MAGNETIC MEASUREMENTS
or ANTIFERROMAGNEI‘IC
KMnCla" 2H2 0 AND LICuI313° 2H2 o

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
PETER TONE BAILEY
1971

‘pcb.l

   
 

LIBRARY

Michigan State
University

—-—-

This is to certify that the

thesis entitled

MAGNETIC MEASUREMENTS 0F ANTI-
FERBOMAGNETIC KMnC13o2H20 AND

L1Cu013o2H20 '
presented by

Peter Tone Bailey

has been accepted towards fulfillment
of the requirements for

Pth . degree in P2118108

 

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Major professor

Date é/g//7/

 

 

 

  
    

amoma BY 3
"MG & SUNS'
300K BINDERY INC.

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ABSTRACT

MAGNETIC MEASUREMENTS OF ANTIFERROMAGNETIC

KMnCl3-2H20 AND LiCuCl3-2H20

BY
Peter Tone Bailey

A molecular field theory for calculating the ani-
sotropy and exchange fields from the magnetic phase
boundaries is given. The theory for LiCuCl3-2H20 assumes
an intersublattice exchange, J1, and an intrasublattice
exchange, J2, and a uniaxial anisotropy. This gives the
field dependence of the transition temperature in terms
of J and J .

l 2
For KMnClB'ZH O, a two dimensional Ising model

2
with eight sublattices is used. The centers of the
ferromagnetic dimers are coplanar, and the dimers have
an antiferromagnetic arrangement with each other. Assum-
ing that a particular sublattice f10ps 180° at each of
the four regions above the antiferromagnetic state, four
spin interactions are calculated from the four boundary

values at T=O°K. A brief theory of the effects of a non-

uniaxial anisotropy is mentioned.

Peter Tone Bailey

Specific heat measurements, field rotations, and
field sweeps were done adiabatically to determine the
magnetic phase boundaries. The apparatus and methods
are described. For LiCuCl3-2H20, the results show a
triple point at 4.2°K and 12.7 kG. The ratio of JZ/Jl is
17, implying a relatively large intrasublattice exchange.
The total exchange field is 12.5 kG, and the anisotropy
field is 3.1 kG. The rotations show that the spins flop
in the AC plane, and the variation of the paramagnetic
boundary for different field orientations indicates a
non-uniaxial anisotropy.

The results for KMnCl3°2H20 show five distinct
magnetic phases. Using field sweeps, two boundaries are
found near 12 RG and 14 RG that look like first order
spin flop boundaries. The other boundaries appear to be
second order phase transititions since the specific heat
shows an anomaly as the boundary is crossed. The magne-
tic field rotations show that in the first phase above
the antiferromagnetic state, some or all of the spins
may flOp in one plane. Then in the second phase above
the antiferromagnetic state, the spins may flop in an-
other plane that is nearly perpendicular to the first one.

With the field perpendicular to the easy axis,
specific heat measurements indicate an antiferromagnetic
to paramagnetic boundary with less curvature than for

the case with the field parallel to the easy axis. In a

Peter Tone Bailey

second perpendicular position (90° from the above perpen-
dicular position), two boundaries were observed. One
boundary was similar to the other perpendicular one, and
the other was somewhat more curved.

After extrapolating the phase diagram boundaries
to T=O°K, the zero temperature field splitting for the
lower magnetic phases is comparable to that for the upper

magnetic phases as predicted from the Ising theory.

MAGNETIC MEASUREMENTS OF ANTIFERROMAGNETIC

‘2H 0 AND LiCuCl ‘2H 0

KMnCl3 2 3 2

BY

Peter Tone Bailey

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1971

 

 

PLEASE NOTE:

Some Pages have indistinct
print. Filmed as received.

UNIVERSITY MICROFILMS

ACKNOWLEDGMENTS

The author wishes to express his thanks to all
those who have helped in this study: to Dr. H. Forstat
for invaluable assistance during the research; to Dr. J.
A. Cowen and Dr. R. D. Spence for helpful discussions;
to Mr. John R. Ricks for help with the experiments and
data processing; and especially to the Air Force Office
of Scientific Research, Office of Aerospace Research,
United States Air Force for their financial support of

this work.

ii

Chapter

TABLE OF CONTENTS

LIST OF TABLES O O O I O O O O O O O O O O 0

LIST OF FIGUES O O O O O O O O O O O O O 0

INTRODUCTION 0 O O O I O I O O O I O O O O O

I.

II.

III.

IV.

GENE RAL TH E0 RY . O O I O O I I O O O O

A. Models for Antiferromagnetism . .
B. The Anisotropy Field . . . . . . .
C. Spin Flopping . . . . . . . . . .
D. Observing Spin Flop Boundaries . .
E. Observing Paramagnetic Boundaries
F. Demagnetization Corrections . . .
LicuC13 . 2H20 THEORY O O O O I O O O O
KMIIC13 . 2H20 THEORY O O O O I O O O O O
A. ISing MOdel O O O O O O O O O O O
B. Dipolar Anisotropy . . . . . . . .
C. The AF-P Perpendicular Boundary .

EXPERIMENTAL METHODS . . . . . . . . .

A.

Experimental Apparatus . . . . . .

l. Dewar and Calorimeter . .
2. Sample Holder . . . . . .
3. Vacuum Pumps . . . . . . .
4. Pressure Gauges . . . . .
5. Thermometer Current Supplie
6. Measuring Electronics . .

7. Magnet and Gaussmeter . .

Experimental Procedures . . . . .

1. Sample Preparation . . . . . .
2. Preparing for Experiment . . .

iii

Page

vii

ooquw

10
ll
24

24
31
32

37

37

37
41
43
43
45
46
48

49

49
51

Chapter

. Helium Transfer and Thermometer
Calibration . . . . . . . . .
Adiabatic Field Rotations
Adiabatic Magnetizations .
Specific Heat Measurements
Removing Crystal . . . . .

\lO‘U‘lub w
I O O O

O O O O

O O O O

O O O O 0

Data Analysis . . . . . . . . . . . . .

Converting Pressure to Temperature .
Thermometer Calibration Equation . .
Chart Recorder Calibration . . . . .
Specific Heat, Rotations, Field

Sweeps . . . . . . . . . . . . . . .

wal-J

V. LiC‘JCl . 2H 0 RESULTS 0 O O O O O O O O O O O

3 2

A. Adiabatic Rotations . . . . . . . . . .
B. Adiabatic Magnetizations . . . . . . .
C. Antiferromagnetic-Paramagnetic Boundary

1. Field Parallel to Easy Axis . . . .

2. Field Perpendicular to Easy Axis . .
D. Antiferromagnetic-Spin Flop Boundary . .
E. Spin Flop-Paramagnetic Boundary . . . .
F. Anisotropy Energy . . . . . . . . . . .

VI. KMnCl3'2H20 RESULTS . . . . . . . . . . . .

A. Adiabatic Rotations . . . . . . . . . .
B. "Antiferromagnetic-Spin Flop"

Boundaries . . . . . . . . . . . . . . .
C. Easy Axis Shift . . . . . . . . . . . .
D. Antiferromagnetic-Paramagnetic and

"Spin Flop-Paramagnetic" Boundaries . .
E. Exchange and Anisotropy Fields . . . . .

CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK . . .

REFERENCES

APPENDIX

iv

Page

52
54
55
56
57

58
58

59
62
63

63
63
73

73
76

78
79
80

81
81

94
99

104
105
110
112
114

LIST OF TABLES

Table Page
1. LiCuCl3-2H20 AC plane isentropic rotations . . . 114
2. LiCuCl3'2H20 BC' plane isentrOpic rotations . . 115
3. LiCuC13°2H20 isentrOpic magnetizations . . . . . 116
4. LiCuCl3°2H20 specific heat data . . . . . . . . 117

5. LiCuCl3'2H20 isentropic magnetization
critical fields 0 O O O O O O O I I O O O O O O 118

6. LiCuCl3'2H20 specific heat maxima--
sample 1 O O I I O I O O O O O O O O O O O O O O 119

7. LiCuC13-2H20 angular dependence of
critical field . . . . . . . . . . . . . . . . . 119

8. KMnC13'2H20 plane 1 isentropic
rotations - twin . . . . . . . . . . . . . . . . 120

9. KMnCl3-2H20 plane 2 isentropic

rotations - twin . . . . . . . . . . . . . . . . 121

10. KMnCl3-2H20 plane 1 isentropic

rotations - single . . . . . . . . . . . . . . . 122

ll. KMnC13°2H20 plane 2 isentropic

rotations - single . . . . . . . . . . . . . . . 123

12. KMnC13°2H20 plane 3 isentrOpic

rotations - single . . . . . . . . . . . . . . . 124

13. KMnCl3°2H20 plane 4 isentropic
rotations - single . . . . . . . . . . . . . . . 125

14. KMnC13'2H20 isentropic magnetization
boundary points . . . . . . . . . . . . . . . . 126

Table Page

15. KMnCl '2H 0 specific heat maxima . . . . . . . . 127

3 2
l6. KMnCl3'2H20 isentropic magnetizations . . . . . 128
17. KMnCl3'2H20 critical field angular

dependence, single crystal . . . . . . . . . . . 129

18. KMnCl3'2H20 critical field angular dependence

and rotation minima, twin crystal . . . . . . . 130

19. KMnCl '2H 0 specific heat data - twin . . . . . 131

3 2
20. KMnCl3'2H20 isentropic magnetizations,
2/13/70 0 O I I O I O O O O O O O O O I O O O O 132

vi

LIST OF FIGURES

Figure
1. LiCuC13°2H20 spin arrangement . . . . . . .
2. Definition of angles and vectors . . . . .,.
3. KMnCl3'2H20 8 spin unit . . . . . . . . . .
4. KMnC13‘2H20 two dimensional dimer
arrangement . . . . . . . . . . . . . . . .
5- KMnC13'2H20 J3 interaction in BC plane and
Ising spin model . . . .,. . . . . . . . . .
6. Pyrex helium dewar . . . . . . . . . . . . .
7. Cross section of body of calorimeter . . . .
8. Front and side cross sections of top of
calorimeter . . . . . . . . . . . . . . . .
9. Sample holders . . . . . . . . . . . . . . .
10. Schematic diagram of pumping systems . . . .
11. Diagram of electrical measuring circuits . .
12. Example of data on chart recorder . . . . .
13. IiCuC12'2H20 AC plane isentropic rotations
l4. LiCuCl3'2H20 BC' plane isentropic rotation .
lS. LiCuC13‘2H20 BC' plane isentropic rotations
16. LiCuCl3'2H20 isentropic magnetizations . . .
17. LiCuC13'2H20 specific heat data . . . . . .
18. LiCuC13'2H20 specific heat data . . . . . .
19. LiCuC13'2H20 phase boundaries . . . . . . .

vii

Page

12
16
25

26

28
38

39

40
42
44
47
61
64
65
66
68
69
70
71

Figure

20.

21.
22.
23.
24.

25.

26.

27.

28.

29.

300

31.

32.

33.
34.

35.

LiCuCl3'2H20 angular dependence of

critical field . . . . . . . . . . . . . . .
LiCuC13°2H20 fit to AF-P boundary . . . . .
LiCuC13'2H20 log-log plots for AF-P boundary
KMnCl3'2H20 crystal with rotation planes .

KMnC13-2H20 plane 1 isentropic rotations -

tWin O O I O O O O O O O I O O O O O O O O

KMnC13'2H20 plane 1 isentropic

rotation - twin. . . . . . . . . . . . . . .

KMnC13'2H20 plane 2 isentrOpic
rotations - twin . . . . . . . . . . . . . .

KMnCl3°2H20 plane 1 isentrOpic
rotation - single . . . .-. . . . . . . . .

KMnC13'2H20 plane 1 isentropic

rotations - Single 0 o o o o o o o o o o o o I

KMnC13'2H20 plane 2 isentropic
rotations - single . . . . . . . . . . . . .

KMnC13'2H20 plane 3 isentrOpic
rotations - Single 0 o o o o o o o o o o o o

KMnC13'2H20 plane 4 isentropic
rotations - single . . . . . . . . . . . . .

KMnC13'2H20 plane 4 isentrOpic

rotations - single . . . . . . . . . . . . .-

KMnC13'2H20 phase boundaries . . . . . . . .
KMnCl3'2H20 isentropic magnetizations . . .

KMnCl3'2H20 critical field angular
dependence - single . . . . . . . . . . . .

viii

Page

72
74
77
82

83

84

85

87

88

89

90

91

92
93

95

96

Figure Page

36. KMnC13'2H20 plane 1 critical field

angular dependence and rotation minima - twin . 97
37. KMnC13'2H20 plane 2 critical field angular
dependence and rotation minima - twin . . . . . 98

38. KMnCl3‘2H20 specific heat data - twin . . . . . 100
39. KMnC13'2H20 specific heat data - twin . . . . . 101

40. KMnCl3'2H20 critical field best angle and
rotation extrema - single . . . . . . . . . . . 102

41. KMnCl3'2H20 isentrOpic magnetizations . . . . . 106

ix

INTRODUCTION

The purpose of this study is to further develop
the adiabatic method of studying antiferromagnets and to
investigate some new crystals having the spin flop pro-
perty. The unusual properties of KMnCl3'2H20 necessitated
using some improved methods to investigate the phase
boundaries. The use of a 21 kc magnet with a motor
driven rotating base, a linear sweep circuit for changing
the magnetic field, and a digital gaussmeter made data
taking more precise. This enabled one to record small
temperature changes by observing the output on a linear
chart recorder.

Since the spins of both of the crystals studied
were paired into dimers, modified molecular field theories
were used to account for the spin interactions both within
a dimer and between dimers. One could then estimate the
exchange and anisotropy fields from the results.

'2H 0 indicated that the

3 2
spins flOp 90° in the AC plane. Since KMnCl3'2H20 has an

The results for LiCuCl

unusual number of phase boundaries, it appeared that the
spins could flop 180° (at least at O°K), and an eight

sublattice Ising model was used in the analysis.

I. GENERAL THEORY
A. Models for Antiferromagnetism

Several models have been introduced to explain
the exchange interaction which leads to antiferromagne-
tism. The Heisenberg model uses an isotropic interaction
between a spin, Si, and its neighboring spins, §j' The

interaction potential energy of atom i is,

vi = -2J_S_i'(2j sj), (1.1)

where the exchange integral, J, is negative for antiferro-
magnets.

The Ising model represents a very anisotropic
case since it assumes that the interaction is between
spin components in only one direction. Then the inter-
action potential for atom i is,

Vi = -2JSi Zij . (1.2)
2 Z

The often used Weiss molecular field approxima-
tion expresses the spin interaction in terms of an ex-

change field, H The 2 neighboring spins are represented

E.
by their statistical average, <§j> ,

Vi = -22J_S_i- <~§j> . (1.3)

2

This can also be expressed as,

V. = u.°H

l —1 —E’ (1'4)

where Bi = -guB§i, and the exchange field is thus defined
as,

2

HE = zzlJlgj/(gzibz) = 22IJIMj/(NgzuB ) . (1.5)

Its magnitude at T = 0 is given by

2 2
HE = 22|J|Mo / (Ng “B

)I y (106)
where N is the density of spins per sublattice, Mj is the
magnetization of the j sublattice, and Mo is the satura-

tion sublattice magnetization.
B. The Anisotropnyield

The anistropy energy for a uniaxial antiferro-

magnet may be expressed as,1

EK = -(1/2K(coszei + cos2 Gj), (1.7)

where K is the anisdxopy energy per unit volume and 6i and

ej are the angles that the sublattice magnetization vec-

tors, Mi and Mj, make with the easy axis. Differentiating,

= ' 9 8 8 ' . . . .
dEK K(31n i cos j d i + Sinej cosel den) (1 8)

One can define anisotropy fields, HA and HA ,

1 J

by

EK = -HAi Mi cosei - HA. Mj cosej. (1.9)
3

If one considers that Mi and M5 are equal at zero applied

field and represents them by Mo' then differentiating,

dEK = (HAi SlnGi d8i + HAj Sinej dej) Mo' (1.10)
Comparing this with 1.8, then
HAi== K cosei / Mo’ HAj== K cosej / MO. (1.11)

For small 61 and ej, a general anisotropy field can be

expressed as,

HA = K / MO. (1.12)

C. Spin Flopping

The change of the thermodynamic potential is

found by integrating
d4 = -S dT + V dP - M dH. (1.13)

With the external field, H, parallel to the easy axis in
the antiferromagnetic (AF) state,
H 2

— _ ' '= —..l'.~.
TAFU'I) - IO M(H ) dH 2 X” H , (1.14)

assuming that the parallel susceptibility is independent
of the external field. With the field applied parallel
to the easy axis but with the spins flopped 90° to the
easy axis (SF state), there is an anisotropy energy per

unit volume, K, such that

1 2

(PSFU'I) = K - T XSF H . (1.15)

If the anisotropy energy is very small compared

to the exchange energy, then XSFC'XJJ Then,
_ _ l. 2
¢SF(H) - K '7T' XL.H . (1.16)

The critical field for the AF-SF boundary can be found by

equating these two potentials; then
_ _ 1/2

A slight error is introduced by assuming that x“
is independent of field, since the molecular field approx-

imation shows that x“ increases with H as x is unchanged.

1

The low field values for x” and Kl are usually used, so

that the calculation of HAP-SF may then be slightly dif-

ferent from the experimental value.
The value of the critical field at T = 0 can be

expressed in terms of the anisotropy and exchange fields.

Starting with the perpendicular susceptibility,l

= 1 / (1 + K / 2 M02), (1.18)

X
.L
where A is the molecular field constant for nearest

neighbor interaction (1.6). Using HA = K /MO (eqn. 1.12),

XJ_= 2 Mo / (2HE + HA)' (1.19)

+ HA) K / M0 = 2 K. (1.20)

and x (2 H
.L E

Substituting into 1.17,

_ 2 _ 1/2
HAP-SF - [(2 HE HA + HA ) / (1 XH/xi)] I
(1.21)
or at T = 0,
2 l/2
HAF_SF(0) (2 HEHA + HA ) . (1.22)

To calculate the SF-P boundary field for a two
sublattice model at T = 0, first evaluate the exchange,
magnetic, and anisotropy energies of spins Si and §j'
which are from different sublattices. Using 1.1 and 1.9
and summing over the z nearest neighbors, the spin

energy is,

Z Z
w» = -2 J as. + ayes

- guB [(Ei + §j> ° §.+ I§i°§Al + léj'EAll-

(1.23)
The anistropy energy is expressed in terms of a field
which is parallel to the easy axis.
Just above the AF-SF boundary, the spins are
flopped, perpendicular to the easy axis and antiparallel

to each other. Thus,

E(0)AF_SF = 4 2 J 52. (1.24)

In the paramagnetic region, the spins are parallel to

the external field, and

3(0)P = - 4 2 J 32 - 2 g “B s (H + HA). (1.25)

Thus the field, H P' needed to change the spins from

SF-
the perpendicular (SF) to the parallel (P) orientation is

found by equating the energies in 1.24 and 1.25. Then,

H(0)SF_P = - (4 2 J s / g uB) - HA. (1.26)

Using 1.6 and the fact that J is negative,

H(0)SF_P = 2 HE - HA. (1.27)

D. Observing Spin Flop Boundaries

As the external field is changed adiabatically,

the temperature variation can be calculated as,2

(dT/dH) = (8T/3H)S = -(BS/3H)T
-T§§7§TT; I (1.28)

where S is the entropy. Using the Maxwell relation
(BS/8H)T = (8M/3T)H, (1.29)

and T (BS/8T)H = CH(T,H), (1.30)

where C is the constant field specific heat, then 1.28 is

H

(dT/dH)

”(T/CH)(3M/3T)H, (1.31)

or (dT/dH)

- (T H/CH)(8x/3T)H. (1.32)

In the antiferromagnetic state with x = x” and
(ax/3T) > 0, (dT/dH) is negative. In the spin flop state
with x 3 X1 and (ax/3T) = 0, (ET/3H) = 0. Thus with an
adiabatically increasing field aligned along the easy axis,
the sample temperature should decrease until the spins
flop and then should remain relatively constant.

It can be shown2 that an adiabatic rotation should
show a minimum in the AF state when the field is along the
easy axis. Also, if the spins flop in the plane of rota-
tion, there should be a relative maximum in the SF state
at the easy axis position. If the spins flop perpendicular
to the plane of rotation, there should be no temperature
change on rotating near the easy axis for fields slightly
above the AF-SF boundary. There the spins have almost no
component along the easy axis or in the plane of rotation.

Thus there is no change in the magnetic energy on rotating,

as long as the spins remain flopped.
E. Observing Paramagnetic Boundaries

The AF-P and SF-P transitions are of second order.
They can be observed by finding the discontinuity in the
specific heat as a function of temperature in a constant
field. Sometimes an isentrope can be used to denote the
crossing of the paramagnetic boundary. Schelling and

Friedberg3 noticed for MnBr2-4HZO that the intersection of

the isentrOpe with the AF-P phase boundary coincides with
an inflection point in the isentrOpe.

It can be shown4 that the isentropes cross the
paramagnetic boundaries tangentially. If Sb(H) is the
entropy on the AF-P boundary, the entropy can be repre-

sented by a new variable,
5 = S - Sb(H). (1.33)
Taking partial derivatives,
(B/BH)S = (a/BH)S - (dSb(H)/dH)(3/BS)H. (1.34)

Then,

(ET/3H)S = (3T/3H)S + (dSb(H)/dH)(3T/BS)H. (1.35)

From 1.30, (GT/BS)H = T/CH, and if the specific heat, CH’
diverges at the phase boundary, then the last term of 1.35
vanishes as s approaches zero. Then,

lim (GT/3H)S = (8T/3H)b, (1.36)
5+0

where b refers to differentiation along the phase boundary.
Thus an isentrope near the boundary has the slope

of the boundary. Under perfectly adiabatic conditions,

the isentrOpe would continue along the boundary and would

not cross it. However, background temperature effects in

our experiments caused the isentrope to cross the boundary,

and an inflection point was observed.

10

F. Demagnetization Corrections

 

The field, H', inside a sample is different from
the applied field, H, due to the magnetization of the

sample;

H. = 31.224— p (1.37)

where D is the tensor demagnetizing factor. A sphere has
the lowest factor, a scalar 4n/3.l It would be difficult
to shape the samples into spheres and then align them
properly or to make demagnetization calculations for their
odd shapes.

These corrections might be significant in the re—

gion near T where the magnetization is changing rapidly

N
with temperature. In particular, the slopes of the isen-
tropes are field dependent (1.32) and might be affected
in this region. Since specific heat measurements were

used to find most of the boundaries near T the correc-

NI
tions were not used. All of the fields measured may be
slightly higher than the actual field inside the crystal,
especially in the low temperature region where M_approaches

its saturation value.

'2H 0 THEORY

II. LICuCl3 2

For LiCuC13°2HZO, it is proposed that the copper
ions are arranged in pairs with their spins parallel in
each such dimer.S The spins of the dimer are then anti-
parallel to those of the four nearest neighbor dimers
(see Fig. 1). This model could be represented by an ef-
fective molecular field (in addition to the applied field,
H) which represented the interaction between the spins of
the (+) and (-) sublattices. Following Heller,6 one may
write,

EE 2‘. = -2-M - 914 . <2-1)

The tensor, 3, corresponds to the antiferromagnetic ex-
change interaction between spins on different sublattices.
The tensor, 2, refers to the ferromagnetic exchange inter-

action within a sublattice, and M+

refers to the magneti-
zation on respective sublattices.

The anisotrOpy is included in these tensor coupl-
ing constants, and the contribution of the anisotropy to
the molecular field is assumed to be a linear function of

the sublattice magnetization. It is assumed that a and c

have tetragonal symmetry about the preferred axis with
values a” and c” along the axis and ap and cp perpendicular

to the axis.

11

12

Li Cu CI3'2H O

2’

 

 

 

 

 

Figure 1. LiCuClB-ZHZO spin arrangement.

13

For each sublattice there are N spins, S, per
unit volume, each with magnetic moment SguB where “B is
the Bohr magneton and g g 2, as determined by Date.-7

With an applied field, H,

M = M B (lg - 2°M - g-M ISguB/ kT), (2.2)

i 00 S :- —:
where BS is the Brillouin functionfiland
Moo = NSguB (2.3)

is the saturated sublattice magnetization. It is also

required that

M+ be parallel to (fi.- §f§_ - g ). (2.4)

-M
_ + d:

The reduced paramagnetic and antiferromagnetic parts of

the magnetization are defined as respectively

g (”—4- + 1~_a )/2MO (2.5)

0

5. (13+ - II_4_)/2MO . (2.6)

O

The assumption that the contribution of the aniso-
tropy to the molecular field is linear in the magnetization
is not necessary for the cases with the field parallel or
perpendicular to the easy axis. For a parallel field, the

perpendicular components of and M| are zero, and thus

31+

a and c are not used.
P P

If one assumes this linearity for the case of a

perpendicular field, then by equation 2.4,

14

M W M - and 2.7
+H +H (all 0"): ( )
M N M (-a - c ) + H, (2.8)
+ +
p P P P
since M_ = M+ and M_ = -M+H by symmetry. By looking
P P

at 2.7 and 2.8, it can be seen that the contribution of
anisotropy to the molecular field must be linear in the
magnetization. Otherwise 2.4 would not hold.

Using the inverse Brillouin function, a new func-

tion is defined:

G(x) = Bs’l(x)(s + 1)/3s. (2.9)
It can be shown that

G'(O) = 1, G"(0) = 0, and

G” (0) = -Bsm (0)/[BS'(0)]3 = (27/15)(282+zs+1)/(s+1)2.
(2.10)
For a vector, Z, G(!) is defined as,
C(y) = y_G(|yj)AyI (2.11)

Using the unit vectors 1“ and lp which are respectively

parallel and perpendicular to the preferred axis,

v = V 1 + V 1 (2.12)
- II --II p—p

G(g_+ 5)-G(g:§) = (1+B;1|g+éj - 1_B;l|PfA|)(S+l)/3S,

(2.13)

15

where 1+ = M+/IM+I, and l_ = M_/IM_I. For purposes of

simplification, write K as,

5 [sew - G(§-§)](3kT)/[gIJB(S+l)] (2.14)

= l IH - a'M - c-M I - 1 [H - a-M - C'M I.
-4 — 1: — ::—+ —; —’ ::—+ :'—-

The plane of spins is defined by the unit vectors 1” and

l as shown in Fi . 2.
_p g
From 2.4, the only non-zero components of

(§'- an} - c-M ) are those parallel to 1 . Then from

2.14,

-a M_ - c M )]

K=l 611- M- +‘6H
- -+ [COS +( II allncliMql) Sin +(p p p p+p

+1 eH- - +'8-+M+M
__ [cos _( H aHM+H CHM—H) Sln _( Hp ap +p cp _P)],
(2.15)

Parallel and perpendicular components of K are taken using,

14 = cos 0+ 1 + sin 8+ 1p
1_ = -cos 8_ 1 + sin 9_ 1p
Then,
K” = cos2 6+(HH- alfl_“- CHM+”)+cos 8+ sin 8+(Hp-apM_p-cpM+p)

2
- cos 8 H - - - 8 ' 8 -H + M + M
_( H aHM+H c”M_“) cos _ Sin _( p ap +p cp _p),

(2.17)

16

 

 

 

 

 

i '0)

H
“U

 

Figure 2. Definition of angle- and vectors.

17

and,
KP = COS 9+ sin 9+ (HH - a“ M-H - C” M+|l )
+ sin2 0+ (Hp - apM_ - cp M+ )
P P
+ COS 9_ sin 9_ (HH - a” M+H - C“ M_” )
. 2
+ Sln 9_ (-Hp + ap M+ + cp M_ ) (2.18)
P P
Using 2.4 it is found that,
H - M - M =t 9 H - M - M
‘p ap-p Cp+p’ 8“ + ‘ II all -,, °|| “'n’
H - M - M =- 9 H - M - M .
‘p ap+p ap-p’ tan -‘ II an +H °|I -H’
(2.19)
Finally,
K = (c0326 +sin26 )(H -a|M -c|M)
H + "' H I “H '4'“
- (0082 9_ + Sin2 9_)(HH ‘ a” M+l- C” M_H)
= (a” - C” )(M+ - M_ ). (2.20)
II II
Similarly,
.K = (cosze + sin20 ) (H - a M - c M )
p + + p p- p+
P P
- (c0328 + sin26 ) (H - a M - c M )
. ' ‘ P P +p P 'p
= (a - C) (M -M ). (2.21)
+ —
P P p p

Then returning to 2.14,

18

 

 

 

 

-—%— [G(§_+ 5) - G(g_- 5)]
(s + l)guB
== 6kT [(aH- C”)(§+”‘§_H) + (ap ‘ Cp)(fl+p ' fl_p)]
2 2 M - M M - M
NS(S + l)g “B (a - c ) —+“ ——H a -c —+ -—
= 3kTII H H 2Moo + aH-c“) pZMOOP
(2.22)

since Moo = NguBs. To simplify the constant coefficient,

consider the case with H along the preferred axis giving

 

 

LN; = 0 = M_ . Then the right side of 2.22 is,
P P
NS(S+1)gzuB2
3kT (all-c”) (Ii-4+” " P_4_“)/2MOO. (2.23)

Using 2.10, G(A) can be expanded in a series;
G(A) = A + (A3/6) Gm(0) + . . . (2.24)

.As T approaches Tn from below (assuming a small field, H),

'then the left side of 2.22 may be expanded to yield:

[6(2 + a) - 6 (g - §)]/2 = (gill-E" )/2Moo (2.25)

since M+ and M_ are very small in this case. For 2.23

and 2.25 to be equivalent,
TN = (a - c ) NgzuBz s(s + 1)/3k. (2.26)
H H
Then 2.22 is finally,

[G(g + A) - G(g_- §)]/2 = (5“ + QéP)/t. (2.27)

19
a - c
where t = T/Tn and Q = H .
An expression can similarly be obtained for

[G(P_+ a) + G(g_- A)] as done for equation 2.14:

It"
ll

[6(3 + 5) + G(g - 5)]3kT/(guB(S + 1))

°§+|

=1+|_Ii‘_e_°fl'g +llfl'g'1‘1

- .M o o
+ g __| (2 28)

Proceeding as was done for 2.16,

1 pp p+

£_= l [cose+(HH-a”M_H-CHM+I) + sin8+(H -a M_p-c M p)]

-1 en- -M+'8-+M+M.
_ [cos _( H aHM+|l c” _H) Sln _( Hp ap + cp _ )1

P P
(2.29)
fThen, as was calculated for 2.18,
_ 2 .. _ . - _
L” — cos 6+(HH a“M_H CHM+H)+COSG+ 51n9+(Hp apM_p cpM+p)

2 .
+ 9H- — H- e e-+M+
cos _( H aHM'“ CHM__ll cos _ Sin _( H a I c M_ )

P P
= 2H - + M + M o o
H (all C” )( +H _H) (2 30)
Also,
. . 2
= e e _ .. 6 _ _
LP cos + Sln +(HH a”M_H CHM+”)+31n +(Hp aPM_p CPM+p)

. . 2
-e 63- - -e-+M+
cos _ Sln _( H aHM+H c”M_”) Sln _( Hp ap +p cpM_p)

= 2H - (a + c )(M + M_ ). (2.31)
P P P +p p

20

 

Thus,
(S+l)guB NguBS
[G(Efl‘fl) + G(E‘fln/Z = W— }_H [HM-(aH'I'CH) (M+H+M—|)ITM_]
oo
NguBS
+ 1 H - + M +M
-p [ p (ap Cp)( +p 33m]
1
= ——— F - WP - RWP 2.32
where
a + c a + C
W = aJJ - cH ' R = 32—:_EE" and
II II II II
guB(s + 1)
F = “TIFF—— “-
_ N _

Upon using the expansion 2.24, the left side of
2.32 is equal to P for temperatures not too far below TN

(A<<1) and in moderate applied fields (P<<1). This gives,
P = F t+W P = F t + RW . 2.33

Since isotropic interactions predominate over anisotropic

interactions, ap/aHZ 12 cp/cl . Thus R==1 and P is pro-

 

portional and parallel to E for this case.

The bulk magnetization is,

gmg) + puma) = 3<_(T)-H :x 11 (2.34)

— N

“fliers the tensor, K, is taken as a constant scalar, XN'

for our purposes. Then 2.34 may be rewritten as

g = XNg/zMoo. (2.35)

21

The temperature dependence of A is calculated from 2.27

by regarding P as a known quantity. Define
GD(A_) = [G(g + 5) - G(g - 591/2 (2.36)

and expand according to 2.24.

GD(§) = [(gr§)-(g:§)+(lg+§|2(g+§)-|375I2(gr§))em(0)/61/2
1 2 2 (2.37)
= 5. [(g + 5) [1 + (p + A + 2PA cos))Gm(0)/6]
- (Efé) [1 + (P2 + A2 - 2PA cos1)GM(0)/61]
Or,
GD(A) = 5 [1+(P2+A2)G”(0)/6] + g(g;§)GM(0)/3. (2.38)

'where the angle, A, is defined in Fig. 2.
Taking the components of 2.38 which are in the A

direction,

2 cos21 Gm(0)/3]

A [1 + (A2 + P2)Gm(0)/6 + P
= (Allcosw + QAp sinw)/t

[(1-s1n2w) + Qsinzw]A/t, (2.39)

chere the angle, w, is defined in Fig. 2. The components

<Jf'2.38 which are normal to the A_direction are

PZA cosl sin) sm(0)/3 AP2 sin 21 Gm(0)/6

(-A/t)(l-Q)sin¢ cosw.
(2.40)

= (-sinw + cosw QAp)

22
If the applied field is weak enough so that,
82 G”(0)/6<<(1 - Q). (2.41)

then A is essentially parallel to A“ , i.e., $30.
Then (l-Q)sin2w<<P2(l+2 coszl)Gm(0)/6, and since A2 is

2

small compared to P (1 + 2 coszl), equation 2.39 may be

‘written with l~¢;
A[1 + p2(1 + 2cos2¢)cm(0)/6] = A/t, (2.42)

where ¢ is defined in Fig. 2. From 2.10 and 2.35 this

 

becomes,
2
A/t = A 1+ 3‘25 +25+1> (1+20052¢)X2 H2 . (2.43)
2 2 N
40(s+l) M
00
For TN - TN(H)<<TN, this becomes

 

2
2 X
TN(H) = TN [1 - 3‘23 +23+1) (—E—) H2 (1+2cos2¢)]. (2.44)

40(s+1)2 Moo

When the field, A, is applied perpendicular to
the preferred axis, A+ and A have equal lengths, Mo' as

they turn toward A. Then equation 2.27 yields, (with

8+ = e_)
G(Mo/Moougbr-ggJ/Mo = (g+-g_)/(tmoo). (2.45)
or G(MO/MOO) = Mo/(tMoo) (2.46)

assuming (A| - A )#0. Since 2.46 is independent of A,.A
+ - +

and A_ have a constant length as they turn toward A.

23

In order to check the experimental results with
2.44, in particular, the phase boundary between the para-
magnetic (P) and antiferromagnetic (AF) states, the fol-

lowing molecular field calculations will be used,

2 2

= Nt 9 “B

XN s(s+1)/(6kTN) = thzug/(421Jl) (2.47)

where Nt is the total number of spins per unit volume

and Moo = thuBS/Z. Assuming a general form for TN,
TN =(23(s+l)/(3k))i zilJil (2.48)

where J1 and J2 are the intersublattice and intrasublattice
exchange interactions respectively, and zi refers to the

number of nearest neighbors, then 2.44 becomes,

2 2 2 2 2
g UB(zllJll+z2IJJ)(23 +2s+l)(l+2cos ¢)H

 

 

 

(T -T) , (2.49)
N 80k 212 J12 s(s+l)
or, z IJ I
2 2 2 2 2 2 2
(2s +2s+l) [1+ ElTfilf] g uB H (1+cos ¢)
(T -T) = .
N 2
120 k TN (2.50)
Assuming no intrasublattice interaction,
(232+23+l)g2u:H2(1+2cosz¢)
(TN-T) = 2 , (2.51)
120 k T
N
where TN is written in terms of lJll, or
92“}23(232+23+1)(l+2cosZ¢)H2 (2 52)

 

(TN-T) 80 kzllJ1[s(s+l)

'2H 0 THEORY

III. KMnCl3 2

A. Ising Model

 

To explain some of the results for the phase dia-
gram of KMnC13-2H20 which indicates five distinct magnetic
phases and four boundaries at T = 0, an eight sublattice
Ising model is introduced. An eight spin unit has two
dimers8 with all the spins antiparallelvto those in its
other two dimers (Figs. 3, 4). Oguchi9 has done an analy-
sis using a four spin unit for CoC12-2H20. There are
four different spin interactions assumed, with Jl being
the intradimer exchange interaction between spins 3.85 A°
apart. The J2, J3, J4 interactions are the average of
the interactions between a spin and its two neighbor
dimers with the interdimer separations being respectively
6.5 A°, 6.9 A°, and 7.4 A°. J4 represents the short
diagonal of a parallelogram whose long diagonal is 11.1 A°.
These interactions are shown in Fig. 4 which depicts a
plane of dimers. A dimer in this plane, which is parallel
to the AB plane, is 9.91 A° (along the C axis) from the
closest dimer in a neighboring plane.

It was not necessary to consider all the inter-
actions between a spin and the individual spins of neigh-

boring dimers. If this were done in the calculation

24

curvmocs. x

25
O
- — -'-

 

.ufic: cfiam m ommm.MHoczx .n upswfim

‘/\o

Lat—mumw

 

 

27

below, the interactions from the two neighbor dimers would
be coupled together and thus could be represented by a
single interaction. However, the spin is usually closer
to one neighbor dimer than another (Fig. 5a). Then each
neighbor dimer would probably cause a different interac-
tion, and thus the J value represents the average of the
two interactions.

In summing up all the interactions in Fig. 4, the
interactions between the eight spin unit and neighboring
dimers are halved because only half of the interacting
spins belong to the eight spin unit. In Fig. 5a, one can
check whether the spins of a dimer (kl' k2) have similar
total interactions. Considering a factor of 1/2 for inter-
actions outside the unit (bounded by k1, k2, 11, 12), one
obtains similar totals, remembering that kl and k1' are
of the same sublattice and thus both are involved in the
interaction. A similar figure for the k-j interaction
gives the same result.

There is a Zeeman term in addition to the Ising
spin interaction. For each of the boundary field values
(Fig. 5b), the energies of the two bounding regions are
set equal. The values of the exchange constants can thus
be found from these four fields. Silz’ the 2 component
of Sil’ is represented by il' As shown in Fig. 4, the

sublattices are repeated through the crystal. The energy

is then found from equation 1.2,

 

 

 

 

 

 

H,
H3 “H D
H H
H H (3
H2 H ”B
H. H H

 

HA
0 3o

Figure 5. KMn913-2520 5:3 interaction in BC plane

Ising spin model.

 

29

E = -2 J

1 (11 12 + j1 j2 + k1 k2 + 11 12)

-4 J2[(il + i2)(kl + k2) + (jl +j2)(1l + 12)]
-4 J3[(il + 12)(j1 + j2) + (11 + 12)(k1 + k2)]
-4 J4[(k1 + k2)(j1 + j2) + (11 + 12)(il + 12)]
-guB H (i1 + i2 + jl + j2 + k1 + k2 + 11 + 12).

(3.1)

In the antiferromagnetic state (region A of Fig.

5b):

_ 2 _ _
EA — 8 s ( Jl + 4 J2 + 4 J3 4 J4). (3.2)

For the four other phases,

_ 2 _ _ _
EB — 4 S ( Jl + 4 J2 + 4 J3 4 J4) 2 S guB H (3.3)
(jl changes sign)
EC»: -16 S2 J4 - 4 S guB H, (k1 changes sign) (3.4)
_ 2 _ _ _ _ _
ED - 4 S ( J1 4 J2 4 J3 4 J4) 6 S guB H, (3.5)
(j2 changes sign)
_ _ 2 _
EE - 8 3 (J1 + 4 J2 + 4 J3 + 4 J4) 8 s ng H. (3.6)
(k2 changes sign)
Setting EA = EB at their boundary field, H1,
(2 J1 — 8 J2 — 8 J3 + 8 J4) = guB Hl/S° (3.7)

30

Similarly for EB = BC’ BC = ED, and ED = EE,
(2 J1 - 8 J2 - 8 J3) = guB Hz/S, (3.8)
(-2 J1 - 8 J2 - 8 J3) = guB H3/S (3.9)
(-2 J1 - 8 J2 - 8 J3 - 8 J4) = guB H4/S. (3.10)

From equations 3.7, 3.8, 3.9, and 3.10,
From 3.8 and 3.9,

J1 = - guB (H3 - H2)/4S. (3.12)

'Using 3.11 and 3.12 in 3.7,

J2 + J3 = -guB (H2 + H3)/l6 s. (3.13)

The J2 and J3 interactions are coupled. It is interesting
to note from 3.11 that

- H . (3.14)

H--H=H4 3

If an exchange field is defined as HE = ZzlJIS/guB,

and since the number of neighboring dimers from Fig. 4 is

21 =:1, 22 = z3 = 24 = 2, then

._ 1 _ = 1. = 1. _

31
B. Dipolar Anisotropy

It is believed that an anisotrOpic dipole-dipole
interaction could contribute significantly to the total
anisotropy. Since each spin has only one nearest neighbor,
there is no cancellation of dipole fields by symmetrically
positioned neighbors. The Mn-Mn separation is only 3.85 A°,
relatively short for manganese salts; and the interaction

is proportional to 1/r3:

Hd = Ai°Aj/(rij)3 - 3(Ei-Eij)(Ejo£ij)/(rij)5. (3.16)
The line joining the spins is Eij' and £1 = guBAi is the
spin magnetic movement.

The principal contribution to the anisotropy is
the spin-orbit interaction which results from the crystal-
line electric field quenching the spin-orbit degeneracy.

A change in the magnitude of the dipolar anisotroPy could
shift the easy axis (total anisotropy direction) if they
are not collinear. From equation 3.16, the lowest energy
state is with Bi and Aj parallel to Eij' which is the
dipolar anisotropy direction. For KMnC13-2H20, this di—
rection is almost perpendicular to the easy axis, certainly
not collinear.

Taking the nearest neighbor dipole interactions
for the eight Spin Ising model of Fig. 5b, the interaction

changes as the phase boundaries are crossed. In the

32

antiferromagnetic state (A), the spins make equal angles
(6) of about 90° with g; thus cos20<<1. For regions A
and E,

H =

_ _ 2 2 3
dA HdE -2(2 6cos 8) u /r (3.17)

The other regions give,

H = H (2 - 6 c0328) uZ/r3, (3.18)

dB dD

Hdc = o. (3.19)

If the Ising model is correct for this crystal,
an easy axis shift in going from region A to B should be
duplicated in going from B to C. Then there should be an
equal but opposite shift going from C to D and from D to
E. Thus regions A and E should have the same easy axis,

and likewise for B and D.
C. The AF-P Perpendicular Boundary

If a crystal has uniaxial anisotroPY: one can
calculate an AF-P phase boundary for fields applied per-
pendicular to the easy axis. It is required that the

spins on one sublattice, Al, be parallel to their total

effective field, Aeff ; or
1
Aeff X A1 = 0, (3.20)

33

whereAef = A - HEAz/MO, and H is given by equation

fl E

1.6.
The interaction from the uniaxial anisotroPy.
field, H , is assumed to be proportional to Al. With

-Az

the applied field, Hy’ along the y axis toward which the

Spins rotate, and 6 being the angle made with the z axis

(which is the easy axis), 3.20 becomes

 

 

1. i 15.
0 (Hy-HEMZSinG/MO) (HEM2+HAle)cosé/MO = 0
0 (M181n0) (Mlcosd)
(3.21)
Then,
MlHy - [MlMZHE/Mo - (HEM2+HAZM1)Ml/Mo]81n6 = 0. (3.22)
10

From Shapira and Fonen. M1 = M2 for a perpendicular
field, and sind = 1 (5 = 90°) at the AF-P transition.

Then the AF-P boundary is given by

Hy = (2HE + HA2) M1(T)/MO, (3.23)

where the magnitude of M1 and M2 is assumed to be inde-
pendent of the field, H.10

If an additional small anisotropy field, HAy(HAy<<HAz)'
is assumed to have its axis parallel to y, then applying a

field, Hy, gives a different AF-P boundary. Using 3.20,

34

 

 

i j k
0 = 0 [HY +(HAy Ml -HEM2)sin6/MO ] (H EzM2+HA M1)cos6/Mo .
0 M181n5 Mlcosé
(3.24)
Then,
0 = MlHy + (HAyMl-ZHEMZ-HAle)M181n6/Mo' (3.25)
Since M1 = M2 and sinG = 1 at the AF-P boundary,
Hy = (2HE + HAz HAy)Ml(T)/Mo. (3.26)

The anisotropy assists in aligning the spins with Hy'

and the AF-P intercept at T = 0 occurs at a lower field
that that given by equation 3.23.

With a field, Hx' applied perpendicular to this
HAy axis and the easy 2 axis, the spins rotate toward the

x axis. Then

i j k
(Hx-HEMzsiné/Mo) 0 (H NMZ le)cos(s/Mo = 0
M131n6 0 Mlcos6 (3.27)

 

 

This gives an equation similar to 3.22; and the AF-P

boundary is

Hx = (2HE + HA2) Ml(T)/MO. (3.28)

35

Since the spins have rotated in a plane perpendicular to
HAy' this additional anisotropy has no effect in this
case.

One can calculate the AF-P perpendicular boundary
near TN for a two sublattice Ising or Heisenberg model.
Assuming no anisotropy for the Heisenberg case, a field,
Hy' is applied perpendicular to the easy axis, 2, (as in

3.21) with 6 the angle made with the z axis (Aeff=Ay-AA2).

Then from 3.20,

 

i j k
0 Mlsiné Mlcosé = 0.
0 Hy-AMZSinG AMzcosd (3.29)

 

Then as in 3.22,
siné = Hy/[21M2(0,T)]; (3.30)
or at the AF-P boundary where 6 = 90°,

Hy = ZAM2(O,T). (3.31)

For T just below TN at zero field,10

1/2

M2(0,T)=NguBS[10(S+1)2/(682+68+3)] (1-T/TN)l/2. (3.32)

Then using kTN = (2/3)S(S + 1)zl|J
10

1|, the AF-P boundary

is,

(T -T = 2s2 2‘ 2 2
N ) ( + 25 + 1)g uB H /(120 kZTN). (3.33)

36

With the Ising model, only the z components con-

tribute to the effective field,

H = H - M (3.34)

Then, A. x 0 = (1M M sin6 - MlH)cos6, (3.35)

1 Eeff = 1 2

and sin6 = Hy/AMZ, or at the AF-P boundary,
H = AM2(O.T)- (3.36)
Using this in 3.32 would give
2

(TN-T) = (232 + zs + 1)g2uBZH2/3Ok TN, (3.37)

which is four times the result obtained from equation 3.33.

IV. EXPERIMENTAL METHODS

A. Experimental Apparatus

 

l. Dewar and Calorimeter

The low temperature apparatus consisted of the
pyrex helium dewar shown in Fig. 6, and the calorimeter
shown in Figs. 7 and 8. The dewar was made to specifica—
tions by H. S. Martin and Son, Evanston, Illinois. The
lower portion was tapered to fit between the poles of a
magnet. With the dewar filled with liquid helium and the
outer vacuum can (C) evacuated, a temperature of 1.0° K
could be attained by pumping on the bath in the helium
can (A).

Heat leaks to the helium can (A) were reduced by
several means. German silver, having a low thermal con-
ductivity, was used for the pumping lines from the outer
can flange to the inner can (B) and helium can (A). The
pumping lines leading from the main dewar was used for
the inner can (B) and helium can (A). A nylon spacer was
put in the outer can (C) to prevent this outer can from
touching the inner can (B). The inner can pumping lines,
containing the electrical leads, were surrounded at their

lower ends by the liquid helium in the helium can (A).

37

 

 

 

IOOOmm

475mm

38

   
 
    

 

l75.. .m
0.0

I45mm-
I.D.

ll5mmd

 

 

 

\\ \ \\\\\m\\ \ \\ \\ \\ X\ \\\

a

\\\ \\\\\\\\
\\\\\\\

 

 

\\\\ \

 

\\

 

 

 

 

 

 

 

‘(Y\\\\\

&A\\\ \\\\\\\\\\ \

 

Figure _6.

Pyrex helium

dewar.

M-30640 2mm
BORE STOPCOCK
(90‘ FROM FILL

PORTS)

32 mm 0.0. FILL
\PORTS, I80°
APART, 2" LONG

C ROSS HATCHING

REPRESENTS
VACUUM JACKETS

BOTH DEWARS ARE
SILVERED, WITH A
TAPERED SLI T, I0 mm
WIDE AT THE BOTTOM

‘— 66 mm 0.0.
-— 58 mm 0.0.

50 mm 0-0.
42 mm 0.0.

39

PUMPING LINE

~~>INNER CAN (8)
4, EVACUATION LINEs

I‘I—q‘T ~\ HELIUM CAN (A)
\

 

HELIUM CAN (A)
~— OUTER CAN (C)

CERROLOW Il7
SOLDER JOINT

 

 

 

 

BATH THERMOMETER
L..- EMBEDDED IN BOTTOM
TIP OF HELIUM GAN

./BRASS RADIATION SHIELD

    

 
 

t

4
'pl

 

INNER CAN (B)

GERMAN SILVER
SUPPORT FOR
TERMINAL BOARD AND
SAMPLE HOLDER

NYLON SPACER
<I W/ BAKELITE
V/ TERMINAL BOARD
CERRLOW II?

I r/ SOLDER JOINT
P— |ll_fl

Fig. 7 . Cross section of body of calorimeter.

 

 

 

 

 

 

 

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41

The electrical leads were varnished to the bottom of the
helium can (A) for good thermal contact.

To reduce radiation, right angle bends were put
in the pumping lines to the inner can (B), helium can (A),
and outer can (C). Also, a brass radiation shield was
placed below the ends of the inner can pumping lines, and
an extra shield was placed at the end of the outer can
pumping line. To increase the thermal resistance between
the sample and bath in order to isolate the sample, the
sample was supported by a nylon holder which was attached
to the bakelite terminal board. The terminal board was
connected to the helium can (A) by a strip of German

silver.

2. Sample Holder

 

A nylon c-clamp was made to size for each sample
and was attached to the nylon support (see Fig. 9). Nylon
was used for its low thermal conductivity, and also since
it could be cut easily and tapped for bolt threads.2
Screws had to be tight since the crystal would experience
a torque in the magnetic field. The larger nylon support
was used when it appeared that the KMnCl3-2H20 crystal
may have moved during a rotation experiment. When this

much stronger support was used, no change in crystal

position could be observed.

Figure 9.

#2

 

 

 

 

 

 

 

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Sample holders.

43

The nylon support and c-clamp provided a thermal
path between the bath and sample. On rotating the field,
the sample temperature would change considerably, and the
thermal path would tend to slowly bring the sample tem-
perature back to its original value. This heat leak would
have less effect on a sample with a large heat capacity.
Thus, the smaller support, with a lower thermal conduc-

tivity, was used for the smaller samples.

3. Vacuum Pumps

 

A Welch Duo-Seal pump was used to maintain a vacuum
on the U-tube manometers and to evacuate the McLeod gauge
after a reading. Another such pump was used as a forepump
for the air cooled Veeco EP 2AI 350 watt diffusion pump
which could attain a pressure of 10-6 mm Hg. This system
was used to pump out the inner and outer cans. A high
capacity Stokes vacuum pump was used to pump on the dewar

or the bath in the helium can. The pumping system is

shown schematically in Fig. 10.

4. Pressure Gauges

A mercury filled U-tube manometer was used to mea-
sure helium can pressures above 2.5 cm Hg. Below that
pressure an oil filled U-tube manometer was used to roughly
observe the lowering of the pressure in equal temperature
intervals for thermometer calibration, while a McLeod

gauge measured these pressures accurately.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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45

The high vacuum system for the inner and outer
cans could be read on a NRC 831 vacuum ionization gauge.
This simultaneously gave readings for the ion gauge at
pressures below .001 mm Hg and for two thermocouple gauges.
Both the helium can and the dewar had a U.S.G. pressure
guage which gave rough readings (30 in. vacuum to 15

P.S.I.).

5. Thermometer Current Supplies

 

The sample and bath thermometers were 1/10 watt,
56 ohm, Allen Bradley carbon resistors. One set of leads
carried a constant current of one or ten microamperes,
while another set measured the voltage across the resistor
potentiometrically.

The sample thermometer current supply consisted of
two 28 volt Mallory mercury batteries in series with three
precision resistors, a variable 20 megohm carbon potentio-
meter, and a 100 K ohm precision resistor, all totalling
56 megohms. The current could be adjusted to one micro-
ampere by setting the variable resistor while potentio-
metrically measuring the voltage across the 100 K ohm
resistor. The bath thermometer supply had 5.6 megohms
and 10 microamperes. It was noticed that room temperature
variations would give rise to fluctuations in these cur-

rent supplies. After the supplies for the sample and bath

46

thermometers were enclosed in a l/2-inch thick wooden box,

these fluctuations were substantially reduced.

6. Measuring Electronics

 

The sample thermometer voltage was measured by a
Leeds and Northrup K-3 potentiometer with a galvanometer
system consisting of a Leeds and Northrup 9835-B microvolt
amplifier and a Leeds and Northrup dual pen Speedomax G
recorder with a 5 millivolt range card. The amplifier
could be adjusted to give the amount of sensitivity de-
sired. The bath thermometer voltage was measured by a
similar system using the other pen of the two pen recorder
with a 10 millivolt range card. The circuits are diagrammed
in Fig. 11.

The voltage across the sample heater was measured
with a Data Technology 323 integrating digital voltmeter.
The heater current was measured by using a Leeds and
Northrup Speedomax G single pen recorder to read the
voltage across a precision resistor in series with the
heater. A filtered Lambda LM263 power supply (0 to 32
volts) in series with a set of variable resistors totaling
10 megohms provided the heater current. The heater was
turned on and off by a relay connected to an electronic
timer which was preset to run for a selected time inter-

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48

7. Magnet and Gaussmeter

The magnetic field was provided by a water cooled
Harvey Wells 22 KG magnet with a Harvey Wells DC power
supply providing up to 200 amperes at 80 volts. An elec-
tric motor with two reduction boxes could rotate the
magnet at 2.8, 14, or 70 degrees per minute. Angles were
measured using the 360 gear teeth on the base of the mag-
net. The field could be increased or decreased linearly
at any desired rate up to 3000 gauss per minute. This
sweep circuit compares the output from a stationary coil
on the magnet pole with the set reference voltage. This
voltage difference drives the integrator and high frequency
correcting amplifier, forcing the field to be proportional
to the time integral of the reference voltage. Using a
constant reference voltage gave a constant sweep rate.

The field was measured to within 10 gauss with a
Bell 660 digital gaussmeter using a Hall probe. The probe
was taped to the face of the pole, slightly off center,
where the field strength was equal to that on the mid-
point of the axis of the poles. The calibration of the
gaussmeter was checked with a Rawson-Lush rotating coil
gaussmeter using a Rawson-Lush 501 indicator. A nuclear
magnetic resonance experiment had demonstrated the Rawson-

Lush gaussmeter to be accurate to within 50 gauss.

49

B. Experimental Procedures

1. Sample Preparation

The LlCIlCl3 ' 2H2

aqueous solution containing approximately 14.9 grams of

0 crystals were grown from an

LiCl and 60 grams of CuCl '2H20, both reagent grade chemi-

2

cals. For KMnCl3'2H20, the solution contained approximately

179 grams of MnC12-4H20 and 27.0 grams of KCl, also re-

agent grade chemicals. These amounts were determined by

11 The solutions

examining the solubility phase diagram.
were kept at room temperature in a dry box which had a
small dehumidifier to enhance the crystal growth. The
crystals were generally grown to approximately one gram
in size.

The crystallographic axes were determined from
the prominent crystal faces and, if necessary, by x-ray
diffraction. From proton resonance and x-ray diffraction
data, LiCuCl3'2H20 has its easy magnetization axis in the
AC plane which is represented by a face normal to the B
axis. Since the crystal has the tendency to deteriorate
on handling, the x-ray work was done rapidly in order to
check the B axis orientation.

With KMnCl3°2H20, the crystal faces were not easily
recognized, and the easy magnetization axis was not per-

pendicular to any crystal axis. Thus, x-ray work was

imperative, not only for determining the axes but also to

50

decide whether the crystal was single or twinned. Figs.
1 and 3 show the crystal spin arrangements of LiCuC13'2H20

and KMnC13-2H20 respectively.

After determining the crystal axes, the crystals
were weighed. Then, a heater (approximately 400 ohms of
1.4 mil diameter, double enameled Evanohm* wire) was wound
on the crystal, and a 1/10 watt, 56 ohm Allen Bradley car-
bon resistor thermometer was mounted on the crystal. Both
heater and thermometer were connected at each end with two
six inch coiled manganin wires with nominal resistivity of
thirty ohms per foot. The resistors were cooled by a fan
during soldering to reduce any damage from overheating.

To protect the crystals and to provide mechanical

support for the heater and thermometer, the KMnCl -2H 0

3 2
crystals were coated with a thin layer of clear glyptal
(G.E. 1202 varnish). The LiCuC13‘2H20 samples, however,
were coated with Fluorolube** grease since they were not
sufficiently stable at room temperature to allow varnish
to dry. An 1/8" thick nylon c-clamp was then put on the
crystal and tightened so the crystal would not turn when

placed in a magnetic field.

 

*Obtained from Wilbur B. Driver Co., Newark, N.J.

**Obtained from Cenco.

51

2. Preparing for Experiment

 

Using information on the morphology of the crystal
as well as the direction of easy magnetization, generally
obtained from x-ray work and nuclear magnetic resonance,
the crystals were mounted so that the axis of easy magne-
tization was in the plane of the magnet's rotation. It
was possible to do this within five degrees of the desired
position. Further, the orientation first chosen was gen-
erally the one which was the least ambiguous in its di-
rection. Sometimes there was a small alignment error,
indicated by the fact that the critical fields were un-
usually high. Furthermore, the discernment of this
critical field by the adiabatic method became more and
more difficult at higher temperatures.

During the first run, a plot was made of the
critical field as a function of magnet rotation. This
plot would serve as a guide for insuring proper alignment,
since the minimum critical field generally was obtained
for the best alignment. This plot also served to indi-
cate any alignment error when the sample was rotated
ninety degrees about the magnetization axis. The small
corrections in angle were achieved trigonometrically by
measuring the positions with a cathetometer. This tech-
nique was repeated until the spin flop field was at its

lowest value.

52

After the clamp and sample were mounted on the
nylon holder, the leads were soldered to the terminal
board. Then the inner can (B) was sealed with low tem-
perature Cerrolow 117 solder, and the electrical leads
were checked at the terminals on tOp of the calorimeter.
Lead O-rings were fitted to the outer can (C) which was

then bolted tightly in place. To help keep the LiCuCl -2H 0

3 2
samples from melting during this preparation, the inner
(B) and outer can (C) evacuation lines were shut; and the
cans were submerged in liquid nitrogen.

Next the calorimeter was lowered into the dewar
and the dewar was then evacuated. The outer dewar was
filled with liquid nitrogen, and after about twenty minutes,
helium gas was put into the dewar to cool the calorimeter
overnight. The outer can (C) was usually checked for
leaks by pumping on it; and if vacuum tight, it was then
filled with helium gas.

3. Helium Transfer and
Thermometer CaIIiBration

 

Prior to the liquid helium transfer, the sample
thermometer resistance, atmospheric pressure, and room
temperature readings were taken with the sample at liquid
nitrogen temperature. To further check for any possibility
of vacuum leaks, the inner (B) and outer (C) cans were

evacuated separately to a pressure of 2x10-4mm Hg. If

53

the system appeared to be tight, approximately 2 mm Hg of
helium exchange gas were put into both cans.

The needle valve to the liquid helium can (A) was
closed and four or five liters of liquid helium were
transferred into the helium dewar by pumping on the dewar
to keep its pressure about one pound below the pressurized
storage dewar. The sample and bath thermometers were
monitored to observe their cooling. Using a level sensor
consisting of four carbon resistors placed at different
levels in the liquid helium dewar, it was possible to
check the height of the incoming liquid helium.

After transferring, the needle valve on the liquid
helium can (A) was opened to permit filling. Then the
magnet was rolled into position and the field was brought
up to the value chosen for calibration. The outer vacuum
can (C) was then evacuated in order to thermally isolate
the inner vacuum can (B) and liquid helium can (A) from
the liquid helium dewar. When the sample and bath thermo-
meters had reached equilibrium, the thermometer calibra-
tion was begun. This involved pumping on the liquid in
the helium can (A) which necessitated closing the helium
can needle valve. Generally 10-15 calibration points were
taken in the temperature range 4.2-1.0° K. The calibra-
tion process consisted of reading the vapor pressure of
the liquid on a mercury manometer and a McLeod gauge, and

reading the resistance of the sample thermometer on a

54

Leeds and Northrup K-3 potentiometer with a fixed current

of one microampere through the carbon thermometer.

4. Adiabatic Field Rotations

 

After the calibration had been completed, the
inner vacuum can (B) was evacuated to isolate the sample.
With the field at the calibration value, the magnet was
rotated while the sample resistance was observed. A maxi-
mum in resistance (minimum temperature) was sought, as
this indicated the position of the easy axis. The magnet
was then fixed at this position (equivalent to being
parallel to this easy axis). The field was then increased,
and the temperature of the sample changed along this adia-
batic curve. After the critical field was determined, 180
degree rotations were done in the antiferromagnetic and
spin flop phases. The rotations in the spin flop state
enabled one to determine the direction to which the spins
flOpped.

During a rotation, the bath temperature was held
nearly constant, but the small heat leak through the sam-
ple holder would cause the sample temperature to differ
slightly at positions 180 degrees apart. Because the
sample temperature tended to differ from the bath tempera-
ture during most of the rotation experiment, this caused
an annoying background warming or cooling in the sample.

For this reason any maximum or minimum temperature

55

position could be shifted and had to be corrected for
this heat leak. The background temperature effect could
be observed by stopping the rotation and noting that the
sample had not reached an equilibrium temperature. It
was for this reason that it was necessary to do rotations
of a few degrees, so that the sample could not be very
far from temperature equilibrium, and hence the minimum

position could be found more accurately.

5. Adiabatic Magnetizations

To determine the first order SF-AF (spin flop-
antiferromagnetic) boundary, the field, aligned with the
magnetization axis, was swept at 1000 or 2000 gauss per
minute. It was observed that the sample cooled while
still in the antiferromagnetic state; and when the spin
flop boundary was reached, the sample temperature would
either increase or would remain unchanged in agreement
with the expected effect from equation 1.32. Generally
data were taken at slower field sweeps, over smaller
field increments in order to minimize any possible heat
leaks.

The critical field for the SF-AF boundary was also
checked at several angles. The magnetic field was in-
creased to higher values in order to see whether the SF-P
(spin flop-paramagnetic) boundary could be reached. The

SF-AF boundary was consequently mapped out by taking

56

adiabatic curves for different starting temperatures. For

the case of KMnCl3'2H2

at the SF-P boundary and at the AF-P (antiferromagnetic-

O, inflection points were observed

paramagnetic) boundary by sweeping the field near these
boundaries. Both of these boundaries were also obtained

more accurately from specific heat data.

6. Specific Heat Measurements

 

In order to observe the AF-P boundary and SF-P
boundary, specific heat measurements were taken. With a
uniform slight warming or cooling of the sample thermometer
as a background, a heater current of approximately 10 micro-
amperes was used for twenty to forty seconds. Parameters
recorded were the heater voltage, current, time, and also
the number of divisions the chart recorder pen moved dur-
ing the heating period. Calibration of the chart was also
obtained by recording the pen positions at the extrema of
the chart.

Sometimes it was possible to visually observe on the
recorder the position of the maximum in the specific heat.
This occurred when the number of chart divisions suddenly
increased for a given heat input. These specific heat
measurements were taken for different values on the magnetic
field, thus enabling one to plot out the AF-P and SF-P

boundaries.

57

After a specific heat measurement at a given field,
it was generally necessary to introduce a small amount of
exchange gas into the sample container in order to cool
the sample to lower temperatures (and hence below the phase
boundary) before the start of another specific heat mea-
surement at another field. The inner can (B) was then

evacuated before more data was taken.

7. Removipg Crystal

At the end of the experiment, the helium in the
dewar and helium can (A) was pumped away. If an experiment
was to be rerun with the same sample orientation, the li-
quid nitrogen dewar was filled up so that the sample would
remain cold. Procedures outlined in C. were then repeated
for the next experimental run. When there would be several
days between runs for KMnC13-2H20, the dewars were allowed
to warm up (not adding any liquid nitrogen) with nitrogen
gas in them and in the inner can (B), both of which were
vented to the atmosphere by a one-way valve.

To remove the calorimeter, the liquid helium dewar
was filled with nitrogen gas. The inner vacuum can (B)
was filled with air or nitrogen before it was unsoldered.

Then the sample orientation could be changed or the crystal

could be removed after unsoldering the leads.

58

C. Data Analysis

 

1. Congerting Pressure
to Temperature

The manometer pressures were changed to pressures
at 0°C by correcting for the density change of the mercury,
and a hydrostatic pressure correction was made for all
points above the lambda point of helium to account for the
height of liquid helium above the sample. Below the lambda
point there is no temperature gradient in the liquid helium,
and hence no hydrostatic correction was necessary.

All pressures were read on a mercury manometer, ex-
cept those below 2.5 cm Hg, for which a McLeod gauge was
used. The corrected helium pressure readings were con-

4

verted to temperatures using the "1958 He Scale of Tem-

12

peratures." The first calibration point was taken at

liquid nitrogen temperature for which the temperature can

be calculated using13

255.821
6.49594 - iog10 p

 

T = + 6.6.

Here P is atmospheric pressure read in millimeters and

corrected to 0°C.

2. Thermometer Calibration

Equation

A least squares fit to the calibration points was

performed using a Hewlett Packard 9100A calculator and us-

ing the standard equation14

59

l2%—§ = a + b log R.

This calculation was done immediately after the calibra-

tion was completed so that the results could be used as a

close check on temperatures while running the experiment.
For the M.S.U. CDC 6500 computer, the equation was

modified to give a better fit:

-2
- a + b log R
T - log R 4 n
l-ZCn (log R)
o

 

A fourth degree polynomial gave the best fit to the cali-
bration points below 4.2°K. When extrapolation above these
points was needed (as e.g., for LiCuCl3'2H20), the poly-
nomial was omitted giving the basic two parameter equation.
This linear fit gave the best results when only the six
highest points were used.

3. Chart Recorder
CaIibration

 

For specific heat points, magnet rotations, and
field sweeps, the data was taken with the chart recorder
pen offset from its null position. The thermometer resis-

tance, R could be calculated on the computer from the

TI
potentiometer setting, R0, and the number of divisions,
D, that the recorder pen was to the right of null, using

the following equation:2

60

2 3
RO - [Cl-CZR - 2C3R - 3C4R ]D

1 + [C2 + 2C

 

RT = 2
3R + 3C4R ]D

An iteration process was used to determin RT' by in-
itially setting R equal to R0 to obtain a first approxima-
tion of RT' This new RT value was then substituted for R
in the above equation, to determine a second approximation
to RT’ etc. This process was repeated until two succes-
sive values for RT differed by 0.1%.

The coefficients, C, were found from a least squares
fit to the voltage calibration points using the relation:

R L - R R
O O

_ 2 3
DL - DR ‘ C1 + CZRT + C3(RT) + C4IRT) .

 

A voltage calibration point was taken whenever the increase
or decrease of sample temperature necessitated moving the
recorder pen across the recorder chart by changing the
potentiometer setting. The resistance change in these two
settings is represented by (ROL - ROR), and (DL — DR) is
the corresponding change in divisions (see Fig. 12a). The

resistance value, R used for the calibration point was

TI
approximately by the average resistance, (ROL + ROR)/2,
since the null was at the page center and the pen was

moved from one side of the page to the other.

61

(O)

RECORDER PEN PATH
I NULL
Lev—POSITION

 

 

 

 

 

 

 

 

Figure 12. Example of data on chart recorder.

62

4. Specific Heat, Rotations,
Fie Sweepg

For specific heat points, the heat input was the

 

 

product of the heater voltage, current, and heating time.
The sample mass and molecular weight were used in order
to convert heat capacity measurements to specific heat
results. The heat leaks were subtracted out graphically
as in Fig. 12a. Resistances were calculated for the
start and end of each specific heat point. The tempera-
ture difference was used in the specific heat calculation,
and the average of the two temperatures was taken as the
temperature for that point.

For magnetic rotations and field sweeps, the chart
recorder was marked when a certain angle or field was
reached. As in the specific heat measurements, the tem-
perature was determined from the resistance, which was
calculated from the potentiometer setting and the divisions
from the null position. Voltage calibration points were
taken whenever the pen had to be moved, as with specific

heat points (see Fig. 12b).

V. LiCuCl3'2H20 RESULTS

A. Adiabatic Rotations

In order to align the crystal so that the external
field lies along the direction of easy magnetization, it
is necessary to perform an adiabatic rotation of the ex-
ternal field in the AF state. The easy axis is determined
as the temperature minimum in this rotation. This can be
seen in Fig. 13a for H = 9.0 kG, when the rotation was
done in the AC plane (see Table 1 in Appendix). In fact,
this indication (Fig. 13b) is taken as a Sign that the
spin-flopping has occurred.2 Furthermore, this type of
behavior also indicates that the spins have flopped in
the plane (rather than out of it).

For comparison, rotations in the BC' plane (Figs.
14 and 15, and Table 2 of Appendix) show a small cusp
starting to deve10p at 11.2 kG at the easy axis position.

The cusp gets larger but fades out above 13 kG.
B. Adiabatic Mggnetizations

When the field is increased adiabatically, the
sample cools until the spin flop critical field is reached.

Then the sample temperature would level out or increase

63

61)

 

 

 

 

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65

Li Cu CI3-2HZO
(9.0 kG,BC' PLANE,C'=IIO°)

 

 

0"... ......
LGZI' ’ o
0.. ..
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e o
A LGO" . O
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2 '.'
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I" I540 20 4O 60 80 I00 I20 I40 l60 l80 200
POSITION (DEGREES)
Figure 14. LiCuCl3.2H20 BC' plane isentropic rotation.

66

Li Cu CI3-2HZO
.584. (BC' PLANE,C'=IIO°)

 

 

 

9 O
L.
'I.582
:2
o X
Lu I576 x l2 kG x
(I: - )( )( )< )( )(
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)— l.574-
<1
0: - A
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0- I.572§- A "-5 kG A
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Lu __ A A
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.566 i. I I l+.-I-+l l I I1 I
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POSITION (DEGREES)

Figure 15. LiCuClB-ZHZO BC' plane isentropic rotations.

67

as predicted by equation 1.32. These adiabatics are shown
in Fig. 16 and Table 3 in Appendix. The AF-P and SF-P
boundaries were found by specific heat measurements (Figs.
17, 18) and the magnetic phase diagram is shown in Fig. 19
(Tables 4, 5, and 6 of Appendix).

At the lowest temperatures, the AF-SF boundary
was slightly lower for the AC plane than for the BC' plane
(C' is the easy axis in the AC plane) (see Fig. 19). It
was thought that the easy axis might have been slightly
out of the rotation plane for the BC' case. At higher
temperatures, however, the AF-SF boundary for the AC
orientation occurred at higher fields and was more diffi-
cult to resolve than for the BC' case. Most of the dif-
ficulty lay in the reduced sensitivity of the sample
thermometer near 4°K, causing a lower signal to noise
ratio, and the fact that the temperature change of the
isentropes was smaller than that found for other antifer-
romagnets.

An interesting effect in the BC' plane is that at
the easy axis position there is a maximum for the critical
field as a function of angle (Fig. 20a and Table 7 of Ap-
pendix). However, the sharpest anomaly for a field sweep
occurred at the easy axis position, and the anomalies be-
came broader and less distinct as the angle differed from

the easy axis.

68

.mcofipwufiuocmmE ofimopucomfi ommm.mH050fiA

CE mmakdfiwdzwh

Nod

 

 

 

m and mmfi NON

K.» wk.» 8.» mo.»
6.» on...
_ _ u
.8 8. ......
mmaomhz mm.

 

mm;

mm...

.mH mpswfim

 

 

cu: m .....o 8 ...

_

AS

Loo.

IQN.

 

... 0.».

 

0.!

(S’I) H

69

.OLLI Cu CI3-2Hzo 4

 

 

 

 

 

2 O
-- 9kG
09 0
LG 0 a
- 09 P
I I I I I I 9
"2 4I 4.2 43 4.4 455.
A O
)5 2.0L
0
, _ 5kG 9 0 0
LIJ O
..J [(3_.
O .
E "'9 0° 0 900 0 6)
_I '2 I I ! I. .L I L J I9 J
3 '_ 4| 42 4.3 4.4 45
0
v o9
0.22" O O
0 P OkG
I.8- 9 ‘9
- o
0
L4 I I I I I I I I I3) I
4.0 4.5

4) 4.2 4.3 414
TEMPERATURE (°K)

Figure 17. LiCuCl3oZHZO Ipecific heat data.

2.0 — 7°

 

 

 

 

 

 

0 ., LICuCI3-2H20
r O
O 8
LG" 00 00 2' k6
Co
I” °
89
'2 I I I IQ L I I I I
39 40 4| 42 43
I.8I- 906
A .. 000 o@
g‘ @9996 I7 kG
I _ 0
Lu I.4 @0 0
-' L
8 . “’9
I I I I I I 11m I
\ "O 39 40 4I 42 43
_.I
S _ o
v I.8-— 0 0
0o. _ e o 4 I3 kG
9 O
I.4F- o
O
" (D
'0 I I I I I I I 4 I
' 3.9 4.0 4.I 4.2 4.3

TEM PERATURE (° K)

Figure 18. LiCuC13.2HzO specific heat data.

71

.moaempcsop ommno ommm.mHosofiq .mH whswfim

3.0. mmak<mma2mk

 

. . 0.0. Tm 0.m QN Na 0.. V. 0..
@_¢ .4 NW . . _ a _ . . . _ . . . . _
o
.8
B
E o.
*0» £O.£fl XQ +3.26... «I ‘6‘.“
2.43 .3
. 4
0 .
Aw Aw.v .nv.4u T+ nanfl.++ xv.
8415:19
8.4.3228 ems... $198.05...

.

0.0
0.N

0.0 _

 

0.NN

IO.2
9.8
9.4
o

.36

‘6 9.0
:I:
9.56

9.52

9.48

 

 

72

Li Cu CIB'ZHZO

 

 

 

F o (0)
" (BC' PLANE,C'=IIO°)
Q (T=I.6°K)
" O
__ O
I I I I I I I I I 9
l08 IIO Il2 |I4 ”6 HS
POSITION (DEGREES)
(b)
" (AC PLANE,C'=I54°)
- (T=I.I°K)
_. 0 0
_ O O
I l l I l l l l L .1
I50 l52 l54 I56 l58 |60

Figure 20. LiCuClB-ZHZO angular dependence of

critical field.

73

In the AC plane (Fig. 20b) there is a slight in-
crease in critical field as the angle is varied from the
easy axis. This is a more common effect and can be more
easily understood, since the field strength along the C'

axis is diminished when the field is off axis.

C. Antiferromagnetic-Paramagnetic

Boundary

l. Fielg Parallel to
the Easy Axis

 

 

This boundary is obtained from specific heat mea-
surements, the data for which is given in Figs. 17, 18,

and Table 4 of Appendix. For T close to T one can use

NI
equation 2.52 which neglects any intrasublattice inter-
action and assumes that the anisotropy energy is small

compared to the exchange energy. For the case with the

external field parallel to the easy axis this becomes,

392“82 (2s2 + 2s + 1) H

80kle31I STE + 1)

2
(5.1)

 

 

(TN-T)

A linear least squares fit to T vs. H2 constrained

to pass through TN at H = 0 is shown in Fig. 21. The fit

is,
_ -9 2
(TN—T) - 1.35 X 10 H , (5.2)

with H in gauss. Using equation 5.1, with S = % and

21 = 4, one obtains on comparing coefficients,

74
Li Cu Cla-ZHZO
AF - P BOUNDARY

 

 

0L 4 4 L 1

 

4.2 4.3 4.4
TEMPERATURE (°K)

Figure 21- 141690134320 fit to AF-P boundary.

75

J1
-k—= 0.420K. (5.3)

Then the exchange field can be approximated using 1.5,

28z [J I
HE = )(M 2 —-1-——1— . (5.4)
gIIB

The result is, HE = 12.5 kG.

If one uses the more general form for T (equation

N
2.48), it is possible to calculate an intrasublattice ex-

change interaction, i.e., using equation 2.50,

z J
(232 + 28 + 1) 9211132 [1 + 2 J2]H2

 

 

2
(TN - T) = 2 1 1
40 k TN
(5.5)
This gives,
z J
(TN — T) = .26 x 10 9 1 + H H2. (5.6)
l 1
J2
With z = 4 and z = l, ——-“ l7.
1 2 J1

Such a large intrasublattice exchange is in agree-

ment with the large amount of short range ordering per-

sisting above the Neel temperature.15 As deduced from

the specific heat measurements, 48% of the total magnetic

entropy is recovered above T Although the above calcu-

N.
lations have been made assuming that (TN - T) is propor-

tional to H2, it is of interest to examine how the present

data deviate (if any) from this relationship. For this

76

reason a plot has been made of (TN - T) vs. H, as shown in

Fig. 22a. This result is given analytically as

(TN - T) = 5.38 x 10'9 Hl°38. (5.7)

This deviation is not too suprising since the present data
are extended over a wider range of temperature than is
possibly valid for the two-sublattice molecular field
approximation.

This notion is further substantiated by using the

16

equation of Bienenstock describing the AF-P boundary as

ZSz J
T = TN [1 - (—g—)2] G , H = —1——-1- , (5.8)
c C guB

based on a two sublattice model. Plotting log (-%—) vs.
N
log [1 - {—3—07] , as in Fig. 22b, does not give a
c

straight line. However, a line drawn roughly through the
points gives G ” .02, which is lower than Bienenstock's
values of G = .35 and .36 for the simple cubic and bcc
lattices respectively. Although the crystal structure of
LiCuC13°2H20 is neither simple cubic nor bcc, the differ-
ence given above can not be accounted for merely on the
basis of crystal structure.

2. FielgpPerpendicular to
the Easy Axis

 

 

With the external field aligned along the b axis

and perpendicular to the c' axis, there are four data

 

 

 

 

 

29F
I5?
(OE-
Z3 7}-
i‘, 5—
I _
3—
2 I - I IlIIIIl I III II
.0I .02 .05 .l .2 .5
(TN-T)
".002“ (b) O
"_ 0
F2 .OOGI-
..\_ - O
3 ~ 9
".0I4-
O
I I I I I I I I
'18 “.6 ":4 “.2 2 0.0
LOG(I’(H/HC) I

Figure 22. LiCtClB-ZHZO lag-log plots for AF-P boundary.

78

points (see Fig. 19) which can be approximately fit by

(4.39 - T) = .54 x 10'9 H2. (5.9)

The ratio of this coefficient to that for the field paral-
lel is 0.4. From equations 2.50 and 2.51 we would have
expected a ratio of 0.33.

However, for the field perpendicular to both the b

and c' axes, an approximate fit gives,

(4.42 - T) = 0.04 x 10‘9 H2. (5.10)

Thus it would appear from this brief evidence that the
anisotropy in these two directions is not similar (see
3.23 and 3.26). Consequently the assumption of tetragonal
symmetry about the easy axis may not be valid for
LiCuC13'2H20, so that a modification in the original
equations might be necessary.
D. AnEiferromagnetic-Spin
Flop Boundary
The AF-SF boundary has a slight curvature and can

be fit to the equation,

H = 9.36 + 0.078 T + 0.173 T2. (5.11)

This boundary intersects the paramagnetic boundary at the

triple point, 4.2°K and 12.7 kG. The zero temperature

interCEpt is 9.36 kG (H(0)SF_AF). Equation 1.22,

79

1/2
_ 2
H(0)SF_AF - (2HEHA + HA ) , enables one to calculate an
anisotropy field (HA) of 3100 gauss. Expressing the aniso-

trOpy field as a temperature gives,

T = = 0.21°K. (5.12)

 

This is comparable to the temperature depression
by the external field of the AF-P boundary at the triple
point. This suggests that equation 2.44 is valid for
LiCuCl3°2H20 much closer to TN than has been assumed.

Such a difference also indicates that the anisotropy plays
a larger role in LiCuC13'2H20 than originally expected.
A more detailed comparison would require a modification

of equation 2.51, so that the anisotropy is taken into

account more explicitly.
E. Spin Flop:Paramagnetic Boundary

A linear least squares fit to T vs. H2 gives,

T = 4.323 — 0.654 x 10'9 H2, (5.13)

with H in gauss. The zero temperature intercept is 81.5 kG,
and the zero field intercept is 4.323°K. The latter tem-
perature depression is expected on the basis of the
molecular field approximation. Furthermore, one would
expect that the spin flop case should be similar to the

case with the field perpendicular to the easy axis, save

for the anisotropy energy. One finds that the coefficient,

80

0.654 x 10-9, is not much different than the coefficient,

0.54 x 10-9, in equation 5.9 which represents the case of
the external field perpendicular to the easy axis and

along the b axis.
F. Anisotropy Energy

The anisotroPy energy per unit volume can be found
from 1.17,

_ _ 1/2
HAP-SF — I2K/(>(L x”)] .

17

From a plot of magnetization vs. external field, the

susceptibilities at 1.4°K are,

(AL-X”)=(.026-.0078)emu/mole=.0182emu/mole.

(5.14)
Using the density of .0113 mole/cc, this can be expressed

4

as 2.06 x 10- emu/cc. Since the Spin flop field is 9.7

kG, the anisotropy energy is therefore,
K = 9700 ergs/cc. (5.15)

An alternative calculation can be made for K using

1.6 and 1.12, where A can be written as l/AL. Then,
K = HEHA AL: (5.16)

Using AL=.000294 emu/cc, one obtains K=1l400 ergs/cc,

which is in reasonable agreement with 5.15.

VI. KMnC132H20 RESULTS

A. Adiabatic Rotations

The first crystal tried was twinned, as is not un-

11 Performing magnetic rota-

common for these crystals.
tions in plane 1 (see Fig. 23), two temperature minima
were observed in the AF state (Fig. 24 and Table 8 of
Appendix), corresponding to two magnetization axes about
70° apart. At a field of 12.5 kG (Fig. 24), each minimum
is replaced by a relative maximum which indicates that

the spins have flopped within the plane of rotation. At
18 kG, this spin flopping effect disappears (Fig. 25 and
Table 8 of Appendix).

Rotating in plane 2 (see Fig. 23), an orientation
that was obtained from plane 1 by rotating 90° about the
easy axis, there is only one minimum in the AF state, as
expected (Fig. 26 and Table 9 of Appendix). Moreover, at
13 RG and 15 kG (Fig. 26 and Table 9 of Appendix) the
rounded minimum becomes flattened, which indicates that
the spins may have flopped normal to the plane of rotation
(Chapter I, Part D).

It was desirable to check these results by using a

single crystal. Using x-ray diffraction, a crystal was

81

82

PLANE 2 KMnCls'ZHZO
(.L TO PLANE I)

AB
PLANE

IST EASY ,

XIS A
7 T ”FY 90.
r '1

/' o L. ,l - " '

 

 

 

 

 

 

 

 

 

 

 

 

 

5 LANE I
V (J. TO AB PLANE)

PLANE 5 b 2ND EASY AXIS

(1 TO AB PLANE a .
AND PLANE I) "25

 

 

PLANE 3 IS .1. TO Ist EASY AXIS

PLANE 4 CONTAINS Is' AND 2"d
EASY AXES

Figure 23. Knn013'2H20 crystal with rotation plan...

TEMPERATURE (°K)

33

KMn CI3'2H20

. We I PLANE I I
o
s 9 '

 

 

 

_ O. o
o ' '
94- 0 0'0 . . 599°
_ .0 . o 0
.000 . C o
.92- . .0. o . 9
C
_ °. . 0. 0
0
90i- 9 o
" o o 9
38.. o 9.3 kG - 6w
' I25 kG o
.86 o .
C
.84 0
L ILLL l l _1_Ll 1 I
80 I20 |60 200 240 280

POSITION (DEGREES)

Figure 2“. Knn013-2H20 plan. 1 isentrOpio rotations

-tw1n.

8#

I.38 rKMnCI3°2HzO
(PLANE I, I8.0 kG)

_

l.36 - ' '
L 0

L34 I- 0

TEMPERATURE (°K)
1% ‘6 if:
r I l

in

m
I

O
o

 

I24 I- ' '- -

' ' l I 1 l l l l J J
20 60 I00 I40 l80 220

POSITION (DEGREES)

Figure 25. “11013-21120 plane 1 iuntropic rotation
- twin.

 

85

 

.. (PLANE 2) ,»~.
I.34IL60090. .. a.
I" 00 006 . O. 0y0%
‘. .e 0 . . 0 a" ..
L30" . ¢ . . 0 ..
A ‘ o . o
3‘ P . ° ' o '
"’ C) {III l<C5 . C) .
MLZG- 0 . . 0 0
a: ‘ 9
:> " .9 ‘ ' ° '
I— - . "
<I.22— .0 ‘ , ° '
0: o ‘ 0
LU _ e a. 0 o e
a. e o. ,o o
2 _ e 0.0 0 .
LlJI.I8 0 0 .
I'- _. ' 0 I3 kG o
O o
. G 09 9 e
I.l4 " O 9 O .0
e 0 0 .
— gm .- l5 kG
no .l 1 I I I "I'. I I I I
' 60 I00 I40 I80 220 260

POSITION (DEGREES)

Figure 26. KMnClB-ZHZO plane 2 ilentrOpio rotationl

- twin.

86

found that showed no twinning in plane 1. Now a rotation
in plane 1 confirmed that it was single (Fig. 27 and

Table 10 of Appendix). There was some evidence of spin
flopping in this plane as evidenced by the small hump in
the minimum position at 12.3 kG (Fig. 28, and Table 10 of
Appendix). As before, the small peak disappears at higher
fields. In plane 2 (Fig. 29 and Table 11 of Appendix),
there is a small flat region near the easy axis for 18 kG.

When investigating the AF-P perpendicular boundaries,
a rotation (Fig. 30 and Table 12 of Appendix) was done in
plane 3, which is perpendicular to the easy axis (shown
in Fig. 23). The temperature change on rotating is more
than could be explained by not having the easy axis quite
perpendicular to the rotation plane. This minimum tem-
perature position was designated the second easy axis
which was believed to have a lower anisotropy energy than
the principal (first) easy axis. This was deduced from
the relative temperature changes during the rotations.

The next rotation was in the plane of the first
and second easy axes (plane 4 of Fig. 23). These results
show a definite peak at the easy axis position at and
above 15 kG (Figs. 31,32, and Table 13 of Appendix). The
peak becomes higher and broader as the field increases.

Thus in region B of the magnetic phase diagram
(Fig. 33), some of the spins may flop 90° in plane 1 and

then do not remain flopped in regions C and D. In region

V‘fi rum- .-

I.58 -

87 I

KMnCls-ZHZO
(PLANE I,9.o kG)

I.54- °. I.
C. O.
_ x . . .
I50 -L .
- I...) ° '
O: 0 °
.. 3
L46 I; . .
L E . .
L42 - CL . '
2 o
_ E .
L38” 0
0
L34 I I I I I ..I. I I I I I I

 

 

40

so I20 ISO 200
POSITION (DEGREES)

240

Figure 27. “11013-2820 plane 1 iuntropic rotation

- single .

88

n

 

 

 

 

 

.onCHm l mCOAumpop ofiooppcomfi H mcmfio ommm. Hoczm .wm opswfim
Awwwmowe zo_._._mon_ HmzoS).
00..- 00. on. 0! 0». 0m. 00. on. 03 on. 0N. .
_ J . . _ _ _ _ _ _ _ _ No.
00.. . . . . M134
0 o o. e w
o o o o d
O O 3
«0.. 8 I00._
. . V
l
. e e o n .
3. 00. 0.0. 0.. mm. ASN-8.
3.0 . . . . mm I02
00.0 . . I02
.0. . Inn.
. 0,. 0.0. . . 0,. 0.0 L
8. _ 8:02.23 0NIN . ”.055. 2...

89

 

 

KMn 0I3 . 2 H20
*- (PLANE 2 )
0°00
ISO" 0 0
.. 0 o I8.0 kG
0
I... e'e 0
I052 o ... a. 0
A - 0 .e
0 .
3‘ L44 - ’ °0
V . O
LIJ __ O
s ' '°
0 O
|_ L36 00 0
‘<I e I C)
0: _ 0 e
LIJ 00. 0005099 §
EEEI.2H3HUWQ%59.": ‘QD
E ' I225 kG '
I20" .. .. e
| l2[. I I I I I I I I I J
’ 350 30 70 no I50

POSITION (DEGREES)

Figure 29. KMnCl3-ZHZO plane 2 isentropic rotations
- single.

90

K Mn CI3- 2 H20
IMF I PLANES)“
L- . . I . .0
l.60- '
_ l3.5 kG
§2I56 :.0e '..
LIJI.52_ ' """"
0: .- .
2:) Ir) ,' 9
2 - "
I.48- .
E 1 20.0 kG '.
a. F. .
E l.44I- .
'— e

 

 

#- .
L40 I- . . .0.
I I I I I I I I I "1
60 I 00 I40 I80 220 260

POSITION (DEGREES)

Figure 30. “11013-2320 plane 3 isentropic rotations
- linglO.

91

 

 

KMn Cls-ZHZO
(PLANE 4)
I.56 . ...
%°o '. 0 099
G) e
I.4a- o ‘, . °
A 0 0
VIAOI- 0 '
LIJ . ° 0
0: .. 9 .
:> 0 - 0
I- ' '
<l.32- o ,I3.0 kG , 0
0: 0 0
iii -. C) 9 e G,
2|.24" o o ,. 0
u.) 0 . . 0
I" .. <3 . qf’
O . o
I.IE5" CL) ' " Gf’IEBA:II((3
6
_ #9109
I08 I I I I I I I I I I
° 40 80 |20 I60 200 240

POSITION (DEGREES)

Figure 31. KHn01302820 plane 4 isentropic rotatione
- single.

 

 

92

 

 

 

 

_ KMnCI3°2H20 (PLANE 4)

 

I.27I". LI8 e
_. - ”.8 k6 . _. '59“; .
I.25- '- . ' l.l6-
,I3.5ks ' S
I.2I - , a I.I3 —
. x I5.5 kG '
-. o .—
I.I9 - 0 0 LIJ l,|| Ir- ' . '
O m 0
I- .e 2:) "e
H? — 2 I09— .
. . 0: .
L. I45 kG IéI-J 3
HS - , . ' a l.6| -' '6'5 k6 .
__ . I— _ ' . .
I.I3 - . ' I.59- ' ' ‘ '
I I J I I I I _|___|__
I30 I40 l50 I30 I40 I50

POSITION (DEGREES)

Figure 32. KHn013-2820 plane 4 isentrOpic rotations

- single.

93

K Mn C|3°2 H20

IsymboI l-IAIPIHHVIXIPIQIOIXIHtIsIGIIeIGIBIGIfl
| run |I |2|3|4lsls|7|slsllollIII2|I3II4II5|I61I7IIslIs|2d

 

 

22
A? X
zol: 67 d #-
x x .
Is— x 0
C x D X E
l6— fi 8
‘3 ° I-I.L EASY AXIs
I4— Abagém mA**+% 5% $ ‘7
l2- ~33, .9090. 9“?" a
9
I0- +
A +
._ ‘6‘, . .
0

6" ”v

" e
4,-gg . 4'

3: ‘$
2*- )9

TEMPERATURE (°K)

O I I I I I I I I I .LI

 

 

I.0 I2 L4 I.6 us 2.0 2.2 2.4 2.6 ’2—.s
Figure 33. KKnClB'ZHQO phase boundaries.

94

C, some of the spins may flop 90° in plane 4, and then do
not show flopping in region D.
B. "Antiferromagnetic-Spin Flop"
Boundaries

The boundaries between regions A and B and between
B and C are shown in the magnetic phase diagram (Fig. 33
and Tables 14 and 15 of Appendix). Some of the isentropes
are shown in Fig. 34 (Table 16 of Appendix). As shown in
Fig. 35 (Table 17 of Appendix), the lowest value of Hc2
(upper boundary) was usually at a different angle than
the lowest value of Hcl (lower boundary). In Fig. 35a,
the isentrope lepe change was the sharpest at 137°, and
it was not resolvable below 135°. In Fig. 35b, the slope
change could not be resolved below 162°.

Corresponding diagrams (Figs. 36, 37 and Table 18
of Appendix) are shown for the twinned crystal. The
symmetry of the twin is evidenced by Hc2 in Fig. 36, as
the 140° minimum corresponds to the 224° minimum. In both
figures, the two rotation minima at a given field corre-
spond to the Hcl curve. Thus the region between rotation
minima corresponds to the state B above Hcl'

As shown in the phase diagram (Fig. 33), the HC2
boundary is slightly higher when the angle for lowest HCl
was used. By using the best angle for Hc2’ the slightly

lower boundary was obtained. All the other boundaries

95

_ _ _ -_ _ m _ _

E

on;

N

n

 

 

 

 

.mcofiuwufipmcmmfi oaaoppcmmfi 0 mm. Hoczx .am mpswfim
Ci mmak<deEmk
¢N._ «N. ON. 9.. m... .... .
_ _ _ _ _ _ 1 _ _ _ _ o:
. .... . . . .. led.
.. . . nod.
000.. .0 [00*—
oo o... e
o. owe o
.. . I 0.9
3V 2:
16.2
N n
88:52 0 IN. 6 as;

(9)!) H

 

.mHucfim I wocwccmamc Lwfismcm camfim Havaufipo ommm.mflo:2¥ .mm mpsmfiu
Amwmmwmov ZO_._._mOn.
ON. 00. On. - mm. - Dh— mm. 03 mm. . .
a _ _ _ _ _ _ _ a _ _ _ q _ m:
. . . .. . LNu.
one o e o o o 1 OH
% e o l$.N-)
. XoN._u._. v_oN._u._. $ mm
«xwmzcfirn. N mzfin. .mzfiaI . (
23.. a -9...
E . 1 c .-
e e o e o eee loo¢_
. o~:~.n_o cs; -..:

 

 

 

 

97

N.

m.

E

n.

.cfizp I wEHcHE cofiumuOL

 

 

 

 

ccm mocmpcwawc pmasmcw pfimfiu Hwofipfino H mcmfia ommm.mfioczx .wn mpswfim
Amwmmmmov zo_._._mon.
omw CNN 9N on. o!
_ fl 4 _ _ _ — _ _ _
Q 3@ eagee IN-
” a + a I
As . + 9. Im_.mn
+ 2. a x I )
UH.
xx; _ ”x . ”XISW
x. Xinx. xmxqu*flx x .1
2: . 13 . ......
55.2.2.2 20.255 ... x . .019

x. N. "A ._ mzfin. .o~x~.».o as;

98

KMnCI3-2H20, PLANE 2, T=I.2°K

 

 

“H”, Xch, IROTATION MINIMUM
I'lfi- I
I6 bI, I
AIS -O I I
o - I I I
x
VI4— oi x”IX II
:E’ - I xx" I0
O
I3~ I G,II
.. e I
I2I- “I” o a J,
| I L I I *l I I I I I _l
ISO I60 I70 I80 I90 200
POSITION (DEGREES)
Figure 37. KHnCl -2320 plane 2 critical field angular

depe ence and rotation minima - twin.

99

(for a parallel field) were found using the angle that
gave the lowest Hcl' They were determined by specific
heat measurements as shown in Figs. 38, 39, and Table 19
of Appendix. Runs 1-13 are for the twin crystal, and
runs 14-20 are for the single one (Fig. 33).

C. Easy Axis Shift and
Dipolar Anisotropy

 

 

It is also unusual that the rotation minimum in
the AF State does not correspond to the best angle for
HCl or ch (Fig. 40 and Table 17 of Appendix). It was
thought possible that the nylon sample holder might be
twisting due to the magnetic torque on the crystal. How-
ever, after making a sample holder about ten times as
strong (Fig. 9), identical results were found. The ro-
tation minimum was 16° from the angle for lowest Hcl in
plane 4.

The dipolar anisotropy (Chapter III, B) predicts
that the easy axis shift on going from state A to B would
be equal to the B-C shift, but would be the reverse of
the C-D or D-E shifts. Looking at the rotation minima
and maxima for plane 4 (Figs. 32, 40), the A-B shift is
-4° and likewise for the B-C shift. One then might ex-
pect a + 4° shift for C-D, but a +5.5° shift is observed.

However, the best angles for HCl and ch are rela-
tively far away from these minima. Also, the angular

shifts of the other orientations do not follow a pattern

100

 

 

 

 

 

 

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I I.O kG , ' °
3.4p- . . .
26 2.02 2.06 2. I0 I 2.Il4 2.I8
° TEMPERATURE (‘10
Figure 38. KMnCl‘B- 2820 specific heat data - twin.

101

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103

predicted by dipolar anisotropy. There must be another
reason why the best angles for HCl and ch do not coincide
with the rotation minima.

One explanation is that the observed rotation
minimum is the result of the superposition of minima from
both the first and second easy axes. It is possible that
the second easy axis is not exactly perpendicular to the
first easy axis as was assumed. Then the second easy axis
could slightly shift the rotation minimum from the actual
position of the first easy axis. However, the critical

field, H should still have its best angle at the first

cl’
easy axis.
The effect should be most pronounced in the plane
of the first and second easy axes (plane 4), and it should
be a minimum when the second easy axis makes the greatest
angle with the plane of rotation (thus its projection on
the plane would coincide with the first easy axis).
Surprisingly the results concur, as there is a 16°
shift from the rotation minimum to the best angle for Hcl
in plane 4 (Fig. 40). Moreover, there is a 2° shift in
plane 1, which can be approximately obtained from plane 4
by rotating 90° about the first easy axis. Plane 2, which
is obtained from plane 1 by rotating exactly 90° about the

first easy axis (and is thus not very different from plane

4), shows a 14° shift for the same case.

“"'_'

fie- “DMA‘
V
:1

104

One could conclude that the best angles for Hc1
and ch locate the first easy axis. Then the dipolar (or
other) angle shift is given by these two angles.

D. Antiferromagnetic-Paramagnetic and "Spin
Flop-Paramagnetickaoundaries

The AF-P boundary can be compared to Foner's cal-

10 for a two sublattice Heisenberg model which

culation
assumes an anisotrOpic interaction that is very small
compared to the exchange interaction and assumes that the
intrasublattice exchange interaction is negligible com-

pared to the intersublattice interaction. With the field

parallel to the easy axis, the boundary is

2 2

“32 H2/(40 k

T - T = (262 + 26 + 1) g TN) (6.1)

.0031 H2,

(H in kG), which is similar to equation 2.51. Experimen-
tally, a least squares fit gives T = 2.76 - .0053 H2.
With the field perpendicular to the easy axis,

the curvature predicted from equation 2.51 or 3.33 is l/3

of equation 6.1, or TN - T = .0010 H2. The fit to the
data is T = 2.74 - .00087 H2 which is reasonably close.
An Ising model (3.37) predicts T - T = .0041 H2.

N
Another more curved perpendicular boundary appears

only when the field is near the second easy axis direction,

and this boundary is absent when the field is perpendicular

105

to this axis. This indicates a non-uniaxial anisotropy
component (equation 3.26) as did the rotation (Fig. 30)
with the first easy axis perpendicular to the rotation
plane. Since there are two perpendicular boundaries, this
may suggest the existence of two different sets of sub-
lattices, which might explain the two different "AF-SF"
boundaries (HC1 and H02).

As mentioned in section E of Chapter I, the para-
magnetic boundaries can be observed as an inflection
point in the isentrOpe. This was checked at different
fields as shown in Fig. 41a, b (Table 20 of Appendix).
There was some doubt about the boundary between D and A
since specific heat points did not show a peak there.

The run 9 isentropes (Fig. 410, d) did give the points

between A and D at 12.34 kG and 12.45 kG as shown on the
phase diagram (Fig. 33). This method gives somewhat more
uncertainty in the field value than do the specific heat

points, but the temperature uncertainty is about the same.

E. Exchange and Anisotropy Fields

 

In order to calculate the exchange fields, one
must extrapolate the boundaries to find their values at
T = O. A straight line was found to give a good fit and
the best extrapolation for the boundaries for A-B and
B-C. The values at T = 0 are, H = 10.5 kG and H = 13.2

1 2
kG.

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107

In order to examine the T = 0 separation of the
C-D and D-E boundaries, the extrapolation was done using
the curvature of the AF-P perpendicular boundary. Mole-
cular field theory suggests that the SF-P boundary can be
approximated by this boundary's curvature since the curva-
ture is inversely proportional to the exchange constant
(3.33), or directly pr0portional to IL, where.Il.“ XSF
for small anisotropy (1.16). Also, from the experimental
data, the perpendicular boundary's curvature approximates
the curvature of the two boundaries above 17 kG. The re-
sulting intercepts are H3 = 46.2 kG and H4 = 48.8 kG.

The theory suggests that H2 - H1 = H4 - H3. The
above results show 2.7 kG and 2.6 kG for these differences

which appear to confirm equation 3.14. The exchange

fields are calculated using 3.15. This gives

H = 16.5 kG, H + H = 14.8 kG, H = 1.3 kG.

El E2 E3 E4

(6.2)

Since HEl is the largest of the exchange fields,

one might possibly expect a broad peak in the specific
heat at a slightly higher temperature than TN, represent-

ing a disordering of the intradimer spin alignment, as

occurs with the vertex linked chain in CsMnCl3°2H20.18

19 al-

However, no such anomaly was observed up to 6.8°K,
though it is possible that such an anomaly may occur at

even higher temperatures. Moreover, a calculation from

108

this specific heat data showed that the entropy associated
with the magnetic transition at TN is within 1% of the
expected value for a spin 5/2 manganese ion. This implies
that if an anomaly is present, it is a rather shallow one
to account for this additional 1% of entr0py.

It would seem that the intradimer ordering occurs
at TN and that HEl
sum of the interdimer exchange fields (HEZ’HE3’HE4) as

is the same order of magnitude as the

expected from equation 6.2. In addition, Spence8 has in-
dicated that the coordination octahedra of the dimer are
similar to those of MnC12°2H20 in which the spins are

antiparallel within the dimer. Thus he concludes that

 

the exchange within the dimer is quite small. Since the

16.5 kG value found for HEl is not a very large exchange

field, it may be that such an explanation is valid for
the present sample.

By setting EA = E (equations 3.2, 3.6), the zero

E
_temperature value for the AF-P boundary is 2(HE2 + HEB)

29.6 kG (assuming that regions B, C, and D do not exist).
The same value is found for the small boundary between B

and D by setting EB = E (assuming that C does not exist).

D
Experimentally, the AF-P boundary is fit by T = 2.76-.00533H2,
giving HAE(T = 0) = 22.8 kG. From a least squares fit to
the higher field points of the other boundary, one obtains

T = 2.61 - .00522 H2, which gives H (T = 0) = 22.4 kG.

BD

109

These intercepts are not very different, as the theory
predicts, although they are somewhat lower than predicted.
The very unusual magnetic phase diagram for
KMnCl3°2H20 certainly indicates a rather complicated spin
flopping arrangement. The above analysis is clearly a
first approximation attempt to understand this behavior.
The fact that the Ising model with an eight sublattice
structure at 0°K gives reasonable agreement with the
boundary separations may or may not be fortuitous. The
author realizes that additional considerations for the
case of non-axial symmetry may perhaps be a bit more
fruitful in a more complete understanding of the magnetic

behavior of this compound.

CONCLUSIONS AND SUGGESTIONS

FOR FURTHER WORK

The magnetic phase diagrams of two new compounds

have been studied. For LiCuCl '2H 0, a single AF-SF

3 2
boundary has been observed, as well as information on
both the inter- and intrasublattice exchange constants.
For KMnCl3°2H20, a very unusual phase diagram has been.
observed. This compound appears to pass through several
magnetic phases. A first approximation to understanding
this behavior has been made by assuming an eight sublat-
tice model with an Ising type interaction at 0°K.

Some new techniques have been used for mapping
out a magnetic phase diagram. One is to use isentropic
magnetizations to find the second order AF-P or SF-P
boundary at regions where they make relatively small
angles with the temperature axis, since the specific heat
peak may not be resolvable in this case. Another is to
check the easy axis position by using a sensitive gauss-
meter to find the critical spin flop field at different
angles near the rotation minimum. This is essential for
a crystal like KMnC13-2H20 which has the lowest critical

field at a position that is 16° from the rotation minimum

in one plane.

110

111

The more accurate method of crystal alignment
with a cathetometer was used primarily for changing the
orientation slightly between runs. An improved sample
rotator with at least one axis of rotation that could be
rotated while in the calorimeter would be very helpful so
that one could bring the easy axis closer to the rotation
plane of the magnet in the event of a poor alignment.

Either specific heat or magnetic susceptibility
measurements should be made at higher fields for both
crystals to get a better approximation to the zero tem-
perature intercepts of the paramagnetic boundaries. A
more direct comparison between the theory and the experi-
ment could thus be effected.

Zero field susceptibility measurements for paral-
lel and perpendicular directions in both crystals could
give more insight into the anisotrOpy which does not ap-
pear to be uniaxial. Field dependent susceptibility
measurements should be made on LiCuC13'2H20, for which
our results in one plane show a decrease in the critical
field and less distinct temperature minima in the isen-
tropes when the field direction is varied from the easy
axis position. It would be interesting to observe any
changes in the critical field and in the resolution of
the peak in the susceptibility vs. field at these dif-

ferent angles.

REFERENCES

10.

11.

12.

13.

14.

15.

16.

T.

Y.

J.

REFERENCES
H. Morrish, The Physical Principles of Magnetism
(Wiley, 1965).

N. McElearney, Ph.D. Dissertation, Michigan State
University (1968).

H. Schelling and S. A. Friedberg, Phys. Rev. 185,
728 (1969).

Skalyo, Jr., A. F. Cohen, S. A. Friedberg, and
R. B. Griffiths, Phys. Rev. 164, 705 (1967).

H. Vossus, L. D. Jennings, and R. E. Rundle, J.
Chem. Phys. 32, 1590 (1960).

Heller, Phys. Rev. 146, 403 (1966).
Date and K. Nagata, J. Appl. Phys. 34, 1038 (1963).
D. Spence, J. Chem. Phys. (to be published).
Oguchi, J. Phys. Soc. Japan 29, 2236 (1965).
Shapira and S. Foner, Phys. Rev. B. l, 3083 (1970).

W. Mellor, A Comprehensive Treatige on Inorganic
and Theoretical Chemistry, 13! 367 (1932).

 

G. Brickwedde, H. van Dijk, M. Durieux, J. R.
Clement, and J. K. Logan, J. Res. Nat. Bur. Stand.
64A, 1 (1960).

T. Armstrong, J. Res. Nat. Bur. Stand., 53, 263
(1954).

R. Clement and E. H. Quinnel, Rev. Sci. Instr. 2;
213 (1952).

Forstat and D. R. McNeely, J. Chem. Phys. 35,
1594 (1961).

Bienenstock, J. Appl. Phys. 31, 1459 (1966).

112

113

17. S. C. Abrahams and H. J. Williams, J. Chem. Phys.
32, 2923 (1963).

18. R. D. Spence and K. V. S. Rama Rao, J. Chem. Phys.
53, 2740 (1970).

19. H. Forstat, J. N. McElearney and P. T. Bailey, Phys.
Lett. 27A, 549 (1968).

APPENDIX

 

 

 

 

Table 1. LiCuCl -2H20 AC plane isentropic rotations.
6 T(°K) 8 T(°K) 6 T(°K)
4/24/70 9.0 kG 4/24/70 9.0 kG 4/24/70 12.0 kG
52 1.6330 192 1.6075 184 1.5822
56 1.6333 196 1.6129 182 1.5803
60 1.6331 200 1.6182 180 1.5791
64 1.6321 204 1.6230 178 1.5794
68 1.6306 208 1.6279 174 1.5827
72 1.6285 212 1.6319 170 1.5865
76 1.6257 218 1.6362 166 1.5894
80 1.6224 224 1.6397 162 1.5919
84 1.6189 228 1.6410 158 1.5941
88 1.6150 234 1.6417 152 1.5957
92 1.6106 236 1.6417 148 1.5969
96 1.6058 240 1.6413 144 1.5974
100 1.6009 244 1.6402 140 1.5964
104 1.5951 248 1.6384 136 1.5933
108 1.5895 132 1.5878
112 1.5842 129 1.5838
116 1.5780 4/24/70, 12.0 kG 126 1.5872
120 1.5728 122 1.5985
124 1.5671 250 1.6533 118 1.6102
128 1.5620 246 1.6543 116 1.6164
132 1.5573 242 1.6547 112 1.6257
136 1.5537 238 1.6542 108 1.6352
140 1.5506 234 1.6527 104 1.6434
144 1.5487 230 1.6506 100 1.6519
148 1.5483 226 1.6477 96 1.6482
152 1.5493 222 1.6439 92 1.6639
156 1.5516 218 1.6395 88 1.6684
160 1.5554 214 1.6347 84 1.6726
164 1.5601 210 1.6293 80 1.6753
168 1.5659 206 1.6233 76 1.6777
172 1.5728 202 1.6161 72 1.6792
176 1.5792 198 1.6093 68 1.6796
180 1.5867 194 1.6016 63 1.6792
184 1.5932 190 1.5933 60 1.6780
188 1.5999 186 1.5861 56 1.6763
52 1.6734

 

114

Table 2.

 

 

LiCuCl

'2H

115

0 BC' plane isentropic rotations.

 

 

3 2
6 T(°K) 0 T(°K) 6 T(°K)
4/30/70 9.0 kG 4/30/70 11.0 kG 4/30/70, 12.0 kG

2 1.6250 108.0 1.5673 105 1.5756
5 1.6252 108.5 1.5668 106 1.5749
10 1.6253 109.0 1.5665 107 1.5744
15 1.6244 109.5 1.5663 108 1.5744
20 1.6238 110.0 1.5661 109 1.5748
25 1.6223 110.5 1.5663 110 1.5759
30 1.6204 111.0 1.5665 111 1.5750
35 1.6183 111.5 1.5669 112 1.5747
40 1.6155 112.0 1.5676 113 1.5749
45 1.6124 114 1.5755
50 1.6090 4/30/70, 11.2 kG 115 1.5764
55 1.6054

60 1.6015 113.0 1.5694 4/30/70, 13.0 kG
65 1.5971 112.5 1.5690

70 1.5929 112.0 1.5685 114 1.5828
75 1.5881 111.5 1.5683 112 1.5824
80 1.5824 111.0 1.5681 110 1.5827
85 1.5774 110.5 1.5681 109 1.5827
90 1.5714 110.0 1.5686 108 1.5827
95 1.5650 109.5 1.5686 107 1.5829

100 1.5584 109.0 1.5684 106 1.5833

103 1.5537 108.5 1.5684 105 1.5839

105 1.5509 108.0 1.5688

107 1.5486 107.5 1.5692

109 1.5475

111 1.5484 4/30/70, 11.5 kG

113 1.5513

115 1.5560 105.0 1.5722

120 1.5672 106 1.5713

125 1.5778 107 1.5706

130 1.5858 108 1.5703

135 1.5946 109 1.5708

140 1.5996 110 1.5719

145 1.6051 111 1.5705

150 1.6099 112 1.5706

155 1.6137 113 1.5712

160 1.6173 114 1.5720

165 1.6203 115 1.5731

170 1.6219

175 1.6236

180 1.6247

185 1.6251

190 1.6253

195 1.6250

200 1.6241

 

116

 

 

Table 3. LiCuCl3°2H20 isentrOpic magnetizations.

H(kG) T(°K) H(kG) T(°K) H(kG) T(°K)

4/20/70 AC plane 4/24/70 AC plane 4/30/70, BC' plane
8.65 1.0753 13.00 2.3791 11.00 3.5059
8.70 1.0748 12.60 2.3786 11.10 3.5047
8.80 1.0742 12.20 2.3779 11.20 3.5038
8.90 1.0736 11.80 2.3768 11.30 3.5031
9.00 1.0732 11.40 2.3754 11.40 3.5023
9.10 1.0728 11.00 2.3740 11.50 3.5017
9.20 1.0726 10.80 2.3732 11.60 3.5015
9.25 1.0725 10.60 2.3727 11.70 3.5009
9.30 1.0725 10.40 2.3728 11.80 3.5011
9.35 1.0725 10.30 2.3732 11.90 3.5011
9.40 1.0726 10.20 2.3738 12.00 3.5009
9.45 1.0727 10.00 2.3757 12.10 3.5008
9.50 1.0729 9.80 2.3781 12.20 3.5007
9.55 1.0731 9.60 2.3804 12.30 3.5015
9.60 1.0735 9.40 2.3829 12.40 3.5009

9.20 2.3856 12.50 3.5008
9.00 2.3885 12.60 3.5012

4/21/70,AC plane
9.10 1.5791 4/24/70 AC plane 4/30/70, BC' plane
9.20 1.5776

9.30 1.5763 9.00 3.0894 11.50 3.7678
9.40 1.5751 9.40 3.0855 11.60 3.7659
9.50 1.5741 9.80 3.0817 11.70 3.7643
9.60 1.5733 10.20 3.0784 11.80 3.7638
9.70 1.5727 10.40 3.0768 11.90 3.7631
9.80 1.5726 10.60 3.0752 12.00 3.7622
9.90 1.5729 10.80 3.0742 12.10 3.7619

10.00 1.5736 11.00 3.0730 12.20 3.7622

10.10 1.5746 11.20 3.0723 12.30 3.7621

10.20 1.5759 11.30 3.0722 12.40 3.7619

11.30 1.5775 11.50 3.0722

11.40 1.5790 11.70 3.0723

11.50 1.5806 11.90 3.0726

11.60 1.5820 12.30 3.0732

11.70 1.5833 12.60 3.0739

11.80 1.5844

10.89 1.5855

11.00 1.5863

 

117

Table 4. LiCuCl °2H 0 specific heat data.

 

 

 

 

3 2
O O O
T( K) Cp T( K) Cp T( K) Cp
4/23/70 0.0 kG 4/21/70, 9.0 kG 4/23/70, 17.0 kG
4.145 1.488 4.069 1.312 4.056 1.508
4.175 1.830 4.099 1.347 4.068 1.537
4.307 1.820 4.129 1.529 4.087 1.581
4.318 2.120 4.163 1.548 4.103 1.537
4.354 2.161 4.196 1.670 4.119 1.614
4.373 2.252 4.226 1.651 4.127 1.519
4.386 2.259 4.253 1.950 4.134 1.672
4.404 2.388 4.284 2.088 4.148 1.822
4.429 1.569 4.322 1.656 4.162 1.721
4.456 1.318 4.359 1.272 4.175 1.751
4.408 1.257 4.195 1.323
4.453 1.209 4.216 1.102
4/21/70, 5.0 kG 4.501 1.187 4.235 1.037
4.255 1.200
4.018 1.348 4.269 1.137
4.053 1.362 4/22/70, 13.0 kG
4.080 1.376
4.112 1.422 3.930 1.423 4/23/70, 21.0 kG
4.141 1.414 3.981 1.660
4.152 1.429 4.019 1.582 3.851 1.405
4.177 1.441 4.054 1.866 3.873 1.422
4.210 1.832 4.087 1.712 3.898 1.490
4.251 1.713 4.129 1.673 3.911 1.471
4.286 1.897 4.163 1.744 3.928 1.581
4.317 1.855 4.201 1.892 3.945 1.587
4.345 2.129 4.235 1.540 3.962 1.656
4.377 1.875 4.272 1.233 3.971 1.868
4.418 1.349 4.319 1.184 3.990 1.658
4.452 1.306 4.005 1.903
4.495 1.345 4.023 1.777
4.041 1.563
4.058 1.378
4.071 1.285
4.090 1.314
4.113 1.316

 

118

Table 5. LiCuCl '2H 0 isentropic magnetization critical
fields? 2

 

 

 

 

 

Sample 1 Sample 2 Sample 3
T(°K) H(kG) T(°K) H(kG) T(°K) H(kG)
4/20/70 AC plane 5/8/70 AC plane 6/10/70 BC' plane
1.079 9.49 1.235 9.66 1.370 9.65
1.110 9.52 1.510 9.82 1.550 9.89
1.240 9.62 1.850 10.01 1.770 10.07
1.360 9.66 2.170 10.26 2.270 10.44
1.450 9.72 2.800 10.81 2.950 10.90
1.610 9.82 3.170 11.65 2.980 11.00
1.833 10.00 3.510 11.90
2.210 10.27 3.920 12.50
2.560 10.58 6/5/70, AC plane
3.000 11.32

5/18/70, BC' plane 3.422 11.88
3.426 11.87
4/21/70, AC plane 1.180 9.85 3.760 12.30
1.600 10.08 4.060 12.80
1.574 9.82 2.120 10.38
2.020 10.12 2.600 10.76
2.380 10.47 3.090 11.20
3.440 11.60
3.630 11.80
4/22/70, AC plane 3.800 11.90
3.930 12.30
2.290 10.43 4.090 12.35
2.640 10.75
2.910 11.17
3.164 11.80 5/20/70, BC' plane

4/30/70, BC' plane

1.204
1.414
1.600
1.566
3.090
3.510
3.780

9.95
10.07
10.17
10.13
11.20
11.65
12.10

1.180
1.610
2.590
3.410

9.77
10.13
10.80
11.53

 

 

119

 

 

 

 

 

 

 

Table 6. LiCuC13'2H20 specific heat maxima - sample 1.
H“ easy axis HJ_ easy axis
T(°K) H(kG) T(°K) H(kG)
4/21/70 5/27/70
H” b axis
4.284 9.0 4.23 19.1
4.324 7.0 4.21 15.0
4.346 5.0 4.32 10.0
4.41 5.0
4/22/70
4.256 11.0 H_L b axis
4.201 13.0
4.179 14.0 4.43 10.0
4.146 15.0 4.42 15.0
4.41 19.0
4/23/70 4.42 5.0
4.168 16.0
4.175 17.0
4.107 19.0
4.010 21.0
4.203 14.5
4.385 2.5
4.404 0.0
Table 7. LiCuC13'2H20 angular dependence of critical
field.
BC' plane, C' = 110° AC plane, C' = 154°
T = 1.6°K T = 1.1°K
0 H(kG) 0 H(kG)
108 9.68 151.5 9.51
110 10.13 153.0 9.49
112 9.68 155.0 9.49
114 9.38 158.0 9.52
116 9.06
118 8.87
120 8.70

 

Table 8.

KMnCl

120

0 plane 1 isentr0pic rotations - twin.

 

3 2
6 T(°K) 0 T(°K) 0 T(°K)

1/29/70, 9.3 kG 1/29/70 9.3 kG 2/10/70, 18.0 kG
280 .9364 100 .9757 220 1.2455
276 .9386 95 .9765 216 1.2396
272 .9391 90 .9754 212 1.2365
268 .9383 85 .9723 208 1.2360
265 .9368 80 .9677 204 1.2371
260 .9329 200 1.2409
255 .9278 194 1.2488
250 .9211 1/29/70, 12.5 kG 190 1.2543
245 .9129 185 1.2596
240 .9042 250 .9799 180 1.2656
235 .8957 245 .9622 176 1.2694
230 .8882 240 .9397 172 1.2723
225 .8823 235 .9158 166 1.2751
220 .8791 230 .8873 162 1.2762
218 .8788 225 .8570 158 1.2769
216 .8790 222 .8378 154 1.2780
214 .8800 221 .8435' 150 1.2799
210 .8829 218 .8822 146 1.2828
205 .8895 215 .8864 142 1.2869
200 .8976 210 .8840 138 1.2929
190 .9128 205 .8584 134 1.2986
185 .9190 202 .8738' 130 1.3081
180 .9233 200 .8831 126 1.3186
174 .9263 195 .9074 122 1.3273
172 .9268 190 .9246 118 1.3376
170 .9269» 184 .9400 114 1.3467
169 .9270 180 .9468 110 1.3542
166 .9267 174 .9496 106 1.3602
164 .9263 170 .9476 100 1.3665
160 .9250 165 .9413 96 1.3683
155 .9233 160 .9317 92 1.3682
150 .9222 155 .9167 88 1.3600
148 .9221 152 .9105 84 1.3613
146 .9222 150 .9145 78 1.3506
144 .9229 148 .9268 74 1.3394
142 .9239 146 .9307 70 1.3300
140 .9252 143 .9350 65 1.3145
135 .9300 142 .9364 60 1.2977
130 .9370 140 .9351 56 1.2846
125 .9453 138 .9241 52 1.2717
120 .9535 136 .9093 48 1.2586
115 .9609 134 .9133 40 1.2416
110 .9684 132 .9227 36 1.2350
105 .9727 130 .9319 32 1.2312
128 .9427 28 1.2306

126 .9516 24 1.2313

 

121

Table 9. KMnCl °2H 0 plane 2 isentropic rotations - twin.

3 2

 

 

0 T(°K) 0 T(°K) 6 T(°K)

 

3/19/70, 11.0 kG 3/19/70 13.0 kG 3/19/70 15.0 kG

262 1.3650 64 1.3340 264 1.3137
257 1.3662 70 1.3376 256 1.3171
252 1.3653 74 1.3383 248 1.3138
248 1.3637 80 1.3367 240 1.3036
244 1.3603 88 1.3293 232 1.2849
240 1.3562 96 1.3173 224 1.2598
236 1.3504 104 1.3008 216 1.2258
232 1.3446 112 1.2799 208 1.1823
228 1.3363 120 1.2548 204 1.1563
224 1.3274 128 1.2254 202 1.1460
220 1.3182 136 1.1943 200 1.1396
216 1.3078 144 1.1620 196 1.1356
212 1.2963 150 1.1400 192 1.1278
208 1.2829 155 1.1244 188 1.1199
204 1.2701 158 1.1169 184 1.1099
200 1.2566 160 1.1142 182 1.1082
196 1.2428 164 1.1212 180 1.1085
192 1.2290 170 1.1325 176 1.1100
188 1.2162 176 1.1404 172 1.1132
184 1.2048 182 1.1493 168 1.1145
180 1.1951 186 1.1547 164 1.1154
178 1.1905 188 1.1560 160 1.1157
174 1.1857 190 1.1553 156 1.1155
169 1.1834 192 1.1515 152 1.1150
164 1.1864 194 1.1625 148 1.1169
160 1.1919 196 1.1740 144 1.1258
156 1.1992 200 1.1948 140 1.1403
150 1.2128 204 1.2142 134 1.1663
146 1.2245 208 1.2341 126 1.2038
140 1.2417 212 1.2510 118 1.2351
136 1.2535 216 1.2665 110 1.2623
132 1.2648 220 1.2798 102 1.2849
128 1.2762 228 1.3034 94 1.3026
124 1.2870 236 1.3198 86 1.3133
120 1.2969 244 1.3296 80 1.3176
116 1.3067 254 1.3338 76 1.3185
112 1.3166 264 1.3297 72 1.3179
106 1.3272 67 1.3144
102 1.3341

98 1.3400

94 1.3445

90 1.3485

86 1.3517

82 1.3535

77 1.3547

72 1.3540

 

122

Table 10. KMnC13'2H 0 plane 1 isentrOpic rota—
tions - sfngle.

 

 

 

8 T(°K) e T(°K)
8/27/70 9.0 kG 8/31/70 12.3 RG
35 1.5726 154 1.0795
40 1.5747 150 1.0535
45 1.5754 145 1.0503
50 1.5741 140 1.0515
55 1.5705 138 1.0500
60 1.5646 136 1.0350
65 1.5560 135 1.0255
70 1.5459 133 1.0320
75 1.5338 130 1.0510
80 1.5206 125 1.0858
85 1.5053
90 1.4890
95 1.4714 8/31/70 18.0 kG
100 1.4524
105 1.4342 120 1.0217
110 1.4145 125 1.0020
115 1.3960 130 0.9855
120 1.3790 135 0.9727
125 1.3632 140 0.9626
130 1.3510 145 0.9743
135 1.3440 150 0.9975
140 1.3422 155 1.0256
145 1.3468
150 1.3566
155 1.3712 8/31/70 15.0 kG
160 1.3885
165 1.4060 152 1.6368
170 1.4226 150 1.6156
175 1.4420 148 1.6008
180 1.4586 146 1.5897
185 1.4742 144 1.5844
190 1.4883 142 1.5832
195 1.5006 140 1.5915
200 1.5116 138 1.5995
205 1.5211 136 1.6063
210 1.5284 134 1.6156
215 1.5324 132 1.6261
220 1.5352 130 1.6394
225 1.5359
230 1.5351
235 1.5313

245 1.5180

 

123

 

 

 

 

Table 11. KMnC13'2H 0 plane 2 isentropic rota—
tions - s ngle.

0 T(°K) 0 T(°K)
9/8/70 12.25 kG 9/8/70 18.0 kG
170 1.1974 175 1.2849
165 1.1738 170 1.2712
160 1.1720 168 1.2697
155 1.1777 166 1.2747
150 1.1924 160 1.2818
145 1.2111 155 1.2840
140 1.2338 150 1.2850
130 1.2857 145 1.2856
125 1.3122 140 1.2884
120 1.3275 135 1.3013
115 1.3695 130 1.3270
110 1.3955 125 1.3559
105 1.4241 120 1.3830
100 1.4473 115 1.4090
95 1.4687 110 1.4265
90 1.4858 105 1.4498
85 1.5009 98 1.5036
80 1.5111 95 1.5261
75 1.5168 90 1.5568
70 1.5191 85 1.5818
65 1.5184 80 1.6016
60 1.5126 75 1.6130
55 1.5019 70 1.6178
50 1.4874 65 1.6164
45 1.4709 60 1.6074
40 1.4481 55 1.5905
35 1.4223 50 1.5662
30 1.3940 45 1.5356
25 1.3575 40 1.4955
20 1.3269 35 1.4516
13 1.2755 30 1.3942
10 1.2492 25 1.3662
4 1.2232 20 1.3381
0 1.2160 15 1.3118
353 1.1839 10 1.3016
350 1.1735 5 1.2886
345 1.1474 0 1.2742
340 1.1445 355 1.2747
335 1.1504 350 1.2600
330 1.1625 345 1.2680
340 1.2729
335 1.2753

325 1.2768

 

124

Table 12. KMnCl3
tions - single.

°2H20 plane 3 isentropic rota-

 

 

8 T(°K) e T(°K)
12/9/70 13.5 kG 12/9/70 20.0 kG
266 1.5754 64 1.4567
262 1.5733 72 1.4451
258 1.5722 76 1.4433
254 1.5722 80 1.4443
250 1.5734 88 1.4545
246 1.5759 96 1.4692
238 1.5824 104 1.4843
230 1.5913 112 1.4974
222 1.6003 120 1.5080
214 1.6091 128 1.5164
206 1.6172 136 1.5222
198 1.6238 144 1.5251
190 1.6285 152 1.5260
182 1.6320 160 1.5242
174 1.6336 168 1.5203
166 1.6334 176 1.5140
158 1.6314 184 1.5059
152 1.6288 192 1.4955
144 1.6241 200 1.4831
136 1.6175 208 1.4696
128 1.6097 216 1.4544
120 1.6008 224 1.4374
112 1.5913 230 1.4247
104 1.5813 238 1.4089
96 1.5722 246 1.3953
88 1.5648 250 1.3905
80 1.5608 254 1.3887
72 1.5613 258 1.3894
64 1.5659 262 1.3938

268 1.4041

 

125

frable 13. KMnC13°2H20 plane 4 isentropic rotations -

single.

0 T(°K) 0 T(°K) 0 T(°K)

 

12/11/70 13.0 kG 12/11/70 18.0 kG 12/11/70 14.5 RG

256 1.5062 244 1.4862 130 1.1732
248 1.5320 236 1.5016 132 1.1608
240 1.5320 232 1.5026 134 1.1481
234 1.5474 224 1.4908 136 1.1363
230 1.5454 216 1.4608 138 1.1283
222 1.5328 208 1.4150 140 1.1310
214 1.5078 200 1.3478 142 1.1359
206 1.4730 192 1.2731 144 1.1416
196 1.4150 184 1.2243 146 1.1468
188 1.3595 176 1.1984 148 1.1543
182 1.3143 168 1.1746 150 1.1652
174 1.2618 160 1.1413
166 1.2312 154 1.1245
158 1.1973 152 1.1309 12/11/70 15.0 kG
150 1.1657 148 1.1423
148 1.1483 140 1.1466 150 1.1714
146 1.1238 136 1.1457 148 1.1620
144 1.1150 132 1.1410 144 1.1578
140 1.1219 128 1.1284 142 1.1596
136 1.1410 126 1.1299 140 1.1595
132 1.1647 124 1.1375 138 1.1540
124 1.2181 118 1.1757 136 1.1507
116 1.2657 110 1.2268 134 1.1554
108 1.3095 102 1.2675 132 1.1679
102 1.3498 94 1.3167
94 1.4056 86 1.3671
84 1.4691 78 1.4294 12/11/70 15.5 kG
76 1.5015 70 1.4766
68 1.5269 62 1.5060 130 1.1095
60 1.5406 52 1.5174 134 1.0954
54 1.5431 44 1.5054 138 1.1113
46 1.5370 142 1.1174
37 1.5159 144 1.1150
148 1.1109
12/11/70 11.8 kG 12/11/70 13.5 kG 152 1.1244
132 1.2677 132 1.2244 12/11/70 16.5 kG
136 1.2569 134 1.2123
140 1.2494 136 1.2031 132 1.6127
144 1.2459 138 1.1897 136 1.5987
146 1.2457 140 1.1807 140 1.5933
148 1.2463 142 1.1779 144 1.5915
152 1.2522 144 1.1899 148 1.5961
156 1.2622 146 1.2079 152 1.6060

160 1.2778 148 1.2204

 

126

 

frable 14. KMnCl '2H20 isentropic magnetization boundary
pointg.
m
0 O O
fI( K) Hc1(kG) T( K) HC2(kG) T( K) H(kG)
1/27/70 twin 1/29/70 twin 2/13/70 single
(inflections)
0.997 11.80 1.041 14.46
1.085 11.90 1.083 14.46 1.983 11.8
1.246 12.09 1.143 14.46 1.856 12.34
1.314 12.18 1.206 14.44 1.786 12.45
1.384 12.28 1.260 14.40 2.503 7.0
1.462 12.39 1.314 14.36 1.770 18.1
1.540 12.50 1.358 14.33
1.602 12.62 1.409 14.33
1.570 12.54 1.468 14.36
1.583 12.57 1.516 14.54
1.588 12.59
1.617 12.60 3/5/70 single
1/30/70 twin 1.208 13.84
1.263 13.84
1.540 12.50 1.388 13.86
1.645 12.62
3/10/70 single
2/2/70 twin
1.151 13.85
1.437 12.37 1.495 13.95
1.562 14.12
2/6/70 twin
3/17/70 single
1.620 12.62
1.676 12.64 1.170 13.80
1.480 13.91
8/27/70 single
8/27/70 single
1.234 11.98
1.283 12.02 1.240 13.80
1.396 12.19
1.520 12.41 8/28/70 single
1.637 12.55
1.150 13.80
8/28/70 single 1.252 13.85
1.353 13.85
1.142 11.83 1.432 13.90

 

Table 15.

KMnCl

127

0 specific heat maxima.

 

 

 

3
H easy axis H easy axis KL easy axis
T(°K) H(kG) T(°K) H(kG) T(°K) H(kG)
1/30/70 twin 2/13/70 twin 12/2/70 single
H in plane 1
1.681 13.50 1.706 12.80 g
2.120 11.00 1.865 12.80 2.698 8.00
2.039 11.50 2.585 14.00
1.945 12.00 2/25/70 twin 2.416 20.00
2/2/70 twin 1.574 16.30 12/7/70 single
1.788 16.30 H in plane 2
1.603 14.00 1.784 15.03
1.567 14.20 1.797 14.06 2.578 8.00
1.715 13.00 2.675 8.00
1.89 12.50 2/27/70 twin 2.278 14.00
2.192 10.50 2.569 14.00
2.298 9.50 1.786 17.40 2.490 17.00
1.754 19.00 2.386 20.00
2/3/70 twin 1.726 20.20
1.683 21.40 12/9/70 single
2.407 8.50 HHto second easy
2.485 7.50 8/27/70 single axis
2.578 6.00
2.650 4.50 2.102 11.00 2.263 14.00
2.679 3.50 2.568 14.00
2.724 2.00 8/28/70 single 2.383 20.00
2.737 0.00
1.597 16.00
2/6/70 twin 1.519 20.00
2.422 8.00
1.500 14.50
1.894 12.60 8/31/70 single
2/9/70 twin 1.789 13.50
2.620 5.00
1.558 15.64
1.574 17.00 9/8/70 single
1.563 18.00
1.547 19.00 1.644 13.50
2/10/70 twin
1.536 20.20
1.493 21.40
1.552 15.00

 

Table 16.

KMnCl

128

-2H20 isentropic magnetizations.

 

3

H(kG) T(°K) H(kG) T(°K) H(kG) T(°K)
8/28/70 single 3/27/70 twin 3/20/70 twin.
11.64 1.1491 14.90 1.2177 12.00 1.5407
11.70 1.1458 14.80 1.2111 12.10 1.5367
11.76 1.1432 14.70 1.2106 12.20 1.5330
11.82 1.1491 14.60 1.2100 12.30 1.5293
11.88 1.1432 14.50 1.2095 12.35 1.5273
11.94 1.1485 14.40 1.2095 12.40 1.5255
12.00 1.1538 14.35 1.2099 12.45 1.5238
12.12 1.1576 14.30 1.2105 12.50 1.5232
12.30 1.1547 14.20 1.2122 12.55 1.5241
12.48 1.1503 14.10 1.2143 12.60 1.5253
12.72 1.1445 14.00 1.2163 12.65 1.5257
12.96 1.1378 13.90 1.2185 12.70 1.5254
13.20 1.1325 13.80 1.2203 12.80 1.5250
13.44 1.1263 13.70 1.2223 12.90 1.5244
13.68 1.1212 13.60 1.2241 13.00 1.5230
13.80 1.1194 13.50 1.2261 13.10 1.5217
14.04 1.1184 13.40 1.2279 13.20 1.5201
14.28 1.1185 13.30 1.2296 13.30 1.5183
14.52 1.1189 13.20 1.2313 13.40 1.5165
14.76 1.1194 13.10 1.2329 13.50 1.5148
15.00 1.1199 13.00 1.2346 13.60 1.5130
15.24 1.1204 12.90 1.2362 13.70 1.5109
12.80 1.2378 13.80 1.5086
12.70 1.2393 13.90 1.5071
12.60 1.2406 14.00 1.5049
12.50 1.2421 14.10 1.5029
12.40 1.2430 14.20 1.5011
12.35 1.2430 14.30 1.4997
12.30 1.2421 14.40 1.4990
12.25 1.2400 14.45 1.4987
12.20 1.2349 14.50 1.4987
12.15 1.2279 14.55 1.4993
12.10 1.2238 14.60 1.5002
12.05 1.2235 14.65 1.5011
12.00 1.2252 14.70 1.5019
11.95 1.2279 14.80 1.5029
11.85 1.2323 14.90 1.5036
11.80 1.2347 15.00 1.5043

11.75 1.2374

 

129

Table 17. KMnC13'2H20 critical field angular dependence,
single crystal.

0 Hcl(kG) Hc2(kG) 0 HCl(kG) HC2(kG)

 

8/27/70, plane 1 12/11/70, plane 4
T = 1.2°K T = 1.3°K
135 13.80 145 14.31
137 12.38 13.88 147 14.17
138 12.16 13.96 149 14.06
139 14.03 152 13.97
140 11.99 153 12.60
141 11.98 14.03 155 13.96
142 11.99 156 12.47
143 12.02 ' 158 14.05
145 12.13 159 12.40
145.5 12.17 160 14.14
162 12.37 14.28
164 12.39 14.45
9/8/70, plane 2 166 12.42
T = 1.2°K
162 12.65 13.49
164 12.42 13.48
166 12.29 13.50
168 12.14 13.54
170 12.04 13.61
172 11.98 13.71
173 11.95
174 11.94 13.83
175 11.94
176 11.93 14.00
177 11.93
178 11.94 14.18
179 11.96
180 14.38
181 12.00
183 12.06
185 12.16
187 12.26

189 12.43

 

 

130

vm.va omN

mN.wH mNN
mm.ma mNN
mm.ma qNN

oo.vH om.NH NNN
.mm.NH HNN
om.va mN.NH CNN
mv.¢a mHN
mo.NH bHN
mv.vH mo.NH mHN
mo.NH mHN
Nm.vH no.NH vHN
NH.vH mN.NH NHN

 

 

 

omH.m¢H o.hH me.~a Ham
nma.oma c.6H «H.4H cam
HmH.mmH o.mH on.va mom
maa.mma m.¢H
mma.mha.bma o.qH M.N.H u a
¢mH.mmH m.ma H mamam .on\~m\m
HmH.~mH o.mH mH.mH «ma
mmfl.mmfl m.~H oH.~H oma «6.4H om.~a mma
vma.mma m~.~H mo.~a was m¢.¢a Hm.~H mma
ops o.~H mo.~a baa Hm.¢a ~m.~a Hma
mo.~a was mm.¢a omH
Oaxaaxm .M.N.H u a .N 88638 ma.qa mo.~H mas nm.¢a oa.ma mva
mo.mH was «4.8H mo.~H eve
mam 54H o.mH mm.ma mo.~a «as mu.va ma.~H m¢H
58H m.eH mu.MH ops o¢.¢a m~.~H mva
www.mam HmH.H¢H o.vH mH.~H mma 6H.¢H mm.~a sea
mm~.oam NmH.o¢H m.ma H5.ma mos mm.ma ~m.~H sea
mmm.oam mma.mma o.ma mo.ma 66H OH.QH mma
H-.HH~ mmH.HwH m.ma 6m.MH 46H mm.va and
on\h~\m ehxmmxm M.~.H u a s.~.H u a
x.~.a n a .H mamam m mamam .oa\ma\m H mamam .on\m~\m
8525855 a Augie hegemom onvaom o wavmom onvaom o
.Hmumhuo

CH3» .mEHcflE coaumuou paw mocmpcmmmp Hmaswcm pawflm HMUHMHHO ONmN. HUGZM .mH wands

 

Table 19. KMnCl °2H20 specific heat data - twin.
0 O O
T( K) Cp T( K) Cp T( K) Cp
2/3/70 0.0 kG 2/2/70, 14.0 kG 2/25/70, 16.3 kG
2.665 6.390 1.573 2.074 1.531 2.049
2.676 6.395 1.578 2.130 1.536 2.105
2.685 6.850 1.582 2.096 1.542 2.154
2.696 6.979 1.587 2.138 1.547 2.288
2.706 7.428 1.591 2.216 1.551 2.248
2.715 8.185 1.595 2.163 1.556 2.401
2.724 8.954 1.598 2.281 1.560 2.484
2.731 8.881 1.602 2.415 1.564 2.709
2.737 9.732 1.606 2.389 1.568 2.896
2.746 7.671 1.610 2.364 1.571 2.791
2.758 4.275 1.613 2.282 1.574 3.083
2.774 3.627 1.617 2.259 1.578 2.894
1.621 2.255 1.581 2.889
2/3/70, 6.0 kG 1.625 2.161 1.585 2.883
1.629 2.123 1.588 2.717
2.511 5.032 1.632 2.096 1.592 2.501
2.519 5.198 1.596 2.464
2.529 5.427 2/25/70, 14.0 kG 1.600 2.506
2.538 5.647 1.604 2.437
2.547 5.672 1.763 2.392 1.608 2.544
2.555 5.924 1.767 2.471 1.611 2.470
2.563 6.110 1.771 2.550 1.615 2.400
2.571 6.534 1.776 2.493 1.619 2.409
2.578 6.716 1.780 2.493 1.756 2.524
2.585 6.272 1.784 2.562 1.763 2.542
2.593 5.054 1.789 2.550 1.767 2.444
2.603 4.790 1.793 2.527 1.770 2.554
1.797 2.603 1.774 2.573
1.800 2.530 1.778 2.662
1/30/70 11.0 kG 1.804 2.556 1.782 2.652
1.808 2.427 1.785 2.771
2.042 3.104 1.812 2.474 1.788 2.795
2.053 3.164 1.816 2.384 1.792 2.749
2.064 3.229 1.820 2.419 1.796 2.728
2.075 3.241 1.824 2.352 1.803 2.602
2.086 3.284 1.829 2.352 1.807 2.480
2.096 3.505 1.833 2.436 1.811 2.498
2.106 3.724 1.837 2.425 1.815 2.421
2.116 4.028 1.819 2.476
2.125 3.933 1.823 2.455
2.136 3.708 1.827 2.445
2.147 3.286
2.159 3.109
2.179 2.978

 

132

 

Table 20. KMnC13°2H20 isentropic magnetizations,
2/13/70.
H(kG) T(°K) H(kG) T(°K)
(a) (C)
5.6 2.539 11.5 1.897
5.8 2.534 11.6 1.893
6.0 2.530 11.7 1.888
6.2 2.525 11.8 1.884
6.4 2.520 11.9 1.880
6.6 2.515 12.0 1.875
6.8 2.510 12.1 1.870
7.0 2.503 12.2 1.865
7.2 2.496 12.3 1.859
7.4 2.490 12.4 1.852
7.6 2.485 12.5 1.847
7.8 2.482 12.6 1.843
8.0 2.478 12.7 1.839
8.2 2.474 12.8 1.836
8.4 2.471 12.9 1.833
8.6 2.469 13.0 1.829
8.8 2.467 13.1 1.827
9.0 2.464 13.2 1.825
9.2 2.462 13.3 1.822
(b) (d)

12.6 1.960 11.6 1.822
12.5 1.962 11.7 1.818
12.4 1.963 11.8 1.814
12.3 1.965 11.9 1.810
12.15 1.968 12.0 1.806
12.0 1.973 12.1 1.802
11.9 1.977 12.2 1.798
11.8 1.983 12.3 1.793
11.7 1.989 12.4 1.789
11.6 1.993 12.5 1.783
11.5 1.997 12.6 1.778
11.4 2.001 12.7 1.773
11.3 2.004 12.8 1.770
11.2 2.008 12.9 1.767
11.1 2.011 13.0 1.765
11.0 2.014 13.1 1.762
13.2 1.760
13.3 1.758