESF(CQHQHKQHLQHD)“SSFH(C9H9HK9HL0HDI'ESFH(CQHOHK7HL9HD)P
ISSF(C9H9HK9HL9HD)I/(ESFC(C9HOHK9HL9HD)*SSFHICOHQHKOHLCHD)'ESFH(
ZCQHOHKOHLOHDI*SSECICQHQHKQHLOHD)I
H=H-(ESFC(C9H0HK9HL9H0)‘SSF(CQHQHKQHLOHDI-SSFC(CQHQHKIHLOHDIO
1ESF(C9H9HK9HLOHD)I/(ESFC(COHOHK0HL0HD)'SSFH(C9H9HK9HL9HDI-ESFH(
ZCIHOHKQHLOHDI’SSFC(CvHoHKoHLvHDII

K=K01

S 3 ABS(ESF(C9H9HK9HL9HD)I’ABS(SSF(C9H9HK9HL9HD)I

IF(K.GT.50IGO T0 35

IF‘SOGEOOOOOOOOI’GO T0 10

SE57 3 ESFICOHOHKOHLIHD)

SSSF 3 SSF(C9H0HK9HL9HOI

SUSF 3 USF‘COHOHKDHLOHDI

SCSF=CPIBOoOIPI

KSF =K

SH 3H

WRITE OPERATIONS

L 3 L'1

142

200 URITE(619210)

210 FORMAT(1H1 ' HK HL HD H ESF SSF USF
1‘)
L = 60

300 HRITEIbloJIOI(SHKoSHLoSHDoSHvSESFoSSSFoSUSFoKSFoSCSF)
310 FORMAT(X9F4.202F6.30F6.403F8.49139F863)
110 CONTINUE
120 CONTINUE
END
0000000000000000000000

143

SPENCEo T40 o CM441009 P2. FAIRALL
MAPIOFFI
RUN(SI

LGU.

0000000000000000000000

NO LIST

PROGRAM HSHIINPUTOOUTPUTOTAPE613OUTPUTI
EBAF(AoBoHoHKoHLoHDI=(ZOHKOHL’COS(?’AII‘SIN(2’RI-2'HD'COS(2'II
1‘2“H‘$IN(AI‘COS(B)

SAAFIAOH¢H7HK0HL~HDI3((HKOHL‘COSIZ’BII'SINIAIO H'SIN(B))'COSIAI
EBAFB(AOBOHOHKOHLOHD’=Z.I(ZOHKOHL.CUS(2.AII.COSIZ.UI.2.HD.SIN(2.BI
I'H*SIN(AI'SIN(B)I
EAAFB(AOBOHOHKOHLOHD)=COS(AI'(H'COS(8)'2'HL’SIN(2'BI'SIN(AII
FRAEA:AoBoHoHKoHLoHDI32'(H‘COS(A)‘COS(BI’HL’SIN(2’BI*SINIZ'A)I
EAAFA(AoBoHoHKoHLoHDI3COS(AI' (HKOHL'COS(2'BII'COSIAI-SIN(AI*((
IHKOHL'COS(2‘BII'SINIAIOH'SIN(BII
UAF(AOBOHOHKaHLoHOI3-COSI2'BI-HL'(COS(2‘BI'COS(AI’COS(AIOSINIBI'
ISIN(BII-HK'(COSIAI’COS(A)-SIN(B)'SINIB)I'HD'SIN(2.BI
2‘2“H’SIN(AI‘SIN(BI

EBAFH(AOBI=2’SINIAI’COS(BI

EAAFHIAIBI=SIN(B)’COS(AI
SIAF(AoBvHoHKoHLvHD)3(20HKOHL'COSIZPAI)‘COSIZ'BIOZPHDPSIN(2'II
1‘H’51NIAI'SIN(BI
SIAFA(AQBoHchoHLoHDI3’2'HL‘SINI2'A)*COS(2*BI-H*COS(AI'SIN(BI
SIAFB(A980H0HK9HL9HDI3'2“(ZOHKOHL’COS(2‘AII’SIN(2*BI04‘HD'COS(
IZ’BI-H*SIN.AI'COS(BI

SIAFH(AIRIz-SIN(AI*SIN(HI
SZAF(A989H9HK9HL9HDI3(HK0HL8CUS(2'8)I’COS(2'AI-H'SIN(AI*SINIDI
SZAFA(AoBoHoHKoHLoHD)='Z'(HKOHL’COS(2’RI)‘SINIZ‘AI-H'COS(AI’SINIBI
SZAFB(AOBOHQHKOHLQHD)3'2’HL’SIN(2'BI’COS(2*AI-H“SIN(AI‘COS(B)
SZAFHIAOB)=-SIN(AI*SINIB)
S3AF(AQBOH9HKOHL9HOI=HL.SIN(2.A).SIN(2.B,-H.COS(AI.COS(B)
S3AFA(AvBoHcHKcHLoHDI=2“HL“COS(2'AI’SIN(2‘BIOH“SIN(A)'COS(B)
SJAFB(AQBPHQHKQHLoHDI=2'HL'SINIZPAI’COSIZ'RIOH“COS(A)'SIN(BI
53AFH(AOB)=-COS(AI*COS(BI
SAF(AoBoHoHKoHLvHDI=SIAF(AQHOHOHKOHLOHDI'SZAF(AOBOHOHKOHLOHDI-

1 S3AF(AOBSHOHKQHLOHDI..2
SAFA(AOBOH0HK0HL9HDI3SIAFAIAOB.H9HK9HL9HD)*SZAF(AOBOHOHKOHLOHD)

1 9SIAF(AQBOHOHKOHLOHD).SZAFA(AOBOHOHKOHLOHDI-2.S3AF‘A98939HK9HL9
2 H0).S3AFR(AQB’HCHKOHLOHD)
SAFB(AIBOHOHKQHLQHU)3SIAFBIA980H9HK9HL9HD)'SZAF(AoBoHcHKoHLoHDI
IOSIAF(APBOH9HK9HL9HDI*SZAFB(AQBOHOHKOHLQHDI'Z'S3AF(AQBOHQHKOHLQHD)
2’S3AFUIAOBOH9HK9HL9HD)
SAFHIAIROHQHKOHLOHD)=SIAFH(A08I*SZAFIAOBOHOHKOHLOHD)
JOSIAF(AOBQHOHKOHLOHD)“SZAFH(AOBI ‘2’53AF(A980H9HK9HL9HDI
2 '53AFHIAOBI
DELTA(AOBQHOHKOHLQHOI=EAAEA(AOBOHOHKQHLOHDI’(EBAFB(AOBOHOHK9HL9HD)
I’SAFH(AoBOHoHKoHLoHDI'SAFB(A989H9HK0HL0HDI*EBAFH(AODII-EBAFA(A9
289H9HK9HL9HDI“(EAAFB(AQBOH9HK9HL9HDI’SAFH(A989HOHK9HL9HDI-SAFB(
3A989H9HK9HL0HOI'EAAFH(AOBIIOSAFA(AoBoHoHKcHLvHD)'(EAAFB(AOBOH9HK9
QHLIHD)’EHAFH(AQB)'EBAFBIAchHoHKoHLQHD)”EAAFHIAIBII

L30

PI 3 3.1415926536

HK=001

144

HL80.3
SHK3HK
SHL SHL
H 3 ((29HK0HL).(HKOHLI I..0o5
A80
880
C80
“03‘00005
DO 100 J31910
H8H60.01
H03H0300005
SHD 3 HD
K80
IO CONTINUE
A3A-(EAAF(AQBOHOHKOHLOHDI.(EBAFBIAOBOHOHKOHLOHOI.SAFH(AOBCHCHKOHLO
IHDI-SAFB(AoaoHoHKvHLoHD)'EBAFH(AQB)I'EBAF(A980H9HK0HL9HD)'(EAAF8(
ZAOBOHOHKQHLQHD).SAFH(A.BOHOHKIHLOHDI‘S‘FB(AQBOHOHKOHLQHD,.EA‘FHI
3A98IIOSAF(AOBOHOHKOHLQHD)'(EAAFB(A989H9HK0HL0HDI.EBAFH(A98I'
4 EBAFB(AOBQHOHKOHLQHO).EAAFH(AOBI’I/DELIA(AOBOHOHKOHLOHDI
838’(EAAFAIAOBOHOHKOHLOHDI'(EBAF(AoBvHoHKoHLOHD)'SAFHIAOBOHOHKOHLQ
IHDI -SAF(AoBoHoHKoHLoHDI'EBAFH(AOBII'EBAFA(AQBOHOHKOHLOHDIPI
ZEAAF(AOBOHQHKQHLOHDI’SAFH(A089H9HK9HLOHDI-SAF(AOBQHOHKOHLOHD).
3EAAFH(A08)IOSAFA(A980H9HK9HL9HDI.(EAAF(AOBOHOHKOHLOHOI'EBAFH(A98)
h‘EBAF(AoBoHoHKOHLoHDI'EAAFH(A08I)I/DELTA(AOBOHIHK9HL9HDI
HsH-(EAAFA(A080H0HKOHL¢HDI'(EBAFB(AOBOHOHKOHLOHDIPSAF(AOBOHIHKQHLO
IHDI-SAFB(AOBOHOHKOHLOHDI‘EBAF(AOBOHOHKOHLQHD))-EBAFA(A989H9HK9HLO
2H0)’(EAAFB(AoBoHoHKoHLcHDI'SAF(AOBOHOHKOHLOHD)-SAFB(AOBOH9HK9HL0HD
BI'EAAFIAOBOHIHKOHLOHDIIOSAFA(AoBoHoHKoHLvHDI'(EAAFB(AOBOH9HK9HL0
“HDI’EBAF(AoBoHoHKoHLoNDI-EBAFB(AOBOHOHKOHLQHD)'EAAF(A989H9HKOHL9
5H0)II/DELTA(AoBoH9HK9HLoHDI
K8K01
SIABSIEAAF(AIBOHOHKQHLIHDIIOABS(EBAF(A089H7HK0HL9HDI)OABS(SAF(A0
IBOHOHKOHLOHD),
IFIK.GT.50IGO T0 35
IF(S.GE.0.0000001IGO T0 10
35 SS 85
KSK 8K
SH 8H
AsASINISINIAII
SR".18000/PI
5838.18000/PI
SC'C’IOOoOIPI
HRITE OPERATIONS
L 3 L-I
IFILoLEoOIZOOO300
200 HRITE(619210I
210 FORMATIIHI ' HK HL HD H S A 8 C
1 K.)
L860
300 URITE(619310I(SHKoSHLOSHDOSHOSSOSAQSBQSCOKI
310 FORMAT(X0F50392F603978069F8.603F8030I3)
100 CONTINUE
END

145

RUNISI

LGO.
0000000000000000000000

C

10

PROGRAM HTHIINPUTOOUTPUT9TAPE61=OUTPUTI

AF STATE

EAF(89HOHK9HL9HD) 3((ZOHKOHLI-H'H'(HKOHLI/(HKOHL'COS(2'BII'.2I.
ISIN(2'8)°2'HD*COS(2.B)

UAFIBOHOHKOHLOHD) 3-(10HKOHLIO(ZOHKOHL)'SINIB)'SIN(BI’H.H'SIN(BI
I'SIN(8)/(HKOHL’COS(2’BII’HDPSIN(2'B)

EAFHIBOHOHKOHLOHD) 3 'ZPH’(HKOHLI’SIN(2‘81/(HKOHL’COSIZ'BI1"2

UAFH(BOHOHK9HL0HDI 3 ‘Z'H'SINIBI'SIN(BI/(HKOHL'COS(2’BII_

EAFB(BoHoHKoHLoHDI3-k'SINIZPB)’SIN(2*B)'H'H'HL'(HKOHLI/(HKOHL’COS
1(2’B)I"392'((ZOHKOHLI-H'H'(HKOHLI/(HKOHL’COS(2.BII"2)'COS(2'BI
206*HD’SIN(2‘B)

DH(89C9HOHK0HL9HDI 3 '2'H‘SIN(BI*SIN(B)/(HKOHL‘COS(2'BIIOZRSIN(C)

SF STATE

USE(COHOHKOHLOHDI3-IOSIN(CI3((ZOHK-HL)'SINICI-Z’H-Z’HO'COS(CII

ESF(C9H9HK9HL9HDI 3 (2 0 HK - HLI'SIN(2‘CI ' Z'HD'COSIZPC) - 2”
IH'COSIC)

ESFH(C9H9HK9HL9HDI 3 ’2’COS(C)

ESFC(C9H9HK9HL9HD) 3 25(2 9 HK - HLI’COS(2'CI 9 4*HD'SIN(2‘CI
19 2'H'SIN(CI

USFH(C9H9HK9HL9HO) 3 -2.51N(C)

L30

P133.1415926536

HK=0

HL80.001

SHD3HD

SHK’HK

H8I(ZoHK-HLI‘IHKOHLII'0.S

8:000

C3000

B39000.PI/18000

HL30.0

DO 100 J31910

HL3HL90.001

“SHOOOOOS

SHL3HL

K80

D3 UAF(BOHQHKQHLOH0)-USFICOHOHKOHLOHD)

DELTA 3 EAFBIBQHOHKQHLoHDI'(ESFC(CoHcHKoHLoHD)'DH(BOC0H9HKOHL9HOI
1*ESF(COHOHKOHLOHOI'ESFH(C0H0HK9HL9HDII-EAF(BOHOHKOHLOHD)'ESFC(COH9
ZHKOHLQHDI*EAFH(89H9HK9HL9HD)

B3B-(EAF(BOH9HK9HL9HOI*(ESFC(COHOHKOHLOHO)*OH(BOCOHOHK9HL9HDIO
ZESF(COHOHKOHLOHD)'ESFH(C9H9HK9HLOHD)IOEAFH(BOHOHKOHLOHD)'(ESF(CO
3H9HK9HL9HDI"2-D'ESFCICOH9HK0HL9HO)I)lDELTA

C3C‘(EAFB(B¢H9HK9HL9HOI'(ESF(COHQHKOHLOHDI'OH(B!COH9HK9HL9HDI'
ID'ESFH(C9H9HK9HLIHDII'EAF(BOHOHKOHLOHD)*(EAF(BCHOHKOHLOHDIPESFH(C
ZTHOHKOHLOHDI'ESF(COHOHKQHLOHDI'EAFH(BOH9HK9HL9HDIII/DELTA

H3H-(EAFBIBQHQHKOHLOHD)'(ESFC(COHOHKOHLOHDI'DOESF(COHOHKOHLOHDI'.2
1I-EAF(BQHQHKOHLOHD)"2'ESFC(C9HOHKOHLOHDII/DELTA

D3 UAFIBOHOHKOHLOHOI-USF(CQHOHKOHLOHO)

146

S3ABS(EAF(BOH9HKOHLOHD))OABS(ESF(C0H9HK9HL9HD)IOABS(D)
K8K01
IF(K.GT.50IGO TO 35
IF(S.GE.0.000000IIGO TO 10

35 5535
A=-9110’PI/18010
X3-HPSIN(BI/(HK0HLECOS(2.B)I
IE(ABS(XI.GT01.OIGO TO 20
A3ASIN(XI

20 CONTINUE
KSK 8K
SH 3H
SA3A'18000/PI
5838.18000/PI
SC3C'18060/P1

C NRITE OPERATIONS

L 3 L'l
IF(L.LE.0)2009300

200 NRITE(619210)

210 FORMAT(1HI ' HK HL HD H S A
1 K’I
L860

300 NRITE(610310I(SHKOSHL’SHDOSHOSSOSAQSBQSCQKI

310 FORMAT(X9F5.392E6.39F8.49F8.693E8039I3)

100 CONTINUE
END

0000000000000000000000

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