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Experimental studies of integer spin antiferromagnetic chains

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Experimental studies of integer spin antiferromagnetic chains
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Granroth, Garrett E., 1971-
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Thesis (Ph.D.)--University of Florida, 1998.
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Includes bibliographical references (leaves 231-239).
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by Garrett E. Granroth.

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EXPERIMENTAL STUDIES OF INTEGER SPIN ANTIFERROMAGNETIC
CHAINS









By

GARRETT E. GRANROTH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1998














ACKNOWLEDGMENTS


I am sincerely grateful for the guidance and encouragement of my thesis ad-

visor Professor Mark Meisel. Many thanks go to Professors Selman Hershfield,

Gary Ihas, Kevin Ingersent, Stephen Nagler, Fred Sharifi, and Dan Talham for

serving on my supervisory committee at one time or another. Special thanks go

to the materials wizards Brian Ward and Liang-Keui Chou and their supervisor

Professor Dan Talham for synthesizing the samples. Several people have provided

help with various measurements. Thanks to Dr. Murali Chaparala for help with

the cantilever magnetometer and interpretation of the resultant data. Thanks to

Dr. Brian Chakoumakos for assistance with the powder neutron diffractometer

and for advice concerning Rietveld refinement. Thanks to Dr. Jeff Lynn and

Dr. Ross Erwin for assistance with the time of flight neutron diffraction experi-

ments. Thanks to Brian Ward, Liang-Keui Chou, and Candace Seip for operating

the 9 GHz ESR spectrometer. Thanks to Gail Fanucci for running the NINAZ

on the high field ESR spectrometer. Thanks to Nelson Bell and his advisor

Professor Jim Adair for preparing the ball milled samples and for characterizing

their particle size. Thanks to Dr. Thierry Jolicceur for help with many aspects

of the theory for S = 2 chains and for providing the high temperature series

expansions for S = 1 chains. I am grateful for several fruitful discussions with

Kingshuk Majumdar and Dr. Arnold Sikkema that have helped me to understand

non-linear sigma models and other theoretical obfuscations. Several individuals












have provided tutelage along the path to this degree. Professor Stephen Nagler

helped with the neutron scattering experiments. Professor Satoru Maegawa gave

stimulating input on the NINAZ work. Dr. Philippe Signore provided the es-

sential knowledge that is passed down the line of graduate students. Professor

Kevin Riggs introduced me to experimental studies of magnetic materials us-

ing microwave resonance techniques. I would also like to thank my high school

chemistry teacher Mrs. Dorris Wasson for encouraging me to choose a career in

science.

I am always grateful for the encouragement of my parents and family that

has helped me along the way. Additionally, I would like to acknowledge the

Creator who provides endless scientific problems to study. Finally, I am indebted

to Amelia Helmus for grammatical editing of this thesis.















TABLE OF CONTENTS


ACKNOWLEDGMENTS ........................... ii

LIST OF FIGURES .............................. vii

ABSTRACT .................................. xi

CHAPTERS

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

2 REVIEW OF THEORETICAL STUDIES ................... 3
2.1 Theoretical Models ..................... 4
2.1.1 The Non-Linear Sigma Model ................... 4
2.1.2 Valence Bond Solids ......................... 6
2.2 An Intuitive Picture of the Haldane Gap ................ 9
2.3 Numerical Studies of Quantum Spin Chains ............. 10
2.3.1 Realistic Hamiltonians ...................... 18
2.4 Measurement Theory ......................... 22
2.4.1 Electron Spin Resonance ..................... 24

3 REVIEW OF EXPERIMENTAL STUDIES ................. 30
3.1 Haldane Gap Materials ........................ 30
3.2 Experimental Tests of the Haldane Gap ................. 32
3.3 Studies of End-chain Spins ...................... 40
3.4 Other Experimental Results ....................... 45

4 EXPERIMENTAL MEASUREMENT TECHNIQUES ......... 48
4.1 SQUID Magnetometer ........................ 48
4.2 Cantilever Magnetometer ....................... 54
4.3 Electron Spin Resonance ....................... 61
4.4 Neutron Scattering .......................... 68













Mechanical Ball Milling . .
Centripetal Sedimentation . .
Inductively Coupled Plasma Mass Spectrometry


5 T M N IN . . . . .

5.1 Synthesis and Structure of TMININ . . .
5.2 Magnetization Measurements on TMNIN . .
5.3 Comparison with Other Experiments . .


6 NINAZ . . .

6.1 Material Description . .
6.2 Other Experiments . .
6.3 Experimental Studies . .
6.3.1 Samples . .
6.3.2 [Ni(C4H12N2)2(/ -N3)]n(ClO4)n
6.3.3 Macroscopic Measurements .


. . 92

. . 93
. . 96
. . 99
. . 99
. . 103
. .. 108


6.3.4 Microscopic Measurements: Electron Spin Resonance 117


7 THE S = 2 HALDANE GAP IN MnC13(bipy) . .

7.1 S = 2 Quasi-linear Chain Materials . . .
7.1.1 MnCl3(phen) . . . .
7.1.2 Mn(acac)2N3 . . . .
7.1.3 Mn(salpn)OAc . . .
7.1.4 MnCl3(bipy) . . . .
7.1.5 Synthesis . . . .
7.2 Macroscopic Magnetic Measurements of MnCl3(bipy) .
7.3 Low Temperature Crystal Structure: Preparation for Microscopic
Measurements . . . .
7.4 Inelastic Neutron Scattering Measurements . .


8 CONCLUSIONS . .

8.1 TMNIN . .
8.2 NINAZ . .
8.3 MnCl3(bipy) . .


153
154
154
154
157
161
161
164

175
180

196

196
197
198


. . .O . .
. . . O
. . .












APPENDICIES

A High Temperature Expansions ....................... 200
A.1 Specific Heat ............................. 200
A.2 Perpendicular Magnetic Susceptibility . ... .. 214
A.3 Parallel Magnetic Susceptibility . . ... ..220

B Low Temperature Crystallographic data for MnC13(d-bipy) ...... .. .226

BIBLIOGRAPHY . . . ... .. 232

BIOGRAPHICAL SKETCH . . ... ..240















LIST OF FIGURES


2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11

3.1
3.2
3.3
3.4

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11


5.1 TMNIN Crystal Structure . .
5.2 SQUID M vs. B for TMNIN . .


79
.. 81


F and M vs. B1 for TMNIN packet 16 cm off the center of the field
F and M vs. B11 for TMNIN packet 16 cm off the center of the field
F and M vs. B1 for TMNIN packet 5 cm off the center of the field
F and M vs. B1 for TMNIN single crystal . .


A single valence bond .........................
An S = 1/2 chain in the RVB picture ................
An S = 1 chain according to a VBS model . .
The VBS state for an S = 2 chain . . .
S = 1 chain dispersion curve . . .
Staggered magnetization for S = 1 chains . .
l/X vs. T for end-chain spins from QMC work . .
S = 1 phase diagram . . . .
S = 2 phase diagram . . . .
Energy level diagram showing the Haldane gap . .
Allowed transitions between end-chain spin states . .

M(H) for NENP . . . .
Inelastic neutron scattering intensity for Y2BaNiO5 .........
X(400 pK < T < 300 K) for pure NENP . .
The crystal structure of NENP . . .

M(T) SQUID background . . .
M(H) SQUID background . . .
Cantilever magnetometer diagram . . .
Cantilever magnetometer electronics schematic . .
Schematic diagram of the X-band ESR spectrometer .
Cross-sectional schematic view of the ESR flow cryostat .
dI/dH vs. H for a 0.1 mg sample of DPPH . .
I vs. H for a 0.1 mg sample of DPPH . .
NIST TOF spectrometer . . .
A normalized log-normal distribution . .
Schematic diagram of an ICP-MS system . .


6
7
7
8
13
16
17
19
21
23
27

34
36
44
46












5.7 F and M vs. B11 for TMNIN single crystal .............. 88

6.1 Room temperature NINAZ crystal structure . .... ..94
6.2 DC x(T) of NINAZ . . ... .. 97
6.3 Constant Q z 7r inelastic neutron scattering data for deuterated
NINA Z . . . .. .. 98
6.4 Powder NINAZ particle size distributions . ... .. 101
6.5 Ultrafine powder NINAZ particle size distributions ... .102
6.6 The crystal structure of [Ni(C4H12N2)2(M-N3)]n(C104)n .. 105
6.7 X-(2K 6.8 xii(2K 6.9 X(50 mK < T < 2 K) of polycrystalline and powder NINAZ .. 109
6.10 X(T) for powder and ultrafine powder NINAZ . ... ..110
6.11 M(H < 5 T, 2 K) for polycrystalline, powder, and ultrafine powder
NINAZ samples . . . ... .. 112
6.12 x(T) for doped NINAZ samples . . ... .. 114
6.13 M(H < 5 T, 2 K) for doped NINAZ samples . .... ..116
6.14 Typical ESR lines for polycrystalline, powder and ultrafine powder
samples of NINAZ . . ... .. 119
6.15 ESR line for a polycrystalline sample of NINAZ ... ..120
6.16 ESR spectra for a polycrystalline NINAZ sample as a function of
angle . . . . .. 122
6.17 ESR peak g values as function of angle in a polycrystalline sample 123
6.18 Hg doped NINAZ ESR spectrum . ... .. 125
6.19 Hg doped NINAZ ESR peak g values as function of angle 126
6.20 T dependence of ESR lines for H I to the chains of a NINAZ
polycrystalline sample from Batch 1 . ... .. 129
6.21 T dependence of ESR lines for H 11 to the chains of a NINAZ
polycrystalline sample from Batch 1 . ... .. 130
6.22 T dependent ESR spectra for H 11 to the chains of a NINAZ poly-
crystalline sample . . . ... .. 131
6.23 T dependence of ESR lines for H 1 to the chains of a NINAZ
polycrystalline sample from Batch 2 . ... .. 132
6.24 ESR I(T) for a NINAZ polycrystalline sample from Batch 1 133
6.25 ESR I(T) for a NINAZ polycrystalline sample from Batch 2 134
6.26 T dependence of the ESR line for a powder sample of NINAZ 136
6.27 T dependence of the ESR line for an ultrafine powder sample of
NIN A Z . . . .. .. 137
6.28 I(T) for powder and ultrafine powder NINAZ samples ....... ..138












6.29 Energy level diagram for powder NINAZ . ... .. 140
6.30 ESR FWHM vs. T for powder and ultrafine powder NINAZ 142
6.31 1/x vs. T for powder and ultrafine powder NINAZ ... .144
6.32 The ratio of the area under ultrafine powder data to the area under
the powder data for NINAZ . . ... .. 145
6.33 x(T) T for the powder and ultrafine powder samples of NINAZ
from Batch 1 . . . ... .. 147
6.34 NINAZ ESR spectrum at 93.934 GHz, increasing H ... ..148
6.35 NINAZ ESR spectrum at 93.934 GHz, decreasing H ... ..149
6.36 NINAZ ESR spectrum at 189.866 GHz, increasing H ...... .. .150
6.37 NINAZ ESR spectrum at 189.866 GHz, decreasing H ...... .. .151
6.38 g vs. v for ESR in a NINAZ powder sample . .... ..152
7.1 Approximate crystal structure of MnCl3(phen) . ... ..155
7.2 M(T) for MnCl3(phen) . . ... .. 156
7.3 Crystal structure of Mn(acac)2N3 . ... .. 158
7.4 M(T) of Mn(acac)2N3 . . ... .. 159
7.5 Crystal structure of Mn(salpn)OAc . ... .. 160
7.6 x(T) of Mn(salpn)OAc . . ... .. 162
7.7 Crystal structure of MnCl3(bipy) . ... .. 163
7.8 x(T) of MnCl3(bipy) . . ... .. 166
7.9 9 GHz ESR signal intensity vs. T for MnCl3(bipy) ... ..168
7.10 SQUID M vs. H measurement of MnCl3(bipy) . ... ..170
7.11 Raw M vs. H data for three crystals of MnCl3(bipy) ...... .. .171
7.12 Raw M vs. H data for one crystal of MnCl3(bipy) ... ..172
7.13 M vs. H data for two oriented crystals of MnCl3(bipy) 174
7.14 Powder diffraction pattern, Rietveld refinement, and difference
plot for MnCl3(d-bipy) . . ... .. 177
7.15 Crystal structure near a Mn atom in MnCl3(d-bipy) ... ..178
7.16 a vs. T for MnC13(d-bipy) . . ... .. 181
7.17 b vs. T for MnCl3(d-bipy) . . ... .. 182
7.18 c vs. T for MnCl3(d-bipy) . . ... .. 183
7.19 7-y vs. T for MnCl3(d-bipy) . . ... .. 184
7.20 V vs. T for MnCl3(d-bipy) . . ... .. 185
7.21 Estimated S = 2 dispersion for MnCl3(bipy) and theoretical TOF
spectrometer behavior . . ... .. 186
7.22 Incoherent scattering from a vanadium standard ... ..188
7.23 Incoherent scattering from the MnCl3(d-bipy) sample ...... .. .189
7.24 Inelastic neutron scattering intensity at T = 0.4 K and Q r 191











7.25 Inelastic neutron scattering intensity at T = 0.4 K and Q < r 192
7.26 Inelastic neutron scattering intensity near Q = 7r at T = 10 K 194












Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


EXPERIMENTAL STUDIES OF INTEGER SPIN ANTIFERROMAGNETIC
CHAINS


By


Garrett E. Granroth


May 1998

Chairman: Mark W. Meisel
Major Department: Physics
Experimental studies of several S = 1 and S = 2 antiferromagnetic chain ma-
terials have uncovered novel phenomena. Firstly, measurements on
MnCla3(CloHsN2) provided the first evidence of an S = 2 Haldane gap. Secondly,
studies of the S = 1 material Ni(C3Ho10N2)2N3(C104), NINAZ, demonstrated the
presence of S = 1/2 end-chain spins and provided evidence of their interactions
with the magnetic excitations on the chain. Thirdly, investigations of the S = 1
material (CH3)4N[Ni(N02)3], TMNIN, revealed a small single-ion anisotropy, D.
Finally, the magnetic properties of MnCl3(C12H8N2), Mn(C5H702)2N3,
Mn(C17H1503N2)CH3COO, and Ni(C4H12N2)2(p-N3)]n(C104)n were studied in
a search for additional systems with properties similar to the aforementioned
materials.












Magnetic field, H, dependent magnetization, M, measurements at T = 30 mK

for H < 16 T have demonstrated MnC13(CioH8N2) to be the first known S = 2

material to posses a Haldane gap. Specifically, M 0 until critical fields

H, = 1.8 0.2 T and Hc11 = 1.2 0.2 T are reached for H oriented perpen-

dicular and parallel to the chains respectively. These critical fields and the intra-

chain exchange J = 34.8 1.6 K reveal a Haldane gap of A = 2.3 0.8 K and

D/J = 0.010 0.003, which are consistent with theoretical predictions for S = 2

antiferromagnetic chains. Furthermore, preliminary inelastic neutron scattering

data possess a possible magnetic scattering peak at an energy consistent with the

Haldane gap.

Polycrystalline, two powders with different particle sizes, and doped samples

of NINAZ were used to study end-chain spin effects. Fits to M(H < 5 T, T = 2 K),

for all the samples, confirm that the end-chain spins are S = 1/2. The electron

spin resonance, ESR, intensity as a function of temperature, provided a micro-

scopic comparison of the chain lengths to the macroscopic particle size distribu-

tions. Finally, the ESR line widths of the powder samples are consistent with

a model describing interactions between the magnetic excitations on the chains

and the end-chain spins.

Measurements of M(H < 8 T, T = 25 mK) performed on both an oriented

packet of single crystals and individual single crystals of TMNIN have revealed

H, = 2.60 0.15 T and H |1 = 2.40 0.15 T. These critical fields, along with

J = 10 K, determine A = 3.5 2 K and D/J = 0.06 0.03.














CHAPTER 1
INTRODUCTION


Ever since the pioneering theoretical work by Bethe [1], studies of antifer-
romagnetic chains have illuminated new phenomena related to quantum mag-
netism. A particularly intriguing theoretical prediction, made by Haldane [2,3],

is that Heisenberg antiferromagnetic chains have a gap between a singlet ground
state and a triplet excited state for integer spin, in contrast to the half-integer
spin case that has no gap. This prediction sparked a flurry of theoretical and
experimental work that is reviewed in Chapters 2 and 3, respectively. The ma-
jority of this dissertation presents experimental studies on three materials that
provide insight into several different phenomena related to the Haldane state.

Chapter 4 explains the experimental techniques used to study these samples.
Chapter 5 discusses magnetization measurements identifying the Haldane gap
and a small single-ion anisotropy in the S = 1 material (CH3)4N[Ni(N02)3],
TMNIN. Chapter 6 details our extensive studies of S = 1/2 end-chain spins in the

S = 1 system Ni(C3Ho10N2)2N3(C104), NINAZ, using magnetization, magnetic

susceptibility, and electron spin resonance. In addition to the intrinsic crystal

shattering processes, chain breaks were generated in NINAZ primarily by pulver-
ization techniques, and in a few cases by doping. Chapter 7 covers the search and
discovery of the S = 2 Haldane gap. The discussion is focused on MnCl3(bipy),
where bipy = C10H8N2, for which our magnetic field dependent magnetization









2


measurements at T = 30 mK provide the first evidence of the S = 2 Haldane

gap. In addition, preliminary neutron scattering results, which microscopically

test the S = 2 Haldane gap, are presented. Finally, Chapter 8 will review the

conclusions of the previous chapters and provide suggestions for further study.














CHAPTER 2
REVIEW OF THEORETICAL STUDIES


Since the nascence of quantum mechanics, quantum antiferromagnetism has

provided puzzling theoretical problems. The Heisenberg Hamiltonian


"/= g- S- Si+1 S(2.1)
i


describes the system where J, the exchange, characterizes the strength of the

interaction between nearest neighbor spins and Si is the spin at the ith site.

The ground state of this system is inherently difficult to find, so several approx-

imations have been used. A common approximation is the classical limit, where
S -+* oo. In this limit, the three-dimensional ground state is long-range ordered

with no net magnetization because each spin is oriented opposite its nearest neigh-

bors. This state can be represented by I t4"t4It4"t4">. However, when the spins

are quantized, it is easily shown that a state exists below the classical ground

state [4]. Nevertheless, the true ground state remains unsolved. Consequently,

several other simplifying assumptions have been made, and the one pertinent to

this thesis is to reduce the system to one dimension. Even the one-dimensional,

1-D, case is difficult to handle theoretically. Attempts at using the classical ap-

proximation reveal a negatively divergent correlation length, known as the long

wavelength divergence. Most of the theoretical progress on 1-D antiferromagnets

has involved creative methods of avoiding this divergence. Bethe in 1931 [1,5]












successfully circumvented this divergence for the S = 1/2 Heisenberg model by

a technique know as the Bethe ansatz. Unfortunately, the resultant ground state

is hard to manipulate, so further predictions about the dispersion were not made

until 1962 [6]. The difficulty of the theory discouraged theoretical work for higher

spins values. It was not until Haldane [2,3] used clever limits on an approximate

model that evidence of a gap in the integer spin case was observed. Since then,

several approximate models have been used to describe the properties of integer

spin antiferromagnetic chains. Therefore, this chapter begins with an overview

of the two primary theoretical models used: the non-linear sigma model (NLaM)

and the valence bond solid (VBS) model. Since the details of these models may

obfuscate the physical processes behind the predicted phenomena, a physically

intuitive argument for gapped integer and gapless half integer spin chains will

be given next. Then, a discussion of realistic Hamiltonians and the numerical

studies that illuminated their properties will be provided to connect the theory

to the experiment. The chapter will finish with several theoretical details on the

experimental techniques used.


2.1 Theoretical Models


2.1.1 The Non-Linear Sigma Model

Haldane proposed, in 1983, that integer and half integer Heisenberg spin

chains would have a fundamentally different excitation spectrum [2,3]. By mak-

ing an approximation, to be described below, he proposed that the integer spin

case has a singlet ground state and a triplet excited state separated by an energy












gap, A, and the half integer spin case is gapless. The half integer spin case is
completely consistent with previous exact work, but the integer case was an un-
expected prediction that started a flurry of theoretical and experimental research.
In his approach, he assumed that order existed for some distance larger than the

lattice spacing but smaller than the long-range ordered domains. This assump-

tion of short-range order circumvents long wave length divergences found in the
classical spin wave approach [5]. Next, the system was mapped from a statistical
mechanical model to a field theory in (1+1) dimensions, namely a NLaOM. Once
the mapping is complete, the S -+ oc limit [7-9] reveals a gap in the excitation
spectrum for the integer spin chain and no gap for the half integer spin chain.
More specifically, the Haldane gap, A, is proportional to e-'s, and the correlation
length between spins, is proportional to e's [7]. There is only one problem
with this approach. Spins of value one and one half can hardly be called large,
and in fact, this technique [8,10] provides inconclusive results for S < 1. Studies
of the NLaM are well documented in the literature [7-9,11,12], so further details
are beyond the scope of this thesis.
Non-linear sigma models have been extended beyond arguing the existence of

the Haldane gap. Jolicoeur and Golinelli used the N -+ oo limit to study A(T)
for T < A. The result is an "anti-BCS" gap of the form [10]


A(T) ; A(0) + 2 TrA()Te-A()T. (2.2)











Furthermore, they calculated the free energy and used it to derive expressions for
the low temperature specific heat

_3e-()/T (2.3)
27r T )

and magnetic susceptibility

X(T) ) 2e- (2.4)


2.1.2 Valence Bond Solids

Another approach that has been both theoretically and experimentally illu-
minating is the valence bond solid (VBS) model, also called the AKLT model,
introduced by Affleck, Kennedy, Lieb, and Tasaki [13,14]. Anderson [15,16] ini-
tially introduced valence bonds in the resonating valence bond (RVB) model to
suggest the existence of a quantum disordered ground state in an S = 1/2 two-
dimensional antiferromagnet. This model has been used to explain some features
of high-T, superconductivity [7,16] and may accurately describe spin ladders [17].
A valence bond is a bond between two S = 1/2 spins at two sites [7]. The bond
is commonly represented [7,14] as a solid line between the S = 1/2 sites given
as points. Figure 2.1 shows a valence bond for two S = 1/2 entities. It is easily
determined that for S = 1/2 spins on any lattice, there are multiply degenerate

p a


Figure 2.1: A single valence bond.












configurations of valence bonds, hence the name resonating valence bond [7,15].

For example, Figure 2.2 shows an RVB configuration for an S = 1/2 chain where

the dashed lines are the unoccupied bonds, but an equally probable state is one

where the dashed lines are the valence bonds and the solid lines are not; there-

fore, these two states are in resonance. Affleck, Kennedy, Lieb and Tasaki [13,14]

-- 4- -. -- W -


Figure 2.2: An S = 1/2 chain in the RVB picture.


extended the theory by examining valence bonds on lattices with higher spin.

They consider an S = 1 site as two S = 1/2 entities, represented by two points

enclosed by a circle. Therefore, Figure 2.3 shows an S = 1 chain coupled by va-

lence bonds. Clearly, there is no degeneracy, so the singlet ground state is "solid"






Figure 2.3: An S = 1 chain according to a VBS model.


when compared to the resonating bonds of the S = 1/2 chain. A continuation of

the theory in Figure 2.4 shows an S = 2 chain can also be modeled as a valence

bond solid. Another look at Figures 2.3 and 2.4 shows an interesting property of

VBS models, namely that finite chains terminate with spins of value S/2. This

feature will be discussed more below.

Up to this point in the discussion, nothing has been said about the system

for which the VBS state is the ground state nor how closely related the system is






















Figure 2.4: The VBS state for an S = 2 chain.

to a Heisenberg model. By assuming the VBS state is the ground state of some

Hamiltonian, and working backward to find it, the S = 1 Hamiltonian


= Si S.+ i- (Si. S5+i)2 (2.5)
i

is obtained where, for now, 3 = -1/3, but it will be varied later in the discus-

sion [13,14]. This system exhibits a gap [7,14], albeit with an energy, 0.350J,

significantly smaller than the Haldane gap. To compare the properties between

this system and the Heisenberg system, Kennedy [18] performed exact diagonal-

ization studies on Equation 2.5 for a range of 0 values. He confirmed a continuous

region of phase space in 3 from -1/3 through 0, so the gap and the end-chain

spins should be realized for the Heisenberg model as well. Equation 2.5 is for

S = 1 chains. A different Hamiltonian is constructed when a similar procedure
is carried out for an S = 2 chain, and the result is not as closely related to the

Heisenberg Hamiltonian [7,14] as Equation 2.5. Nevertheless, the VBS model

provides some physical intuition to help understand the experimental results in
an S = 2 system.












2.2 An Intuitive Picture of the Haldane Gap

Though the VBS model produces a physically intuitive explanation for the

presence of S = 1/2 spins at the ends of a finite S = 1 spin chain, it does not
provide similar intuition for the presence of a gap for the integer spin case and

the absence of it for the 1/2 integer spin case. The NLaM is even less intuitive.
Nevertheless, the following argument, derived from the large body of theoretical

and numerical work, lucidates the physical origin of the Haldane gap [19]. To
start from a long-range ordered ground state


I 14t4 '>t, (2.6)


assume an Ising system
S E= JE Si+1. 1(2.7)

Next, make the spins more isotropic, thereby approaching the Heisenberg model


t = J- S+S1+1 + SISS- + SS+, (2.8)
i

where S+ = Sx + iSy and S- = S iSy. Clearly, the added interactions destabi-
lize the long-range ordered state, so there is a lower ground state. Nevertheless,
for the ground state to remain antiferromagnetic, < M > = 0. Therefore, the
ground state must at least be the superposition of short-range ordered antiferro-

magnetic states. If a source of energy (i.e. T, H) is added to the system so as to

cause one spin to locally increase above the ground state, the new state for the
S = 1/2 chain is












I 44Mi"U > (2.9)

that has < M > 0 0 which is indicative of an excited state. In sharp contrast, a
spin on the S = 1 chain has the Sz = 0 state as its next excited state. Therefore,
the many-body state of the S = 1 chain with an excitation on it would look like


10 44W 0 4t>. (2.10)


Clearly, < M > = 0, so this state is degenerate with the ground state. If more
energy is added to the system, only when the AS, = 1 transition becomes the
most probable for each site does < M > 0 0, and the ground state is destroyed.

Since a finite quantum of energy must be added to the system for this transition
to occur, an energy gap in the excitation spectrum is observed.
A further interesting theoretical observation can be made from the above
discussion. First, notice that as long as two of the spins are arranged in the zero
state, they can be anywhere along the chain. Second, notice that if the zeros
were simply removed from the chain, the spins across the empty space would be
antiferromagnetically arranged. This result is know as hidden (or string) order
and was first identified by Tasaki [20].


2.3 Numerical Studies of Quantum Spin Chains

Other tools useful for characterizing spin chain behavior and for connecting
the theory to experimentally studied systems are numerical techniques. Four

techniques have been used, and they are high temperature series expansions,











exact diagonalization, quantum Monte Carlo (QMC), and density matrix renor-
malization group (DMRG). High temperature series expansions have produced
expressions for the directly measurable quantities Cv(T), the specific heat, and
X(T), the magnetic susceptibility. Weng [21] performed expansions of X(T) for
S = 1/2, 1, and co. Furthermore, he extrapolated between the S = 1 and S = o00
results to determine X(T) for S = 2 and S = 3/2. Meyer et al. [22] have generated
a rational function representation of Weng's S = 1 curve, namely


X(T) = Ng' 2 + 0.0194X +.777X2 1(2.11)
kBT 13 + 4.346X + 3.232X2 + 5.834X3

where X = J/kBT. In a similar manner, Hiller et al. [23] arrived at a more
versatile group of rational functions which can be used for many different values
of S as given by
N(g2 [ A+Bx2 1(212)
X(T) kBT [l+ Cx + Dx3j (2.12)

where x = J/(2kBT) and the coefficients are given in Table 2.1. All of these

S A B C D
1/2 0.2500 .18297 1.5467 3.4443
1 .6667 2.5823 3.605 39.558
3/2 1.2500 17.041 6.7360 238.47
2 2.0000 71.938 10.482 9555.56
Table 2.1: Coefficients as a function of S for Equation 2.12 from Reference [23].


expressions are only for the Heisenberg model. Recent extensions to Hamiltonians
including single-ion anisotropy will be discussed below. In recent QMC studies,











Yamamoto and Miyashita [24] calculated the temperature dependent magnetic
susceptibility and specific heat of open and closed chains for O.1J < T < 5J.
Their work agrees well with the high temperature expansions above J and with
the exponential excitation of a gap below J, until end-chain spin effects become
important. Further discussion of the end-chain spin effects will be given below.
Furthermore, QMC has been used to determine the dispersion curves of linear
chain systems. Figure 2.5 shows the S = 1 dispersion as Q is varied from 0 to
S[25,26]. One interesting feature is that the points suggest the gap at Q = 0
is larger than the one at Q = ir. This feature has been interpreted as evidence
of a two particle continuum of excited states near Q = 0, instead of the triplet
excitations at Q = 7r [25-29]. The solid and dashed lines in Figure 2.5 show
the agreement between the theoretically predicted curves and the QMC data
for the triplet excitations at Q = 0 and the two particle excitations at Q = 7,
respectively. The net result is a gap of 2A at Q = 0 and A at Q = 7r. Meshkov [29]
performed QMC work on an S = 2 chain to determine the dispersion. Similar
to the S = 1 case, the Q = 0 gap is twice the Q = 7r gap. In addition for both
the S = 1 and S = 2 cases, S(Q, w) looses intensity and broadens as Q -4 0 [29].
This loss of intensity explains why the 2A gap has not been observed. Detailed
discussion of the shape of S(Q, w) is not required for this thesis and is discussed
at length in the literature, so no further comments will be made here. The
DMRG technique has been shown to determine very precise results for relatively
long chains (-, 60 sites) [30-33], but unfortunately it is limited to T = 0. A
combination of QMC, exact diagonalization, and DMRG studies produced the
















3.0 1 1 1 1 1I--|
.oo O........ ...

2.5 ,..


2.0


- 1.5
w

1.0


0.5


0.0 I I
0.0 0.2 0.4 0.6 0.8 1.0

Q/7K

Figure 2.5: S = 1 chain dispersion curve. The squares are the dispersion for a
1-D S = 1 Heisenberg antiferromagnet calculated by QMC [25,26]. The solid
line is the expected dispersion for single particle excitations and the dotted line
is the expected dispersion for the two particle continuum [28].













A/ J _Reference
0.08 Deisz et al. [34]
0.055 0.015 Nihiyama et al. [35]
0.05 Sun [36]
0.085 0.005 Schollw6ck et al. [37,38]
0.049 0.018 Yamamoto et al. [39]

Table 2.2: Values for the S = 2 Haldane gap calculated by different groups.


result of A = 0.41J for S = 1, and work is converging for the S = 2 chains where

the present published values are given in Table 2.2.
Numerical studies have also provided information on how the Haldane gap

behaves as a function of magnetic field, H. The triplet excited state is split, and

at a certain field, H, = A, the non-magnetic singlet is replaced by one of the
magnetic states as the ground state. A detailed description of the splitting will
be provided in relationship to more realistic Hamiltonians.
The VBS model prediction of S = 1/2 end-chain spins is really quite in-

triguing, and an extensive body of numerical work [18,24,30-33,40-43] has been

performed to compare this model to the exact Heisenberg case. Initially, exact

diagonalization [18] results demonstrated the presence of S = 1/2 degrees of free-

dom for models where f/ in Equation 2.5 is varied through zero. Furthermore,

Kennedy [18] applied the general result of Lieb and Mattis [44], that implies a

difference between even and odd length chains, to the integer spin Heisenberg

antiferromagnetic chain. More specifically, he found that the ground state of a

finite chain is a four-fold degenerate ground state, made of two levels from each
S = 1/2 end-chain spin. If an interaction along the chain is permitted between












the end-chain spins, the ground state is divided into a singlet and a triplet.

In other words, for chains with even and odd numbers of spins, the singlet and

the triplet are the ground states, respectively. Therefore, the chain with an odd

number of spins has a magnetic ground state while the even case does not. Both

exact diagonalization [18] and QMC [41] on short chains support this difference,
but similar techniques on longer chains do not. Figure 2.6 shows < Sz > for

chains of 60, 65, and 97 units. The shortest chain was calculated by DMRG [31]

and the longer chains were calculated by QMC [41]. Clearly within the size of the

data points, whose size is an estimate of the uncertainty, the staggered magne-

tization is independent of the number of spin sites. This result is not surprising
since the correlation length of 6 units is so short that any parity effects are aver-
aged out over several correlation lengths. More directly related to experimental

measurements, Yamamoto and Miyashita [24] derived information on the temper-

ature dependent properties of end-chain spins from QMC results. By subtracting
the magnetic susceptibility for chains with closed ends from open chains with

odd numbers of sites, they can determine the end-chain spin contribution to the
magnetic susceptibility alone. At first glance, their results are well fit by a Curie

law, but a closer examination, performed by plotting 1/X vs. T (Figure 2.7), sug-

gests otherwise. Discontinuous features near T = J suggest that the subtraction

did not completely remove the contribution from the chain. One could argue
that below T z J the data obeys a Curie law, but this range is smaller than

one decade in temperature. Therefore, a fit can not be performed to determine

a definitive functional form. Nevertheless, these results are directly compared to

experimental results in Chapter 6.

























A
N
C)
v


0.6


0.4


0.2


0.0


-0.2


-0.4


' I I I I I I I I '


0l DMRG 60 sites A3
0 QMC 61 sites
A QMC 97 sites A3



tt I------------DO --
"6 ~DO -
no A








I ..o A
00 A
0
%A
0A-


0 10 20 30 40 50 60 70 80 90 100



Figure 2.6: Staggered magnetization for S = 1 chains of 60, 65, and 97 spins.
The 60 spin chain was calculated by DMRG [31], and the two longer chains were
calculated by QMC [41].














I I I I I I


I


IF
/-



J


m
m

* U


4.0
3.5
3.0
2.5
e 2.0
1.5
1.0
0.5


u.u "
0.0 0.2


0.4 0.6
T/J


0.8 1.0 1.2


L I I I *
0 1 2 3 4 5
T/J

Figure 2.7: l/x vs. T for end-chain spins from QMC work of Yamamoto and
Miyashita [24]. Several features suggesting non-Curie like behavior are mentioned
in the text. Although, at the lowest temperatures, the inset shows reasonable
agreement with a Curie-law.


V =-
U


Pe
mU

.,
*i'


. K 7












2.3.1 Realistic Hamiltonians

The above discussion has been for systems described by Equation 2.1, but

in a real system, next order terms relating to the symmetry of the environment

around the magnetic ion must be considered. More specifically, the Hamiltonian

should include single-ion anisotropy D, in some cases orthorhombic anisotropy

E, and the degree of symmetry in the exchange 7/. As these terms break some of

the symmetries of the system, they become very difficult to handle analytically.

Nevertheless, numerical studies have revealed many properties of these systems.

The Hamiltonian which takes all these effects into account is


W = J_ SE ST+ + SS'+, + 7S1S ,+ + D(Sz)2 + E [()2 (S)2]. (2.13)
i

Since E is small, it is usually negligible. However, splitting in inelastic neu-

tron scattering data [45] demonstrates that this term is measurable for NENP.

The remainder of this discussion will consider E = 0. The best way to describe

the overall behavior of Equation 2.13 is with the semi-quantitative phase dia-

grams [20,37,46] shown in Figures 2.8 and 2.9. First, examine the S = 1 case in

Figure 2.8. Note that a rather large region of phase space is the Haldane phase,

and the XY phase does not take over until 77 < 0. Two of the extreme bound-

aries of the phase diagram are long-range antiferromagnetic order for 77 0 and

ferromagnetic order for 77 < 0, i.e. the Ising limit. If D is sufficiently large, a

singlet phase, which is gaped but is not the Haldane gap, develops. Finally, for

the S = 2 case, Figure 2.9 shows a greatly reduced Haldane phase region. This

result is reasonable because higher spin means more possible S components in









































Ferromagnetic


Large-D
Singlet


Antiferromagnetic


Figure 2.8: S = 1 phase diagram. The single-ion anisotropy vs. exchange
anisotropy phase diagram for an S = 1 linear chain antiferromagnet described by
the Hamiltonian 2.13 [20,37,46].











the x and y direction to amplify the anisotropies. Therefore, it can be said that
the Haldane phase in the S = 1 system is lost to the XY phase as the spin is
increased.
Finite values of D affect antiferromagnetic chains in the Haldane phase region
as well. As was stated earlier, A is a gap between the singlet ground state and a
triplet excited state (see Figure 2.10a). A finite D term splits the triplet to form
two gaps (see Figure 2.10b), Aj for directions perpendicular to the chains and
All for directions parallel to the chains, so at Q = 7r, Aj < All [28,47-50]. As
Q is reduced to zero, the numerical work of Golinelli et al. [28] shows that the
dispersion curves I and 11 to the chains cross so at Q = 0, Aj > All. To directly
compare results with several experiments (see Chapters 3, 5, and 7), extensive
work [47-50] has described the magnetic field dependence of the excited states
split by D at Q = 7r. Figure 2.10c shows that the doublet splits, with the lower
state moving towards, and finally crossing, the ground state at critical fields Hci\
and H,, depending on the orientation of the field. The energy gaps are related
to the critical fields by


g11iBHcll = Al (2.14)

gLILBH = 'A||A (2.15)









































Ferromagnetic


Large-D
Singlet


Haldane


Antiferromagnetic


Figure 2.9: S = 2 phase diagram. The single-ion anisotropy vs. exchange
anisotropy phase diagram for an S = 2 linear chain antiferromagnet for the
Hamiltonian 2.13 [37].












where gll and gz are the Land6 g factors for each orientation [48,50]. Then, to

correlate A and All with A and D, the following expressions [50,51] are used


All = A + 2KD (2.16)

A1 = A KD (2.17)


where K = 1/3 for S = 1 and K = 2 for S = 2.

Yamamoto and Miyashita extended their QMC studies, mentioned earlier, to

include finite anisotropy. At T = 0, their staggered magnetization results show

an increased or decreased correlation length depending on if the magnetic field

orientation is parallel or antiparallel to the easy axis, respectively. The easy axis

is defined by the sign of D. Furthermore, Cv(T) and x(T) results show effects of

finite D, but they did not fit a function to their results, making comparisons with

experimental data difficult. Nevertheless, Jolicoeur [51] performed high temper-

ature series expansions to arrive at the expressions given in Appendix A. Since

the expressions are expansions near a phase transition, minute errors in the fit

parameters cause huge discrepancies, so the method of Pad6 approximants must

be used for each value of D.


2.4 Measurement Theory

One challenging part of any research program is connecting the theory to

experimentally measured quantities. Some less than obvious theoretical work

has been used to connect the experimental technique of ESR and the above

theory. This section overviews the essential theory needed to understand the

measurements.















(a) (b) (c)
D=0 D>O E
H=O H=O /
11,0> ../ '"

(T) 2KD
(T ) -------- ........................................

A
\ 11,+1> KD /






10,0>
(S) -
H I I HcL




Figure 2.10: Energy level diagram showing the Haldane gap for (a) H = 0, D = 0
(b) H = 0, D > 0 (c) H 5 0, D > 0. The values of the splitting are explained in
the text. Please note that this is a generalized and corrected version of Figure 4
appearing in Reference [97].












2.4.1 Electron Spin Resonance

Part of this thesis reports on measurements involving end-chain spins in
NINAZ. One way to detect free spins is through electron spin resonance (ESR).
If a paramagnetic spin is placed in a DC magnetic field, H, it will align itself
such that the magnetic moment of the ion processes about H in a manner similar
to a toy top in the gravitational field. Since the system is a damped harmonic
oscillator, the system will absorb a large amount of energy when driven at its
resonance frequency, w0. This resonant frequency is dependent on the size of the
applied magnetic field as
hwo = gIBH (2.18)

where g, h and AB follow their standard definitions. A convenient driving field for
an electronic spin is the magnetic field component of electromagnetic radiation in
the microwave (X-band) frequency regime. Several microscopic phenomena can
be interpreted from the characteristics of the observed absorption peak. The area
under the absorption peak is one property of interest for this thesis. Any spin
resonance technique measures energy absorbed by the change in magnetization,
AM, of the sample for the oscillating field, AH, as a function of H. Since X is

defined by '-6 in the limit where AH < H, this technique measures the lossy
component X" of the complex magnetic susceptibility, XAC = X' + ix". Since
X" is sharply peaked near the resonance frequency, X' can be determined at any
frequency through the Kramers-Kronig relations [52-54]. Specifically, the DC
magnetic susceptibility is [53]











00
XDC = X'(w = 0) oc J x"(W)d. (2.19)


Experimentally, ESR is performed by holding w constant and varying H; there-
fore, by Equation 2.18, XDC becomes [54]


XDC = X(W= 0) Oc f x"(H')dH'. (2.20)


This result means that the area under the resonance line is simply proportional
to XDC [53,54]. This completes a short introduction to the ESR technique. There
are many excellent texts on ESR, if more detailed theory is required [52,53,55,56].
The above discussion has been a general overview of ESR for standard para-
magnetic spins. There are several features unique to end-chain spins that are
observed with ESR. Mitra, Halperin, and Affleck propose a simple model to ex-
plain the ESR response of the end-chain spins in NENP [57]. They begin by
assuming chains with fixed ends which reflect a magnetic excitation back to the
other end of the chain. These excitations are described by bosons in agreement
with NLaM predictions. When one of these excitations interacts with an end-
chain spin, either the excitation can change energy levels or there will be a slight
phase shift that would cause a small change in energy. Generally, their expression
for the intensity of the ESR spectra is












I(w) = tanh (e') -2r6 W WO)

SlO5kB() e2-Z _1
+ e1- eIT -- -e-^
<< ;8 cosh^

x B [i27rb (U O+ + I < n, +1, > 2

+[+ -]]... (2.21)


where the first term is the ESR line for the end-chain spin alone, and the second
term includes effects resulting from interactions with the bosons. In the second
term, the + indicates that the boson spin and the end-chain spin are parallel
and the indicates that they are antiparallel. Figure 2.11 shows the energy
level diagram for three boson levels. The heavy solid line shows the transition
corresponding to the ESR line when the bosons do not change energy levels.
The thin lines show typical transitions between energy levels where the open
arrows indicate transitions to higher energy levels and the filled arrows represent
transitions with lower energy levels. Notice that the transitions would produce
symmetric lines around the peak if the frequency of the microwaves is large enough
to see the transition at lower fields. Nevertheless, the higher field transition should
always be observable if a sufficient magnetic field is applied to the sample. The
processes where the end-chain spin does not flip are forbidden transitions. If the
boson changes energy levels, then the change in energy for a chain of length, L,
is
= 6: (h C)2 (2.22)
+ 2AL2























+60

0


-861' -


Figure 2.11: Allowed transitions between end-chain spin states. Energy level
diagram showing the allowed transitions that should be observed in ESR. Each
transition is signified as a line with an arrow on each end. The heavy line signifies
a resonance signal where a boson does not change energy levels. The open ended
arrows indicate transitions with a higher boson energy level and the closed arrows
signify transitions with a lower boson energy level. All arrows are of a length
E = hwo = constant.










where c is the spin wave velocity. If L is too short (e.g. 100 spins for
NENP), the side peaks are not within the observation limits of the typical X-
band (v = 9 GHz) ESR experiment. Therefore, to calculate the intensity of the
ESR line, the second term in Equation 2.21 can be neglected. Therefore, the
chains that significantly contribute to the central intensity are those without
bosons on them. Since the probability of a magnetic excitation existing on a
chain at any one time is an exponential function of temperature, the number or
spins participating in the ESR line is temperature dependent. As a result, the
ESR line intensity increases faster than a Curie law with decreasing temperature.
The temperature dependence of these intermediate length chains is given by

I(T) = JO tanh ( e- (2.23)
I(T =Io an \2-kT]

where < Z1 > is the partition function averaged over the distribution of chain
lengths as given by
Z -=[< 1 e-] ". (2.24)

The de Broglie thermal wavelength of a boson is
r 1 11/2
AT = hc 2kBT1 "2 (2.25)

Mitra, Halperin and Affleck [57] finish their calculation by assuming a distribution
of P(L) oc exp(-L/Lo) with a cutoff of Lmin. This distribution provides the
analytical solution











( t ( ) exp [- (l2LiAT1 1) ekB
I(T) = Io tanh 1 1/2LoAe kA. (2.26)
112kBT) 1P 1+ 7r-1/LoAT'eB

Calculating other quantities for the ESR end-chain spin is less trivial because
the second term in Equation 2.21 contributes to the line width of the central
peak even when the boson does not change energy levels. To approximate the
line width, consider the potential that a boson sees as it traverses the chain.
At distances far away from an end-chain spin, the potential is essentially flat,
but as the boson approaches an end-chain spin, it sees a finite potential, and
therefore, its wave vector shifts slightly. This shift of the boson's wave vector can
be characterized by a phase shift 6(k) defined by

k- nr 6(k) (2.27)
L L

To parameterize this effect, Mitra, Halperin, and Affleck used the expression


6(k) ; Vk, (2.28)


where 2D is a constant of order 1. In their paper, they conclude that this effect
is not observable because the line width would be controlled by the very short
chains in the sample. It is important to recall that their discussion focused only
on NENP where several phenomena (i.e. staggered magnetization and thermally
excited bosons) could effect the linewidth. For this thesis, we will apply this
analysis to NINAZ where there is no staggered magnetization and J is large
enough that the quantum limit is easily accessible. The line widths in NINAZ
are consistent with this phase shift, see Chapter 6.














CHAPTER 3
REVIEW OF EXPERIMENTAL STUDIES


This chapter reviews the status of experimental studies of phenomena related

to the Haldane state. The main emphasis is placed on materials that are not part

of this thesis since these systems will be discussed in their own chapters. This

chapter begins with a discussion of the material properties of Haldane gap sys-

tems. Next, a discussion of the different techniques used to measure the Haldane

gap, and other information gained by these methods, will be given. Third, exper-

imental results from the study of end-chain spins will be provided. To complete

the review, a few relevant subtopics will be discussed.


3.1 Haldane Gap Materials

In order to study the phenomena related to the Haldane state, materials

are needed which are closely approximated by the theoretical model. There are

two main properties that must be met for a material to exhibit a Haldane gap.

Firstly, the system must have a coupling along the chains much stronger than

a coupling between the chains so the material exhibits 1-D behavior. Secondly,

any anisotropies, that give the spins a preferential orientation (i.e. single-ion

anisotropy or exchange anisotropy), must be small enough so the material re-

mains in the Haldane phase, see Figures 2.8 and 2.9. The first part of this section












discusses issues related to fabricating antiferromagnetic chain materials and the

second part deals with the anisotropy.

Materials that provide a realization of integer spin antiferromagnetic chains

have been available for several decades [58]. Chemically these systems are charac-

terized by their composition as being either metal-organic compounds or purely

inorganic compounds. For both groups of materials, each metal ion has strong

S180 bond overlap with the bridging ligand that mediates the superexchange

along the chain. Bond angles near 180 are characteristic of antiferromagnetic

compounds since the strength of the superexchange is controlled by the bond

angle [54,59]. There are two ways that the chains are separated from each other.
For some materials, the chains have a slight net positive charge which is balanced

by a counter ion between the chains. Therefore, no bond overlap exists to pro-

vide a superexchange pathway between chains. Thus, only dipolar interactions

magnetically couple the chains. The other class of chain materials relies on the

contrast between strong antiferromagnetic bonds along the direction of the chain

and very weak 90 bonds, ferromagnetic in nature, between the chains. Both

types of materials posses an interchain coupling that is four orders of magnitude

smaller than the intrachain coupling. The local environment around the mag-

netic ion determines the anisotropy of the chain material. Generally, if chemical

bonds are uniformly distributed around the magnetic ion, the anisotropy is low.

However, completely isotropic bonding is rare because Jahn-Teller distortions fa-

vor at least some slight anisotropy. Fortunately, there are many materials where












material S J D/J E/J A references
~ (K) (K)
CsNiC13 1 16.6 0.003 ______5.25 [60]
NENP 1 46 2 0.18 + 0.01 0.02 0.01 14 [22,45,61-70]
NINO 1 47 0.34 0.03 9.81 [62,65]
TMNIN 1 10 0.003 ______ 3 [71-77]
AgVP2S6 1 400 0.01 < 3 x 10-4 300 [62,78-80]
NINAZ 1 125 0.16 _____41.9 [73,77,81-83]
NiC204.2MINz 1 39.7 20.3 [84]
NiC204-2DMIz 1 42.9 ______19.0 [84]
Y2BaNiO5 1 285 0.16 ______100 [85-94]
MnCl3(bipy) 2 35 0.010 0.003 ______2.3 + 0.8 [95-97]

Table 3.1: The Haldane gap materials and their properties. The parameters that
describe each system are the intrachain exchange, J, the single-ion anisotropy,
D, the orthorhombic anisotropy, E, and the Haldane Gap, A. The empty spaces
indicate that the parameter has not been experimentally determined. Note that
CsNiC13 has a transition to an antiferromagnetic long-range ordered state at
TN = 4.6 K.

the bonds are sufficiently isotropic for the Haldane phase to be realized, as seen
by Table 3.1.


3.2 Experimental Tests of the Haldane Gap

A large portion of this thesis is concerned with measurement of the Haldane
gap, and this section overviews experimental tests of this state. The macro-
scopic magnetic susceptibility is usually the first property tested for a new 1-D
antiferromagnetic material. The standard SQUID magnetometer has sufficient
sensitivity to test for undesired magnetic properties, i.e. long-range order or gross
impurities, even in very small samples (- 10 mg). Such tests are necessary to de-
termine if the system merits further exploration. All of the samples in Table 3.1












display, with decreasing temperature, a broad peak, demonstrating 1-D antifer-

romagnetism, followed by a sharp decrease, consistent with an activated gap. In

addition, many materials display a low temperature paramagnetic tail consistent

with either end-chain spins or impurities in the sample. Another macroscopic

measurement, better suited to measure the size of the Haldane gap, is the mag-

netic field, H, dependent magnetization, M, at a temperature well below the gap.

While the material is in the Haldane state, the magnetization is zero, and as H

is increased such that the lower branch of the triplet excited state crosses the

ground state (Figure 2.10), a finite magnetization develops. This technique has

been used in many of the materials [63,64,75,97]. The experiments in TMNIN

and MnCl3(bipy), as part of this thesis, will be discussed in Chapters 5 and 7,

respectively. The first M vs. H measurement to identify the Haldane gap was

performed on NENP [63,64]. As can be seen in Figure 3.1, there are critical fields

defined by the intersection of a constant, weakly magnetic state with an increas-

ing magnetic state. These critical fields provide values of the Haldane gap, A,

and the single-ion anisotropy, D, consistent with the values given in Table 3.1.

Notice that for a magnetic field oriented parallel to the a axis (b is the chain

axis), a finite magnetization was observed below the gap. Katsumata et al. [63]

attributed this property to impurities. Subsequent NMR [98] and high field ESR

measurements [19,99,100] show that it results from non-equivalent magnetic sites

as will be discussed in Section 3.4.

Though macroscopic measurements can detect the presence of a gap and their

careful use can eliminate any possibility of other physical processes opening a gap,










34











0.10 -,
#I
to


0.08
/
f

~-0.08 b___





=- 0.04



0.02.... .
^...................... ... ............


0.00
0 2 4 6 8 10 12 14

H(T)


Figure 3.1: M(H) for NENP. There are clearly defined critical fields for all three
orientations. The finite magnetization for H 11 a results from a staggered magne-
tization of the Ni sites. The data are from Reference [63].











a direct measure of the gap is needed to test additional properties. Inelastic neu-
tron scattering is the probe most often used and has had great success. The
general idea behind inelastic neutron scattering is to measure the energy and
momentum change for neutrons after they have scattered from the sample. Since
the Haldane gap is a low temperature phenomenon, any energy and momentum
changes by phonons are negligible. Therefore, all energy and momentum changes
correspond to the creation or annihilation of magnetic excitations. When neu-
trons of the energy equivalent to the gap at a certain value of Q are sent into
the sample, the neutrons excite the system over the Haldane gap, producing res-
onant scattering. For neutron energies below the gap, the system is not excited,
so magnetic scattering is not observed. When energies above the gap are mea-
sured, the magnetic scattering is not observed. Furthermore, neutron scattering
is a Q dependent measurement. Therefore, by taking energy sweeps at different
values of Q, the dispersion can be measured. As an example, Figure 3.1 shows
the magnetic neutron scattering near Q = 7r corresponding to a Haldane gap in
Y2BaNiO5 [90]. The most prominent feature, i.e. the peak around zero energy,
originates from incoherent elastic scattering, and this feature is always observed.
The first unambiguous measurement of the Haldane gap was in NENP by
Renard et al. [61] from which A and D could be determined. Later measure-
ments [45], with better resolution, were able to determine E as well. Near Q = 7r,
several measurements have determined the dispersion. Two groups [45,67] mea-

sured the dispersion for a larger range of Q, but the scattering intensity de-
creased rapidly below Q = 0.3, so the Q = 0 gap of 2A has eluded confirmation.

















70 1 32K

S60 117K

S50
0

40
cf)


0 I I ," I
5,30
C:
C20 00 0
o- .0.1:31
4-. 0
10 0
(D)
z -

-10 -5 0 5 10 15 20

E (meV)

Figure 3.2: Inelastic neutron scattering intensity for Y2BaNiOs. Constant Q 7r
scan showing the peak due to incoherent elastic scattering and the peak resulting
from the Haldane gap for T > A and T < A. The data are from [90].












Nevertheless, this loss of scattering intensity for decreasing Q is consistent with
the model of a two particle continuum near Q = 0 [26]. In addition, these ex-
periments do not have sufficient resolution to observe the crossing of the All and
AI. components of the dispersion [28]. Nevertheless, the empirically determined

dispersion agrees with the QMC predictions of Takahashi [26].
NMR is another microscopic probe useful to test for the existence of the
Haldane gap. Of course, if the gap is present, additional information can be
obtained. Since the purpose of this chapter is to demonstrate how NMR is used
for probing the Haldane gap, only spin-lattice relaxation times and the Knight
shift will be discussed. In an NMR experiment, a uniform magnetic field is applied
to a sample, resulting in the precession of the nuclear spins. When an rf pulse is
applied to the sample to flip the nuclear spins, the local magnetic environment
affects how long it takes for the spin to return to equilibrium. The contribution to
this relaxation time by the crystal lattice is known as the spin-lattice relaxation
time, T1. In magnetic systems, where the electronic moments are large and

long-ranged (compared to the nuclear moments), T1 is the dominant relaxation
process. Clearly T1 is related to how correlated the local moments are with
each other or, in other words, are related to the spin-spin correlation function.
Since the experiment is performed in frequency space, the Fourier transform of
the correlation function, or the dynamic structure factor S(q, w), is the quantity

measured. It has been shown [10,12,101,102] that

1 4
T c S(q,w) c( e-T (3.1)












for T < A. Therefore, the slope of a ln(T'-1) vs. T-1 plot of the data provides

a microscopic measure of A. If the sample has conduction electrons, the Knight

shift can be measured as well. This effect is a shift in frequency of the resonance
line due to the nuclei observing a background of mobile spin polarized electrons.

A detailed discussion [53] shows that the Knight shift is proportional to the local

spin susceptibility of the sample. Therefore, it is a microscopic measure of the

magnetic susceptibility of the chain.
Since there are numerous hydrogen atoms in typical organo-metallic com-
pounds, proton NMR is often performed. Several groups [66,69,103,104] have
measured the proton spin-lattice relaxation time in NENP, and all of their results

are consistent with Equation 3.1. Nevertheless, these proton sites are somewhat
removed from the magnetic nickel site. To use a probe closer to this site, Reyes

et al. [102] performed NMR using the naturally abundant 3C of NENP. Besides
measuring the gap, they also observed features in Tj-1(T) that are absent in the
proton NMR data [66,69,103,104]. These observed features may be interpreted by

adding a temperature dependent gap to the standard analysis. In AgVP2S6, "1V

and 31p were used as probing nuclei, whereas 81Y was used in Y2BaNiO5 [105]. In

addition, both of these samples possess conduction electrons, so the Knight shift
was the measured quantity. Since the Knight shift measures the local magnetic
susceptibility, any end-chain spin effect is negligible, and it is a good measure

of the chain contribution alone. Takigawa et al. [80] fit their Knight shift data
for AgVP2S6 with the expressions of Jolicceur et al. [10] which produced gap val-

ues consistent with the inelastic neutron scattering results [79]. Furthermore, by












examining the orientational dependence of the Knight shift, they measured D and

found an upper bound on E. Similar techniques were applied to Y2BaNiO5 [105],

but these workers interpreted their results as being inconsistent with the energy

level diagram including anisotropy as described by theory [47,48].

Finally, ESR has been used to examine the microscopic magnetic susceptibil-

ity and to map out portions of the energy-magnetic field diagram. As explained

in Section 2.4.1, the area under the ESR absorption curve is proportional to the

static magnetic susceptibility. Unfortunately, as antiferromagnetic correlations

increase, the ESR line broadens and loses intensity, resulting in a loss of reso-

lution. Therefore, if conduction electrons exist in the system, the NMR Knight

shift is a better probe of the local magnetic susceptibility because the NMR

line is not degraded by the antiferromagnetic correlations. Nevertheless, in the

organo-metallic compounds, there are no conduction electrons, so there is no

NMR Knight shift. Since an ESR line is the result of the resonant absorption of

energy between two states split by the magnetic field, high field ESR has been

an effective tool for mapping out the energy vs. magnetic field diagram. This ex-

perimentally determined diagram has confirmed the theory used to assign critical

fields in M vs. H experiments.

Date and Kindo [65] measured the ESR line at a frequency of 47 GHz as a

function of temperature in NENP and NINO. They found that the area under

the absorption curve as a function of temperature fits the expected magnetic

susceptibility with the exponential activation. Recent measurements have been

performed on Y2BaNiO5 [106] with similar results. Several groups have examined












the energy-magnetic field diagram using ESR [19,99,100,107]. These results for

both NENP and NINO are consistent with the theory for finite anisotropy [47,48],

assuming a staggered magnetic field is considered in the analysis. Discussion of

the staggered magnetic field is given below.


3.3 Studies of End-chain Spins

The other main topic of this thesis involves the properties of end-chain spins,

and therefore, this section will provide an overview of the research of other work-

ers. As predicted by the VBS model [13,14] described in Section 2.1.2, the spin

value at the termination of a chain is expected to be one half the value of a spin

site. To test this prediction, finite chains are needed. The first and most predom-

inant way of breaking the chains was by doping the materials. Initial evidence

of the presence of end-chain spins was an enhanced impurity tail in the magnetic

susceptibility of NENP doped with Cu2+ magnetic impurities [62]. Later, using

X-band ESR on a similar sample, Hagiwara et al. [108] observed hyperfine inter-

actions that could be attributed to the interaction between S = 1/2 variables and

the Cu2+ S 1/2 impurities. Furthermore, the temperature dependence of the

ESR peak, i.e. the peak attributable to the end-chain spins, was fit with an em-

pirical equation whose form was later confirmed by Mitra et al. [57]. To study an

even simpler system by avoiding magnetic impurities, Glarum et al. [109] studied

samples of NENP doped with Cd, Zn, and Hg. These workers observed a sin-

gle asymmetric ESR peak, and this asymmetry was attributed to the staggered

magnetization present in the sample. Ajiro et al. [110] recently extended this












work with a detailed study of the same system. All of the samples mentioned
so far in this section were lightly doped, but heavily doped samples exhibited

Curie tails [111] that increased faster than if each dopant caused only one chain

break. Several possible explanations of this effect have been tested in different

ways. First, NENP does not have a non-magnetic isomorph, so naturally, beyond

a certain doping level, the sample being doped is not necessarily NENP. To avoid

these problems, primarily two materials have been tested, NiC2O4.2DMIZ [112]

and Y2BaNiO5 [91,113].

The sample NiC204.2DMIZ is isomorphically related to ZnC204.2DMIZ.

Therefore, Zn dopants in NiC204.2DMIZ should enter the crystal uniformly.

Nevertheless, magnetic susceptibility measurements reveal paramagnetic tails

that increase faster than is consistent with the doping level [114]. Kikuchi

et al. [114] explain these variations in terms of an energy gap between states

inside the Haldane gap. They explain the existence of states in the gap as re-

sulting from interactions, mediated through chain, between end-chain spins as

described by the numerical work of Yamomoto and Miyashita [24]. Nevertheless,

their chains are too long to observe in-gap states, and the simple explanation

may be that the Zn dopants are not entering the crystal uniformly. Careful X-

ray studies are needed to test the Zn distribution in the crystal. Furthermore,

the analysis has neglected any change in the magnetic susceptibility due to the

interaction of the end-chain spins with magnetic excitations.

The material Y2BaNiO5 can be doped in two ways: either Zn can replace

the Ni, generating chain breaks, or Ca can replace the Ba, creating weak bonds












and charge carrying holes [91,113]. Furthermore, extensive X-ray and electron

scattering studies [88] of doped samples have demonstrated that Y2BaNiO5 can be
uniformly doped. Ramirez et al. measured the specific heat of a Zn doped powder

sample and observed a Schottky anomaly consistent with the presence of S = 1

end-chain spins. They attributed this effect to differences between chains with
odd and even numbers of sites. Once again, these workers have ignored any effect

of the interactions between the magnetic excitations on the chains and the end-
chain spins. However, the temperature dependence of the ESR signal intensity in
NENP [108-110], TMNIN [115], and NINAZ (Section 6.3.4 of this thesis) suggests

that these interactions should be considered. In addition, recent work by Kimura

et al. [106] shows ESR lines consistent with chain breaks caused by Ni3'+ ions
which act as S = 1/2 impurities between two S = 1/2 end-chain spins, and this
situation is similar to the Cu2+ work in NENP [108]. The final evidence that

interactions between magnetic excitations on the chains and the end-chain spins
correctly explain the behavior of Zn doped Y2BaNiO5 is provided by theoretical

work of Hallberg et al. [116] that reconciles the specific heat results in Y2BaNiO5

with ESR results in NENP. DiTusa et al. [91] concentrated on the Ca doped
samples which exhibited a greatly reduced resistivity which is consistent with the
assertion that this material is a 1-D metal. More importantly, inelastic neutron

scattering measurements revealed magnetic scattering inside the Haldane gap for

the Ca doped samples but not for the Zn doped samples. Dagotto et al. [117]
theoretically illuminated these observations by qualitatively describing the in-gap
states as resulting from coupling between the mobile holes and the S = 1 Ni sites.












There are other ways to observe end-chain spins besides breaking chains with

dopants. Two methods have been tried, the first will be discussed here and the

second will be discussed in Chapter 6. The first method does not involve cre-

ating chain breaks, but rather measures the end-chain spins already present in
the sample from crystal imperfections. Hagiwara and Katsumata [118] performed

ESR and magnetization experiments on a quickly grown sample of NENP. Their

results are consistent with the existence S = 1/2 end-chain spins. However,

when compared to samples grown by other groups, this sample has an enormous

number of natural chain breaks. Therefore, a study should be performed on
how crystal growth rate affects the number of crystal imperfections. A tradi-

tionally grown sample was used for the experimental tour de force of measuring

the magnetic susceptibility over six orders of magnitude in temperature space as

shown in Figure 3.3 [119,120]. Three separate experimental apparatuses were

used to map out three regions of temperature space: a SQUID magnetometer

for 300 K > T > 4 K, a mutual inductance coil cooled by a homemade dilution
refrigerator for 4 K > T > 50 mK, and an AC SQUID susceptometer cooled by

the Cu nuclear demagnetization stage of cryostat number 1 at the University of

Florida Microkelvin Facility for 50 mK > T > 400 pK. The portion of Figure 3.3

related to the end-chain spin discussion is the paramagnetic tail observed below

- 200 mK. This tail has a Curie constant consistent with the natural chain length

expected for the material [119,120].

Since the main topics of this thesis are S = 2 Haldane gap materials and end-

chain spin effects in NINAZ, an S = 1 material, it would be amiss not to mention

















0.04



0.03



0.02



0.01



0.00


I I I I oil I I I 11111 I I I 111i 11ii 1 I I 11111 I I 1111111 I I I 111111 I I






-aw-
0h
13


13


1 1 1 22 1 1 111 1 1 111 a li


1E-4 1E-3 0.01


10 100


T(K)


Figure 3.3: X(400 AK <
[119,120].


T < 300 K) for pure NENP. The data are from References












the first evidence for free S = 1 end-chain spins in an S = 2 material. Yamazaki

and Katsumata [121] used ESR, magnetic susceptibility, and heat capacity to
study end-chain spins in CsCrC13 doped with Mg. This material possesses long-

range magnetic order below 16 K, so one of their points was that the presence of

S = 1 end-chain spins constituted the existence of the Haldane state.

Measurements of NINO, NINAZ, and TMNIN have also been performed to

study end-chain spins. The techniques used in NINO [125] are similar to those

discussed above, and the discussion of NINAZ and TMNIN will be given elsewhere
in this thesis.


3.4 Other Experimental Results

Two other topics that have been addressed experimentally must be mentioned.

First, the differing magnetic sites in NENP and NINO will be discussed because a

similar phenomena might explain unresolved issues in TMNIN and MnC13(bipy).

The existence of two different magnetic sites might relate to a small anomaly in
the magnetic susceptibility in MnCl3(bipy) [97]. Both NENP and NINO consist

of Ni sites bridged by an NO2 group. For a specific Ni site, the exchange path
is directed through the nitrogen and one oxygen of the NO2 group on one side

of the Ni. On the other side of the Ni, the exchange path is directed through

the other oxygen of the NO2 group (see Figure 3.4). This asymmetric bonding

causes a slight shift in the magnetization of each site. As a consequence, neigh-

boring Ni sites differ slightly in their transverse magnetization. The existence

of these differing magnetic sites is necessary to explain the NMR line shape of



















































U


Figure 3.4: The crystal structure of NENP. Notice the different oxygen sites
which bridge the Ni [22].












Chiba et al. [98]. This property also explains the finite magnetization observed

below the critical field in M vs. H measurements [63,64] and explains the shape

of the magnetic field diagram as measured by ESR [19,99,100,107].

The second topic of discussion is the coexistence of the Haldane gap and three-

dimensional long-range order. Since this topic is historically significant but is far

afield from the topic of this thesis, only the materials and the references will be

mentioned. The first material to exhibit evidence of a Haldane gap was CsNiC13

[60], but with a gap of 5 K and a Neel temperature of 4.4 K, this assignment

was disputed in the literature [122-124]. Nevertheless, this material provided an

environment to study the coexistence of the Haldane gap and three-dimensional

long-range order [126]. To provide a smooth transition to materials with three-

dimensional long-range order, extensive work has been done on R2BaNiOs, where

R=Y, Nd, Pr or some doping mix of Y and Nd. The pure Pr [127] and Nd

[128-130] compounds have magnetically ordered ground states, and the doped

materials [131] allow a variation of the interchain interaction.

Another technique to force interactions between the samples is to apply pres-

sure. Recent inelastic neutron scattering measurements reveal a reduced single-

ion anisotropy in NENP under a pressure of 2.5 GPa [132]. Furthermore, heat

capacity measurements in NENP are consistent with an increase in A with in-

creased pressure [133]. Nevertheless, pressure studies have been unable to in-

crease the interchain exchange enough to cause the sample to exhibit long-range

magnetic order.














CHAPTER 4
EXPERIMENTAL MEASUREMENT TECHNIQUES


This chapter describes the experimental techniques used to study the materi-

als described in this thesis. Magnetometry using SQUID and cantilever magne-

tometers is discussed first. Next, both high field and X-band ESR are discussed

because they provided a microscopic probe of the end-chain spins. The third

major technique to be reviewed is neutron scattering. I have used both powder

neutron diffraction analyzed by Reitveld refinement to obtain low temperature

crystal structures and inelastic neutron data collected by time of flight methods
to look for the characteristic features of the Haldane gap in MnCl3 (bipy). Finally,

I will discuss several measurement techniques used in the creation and analysis

of our pulverized NINAZ samples.


4.1 SQUID Magnetometer

A SQUID magnetometer is a sensitive device for measuring the magnetiza-

tion of a sample over a range of magnetic fields and temperatures. In addition,

knowledge of the applied magnetic field allows the DC magnetic susceptibility to

be determined as well. Professor J. R. Childress, of the Department of Materials

Science and Engineering at the University of Florida, kindly allowed us to use

his MPMS SQUID magnetometer, made by Quantum Design, for magnetization

and magnetic susceptibility measurements. The main component of this SQUID












magnetometer is an rf SQUID. A SQUID is a small superconducting ring with a

weak link. As magnetic flux is applied to a SQUID, a supercurrent is induced in

the ring to cancel the flux until the phase of the electron wave function matches

across the weak link. Once this happens, a flux quanta enters the loop and the

supercurrent goes to zero. If the flux keeps increasing, the supercurrent increases
again until another flux quanta enters the SQUID. In the magnetometer, an rf

signal on top of an applied flux provides a changing flux in the SQUID with a

bias such that the system oscillates about the point where one flux quanta enters

the ring. As the flux from the sample changes, a feed back loop controls the bias

flux, canceling the flux from the sample, so the SQUID remains near the one flux

quanta point. The flux needed to keep the SQUID at this point is proportional

to the flux from the sample. Therefore, a calibration will provide a measure of

the magnetization. The particularly challenging part about making a practical

device work in a magnetic field is ensuring that the SQUID measures the flux

change of the sample and not of the magnet. Quantum Design conquers this

challenge by coupling the SQUID, which is outside and shielded from the super-

conducting magnet, to the sample through a flux transformer that measures the

second field gradient of the sample [134]. Thus, a signal is only observed in the

SQUID when dm2/dx2 is non-zero. The second derivative is chosen over the first

derivative in order to eliminate noise arising from slow drifts in the magnetic

field. Recovery of the sample magnetization is accomplished by moving the sam-

ple through the gradient coils in a manner to doubly integrate the signal. This

motion requires exact positioning which is accomplished through the use of a












stepper motor and through initial positioning of the sample. Detailed discussion

of the physics behind a SQUID are beyond the scope of this thesis, for a brief

overview see Reference [135].

The samples were mounted in standard number 4 or number 5 gelatin cap-

sules (gel-cap) that were held inside a plastic straw. Depending on the sample,

different mounting arrangements were used. If the sample was a single crystal

or an aligned single crystal packet, it was placed in the gel-cap and oriented in

the desired direction with respect to the magnetic field. Small pieces of gel-cap

were used to secure the sample. If the sample was a powder, it was poured into

the larger diameter portion of the gel-cap, and the smaller diameter portion was

inverted and used to press the sample, thereby stabilizing its position. Usually

S50 mg of sample was measured to provide a signal well above the background,

although samples as small as -~ 2 mg were successfully measured. Several arrange-

ments of this type have been measured sans sample to determine the background

contribution of the gel-cap and straw. A typical H = 0.1 T background is shown

in Figure 4.1, along with fits for three temperature regions. The expressions for

the three temperature regions are given by












-1. 221793 X -5 1.2534219X10-5
-.213x1- + 0.1091171+T
-1.8067205 x 10-8T


1.2156003 x 10-5 1.6642177 x 10-8T

+1.5027333 x 10-10T2 6.5003915 x 10-'3T3

+7.1854421 x 10-14T4


-1.423403 x 10-4 + 2.0092364 x 10-6T

-1.1421511 x 10-8T2 + 2.7916376 x 10-1T3

-2.4700249 x 10-14T4


2K < T < 40K






40K < T < 200K


200K

(4.1)
Similarly, Figure 4.2 shows the background for magnetic field sweeps at T = 2 K

with the fit curve given by


M(H) = -3.5256 x 10-5H 4.3195 x 10-5H2 + 9.0369 x 10-6H3

-6.8631 x 109H4


(4.2)


where H is in units of Tesla. Clearly, the background shows the expected dia-

magnetism and, at the lowest temperatures, a tail showing the presence of some

free spins. Nevertheless, the contributions are on the order of 10 MLemu, which

is negligible when compared to the magnetic response of most samples. Several

specimens consisted of a conglomeration of aligned single crystals that were glued

together with fingernail polish on a piece of weighing paper. To check the back-

ground, similar quantities of weighing paper and fingernail polish were placed









52










-6



-8



E -io



-12



-14
I I I I I
0 100 200 300 400

T(K)

Figure 4.1: M(T) SQUID background for an empty gel-cap and straw in
H = 0.1 T. The solid lines are fits given by Equation 4.1.


















0.0


-1.0x10.4


-2.0xl 04


-3.0xl 04


-4.0xl 04


-5.0xl 04


-6.0xl 04


0 1 2 3 4 5


H(T)


Figure 4.2: M(H) SQUID background for an empty gel-cap and straw at T = 2 K.
The solid line is the fit given in Equation 4.2.












in a gel-cap. This control was then measured, and the resultant background

subtracted from the corresponding data.


4.2 Cantilever Magnetometer

The cantilever was developed by Chaparala et al. [136] to provide a sensi-

tive magnetometer at high magnetic fields, and it is capable of simultaneously

measuring transport properties. The operational technique was refined at the

National High Magnetic Field Laboratory partly through our measurements as

will be discussed in later chapters. The result of this instrumental improvement

is the cantilever magnetometer commercially available from Oxford Instruments.

A schematic diagram of the cantilever is shown in Figure 4.3. There are sev-

eral features to observe. The four leads down the middle are used if transport

measurements are desired along with magnetic measurements. To each side of

the transport leads, other leads form two square loops, which can be used for

calibration if absolute magnetization units are required. The sample is mounted

between the two loops, as shown in the figure. Not shown is the bottom of the

cantilever where a gold plate forms one half of a capacitor. The other half is

a plate fixed below the cantilever. The experimental quantity measured is the

capacitance between the two plates. Typically, the sample was attached to the

cantilever with a small quantity of vacuum grease. This grease has a diamagnetic

background which may need to be subtracted. The determination and removal

of this background will be discussed later with the experimental results.














Lower Capacitance Plate
I


U)


0


1)
I


/ /


Cantilever


Calibration Coil


Figure 4.3: Cantilever magnetometer diagram. This figure is adapted from [137].


E
(0,
CO


To Upper


Capacitance Plate












The cantilever can work either in force mode or in torque mode. Only the

force mode will be discussed here because a discussion of the torque mode [136]

is beyond the scope of this thesis. For the force mode of operation, the plane of

the cantilever is positioned perpendicular to H. The force on a magnetic sample,
F, in a magnetic field gradient, VH, is proportional to MVH. Since the sample

is affixed to the cantilever, both items experience the same force. Following

Hook's law, the cantilever opposes F with a force proportional to the amount it

bends, causing the distance x between the plates to change by a quantity Ax.
This change causes the capacitance, C, to change by an amount AC. Therefore,

any change in the force may follow from AF oc Ax oc AC. In other words,

by knowing AC, the change in the force on the cantilever is known to within a

calibration constant. Furthermore, since VH oc H, the change in magnetization

from a reference is AM oc AF/H oc AC/H. The reference is defined by the fact

that F(H = 0) = 0. The VH needed to apply a force on the sample is obtained

by positioning the sample slightly above the center of the field. Table 4.2 gives

field gradient specifications for the magnet (SCM1) used at the National High

Magnetic Field Laboratory. For typical runs, the cantilever was placed 27.0 mm

or 52.4 mm above the center of the field, but for initial runs, it was placed as far

as 152.4 mm off the center of the field.

Sensitivity is gained by working far off the center of the field because VH in-

creases until approximately 160 mm away from the center of the field. However,

the peak field decreases as the cantilever is moved away from the center of the

field. For example, the 152.4 mm length maximizes the cantilever response



























Table 4.1: Field and gradient at distances off the center of the field along the
axis of the magnet SCM1.

but only provides a field of 8 T on the sample when the center field is 18 T.
The 52.4 mm distance provided the optimal maximum field and sensitivity, al-

though there are occasions when it is experimentally prudent to place the sample

closer to, or farther away from, the center of the field.

As with any technique, certain limitations must be considered to effectively

use the cantilever magnetometer. First, the magnetic response of the apparatus

is difficult to calibrate. A slightly different background response is observed for

every sample, arising from differences in mass and the magnetic response of the

mounting material. Therefore, a calibration must be run in situ for each different

sample mounting. This procedure is time consuming and is often infeasible for

the time allotted at the National High Magnetic Field Laboratory. Furthermore,

our attempts at calibration were too coarse to be useful. Nevertheless, an ab-

solute measure of magnetization is not essential for identifying critical fields.


Z Bz dB/dZ
(cm) (T) (T/cm)
0.000 20.000 0.000
0.500 19.986 0.028
1.000 19.946 0.080
1.500 19.877 0.138
2.000 19.779 0.196
2.500 19.651 0.256
3.000 19.491 0.320
3.500 19.300 0.382
4.000 19.064 0.472
4.500 18.792 0.544












Therefore, the cantilever is an excellent device for studying the Haldane gap.

Second, since the force is divided by the magnetic field to obtain the magne-

tization, the error bars in the magnetization grow quickly at fields below 1 T,

obscuring low field features. In addition at low fields, superconducting magnets,

which have been run above H,1 of the superconducting wire, spontaneously expel

magnetic flux trapped in the wire, and this effect causes noise in the measure-

ment known as flux jumps. Therefore, the issue of a usable low field region is a

combination of the limitation of the technique and the apparatus used. Third,

the force is always measured relative to the zero magnetic field state. Therefore, a

constant force or magnetization can not be distinguished from no magnetization.

For example, at millikelvin temperatures, where paramagnetic spins saturate at

a few mT, a paramagnetic contribution can not be distinguished from the zero

ground state in a Haldane system.

To ensure that the limitations on the experiment result from the cantilever

alone, a low noise set of electronics, shown in Figure 4.4, was used to acquire data.

The dark grey and light grey shading identifies the portions of the apparatus that

are inside the mixing chamber of the dilution refrigerator and immersed in the

liquid helium bath, respectively. The dotted box indicates a device that was

tested, found to provide negligible improvement, and subsequently removed. The

temperature controller used a four wire measurement to regulate the temperature.

At the time of our experiments, the thermometer was calibrated for zero magnetic

field only. Therefore, the resistance of the thermometer at the start of the run

was compared to the resistance at the end of the run to be sure that thermal











































Figure 4.4: Cantilever magnetometer electronics schematic. The dark grey and
light grey shading identifies the portions of the apparatus that are inside the
mixing chamber of the dilution refrigerator and immersed in the liquid helium
bath, respectively.












stability was maintained throughout the run. A heater provided the necessary

temperatures above base temperature, although it was rarely needed. Several

different digital multimeters were used as analog to digital converters, and all of

them added negligible amounts of noise to the measurement. To reduce noise, a

lock-in amplifier was used for phase sensitive detection. The Princeton Applied

Research model 124A lock-in amplifiers, PAR 124A, were chosen since their signal

to noise ratio was at least an order of magnitude better than any other instrument

tested. The PAR 124A has an internal voltage controlled oscillator that was

operated at 10 kHz. In order to reduce coupling from the outside environment

as much as possible, the wires from the General Radio capacitance bridge to the

top of the insert were a twisted pair of coaxes. Finally, all data was collected by

a Macintosh personal computer, using a general purpose interface bus (GPIB)

and the LabView software package.

To cool the samples to temperatures well below the Haldane gap, an Oxford

Instruments Kelvinox top loading dilution refrigerator was used. The top loading

nature of the cryostat allows samples to be mounted while the refrigerator is at

- 4 K. Once the sample is in the mixing chamber, condensation and circulation

can begin, and within 6 hours, the sample is at a base temperature of 30 mK.

The magnet used was SCM1, a superconducting magnet provided by Oxford

instruments. The maximum center field is 18 T at T = 4.2 K, and 20 T can be

reached through the use of a lambda refrigerator that reduces the temperature

to 1.4 K. The current/field ratio is known to be 10 A/T, so the magnetic field

on the sample is determined by the output current of the magnet power supply.












4.3 Electron Spin Resonance

Electron spin resonance is a powerful microscopic probe of magnetic proper-

ties. Its use provided a wealth of information on the end-chain spins in NINAZ.

There were two experimental apparatuses used for the ESR work presented in this

thesis. The most extensively used instrument was a Brueker X-band spectrome-

ter working at 9 GHz. Professor D.R. Talham of the Department of Chemistry

at the University of Florida graciously provided the use of this spectrometer and

its support staff. Figure 4.5 shows a schematic diagram of the apparatus. A

standard (i.e. iron yoke and water cooled) split coil magnet applied a DC field to

the sample in a direction perpendicular to the direction of the microwave prop-

agation. Since this spectrometer uses a resonance cavity and a Klystron for a

microwave source, it is more convenient to vary H instead of w, as is the stan-

dard practice [55]. This resonance cavity is coupled with an Oxford ESR 900

Flow Cryostat, which cools the sample down to T 4 K. The temperature is

measured by a AuFe/Ch thermocouple with the reference junction kept at 77 K.

Figure 4.6 is a schematic diagram illustrating the flow of the helium gas and the

sample and thermometer positions. Single crystal samples were affixed to the end

of a quartz rod using Apiezon L grease and inserted directly into the flow of the

helium gas. On the other hand, powder samples are placed in a quartz tube, so

the sample is thermally coupled to the helium gas through the tube. The heater

is placed well away from the sample to avoid any interference in the measure-

ment. However, this arrangement lengthens the thermal equilibration time of the

sample. Therefore, one should wait at least five minutes, after a temperature














































Figure 4.5: Schematic diagram of the X-band ESR spectrometer.












change, for the system to thermally equilibrate. Additional care should be taken

with powder samples, or samples of low thermal conductivity, as the thermome-

ter directly measures the temperature of the helium gas and not the temperature

of the sample. To ensure proper thermal conductivity between powder samples

and the quartz tube, a syringe was inserted into the rubber stopper at the end

of the tube, and this arrangement was evacuated via a mechanical pump while

initially cooling the cryostat. When the base temperature was reached, pumping

was stopped, and helium exchange gas was injected into the sample tube. Not

shown in Figure 4.6 is the room temperature coupling which allows the sample

to be rotated without leaking air to the cryostat. A homemade goniometer was

used to determine the orientation of the magnetic field with respect to the crystal

axes.

To accommodate for the signal to noise improvements offered by a phase-

locked loop, a small modulation field was added to the DC field. Introduc-

ing the modulation has the added affect of differentiating the absorption signal.

Returning the data to an absorption signal from the derivative signal was accom-

plished using a second-order, Runge-Kutta numerical integration routine [138]

with an automatic baseline adjustment procedure. The code was written in Mi-

crosoft Visual C++ Version 4.0 and ran on a PC running Windows95.

Since a measure of absolute signal intensity was required, a room tempera-

ture calibration against a, a'-diphenyl-fl-picrylhydrazyl (DPPH) was performed

following standard procedures [55,139]. A 0.1 mg single crystal purchased from

the Aldrich Chemical Company was used for the calibration. With a molecular














Resonance
Cavity


Thermometer


Sample












He Return


He Injection


Heat Exchanger
and Heater


Figure 4.6: Cross-sectional schematic view of the ESR flow cryostat. Adapted
from [140]












weight of 394 g/mol and 1 free spin per molecule [139], our sample contained

1.52 x 1017 spins. Figures 4.7 and 4.8 show the derivative signal and the inte-

grated absorption signal, respectively. The incident microwave power used for

the calibration was -30 dB of 300 mW at a frequency of v = 9.5453 GHz. The

area under the intensity, I, vs. H data is 2.279 x 108 Iunits-kG which results

in 6.67 x 108 spins/Iunit.kG. This procedure has effectively calibrated the area

under the ESR spectrum to the number of spins. In order to determine the

intensity of the DPPH signal at lower temperatures, a Curie law was assumed,

and the corresponding number of spins calculated. The procedure avoids the

non-Curie law behavior, observed in DPPH below ,- 50 K [55], because the cal-

ibration was performed at room temperature where correlations between DPPH

spins are negligible.

Preliminary high frequency ESR work was done using the high field trans-

mission spectrometer in the laboratory of Professor L.C. Brunel at the National

High Magnetic Field Laboratory. The spectrometer has four changeable Gunn

oscillator sources with Schottky based harmonic generators and appropriate fil-

ters which allows harmonics of these sources to be used. The available frequencies

are given in Table 4.2. The 25-130 GHz source does not have sufficient power for

use with the harmonic generators. The magnetic field is supplied by an Oxford

Instruments 17 T superconducting magnet oriented with the field longitudinal to

the microwave beam. The spectrometer measures the microwave radiation trans-

mitted through the sample, so the beam comes out of the source, passes through









66






I I I I I


2



1



CO)
c


0






-1




-2 I i l I i I

3380 3390 3400 3410 3420

H(G)

Figure 4.7: dI/dH vs. H for a 0.1 mg sample of DPPH at T = 293 K.










67






i I I I


8





6





24-
.0



2







I I I I I
3380 3390 3400 3410 3420

H(G)


Figure 4.8: I vs. H for a 0.1 mg sample of DPPH at T = 293 K.













Source Harmonics
25-130 -
75 150 225 300 375
95 190 285 380 475
110 220 330 440 550


Table 4.2: Frequencies available for microwave transmission at the National High
Magnetic Field Laboratory, in units of GHz.

the sample, and then is reflected back to a detector, usually a bolometer, which

records the data on a Macintosh personal computer.


4.4 Neutron Scattering

To extend our studies of the Haldane gap, inelastic neutron scattering has

been used as an additional microscopic probe. Precise measurements of the re-

ciprocal lattice vectors are required for inelastic neutron scattering measurements,

so neutron diffraction was used to determine them. In any neutron scattering ex-

periment, both elastic (neutron diffraction) and inelastic scattering is observed.

A full analysis separates the received scattering intensity into four components:

elastic coherent scattering, elastic incoherent scattering, inelastic coherent scat-

tering, and inelastic incoherent scattering [141]. Elastic coherent scattering is

typified by Bragg peaks that are usually the largest contribution to the received

intensity. The next largest contribution results from incoherent scattering. If

a large incoherent scattering peak is observed, it arises from randomly oriented

nuclear magnetic moments. Therefore, careful isotope selection during sample

synthesis will greatly reduce this contribution. For example, the samples studied












for this thesis possess large organic ligands, so hydrogen would be the largest

incoherent scatterer because the single proton has a rapidly moving nuclear mag-

netic moment of I = 1/2. On the other hand, deuterium has no nuclear moment,

so full deuteration of the samples eliminates the largest source of incoherent

scattering. Generally, in properly deuterated samples, the incoherent scattering

component is negligible when compared to the coherent elastic scattering and
can be neglected for structure refinements. In addition, inelastic components are

generally much smaller than elastic components and can be neglected in neutron

diffraction experiments.

When inelastic neutron scattering measurements are performed, coherent elas-

tic scattering is the most troublesome component. The simplest way to eliminate
the contribution due to these Bragg peaks is to avoid measuring regions of Q

space where they exist. In addition, the incoherent elastic component is usu-

ally much larger than the coherent inelastic component and must be avoided.

There are two ways to avoid this contribution. Firstly, reduce the contribution

as much as possible by choosing appropriate isotopes, and secondly, since inco-

herent scattering is always around Q, E = 0, inelastic scattering at sufficiently

large wavevectors and energies are unaffected by the incoherent scattering peak.

A mixed blessing and curse of neutron scattering experiments is that the neu-

trons interact weakly with the material. Therefore, at least 1 g of materials is

needed for sufficient experimental resolution [142]. The metal-organic materials

used for these measurements usually do not grow in single crystals this large.

Therefore, an added complication to the work was that a collection of randomly












oriented microcrystallites was used for the measurement. Fortunately, the mi-

crocrystalites were small enough to effectively act as a symmetric powder as will

be described in Section 7.3. Therefore, to determine the lattice parameters, the

powder neutron diffraction pattern was analyzed by Rietveld refinement.

Rietveld refinement is essentially a non-linear least squares fit between an

assumed crystal structure, whose Bragg peaks are powder averaged, and the

actual data. Since Rietveld, who developed the method, believed strongly in

the free dissemination of information, his computer routines [143-145] have been

freely distributed and used to develop excellent public domain analysis software.

The package used by the author is GSAS from Los Alamos National Laboratory

which has a user friendly interface for the personal computer running Microsoft

Windows. This program runs sufficiently fast on a Pentium based machine with

copious quantities of memory. Insufficient memory space (e.g. 24 megabytes
is not enough) causes program crashes which require restarting the refinement

from the raw data. Generally, the technique is extremely powerful and up to 193

parameters have been refined, 161 of them simultaneously [146,147]. As with

any non-linear least squares fitting, one must have some feel for the right answer

before starting, otherwise the fit of the naive user will be stuck in a local minimum

in the multiparameter hyperspace. The method used to avoid local minima was

to examine the resultant crystal structure for physical plausibility on the basis of

the known chemical composition. Furthermore, several different quantities can

be used to quantify the quality of the refinement [147]. The primary test statistic

used was X2 reduced by the number of parameters being varied, and the total












X2 was used as a secondary test to confirm that changes in the reduced X2 were
significant.

The spectrometer used was the high resolution powder diffractometer on beam

tube HB-4 of the High Flux Isotope Reactor (HFIR) at Oak Ridge National

Laboratory. The wavelength of the neutron beam is selected by orienting a (115)

Ge single crystal. This monochromator provides the sample with wavelengths of

1.0, 1.4, 2.2, or 4.2 A. The beam passes through the sample to 32 'He detectors

that are equally spaced at 2.7 apart. These detectors can be scanned over a range

of 40 so the total angle covered is 11 to 135. A collimator in the beam reduces

the beam size from the maximum 3.75 x 5 cm2 to the desired size. Typically, the

flux at the sample is 2 x 105 neutron [148]. Using a glove box, the sample is placed
CM2s
inside a vanadium can in a helium atmosphere. The can is sealed with an indium

o-ring before removing it from the glove box. If two sample batches need to be

kept separate inside the vanadium can, each batch is placed in an aluminum foil

pouch. For room temperature measurements, the sample is rotated while in the

neutron beam to average out any asymmetry in the powder. For temperature

dependent measurements, a displex was used with aluminum vacuum cans, and

this refrigerator was capable of achieving 11 K.

The main goal of the neutron scattering study is to examine the magnetic

excitation spectrum with inelastic neutron scattering. There are several factors

to consider when choosing an instrument for inelastic neutron scattering. Firstly,

the incident neutron energy must be selected so that the desired features are

resolvable. As a general rule of thumb, the resolution is related to the incident












energy by AE 0.02Ei [142]. Therefore, neutrons from a water moderated

reactor (thermal neutrons) have A/E a 0.5 meV, but the Haldane gap is expected

to be around 0.2 meV. Consequently a source of lower energy neutrons (cold

source) is required. Once the cold source is identified as necessary, there are

several instruments from which to choose to make the measurement.
Since the sample used is a powder sample, a technique which could collect

data for a broad range of Q values simultaneously is desirable. This requirement

means a time of flight spectrometer is ideal. For inelastic neutron scattering,

four quantities must be known: the initial energy, the initial momentum, the

final energy, and the final momentum. Assuming the initial quantities can be

tuned, only the final energy and momentum have to be measured. For time of

flight, the energy and the magnitude of the momentum are determined by the time

required for the neutron to travel a known distance. A pulsed source of neutrons
provides a t = 0 reference for the time measurement. To determine the final angle

of the momentum, the detectors are equally spaced at known angles. Since the

detectors are fixed, data for multiple Q values is collected simultaneously.

The time of flight spectrometer (TOF) on the cold source at the National

Institute of Standards and Technology was chosen for the measurement. A liquid

H2 moderator is used as the cold source to provide a distribution of neutrons

peaked around an energy of 5 meV. Neutrons from the cold source are directed

to the instrument in neutron guide NG-6. Once at the instrument (Figure 4.9),

two pyrolytic graphite monochromators select neutrons of the desired energy.

Next, a cooled Be and pyrolytic graphite filter removed neutron beam compo-










73













S00000000000 Detectors where
0 0 o0000 000 possible Haldane
00 / -o/, gap is observed
0o Sample Do
^ Area /
Monochromator ".
Section -^ \/ /

















PG Section
Filter

Figure 4.9: NIST TOF spectrometer from Reference [149]. Notice the region of
detectors used to identify hints of the Haldane gap in MnC13(d-bipy).












nents remaining from higher order reflections in the monochromator crystals. In

addition, a Fermi chopper sets the length of the time pulse. Finally, a flux of

2.4 x 104 neutron interacts with the sample, passes through an oscillating radial
cmo2s

collimator, and is detected by 3He detectors. The detectors are equally spaced

at 2.546 for angles from 1.7 to 130. Unfortunately, the detectors for angles

less than 22 are too noisy for use on this experiment. For future reference, the

region of detectors used to identify hints of the Haldane gap in MnC13(d-bipy) is

indicated in Figure 4.9.


4.5 Mechanical Ball Milling

For part of this thesis, the antiferromagnetic chains were broken by mechanical

methods. Mechanical ball milling was used to produce the finest ground powder

using the ball mill of Professor J. H. Adair in the Department of Materials Science

and Engineering at the University of Florida. To perform the milling, the material

is placed in a polypropylene bottle with zicronia balls and fluid, hexane in this

case, which efficates the crushing process. The bottle is then rotated for 36 hours

to perform the pulverization. The resulting particle size distribution usually

follows a log-normal distribution (Figure 4.10) [150].


4.6 Centripetal Sedimentation

A method was needed to determine the characteristic particle size of various

samples. To this end, centripetal sedimentation was performed in the laboratory

of Professor J. H. Adair in the Department of Materials Science and Engineering
















0.5 *,i



0.4




0.3



t. 0.2




0.1




0.0
I I I I I
0 5 10 15 20

X

Figure 4.10: A normalized log-normal distribution of width 1 centered at x = 1.
This is the theoretically predicted particle size distribution for a ball milled
sample.












at the University of Florida. The main idea is to measure the spatial distribution

of the particles in a medium of known viscosity when a known force is applied to

them for a certain length of time. In our case, the viscous medium was hexane

as it did not react with the material being studied. To measure the spatial dis-

tribution of distances, a laser light is passed through the particle-fluid mixture,

and the relative intensity is directly proportional to the particle size distribu-

tion. The applied force arises from the centripetal acceleration from spinning the

mixture [151].


4.7 Inductively Coupled Plasma Mass Spectrometry

Since this thesis claims that the end-chain spins are intrinsic impurities, a

technique was needed to ensure that the extrinsic impurities in the sample were

negligible. Therefore, inductively coupled plasma mass spectrometry, ICP-MS,

was performed in the laboratory of Dr. D. H. Powell in the Department of Chem-

istry at the University of Florida. Figure 4.11 is a schematic of a typical ICP-MS

system [152,153]. It consists of a silica plasma torch which maintains the plasma

by a continuous feed of gas (usually Argon) through radio frequency coils which

couple to the plasma. The sample to be measured is introduced directly into the

plasma and is ionized by the intense heat. Two small orifice cones are placed in

series, approximately 10 mm from the edge of the torch, to remove ions from the

useful part of the plasma. Once the ions are removed, they are sent into a stan-

dard quadrapolar mass spectrometer for analysis. Then, the ions are deflected by

the charged plates in the quadrapolar mass spectrometer providing a distribution

of the elements according to their molecular weight.


















Quadrapolar
mass
spectrometer


Sampling
Cones


RF Coils
/ Sample
Feed
c)(-

(K__ ^-


Ar Feed


Figure 4.11: Schematic diagram of an ICP-MS system. The size of the torch
and the cones is greatly exaggerated with respect to the mass spectrometer for
illustrative purposes.














CHAPTER 5
TMNIN


This chapter covers magnetic measurements on (CH3)4N[Ni(N02)3], com-
monly known as TMNIN. Initial magnetic susceptibility measurements displayed

the characteristics of a 1-D chain with a small exchange energy. Furthermore,
magnetization, M, vs. magnetic field, B, measurements revealed a Haldane gap

small enough to be reached by high resolution NMR magnets. Therefore, a pro-

gram was started to grow and characterize single crystal specimens for NMR

experiments. To this end, M vs. B measurements were performed at millikelvin
temperatures in order to characterize the gap and single-ion anisotropy. These
measurements revealed a small value of D and are the main topic of this chapter.


5.1 Synthesis and Structure of TMNIN

Tetramethylammonium nickel nitrate (TMNIN) was first grown by Goodgame

and Hitchman [71], and initial attempts at the crystal structure were made by

Gadet et al. [72] using twined single crystals. Their results confirm that TMNIN
is an antiferromagnetic chain with adjacent Ni2+ ions bridged by three NO2

groups. The tetramethylammonium group, [(CH3)4N]+, sits between the chains
to space them and provide charge neutrality. These features are clearly observed

in Figure 5.1. Nevertheless, one feature of the crystal structure remained il-

lusive until the single crystal X-ray diffraction studies by Chou et al. [74,83].






























Nil






02










Figure 5.1: TMNIN Crystal Structure [74,83]. Note that Nil is octahedrally co-
ordinated with nitrite nitrogens, and Ni2 is octahedrally coordinated with nitrite
oxygens.










More specifically, the Nil is octahedrally coordinated with six nitrite nitrogens,
and Ni2 is octahedrally coordinated with six nitrite oxygens. Different Ni sites

may slightly affect the magnetic properties of the chain, as will be discussed later

in the chapter.

The single crystals used for the X-ray studies and the magnetization measure-

ments, to be detailed below, were produced following the synthesis procedure of

Goodgame and Hitchman [71], but modified to allow the crystals to grow undis-

turbed in a 284 K environment for 4 months. The synthesis procedure starts by

slowly adding solutions of NiBr2 and (CH3)4NBr to a solution of NaNO2. After

4 months, large single crystals were harvested from the sides of the container.

Fifty of these single crystals were oriented and glued together with clear finger

nail polish to produce a 2.1 mg packet. Both this packet and a few single crystals

of approximately 1.1 mm in length and 0.11 mm in diameter were used in the

magnetization studies to be described below.


5.2 Magnetization Measurements on TMNIN

Initial magnetization measurements on the TMNIN packet were performed

on a SQUID magnetometer at 1.8 K and 30 K for magnetic fields from 0 to

5.5 T oriented parallel to the chains. As can be seen in Figure 5.2, a hint of the

critical field associated with the Haldane gap is observed at 2.7 T. However,

since the sample temperature is close to the size of the gap, the signature of

the critical field is thermally smeared. The T = 30 K (T A) data provide

additional circumstantial evidence for the presence of the gap since no critical field







81


1.5 1


TMNIN x
B II Ni-chain o

1 -o 30 K
CO 0
x 1.8K x
3 0
.-"
o0 x
0


xx
0.5 -




00 x
0 X X
00
X
0 X
0 1,- ix x x
I I I I I
0 2 4 6
B(T)

Figure 5.2: SQUID M vs. B for TMNIN at temperatures of 1.8 K and 30 K [75].

is observed. For both temperatures, the chain axis of the packet was oriented

parallel to the magnetic field. Similar results were obtained by other workers

[72,77] for a powder sample.

To better resolve the critical magnetic field, subsequent measurements were

performed at lower temperatures. The facilities of the National High Mag-

netic Field Laboratory were ideal for this purpose. Magnetization as a function

of magnetic field was measured with the cantilever magnetometer described in

Section 4.2. Millikelvin temperatures at high magnetic fields were provided by a








top loading dilution refrigerator in the superconducting magnet SCM1. Since our
only concern was the value of the critical field, a quantity not requiring an ab-

solute determination of the magnetization, the complicated and time consuming

calibration process was bypassed. Two runs were made, the first on the above

described packet and the second on a single TMNIN crystal. The packet was

attached to the cantilever with silver paint, and the cantilever and sample were

placed a distance of 16 cm from the center of the field. This position produced an

ample field gradient for the cantilever to operate but reduced the maximum mag-

netic field at the sample by a factor of 2.5. Figure 5.3 shows both the force on the

cantilever and the magnetization of the sample for sweeps from 0 to 8 T with the

magnetic field oriented perpendicular to the chain axis. A critical field, associated

with the Haldane gap, can be extracted as Bej = 2.90 0.15 T. In addition, the

uncertainty in the magnetization data increases greatly below 1 T. This increase

is an artifact of the technique since magnetization is derived from the data using

M = F/B, as described in Section 4.2. This effect emphasizes the need to exam-

ine the raw force data. Figure 5.4 shows data for the same sample with the field

oriented perpendicular to the chain axis. Several sweeps are displayed. Again the

critical field can be extracted as B\11 = 2.40 0.15 T. The other striking feature

in Figure 5.4 is the continuous bend in the data, starting near 4 T and continuing

to the end of the data. This experimental artifact results from the diamagnetic

contribution of the silver paint. For low and high fields, the sample and the

diamagnetic background control the cantilever response, respectively. Therefore,

at intermediate fields the observed data result from a competition between the







83




80

TMNIN
g60 B I Ni-chain

I- x 25mK -
S40 + 100 mK
(a)
0
LL 20


0

8
L. (b)
.- 6 7-

n4


2-


0
0 2 4 6 8
B(T)
Figure 5.3: (a) F and (b) M vs. B1 for TMNIN packet 16 cm off the center of
the field [75]. The packet weighed 2.1 mg and was tested at T = 25 and 100 mK
using the cantilever magnetometer.









sample behavior and the diamagnetic background. A couple of unique features

are observed when the data in Figures 5.3 and 5.4 are compared. First, the data

in Figure 5.3 exhibit a finite magnetization below the critical field. To see if this

finite magnetization resulted from paramagnetic impurities, scans were performed

at temperatures of 25 + 5 mK and 100 5 mK. There is no discernible difference

between the data at the two temperatures, suggesting the absence of paramag-

netic impurities. However, paramagnetic impurities saturate at a few mT for

temperatures in the millikelvin range, and, therefore, a finite magnetization be-

low the gap would result. Nevertheless, since this finite magnetization is absent

for the other field orientation and any effect from paramagnetic spins should be

isotropic, another explanation is needed for this effect. By comparisons of several

runs on TMNIN, and further studies on other samples, we have determined this

effect originates from slight differences in the mounting material from one run to

the next. Later, the same sample packet and the cantilever were moved closer

to the center of the field to improve the response of the system. The results are

shown in Figure 5.5. Clearly there is a strong reaction to the magnetic field,

such that above 6 T the cantilever responds nonlinearly. Nevertheless, a critical

field of BI = 2.60 0.15 T can be extracted. To reduce the nonlinear effects,

a second run on a single crystal of TMNIN was performed. The data are dis-

played in Figures 5.6 and 5.7. These data are consistent with the packet data,

albeit with slightly different critical fields. This difference might be caused by

internal stresses present in the packet, due to thermal contraction differences be-

tween the fingernail polish and the sample, that are absent in the single crystal.


















TMNIN
SB II Ni-chain
25mK
.oupl
C
'" +up2
-Q xdn2
Ca 2:J -
B(T
0
u-
0
0 (a)






0.5



Ca
M 0
,, (b)



-0.5
0 2 4 6 8
B (T)

Figure 5.4: (a) F and (b) M vs. BI, for TMNIN packet 16 cm off the center of
the field [75]. The results of several magnetic field up and down sweeps for this
2.1 mg packet using the cantilever at T = 25 mK are shown.








86





2 I I I I I I I

3

2-
C

I I


B(T)

L.L



0


0 2 4 6 8
B(T)



Figure 5.5: (a) F and (b) M vs. B for TMNIN packet 5 cm off the center of the
field [76]. The cantilever magnetometer made the measurement on this 2.1 mg
packet at T = 60 mK.








87






6

4

2
U.
~(a) -
0



2 0.2-



0(b)

-0.2
0 5 10 15 20
B(T)

Figure 5.6: (a) F and (b) M vs. B1 for TMNIN single crystal using the cantilever
magnetometer placed 5 cm off the center of the field [76].








88






1

0.5

c-
"- (a)




0.2
4W.

0.1-?
-





2 3 4 5
B (T)

Figure 5.7: (a) F and (b) M vs. BII for TMNIN single crystal using the cantilever
magnetometer placed 5 cm off the center of the field [76].




Full Text

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PAGE 255

B f} A ILOLO


EXPERIMENTAL STUDIES OF INTEGER SPIN ANTIFERROMAGNETIC
CHAINS
By
GARRETT E. GRANROTH
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1998

ACKNOWLEDGMENTS
I am sincerely grateful for the guidance and encouragement of my thesis ad¬
visor Professor Mark Meisel. Many thanks go to Professors Selman Hershfield,
Gary Ihas, Kevin Ingersent, Stephen Nagler, Fred Sharifi, and Dan Talham for
serving on my supervisory committee at one time or another. Special thanks go
to the materials wizards Brian Ward and Liang-Keui Chou and their supervisor
Professor Dan Talham for synthesizing the samples. Several people have provided
help with various measurements. Thanks to Dr. Murali Chaparala for help with
the cantilever magnetometer and interpretation of the resultant data. Thanks to
Dr. Brian Chakoumakos for assistance with the powder neutron diffractometer
and for advice concerning Rietveld refinement. Thanks to Dr. Jeff Lynn and
Dr. Ross Erwin for assistance with the time of flight neutron diffraction experi¬
ments. Thanks to Brian Ward, Liang-Keui Chou, and Candace Seip for operating
the 9 GHz ESR spectrometer. Thanks to Gail Fanucci for running the NINAZ
on the high field ESR spectrometer. Thanks to Nelson Bell and his advisor
Professor Jim Adair for preparing the ball milled samples and for characterizing
their particle size. Thanks to Dr. Thierry Jolicoeur for help with many aspects
of the theory for S — 2 chains and for providing the high temperature series
expansions for S = 1 chains. I am grateful for several fruitful discussions with
Kingshuk Majumdar and Dr. Arnold Sikkema that have helped me to understand
non-linear sigma models and other theoretical obfuscations. Several individuals
li

have provided tutelage along the path to this degree. Professor Stephen Nagler
helped with the neutron scattering experiments. Professor Satoru Maegawa gave
stimulating input on the NINAZ work. Dr. Philippe Signore provided the es¬
sential knowledge that is passed down the line of graduate students. Professor
Kevin Riggs introduced me to experimental studies of magnetic materials us¬
ing microwave resonance techniques. I would also like to thank my high school
chemistry teacher Mrs. Dorris Wasson for encouraging me to choose a career in
science.
I am always grateful for the encouragement of my parents and family that
has helped me along the way. Additionally, I would like to acknowledge the
Creator who provides endless scientific problems to study. Finally, I am indebted
to Amelia Helmus for grammatical editing of this thesis.
in

TABLE OF CONTENTS
ACKNOWLEDGMENTS ii
LIST OF FIGURES vii
ABSTRACT xi
CHAPTERS
1 INTRODUCTION 1
2 REVIEW OF THEORETICAL STUDIES 3
2.1 Theoretical Models 4
2.1.1 The Non-Linear Sigma Model 4
2.1.2 Valence Bond Solids 6
2.2 An Intuitive Picture of the Haldane Gap 9
2.3 Numerical Studies of Quantum Spin Chains 10
2.3.1 Realistic Hamiltonians 18
2.4 Measurement Theory 22
2.4.1 Electron Spin Resonance 24
3 REVIEW OF EXPERIMENTAL STUDIES 30
3.1 Haldane Gap Materials 30
3.2 Experimental Tests of the Haldane Gap 32
3.3 Studies of End-chain Spins 40
3.4 Other Experimental Results 45
4 EXPERIMENTAL MEASUREMENT TECHNIQUES 48
4.1 SQUID Magnetometer 48
4.2 Cantilever Magnetometer 54
4.3 Electron Spin Resonance 61
4.4 Neutron Scattering 68
IV

4.5 Mechanical Ball Milling 74
4.6 Centripetal Sedimentation 74
4.7 Inductively Coupled Plasma Mass Spectrometry 76
5 TMNIN 78
5.1 Synthesis and Structure of TMNIN 78
5.2 Magnetization Measurements on TMNIN 80
5.3 Comparison with Other Experiments 89
6 NINAZ 92
6.1 Material Description 93
6.2 Other Experiments 96
6.3 Experimental Studies 99
6.3.1 Samples 99
6.3.2 [Ni(C4H12N2)2(//-N3)]n(C104)n 103
6.3.3 Macroscopic Measurements 108
6.3.4 Microscopic Measurements: Electron Spin Resonance ... 117
7 THE 5 = 2 HALDANE GAP IN MnCl3(bipy) 153
7.1 5 = 2 Quasi-linear Chain Materials 154
7.1.1 MnCl3(phen) 154
7.1.2 Mn(acac)2N3 154
7.1.3 Mn(salpn)OAc 157
7.1.4 MnCl3(bipy) 161
7.1.5 Synthesis 161
7.2 Macroscopic Magnetic Measurements of MnCl3(bipy) 164
7.3 Low Temperature Crystal Structure: Preparation for Microscopic
Measurements 175
7.4 Inelastic Neutron Scattering Measurements 180
8 CONCLUSIONS 196
8.1 TMNIN 196
8.2 NINAZ 197
8.3 MnCl3(bipy) 198
v

APPENDICIES
A High Temperature Expansions 200
A.l Specific Heat 200
A.2 Perpendicular Magnetic Susceptibility 214
A.3 Parallel Magnetic Susceptibility 220
B Low Temperature Crystallographic data for MnCl3(d-bipy) 226
BIBLIOGRAPHY 232
BIOGRAPHICAL SKETCH 240
vi

LIST OF FIGURES
2.1 A single valence bond 6
2.2 An 5 = 1/2 chain in the RVB picture 7
2.3 An 5 = 1 chain according to a VBS model 7
2.4 The VBS state for an 5 = 2 chain 8
2.5 5 = 1 chain dispersion curve 13
2.6 Staggered magnetization for 5 = 1 chains 16
2.7 1/x vs. T for end-chain spins from QMC work 17
2.8 5 = 1 phase diagram 19
2.9 5 = 2 phase diagram 21
2.10 Energy level diagram showing the Haldane gap 23
2.11 Allowed transitions between end-chain spin states 27
3.1 M(H) for NENP 34
3.2 Inelastic neutron scattering intensity for Y2BaNi05 36
3.3 x(400 pK < T < 300 K) for pure NENP 44
3.4 The crystal structure of NENP 46
4.1 M(T) SQUID background 52
4.2 M(H) SQUID background 53
4.3 Cantilever magnetometer diagram 55
4.4 Cantilever magnetometer electronics schematic 59
4.5 Schematic diagram of the X-band ESR spectrometer 62
4.6 Cross-sectional schematic view of the ESR flow cryostat 64
4.7 dl/dH vs. H for a 0.1 mg sample of DPPH 66
4.8 I vs. H for a 0.1 mg sample of DPPH 67
4.9 NIST TOF spectrometer 73
4.10 A normalized log-normal distribution 75
4.11 Schematic diagram of an ICP-MS system 77
5.1 TMNIN Crystal Structure 79
5.2 SQUID M vs. B for TMNIN 81
5.3 F and M vs. B±_ for TMNIN packet 16 cm off the center of the field 83
5.4 F and M vs. B\\ for TMNIN packet 16 cm off the center of the field 85
5.5 F and M vs. B± for TMNIN packet 5 cm off the center of the field 86
5.6 F and M vs. B± for TMNIN single crystal 87
vii

5.7F and M vs. B\\ for TMNIN single crystal 88
6.1 Room temperature NINAZ crystal structure 94
6.2 DC X(T) of NINAZ 97
6.3 Constant Q « tt inelastic neutron scattering data for deuterated
NINAZ 98
6.4 Powder NINAZ particle size distributions 101
6.5 Ultrafine powder NINAZ particle size distributions 102
6.6 The crystal structure of [Ni(C4Hi2N2)2(//—N3)]n(C104)n 105
6.7 x±(2K < T < 300 K) for [Ni(C4H12N2)2(/i-N3)j n(C104)„ 106
6.8 X||(2K 6.9 x(50 mK < T < 2 K) of polycrystalline and powder NINAZ .... 109
6.10 x(T) for powder and ultrafine powder NINAZ 110
6.11 M(H < 5 T, 2 K) for polycrystalline, powder, and ultrafine powder
NINAZ samples 112
6.12 x(T) for doped NINAZ samples 114
6.13 M(H < 5T, 2K) for doped NINAZ samples 116
6.14 Typical ESR lines for polycrystalline, powder and ultrafine powder
samples of NINAZ 119
6.15 ESR line for a polycrystalline sample of NINAZ 120
6.16 ESR spectra for a polycrystalline NINAZ sample as a function of
angle 122
6.17 ESR peak g values as function of angle in a polycrystalline sample 123
6.18 Hg doped NINAZ ESR spectrum 125
6.19 Hg doped NINAZ ESR peak g values as function of angle 126
6.20 T dependence of ESR lines for HA. to the chains of a NINAZ
polycrystalline sample from Batch 1 129
6.21 T dependence of ESR lines for H || to the chains of a NINAZ
polycrystalline sample from Batch 1 130
6.22 T dependent ESR spectra for H || to the chains of a NINAZ poly¬
crystalline sample 131
6.23 T dependence of ESR lines for HA. to the chains of a NINAZ
polycrystalline sample from Batch 2 132
6.24 ESR I(T) for a NINAZ polycrystalline sample from Batch 1 . . . 133
6.25 ESR I(T) for a NINAZ polycrystalline sample from Batch 2 . . . 134
6.26 T dependence of the ESR line for a powder sample of NINAZ . . 136
6.27 T dependence of the ESR line for an ultrafine powder sample of
NINAZ 137
6.28 I(T) for powder and ultrafine powder NINAZ samples 138
viii

6.29 Energy level diagram for powder NINAZ 140
6.30 ESR FWHM vs. T for powder and ultrafine powder NINAZ . . . 142
6.31 l/x vs. T for powder and ultrafine powder NINAZ 144
6.32 The ratio of the area under ultrafine powder data to the area under
the powder data for NINAZ 145
6.33 x{T) • T for the powder and ultrafine powder samples of NINAZ
from Batch 1 147
6.34 NINAZ ESR spectrum at 93.934 GHz, increasing H 148
6.35 NINAZ ESR spectrum at 93.934 GHz, decreasing H 149
6.36 NINAZ ESR spectrum at 189.866 GHz, increasing H 150
6.37 NINAZ ESR spectrum at 189.866 GHz, decreasing H 151
6.38 g vs. v for ESR in a NINAZ powder sample 152
7.1 Approximate crystal structure of MnCl3(phen) 155
7.2 M(T) for MnCl3(phen) 156
7.3 Crystal structure of Mn(acac)2N3 158
7.4 M(T) of Mn(acac)2N3 159
7.5 Crystal structure of Mn(salpn)OAc 160
7.6 x(T) of Mn(salpn)OAc 162
7.7 Crystal structure of MnCl3(bipy) 163
7.8 x(T) of MnCl3(bipy) 166
7.9 9 GHz ESR signal intensity vs. T for MnCl3(bipy) 168
7.10 SQUID M vs. H measurement of MnCl3(bipy) 170
7.11 Raw M vs. H data for three crystals of MnCl3(bipy) 171
7.12 Raw M vs. H data for one crystal of MnCl3(bipy) 172
7.13 M vs. H data for two oriented crystals of MnCl3(bipy) 174
7.14 Powder diffraction pattern, Rietveld refinement, and difference
plot for MnCl3(d-bipy) 177
7.15 Crystal structure near a Mn atom in MnCl3(d-bipy) 178
7.16 a vs. T for MnCl3(d-bipy) 181
7.17 b vs. T for MnCl3(d-bipy) 182
7.18 c vs. T for MnCl3(d-bipy) 183
7.19 7 vs. T for MnCl3(d-bipy) 184
7.20 V vs. T for MnCl3(d-bipy) 185
7.21 Estimated 5 = 2 dispersion for MnCl3(bipy) and theoretical TOF
spectrometer behavior 186
7.22 Incoherent scattering from a vanadium standard 188
7.23 Incoherent scattering from the MnCl3(d-bipy) sample 189
7.24 Inelastic neutron scattering intensity at T = 0.4 K and Q « n . . 191
IX

7.25 Inelastic neutron scattering intensity at T = 0.4 K and Q 7.26 Inelastic neutron scattering intensity near Q = it at T = 10 K . . 194
x

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
EXPERIMENTAL STUDIES OF INTEGER SPIN ANTIFERROMAGNETIC
CHAINS
By
Garrett E. Granroth
May 1998
Chairman: Mark W. Meisel
Major Department: Physics
Experimental studies of several 5=1 and 5 = 2 antiferromagnetic chain ma¬
terials have uncovered novel phenomena. Firstly, measurements on
MnCl3(CioH8N2) provided the first evidence of an 5 = 2 Haldane gap. Secondly,
studies of the 5=1 material Ni(C3Hi0N2)2N3(ClO4), NINAZ, demonstrated the
presence of 5 = 1/2 end-chain spins and provided evidence of their interactions
with the magnetic excitations on the chain. Thirdly, investigations of the 5=1
material (CH3)4N[Ni(N02)3], TMNIN, revealed a small single-ion anisotropy, D.
Finally, the magnetic properties of MnCl3(Ci2H8N2), Mn(C5H702)2N3,
Mn(Ci7Hi503N2)CH3C00, and Ni(C4Hi2N2)2(/i—N3)]n(C104)n were studied in
a search for additional systems with properties similar to the aforementioned
materials.
xi

Magnetic field, H, dependent magnetization, M, measurements at T = 30 mK
for H < 16 T have demonstrated MnCl3(CioH8N2) to be the first known 5 = 2
material to posses a Haldane gap. Specifically, M « 0 until critical fields
Hc_l = 1.8 ±0.2 T and Hcy = 1.2 ± 0.2 T are reached for H oriented perpen¬
dicular and parallel to the chains respectively. These critical fields and the intra¬
chain exchange J = 34.8 ± 1.6 K reveal a Haldane gap of A = 2.3 ± 0.8 K and
D/ J — 0.010 ± 0.003, which are consistent with theoretical predictions for 5 = 2
antiferromagnetic chains. Furthermore, preliminary inelastic neutron scattering
data possess a possible magnetic scattering peak at an energy consistent with the
Haldane gap.
Polycrystalline, two powders with different particle sizes, and doped samples
of NINAZ were used to study end-chain spin effects. Fits to M(H < 5 T, T = 2 K),
for all the samples, confirm that the end-chain spins are 5 = 1/2. The electron
spin resonance, ESR, intensity as a function of temperature, provided a micro¬
scopic comparison of the chain lengths to the macroscopic particle size distribu¬
tions. Finally, the ESR line widths of the powder samples are consistent with
a model describing interactions between the magnetic excitations on the chains
and the end-chain spins.
Measurements of M(H < 8T,T = 25 mK) performed on both an oriented
packet of single crystals and individual single crystals of TMNIN have revealed
Hc_L = 2.60 ± 0.15 T and Hc\\ — 2.40 ± 0.15 T. These critical fields, along with
J = 10 K, determine A = 3.5 ± 2 K and D/ J = 0.06 ± 0.03.
Xll

CHAPTER 1
INTRODUCTION
Ever since the pioneering theoretical work by Bethe [1], studies of antifer¬
romagnetic chains have illuminated new phenomena related to quantum mag¬
netism. A particularly intriguing theoretical prediction, made by Haldane [2,3],
is that Heisenberg antiferromagnetic chains have a gap between a singlet ground
state and a triplet excited state for integer spin, in contrast to the half-integer
spin case that has no gap. This prediction sparked a flurry of theoretical and
experimental work that is reviewed in Chapters 2 and 3, respectively. The ma¬
jority of this dissertation presents experimental studies on three materials that
provide insight into several different phenomena related to the Haldane state.
Chapter 4 explains the experimental techniques used to study these samples.
Chapter 5 discusses magnetization measurements identifying the Haldane gap
and a small single-ion anisotropy in the 5 = 1 material (CH3)4N[Ni(N02)3],
TMNIN. Chapter 6 details our extensive studies of 5 = 1/2 end-chain spins in the
5 = 1 system Ni(C3HioN2)2N3(C104), NINAZ, using magnetization, magnetic
susceptibility, and electron spin resonance. In addition to the intrinsic crystal
shattering processes, chain breaks were generated in NINAZ primarily by pulver¬
ization techniques, and in a few cases by doping. Chapter 7 covers the search and
discovery of the 5 = 2 Haldane gap. The discussion is focused on MnC^bipy),
where bipy = CioH8N2, for which our magnetic field dependent magnetization
1

2
measurements at T = 30 mK provide the first evidence of the 5 = 2 Haldane
gap. In addition, preliminary neutron scattering results, which microscopically
test the 5 = 2 Haldane gap, are presented. Finally, Chapter 8 will review the
conclusions of the previous chapters and provide suggestions for further study.

CHAPTER 2
REVIEW OF THEORETICAL STUDIES
Since the nascence of quantum mechanics, quantum antiferromagnetism has
provided puzzling theoretical problems. The Heisenberg Hamiltonian
H = J'£Si-Si+1 (2-1)
i
describes the system where J, the exchange, characterizes the strength of the
interaction between nearest neighbor spins and S¿ is the spin at the zth site.
The ground state of this system is inherently difficult to find, so several approx¬
imations have been used. A common approximation is the classical limit, where
S —> oo. In this limit, the three-dimensional ground state is long-range ordered
with no net magnetization because each spin is oriented opposite its nearest neigh¬
bors. This state can be represented by | However, when the spins
are quantized, it is easily shown that a state exists below the classical ground
state [4]. Nevertheless, the true ground state remains unsolved. Consequently,
several other simplifying assumptions have been made, and the one pertinent to
this thesis is to reduce the system to one dimension. Even the one-dimensional,
1-D, case is difficult to handle theoretically. Attempts at using the classical ap¬
proximation reveal a negatively divergent correlation length, known as the long
wavelength divergence. Most of the theoretical progress on 1-D antiferromagnets
has involved creative methods of avoiding this divergence. Bethe in 1931 [1,5]
3

4
successfully circumvented this divergence for the 5=1/2 Heisenberg model by
a technique know as the Bethe ansatz. Unfortunately, the resultant ground state
is hard to manipulate, so further predictions about the dispersion were not made
until 1962 [6]. The difficulty of the theory discouraged theoretical work for higher
spins values. It was not until Haldane [2,3] used clever limits on an approximate
model that evidence of a gap in the integer spin case was observed. Since then,
several approximate models have been used to describe the properties of integer
spin antiferromagnetic chains. Therefore, this chapter begins with an overview
of the two primary theoretical models used: the non-linear sigma model (NLaM)
and the valence bond solid (VBS) model. Since the details of these models may
obfuscate the physical processes behind the predicted phenomena, a physically
intuitive argument for gapped integer and gapless half integer spin chains will
be given next. Then, a discussion of realistic Hamiltonians and the numerical
studies that illuminated their properties will be provided to connect the theory
to the experiment. The chapter will finish with several theoretical details on the
experimental techniques used.
2.1 Theoretical Models
2.1.1 The Non-Linear Sigma Model
Haldane proposed, in 1983, that integer and half integer Heisenberg spin
chains would have a fundamentally different excitation spectrum [2,3]. By mak¬
ing an approximation, to be described below, he proposed that the integer spin
case has a singlet ground state and a triplet excited state separated by an energy

5
gap, A, and the half integer spin case is gapless. The half integer spin case is
completely consistent with previous exact work, but the integer case was an un¬
expected prediction that started a flurry of theoretical and experimental research.
In his approach, he assumed that order existed for some distance larger than the
lattice spacing but smaller than the long-range ordered domains. This assump¬
tion of short-range order circumvents long wave length divergences found in the
classical spin wave approach [5]. Next, the system was mapped from a statistical
mechanical model to a field theory in (1+1) dimensions, namely a NLerM. Once
the mapping is complete, the -S' —>• oo limit [7-9] reveals a gap in the excitation
spectrum for the integer spin chain and no gap for the half integer spin chain.
More specifically, the Haldane gap, A, is proportional to e~nS, and the correlation
length between spins, £, is proportional to enS [7]. There is only one problem
with this approach. Spins of value one and one half can hardly be called large,
and in fact, this technique [8,10] provides inconclusive results for S < 1. Studies
of the NLerM are well documented in the literature [7-9,11,12], so further details
are beyond the scope of this thesis.
Non-linear sigma models have been extended beyond arguing the existence of
the Haldane gap. Jolicoeur and Golinelli used the N —>■ oo limit to study A(T)
for T < A. The result is an “anti-BCS” gap of the form [10]
A(T) « A(0) + y/2nA(0)Te~AW/T. (2.2)

6
Furthermore, they calculated the free energy and used it to derive expressions for
the low temperature specific heat
Cv(T)
Nm (mV'2,-amt
s/2i { T )
(2.3)
and magnetic susceptibility
xfT)w
(2.4)
2.1.2 Valence Bond Solids
Another approach that has been both theoretically and experimentally illu¬
minating is the valence bond solid (VBS) model, also called the AKLT model,
introduced by Affleck, Kennedy, Lieb, and Tasaki [13,14]. Anderson [15,16] ini¬
tially introduced valence bonds in the resonating valence bond (RVB) model to
suggest the existence of a quantum disordered ground state in an 5 = 1/2 two-
dimensional antiferromagnet. This model has been used to explain some features
of high-Tc superconductivity [7,16] and may accurately describe spin ladders [17].
A valence bond is a bond between two 5=1/2 spins at two sites [7]. The bond
is commonly represented [7,14] as a solid line between the 5 = 1/2 sites given
as points. Figure 2.1 shows a valence bond for two 5 = 1/2 entities. It is easily
determined that for 5 = 1/2 spins on any lattice, there are multiply degenerate
Figure 2.1: A single valence bond.

7
configurations of valence bonds, hence the name resonating valence bond [7,15].
For example, Figure 2.2 shows an RVB configuration for an 5 = 1/2 chain where
the dashed lines are the unoccupied bonds, but an equally probable state is one
where the dashed lines are the valence bonds and the solid lines are not; there¬
fore, these two states are in resonance. Affleck, Kennedy, Lieb and Tasaki [13,14]
• • • • • • • 1 •
Figure 2.2: An 5 = 1/2 chain in the RVB picture.
extended the theory by examining valence bonds on lattices with higher spin.
They consider an 5 = 1 site as two 5 = 1/2 entities, represented by two points
enclosed by a circle. Therefore, Figure 2.3 shows an 5 = 1 chain coupled by va¬
lence bonds. Clearly, there is no degeneracy, so the singlet ground state is “solid”
Figure 2.3: An 5 = 1 chain according to a VBS model.
when compared to the resonating bonds of the 5 = 1/2 chain. A continuation of
the theory in Figure 2.4 shows an 5 = 2 chain can also be modeled as a valence
bond solid. Another look at Figures 2.3 and 2.4 shows an interesting property of
VBS models, namely that finite chains terminate with spins of value 5/2. This
feature will be discussed more below.
Up to this point in the discussion, nothing has been said about the system
for which the VBS state is the ground state nor how closely related the system is

8
Figure 2.4: The VBS state for an 5 = 2 chain.
to a Heisenberg model. By assuming the VBS state is the ground state of some
Hamiltonian, and working backward to find it, the 5=1 Hamiltonian
'H = J'£Si-Si+1-(3(Si-Si+l)2 (2.5)
i
is obtained where, for now, f3 = —1/3, but it will be varied later in the discus¬
sion [13,14]. This system exhibits a gap [7,14], albeit with an energy, ~ 0.350J,
significantly smaller than the Haldane gap. To compare the properties between
this system and the Heisenberg system, Kennedy [18] performed exact diagonal-
ization studies on Equation 2.5 for a range of (3 values. He confirmed a continuous
region of phase space in (3 from —1/3 through 0, so the gap and the end-chain
spins should be realized for the Heisenberg model as well. Equation 2.5 is for
5 = 1 chains. A different Hamiltonian is constructed when a similar procedure
is carried out for an 5 = 2 chain, and the result is not as closely related to the
Heisenberg Hamiltonian [7,14] as Equation 2.5. Nevertheless, the VBS model
provides some physical intuition to help understand the experimental results in
an 5 = 2 system.

9
2.2 An Intuitive Picture of the Haldane Gap
Though the VBS model produces a physically intuitive explanation for the
presence of 5 = 1/2 spins at the ends of a finite S = 1 spin chain, it does not
provide similar intuition for the presence of a gap for the integer spin case and
the absence of it for the 1/2 integer spin case. The NLctM is even less intuitive.
Nevertheless, the following argument, derived from the large body of theoretical
and numerical work, lucidates the physical origin of the Haldane gap [19]. To
start from a long-range ordered ground state
l (2.6)
assume an Ising system
H = JY.S;S'+l- (2.7)
Next, make the spins more isotropic, thereby approaching the Heisenberg model
H = j'£s?s-+1+s&1sr+s;sul, (2.8)
i
where S+ = Sx + iSy and S~ = Sx — iSy. Clearly, the added interactions destabi¬
lize the long-range ordered state, so there is a lower ground state. Nevertheless,
for the ground state to remain antiferromagnetic, < M > = 0. Therefore, the
ground state must at least be the superposition of short-range ordered antiferro¬
magnetic states. If a source of energy (i.e. T, H) is added to the system so as to
cause one spin to locally increase above the ground state, the new state for the
5 = 1/2 chain is

10
i imun> (2.9)
that has < M > / 0 which is indicative of an excited state. In sharp contrast, a
spin on the 5=1 chain has the Sz = 0 state as its next excited state. Therefore,
the many-body state of the 5 = 1 chain with an excitation on it would look like
|0 |Ut 0 |t> • (2-10)
Clearly, < M > = 0, so this state is degenerate with the ground state. If more
energy is added to the system, only when the A5Z = 1 transition becomes the
most probable for each site does < M > ± 0, and the ground state is destroyed.
Since a finite quantum of energy must be added to the system for this transition
to occur, an energy gap in the excitation spectrum is observed.
A further interesting theoretical observation can be made from the above
discussion. First, notice that as long as two of the spins are arranged in the zero
state, they can be anywhere along the chain. Second, notice that if the zeros
were simply removed from the chain, the spins across the empty space would be
antiferromagnetically arranged. This result is know as hidden (or string) order
and was first identified by Tasaki [20].
2.3 Numerical Studies of Quantum Spin Chains
Other tools useful for characterizing spin chain behavior and for connecting
the theory to experimentally studied systems are numerical techniques. Four
techniques have been used, and they are high temperature series expansions,

11
exact diagonalization, quantum Monte Carlo (QMC), and density matrix renor¬
malization group (DMRG). High temperature series expansions have produced
expressions for the directly measurable quantities Cy(T), the specific heat, and
x(T), the magnetic susceptibility. Weng [21] performed expansions of x{T) for
5 = 1/2, 1, and oo. Furthermore, he extrapolated between the 5 = 1 and 5 = oo
results to determine x{T) for 5 = 2 and 5 = 3/2. Meyer et al. [22] have generated
a rational function representation of Weng’s 5 = 1 curve, namely
X(T) =
NnW \ 2 + 0.0194X + .777X2
kBT [3 + 4.346X + 3.232X2 + 5.834AT3
(2.11)
where X = J/kBT. In a similar manner, Hiller et al. [23] arrived at a more
versatile group of rational functions which can be used for many different values
of 5 as given by
NnW
kBT
A + Bx2
1 + Cx + Dx3
(2.12)
where x — J/(2kBT) and the coefficients are given in Table 2.1. All of these
5
A
B
C
D
1/2
0.2500
.18297
1.5467
3.4443
1
.6667
2.5823
3.605
39.558
3/2
1.2500
17.041
6.7360
238.47
2
2.0000
71.938
10.482
9555.56
Table 2.1: Coefficients as a function of 5 for Equation 2.12 from Reference [23].
expressions are only for the Heisenberg model. Recent extensions to Hamiltonians
including single-ion anisotropy will be discussed below. In recent QMC studies,

12
Yamamoto and Miyashita [24] calculated the temperature dependent magnetic
susceptibility and specific heat of open and closed chains for 0.1 J < T < 5 J.
Their work agrees well with the high temperature expansions above J and with
the exponential excitation of a gap below J, until end-chain spin effects become
important. Further discussion of the end-chain spin effects will be given below.
Furthermore, QMC has been used to determine the dispersion curves of linear
chain systems. Figure 2.5 shows the 5 = 1 dispersion as Q is varied from 0 to
7T [25,26]. One interesting feature is that the points suggest the gap at Q = 0
is larger than the one at Q = 7r. This feature has been interpreted as evidence
of a two particle continuum of excited states near Q = 0, instead of the triplet
excitations at Q = tt [25-29]. The solid and dashed lines in Figure 2.5 show
the agreement between the theoretically predicted curves and the QMC data
for the triplet excitations at Q — 0 and the two particle excitations at Q = tt,
respectively. The net result is a gap of 2A at Q = 0 and A at Q = n. Meshkov [29]
performed QMC work on an 5 = 2 chain to determine the dispersion. Similar
to the 5 = 1 case, the Q = 0 gap is twice the Q = tt gap. In addition for both
the 5 = 1 and 5 = 2 cases, S(Q,u>) looses intensity and broadens as Q —» 0 [29].
This loss of intensity explains why the 2A gap has not been observed. Detailed
discussion of the shape of S(Q,u) is not required for this thesis and is discussed
at length in the literature, so no further comments will be made here. The
DMRG technique has been shown to determine very precise results for relatively
long chains (~ 60 sites) [30-33], but unfortunately it is limited to T = 0. A
combination of QMC, exact diagonalization, and DMRG studies produced the

13
Q/n
Figure 2.5: 5 = 1 chain dispersion curve. The squares are the dispersion for a
1-D 5 = 1 Heisenberg antiferromagnet calculated by QMC [25,26]. The solid
line is the expected dispersion for single particle excitations and the dotted line
is the expected dispersion for the two particle continuum [28].

14
A/J
Reference
0.08
Deisz et al. [34]
0.055 ±0.015
Nihiyama et al. [35]
0.05
Sun [36]
0.085 ± 0.005
Schollwóck et al. [37,38]
0.049 ±0.018
Yamamoto et al. [39]
Table 2.2: Values for the S' = 2 Haldane gap calculated by different groups.
result of A = 0.41 J for 5 = 1, and work is converging for the 5 = 2 chains where
the present published values are given in Table 2.2.
Numerical studies have also provided information on how the Haldane gap
behaves as a function of magnetic field, H. The triplet excited state is split, and
at a certain field, Hc — A, the non-magnetic singlet is replaced by one of the
magnetic states as the ground state. A detailed description of the splitting will
be provided in relationship to more realistic Hamiltonians.
The VBS model prediction of 5 = 1/2 end-chain spins is really quite in¬
triguing, and an extensive body of numerical work [18,24,30-33,40-43] has been
performed to compare this model to the exact Heisenberg case. Initially, exact
diagonalization [18] results demonstrated the presence of 5 = 1/2 degrees of free¬
dom for models where /3 in Equation 2.5 is varied through zero. Furthermore,
Kennedy [18] applied the general result of Lieb and Mattis [44], that implies a
difference between even and odd length chains, to the integer spin Heisenberg
antiferromagnetic chain. More specifically, he found that the ground state of a
finite chain is a four-fold degenerate ground state, made of two levels from each
5 = 1/2 end-chain spin. If an interaction along the chain is permitted between

15
the end-chain spins, the ground state is divided into a singlet and a triplet.
In other words, for chains with even and odd numbers of spins, the singlet and
the triplet are the ground states, respectively. Therefore, the chain with an odd
number of spins has a magnetic ground state while the even case does not. Both
exact diagonalization [18] and QMC [41] on short chains support this difference,
but similar techniques on longer chains do not. Figure 2.6 shows < Sz > for
chains of 60, 65, and 97 units. The shortest chain was calculated by DMRG [31]
and the longer chains were calculated by QMC [41]. Clearly within the size of the
data points, whose size is an estimate of the uncertainty, the staggered magne¬
tization is independent of the number of spin sites. This result is not surprising
since the correlation length of 6 units is so short that any parity effects are aver¬
aged out over several correlation lengths. More directly related to experimental
measurements, Yamamoto and Miyashita [24] derived information on the temper¬
ature dependent properties of end-chain spins from QMC results. By subtracting
the magnetic susceptibility for chains with closed ends from open chains with
odd numbers of sites, they can determine the end-chain spin contribution to the
magnetic susceptibility alone. At first glance, their results are well fit by a Curie
law, but a closer examination, performed by plotting 1/x vs. T (Figure 2.7), sug¬
gests otherwise. Discontinuous features near T — J suggest that the subtraction
did not completely remove the contribution from the chain. One could argue
that below T « J the data obeys a Curie law, but this range is smaller than
one decade in temperature. Therefore, a fit can not be performed to determine
a definitive functional form. Nevertheless, these results are directly compared to
experimental results in Chapter 6.

16
A
N
C/)
V
0.6 -
0.4 -a
0.2 -
0.0 -
-0.2 -
-0.4
0 10
Figure 2.6: Staggered magnetization for S = 1 chains of 60, 65, and 97 spins.
The 60 spin chain was calculated by DMRG [31], and the two longer chains were
calculated by QMC [41].

17
0 1 2 3 4 5
T/J
Figure 2.7: 1/x vs. T for end-chain spins from QMC work of Yamamoto and
Miyashita [24]. Several features suggesting non-Curie like behavior are mentioned
in the text. Although, at the lowest temperatures, the inset shows reasonable
agreement with a Curie-law.

18
2.3.1 Realistic Hamiltonians
The above discussion has been for systems described by Equation 2.1, but
in a real system, next order terms relating to the symmetry of the environment
around the magnetic ion must be considered. More specifically, the Hamiltonian
should include single-ion anisotropy D, in some cases orthorhombic anisotropy
E, and the degree of symmetry in the exchange ry. As these terms break some of
the symmetries of the system, they become very difficult to handle analytically.
Nevertheless, numerical studies have revealed many properties of these systems.
The Hamiltonian which takes all these effects into account is
« = ./£sfSf+1 + S*SI, + vs;s-+i + D(s‘f + e[(s*)2 - (s?)2]. (2.13)
i
Since E is small, it is usually negligible. However, splitting in inelastic neu¬
tron scattering data [45] demonstrates that this term is measurable for NENP.
The remainder of this discussion will consider E = 0. The best way to describe
the overall behavior of Equation 2.13 is with the semi-quantitative phase dia¬
grams [20,37,46] shown in Figures 2.8 and 2.9. First, examine the 5 = 1 case in
Figure 2.8. Note that a rather large region of phase space is the Haldane phase,
and the XY phase does not take over until 77 < 0. Two of the extreme bound¬
aries of the phase diagram are long-range antiferromagnetic order for 77 0 and
ferromagnetic order for r¡ singlet phase, which is gaped but is not the Haldane gap, develops. Finally, for
the 5 = 2 case, Figure 2.9 shows a greatly reduced Haldane phase region. This
result is reasonable because higher spin means more possible S components in

19
Figure 2.8: 5 = 1 phase diagram. The single-ion anisotropy vs. exchange
anisotropy phase diagram for an 5 = 1 linear chain antiferromagnet described by
the Hamiltonian 2.13 [20,37,46].

20
the x and y direction to amplify the anisotropies. Therefore, it can be said that
the Haldane phase in the S — 1 system is lost to the XY phase as the spin is
increased.
Finite values of D affect antiferromagnetic chains in the Haldane phase region
as well. As was stated earlier, A is a gap between the singlet ground state and a
triplet excited state (see Figure 2.10a). A finite D term splits the triplet to form
two gaps (see Figure 2.10b), Aj_ for directions perpendicular to the chains and
Ay for directions parallel to the chains, so at Q = 7r, Ax < Ay [28,47-50]. As
Q is reduced to zero, the numerical work of Golinelli et al. [28] shows that the
dispersion curves ± and || to the chains cross so at Q = 0, Ax > An. To directly
compare results with several experiments (see Chapters 3, 5, and 7), extensive
work [47-50] has described the magnetic field dependence of the excited states
split by D at Q = 7r. Figure 2.10c shows that the doublet splits, with the lower
state moving towards, and finally crossing, the ground state at critical fields Hc\\
and i7cx, depending on the orientation of the field. The energy gaps are related
to the critical fields by
a\\^BHc\\ = Ax
QlVbHc±_ = ^/A||Ax
(2.14)
(2.15)

21
Figure 2.9: 5 = 2 phase diagram. The single-ion anisotropy vs. exchange
anisotropy phase diagram for an 5 = 2 linear chain antiferromagnet for the
Hamiltonian 2.13 [37].

22
where g\\ and g± are the Landé g factors for each orientation [48,50]. Then, to
correlate Aj_ and A|| with A and D, the following expressions [50,51] are used
A,, = A + 2kD (2.16)
Ax = A -kD (2.17)
where n = 1/3 for S — 1 and /c = 2 for S = 2.
Yamamoto and Miyashita extended their QMC studies, mentioned earlier, to
include finite anisotropy. At T = 0, their staggered magnetization results show
an increased or decreased correlation length depending on if the magnetic field
orientation is parallel or antiparallel to the easy axis, respectively. The easy axis
is defined by the sign of D. Furthermore, Cy(T) and x(T) results show effects of
finite D, but they did not fit a function to their results, making comparisons with
experimental data difficult. Nevertheless, Jolicoeur [51] performed high temper¬
ature series expansions to arrive at the expressions given in Appendix A. Since
the expressions are expansions near a phase transition, minute errors in the fit
parameters cause huge discrepancies, so the method of Padé approximants must
be used for each value of D.
2.4 Measurement Theory
One challenging part of any research program is connecting the theory to
experimentally measured quantities. Some less than obvious theoretical work
has been used to connect the experimental technique of ESR and the above
theory. This section overviews the essential theory needed to understand the
measurements.

23
ía) (b) (c)
Figure 2.10: Energy level diagram showing the Haldane gap for (a) H = 0, D — 0
(b) H = 0, D > 0 (c) H ± 0, D > 0. The values of the splitting are explained in
the text. Please note that this is a generalized and corrected version of Figure 4
appearing in Reference [97].

24
2.4.1 Electron Spin Resonance
Part of this thesis reports on measurements involving end-chain spins in
NINAZ. One way to detect free spins is through electron spin resonance (ESR).
If a paramagnetic spin is placed in a DC magnetic field, H, it will align itself
such that the magnetic moment of the ion precesses about H in a manner similar
to a toy top in the gravitational field. Since the system is a damped harmonic
oscillator, the system will absorb a large amount of energy when driven at its
resonance frequency, u;0. This resonant frequency is dependent on the size of the
applied magnetic field as
hu o = guBH (2.18)
where g, h and hb follow their standard definitions. A convenient driving field for
an electronic spin is the magnetic field component of electromagnetic radiation in
the microwave (X-band) frequency regime. Several microscopic phenomena can
be interpreted from the characteristics of the observed absorption peak. The area
under the absorption peak is one property of interest for this thesis. Any spin
resonance technique measures energy absorbed by the change in magnetization,
AM, of the sample for the oscillating field, AH, as a function of H. Since \ is
defined by AM. jn the Hmit where AH component x" of the complex magnetic susceptibility, xac — x' + Since
x" is sharply peaked near the resonance frequency, x! can be determined at any
frequency through the Kramers-Kronig relations [52-54]. Specifically, the DC
magnetic susceptibility is [53]

25
roo
Xdc = xV = 0) a / x"(a/)du/.
Jo
(2.19)
Experimentally, ESR is performed by holding lo constant and varying H\ there¬
fore, by Equation 2.18, Xdc becomes [54]
roo
XDC = = 0) a / x!’(H')dH'.
Jo
(2.20)
This result means that the area under the resonance line is simply proportional
to xdc [53,54]. This completes a short introduction to the ESR technique. There
are many excellent texts on ESR, if more detailed theory is required [52,53,55,56].
The above discussion has been a general overview of ESR for standard para¬
magnetic spins. There are several features unique to end-chain spins that are
observed with ESR. Mitra, Halperin, and Affleck propose a simple model to ex¬
plain the ESR response of the end-chain spins in NENP [57]. They begin by
assuming chains with fixed ends which reflect a magnetic excitation back to the
other end of the chain. These excitations are described by bosons in agreement
with NL chain spin, either the excitation can change energy levels or there will be a slight
phase shift that would cause a small change in energy. Generally, their expression
for the intensity of the ESR spectra is

26
I(uj) = \tanh e Zl2iró (w - u0)
fiu»n
4
+
2kBTJ
hui \ e2kBT
(l-e
o-z 1
xE
n,m L
^27r5
e
Wo +
,n
e+ e-
h
8 cosh (^¿r)
) I < n,+\m,~ > |
+[+ *+ -]]
(2.21)
where the first term is the ESR line for the end-chain spin alone, and the second
term includes effects resulting from interactions with the bosons. In the second
term, the + indicates that the boson spin and the end-chain spin are parallel
and the — indicates that they are antiparallel. Figure 2.11 shows the energy
level diagram for three boson levels. The heavy solid line shows the transition
corresponding to the ESR line when the bosons do not change energy levels.
The thin lines show typical transitions between energy levels where the open
arrows indicate transitions to higher energy levels and the filled arrows represent
transitions with lower energy levels. Notice that the transitions would produce
symmetric lines around the peak if the frequency of the microwaves is large enough
to see the transition at lower fields. Nevertheless, the higher field transition should
always be observable if a sufficient magnetic field is applied to the sample. The
processes where the end-chain spin does not flip are forbidden transitions. If the
boson changes energy levels, then the change in energy for a chain of length, L,
is
fn _
(hnc)2
2AL2
(2.22)

27
H
Figure 2.11: Allowed transitions between end-chain spin states. Energy level
diagram showing the allowed transitions that should be observed in ESR. Each
transition is signified as a line with an arrow on each end. The heavy line signifies
a resonance signal where a boson does not change energy levels. The open ended
arrows indicate transitions with a higher boson energy level and the closed arrows
signify transitions with a lower boson energy level. All arrows are of a length
E = hu> o = constant.

28
where c is the spin wave velocity. If L is too short (e.g. ~ 100 spins for
NENP), the side peaks are not within the observation limits of the typical X-
band (is = 9 GHz) ESR experiment. Therefore, to calculate the intensity of the
ESR line, the second term in Equation 2.21 can be neglected. Therefore, the
chains that significantly contribute to the central intensity are those without
bosons on them. Since the probability of a magnetic excitation existing on a
chain at any one time is an exponential function of temperature, the number or
spins participating in the ESR line is temperature dependent. As a result, the
ESR line intensity increases faster than a Curie law with decreasing temperature.
The temperature dependence of these intermediate length chains is given by
(2.23)
where < Z\ > is the partition function averaged over the distribution of chain
lengths as given by
The de Broglie thermal wavelength of a boson is
(2.25)
Mitra, Halperin and Affleck [57] finish their calculation by assuming a distribution
of P(L) oc exp(—L/L0) with a cutoff of Lmin. This distribution provides the
analytical solution

29
I(T) = Iq tank
/ huj0 \
exp
[-
7T l/2Lmin\T1 — 1
e &
\2kBT)
1 + 7T~ll2Lo\Tle kB
T
(2.26)
Calculating other quantities for the ESR end-chain spin is less trivial because
the second term in Equation 2.21 contributes to the line width of the central
peak even when the boson does not change energy levels. To approximate the
line width, consider the potential that a boson sees as it traverses the chain.
At distances far away from an end-chain spin, the potential is essentially flat,
but as the boson approaches an end-chain spin, it sees a finite potential, and
therefore, its wave vector shifts slightly. This shift of the boson’s wave vector can
be characterized by a phase shift S(k) defined by
nir 5(k)
k=T~ —
(2.27)
To parameterize this effect, Mitra, Halperin, and Affleck used the expression
6{k) « Vk£,
(2.28)
where V is a constant of order 1. In their paper, they conclude that this effect
is not observable because the line width would be controlled by the very short
chains in the sample. It is important to recall that their discussion focused only
on NENP where several phenomena (i.e. staggered magnetization and thermally
excited bosons) could effect the linewidth. For this thesis, we will apply this
analysis to NINAZ where there is no staggered magnetization and J is large
enough that the quantum limit is easily accessible. The line widths in NINAZ
are consistent with this phase shift, see Chapter 6.

CHAPTER 3
REVIEW OF EXPERIMENTAL STUDIES
This chapter reviews the status of experimental studies of phenomena related
to the Haldane state. The main emphasis is placed on materials that are not part
of this thesis since these systems will be discussed in their own chapters. This
chapter begins with a discussion of the material properties of Haldane gap sys¬
tems. Next, a discussion of the different techniques used to measure the Haldane
gap, and other information gained by these methods, will be given. Third, exper¬
imental results from the study of end-chain spins will be provided. To complete
the review, a few relevant subtopics will be discussed.
3.1 Haldane Gap Materials
In order to study the phenomena related to the Haldane state, materials
are needed which are closely approximated by the theoretical model. There are
two main properties that must be met for a material to exhibit a Haldane gap.
Firstly, the system must have a coupling along the chains much stronger than
a coupling between the chains so the material exhibits 1-D behavior. Secondly,
any anisotropies, that give the spins a preferential orientation (i.e. single-ion
anisotropy or exchange anisotropy), must be small enough so the material re¬
mains in the Haldane phase, see Figures 2.8 and 2.9. The first part of this section
30

31
discusses issues related to fabricating antiferromagnetic chain materials and the
second part deals with the anisotropy.
Materials that provide a realization of integer spin antiferromagnetic chains
have been available for several decades [58]. Chemically these systems are charac¬
terized by their composition as being either metal-organic compounds or purely
inorganic compounds. For both groups of materials, each metal ion has strong
~ 180° bond overlap with the bridging ligand that mediates the superexchange
along the chain. Bond angles near 180° are characteristic of antiferromagnetic
compounds since the strength of the superexchange is controlled by the bond
angle [54,59]. There are two ways that the chains are separated from each other.
For some materials, the chains have a slight net positive charge which is balanced
by a counter ion between the chains. Therefore, no bond overlap exists to pro¬
vide a superexchange pathway between chains. Thus, only dipolar interactions
magnetically couple the chains. The other class of chain materials relies on the
contrast between strong antiferromagnetic bonds along the direction of the chain
and very weak 90° bonds, ferromagnetic in nature, between the chains. Both
types of materials posses an interchain coupling that is four orders of magnitude
smaller than the intrachain coupling. The local environment around the mag¬
netic ion determines the anisotropy of the chain material. Generally, if chemical
bonds are uniformly distributed around the magnetic ion, the anisotropy is low.
However, completely isotropic bonding is rare because Jahn-Teller distortions fa¬
vor at least some slight anisotropy. Fortunately, there are many materials where

32
material
S
J
D/J
E/J
A
references
(K)
(K)
CsNiCl3
1
16.6
0.003
5.25
[60]
NENP
1
46 ±2
0.18 ±0.01
0.02 ±0.01
14
[22,45,61-70]
NINO
1
47
0.34
0.03
9.81
[62,65]
TMNIN
1
10
0.003
3
[71-77]
AgVP2S6
1
400
0.01
< 3 x 10~4
300
[62,78-80]
NINAZ
1
125
0.16
41.9
[73,77,81-83]
NiC204-2MIz
1
39.7
20.3
[84]
NiC204-2DMIz
1
42.9
19.0
[84]
Y2BaNi05
1
285
0.16
100
[85-94]
MnCl3(bipy)
2
35
0.010 ±0.003
2.3 ±0.8
[95-97]
Table 3.1: The Haldane gap materials and their properties. The parameters that
describe each system are the intrachain exchange, J, the single-ion anisotropy,
D, the orthorhombic anisotropy, E, and the Haldane Gap, A. The empty spaces
indicate that the parameter has not been experimentally determined. Note that
CSNÍCI3 has a transition to an antiferromagnetic long-range ordered state at
Tn = 4.6 K.
the bonds are sufficiently isotropic for the Haldane phase to be realized, as seen
by Table 3.1.
3.2 Experimental Tests of the Haldane Gap
A large portion of this thesis is concerned with measurement of the Haldane
gap, and this section overviews experimental tests of this state. The macro¬
scopic magnetic susceptibility is usually the first property tested for a new 1-D
antiferromagnetic material. The standard SQUID magnetometer has sufficient
sensitivity to test for undesired magnetic properties, i.e. long-range order or gross
impurities, even in very small samples (~ 10 mg). Such tests are necessary to de¬
termine if the system merits further exploration. All of the samples in Table 3.1

33
display, with decreasing temperature, a broad peak, demonstrating 1-D antifer¬
romagnetism, followed by a sharp decrease, consistent with an activated gap. In
addition, many materials display a low temperature paramagnetic tail consistent
with either end-chain spins or impurities in the sample. Another macroscopic
measurement, better suited to measure the size of the Haldane gap, is the mag¬
netic field, H, dependent magnetization, M, at a temperature well below the gap.
While the material is in the Haldane state, the magnetization is zero, and as H
is increased such that the lower branch of the triplet excited state crosses the
ground state (Figure 2.10), a finite magnetization develops. This technique has
been used in many of the materials [63,64,75,97]. The experiments in TMNIN
and MnCl3(bipy), as part of this thesis, will be discussed in Chapters 5 and 7,
respectively. The first M vs. H measurement to identify the Haldane gap was
performed on NENP [63,64]. As can be seen in Figure 3.1, there are critical fields
defined by the intersection of a constant, weakly magnetic state with an increas¬
ing magnetic state. These critical fields provide values of the Haldane gap, A,
and the single-ion anisotropy, D, consistent with the values given in Table 3.1.
Notice that for a magnetic field oriented parallel to the a axis (b is the chain
axis), a finite magnetization was observed below the gap. Katsumata et al. [63]
attributed this property to impurities. Subsequent NMR [98] and high field ESR
measurements [19,99,100] show that it results from non-equivalent magnetic sites
as will be discussed in Section 3.4.
Though macroscopic measurements can detect the presence of a gap and their
careful use can eliminate any possibility of other physical processes opening a gap,

M (hb/N¡
34
H (T)
Figure 3.1: M(H) for NENP. There are clearly defined critical fields for all three
orientations. The finite magnetization for H || a results from a staggered magne¬
tization of the Ni sites. The data are from Reference [63].

35
a direct measure of the gap is needed to test additional properties. Inelastic neu¬
tron scattering is the probe most often used and has had great success. The
general idea behind inelastic neutron scattering is to measure the energy and
momentum change for neutrons after they have scattered from the sample. Since
the Haldane gap is a low temperature phenomenon, any energy and momentum
changes by phonons are negligible. Therefore, all energy and momentum changes
correspond to the creation or annihilation of magnetic excitations. When neu¬
trons of the energy equivalent to the gap at a certain value of Q are sent into
the sample, the neutrons excite the system over the Haldane gap, producing res¬
onant scattering. For neutron energies below the gap, the system is not excited,
so magnetic scattering is not observed. When energies above the gap are mea¬
sured, the magnetic scattering is not observed. Furthermore, neutron scattering
is a Q dependent measurement. Therefore, by taking energy sweeps at different
values of Q, the dispersion can be measured. As an example, Figure 3.1 shows
the magnetic neutron scattering near Q = n corresponding to a Haldane gap in
Y2BaNi05 [90]. The most prominent feature, i.e. the peak around zero energy,
originates from incoherent elastic scattering, and this feature is always observed.
The first unambiguous measurement of the Haldane gap was in NENP by
Renard et al. [61] from which A and D could be determined. Later measure¬
ments [45], with better resolution, were able to determine E as well. Near Q = it,
several measurements have determined the dispersion. Two groups [45,67] mea¬
sured the dispersion for a larger range of Q, but the scattering intensity de¬
creased rapidly below Q = 0.3, so the Q = 0 gap of 2A has eluded confirmation.

36
Figure 3.2: Inelastic neutron scattering intensity for Y2BaNiOs. Constant Q « n
scan showing the peak due to incoherent elastic scattering and the peak resulting
from the Haldane gap for T > A and T < A. The data are from [90].

37
Nevertheless, this loss of scattering intensity for decreasing Q is consistent with
the model of a two particle continuum near Q = 0 [26]. In addition, these ex¬
periments do not have sufficient resolution to observe the crossing of the Ay and
A_l components of the dispersion [28]. Nevertheless, the empirically determined
dispersion agrees with the QMC predictions of Takahashi [26].
NMR is another microscopic probe useful to test for the existence of the
Haldane gap. Of course, if the gap is present, additional information can be
obtained. Since the purpose of this chapter is to demonstrate how NMR is used
for probing the Haldane gap, only spin-lattice relaxation times and the Knight
shift will be discussed. In an NMR experiment, a uniform magnetic field is applied
to a sample, resulting in the precession of the nuclear spins. When an rf pulse is
applied to the sample to flip the nuclear spins, the local magnetic environment
affects how long it takes for the spin to return to equilibrium. The contribution to
this relaxation time by the crystal lattice is known as the spin-lattice relaxation
time, T\. In magnetic systems, where the electronic moments are large and
long-ranged (compared to the nuclear moments), 7\ is the dominant relaxation
process. Clearly T\ is related to how correlated the local moments are with
each other or, in other words, are related to the spin-spin correlation function.
Since the experiment is performed in frequency space, the Fourier transform of
the correlation function, or the dynamic structure factor S(q,uj), is the quantity
measured. It has been shown [10,12,101,102] that
Jr °c S{q,u) a e~r
(3.1)

38
for T < A. Therefore, the slope of a ln(T{1) vs. T-1 plot of the data provides
a microscopic measure of A. If the sample has conduction electrons, the Knight
shift can be measured as well. This effect is a shift in frequency of the resonance
line due to the nuclei observing a background of mobile spin polarized electrons.
A detailed discussion [53] shows that the Knight shift is proportional to the local
spin susceptibility of the sample. Therefore, it is a microscopic measure of the
magnetic susceptibility of the chain.
Since there are numerous hydrogen atoms in typical organo-metallic com¬
pounds, proton NMR is often performed. Several groups [66,69,103,104] have
measured the proton spin-lattice relaxation time in NENP, and all of their results
are consistent with Equation 3.1. Nevertheless, these proton sites are somewhat
removed from the magnetic nickel site. To use a probe closer to this site, Reyes
et al. [102] performed NMR using the naturally abundant 13C of NENP. Besides
measuring the gap, they also observed features in Tf 1(T) that are absent in the
proton NMR data [66,69,103,104]. These observed features may be interpreted by
adding a temperature dependent gap to the standard analysis. In AgVP2S6, 51V
and 31P were used as probing nuclei, whereas 89Y was used in Y2BaNi05 [105]. In
addition, both of these samples possess conduction electrons, so the Knight shift
was the measured quantity. Since the Knight shift measures the local magnetic
susceptibility, any end-chain spin effect is negligible, and it is a good measure
of the chain contribution alone. Takigawa et al. [80] fit their Knight shift data
for AgVP2S6 with the expressions of Jolicoeur et al. [10] which produced gap val¬
ues consistent with the inelastic neutron scattering results [79]. Furthermore, by

39
examining the orientational dependence of the Knight shift, they measured D and
found an upper bound on E. Similar techniques were applied to Y2BaNiOs [105],
but these workers interpreted their results as being inconsistent with the energy
level diagram including anisotropy as described by theory [47,48].
Finally, ESR has been used to examine the microscopic magnetic susceptibil¬
ity and to map out portions of the energy-magnetic field diagram. As explained
in Section 2.4.1, the area under the ESR absorption curve is proportional to the
static magnetic susceptibility. Unfortunately, as antiferromagnetic correlations
increase, the ESR line broadens and loses intensity, resulting in a loss of reso¬
lution. Therefore, if conduction electrons exist in the system, the NMR Knight
shift is a better probe of the local magnetic susceptibility because the NMR
line is not degraded by the antiferromagnetic correlations. Nevertheless, in the
organo-metallic compounds, there are no conduction electrons, so there is no
NMR Knight shift. Since an ESR line is the result of the resonant absorption of
energy between two states split by the magnetic field, high field ESR has been
an effective tool for mapping out the energy vs. magnetic field diagram. This ex¬
perimentally determined diagram has confirmed the theory used to assign critical
fields in M vs. H experiments.
Date and Kindo [65] measured the ESR line at a frequency of 47 GHz as a
function of temperature in NENP and NINO. They found that the area under
the absorption curve as a function of temperature fits the expected magnetic
susceptibility with the exponential activation. Recent measurements have been
performed on Y2BaNi05 [106] with similar results. Several groups have examined

40
the energy-magnetic field diagram using ESR [19,99,100,107]. These results for
both NENP and NINO are consistent with the theory for finite anisotropy [47,48],
assuming a staggered magnetic field is considered in the analysis. Discussion of
the staggered magnetic field is given below.
3.3 Studies of End-chain Spins
The other main topic of this thesis involves the properties of end-chain spins,
and therefore, this section will provide an overview of the research of other work¬
ers. As predicted by the VBS model [13,14] described in Section 2.1.2, the spin
value at the termination of a chain is expected to be one half the value of a spin
site. To test this prediction, finite chains are needed. The first and most predom¬
inant way of breaking the chains was by doping the materials. Initial evidence
of the presence of end-chain spins was an enhanced impurity tail in the magnetic
susceptibility of NENP doped with Cu2+ magnetic impurities [62]. Later, using
X-band ESR on a similar sample, Hagiwara et al. [108] observed hyperfine inter¬
actions that could be attributed to the interaction between 5 = 1/2 variables and
the Cu2+ 5 = 1/2 impurities. Furthermore, the temperature dependence of the
ESR peak, i.e. the peak attributable to the end-chain spins, was fit with an em¬
pirical equation whose form was later confirmed by Mitra et al. [57]. To study an
even simpler system by avoiding magnetic impurities, Glarum et al. [109] studied
samples of NENP doped with Cd, Zn, and Hg. These workers observed a sin¬
gle asymmetric ESR peak, and this asymmetry was attributed to the staggered
magnetization present in the sample. Ajiro et al. [110] recently extended this

41
work with a detailed study of the same system. All of the samples mentioned
so far in this section were lightly doped, but heavily doped samples exhibited
Curie tails [111] that increased faster than if each dopant caused only one chain
break. Several possible explanations of this effect have been tested in different
ways. First, NENP does not have a non-magnetic isomorph, so naturally, beyond
a certain doping level, the sample being doped is not necessarily NENP. To avoid
these problems, primarily two materials have been tested, NiC204-2DMIZ [112]
and Y2BaNi05 [91,113].
The sample NiC204-2DMIZ is isomorphically related to ZnC204-2DMIZ.
Therefore, Zn dopants in NiC204-2DMIZ should enter the crystal uniformly.
Nevertheless, magnetic susceptibility measurements reveal paramagnetic tails
that increase faster than is consistent with the doping level [114]. Kikuchi
et al. [114] explain these variations in terms of an energy gap between states
inside the Haldane gap. They explain the existence of states in the gap as re¬
sulting from interactions, mediated through chain, between end-chain spins as
described by the numerical work of Yamomoto and Miyashita [24]. Nevertheless,
their chains are too long to observe in-gap states, and the simple explanation
may be that the Zn dopants are not entering the crystal uniformly. Careful X-
ray studies are needed to test the Zn distribution in the crystal. Furthermore,
the analysis has neglected any change in the magnetic susceptibility due to the
interaction of the end-chain spins with magnetic excitations.
The material Y2BaNiC>5 can be doped in two ways: either Zn can replace
the Ni, generating chain breaks, or Ca can replace the Ba, creating weak bonds

42
and charge carrying holes [91,113]. Furthermore, extensive X-ray and electron
scattering studies [88] of doped samples have demonstrated that Y2BaNiOs can be
uniformly doped. Ramirez et al. measured the specific heat of a Zn doped powder
sample and observed a Schottky anomaly consistent with the presence of S = 1
end-chain spins. They attributed this effect to differences between chains with
odd and even numbers of sites. Once again, these workers have ignored any effect
of the interactions between the magnetic excitations on the chains and the end-
chain spins. However, the temperature dependence of the ESR signal intensity in
NENP [108-110], TMNIN [115], and NINAZ (Section 6.3.4 of this thesis) suggests
that these interactions should be considered. In addition, recent work by Kimura
et al. [106] shows ESR lines consistent with chain breaks caused by Ni3+ ions
which act as S = 1/2 impurities between two S = 1/2 end-chain spins, and this
situation is similar to the Cu2+ work in NENP [108]. The final evidence that
interactions between magnetic excitations on the chains and the end-chain spins
correctly explain the behavior of Zn doped Y2BaNi05 is provided by theoretical
work of Hallberg et al. [116] that reconciles the specific heat results in Y2BaNiOs
with ESR results in NENP. DiTusa et al. [91] concentrated on the Ca doped
samples which exhibited a greatly reduced resistivity which is consistent with the
assertion that this material is a 1-D metal. More importantly, inelastic neutron
scattering measurements revealed magnetic scattering inside the Haldane gap for
the Ca doped samples but not for the Zn doped samples. Dagotto et al. [117]
theoretically illuminated these observations by qualitatively describing the in-gap
states as resulting from coupling between the mobile holes and the S = 1 Ni sites.

43
There are other ways to observe end-chain spins besides breaking chains with
dopants. Two methods have been tried, the first will be discussed here and the
second will be discussed in Chapter 6. The first method does not involve cre¬
ating chain breaks, but rather measures the end-chain spins already present in
the sample from crystal imperfections. Hagiwara and Katsumata [118] performed
ESR and magnetization experiments on a quickly grown sample of NENP. Their
results are consistent with the existence S = 1/2 end-chain spins. However,
when compared to samples grown by other groups, this sample has an enormous
number of natural chain breaks. Therefore, a study should be performed on
how crystal growth rate affects the number of crystal imperfections. A tradi¬
tionally grown sample was used for the experimental tour de force of measuring
the magnetic susceptibility over six orders of magnitude in temperature space as
shown in Figure 3.3 [119,120]. Three separate experimental apparatuses were
used to map out three regions of temperature space: a SQUID magnetometer
for 300 K > T > 4 K, a mutual inductance coil cooled by a homemade dilution
refrigerator for 4 K > T > 50 mK, and an AC SQUID susceptometer cooled by
the Cu nuclear demagnetization stage of cryostat number 1 at the University of
Florida Microkelvin Facility for 50 mK > T > 400 /xK. The portion of Figure 3.3
related to the end-chain spin discussion is the paramagnetic tail observed below
~ 200 mK. This tail has a Curie constant consistent with the natural chain length
expected for the material [119,120].
Since the main topics of this thesis are S' = 2 Haldane gap materials and end-
chain spin effects in NINAZ, an 5 = 1 material, it would be amiss not to mention

44
1E-4 1E-3 0.01 0.1 1 10 100
T (K)
Figure 3.3: x(400 nK < T < 300 K) for pure NENP. The data are from References
[119,120].

45
the first evidence for free 5=1 end-chain spins in an 5 = 2 material. Yamazaki
and Katsumata [121] used ESR, magnetic susceptibility, and heat capacity to
study end-chain spins in CsCrCl3 doped with Mg. This material possesses long-
range magnetic order below 16 K, so one of their points was that the presence of
5 = 1 end-chain spins constituted the existence of the Haldane state.
Measurements of NINO, NINAZ, and TMNIN have also been performed to
study end-chain spins. The techniques used in NINO [125] are similar to those
discussed above, and the discussion of NINAZ and TMNIN will be given elsewhere
in this thesis.
3.4 Other Experimental Results
Two other topics that have been addressed experimentally must be mentioned.
First, the differing magnetic sites in NENP and NINO will be discussed because a
similar phenomena might explain unresolved issues in TMNIN and MnC^bipy).
The existence of two different magnetic sites might relate to a small anomaly in
the magnetic susceptibility in MnC^bipy) [97]. Both NENP and NINO consist
of Ni sites bridged by an NO2 group. For a specific Ni site, the exchange path
is directed through the nitrogen and one oxygen of the NO2 group on one side
of the Ni. On the other side of the Ni, the exchange path is directed through
the other oxygen of the N02 group (see Figure 3.4). This asymmetric bonding
causes a slight shift in the magnetization of each site. As a consequence, neigh¬
boring Ni sites differ slightly in their transverse magnetization. The existence
of these differing magnetic sites is necessary to explain the NMR line shape of

46
Figure 3.4: The crystal structure of NENP. Notice the different oxygen sites
which bridge the Ni [22].

47
Chiba et al. [98]. This property also explains the finite magnetization observed
below the critical field in M vs. H measurements [63,64] and explains the shape
of the magnetic field diagram as measured by ESR [19,99,100,107].
The second topic of discussion is the coexistence of the Haldane gap and three-
dimensional long-range order. Since this topic is historically significant but is far
afield from the topic of this thesis, only the materials and the references will be
mentioned. The first material to exhibit evidence of a Haldane gap was CsNiCla
[60], but with a gap of 5 K and a Néel temperature of 4.4 K, this assignment
was disputed in the literature [122-124]. Nevertheless, this material provided an
environment to study the coexistence of the Haldane gap and three-dimensional
long-range order [126]. To provide a smooth transition to materials with three-
dimensional long-range order, extensive work has been done on R2BaNi05, where
R=Y, Nd, Pr or some doping mix of Y and Nd. The pure Pr [127] and Nd
[128-130] compounds have magnetically ordered ground states, and the doped
materials [131] allow a variation of the interchain interaction.
Another technique to force interactions between the samples is to apply pres¬
sure. Recent inelastic neutron scattering measurements reveal a reduced single¬
ion anisotropy in NENP under a pressure of 2.5 GPa [132]. Furthermore, heat
capacity measurements in NENP are consistent with an increase in A with in¬
creased pressure [133]. Nevertheless, pressure studies have been unable to in¬
crease the interchain exchange enough to cause the sample to exhibit long-range
magnetic order.

CHAPTER 4
EXPERIMENTAL MEASUREMENT TECHNIQUES
This chapter describes the experimental techniques used to study the materi¬
als described in this thesis. Magnetometry using SQUID and cantilever magne¬
tometers is discussed first. Next, both high field and X-band ESR are discussed
because they provided a microscopic probe of the end-chain spins. The third
major technique to be reviewed is neutron scattering. I have used both powder
neutron diffraction analyzed by Reitveld refinement to obtain low temperature
crystal structures and inelastic neutron data collected by time of flight methods
to look for the characteristic features of the Haldane gap in MnC^bipy). Finally,
I will discuss several measurement techniques used in the creation and analysis
of our pulverized NINAZ samples.
4.1 SQUID Magnetometer
A SQUID magnetometer is a sensitive device for measuring the magnetiza¬
tion of a sample over a range of magnetic fields and temperatures. In addition,
knowledge of the applied magnetic field allows the DC magnetic susceptibility to
be determined as well. Professor J. R. Childress, of the Department of Materials
Science and Engineering at the University of Florida, kindly allowed us to use
his MPMS SQUID magnetometer, made by Quantum Design, for magnetization
and magnetic susceptibility measurements. The main component of this SQUID
48

49
magnetometer is an rf SQUID. A SQUID is a small superconducting ring with a
weak link. As magnetic flux is applied to a SQUID, a supercurrent is induced in
the ring to cancel the flux until the phase of the electron wave function matches
across the weak link. Once this happens, a flux quanta enters the loop and the
supercurrent goes to zero. If the flux keeps increasing, the supercurrent increases
again until another flux quanta enters the SQUID. In the magnetometer, an rf
signal on top of an applied flux provides a changing flux in the SQUID with a
bias such that the system oscillates about the point where one flux quanta enters
the ring. As the flux from the sample changes, a feed back loop controls the bias
flux, canceling the flux from the sample, so the SQUID remains near the one flux
quanta point. The flux needed to keep the SQUID at this point is proportional
to the flux from the sample. Therefore, a calibration will provide a measure of
the magnetization. The particularly challenging part about making a practical
device work in a magnetic field is ensuring that the SQUID measures the flux
change of the sample and not of the magnet. Quantum Design conquers this
challenge by coupling the SQUID, which is outside and shielded from the super¬
conducting magnet, to the sample through a flux transformer that measures the
second field gradient of the sample [134]. Thus, a signal is only observed in the
SQUID when dm2/dx2 is non-zero. The second derivative is chosen over the first
derivative in order to eliminate noise arising from slow drifts in the magnetic
field. Recovery of the sample magnetization is accomplished by moving the sam¬
ple through the gradient coils in a manner to doubly integrate the signal. This
motion requires exact positioning which is accomplished through the use of a

50
stepper motor and through initial positioning of the sample. Detailed discussion
of the physics behind a SQUID are beyond the scope of this thesis, for a brief
overview see Reference [135].
The samples were mounted in standard number 4 or number 5 gelatin cap¬
sules (gel-cap) that were held inside a plastic straw. Depending on the sample,
different mounting arrangements were used. If the sample was a single crystal
or an aligned single crystal packet, it was placed in the gel-cap and oriented in
the desired direction with respect to the magnetic field. Small pieces of gel-cap
were used to secure the sample. If the sample was a powder, it was poured into
the larger diameter portion of the gel-cap, and the smaller diameter portion was
inverted and used to press the sample, thereby stabilizing its position. Usually
~ 50 mg of sample was measured to provide a signal well above the background,
although samples as small as ~ 2 mg were successfully measured. Several arrange¬
ments of this type have been measured sans sample to determine the background
contribution of the gel-cap and straw. A typical H = 0.1 T background is shown
in Figure 4.1, along with fits for three temperature regions. The expressions for
the three temperature regions are given by

51
-1.221793 x 10-5 +
1.2534219xl0~5
0.1091171+T
-1.8067205 x 10_8T
2 K < T < 40 K
M{T) = <
1.2156003 x 10~5 - 1.6642177 x 10~8T
+1.5027333 x 10"10T2 - 6.5003915 x 10“13T3
+7.1854421 x io~14T4
40 K < T < 200 K
-1.423403 x 10"4 + 2.0092364 x 10-6T
-1.1421511 x 10~8T2 + 2.7916376 x 10-UT3
-2.4700249 x 10~14T4
200 K < T < 380 K.
(4.1)
Similarly, Figure 4.2 shows the background for magnetic field sweeps at T = 2 K
with the fit curve given by
M(H) = -3.5256 x 10~bH - 4.3195 x 10~hH2 + 9.0369 x lO”6#3
-6.8631 x 109H4
(4.2)
where H is in units of Tesla. Clearly, the background shows the expected dia¬
magnetism and, at the lowest temperatures, a tail showing the presence of some
free spins. Nevertheless, the contributions are on the order of 10 /¿emu, which
is negligible when compared to the magnetic response of most samples. Several
specimens consisted of a conglomeration of aligned single crystals that were glued
together with fingernail polish on a piece of weighing paper. To check the back¬
ground, similar quantities of weighing paper and fingernail polish were placed

(nw8TÍ) iai
52
O 100 200 300 400
T (K)
Figure 4.1: M(T) SQUID background for an empty gel-cap and straw in
H = 0.1 T. The solid lines are fits given by Equation 4.1.

M (emu)
53
H(T)
Figure 4.2: M(H) SQUID background for an empty gel-cap and straw at T = 2 K.
The solid line is the fit given in Equation 4.2.

54
in a gel-cap. This control was then measured, and the resultant background
subtracted from the corresponding data.
4.2 Cantilever Magnetometer
The cantilever was developed by Chaparala et al. [136] to provide a sensi¬
tive magnetometer at high magnetic fields, and it is capable of simultaneously
measuring transport properties. The operational technique was refined at the
National High Magnetic Field Laboratory partly through our measurements as
will be discussed in later chapters. The result of this instrumental improvement
is the cantilever magnetometer commercially available from Oxford Instruments.
A schematic diagram of the cantilever is shown in Figure 4.3. There are sev¬
eral features to observe. The four leads down the middle are used if transport
measurements are desired along with magnetic measurements. To each side of
the transport leads, other leads form two square loops, which can be used for
calibration if absolute magnetization units are required. The sample is mounted
between the two loops, as shown in the figure. Not shown is the bottom of the
cantilever where a gold plate forms one half of a capacitor. The other half is
a plate fixed below the cantilever. The experimental quantity measured is the
capacitance between the two plates. Typically, the sample was attached to the
cantilever with a small quantity of vacuum grease. This grease has a diamagnetic
background which may need to be subtracted. The determination and removal
of this background will be discussed later with the experimental results.

Transport Terminals
55
Lower Capacitance Plate
To Upper Cantilever Calibration Coil
Capacitance Plate
Figure 4.3: Cantilever magnetometer diagram. This figure is adapted from [137].
Sample

56
The cantilever can work either in force mode or in torque mode. Only the
force mode will be discussed here because a discussion of the torque mode [136]
is beyond the scope of this thesis. For the force mode of operation, the plane of
the cantilever is positioned perpendicular to H. The force on a magnetic sample,
F, in a magnetic field gradient, VH, is proportional to MVH. Since the sample
is affixed to the cantilever, both items experience the same force. Following
Hook’s law, the cantilever opposes F with a force proportional to the amount it
bends, causing the distance x between the plates to change by a quantity Ax.
This change causes the capacitance, C, to change by an amount AC. Therefore,
any change in the force may follow from AF oc Ax oc AC. In other words,
by knowing AC, the change in the force on the cantilever is known to within a
calibration constant. Furthermore, since VH oc H, the change in magnetization
from a reference is AM oc AF/H oc AC/H. The reference is defined by the fact
that F(H — 0) = 0. The VH needed to apply a force on the sample is obtained
by positioning the sample slightly above the center of the field. Table 4.2 gives
field gradient specifications for the magnet (SCM1) used at the National High
Magnetic Field Laboratory. For typical runs, the cantilever was placed 27.0 mm
or 52.4 mm above the center of the field, but for initial runs, it was placed as far
as 152.4 mm off the center of the field.
Sensitivity is gained by working far off the center of the field because VH in¬
creases until approximately 160 mm away from the center of the field. However,
the peak field decreases as the cantilever is moved away from the center of the
field. For example, the 152.4 mm length maximizes the cantilever response

57
z
(cm)
Bz
(T)
dB/dZ
(T/cm)
0.000
20.000
0.000
0.500
19.986
0.028
1.000
19.946
0.080
1.500
19.877
0.138
2.000
19.779
0.196
2.500
19.651
0.256
3.000
19.491
0.320
3.500
19.300
0.382
4.000
19.064
0.472
4.500
18.792
0.544
Table 4.1: Field and gradient at distances off the center of the field along the
axis of the magnet SCM1.
but only provides a field of 8 T on the sample when the center field is 18 T.
The 52.4 mm distance provided the optimal maximum field and sensitivity, al¬
though there are occasions when it is experimentally prudent to place the sample
closer to, or farther away from, the center of the field.
As with any technique, certain limitations must be considered to effectively
use the cantilever magnetometer. First, the magnetic response of the apparatus
is difficult to calibrate. A slightly different background response is observed for
every sample, arising from differences in mass and the magnetic response of the
mounting material. Therefore, a calibration must be run in situ for each different
sample mounting. This procedure is time consuming and is often infeasible for
the time allotted at the National High Magnetic Field Laboratory. Furthermore,
our attempts at calibration were too coarse to be useful. Nevertheless, an ab¬
solute measure of magnetization is not essential for identifying critical fields.

58
Therefore, the cantilever is an excellent device for studying the Haldane gap.
Second, since the force is divided by the magnetic field to obtain the magne¬
tization, the error bars in the magnetization grow quickly at fields below 1 T,
obscuring low field features. In addition at low fields, superconducting magnets,
which have been run above Hci of the superconducting wire, spontaneously expel
magnetic flux trapped in the wire, and this effect causes noise in the measure¬
ment known as flux jumps. Therefore, the issue of a usable low field region is a
combination of the limitation of the technique and the apparatus used. Third,
the force is always measured relative to the zero magnetic field state. Therefore, a
constant force or magnetization can not be distinguished from no magnetization.
For example, at millikelvin temperatures, where paramagnetic spins saturate at
a few mT, a paramagnetic contribution can not be distinguished from the zero
ground state in a Haldane system.
To ensure that the limitations on the experiment result from the cantilever
alone, a low noise set of electronics, shown in Figure 4.4, was used to acquire data.
The dark grey and light grey shading identifies the portions of the apparatus that
are inside the mixing chamber of the dilution refrigerator and immersed in the
liquid helium bath, respectively. The dotted box indicates a device that was
tested, found to provide negligible improvement, and subsequently removed. The
temperature controller used a four wire measurement to regulate the temperature.
At the time of our experiments, the thermometer was calibrated for zero magnetic
field only. Therefore, the resistance of the thermometer at the start of the run
was compared to the resistance at the end of the run to be sure that thermal

59
Figure 4.4: Cantilever magnetometer electronics schematic. The dark grey and
light grey shading identifies the portions of the apparatus that are inside the
mixing chamber of the dilution refrigerator and immersed in the liquid helium
bath, respectively.

60
stability was maintained throughout the run. A heater provided the necessary
temperatures above base temperature, although it was rarely needed. Several
different digital multimeters were used as analog to digital converters, and all of
them added negligible amounts of noise to the measurement. To reduce noise, a
lock-in amplifier was used for phase sensitive detection. The Princeton Applied
Research model 124A lock-in amplifiers, PAR 124A, were chosen since their signal
to noise ratio was at least an order of magnitude better than any other instrument
tested. The PAR 124A has an internal voltage controlled oscillator that was
operated at ~ 10 kHz. In order to reduce coupling from the outside environment
as much as possible, the wires from the General Radio capacitance bridge to the
top of the insert were a twisted pair of coaxes. Finally, all data was collected by
a Macintosh personal computer, using a general purpose interface bus (GPIB)
and the LabView software package.
To cool the samples to temperatures well below the Haldane gap, an Oxford
Instruments Kelvinox top loading dilution refrigerator was used. The top loading
nature of the cryostat allows samples to be mounted while the refrigerator is at
~ 4 K. Once the sample is in the mixing chamber, condensation and circulation
can begin, and within 6 hours, the sample is at a base temperature of ~ 30 mK.
The magnet used was SCM1, a superconducting magnet provided by Oxford
instruments. The maximum center field is 18 T at T = 4.2 K, and 20 T can be
reached through the use of a lambda refrigerator that reduces the temperature
to ~ 1.4 K. The current/field ratio is known to be 10 A/T, so the magnetic field
on the sample is determined by the output current of the magnet power supply.

61
4.3 Electron Spin Resonance
Electron spin resonance is a powerful microscopic probe of magnetic proper¬
ties. Its use provided a wealth of information on the end-chain spins in NINAZ.
There were two experimental apparatuses used for the ESR work presented in this
thesis. The most extensively used instrument was a Brueker X-band spectrome¬
ter working at ~ 9 GHz. Professor D.R. Talham of the Department of Chemistry
at the University of Florida graciously provided the use of this spectrometer and
its support staff. Figure 4.5 shows a schematic diagram of the apparatus. A
standard (i.e. iron yoke and water cooled) split coil magnet applied a DC field to
the sample in a direction perpendicular to the direction of the microwave prop¬
agation. Since this spectrometer uses a resonance cavity and a Klystron for a
microwave source, it is more convenient to vary H instead of u>, as is the stan¬
dard practice [55]. This resonance cavity is coupled with an Oxford ESR 900
Flow Cryostat, which cools the sample down to T ~ 4 K. The temperature is
measured by a AuFe/Ch thermocouple with the reference junction kept at 77 K.
Figure 4.6 is a schematic diagram illustrating the flow of the helium gas and the
sample and thermometer positions. Single crystal samples were affixed to the end
of a quartz rod using Apiezon L grease and inserted directly into the flow of the
helium gas. On the other hand, powder samples are placed in a quartz tube, so
the sample is thermally coupled to the helium gas through the tube. The heater
is placed well away from the sample to avoid any interference in the measure¬
ment. However, this arrangement lengthens the thermal equilibration time of the
sample. Therefore, one should wait at least five minutes, after a temperature

62
Personal
Computer
486
•GPIB'
Figure 4.5: Schematic diagram of the X-band ESR spectrometer

63
change, for the system to thermally equilibrate. Additional care should be taken
with powder samples, or samples of low thermal conductivity, as the thermome¬
ter directly measures the temperature of the helium gas and not the temperature
of the sample. To ensure proper thermal conductivity between powder samples
and the quartz tube, a syringe was inserted into the rubber stopper at the end
of the tube, and this arrangement was evacuated via a mechanical pump while
initially cooling the cryostat. When the base temperature was reached, pumping
was stopped, and helium exchange gas was injected into the sample tube. Not
shown in Figure 4.6 is the room temperature coupling which allows the sample
to be rotated without leaking air to the cryostat. A homemade goniometer was
used to determine the orientation of the magnetic field with respect to the crystal
axes.
To accommodate for the signal to noise improvements offered by a phase-
locked loop, a small modulation field was added to the DC field. Introduc¬
ing the modulation has the added affect of differentiating the absorption signal.
Returning the data to an absorption signal from the derivative signal was accom¬
plished using a second-order, Runge-Kutta numerical integration routine [138]
with an automatic baseline adjustment procedure. The code was written in Mi¬
crosoft Visual C++ Version 4.0 and ran on a PC running Windows95.
Since a measure of absolute signal intensity was required, a room tempera¬
ture calibration against a, a'-diphenyl-/?-picrylhydrazyl (DPPH) was performed
following standard procedures [55,139]. A 0.1 mg single crystal purchased from
the Aldrich Chemical Company was used for the calibration. With a molecular

64
Resonance
Cavity
Figure 4.6: Cross-sectional schematic view of the ESR flow cryostat. Adapted
from [140]

65
weight of 394 g/mol and 1 free spin per molecule [139], our sample contained
1.52 x 1017 spins. Figures 4.7 and 4.8 show the derivative signal and the inte¬
grated absorption signal, respectively. The incident microwave power used for
the calibration was -30 dB of 300 mW at a frequency of v = 9.5453 GHz. The
area under the intensity, /, vs. H data is 2.279 x 108 Iunits-kG which results
in 6.67 x 108 spins/Iunit-kG. This procedure has effectively calibrated the area
under the ESR spectrum to the number of spins. In order to determine the
intensity of the DPPH signal at lower temperatures, a Curie law was assumed,
and the corresponding number of spins calculated. The procedure avoids the
non-Curie law behavior, observed in DPPH below ~ 50 K [55], because the cal¬
ibration was performed at room temperature where correlations between DPPH
spins are negligible.
Preliminary high frequency ESR work was done using the high field trans¬
mission spectrometer in the laboratory of Professor L.C. Brunei at the National
High Magnetic Field Laboratory. The spectrometer has four changeable Gunn
oscillator sources with Schottky based harmonic generators and appropriate fil¬
ters which allows harmonics of these sources to be used. The available frequencies
are given in Table 4.2. The 25-130 GHz source does not have sufficient power for
use with the harmonic generators. The magnetic field is supplied by an Oxford
Instruments 17 T superconducting magnet oriented with the field longitudinal to
the microwave beam. The spectrometer measures the microwave radiation trans¬
mitted through the sample, so the beam comes out of the source, passes through

66
Figure 4.7: dl/dH vs. H for a 0.1 mg sample of DPPH at T = 293 K.

67
H (G)
Figure 4.8: / vs. H for a 0.1 mg sample of DPPH at T = 293 K.

68
Source
Harmonics
25-130
-
-
-
-
75
150
225
300
375
95
190
285
380
475
110
220
330
440
550
Table 4.2: Frequencies available for microwave transmission at the National High
Magnetic Field Laboratory, in units of GHz.
the sample, and then is reflected back to a detector, usually a bolometer, which
records the data on a Macintosh personal computer.
4.4 Neutron Scattering
To extend our studies of the Haldane gap, inelastic neutron scattering has
been used as an additional microscopic probe. Precise measurements of the re¬
ciprocal lattice vectors are required for inelastic neutron scattering measurements,
so neutron diffraction was used to determine them. In any neutron scattering ex¬
periment, both elastic (neutron diffraction) and inelastic scattering is observed.
A full analysis separates the received scattering intensity into four components:
elastic coherent scattering, elastic incoherent scattering, inelastic coherent scat¬
tering, and inelastic incoherent scattering [141]. Elastic coherent scattering is
typified by Bragg peaks that are usually the largest contribution to the received
intensity. The next largest contribution results from incoherent scattering. If
a large incoherent scattering peak is observed, it arises from randomly oriented
nuclear magnetic moments. Therefore, careful isotope selection during sample
synthesis will greatly reduce this contribution. For example, the samples studied

69
for this thesis possess large organic ligands, so hydrogen would be the largest
incoherent scatterer because the single proton has a rapidly moving nuclear mag¬
netic moment of I — 1/2. On the other hand, deuterium has no nuclear moment,
so full deuteration of the samples eliminates the largest source of incoherent
scattering. Generally, in properly deuterated samples, the incoherent scattering
component is negligible when compared to the coherent elastic scattering and
can be neglected for structure refinements. In addition, inelastic components are
generally much smaller than elastic components and can be neglected in neutron
diffraction experiments.
When inelastic neutron scattering measurements are performed, coherent elas¬
tic scattering is the most troublesome component. The simplest way to eliminate
the contribution due to these Bragg peaks is to avoid measuring regions of Q
space where they exist. In addition, the incoherent elastic component is usu¬
ally much larger than the coherent inelastic component and must be avoided.
There are two ways to avoid this contribution. Firstly, reduce the contribution
as much as possible by choosing appropriate isotopes, and secondly, since inco¬
herent scattering is always around Q,E = 0, inelastic scattering at sufficiently
large wavevectors and energies are unaffected by the incoherent scattering peak.
A mixed blessing and curse of neutron scattering experiments is that the neu¬
trons interact weakly with the material. Therefore, at least 1 g of materials is
needed for sufficient experimental resolution [142], The metal-organic materials
used for these measurements usually do not grow in single crystals this large.
Therefore, an added complication to the work was that a collection of randomly

70
oriented microcrystallites was used for the measurement. Fortunately, the mi-
crocrystalites were small enough to effectively act as a symmetric powder as will
be described in Section 7.3. Therefore, to determine the lattice parameters, the
powder neutron diffraction pattern was analyzed by Rietveld refinement.
Rietveld refinement is essentially a non-linear least squares fit between an
assumed crystal structure, whose Bragg peaks are powder averaged, and the
actual data. Since Rietveld, who developed the method, believed strongly in
the free dissemination of information, his computer routines [143-145] have been
freely distributed and used to develop excellent public domain analysis software.
The package used by the author is GSAS from Los Alamos National Laboratory
which has a user friendly interface for the personal computer running Microsoft
Windows. This program runs sufficiently fast on a Pentium based machine with
copious quantities of memory. Insufficient memory space (e.g. 24 megabytes
is not enough) causes program crashes which require restarting the refinement
from the raw data. Generally, the technique is extremely powerful and up to 193
parameters have been refined, 161 of them simultaneously [146,147]. As with
any non-linear least squares fitting, one must have some feel for the right answer
before starting, otherwise the fit of the naive user will be stuck in a local minimum
in the multiparameter hyperspace. The method used to avoid local minima was
to examine the resultant crystal structure for physical plausibility on the basis of
the known chemical composition. Furthermore, several different quantities can
be used to quantify the quality of the refinement [147]. The primary test statistic
used was \2 reduced by the number of parameters being varied, and the total

71
X2 was used as a secondary test to confirm that changes in the reduced x2 were
significant.
The spectrometer used was the high resolution powder diffractometer on beam
tube HB-4 of the High Flux Isotope Reactor (HFIR) at Oak Ridge National
Laboratory. The wavelength of the neutron beam is selected by orienting a (115)
Ge single crystal. This monochromator provides the sample with wavelengths of
1.0, 1.4, 2.2, or 4.2 Á. The beam passes through the sample to 32 3He detectors
that are equally spaced at 2.7° apart. These detectors can be scanned over a range
of 40° so the total angle covered is 11° to 135°. A collimator in the beam reduces
the beam size from the maximum 3.75 x 5 cm2 to the desired size. Typically, the
flux at the sample is 2 x 105 [148]. Using a glove box, the sample is placed
inside a vanadium can in a helium atmosphere. The can is sealed with an indium
o-ring before removing it from the glove box. If two sample batches need to be
kept separate inside the vanadium can, each batch is placed in an aluminum foil
pouch. For room temperature measurements, the sample is rotated while in the
neutron beam to average out any asymmetry in the powder. For temperature
dependent measurements, a displex was used with aluminum vacuum cans, and
this refrigerator was capable of achieving 11 K.
The main goal of the neutron scattering study is to examine the magnetic
excitation spectrum with inelastic neutron scattering. There are several factors
to consider when choosing an instrument for inelastic neutron scattering. Firstly,
the incident neutron energy must be selected so that the desired features are
resolvable. As a general rule of thumb, the resolution is related to the incident

72
energy by AE ~ 0.02Ei [142]. Therefore, neutrons from a water moderated
reactor (thermal neutrons) have AE ~ 0.5 meV, but the Haldane gap is expected
to be around 0.2 meV. Consequently a source of lower energy neutrons (cold
source) is required. Once the cold source is identified as necessary, there are
several instruments from which to choose to make the measurement.
Since the sample used is a powder sample, a technique which could collect
data for a broad range of Q values simultaneously is desirable. This requirement
means a time of flight spectrometer is ideal. For inelastic neutron scattering,
four quantities must be known: the initial energy, the initial momentum, the
final energy, and the final momentum. Assuming the initial quantities can be
tuned, only the final energy and momentum have to be measured. For time of
flight, the energy and the magnitude of the momentum are determined by the time
required for the neutron to travel a known distance. A pulsed source of neutrons
provides a t = 0 reference for the time measurement. To determine the final angle
of the momentum, the detectors are equally spaced at known angles. Since the
detectors are fixed, data for multiple Q values is collected simultaneously.
The time of flight spectrometer (TOF) on the cold source at the National
Institute of Standards and Technology was chosen for the measurement. A liquid
H2 moderator is used as the cold source to provide a distribution of neutrons
peaked around an energy of 5 meV. Neutrons from the cold source are directed
to the instrument in neutron guide NG-6. Once at the instrument (Figure 4.9),
two pyrolytic graphite monochromators select neutrons of the desired energy.
Next, a cooled Be and pyrolytic graphite filter removed neutron beam compo-

73
Detectors where
possible Haldane
gap is observed
Monochromator
Section
Chopper
PG Section
Filter
Figure 4.9: NIST TOF spectrometer from Reference [149]. Notice the region of
detectors used to identify hints of the Haldane gap in MnCl3(d-bipy).

74
nents remaining from higher order reflections in the monochromator crystals. In
addition, a Fermi chopper sets the length of the time pulse. Finally, a flux of
2.4x 104 n^2°n interacts with the sample, passes through an oscillating radial
collimator, and is detected by 3He detectors. The detectors are equally spaced
at 2.546° for angles from 1.7° to 130°. Unfortunately, the detectors for angles
less than 22° are too noisy for use on this experiment. For future reference, the
region of detectors used to identify hints of the Haldane gap in MnCls(d-bipy) is
indicated in Figure 4.9.
4.5Mechanical Ball Milling
For part of this thesis, the antiferromagnetic chains were broken by mechanical
methods. Mechanical ball milling was used to produce the finest ground powder
using the ball mill of Professor J. H. Adair in the Department of Materials Science
and Engineering at the University of Florida. To perform the milling, the material
is placed in a polypropylene bottle with zicronia balls and fluid, hexane in this
case, which efficates the crushing process. The bottle is then rotated for 36 hours
to perform the pulverization. The resulting particle size distribution usually
follows a log-normal distribution (Figure 4.10) [150].
4.6Centripetal Sedimentation
A method was needed to determine the characteristic particle size of various
samples. To this end, centripetal sedimentation was performed in the laboratory
of Professor J. H. Adair in the Department of Materials Science and Engineering

75
Figure 4.10: A normalized log-normal distribution of width 1 centered at x = 1.
This is the theoretically predicted particle size distribution for a ball milled
sample.

76
at the University of Florida. The main idea is to measure the spatial distribution
of the particles in a medium of known viscosity when a known force is applied to
them for a certain length of time. In our case, the viscous medium was hexane
as it did not react with the material being studied. To measure the spatial dis¬
tribution of distances, a laser light is passed through the particle-fluid mixture,
and the relative intensity is directly proportional to the particle size distribu¬
tion. The applied force arises from the centripetal acceleration from spinning the
mixture [151].
4.7 Inductively Coupled Plasma Mass Spectrometry
Since this thesis claims that the end-chain spins are intrinsic impurities, a
technique was needed to ensure that the extrinsic impurities in the sample were
negligible. Therefore, inductively coupled plasma mass spectrometry, ICP-MS,
was performed in the laboratory of Dr. D. H. Powell in the Department of Chem¬
istry at the University of Florida. Figure 4.11 is a schematic of a typical ICP-MS
system [152,153]. It consists of a silica plasma torch which maintains the plasma
by a continuous feed of gas (usually Argon) through radio frequency coils which
couple to the plasma. The sample to be measured is introduced directly into the
plasma and is ionized by the intense heat. Two small orifice cones are placed in
series, approximately 10 mm from the edge of the torch, to remove ions from the
useful part of the plasma. Once the ions are removed, they are sent into a stan¬
dard quadrapolar mass spectrometer for analysis. Then, the ions are deflected by
the charged plates in the quadrapolar mass spectrometer providing a distribution
of the elements according to their molecular weight.

77
RF Coils
Quadrapolar l glslgl
mass ^
v
spectrometer
/
~ww
Sampling
Cones
Ar Feed
Sample
Feed
Figure 4.11: Schematic diagram of an ICP-MS system. The size of the torch
and the cones is greatly exaggerated with respect to the mass spectrometer for
illustrative purposes.

CHAPTER 5
TMNIN
This chapter covers magnetic measurements on (CH3)4N[Ni(N02)3], com¬
monly known as TMNIN. Initial magnetic susceptibility measurements displayed
the characteristics of a 1-D chain with a small exchange energy. Furthermore,
magnetization, M, vs. magnetic field, B, measurements revealed a Haldane gap
small enough to be reached by high resolution NMR magnets. Therefore, a pro¬
gram was started to grow and characterize single crystal specimens for NMR
experiments. To this end, M vs. B measurements were performed at millikelvin
temperatures in order to characterize the gap and single-ion anisotropy. These
measurements revealed a small value of D and are the main topic of this chapter.
5.1 Synthesis and Structure of TMNIN
Tetramethylammonium nickel nitrate (TMNIN) was first grown by Goodgame
and Hitchman [71], and initial attempts at the crystal structure were made by
Gadet et al. [72] using twined single crystals. Their results confirm that TMNIN
is an antiferromagnetic chain with adjacent Ni2+ ions bridged by three NO2
groups. The tetramethylammonium group, [(CH3)4N]+, sits between the chains
to space them and provide charge neutrality. These features are clearly observed
in Figure 5.1. Nevertheless, one feature of the crystal structure remained il¬
lusive until the single crystal X-ray diffraction studies by Chou et al. [74,83].
78

79
Figure 5.1: TMNIN Crystal Structure [74,83]. Note that Nil is octahedrally co¬
ordinated with nitrite nitrogens, and Ni2 is octahedrally coordinated with nitrite
oxygens.

80
More specifically, the Nil is octahedrally coordinated with six nitrite nitrogens,
and Ni2 is octahedrally coordinated with six nitrite oxygens. Different Ni sites
may slightly affect the magnetic properties of the chain, as will be discussed later
in the chapter.
The single crystals used for the X-ray studies and the magnetization measure¬
ments, to be detailed below, were produced following the synthesis procedure of
Goodgame and Hitchman [71], but modified to allow the crystals to grow undis¬
turbed in a 284 K environment for 4 months. The synthesis procedure starts by
slowly adding solutions of NiBr2 and (CH3)4NBr to a solution of NaN02. After
4 months, large single crystals were harvested from the sides of the container.
Fifty of these single crystals were oriented and glued together with clear finger
nail polish to produce a 2.1 mg packet. Both this packet and a few single crystals
of approximately 1.1 mm in length and 0.11 mm in diameter were used in the
magnetization studies to be described below.
5.2 Magnetization Measurements on TMNIN
Initial magnetization measurements on the TMNIN packet were performed
on a SQUID magnetometer at 1.8 K and 30 K for magnetic fields from 0 to
5.5 T oriented parallel to the chains. As can be seen in Figure 5.2, a hint of the
critical field associated with the Haldane gap is observed at ~ 2.7 T. However,
since the sample temperature is close to the size of the gap, the signature of
the critical field is thermally smeared. The T = 30 K (T A) data provide
additional circumstantial evidence for the presence of the gap since no critical field

81
1.5
_ 1
w
-»-«
c
ri
k.
0.5
0
0 2 4 6
B (T)
Figure 5.2: SQUID M vs. B for TMNIN at temperatures of 1.8 K and 30 K [75].
is observed. For both temperatures, the chain axis of the packet was oriented
parallel to the magnetic field. Similar results were obtained by other workers
[72,77] for a powder sample.
To better resolve the critical magnetic field, subsequent measurements were
performed at lower temperatures. The facilities of the National High Mag¬
netic Field Laboratory were ideal for this purpose. Magnetization as a function
of magnetic field was measured with the cantilever magnetometer described in
Section 4.2. Millikelvin temperatures at high magnetic fields were provided by a
1 1
TMNIN
â–  1 1 l ' 1 '
X
B II Ni-chain
o
—
o 30 K
—
x 1.8 K
o
X
0
â– 
o X
o
X
o
o
X
X
o
X
° X
o x
9 x x
X
_J I 1 1
—I 1 1 1 1 1 1

82
top loading dilution refrigerator in the superconducting magnet SCM1. Since our
only concern was the value of the critical field, a quantity not requiring an ab¬
solute determination of the magnetization, the complicated and time consuming
calibration process was bypassed. Two runs were made, the first on the above
described packet and the second on a single TMNIN crystal. The packet was
attached to the cantilever with silver paint, and the cantilever and sample were
placed a distance of 16 cm from the center of the field. This position produced an
ample field gradient for the cantilever to operate but reduced the maximum mag¬
netic field at the sample by a factor of 2.5. Figure 5.3 shows both the force on the
cantilever and the magnetization of the sample for sweeps from 0 to 8 T with the
magnetic field oriented perpendicular to the chain axis. A critical field, associated
with the Haldane gap, can be extracted as Bcl_ = 2.90 ± 0.15 T. In addition, the
uncertainty in the magnetization data increases greatly below 1 T. This increase
is an artifact of the technique since magnetization is derived from the data using
M = F/B, as described in Section 4.2. This effect emphasizes the need to exam¬
ine the raw force data. Figure 5.4 shows data for the same sample with the field
oriented perpendicular to the chain axis. Several sweeps are displayed. Again the
critical field can be extracted as jBc¡| = 2.40 ± 0.15 T. The other striking feature
in Figure 5.4 is the continuous bend in the data, starting near 4 T and continuing
to the end of the data. This experimental artifact results from the diamagnetic
contribution of the silver paint. For low and high fields, the sample and the
diamagnetic background control the cantilever response, respectively. Therefore,
at intermediate fields the observed data result from a competition between the

83
0 2 4 6 8
B(T)
Figure 5.3: (a) F and (b) M vs. B± for TMNIN packet 16 cm off the center of
the field [75]. The packet weighed 2.1 mg and was tested at T = 25 and 100 mK
using the cantilever magnetometer.

84
sample behavior and the diamagnetic background. A couple of unique features
are observed when the data in Figures 5.3 and 5.4 are compared. First, the data
in Figure 5.3 exhibit a finite magnetization below the critical field. To see if this
finite magnetization resulted from paramagnetic impurities, scans were performed
at temperatures of 25 ± 5 mK and 100 ± 5 mK. There is no discernible difference
between the data at the two temperatures, suggesting the absence of paramag¬
netic impurities. However, paramagnetic impurities saturate at a few mT for
temperatures in the millikelvin range, and, therefore, a finite magnetization be¬
low the gap would result. Nevertheless, since this finite magnetization is absent
for the other field orientation and any effect from paramagnetic spins should be
isotropic, another explanation is needed for this effect. By comparisons of several
runs on TMNIN, and further studies on other samples, we have determined this
effect originates from slight differences in the mounting material from one run to
the next. Later, the same sample packet and the cantilever were moved closer
to the center of the field to improve the response of the system. The results are
shown in Figure 5.5. Clearly there is a strong reaction to the magnetic field,
such that above 6 T the cantilever responds nonlinearly. Nevertheless, a critical
field of Bc_l = 2.60 ±0.15 T can be extracted. To reduce the nonlinear effects,
a second run on a single crystal of TMNIN was performed. The data are dis¬
played in Figures 5.6 and 5.7. These data are consistent with the packet data,
albeit with slightly different critical fields. This difference might be caused by
internal stresses present in the packet, due to thermal contraction differences be¬
tween the fingernail polish and the sample, that are absent in the single crystal.

85
6
'5T 4
'c
xi
S 2
o
O
0
» 0.5
'c
xi
k.
3
^ 0
-0.5
0 2 4 6 8
B(T)
Figure 5.4: (a) F and (b) M vs. F|| for TMNIN packet 16 cm off the center of
the field [75]. The results of several magnetic field up and down sweeps for this
2.1 mg packet using the cantilever at T = 25 mK are shown.

86
B (T)
Figure 5.5: (a) F and (b) M vs. B± for TMNIN packet 5 cm off the center of the
field [76]. The cantilever magnetometer made the measurement on this 2.1 mg
packet at T = 60 mK.

M (arb. units) F (arb. units)
87
-0.2
0 5 10 15 20
B (T)
Figure 5.6: (a) F and (b) M vs. B± for TMNIN single crystal using the cantilever
magnetometer placed 5 cm off the center of the field [76].

M (arb. units) F (arb. units)
88
Figure 5.7: (a) F and (b) M vs. B\\ for TMNIN single crystal using the cantilever
magnetometer placed 5 cm off the center of the field [76].

89
The validity of this assertion is strengthened by the experimental results on NENP
that demonstrate an increased gap with increased pressure [133].
As mentioned in Sections 2.3.1 and 3.2, the critical fields can be used to
determine the Haldane gap and the single-ion anisotropy. The critical fields with
the least uncertainty were obtained from Figures 5.5 and 5.4. Using Equations
2.14, 2.15, and g — 2.09 from previous work [75,83], the corresponding gaps are
Aj_ = 3.34 ± 0.21 K and Ay = 4.88 ± 0.28 K. Solving Equations 2.17 and 2.16
simultaneously, with the above values and k = 1/3, provides A = 3.5 ± 0.2 K
and D = 0.6 ± 0.3 K. The ratio D/J = 0.06, where J = 10.11 ± 0.05 K, shows
that TMNIN has the lowest single-ion anisotropy of any 5 = 1 material known
to posses a Haldane gap, see Table 3.1.
5.3 Comparison with Other Experiments
Several other groups have performed measurements on TMNIN using several
techniques. Comparisons between all of the studies provide insight into global
features of the system.
Most of these experiments also provide measurements of A. Gadet et al. [72]
made several tests on powdered specimens of TMNIN, all of them consistent with
a Haldane gap of 3.5 K. Notably, they performed proton NMR, and the tempera¬
ture dependence of Tx behaves in a manner consistent with similar measurements
in NENP (Section 3.2). Nevertheless, their resolution of the gap was greatly
reduced by thermal smearing since their lowest temperature was 2 K. The mag¬
netic field dependent magnetization measurements of Takeuchi et al. [73] exhibit

90
a critical field of ~ 2.5 T, which is also consistent with our results. Furthermore,
using magnetic fields up to 40 T, they saturated the spins of the chain. TMNIN
has been studied by several groups using high field ESR [154,155]. The results
for powder samples show an asymmetric line shape with a large peak and sev¬
eral small features. The large peak is clearly attributable to the Haldane phase
and is consistent with our measured value for the gap. However, the line shape
and additional peaks are consistent with a picture of large single-ion anisotropy,
D « 0.5, in sharp disagreement with the magnetization results presented here
for single crystals. A simple explanation could be provided by sample differences
because the strongest point for large single-ion anisotropy is a peak attributable
to a transition between Axy and Az, but this line is small. Therefore, it could
be due to an impurity phase in the powder. The existence of an impurity phase
in powder samples is not unprecedented since early growth batches by our re¬
search group displayed impurities when checked by magnetic susceptibility mea¬
surements [156]. However, if good quality is assumed for all the samples, the
existence of two magnetically different Ni sites is another possible cause for this
discrepancy. The alternating coordination of the Ni sites would be the likely
origin of these non-equivalent sites. Though no measurements have been made
to definitively test this assumption, unexplained details of several experiments
suggest that this effect warrants further investigation. Magnetically different Ni
sites could produce additional local magnetic fields that would cause additional
ESR lines and an asymmetric line shape. Another detail to examine is the ESR
spectrum for Zn doped samples of TMNIN. Deguchi et al. [115] doped TMNIN

91
with Zn to study end-chain spin effects by ESR and magnetization. The sig¬
nificant feature of their data is the unexplained ESR line shape. Clearly their
1.7 K trace shows two peaks. These two peaks could originate from the two
distinct Ni sites. However, further analysis is needed to eliminate other sources,
e.g. the dopants, of multiple lines. Neither of these reasons alone or combined
are convincing evidence for magnetically different Ni sites in TMNIN, yet they
provide a sufficient argument to pursue further tests. One possible test would be
to continue this work with proton NMR. The observed line shape below the gap
could confirm or deny this staggered magnetization as was the case in NENP [98].
However, the protons are far removed from the Ni sites in TMNIN, so the effects
of these sites on the line shape may be unresolvable.

CHAPTER 6
NINAZ
After the magnetic susceptibility results on NENP showed a possible magnetic
transition at approximately 4 mK, a search for a material with a larger exchange
energy was commenced. The material chosen was Ni(C3H10N2)2N3(ClO4), which
is commonly known as NINAZ. Although this material has a larger J, to date
no evidence of a transition, reminiscent of the possible transition in NENP (see
Section 3.3), has been observed down to T « 40 mK. Nevertheless, NINAZ
is ideal for the study of end-chain spins because it has a structural transition
that naturally causes chain breaks. In addition to these numerous natural chain
breaks, NINAZ posses several other qualities that make it a good model system.
For example, the interaction energy scale is large enough to study the quantum
properties of end-chain spins at 4 K, and there is no staggered magnetization to
affect the symmetry of the ESR line shape as was observed in NENP [109,110].
The primary focus of this chapter is a description of the behavior of end-chain
spins as studied by macroscopic and microscopic probes. To introduce this topic,
a description of the material will be given, followed by a discussion of experimental
work on NINAZ prior to this thesis.
92

6.1 Material Description
The sample NINAZ, properly called 6¿s(propylenediamine)azidonickel(II) per¬
chlorate, is an ionic-covalent salt where the Ni chain is the cation and the perchlo¬
rate is the counterion. The antiferromagnetic chain is made of Ni2+ ions bridged
by azido ligands (N3 bridge) providing a Ni-Ni distance of 5.849 Á. The N atoms
of two propylenediamine ligands fill the four remaining bonding sites of each Ni
ion. The room temperature crystal structure is shown in Figure 6.1. Gadet et
al. [81] were the first researchers to resolve this crystal structure. At approxi¬
mately 255 K, NINAZ has an irreversible structural transition which causes the
sample to shatter. The room temperature phase has an orthorhombic unit cell
with a = 5.86 Á, b = 8.28 Á, and c = 15.15 Á. For the lower temperature phase,
the structure changes to monoclinic, where the b axis becomes a = 16.22 Á, the
a axis becomes the c = 31.04 A, the c axis becomes the b axis with no change in
magnitude, and ¡3 ~ 95°. Since the chain axis is the low temperature b axis (high
temperature c axis), the structural transition does not cause any dimerization.
Neutron scattering studies of a deuterated crystal cooled through the transition
temperature reveal that the low temperature phase has two domains [82]. A
detailed crystal structure of these domains is still unavailable. The structural
transition is an order to disorder transition. At room temperature, both the per¬
chlorate counter ion and the propylenediamine ligands are disordered. Therefore,
there are two possible order to disorder transitions to cause the observed struc¬
tural transition. For similar salts, the counter ion orders, and this assignment
is made by Gadet et al. [81]. Nevertheless, our research with similar materials

94
Figure 6.1: Room temperature NINAZ crystal structure [81]. The N3 ligand
provides the superexchange pathway between Ni sites. The (CaHxol^) ligands
and the (CIO4) ions, which are not shown for clarity, space the chains.

95
suggests ordering in the propylenediamine group. For example, the material
Ni(C3HioN2)2N3(PF6) also shatters when cooled to 77 K, suggesting that the
transition is independent of the counter-ion. Further evidence is provided by the
fact that the related material [Ni(C4H12N2)2(/i— N3)]n(C104)n, where the spacing
organic ligand is different and the Ni sites are bonded with cis rather than trans
bonds, does not shatter. More specific details of these two materials will be given
below.
Chou improved upon the procedure for the formation of NINAZ which is de¬
scribed elsewhere [81,83,157], so the process is only sketched here. During the
synthesis procedure, intermediate materials may be explosive. Therefore, small
amounts of material should be synthesized, and the standard references [81,83]
should be consulted prior to synthesis. A stoichiometric amount of the reac¬
tant H2N(CH2)3NH2 is added in a drop wise fashion to the green solution of
Ni(C104)2 • 6H20 dissolved in water. The product of this reaction is a deep
blue solution of Ni(H2N(CH2)3NH2)2(C104)2. Then a stoichiometric amount of
NaN3- H20 is added to the product. After several days in solution, deep blue crys¬
tals of NINAZ will begin to form and can be collected by filtration. For the doped
samples, 0.5% of the Ni(C104)2 • 6H20 was replaced by X(C104)2 • 6H20, where
X = Hg, Zn, or Cd. In addition, Ward made a batch of Ni(C3Hi0N2)2N3(PF6)
by substituting Ni(C104)2 • 6H20 with Ni(PF6)2 • 6H20.

96
6.2 Other Experiments
Initially the magnetic susceptibility of NINAZ was characterized by Renard
et al. [62] and Gadet et al. [81]. As shown in Figure 6.2, the broad peak, charac¬
teristic of a 1-D antiferromagnetic Heisenberg chain, followed by the sharp drop
which fits an exponential excitation over an energy gap characteristic of a Hal¬
dane gap, are observed by these workers and ourselves. The subtle differences
between the magnetic susceptibility measurements of each of the three groups
will be discussed below. Later, high field M vs. H measurements [73], performed
on a powder sample, identified a critical field of 30.0 T. The inelastic neutron
scattering measurements of Zheludev et al. [82] provided a microscopic confir¬
mation of the Haldane gap. Figure 6.3 shows several constant Q scans, near
Q = tt, which confirm a Haldane gap of A = 41.9 ± 0.3 K. Each scan shows a
relatively sharp increase and a broader drop off for higher energies, consistent
with the predicted scattering function [45,67]. Furthermore, as Q is moved away
from the antiferromagnetic point, the scattering intensity decreases in a manner
which is consistent with the existence of the two particle continuum [45,67,159].
As an aside, constant Q scans with data shaped similar to scans in Figure 6.3,
albeit on a greatly reduced energy scale, are expected for the inelastic neutron
scattering results of MnCl3(d-bipy) to be discussed in Section 7.4. Returning
to NINAZ, in addition to the gap and its temperature dependence, Zheludev et
al. [82] observed the splitting of the Haldane gap due to single-ion anisotropy.
This observation determined D = 21 K. Other important materials parameters
that they measured were the spin wave velocity Cq = (1.30 ±0.05) x 105 m/s and
J = 125 K.

97
T (K)
Figure 6.2: DC x{T) of NINAZ. The results for powder NINAZ samples of Renard
et al. [62], Gadet et al. [81], and this work are given. Notice the discontinuity at
T = 255 K indicative of the structural transition.

98
Figure 6.3: Constant Q ~ 7r inelastic neutron scattering data for deuterated
NINAZ. Below the Haldane gap there is no magnetic scattering. At the gap,
there is a sharp peak characteristic of resonant scattering. The peak intensity
decreases in a manner related to the correlation length of each spin. The data
are from [82].

99
6.3 Experimental Studies
After the identification of NINAZ as a material with a large energy scale, we
made magnetic susceptibility measurements to compare our sample with those
of other groups. Since specimens made by our group and by others all displayed
similar paramagnetic tails (see Figure 6.2), we came to the conclusion that the
shattering processes produce numerous chain breaks as will be explained more
fully below. This realization of non-doped end-chain spins motivated the detailed
study of NINAZ which is discussed here.
6.3.1 Samples
Six types of samples were used in this study: a polycrystalline sample, a
powder, an ultrafine powder, and samples doped with Hg, Cd, and Zn. The
first three samples were all made without doping the material. The last three
samples were measured for comparison with the undoped samples. A polycrys¬
talline sample was a NINAZ single crystal that was cooled through the shatter¬
ing transition. Even though after shattering it is no longer a single crystal, it
remains an ordered conglomeration of microcrystallites with only a 5° mosaic
spread [82]. Therefore, the sample can be oriented with respect to the crys¬
tal axes. Two batches of NINAZ crystals were generated. Batch 1 was made by
Chou and Batch 2 by Ward. This distinction is the same for all samples discussed.
To increase the number of end-chain spins, two techniques were used to grind the
samples. Initial grindings used a pestle and mortar to produce a sample referred
to as powder. Subsequent grindings, using a standard ball mill as described in

100
Section 4.5, produced a sample referred to as ultrafine powder. To examine the
particle sizes of each pulverized sample, centripetal sedimentation as described in
Section 4.6, was performed. There were two powder samples characterized, one
from Batch 1 (ground by Chou) and the other from Batch 2 (ground by Ward).
Figure 6.4 shows that there is a small but observable difference in particle size
distribution of the two batches. Nevertheless, both samples have an average par¬
ticle size of approximately 5 pm. Two ultrafine powder samples were prepared
as well. The first sample (from Batch 1) was powdered before it was ground in
the ball mill. The second sample (from Batch 2) was placed directly into the
ball mill. Figure 6.5 shows the difference between the two samples. The particle
size distribution for the sample that was powdered before entering the ball mill
is peaked at approximately 0.5 pm, whereas the sample that was not powdered
had a particle size distribution broadly peaked around 1.5 pm. Nevertheless, this
slight difference in particle size distribution is not significant enough to produce
differences in the magnetic measurements.
One argument of this thesis is that free spins in the sample can be attributed to
end-chain spins. Therefore, to confirm that the paramagnetic contribution did not
originate from the extrinsic impurities in the samples, inductively coupled plasma
mass spectrometry was performed on all the samples. The observed magnetic
impurity limits are expressed in Table 6.1. The amount of 59Co impurities in the
*
system could not be determined because the Ni line was so large. Nevertheless,
Co impurities are not expected to exist in the starting materials, and none of the
processes should introduce any. In addition, the strength of the Ni line limited
the resolution of the Cu line as well.

% of sample
101
1 1 1 ' ' 1 1 1 â–  ' ' â–  1
........
_
0
-
o
â–  Batch 1
-
â– 
o Batch 2
â– 
0
-
â–  o..
-
° ■ " ■ 1
â– 
-
0 â– 
o
■ °
o
-
o
â– 
o â– 
â– o
OB
a
Cl
B
n
-
0
- .
â–  â–  1 â– 
1 1 1 1 I 1 1 L_
-
0 5 10 15 20
particle size (pm)
Figure 6.4: Powder NINAZ particle size distributions for a one sample ground by
Chou (Batch 1) and another ground by Ward (Batch 2). Though differences are
seen, the most probable particle size is approximately 5 /¿m for both batches.

% of sample
102
60
50
40
30
20
1 1 1 1 1 •—
—i ■ i 1—
â– 
-
â–  Batch 1
o Batch 2
â– 
-
o
o
■ ■ °
n " °
o
u
â– 
O o
A â–  v
O
_ o "
â–  _
â–  â–  â– 
â–  -
i i i i i .
_l i l
J
0
12 3 4
particle size (pm)
Figure 6.5: Ultrafine powder NINAZ particle size distributions. One sample was
ground from powder (Batch 1) and the other was ground from crystals (Batch 2).
The most probable particle size is slightly smaller for the sample ground from
the powder.

103
Element
Concentration
ssMn
< 20 ppm
56Fe
< 24 ppm
65 Cu
< 160 ppm
Table 6.1: Maximum concentration of magnetic impurities in any NINAZ sample.
The shattering transition has an important role in these experiments. There¬
fore, its origin was tested by examining materials with a slightly different chemical
composition. If the shattering transition was caused by ordering of the counter
ion, there is a possibility that changing the counter-ion could eliminate the shat¬
tering process. To test this idea, the material Ni(C3H10N2)2N3(PF6) was made.
Nevertheless, when the sample was immersed in liquid nitrogen, it shattered as
well. Further tests on this material were not made, but recent work by Monfort et
al. [158] agree with our results. Another material with a different organic ligand
will be discussed in the next section.
6.3.2 [Ni(C4H12N2)2(/X-N3)]n(C104)n
(l,2-diamino-2-methylpropane)azidonickel(II) perchlorate or [Ní(C4Hi2N2)2(£í-
N3)]n(C104)„, which we call Ni zig-zag, is closely related to NINAZ, so it will be
discussed in this chapter. The main purpose for studying this material was to
test it as a candidate for an S = 1 spin ladder. A spin ladder is a pair of chains
that are strongly coupled to each other but weakly coupled to other neighboring
pairs. However, the following discussion will show that Ni zig-zag is actually
a spin chain closely related to NINAZ. Therefore, this material lucidates the

104
structural properties of NINAZ. Ribas et al. [160] was the first to synthesize Ni
zig-zag and resolve its crystal structure (Figure 6.6). Furthermore, they measured
the magnetic susceptibility of a powder sample and characterized the results using
the numerical work of Weng [21]. If the sample was actually a spin ladder, subtle
discrepancies between the magnetic susceptibility and fits to linear chain theories
should exist. Therefore, our goal was to improve the measurement resolution
with the SQUID magnetometer to look for these discrepancies [161]. A close
look at the crystal structure (Figure 6.6) shows why the material might be a spin
ladder. The Ni sites are bridged by the azido ligand in a cis fashion. Therefore,
the next nearest neighbor sites might have a sufficiently strong dipolar interac¬
tion to cause the materials to behave like an S = 1 spin ladder. This situation
is similar, but not identical, to the coupling proposed for the 5=1/2 material
Cu2(C5H12N2)2Cl4 [162,163]. Unfortunately, there is no bond overlap and the
next nearest neighbor dipolar interaction is weak, so this material maps to a
linear chain. This determination was made via magnetic susceptibility measure¬
ments on a packet of single crystals oriented with the chains first perpendicular
and then parallel to the magnetic field. Figure 6.7 shows that the data for the
perpendicular orientation and a fit, using the high temperature expansions of
Jolicoeur given in Appendix A, are in excellent agreement. However, Figure 6.8
shows that this agreement breaks down below 60 K for the parallel orientation.
These features are suggestive of an approach to long-range antiferromagnetic or¬
der (with T/v ~ 10 K) consistent with the observations in other materials [164],
Nevertheless, the results of these fits provide J_¡_ = 25.7 ± 0.2 K, g± = 2.16T0.03,

105
Figure 6.6: The crystal structure of [Ni(C4Hi2N2)2(//—N3)]n(C104)n [160]. The
ligands that connect one Ni site to another are in the cis position forming a
zig-zag. There is no bond overlap between next nearest neighbor sites to form a
ladder.

X (memu/mol)
106
T (K)
Figure 6.7: x±(2K < T < 300 K) for [Ni(C4H12N2)a(/i-N3)] „(C104)„ H = 0.1 T.
The solid line is a fit to the numerical expressions of Jolicoeur as discussed in the
text.

107
T(K)
Figure 6.8: *N(2K < T < 300 K) for [Ni(C4H,2N2)2(Ai-N3}]„(C!04)n H = 0.1 T.
The solid line is a fit to the numerical expressions of Jolicoeur as discussed in the
text.

108
D/Jl = 0.033±0.004, J|| = 26.4±0.2 K, = 2.17±0.01, and D/J\\ = 0.0±0.1.
As mentioned Section 2.3.1, the fits require the use of Padé approximates. There¬
fore, the fits were performed using Maple V (release 4) to take advantage of its
Padé approximates routines. A non-linear least squares fitting routine using the
grid-search method [165] was used to semi-automate the process, but many man¬
ual parameter adjustments were required to minimize \2-
6.3.3 Macroscopic Measurements
As mentioned earlier, magnetic susceptibility measurements were performed
on polycrystalline samples of NINAZ to compare them with previously made
materials. There are noticeable differences in the shapes of the curves which will
be explained later, but the similar paramagnetic tails suggest the presence of
end-chain spins. The magnetic susceptibility of both a powder and single crystal
sample was measured down to T « 50 mK, using a standard mutual inductance
technique, to see if the pulverization process created more end-chain spins [156].
The sample was cooled by a homebuilt dilution refrigerator. This technique and
dilution refrigerator are described in detail in Reference [166,167]. Figure 6.9
shows no difference in the paramagnetic tails, suggesting that the powder size
is larger than the chain-length arising from the shattering process. To examine
shorter chains, the magnetic susceptibility for an ultrafine powder sample was
measured for 2K < T < 300 K using the SQUID magnetometer and is shown
in Figure 6.10. The increased paramagnetic tail indicates the presence of more
end-chain spins. Normally a Curie-law is fit to the paramagnetic tail to determine
the number of spins, but the paramagnetic tail is also affected by the magnetic

109
T(K)
Figure 6.9: x(50mK < T < 2K) of polycrystalline and powder NINAZ from
Batch 1 [156]. Notice that within the noise, there is no difference between the
two data sets, implying that the powdering process does not reduce the chain
length below that of the shattering process.

110
3
E
0
£
Figure 6.10: x(T) for powder and ultrafine powder NINAZ from Batch 1. The
ultrafine powder sample shows a increased paramagnetic tail when compared to
the powder sample. In addition, the paramagnetic contribution displays effects
even for T rsj J m

Ill
excitations on the chain. Therefore, an expression incorporating these features
for T < 0.2J, as well as a small contribution resulting from the 5 = 1 chain
components, is
X(T) = % f(T) + <¡\[ 1/2 (6.1)
where f(T) is some function, to be described later, dependent on the number of
thermally excited magnetic excitations and the second term is from Equation 2.4.
This expression leaves too many unknown parameters for a reliable determination
of the number of spins. Nevertheless, qualitative comparisons to the Monte Carlo
work of Yamamoto [24] need to be made. A comparison of the powder and
ultrafine powder data in Figure 6.10 shows a significant difference, even at T ~ J.
This difference is observed, to some degree, in the numerical work of Yamomoto
[24]. However, these numerical chains are at least an order of magnitude shorter
than our observed lengths, so quantitative comparisons are not possible. A similar
change is observed when comparing our measurements to those of Gadet et al.
[81], suggesting that their sample has shorter chain lengths.
Returning to the problem of determining the number of spins, it is clear that
an independent probe is needed, specifically one not affected by thermally excited
magnetic excitations nor the temperature dependence of the gap. Magnetic field
dependent magnetization studies at T = 2 K fulfill this requirement by putting
the sample in an environment such that T « A. The resultant data are shown
in Figure 6.11, and the solid lines are fits using

112
H (T)
Figure 6.11: M(H < 5T, 2K) for polycrystalline, powder, and ultrafine pow¬
der NINAZ samples [168]. Notice each curve shows a successive increase in the
number of end-chain spins.

113
M(H,T) = NAg^B [(1/2) N1/2B1/2(g»BH/kBT)
+NlBl{gnBH/kBT)), (6.2)
where Bi/2(g^BH/kBT) and Bi(gg,BH/kBT) are the Brillouin functions, and
Ni/2 and Ni are the concentrations of 5 = 1/2 and 5 = 1 spins, respectively [4].
Table 6.2 shows the results of the fits with g = 2.174 and T = 2.00 K as fixed
parameters and N\/2 and N\ as determined by the fit. These results demonstrate
Polycrystalline Sample
Powder
Ultrafine powder
iVl/2
1000 ± 60
2840 ± 50
6280 ± 50
Ni
20 ± 10
30 ±9
140 ±8
Table 6.2: Amount (in ppm) of 5 = 1/2 and 5 = 1 in the polycrystalline, powder
and ultrafine powder samples from the M(H,T = 2K) data (Figure 6.11) and
its analysis described in the text.
that the end-chain spins are predominately 5 = 1/2 and that only trace amounts
of 5 = 1, consistent with the presence of some free Ni2+, exist in any of the
samples. Consequently, we conclude all the end-chain spins are 5 = 1/2.
For comparison, the macroscopic properties of the doped samples were mea¬
sured as well. Figure 6.12 shows that the magnetic susceptibility is dominated
by paramagnetic spins. The inset provides a closer examination of the low
temperature regime by plotting 1/x vs. T. This examination shows a non-Curie
like susceptibility above T « 20 K. Nevertheless, below T « 20 K, all data
sets linearly approach zero, suggesting that below a certain temperature the tail
is affected only by free spins. Therefore, in order to characterize the number
and type of spins while ensuring only end-chain spin effects will be observed,

114
Figure 6.12: x(T) for doped NINAZ samples. The doping concentration for each
specimen is 0.5% of Cd, Hg, and Zn, respectively. Clearly, a strong paramagnetic
increase is observed. The inset shows by a plot of 1/x that there is more detail
to the lowest temperatures than a simple Curie law.

115
M{H,T = 2K) was measured. Figure 6.13 shows the data fit by Equation 6.2
in a manner similar to the undoped case, and the concentrations of spins are
given in Table 6.3. The results are astonishing. For example, the magnetization
Hg
Zn
Cd
Nl/2
11.6 ±0.2
31.3 ±0.1
7.0 ±0.2
Ni
0.54 ±0.03
0
0.24 ± 0.03
Table 6.3: Percentage of sample that is the 5 = 1/2 and 5=1 paramagnetic
contribution for the doped samples of NINAZ from the M(H,T = 2K) data
(Figure 6.13) and its analysis described in the text.
of a sample doped with only 0.5% Zn implies that approximately 1/3 of the Ni
sites are paramagnetic 5 = 1/2 entities. This analysis suggests that the dopant
enhances the shattering process, since the simple idea would be that the 0.5%
doping would simply add to the < 0.1% breaks from the shattering process. The
concentrations in Table 6.3 suggest average chain lengths of approximately 30,
17, and 6 sites for Cd, Hg, and Zn doped samples, respectively. If this analysis
is valid, then chains in the doped samples are clearly short enough to observe
interactions between end-chain spins mediated through the chain. In addition,
the Zn doped samples should show evidence that a majority of chains are shorter
than the correlation length. Furthermore, the size of the paramagnetic tails agree
with the tails observed in the numerical work of Yamamoto et al. [24] for short
chains. Another indication for chains shorter than the correlation length is the
destruction of the Haldane phase. No bumps or kinks that can be associated
with the Haldane gap are observed in the inset of Figure 6.12 for the Zn doped
sample, suggesting that the chain length is shorter than the correlation length.

M (emu/mol)
116
H (T)
Figure 6.13: M(H < 5T, 2K) for doped NINAZ samples. The doping concen¬
tration for each sample is 0.5% of Cd, Hg, and Zn, respectively.

117
However, for the Hg doped sample, where chains are supposedly a factor of 5
longer, a kink is observed around 20 K, consistent with a hint of the Haldane
gap. Nevertheless, caution should be taken when assigning any effects in these
samples to end-chain spins because a small amount of dopant causes a large in¬
crease in the number of paramagnetic spins, and neither the Cd, Hg, nor Zn
materials are a non-magnetic isomorph of NINAZ. Consequently, the simple ex¬
planation of the above features is that the material is not a chain system. Further
clarification of these points is needed before a definite statement can be made.
Some helpful information is provided by the ESR measurements, but careful
crystal structure refinement of the low temperature phases of doped NINAZ are
required. Unfortunately, this test is extremely difficult to perform as even the
low temperature phases of the pure material have not yet been identified.
6.3.4 Microscopic Measurements: Electron Spin Resonance
Macroscopic measurements alone are insufficient to confirm the existence of
the end-chain spins predicted by Affleck, Kennedy, Leib and Tasaki [13]. We used
the microscopic measurement of electron spin resonance (ESR) to provide the
needed additional information. Most of the work was performed with a resonant
spectrometer working at v = 9.25 GHz, and preliminary studies were performed
with a transmission spectrometer at 98 and 189 GHz. These frequencies avoid
transitions from other physical phenomena besides end-chain spins because the
smallest energy scale for the chains in NINAZ is D « 440 GHz.
Central to an ESR experiment is the absorption spectrum. The simplest spec¬
trum contains a single peak, but if there is an anisotropic magnetic or electric

118
environment, then subtle energy shifts may cause multiple peaks. In the measure¬
ments on NINAZ, the polycrystalline samples exhibited three peaks, the powder
samples exhibited a single peak, and the doped samples exhibited five peaks.
Therefore, a discussion of the origin of the observed peaks is provided before the
properties of the peak associated with the end-chain spins are discussed.
Figure 6.14 shows ESR lines at 4 K for polycrystalline, powder and ultra-
fine powder samples. The predominant feature is the central peak which is at¬
tributable to end-chain spins with an observed g value of 2.174. This result agrees
with g values of Ni2+ ions in a paramagnetic environment [169]. Further discus¬
sion of the central peak will be provided below. The other dominant feature in
Figure 6.14 is an additional peak in the polycrystalline sample, observed as a
subtle shoulder at fields slightly above center field, that is absent in the pow¬
der and ultrafine powder samples. This peak is attributed to a few neighboring
end-chain spins located close enough to each other to permit dipolar coupling.
To understand the behavior of each peak independent of the shape of the whole
spectra, the polycrystalline line was fit to a sum of standard spectral functions.
The best fit was provided by a sum of three Lorentzians as shown in Figure 6.15.
The existence of the peak to the left of the central line is debatable since the
line shape for end-chain spins is not well determined [57], but comparison with
the line shapes of doped samples, given below, suggests that this peak is real.
To provide information on the origin of these peaks, the g value of each peak
was measured as a function of magnetic field orientation. The magnetic field was
applied perpendicular to the chains and the sample was rotated 180° about the

119
Figure 6.14: Typical ESR lines for polycrystalline, powder and ultrafine powder
samples of NINAZ at T = 4 K [168]. The central peak is attributed to the end-
chain spins, and the subtle shoulder in the polycrystalline data is attributed to
a few neighboring end-chain spins located close enough to each other to permit
dipolar coupling, as is discussed in the text.

120
Figure 6.15: ESR line for a polycrystalline sample of NINAZ from Batch 1 of
NINAZ at T = 4 K. The fine line is the result of a fit to three Lorentzians
(dotted curves), as described in the text. The arrows indicate the maximum of
each Lorentzian.

121
chain axis in 15° increments, providing the spectra plotted in Figure 6.16. The
angle is measured from an arbitrary zero. If the ESR peaks produced a significant
shift, an appropriate fit would fix the zero. Clearly, there are no large shifts in
the ESR peaks as a function of rotation angle. The g value of each peak as a
function of angle is plotted in Figure 6.17. For the largest g value, there is no
observed shift within the noise of the data. For the other two peaks, the shifts are
< 1% of the resonance field, suggesting nuclear moments as their origin. The ab¬
sence of a shift from electronic moments indicates that the end-chain spins see an
effectively isotropic local magnetic environment, eliminating the possibility that
this line results from dipolar coupling to extrinsic sources. In addition, if the side
peaks originated from a staggered magnetization, as is observed in the spectra of
doped NENP [110,118], the line shape would change substantially when rotated.
Another property of the side peaks is that they are crystal specific and depen¬
dent on the number of thermal cycles through the shattering transition, as can
be seen by comparing the spectra in Figures 6.15, 6.16, and 6.20 — 6.23. These
facts indicate that the side peaks result from the shattering process. The reason
why corresponding spectral weight is not observed in the wings of the powder
or ultrafine powder line shapes is that the pulverizing processes homogenize the
sample, thereby drastically reducing the intensity of the peaks due to interactions
between neighboring end-chain spins. In summary, the only remaining explana¬
tion for the side peak is that it arises from a few neighboring end-chain spins
located close enough to each other to permit dipolar coupling.

122
2500 3000 3500 4000
H (G)
Figure 6.16: ESR spectra for a polycrystalline NINAZ sample as a function of
angle about the chain axis at T = 4 K. The sample is from Batch 2. Any shift
in the ESR peaks can be accounted for by interactions with nuclear moments.

123
0 (degrees)
Figure 6.17: ESR peak g values as function of angle in a polycrystalline sample.
The angle is measured about the chain axis, with an arbitrary zero as explained
in the text. Any slight shift with the 9 can be attributed to interactions with
nuclear magnetic moments. Notice there is no significant line shift with rotation.
The lines are guides for the eyes.

124
Another way to experimentally test the line shape of pure NINAZ is to com¬
pare it to the line shape of doped samples. Figure 6.18 shows the ESR spectra for
Hg doped NINAZ with a fit to five Lorentzians. The reason for choosing this fit
will be provided below. Several features not observed in the polycrystalline sam¬
ples are observed in the doped sample. First, doping has increased the number of
observed peaks in the spectra. This result emphasizes that the dopant affects the
crystal in a more complex way than simply breaking chains. One possible expla¬
nation of the additional lines is that the side peaks in the polycrystalline line are
split by an added dipolar or crystalline field. This explanation is supported by
the fact that peaks A and E are of nearly equal size and position away from the
center line. Furthermore, peaks B and D have similar intensities and are shifted
away from the center line by equal amounts. If the two pairs correspond to a
splitting, then each member of a pair should shift in a complimentary manner
to the other member when the direction of the applied magnetic field is changed
with respect to the local field. To test this explanation, the sample was rotated
about the chain axis which was oriented perpendicular to the magnetic field at
T = 4 K. Figure 6.19 shows the orientational dependence of g. Peak A shifts,
but peak E does not, suggesting that they are of different origins. The behavior
of peaks B and D is less definitive. Peak B shifts, as a function of 9, by amounts
above the scatter of the data. However, the deviations away from the g value at
9 = 0 for peak D may or may not be above the scatter of the data. Therefore,
the only conclusion that can be made from these rotation studies is that there
are several sources of local magnetic fields in the doped samples that are absent

125
H (G)
Figure 6.18: Hg doped NINAZ ESR spectrum at T = 4 K. Also included is the fit
to the five Lorentzian curves. The fit is the narrow line, and each dotted line is
an individual Lorentzian. An arrow indicates the maximum of each Lorentzian,
which is labeled as A, B, C, D, and E.

126
Figure 6.19: Hg doped NINAZ ESR peak g values as function of angle around
the chain axis. Each curve is labeled according to the corresponding peak in
Figure 6.18. The lines are included as guides for the eye. A description of the
shift of the lines with the largest g values is given in the text.

127
from the pure samples. This conclusion provides more evidence that the dopants
have a greater effect than simply breaking the chains.
Another interesting feature of Figure 6.18 is observed by comparing the size
of the resonance spectrum with a similar spectrum in the polycrystalline sample
(Figure 6.15). The same arbitrary units are used in both figures. The size (i.e.
the intensity) is the same to within at least an order of magnitude. However,
the M vs. H measurements described previously demonstrated a difference in
the number of spins by four orders of magnitude. This distinction between the
macroscopic and microscopic measurements can be explained by interactions of
the magnetic excitations on the chains with the end-chain spins. These interac¬
tions shift the ESR line beyond magnetic fields observable by our apparatus. A
full discussion of this effect is provided below. In summary, the doped samples
exhibit numerous ESR lines which may have a variety of origins. Nevertheless, a
significant central peak is present and is consistent with the 5 = 1/2 end-chain
peak in the pure samples. In addition, all the spectral weight attributable to
the number of end-chain spins obtained with M vs. H measurements is not ob¬
served. This result is consistent with the model that the interaction between
end-chain spins and the magnetic excitations has shifted it beyond the limits of
the experiment.
More information of the microscopic properties of the system can be obtained
by the properties of the central ESR peak. There are primarily three properties
used to quantify the interpretation of ESR peaks: the intensity, the area under
the data, and the line width. Typically, the area is related to the number of

128
paramagnetic spins in the system, but for Haldane gap systems, the number of
spins participating in the observed ESR line is related to the number of ther¬
mally excited magnetic excitations on the chain [57]. For this reason, the area
under the curve is only approximately proportional to the number of spins and
provides no better than an order of magnitude estimate when a calibration is
attempted. Furthermore, thermally excited magnetic excitations affect the line
width differently than the intensity which causes the area and the peak maximum
to be disproportionate [57]. Therefore, the intensity of the ESR spectrum should
be quantified by its maximum.
Qualitatively, the ESR signals for polycrystalline, powder, and ultrafine pow¬
der samples demonstrate an increase in intensity and line-width for a decrease
in chain length (Figure 6.14). To provide a quantitative analysis of these fea¬
tures, comparisons are made to the theory of Mitra, Halperin and Affleck [57]
as outlined in Section 2.4.1. Their most easily tested result is the temperature
dependent intensity. To test this prediction, we measured the temperature de¬
pendence of the ESR signal from 4 K up to the temperature where the signal
was unobservable for all the samples. Figures 6.20, 6.21, 6.22 and 6.23 show the
ESR lines for both batches of polycrystalline sample with H parallel and per¬
pendicular to the chains, respectively. Clearly, all lines show a temperature
dependence. Figures 6.24 and 6.25 show the temperature dependent intensity
for both batches of the polycrystalline sample with H parallel and perpendicular
to the chains, respectively. Theoretical fits of these data are not meaningful
because the error bars are so large, and the side ESR peak plays a significant

129
H (G)
Figure 6.20: T dependence of ESR lines for H _L to the chains of a NINAZ
polycrystalline sample from Batch 1.

130
Figure 6.21: T dependence of ESR lines for H || to the chains of a NINAZ poly¬
crystalline sample from Batch 1.

131
H (G)
Figure 6.22: T dependent ESR spectra for H || to the chains of a NINAZ poly¬
crystalline sample from Batch 2.

132
05
C
-Q
CO
; 10 -
4000
Figure 6.23: T dependence of ESR lines for HI to the chains of a NINAZ
polycrystalline sample from Batch 2.

133
T (K)
Figure 6.24: ESR I(T) for a NINAZ polycrystalline sample from Batch 1. Both
orientations of magnetic field are shown. The data are highly affected by the
side peaks, and therefore any assignment of differences between orientations is
questionable.

134
T(K)
Figure 6.25: ESR I(T) for a NINAZ polycrystalline sample from Batch 2. Both
orientations of magnetic field are shown. Again the data are somewhat affected
by the ESR side peak.

135
role. The differences between the data for Batch 1 and Batch 2 show the effect
of the side peak contribution. The increased mass of the powder and ultrafine
powder samples (64 mg in each case) greatly improves the signal to noise ratio,
and the brutal breaking procedures eliminate the side peaks. Figures 6.26 and
6.27 show the ESR lines as a function of temperature for the powder and the
ultrafine powder samples, respectively. Again for easier analysis, Figure 6.28
shows the temperature dependent intensity of the ESR lines. The solid lines show
fits to Equations 2.26 - 2.25, using the materials parameters given in Section 6.2.
The fitting results in average chain lengths of 1600 ± 50 sites (~ 0.9 //m) for the
powder and 920 ± 50 sites (~ 0.5 nm) for the ultrafine powder. Comparison with
the particle size analysis (Figures 6.4 and 6.5) shows that the pestle and mortar
grinding process does not reduce the average chain length below the length of the
polycrystalline domains [156], whereas ball milling produces chains of a length
consistent with the particle size.
The other major feature of an ESR spectrum is its line width. To quan¬
tify the line width, the full width at half maximum (FWHM) is measured from
Figure 6.14, to be 80 ± 10 G, 70 ± 1 G, and 100 ± 1 G, for the polycrystalline,
powder, and ultrafine powder samples, respectively. Clearly the line broadens as
NINAZ is pulverized. As described in Section 2.4.1, Mitra, Halperin, and Affleck
propose two mechanisms [57] which may influence the line width. First, bosons
changing energy levels could affect the line width, where this change in energy
is quantized in units of 5e as given by Equation 2.23. Using the materials pa¬
rameters cited previously, 6e « 1 kG for the powder sample and 3 kG for the

136
H (G)
Figure 6.26: T dependence of the ESR line for a powder sample of NINAZ from
Batch 1.

137
Figure 6.27: T dependence of the ESR line for an ultrafine powder sample of
NINAZ from Batch 1.

138
Figure 6.28: I(T) for powder and ultrafine powder NINAZ samples [168]. The
solid lines are fits to Equation 2.26

139
ultrafine powder sample. These estimates of 6e are two orders of magnitude too
large to explain the line widths shown in Figure 6.14. Nevertheless, this mech¬
anism could cause ESR lines at fields not shown in Figure 6.14. Figure 6.29
shows the energy level diagram as a function of magnetic field for three boson
levels in the powder sample. The arrows indicate all the allowed transitions. The
heavy arrow indicates the transition where the boson does not change energy
levels. The open-ended arrows indicate transitions involving the upper energy
level. The closed-ended arrows indicate transitions involving the lower energy
level. All arrows are of the length v = 9.25 GHz, i.e. the frequency of the mi¬
crowave radiation. If transitions from bosons changing energy levels exist, they
are either near 2 or 4 kG. However, since these processes involve the creation of
a boson on the chain as well as flipping the end-chain spin, they are second order
processes and should have a reduced intensity. Therefore, it is not surprising that
extended magnetic field sweeps through these fields provided no evidence of these
transitions nor did preliminary high field ESR experiments to be discussed below.
Consequently, similar to NENP [57], this mechanism has not been observed in
our study of NINAZ.
The other possible mechanism is a small change in the energy of a boson that
experiences a phase shift, 5(k). As explained in Section 2.4.1, S(k) is defined
by Equations 2.27 and 2.28 with the parameters k, the wave vector, and X>, a
constant of order 1, respectively. Starting with the energy given by the line widths
(Figure 6.14) and the wavevector (n = 1) for the appropriate chain lengths, our
experimental value of V is « 5 for the powder and « 1 for the ultrafine powder.

140
Figure 6.29: Energy level diagram for powder NINAZ showing three boson energy
levels. The heavy arrow indicates the transition where the boson does not change
energy levels. The open-ended arrows indicate transitions involving the upper
energy level. The closed-ended arrows indicate transitions involving the lower
energy level. All arrows are of the length u = 9.25 GHz the frequency of the
microwave radiation. A full description of the physical processes is given in the
text.

141
This analysis is in excellent agreement with the theoretical predictions and is
the first experimental evidence of magnetic excitations affecting the line width
of the end-chain spin. In other words, although typical line broadening processes
are still present, the major contribution to the powder and ultrafine powder line
widths is the interaction between the magnetic excitations and the end-chain
spins. Furthermore, Figure 6.30 shows the FWHM as a function of temperature
for the powder and ultrafine powder samples. For both samples, there are two
distinct behaviors as a function of temperature. The temperature independent
region below 8 K suggests that the excitations that interact with the end-chain
spins are due solely to quantum fluctuations. As the temperature is increased
above 8 K, the FWHM expands as more thermally excited magnetic excitations
are introduced to the chain as is expected by the theory of Mitra, Halperin,
and Affleck [57]. This transition of T* « 8 K is consistent with a picture of the
thermal excitation of magnetic excitations over a barrier of A. One might expect a
difference in T* between the powder and the ultrafine powder samples because the
different chain lengths cause different energies in the excited state. Unfortunately,
this temperature difference should be of the same order as the change in line width
between the powder and ultrafine powder sample (25 G « 3 mK) which is beyond
the temperature resolution of the measurement.
Regardless of the previous analysis, temperature independence alone is insuf¬
ficient to claim the observation of quantum phenomena. The following arguments
will eliminate the other possibilities. The simplest explanation for temperature in¬
dependence is a line width limited by instrumental resolution, but measurements

142
400
350
300
O 250
I 200
§
LL
150
100
50
~~1—
1
“I ' 1 1 1 '
r~
—' 1—
_
â– 
Ultrafine Powder
â– 
-
O
Powder
â– 
O
-
-
H
-
-
o
â– 
-
â– 
-
-
c
)
â– 
â– 
â– 
-
â– 
â– 
â–  o
-
-
o
1
o
1
o
1 . 1 . 1
i
. 1
I
4
6 8 10
12
14
16
T (K)
Figure 6.30: ESR FWHM vs. T for powder and ultrafine powder NINAZ from
Batch 1. Above ~ 8 K, the line width increases as the amount of thermally
excited magnetic excitations increases. Below ~ 8 K, the number of excitations
is limited by quantum fluctuations.

143
of DPPH show that the linewidth resolution limit is less than 2 G (Figure 4.8).
Another possibility is that the sample was no longer coupled to the thermometer,
but if this were true, then a temperature independent region would be observed
for /(T < 8K) (Figure 6.28). Finally, the line width could be limited by the
interaction of end-chain spins mediated by the chain, but to produce line widths
on the order of 100 G, a majority of the chains must be under a length of 100
sites [57]. Nearly all the chains in our samples are at least an order of magnitude
longer than this limit. The only remaining explanation for the temperature in¬
dependent region in Figure 6.30 is the quantum limit. Therefore, these linewidth
results confirm that the M vs. H measurements were performed in the quantum
limit.
Further information about this quantum limit is provided by the area under
the powder and ultrafine powder curves. As was mentioned in Section 2.4.1,
the area under the absorption peak is proportional to the local susceptibility.
Figure 6.31 shows the l/x data for powder and the ultrafine powder samples,
clearly demonstrating that the magnetic susceptibility is more complicated than
a simple Curie law. Nevertheless, Figure 6.32 shows that the ratio of the mag¬
netic susceptibility for the powder and the ultrafine powder samples is a constant.
Therefore, the magnetic susceptibility can be described by a Curie law multi¬
plied by some function, /(T), which considers the number of end-chain spins
not participating in the ESR line because thermally excited magnetic excitations
have excited them beyond the experimental limits. To extract f{T), a plot of
X • T vs. T is shown in Figure 6.33. In a manner similar to the line width in

144
1 •—
-I . 1 1 1 1 1 1 1 1-
—i—1—i—1—r
o
16
—
â–  Ultrafine Powder
^ 12
o Powder
o
(/)
'c
=3
-
4 8
CO
o
â– 
4
o
â– 
0
â–¡ i_
sS â–  â–  â– 
-I 1 1 i 1 i 1 ■ 1 »
â– 
_J 1 1 i c
O 2 4 6 8 10 12 14 16
T(K)
Figure 6.31: l/x vs. T for powder and ultrafine powder NINAZ. Clearly, a Curie
law is not followed.

145
T (K)
Figure 6.32: The ratio of the area under ultrafine powder data to the area under
the powder data for NINAZ as a function of temperature. The value is constant
suggesting that x(T) = Cf(T)/T as discussed in the text.

146
Figure 6.30, the data has a distinct change of slope near 8 K. Therefore, x{T)
can be described by a Curie law below 8 K.
To extend the search for side peaks due to magnetic excitations changing
energy levels when interacting with end-chain spins, high field ESR was per¬
formed at the National High Magnetic Field Laboratory, on the apparatus de¬
scribed in Section 4.3. Magnetic field sweeps were performed at frequencies of
v = 93.934 GHz and v = 189.866 GHz. Figures 6.34 and 6.35 are up and down
sweeps for 93.934 GHz, and Figures 6.36 and 6.37 are the up and down sweeps
for 189.866 GHz. None of the curves show any evidence for other transitions
besides the 5=1/2 line attributable to end-chain spins. Finally, a comparison
of the g values for both the high field and the X-band ESR measurements is
provided by Figure 6.38. As the frequency is increased, there is a decrease in the
g value. This behavior remains unexplained, and more data are needed to verify
it.
Note added in proof: Full field sweeps (0 — 14 T) for high field ESR at both
frequencies show no evidence for 5 = 1 spins. To quantify the resolution of this
experiment, the size of the noise was compared to the peak to peak height of the
ESR lines in Figures 6.34 — 6.37. At 93.934 GHz, if more than 100 ppm of the
sample were 5 = 1 spins, then a peak would have been observed. Since no peak
was observed, the existence of 5 = 1/2 end-chain spins is the only explanation
consistent with all the data.

147
1
—1 1 1 l—
~i—•—i—1 i 1 i ■ r
1
11
—
â–  â–  â– 
-
10
—
-
-
Ultrafine
-
9
Powder
â– 
-
8
-
"c/T
7
â– 
-
c
-
Z5
6
-
-
-Q
-
-
CO
5
-
â– 
1—
4
o
—
O o
â– 
-
3
Powder
o
-
o â– 
-
2
—
o
m
1
_i i i i_
° o
_l i 1 i 1 i 1 i L
-
4 6 8 10 12 14 16
T (K)
Figure 6.33: xCO • T for the powder and ultrafine powder samples of NINAZ
from Batch 1. The high temperature behavior is consistent with the number
participating spins controlled by thermally excited magnetic excitations where
the low temperature T < 8 K behavior is quantum limited.

148
Figure 6.34: NINAZ ESR spectrum at 93.934 GHz, increasing H. The samples
was from Batch 2. Only the 5 = 1/2 peak corresponding to the end-chain spins
is observed. The peaks observed around 3.35 T result from impurities in the
sample holder.

149
Figure 6.35: NINAZ ESR spectrum at 93.934 GHz, decreasing H. The sample
is from Batch 2. Only the 5 = 1/2 peak corresponding to the end-chain spins is
observed.

150
H (T)
Figure 6.36: NINAZ ESR spectrum at 189.866 GHz, increasing H. The sample
is from Batch 2. Only the 5=1/2 peak corresponding to the end-chain spins is
observed.

151
H (T)
Figure 6.37: NINAZ ESR spectrum at 189.866 GHz, decreasing H. The sample
is from Batch 2. Only the S = 1/2 peak corresponding to the end-chain spins is
observed.

152
Figure 6.38: g vs. v for ESR in a NINAZ powder sample from Batch 2 at T = 5 K.
The sharp decrease at the highest frequencies remains unexplained.

CHAPTER 7
THE 5 = 2 HALDANE GAP IN MnCl3(bipy)
Section 2.3.1 described how the 5 = 2 Haldane gap is small (i.e. A « 0.07J)
and highly affected by anisotropies. For our laboratory, the size of the gap is
not an issue since we are capable of measurements down to T « 400 /¿K. On
the other hand, the theoretically predicted bounds of the single-ion anisotropy
required to realize the Haldane state are so small that experimental realiza¬
tion of an 5 = 2 Haldane gap material was doubted. The properties of four
materials were tested as potential Haldane gap materials. Azidobis(pentant-
2,4-dionato)manganese(III), Mn(acac)2N3, and (l,10'-phenanthroline)trichloro-
manganese(III), MnCl3(phen), exhibit long-range order and were excluded from
further study. (N,N,-disalicylidene-2-hydroxy-propylenediamine)manganese(III),
Mn(salpn)OAc, and (2,2,-bipyridine)trichloromanganese(III), MnCl3(bipy), did
not exhibit long-range order. Furthermore, MnCl3(bipy) has a larger J value, so
it was chosen for further tests. These tests demonstrated that MnCl3(bipy) is
the first 5 = 2 Haldane gap material to be identified.
The first part of the chapter will discuss the structure and synthesis of the
aforementioned materials. The rest of the chapter will focus on MnCl3(bipy),
since it has properties consistent with the 5 = 2 Haldane gap. First, the macro¬
scopic measurement of magnetization, M, vs. magnetic field, H, studies will be
discussed, and then the results obtained from the microscopic probe neutron
scattering will be presented.
153

154
7.1S = 2 Quasi-linear Chain Materials
All of the materials studied have crystal structures that demonstrate their
linear chain nature. Each material, which will be discussed individually, was
grown by Brian H. Ward under the direction of Professor Dan R. Talham in the
Department of Chemistry at the University of Florida.
7.1.1MnCl3(phen)
The material, MnCl3(phen), is bridged by Cl atoms, and the chain is spaced
by a phenanthroline group, Ci2H8N2, as shown in Figure 7.1. Unfortunately,
the sample used for the structure refinement was twined, so only an approximate
structure was obtained. Nevertheless, the crystal structure was sufficiently refined
to confirm the existence of linear chains. Goodwin and Sylva [95] originally
measured x(T) in the range of 100 — 300 K and fit their results with a Curie law.
However, their work did not extend to low enough temperatures to identify the
linear chain behavior of the sample. Therefore, we extended the measurements to
T a; 2 K. Figure 7.2 shows the temperature dependent magnetization for a field
cooled and zero field cooled sample, and the difference between the two runs.
All three traces are consistent with the onset of long-range magnetic order at
approximately 22 K. Further work is in progress and will be reported elsewhere.
7.1.2Mn(acac)2N3
The material Mn(acac)2N3 has a crystal structure similar to NINAZ, discussed
in the previous chapter, with the azido bridge and the ligand spacing the chains.

155
Figure 7.1: Approximate crystal structure of MnC^phen) [170]. The quality of
the refinement was hampered by the twined crystal used for the X-ray studies.
Nevertheless, the refinement confirms the linear chain nature of the material.

156
T(K)
Figure 7.2: M(T) for MnCl3(phen). The measurement was performed on a
44.4 mg powder sample in a magnetic field of 0.01 T using the SQUID mag¬
netometer.

157
One notable difference between the two materials is that there are no counter
ions in Mn(acac)2N3. The crystal structure [171] is presented in Figure 7.3 which
shows a linear chain of Mn ions bridged by the azide ligands and spaced by the
acac groups, C5H702. These same researchers measured the magnetic suscepti¬
bility from room temperature to 4 K. They observed broad features characteristic
of a 1-D antiferromagnet at high temperatures, but at low temperatures, they
saw a few points consistent with a transition to long-range antiferromagnetic or¬
der. In an attempt to improve upon their work, we measured the temperature
dependence of the magnetization, in a magnetic field of 0.1 T, to test if the fea¬
tures attributed to long-range antiferromagnetic order were sample dependent.
Figure 7.4 shows a kink around 11 K in the magnetization of Mn(acac)2N3. This
signature is characteristic of a transition to long-range order and is completely
consistent with previous work [171]. No further work is presently planned for this
material.
7.1.3 Mn(salpn)OAc
The material Mn(salpn)OAc forms linear chains of manganese ions bridged
by an OAc group, CH3COO-, and spaced with salpn ligands, Ci7H15N203, as
shown in Figure 7.5. This crystal structure was solved by Bonadies et al. who also
performed temperature dependent magnetic susceptibility measurements to 4 K
with no evidence of long-range order [172], Figure 7.6 shows our magnetic suscep¬
tibility measurements to T « 2 K. The inset shows a peak characteristic of 1-D
antiferromagnetic correlations at 12 K, which suggests a J ~ 6 K. Furthermore,
no change between field cooled and zero field cooled measurements supports the

158
#
c
Figure 7.3: Crystal structure of Mn(acac)2N3 [171].

159
4 6 8 10 12 14 16 18 20 22
T(K)
Figure 7.4: M(T) of Mn(acac)2N3 The measurement was performed on a
40.68 mg powder sample in a magnetic field of 0.1 T using a SQUID magne¬
tometer.

160
Figure 7.5: Crystal structure of Mn(salpn)OAc [172].

161
assertion of no long-range magnetic order. Nevertheless, J is quite small, making
tests of the gap difficult. Therefore, tests for the gap were first performed with
MnCl3(bipy) which, as will be shown below, has a larger J. Further research on
Mn(salpn)OAc is planned for the future.
7.1.4 MnCl3(bipy)
The crystal structure of MnC^bipy) was first determined by Perlepes et al.
[96]. Figure 7.7 shows that the structure, as checked by Ward [170], is very similar
to MnCl3(phen). The main difference is that the organic ligand, which spaces
the chains, is bipy, CioH8N2, rather than phen. A careful examination of Figure
7.7 shows a slight staggering of the Mn ions. A comparison of the two structures
demonstrates that long-range magnetic order is not a simple function of the size
of the organic ligand that spaces the chains since MnC^phen), with the larger
organic group, exhibits long-range magnetic order while MnCl3(bipy), with the
smaller organic group, does not. Extensive work and analysis was performed
on MnCl3(bipy); therefore, the magnetic characterization will be discussed in
Section 7.2.
7.1.5 Synthesis
The materials Mn(salpn)OAc, Mn(acac)2N3, and MnCl3(phen) were synthe¬
sized according to the standard procedures in References [95,171,172], and no
further discussion will be provided here. On the other hand, MnCl3(bipy) was
prepared by the method of Goodwin and Sylva [95], with modifications by Ward
[170], Trimethylchlorosilane was added dropwise to a solution of

X (emu/mol)
162
0.10
0.08
0.06
0.04
0.02
0.00
Figure 7.6: x(T) of Mn(salpn)OAc. A 30.17 mg powder sample was measured in
a magnetic field of 0.1 T using the SQUID magnetometer. The solid black circles
are for a zero field cooled measurement. The inset expands the low temperature
region for both the field cooled and the zero field cooled runs. No difference is
observed between the two runs, suggesting no long-range order. Furthermore,
the broad peak reveals a J ~ 6 K.
0.10
_ 0.08
o
E
E
CD
— 0.06
0.04
"I"’ I' T-t | ■
o
§
§
â–¡ Zero Field Cooled
o Field Cooled
V
°°a0)
D°a0
an
1 i . i i 1 . i . . 1 i « i i 1
5 10 15 20 25 30
T (K)
• •
• • •
50
100 150 200 250
300
T (K)

163
Figure 7.7: Crystal structure of MnCl3(bipy) [96,170].

164
Mn120i2(CH3C00)i6(H20)4 • 2CH3C00H-4H20 in acetonitrile. The resulting
solution was filtered, and the filtrate retained. Next, a 2,2'-bipyridine in ace¬
tonitrile solution was layered on top of the filtrate. After seven days of undis¬
turbed growth, red crystals of MnC^bipy) were filtered out of the solution. The
MnCl3(bipy) material was grown first in a protonated form which was used in
the macroscopic measurements. For neutron scattering measurements, a fully
deuterated specimen is required to reduce the incoherent scattering, as described
in Section 4.4. Therefore, starting with 99.98% deuterated materials, several
batches of MnCl3(d-bipy) were grown to produce a total of 3 g of material. The
magnetic susceptibility as a function of temperature was measured for each batch
as a quality check. The typical increase from room temperature down to a value
of J was exhibited in all samples although in some samples the low temperature
paramagnetic tail became the dominant feature before the peak associated with J
was observed. Nevertheless, none of the batches exhibited evidence of long-range
magnetic order.
7.2 Macroscopic Magnetic Measurements of MnC^bipy)
The macroscopic magnetic measurements of MnC^bipy) provided the first
evidence of a Haldane gap in an S = 2 material. Similar to the measurements
of the other materials studied, the initial measurements were performed on the
SQUID magnetometer as described in Section 4.1. Figure 7.8 shows the tem¬
perature dependence of the magnetic susceptibility of a 2.4 mg packet of 90
oriented single crystals for an applied field of 0.1 T. The x(T) results show a

X (memu/mol)
165
0 50 100 150 200 250 300
T (K)
Figure 7.8: x(T) of MnC^bipy) [97]. A 2.4 mg packet of oriented single crystals
was sample used in the measurement which was performed in an applied field of
0.1 T using the SQUID magnetometer. The solid line is a fit to Equation 7.1,
where the details are given in the text.

166
broad peak near 100 K, anisotropy for T < 80 K, and a strong upturn at the
lowest temperatures. While the broad peak is the expected behavior for lin¬
ear chain Heisenberg antiferromagnets, the Curie-like increase may be associated
with impurities. The anisotropy is such that x± > X|| which is in contrast to the
Cr2+ compounds [164,173] where Xx < X||- Therefore, the anisotropy is not a
signature of long-range magnetic order. One possible explanation for this feature
is a slight staggered magnetization originating from the staggered Mn positions.
Further experimental work is needed to test this hypothesis. Since no explicit
expressions exist for x{T) of an S = 2 linear chain Heisenberg antiferromagnet
over a broad temperature range that includes anisotropy terms, we have fit the
data of Figure 7.8 to
X(T) = x(0) + C/T + Xlcha(S = 2, g, J, T), (7.1)
where Xlcha{S = 2,g,J,T) is given by Equation 2.12. Another possible ex¬
pression for Xlcha(S = 2,g,J,T) is given by a high temperature expansion of
Yamamoto [174]. Unfortunately, he only calculated two terms in the expan¬
sion which are not enough for a good fit below 7J, as shown in Reference [174].
Furthermore, to take advantage of the Padé approximates technique, at least one
more term is required. Therefore, Equation 2.12 is the best expression available
to date. Since this expression for XlchaÍ'S' = 2,g,J,T) is not expected to be
valid when significant anisotropy is present or in a region where a gapped phase
might exist (i.e. T < J), the fitting procedure focused primarily on the region
T > 80 K, with the exception that the Curie constant C was adjusted to the low

167
temperature data. The results of the fit, shown by the solid line in Figure. 7.8,
yield x(0) = 0.0±0.5 memu/mol, C = 47.5±0.5 memu K/mol, J = 34.8 ± 1.6 K
and g = 2.04 ± 0.04. The Curie constant could be explained by a small con¬
centration of impurity spins. However, we want to be careful about making this
assignment and trying to subtract this “Curie-tail.” For example, as previously
mentioned, we know that Xlcha{S = 2,g,J,T) is an inadequate description of
x(T) in this region. Nevertheless, it is noteworthy that various attempts to sub¬
tract a reasonable Curie-like contribution always give x(T) —» 0 as T —» 0. To
further explore the nature of the magnetic signal at the lowest temperatures,
standard 9 GHz ESR was performed on a packet of 5 oriented crystals from 4 to
60 K. The observed signal at 4 K is consistent with a concentration of approxi¬
mately 0.05 ±0.03% Mn2+ spins (S = 5/2, g = 2) that follow a Curie temperature
dependence as shown in Figure 7.9. The signal may also contain contributions
from trace amounts (at the ppm level) of S = 3/2 and 5=1/2 extrinsic impuri¬
ties. However, since the concentration of ESR visible spins is more than an order
of magnitude smaller than needed to explain the static susceptibility data, we
consider isolated Mn3+ ions not in the chain environment and 5=1 end-chain
spins [35,37] as the most likely sources of the low temperature behavior.
Since the magnetic susceptibility reveals no evidence for long-range magnetic
order in MnCl3(bipy), the next step is to test for the existence of the Haldane gap.
To this end, field dependent magnetization measurements were performed at the
lowest attainable stable temperature on the SQUID magnetometer. The mag¬
netization for the sample oriented both perpendicular and parallel to the chains

X (arb. units)
168
T(K)
Figure 7.9: 9 GHz ESR signal intensity vs. T for MnCl3(bipy). The solid line
shows, that within the error bars, the data follow a Curie law. The sample
consisted of 5 oriented single crystals providing a total mass of approximately
500 /ig.

169
(Figure 7.10) shows strong backgrounds consistent with free spins that saturate
at high magnetic field but no clear evidence of a Haldane gap. Nevertheless, if
the two sets of data are subtracted from each other, a small hint of a thermally
smeared Haldane gap appears (inset of Figure 7.10), but to obtain convincing
evidence, measurements at lower temperatures were required.
The cantilever magnetometer at the National High Magnetic Field Laboratory
in the top-loading dilution refrigerator was a clear choice for this measurement.
The sample was mounted on the cantilever with a small amount of vacuum grease.
The cantilever and sample were inserted into a top loading dilution refrigerator to
run at a temperature of 30 mK as described in Section 4.2. In an attempt to max¬
imize the signal, initial sweeps were made on three single crystals. Unfortunately,
the magnetic response of the sample was so strong that the cantilever was driven
into a non-linear region at fields close enough to the critical field to obscure any
gap features, as evidenced by the continuous curving of the data from point A to
point B as shown in Figure 7.11. Furthermore, point B labels the feature charac¬
teristic of the cantilever behaving so nonlinearly as to enter a “blocked” region. To
circumvent this problem, a second run was performed on a single crystal. Figure
7.12 shows that the resultant data were dominated by a diamagnetic background.
Nevertheless, there is a discontinuity between 1 and 2 T, where the diamagnetic
contribution becomes less pronounced, suggesting that a magnetic state above the
Haldane gap is competing with the background. Finally, two single crystals were
used and provided data that were dominated by neither background nor nonlin¬
ear effects. Nevertheless, a diamagnetic term had to be subtracted from the data.

170
Figure 7.10: SQUID M vs. H measurement of MnCl3(bipy). H was oriented
both parallel and perpendicular to the chain axis at T = 2 K. The inset shows
AM = Mx - M,|.

171
H (T)
Figure 7.11: Raw M vs. H data for three oriented single crystals of MnC^bipy)
measured with the cantilever magnetometer. Notice the continuous curve be¬
tween points A and B showing non-linear behavior over the whole field region.
Furthermore, point B labels the characteristic feature of a blocked region caused
by extreme nonlinearities in the system.

M (arb. units)
172
8
6
4
2
0
0 2 4 6 8 10 12 14 16 18
H (T)
Figure 7.12: Raw M vs. H data for one crystal of MnCl3(bipy) using the can¬
tilever magnetometer at T = 30 mK. The data are clearly dominated by a strong
diamagnetic background. Nevertheless, the discontinuity between 1 and 2 T is
suggestive of a Haldane gap.

173
H (T)
Figure 7.13: M vs. H data for two oriented crystals of MnC^bipy) from the
cantilever magnetometer at T = 30 mK [97]. A diamagnetic background was
removed by the procedure described in the text. Data are presented for magnetic
field oriented both parallel (||) and perpendicular (_L) to the chain. For each
orientation, a critical field is observed providing evidence of the Haldane gap in
an S = 2 antiferromagnetic chain. The inset shows the data for Hi. chains to 16
T. This trace shows no bumps or kinks that would indicate non-Haldane phase
behavior. A full discussion as to why these features can only be associated with
a Haldane gap is given in the text.

174
Since the region below the critical field should exhibit a constant magnetization,
any low field slope could be assigned to the diamagnetic background alone. A fit
to all the 30 mK data runs using the two crystals provides a diamagnetic term
of Md{H) = —2.968H, where the units are the same arbitrary units given in
Figure 7.13. The consistency of this diamagnetic correction for the two crystal
orientations suggests that this term is independent of the sample since it does
not change scale like the properties associated with the Haldane gap. With this
diamagnetic term removed, Figure 7.13 clearly shows evidence of critical fields
for magnetic fields applied both parallel (Hcy = 1.2 ± 0.2 T) and perpendicular
(Hcx = 1.8 ± 0.2 T) to the magnetic fields. These critical fields are the first evi¬
dence for a Haldane gap in an S = 2 material. Using the values for these critical
fields, Equations 2.14 and 2.15, and g — 2.04 ±0.04, the two gaps are determined
to be A|| = 3.7±0.9 K and Aj_ = 1.6±0.3 K. These orientationally dependent gap
values, S = 2, k = 2, and Equations 2.16 and 2.17 yield an isotropic Haldane gap
of A = 2.3 ±0.8 K and a single-ion anisotropy of D = 0.3 ±0.1 K. For comparison
to the theoretical predictions, the gap and single-ion anisotropies are best written
in units of J. For MnCl3(bipy), A/J — 0.07 ± 0.02 and D/J = 0.010 ± 0.003,
when J = 34.8 ±1.6 K. The gap value is in excellent agreement with the numer¬
ical predictions [34,35,37-39], and the value of the single-ion anisotropy places
MnCl3(bipy) inside the Haldane phase region (see Figure 2.9). Further compari¬
son with other regions of this diagram show that the Haldane region is the only
one consistent with all the experimental data. First, the XY phase is gapless, so
the existence of critical fields demonstrates that the system is not in this region.

175
Second, if the system was in the large D phase, critical fields would be observed in
both orientations, but they would be much farther apart in magnetic field. Third,
if the material was in a long-range antiferromagnetically ordered state, one of the
traces would remain zero until a spin-flop field Hsf was achieved [54,58]. From
standard theory [54],
Hsf = —V4SDJ. (7.2)
9»b
Therefore, assuming the single-ion anisotropy is the value provided from the Hal¬
dane analysis, Hsf ~ 14 T. Clearly, this critical field would have been observed
in the inset of Figure 7.13, and if D was any larger, evidence of long-range order
would have been observed in the SQUID results. The last alternative explanation
for discontinuities in M(H) is an effect resulting from saturation of the chain.
The saturation field is given by Hs = gg,BSJ, which for MnCl3(bipy) is 210 T.
Clearly, this field is too high to cause an effect at 2 T. None of these alternative
explanations are consistent with the experimental data. Therefore, magnetization
as a function of magnetic field provides the first confirmation of the Haldane gap
in an S = 2 system.
7.3 Low Temperature Crystal Structure: Preparation for Microscopic Measurements
To provide a direct measure of the Haldane gap in MnC^bipy), a microscopic
probe is required. The microscopic probe chosen for this work was inelastic
neutron scattering which has the added benefit of providing information on the
dispersion and the dynamics of the Haldane phase. However, in order to know
the proper momentum transfer, Q, the reciprocal lattice vectors must be known

176
precisely. Therefore, neutron diffraction was used to measure the low temperature
crystal structure, thus determining the reciprocal lattice vectors. In addition, the
background and temperature dependence of the diffraction pattern provide a test
of the deuteration and a check for structural transitions, respectively.
Since large single crystals are not available, Rietveld refinement on a powder
sample was required. The sample was a 1 g conglomeration of microcrystalites,
packed in a vanadium can in a He atmosphere to provide a means of thermal
contact between the sample and the displex used to cool the sample. Data were
acquired on the HB-4 powder diffractometer associated with the High Flux Iso¬
tope Reactor at Oak Ridge National Laboratory. Figure 7.14 shows the powder
diffraction pattern and final refinement at 11 K for MnCl3(d-bipy). The refined
structure around a Mn site is shown in Figure 7.15, and the quality of the refine¬
ment is characterized by a reduced x2 of 3.4. To arrive at the final result, suc¬
cessive refinements on groups of approximately ten parameters were performed.
After each refinement step, the physical validity of the crystal structure near the
Mn site was checked. During this process, there were several local minima in
the least squares hyperspace, identified by a carbon or deuterium atom that was
shifted unphysically. The procedure of letting the parameters of one of the C or
N atoms and all of its nearest neighbors, except the deuterium atoms, vary, and
subsequently varying each C with its corresponding deuterium atom, massaged
the least squares hyperspace to avoid these local minima. This procedure greatly
reduced the difference between the powder pattern and the calculated pattern
at several of the Bragg peaks. Further refinement would require the next order

Counts X10
177
WlfyjpiwJ
0.2
0.4
0.6
0.8
1.0
1.2
20(deg xlO2)
Figure 7.14: Powder diffraction pattern, Rietveld refinement, and difference plot
for MnCl3(d-bipy) at T = 11 K with neutrons of wavelength A = 1.4997 Á.

178
C\ Cl
Figure 7.15: Crystal structure near a Mn atom in MnCl3(d-bipy) at 11 K, deter¬
mined by Rietveld refinement.

179
effect to appropriately tweak the deuterium atoms slightly out of the plane of the
bipyridene group. Nevertheless, the present refinement is more than sufficient to
determine the low temperature lattice parameters. The 11 K unit cell parame¬
ters are a = 8.0872 ± 0.0002 Á, b = 16.1394 ± 0.0006 Á, c = 9.5517 ± 0.00024 Á,
7 = 109.469 ± 0.002°, and the distance between Mn sites is 4.80698 ± 0.00008 Á.
A full presentation of the low temperature crystallographic data are given in
Appendix B.
Besides the low temperature crystal structure, other useful information is pro¬
vided by the neutron diffraction experiments. First, they demonstrate that the
sample was fully deuterated. The negligible incoherent scattering background
(low angles in Figure 7.14) qualitatively shows the absence of hydrogen in the
system. Furthermore, initial refinements were performed with additional param¬
eters to account for a small concentration of hydrogen in the system. As these
refinements progressed, this concentration tended to zero within its uncertainty.
Therefore, any hydrogen contribution was deemed negligible and excluded from
further refinements. The uniformity of the powder was tested in a similar man¬
ner. The preferred orientation parameters were allowed to vary during early
refinement steps. Successive refinements demonstrated that this effect is negli¬
gible. Therefore, it was excluded from the final refinement. Several quick scans
(30 minutes in duration) were performed between 11 K and room temperature
to examine the temperature dependence of the lattice parameters. These data
provide another test for dimerization or other structural transitions which might
open a gap in the magnetic excitation spectrum. Figures 7.16 - 7.20 show the

180
temperature dependence of the lattice parameters and the volume as a function
of temperature. The absence of any discontinuities confirms that the chains do
not dimerize or experience any other structural transitions.
7.4 Inelastic Neutron Scattering Measurements
With the Haldane gap macroscopically confirmed and the low temperature
lattice constants determined, the next step is to perform the microscopic mea¬
surement of inelastic neutron scattering to directly measure the gap. Time of
flight was the measurement technique chosen so that the signal to noise ratio
could be increased by adding all detectors corresponding to Q values near the
antiferromagnetic point. This technique should provide a sufficient signal to ob¬
serve the Haldane gap. Furthermore, comparison with scattering from detectors
that correspond to Q near zero can provide insight on the Q dependence of the
Haldane gap. Therefore, Figure 7.21 was calculated to provide guidance as to
which detectors to sum. The detectors corresponding to angles between 23° and
38°, represented by the area between the dashed and the solid lines in Figure
7.21, should provide the best resolution of the gap.
The instrument chosen was the time of flight spectrometer (TOF) on the cold
source at the NIST reactor discussed in Section 4.4. The 3 mg sample consisted
of three packets of powder wrapped in aluminum foil and mounted in an alu¬
minum can flushed with He to remove any air and provide thermal contact to the
cryostat. A hefty humdinger of a homemade 3He cryostat, appropriately named
“Big Blue,” was used to cool the sample to approximately 400 mK. To remove

181
Figure 7.16: a vs. T for MnCl3(d-bipy).

182
T(K)
Figure 7.17: b vs. T for MnCl3(d-bipy).

183
T (K)
Figure 7.18: c vs. T for MnCl3(d-bipy).

184
Figure 7.19: 7 vs. T for MnCl3(d-bipy).

185
T (K)
Figure 7.20: V vs. T for MnCl3(d-bipy).

E (meV)
186
Figure 7.21: Estimated S — 2 dispersion for MnC^bipy) and theoretical TOF
spectrometer behavior. The heavy solid line represents the expected dispersion
curve. The dashed line, thin solid line, and dotted line represent the Q-E curves
for angles of 23°, 40°, and 50°, respectively.

187
background scattering, Cd shielding was placed on the outside of the vacuum
can in locations that should not have blocked any of the detectors. Data were
collected on the sample for 40 hours at this temperature using neutrons with a
wavelength of 6.0 Á. To be able to compare groups of detectors, the response of
these groups had to be normalized against a standard. A good feature of the
scattering function to compare for the normalization is the incoherent scattering
peak. Consequently, vanadium is an appropriate normalization standard as it is
nearly a pure incoherent scatterer. A setup for the normalization was made of
a 1/4 inch diameter vanadium rod with precisely positioned Cd shields, placed
in an aluminum vacuum can. Figure 7.22 reveals little difference in the incoher¬
ent peaks between the different groups of detectors for the vanadium standard.
However, for the MnCl3(d-bipy) sample, Figure 7.23 shows that the incoherent
scattering peak is significantly smaller for the detectors spanning 23° to 38° when
compared to other regions. To quantify the difference between the groups of de¬
tectors, the area under the inelastic scattering peak is shown in Table 7.1 for both
the vanadium standard and the MnCl3(d-bipy) sample. Furthermore, the ratio
between the sample and the standard is also given. For a proper normalization,
Detector
Grouping
V
MnCl3(d-bipy)
MnCl3(d-bipy)/V
118-135°
0.336
0.632
1.88
o
O
in
1
o
0.325
0.440
1.35
23 - 38°
0.258
0.179
0.694
Table 7.1: Normalization constants for three groups of detectors. Notice only
the 23 — 38° range is less than one, indicating problems with the measurement
as discussed in the text.

188
Figure 7.22: Incoherent scattering from a vanadium standard for several groups
of detectors on the NIST TOF Spectrometer. The lines are guides for the eye.

189
E (meV)
Figure 7.23: Incoherent scattering from the MnCl3(d-bipy) sample for several
groups of detectors on the NIST TOF spectrometer. The lines are guides for the
eye.

190
the ratios should be consistently greater or less than one. The 23 — 38° detectors
for the MnCl3(d-bipy) are inconsistent with the other detectors. Therefore, these
detectors were probably shadowed by the cadmium shielding. Nevertheless, the
range of detectors that covers the angles of 40° to 50° degrees provides hints of
the Haldane gap. These detectors are not physically excluded from observing
some contribution from the gap since the powder sample provides a spherical
average of resultant Q values, which smears the boundary regions of Figure 7.21.
For comparison, Q values distant to the antiferromagnetic point, measured by
detectors which represent angles of 118° to 135°, were checked as well. Figures
7.24 and 7.25 show the scattering intensity for Q values near and distant to the
antiferromagnetic point, respectively. Figure 7.24 shows a feature at ~ 0.3 meV
which is consistent in energy with the Haldane gap. To test this peak for statis¬
tical significance, a sum of two Gaussian functions, one function for the inelastic
contribution and the other for this subtle feature, was fit to the data as rep¬
resented by the heavy curve in Figure 7.24. The uncertainty in the curve is
represented by the lighter lines above and below the heavy line. Figure 7.25 does
not show any increased scattering at ~ 0.3 meV which is consistent with the
Q dependence of the Haldane gap discussed in Section 2.3. Finally, data were
collected at T = 10 K in order to confirm that the ~ 0.3 meV feature disappears
at temperatures well above the Haldane gap. Figure 7.26 shows no statistically
significant features consistent with the Haldane gap. To quantify the statisti¬
cal significance of the 0.3 meV feature, an F-test was performed comparing the
variance of a fit with two Gaussian peaks to a fit with one Gaussian peak for

I (arb. units)
191
E (meV)
Figure 7.24: Inelastic neutron scattering intensity at T = 0.4 K and Q « 7r from
MnCl3(d-bipy). The heavy line shows a best fit to two Gaussians suggesting
magnetic scattering consistent with the Haldane gap. The thin lines show the
uncertainty in the fit. Further discussion of the statistical significance of the
0.3 meV peak is given in the text.

192
Figure 7.25: Inelastic neutron scattering intensity at T = 0.4 K and Q MnCl3(d-bipy).

I (arb. units)
193
Figure 7.26: Inelastic neutron scattering intensity near Q = 7r at T = 10 K from
MnCl3(d-bipy).
0.4 K data. Attempts were also made for the 10 K data, but the non-linear least
squares fitting routine would not converge with a finite second peak. Unfortu¬
nately, the noise in the measurement is too strong to provide an excellent test, as
is demonstrated by the fact that F is close to one. For the 0.4 K data, F = 1.05
which means the 0.3 meV feature exists above the noise with a 65% confidence
level. Clearly, this analysis indicates that better signal to noise is needed to con¬
firm the Haldane gap, but there is a hint that the feature exists. In summary, the
inelastic neutron scattering data for Q values near the antiferromagnetic point
reveal a subtle shoulder that is consistent with the microscopic presence of the

194
Haldane gap. Furthermore, comparisons between the scattering intensity in the
region of ~ 0.3 meV, for Q values near to and far from the antiferromagnetic
point, and for temperatures well below and well above the gap reveal behavior
consistent with a Haldane gap. Nevertheless, the signal to noise is not sufficient
to model the magnetic scattering according to theoretical models for S(Q, u>) and,
therefore, further work remains in progress.

CHAPTER 8
CONCLUSIONS
The experimental work in this thesis has covered unique aspects of three
integer spin antiferromagnetic chains. Chapters 5, 6, and 7 described the results
on TMNIN, NINAZ, and MnC^bipy) respectively. This chapter will review
the major results for each material and provide suggestions for future scientific
studies.
8.1 TMNIN
Measurements of the magnetization as a function of magnetic field on the
S = 1 antiferromagnetic chain TMNIN revealed critical fields (see Figures
5.3 — 5.7) corresponding to A = 3.5 ± 2 K and D/ J = 0.06 ± 0.03, where
J = 10.11 ±0.05 K. Therefore, this material not only exhibits a Haldane gap,
but also is very close to a pure Heisenberg model. However, high field ESR
spectra [154,155] possess features that have been associated with D/J « 0.5.
Section 5.3 described the comparisons between the results from magnetization
and the ESR studies and introduced a possible solution to the discrepancy,
namely the presence of two slightly different magnetic sites, similar to the case
of NENP [98]. Although the results from several experiments are consistent
with the proposed picture, an experiment is needed to unambiguously confirm it.
195

196
Investigations on the NMR line shape at T < A may resolve this issue by pro¬
viding the information about the local magnetic environment.
8.2 NINAZ
The 5 = 1 material NINAZ has been an excellent model system to test end-
chain spin effects for several reasons. Firstly, J « 125 K is sufficiently large
so fully developed quantum properties can be tested at T — 4 K. Secondly, the
material has a structural transition which naturally causes chain breaks. Thirdly,
our experimental tests demonstrated that more chain breaks can be introduced
by pulverizing the material. Magnetization measured as a function of magnetic
field (see Figure 6.11) not only provided macroscopic evidence of this increase,
but also demonstrated that the end-chain spins are 5 = 1/2.
Electron spin resonance investigations provided additional information on the
system. Firstly, the existence of the X-band ESR signal is another demonstra¬
tion that the end-chain spins are 5 = 1/2. Secondly, the temperature depen¬
dence of the ESR intensity (see Figure 6.28) was fit to the expressions of Mitra,
Halperin, and Affleck [57], providing a microscopic measurement of the chain
length. Comparisons of this measurement to the macroscopic particle size distri¬
butions in Figures 6.4 and 6.5 show that the characteristic particle size for the
ultrafine powder and the powder are smaller and larger than the characteristic
polycrystalline chain length, respectively. Finally, the line width of the powder
and ultrafine powder samples shows the first evidence of an interaction of the
magnetic excitations on the chain with the end-chains spins. Furthermore, by

197
comparison with the theory of Mitra, Halperin, and Affleck explained in Section
2.4.1, the interaction does not cause the magnetic excitations to change energy
levels.
Another method used to study the end-chain spins in NINAZ was to com¬
pare the response of pure and doped samples. Magnetization as a function of
magnetic field (see Figure 6.13) for samples doped with 0.5% of Cd, Hg, and Zn
suggests as much as 1/3 of the sample consists of paramagnetic 5 = 1/2 spins. On
the other hand, ESR resonance studies observed quantities of spins four orders
of magnitude smaller. This result suggests that the majority of the end-chain
spin spectral weight has been shifted to fields inaccessible by the X-band spec¬
trometer. To confirm this assumption, high field ESR measurements should be
performed on the doped samples. However, since the doping has caused such a
large increase in the number of paramagnetic spins observed in the system, the
consistency of the crystal structure with the structure of the pure samples is ques¬
tionable. Therefore, before any other experiments are attempted on the doped
samples, careful X-ray studies should be performed to ensure that the system is
still NINAZ.
8.3 MnCl3(bipy)
The first identification of an 5 = 2 Haldane gap was provided by our magne¬
tization measurements at T = 30 mK in MnCl3(bipy). Studies with the magnetic
field applied both parallel and perpendicular to the chains exhibited critical fields
that provide A = 2.3 ± 0.8 K and D/J = 0.010 ± 0.003, where J = 34.8 ± 1.6 K.

198
This assignment resulted from a careful elimination of other possible sources
(e.g., spin flops, dimerizations, large D transitions) of critical fields. Preliminary
attempts to find the Haldane gap using inelastic neutron scattering (see Fig¬
ure 7.24) provide hints of magnetic scattering at an energy consistent with the
magnetization analysis, but further measurements are needed to confirm its exis¬
tence. Besides additional inelastic neutron scattering studies, other microscopic
measurements can provide useful information. High field ESR at temperatures
below the gap can map out the energy vs. magnetic field diagram similar to the
study performed in NENP [19,99,100,107]. Finally, NMR can provide another
identification of the gap by a measurement of T\ vs. T, and the line shape would
check for a staggered magnetization similar to NENP [98].

APPENDIX A
High Temperature Expansions
This appendix contains the high temperature series expansions of Jolicoeur
[51] performed on the 5=1 linear antiferromagnetic chain described by the
Hamiltonian
n = J'£Si-Si+i+D(Sz)2. (A.l)
When a Padé approximate method is used to evaluate the series, the fit is reliable
down to T/J = 0.2. In this thesis, Maple V (release 4) was used for the fits as
it has a robust Padé approximate routine and numerical precision can be kept to
as large as necessary, which is crucial in intermediate steps.
A.l Specific Heat
The series for the specific heat is
C(T) = j:an/(3"n!)(l/7T (A.2)
where the term an is a polynomial in D, and T is measured in units of J. The
coefficients an are:
199

200
1/T-order : 0
D-order :
0 0.00000000000000000000000000000000000
1/T-order : 1
D-order :
0 0.00000000000000000000000000000000000
1 0.00000000000000000000000000000000000
1/T-order : 2
D-order :
0 24.000000000000000000000000000000000
1 0.00000000000000000000000000000000000
2 4.0000000000000000000000000000000000
1/T-order : 3
D-order :
0 108.00000000000000000000000000000000
1 0.00000000000000000000000000000000000
2 0.00000000000000000000000000000000000
3 12.000000000000000000000000000000000

201
1/T-order : 4
D-order :
0 -4320.0000000000000000000000000000000
1 0.00000000000000000000000000000000000
2 -864.00000000000000000000000000000000
3 0.00000000000000000000000000000000000
4-72.000000000000000000000000000000000
1/T-order : 5
D-order :
0 -48600.000000000000000000000000000000
1 -0.68649976253089185231777556444485206E-30
2 -7199.9999999999999999999999999999990
3 -7200.0000000000000000000000000000003
4 0.00000000000000000000000000000000000
5 -600.00000000000000000000000000000000
1/T-order : 6
D-order :
0 1496880.0000000000000000000000000000
1 0.29209713519247226294134809815478875E-28
2 534599.99999999999999999999999999995
3 -25919.999999999999999999999999999974

202
4 29159.999999999999999999999999999996
5 0.00000000000000000000000000000000000
6 1260.0000000000000000000000000000000
1/T-order : 7
D-order :
0 32720436.000000000000000000000000000
1 0.32674430282633398670106147674185958E-25
2 7810991.9999999999999999999999999345
3 4413528.0000000000000000000000000430
4 571535.99999999999999999999999998882
5 920808.00000000000000000000000000098
6 0.00000000000000000000000000000000000
7 37044.000000000000000000000000000000
1/T-order : 8
D-order :
0 -845474112.00000000000000000000000000
1 -0.64116053038295958919011396159358090E-22
2 -459188351.99999999999999999999985512
3 51383807.999999999999999999999882993
4 -57516479.999999999999999999999956877
5 5322239.9999999999999999999999926599

203
6 2346624.0000000000000000000000004640
7 0.00000000000000000000000000000000000
8 53424.000000000000000000000000000000
1/T-order : 9
D-order :
0 -32830719120.000000000000000000000000
1 -0.38819088759895519517368477375382480E-19
2 -11557017791.999999999999999999903733
3 -4241030400.0000000000000000000899849
4 -1171532159.9999999999999999999586511
5 -1276928064.0000000000000000000099514
6 -47589119.999999999999999999998804783
7 -110014848.00000000000000000000005612
8 0.00000000000000000000000000000000000
9 -2870640.0000000000000000000000000000
1/T-order : 10
D-order :
0 675493966080.00000000000000000000000
1 0.11137744414028595129240481140894957E-17
2 529920160199.99999999999999999714654
3 -94583375999.999999999999999997182494

204
4 107062397999.99999999999999999858224
5 -20270865599.999999999999999999602390
6 929912399.99999999999999999993734908
7 -1028764799.9999999999999999999948383
8 -1088688600.0000000000000000000001721
9 0.00000000000000000000000000000000000
10 -18356220.000000000000000000000000000
1/T-order : 11
D-order :
0 46217135541060.000000000000000000000
1 0.74947242356798746358786274640036187E-17
2 22067612479440.000000000000000071546
3 5621958274679.9999999999999997933416
4 3317138256960.0000000000000002036914
5 2471736144239.9999999999999999015087
6 121897781280.00000000000000002589146
7 297137309039.99999999999999999625685
8 2359296720.0000000000000000002773223
9 12122268839.999999999999999999991853
10 0.00000000000000000000000000000000000
11 253061820.00000000000000000000000000

205
1/T-order : 12
D-order :
0 -658155004927200.00000000000000000000
1 0.98656140527047102541390713938654208E-13
2 -774510315147360.00000000000016634030
3 212199802379136.00000000000001550494
4 -228063139449551.99999999999987449320
5 66863279070719.999999999999893112749
6 -11844806136767.999999999999958526408
7 6531390049535.9999999999999910549341
8 2583589005360.0000000000000011015880
9 212671578239.99999999999999992759574
10 305088179616.00000000000000000196849
11 0.00000000000000000000000000000000000
12 3917484648.0000000000000000000000000
1/T-order : 13
D-order :
0 -86589623748617496.000000000000000000
1 0.16691726314776550857258894622257804E-12
2 -52949249647695648.000000000000158118
3 -9537683865454368.0000000000002814010
4 -10939408222245983.999999999999497409

206
5 -5811190813634544.0000000000003157060
6 -654690366315647.99999999999989743152
7-1057737434869440.0000000000000186113
8 20932078561056.000000000000001845458
9 -57774974907312.000000000000000087719
10 833185694016.00000000000000000128083
11 -500353757279.99999999999999999999873
12 0.00000000000000000000000000000000000
13 -18869465160.000000000000000000000000
1/T-order : 14
D-order :
0 562412828945822400.00000000000000000
1 0.11850215357582634283041556985597184E-09
2 1328006212481717495.9999999998948148
3 -585447359184507456.00000000012492762
4 561219441214651128.00000000013708412
5 -250575681395134368.00000000000284687
6 55329600291081479.999999999967202634
7 -35218565624092031.999999999989725929
8 -7212097967275655.9999999999994161914
9 -1975748425743456.0000000000008442731
10 -1418333640318503.9999999999998279087

207
11 -46744454877120.000000000000014911379
12 -81469060447319.999999999999999513042
13 0.00000000000000000000000000000000000
14 -847702120356.00000000000000000000000
1/T-order : 15
D-order :
0 207094862857112020980.00000000000273
1 0.95164090034462402448141890716724089E-10
2 156234221526848774399.99999999997272
3 19292203833996529799.999999999853117
4 41788935394790648400.000000000050022
5 16098137687636991000.000000000055252
6 3752774819450873999.9999999999737099
7 4104853966504279799.9999999999967599
8 -87072381012355199.999999999996527829
9 375185375401699799.99999999999958078
10 -26373476928632400.000000000000108126
11 4519518512833800.0000000000000354381
12 -563922660798000.00000000000000365076
13 -511631087952599.99999999999999987120
14 0.00000000000000000000000000000000000
15 -1482426909180.0000000000000000000000

208
1/T-order : 16
D-order :
0 630879625505013160319.99999999999272
1 -0.93750006531211146796575178625273501
2 -2307858870873010060798.2812499681022
3 1957964740817863127039.0625000901055
4 -1544250744592008756479.8437500460423
5 1076745296017426913279.9999999732536
6 -241172007857181043199.99999998394560
7 206575351164445132800.00000000189038
8 21621413448277799039.999999998013209
9 16214802798054850560.000000000087224
10 7965880526486868480.0000000000939053
11 531042160467363839.99999999998826682
12 598629070523301119.99999999999867962
13 10068606925977600.000000000000304258
14 21357039422937599.999999999999985771
15 0.00000000000000000000000000000000000
16 193916304689760.00000000000000000000

209
1/T-order : 17
D-order :
0 -611391451107142135739232.00000000000
1 170.83647410036269159424402838873882
2 -555542879294109646242852.66717503592
3 -42055172921293277231934.789663109928
4 -184863257614180875124501.09291181341
5 -50598464162370659728047.265661587939
6 -22821531762570616968186.157444739481
7 -17553777522564556251660.565531834902
8 165379151591294909188.16183909051142
9 -2332525596679102819968.4650002297203
10 238660913931373421568.00570257517029
11 -88578617135744927232.000618769507128
12 16895283750395352575.999990645846879
13 5185291917152994048.0000007161134761
14 271564358604788735.99999999412750999
15 363794918632487424.00000000039150466
16 0.00000000000000000000000000000000000
17 1813660872084000.0000000000000000000

210
1/T-order : 18
D-order :
0 -9149886935001318386336976.0000000000
1 -104183.49239789221226012403603562491
2 1888478511240025923878117.7024842501
3 -7804306068665003445108120.3967514634
4 4372711533869579385360954.0441572070
5 -5307924935348359636310439.6859882474
6 1068364827153506101530969.6268084273
7-1316059498241853198364699.7804109380
8 -63187314070205789537870.847942554741
9 -142812438689065747385703.50490794890
10 -48235380437888763074055.549985578749
11 -6205492715259562175232.1392495080363
12 -5490123811804972288799.9755676076747
13 -75074387239658325888.000281999632080
14 -215513382145316093279.99996624477535
15 -1493759328807446783.9999997223833116
16 -4860576923259644568.0000000284355224
18 -44540376825659148.000000000000000000

211
1/T-order : 19
D-order :
0 2165142755124992938943759579.9999847
1 6362561.4750850828031001338302985459
2 2335609052750318409888355218.5256958
3 73800566091717230983652163.613878727
4 941658115694904121375956367.19382477
5 169665476584916097821667095.13447475
6 151009040284465617563306225.24403191
7 81576694161898114661018105.546412468
8 1767159042015755989160948.0691311806
9 14967138701565360744465994.661488563
10 -2109061916567534176705480.4004050270
11 955467798876662717763012.37924189866
12 -224598125967406702392959.75556565262
13 -22762795562944913167775.685135310283
15 -5358495235921729606944.0004995671043
16 -124543401156142067376.00000001676744
17 -198012090204084168215.99999985099930
18 0.00000000000000000000000000000000000
19 -966836320351010676.00000000000000000

212
1/T-order : 20
D-order :
0 65414053028901744339879146780.000000
1 -1645730233.9224603463836160699429456
2 22857867846598172311640275903.581543
3 36462803689119376954310641857.098145
4 -9519871925535162142741297331.4746704
5 29885521010887589936726636581.248779
6 -4669090331979863169130648499.1762085
7 9209218694914395992756282345.0463257
8 150414403489428631013848646.57219696
9 1306943627402657146343379144.3957672
10 308819891359757588757530216.55378151
11 81270845855369479465627474.792070389
12 51210550941140087678555938.085169315
13 1053815698044180173951983.3968225271
14 3123342386115456229776005.8624264002
15 -64307863926090059558400.056661920622
17-472689194184185328000.00004387601803
18 279280950742909533600.00001169103780
19 0.00000000000000000000000000000000000

A.2 Perpendicular Magnetic Susceptibility
For the magnetic susceptibility with the magnetic field oriented perpendicular
to the chain the series is
3rx« = Eon/(3”n!)(l/r)”. (A.3)
The coefficients an are polynomials in D whose coefficients are:
1/T-order : 0
D-order :
0 2.0000000000000000000000000000000000
1/T-order : 1
D-order :
0 -8.0000000000000000000000000000000000
1 4.0000000000000000000000000000000000
1/T-order : 2
D-order :
0 20.000000000000000000000000000000000
1 -16.000000000000000000000000000000000
2 4.0000000000000000000000000000000000

214
1/T-order : 3
D-order :
0 96.000000000000000000000000000000000
1 72.000000000000000000000000000000000
2 36.000000000000000000000000000000000
3-12.000000000000000000000000000000000
1/T-order : 4
D-order :
0 -780.00000000000000000000000000000000
1 -456.00000000000000000000000000000000
2 -576.00000000000000000000000000000000
3 336.00000000000000000000000000000000
4 -60.000000000000000000000000000000000
1/T-order : 5
D-order :
0 -14688.000000000000000000000000000000
1 3240.0000000000000000000000000000000
2 -7800.0000000000000000000000000000000
3 -4656.0000000000000000000000000000000
4 -84.000000000000000000000000000000000
5 84.000000000000000000000000000000000

215
1/T-order : 6
D-order :
0 101916.000000000000000000000000000000
1 76176.000000000000000000000000000000
2 153216.00000000000000000000000000000
3 -3600.0000000000000000000000000000000
4 16200.000000000000000000000000000000
5 -12312.000000000000000000000000000000
6 1764.0000000000000000000000000000000
1/T-order : 7
D-order :
0 5446440.0000000000000000000000000000
1 -1868471.9999999999999999999999999952
2 3465827.9999999999999999999999999879
3 458064.00000000000000000000000000969
4 568331.99999999999999999999999999657
5 373248.00000000000000000000000000056
6 -35676.000000000000000000000000000032
71908.0000000000000000000000000000000

216
1/T-order : 8
D-order :
0 -16226028.000000000000000000000000000
1 -38953224.000000000000000000000001137
2 -65471183.999999999999999999999997699
3 7226495.9999999999999999999999984490
4 -17502047.999999999999999999999999612
6 180287.99999999999999999999999998829
7 635616.00000000000000000000000000121
8 -79740.000000000000000000000000000000
1/T-order : 9
D-order :
0 -3552744239.9999999999999999999999992
1 1281675960.0000000000000000000000281
2 -2755937520.0000000000000000000000546
3 16722288.000000000000000000000031019
4 -639975816.00000000000000000000000414
5 -194293728.00000000000000000000000134
6 -37250927.999999999999999999999999612
7 -36477216.000000000000000000000000026
8 5331420.0000000000000000000000000000
9 -407916.00000000000000000000000000000

217
1/T-order : 10
D-order :
0 -11465749259.999999999999999999999974
1 36909350352.000000000000000000193150
2 38433358223.999999999999999999473834
3 -5069386943.9999999999999999994314953
4 20139342335.999999999999999999675626
5 -6320484431.9999999999999999998917460
6 1370061215.9999999999999999999781690
7 -682336223.99999999999999999999738508
8 -199814040.00000000000000000000017099
9 -36923687.999999999999999999999995308
10 4601124.0000000000000000000000000000
1/T-order : 11
D-order :
0 3508721851967.9999999999999999999966
1 -1203821801711.9999999999999999990378
2 3322825109819.9999999999999999973001
3 -323755669007.99999999999999999688885
4 975639189383.99999999999999999800439
5 192893691600.00000000000000000080320

218
6 97329853367.999999999999999999787659
7 48663506592.000000000000000000036806
8 175612211.99999999999999999999598983
9 4033653552.0000000000000000000002482
10 -787587948.00000000000000000000000662
11 59355828.000000000000000000000000000
1/T-order : 12
D-order :
0 42078240698508.000000000000000000027
1 -52611861285432.000000000000004787620
2 -21973888468079.999999999999985734014
3 3290087461967.9999999999999825139738
4 -26377344970847.999999999999988193235
5 10835510605631.999999999999995083119
6 -4540663595087.9999999999999986703175
7 2381295995999.9999999999999997621968
8 146680223760.00000000000000002788602
9 116123545295.99999999999999999794224
10 55352491776.000000000000000000086417
11 876537935.99999999999999999999843073
12 -241916220.00000000000000000000000000

219
A.3 Parallel Magnetic Susceptibility
For the magnetic susceptibility with the magnetic field oriented parallel to
the chain the series is
3TX, = £a„/(3"n!)(l/r)“. (A.4)
The coefficients an are polynomials in D whose coefficients are:
1/T-order : 0
D-order :
0 2.0000000000000000000000000000000000
1/T-order : 1
D-order :
0 -8.0000000000000000000000000000000000
1 -2.0000000000000000000000000000000000
1/T-order : 2
D-order :
0 20.000000000000000000000000000000000
1 32.000000000000000000000000000000000
2 -2.0000000000000000000000000000000000

220
1/T-order : 3
D-order :
0 96.000000000000000000000000000000000
1 -252.00000000000000000000000000000000
2 0.0000000000000000000000000000000000
3 6.0000000000000000000000000000000000
1/T-order : 4
D-order :
0 -780.00000000000000000000000000000000
1 480.00000000000000000000000000000000
2 936.00000000000000000000000000000000
3 -384.00000000000000000000000000000000
4 30.000000000000000000000000000000000
1/T-order : 5
D-order :
0 -14688.000000000000000000000000000000
1 11340.000000000000000000000000000000
2 -21840.000000000000000000000000000000
3 6360.0000000000000000000000000000000
4 -480.00000000000000000000000000000000
5 -42.000000000000000000000000000000000

221
1/T-order : 6
D-order :
0 101916.000000000000000000000000000000
1 7056.0000000000000000000000000000000
2 115092.00000000000000000000000000000
3 36576.000000000000000000000000000000
4 -41148.000000000000000000000000000000
5 12096.000000000000000000000000000000
6 -882.00000000000000000000000000000000
1/T-order : 7
D-order :
0 5446440.0000000000000000000000000000
1 -2412899.9999999999999999999999999742
2 4474007.9999999999999999999999999418
3 -3146219.9999999999999999999999999540
4 2022551.9999999999999999999999999834
5 -460403.99999999999999999999999999717
6 50399.999999999999999999999999999823
7-954.00000000000000000000000000000000

222
1/T-order : 8
D-order :
0 -16226028.000000000000000000000000000
1 -29102976.000000000000000000000000517
2 -37156175.999999999999999999999998733
3 22410431.999999999999999999999998798
4 -15662807.999999999999999999999999447
5 -1005120.00000000000000000000000012602
6 2198160.0000000000000000000000000137
8 39870.000000000000000000000000000000
1/T-order : 9
D-order :
0 -3552744240.0000000000000000000000000
1 969449148.00000000000000000000000662
2 -2712940128.0000000000000000000000265
3 1198075536.0000000000000000000000352
4 -1227855024.0000000000000000000000215
5 580839336.00000000000000000000000651
6 -259928352.00000000000000000000000098
7 51247728.000000000000000000000000058
8 -5824224.0000000000000000000000000000
9 203958.00000000000000000000000000000

223
1/T-order : 10
D-order :
0 -11465749260.000000000000000000000026
1 35607476879.999999999999999999790042
2 12536744220.000000000000000000574712
3 -11133858240.000000000000000000624779
4 20183570280.000000000000000000358910
5 -11997302688.000000000000000000120557
6 4642709400.0000000000000000000244300
7-709871040.00000000000000000000293173
8 -71320499.999999999999999999999808650
9 33773759.999999999999999999999994778
10 -2300562.0000000000000000000000000000
1/T-order : 11
D-order :
0 3508721851968.0000000000000000000000
1 -526107590315.99999999999999997153630
2 2882974449959.9999999999999999185781
3 -900694903283.99999999999999990588151
4 1178670084767.9999999999999999413815
5 -428431112999.99999999999999997813289

224
6 303064868303.99999999999999999491230
7 -113849549351.99999999999999999926110
8 41197238279.999999999999999999935539
9 -7332876539.9999999999999999999969518
10 804751199.99999999999999999999994106
11 -29677914.000000000000000000000000000
1/T-order : 12
D-order :
0 42078240698508.000000000000000000054
1 -51430862185631.999999999999995972406
2 9291597603767.9999999999999880875997
3 1894921581600.0000000000000144841143
4 -18829421828604.000000000000009711294
5 17630500756608.000000000000004025961
6 -11126075134032.000000000000001087792
7 5488114253760.0000000000000001952284
8 -1647970515324.0000000000000000230867
9 326382740064.00000000000000000172742
10 -22864758408.000000000000000000074023
11 -766184831.99999999999999999999861819
12120958110.00000000000000000000000000

APPENDIX B
Low Temperature Crystallographic data for MnCl3(d-bipy)
Unit Cell Parameters
Space group
Cc
T(K)
11
a(Á)
8.0872(2)
6(A)
16.1394(6)
C(A)
9.5517(2)
a(deg)
90
/?(deg)
90
7(deg)
109.469(2)
225

226
Atomic positions
atom
X
Y
Z
Mn
0.7725
0.0169
0.7158
Cl
0.7750
0.0605
0.9760
Cl
0.6236
0.1237
0.5989
Cl
1.0374
0.0774
0.7437
N
0.5597
-0.0519
0.7055
Cl
0.3990
-0.0261
0.6549
C2
0.2596
-0.0840
0.6217
C3
0.2900
-0.1628
0.6490
C4
0.4560
-0.1892
0.7034
C5
0.5927
-0.1359
0.7267
C6
0.7781
-0.1554
0.7885
C7
0.8449
-0.2365
0.8289
C8
1.0216
-0.2500
0.8960
C9
1.1359
-0.1856
0.9206
CIO
1.0661
-0.1054
0.8823
N
0.8978
-0.0916
0.8186
HI
0.3748
0.0375
0.6473
H2
0.1185
-0.0578
0.5794
H3
0.1854
-0.2091
0.6185
H4
0.4864
-0.2610
0.7213
H5
0.7411
-0.2876
0.8034
H6
1.0761
-0.3139
0.9180
H7
1.2764
-0.1863
0.9738
H8
1.1489
-0.0499
0.9019

227
Bond Lengths
Bond
Length (Á)
Mn-Mn
4.80698(8)
Mn-Cl
2.57743(4)
Mn-Cl
2.61529(4)
Mn-Cl
2.18371(5)
Mn-Cl
2.28781(5)
Mn-N
2.02353(4)
Cl-Mn
2.18371(5)
N-Cl
1.29436(3)
N-C5
1.38428(5)
C1-C2
1.41633(3)
Cl-Hl
1.04402(3)
C2-C3
1.30515(4)
C2-H2
1.15669(3)
C3-C4
1.33747(3)
C3-H3
1.09282(2)
C4-C5
1.35881(3)
C4-H4
1.18513(4)
C5-C6
1.45094(4)
C6-C7
1.42001(4)

228
Bond
Length(Á)
C7-C8
1.37524(4)
C7-H5
1.14319(2)
C8-C9
1.35908(3)
C8-H6
1.11209(3)
C9-C10
1.41103(4)
C9-H7
1.07881(3)
C10-H8
1.09629(3)
N-C6
1.37622(3)
N-C10
1.31114(3)

229
Bond Angles
Bond
Angle (deg.)
Cl-Mn-Cl
167.270(0)
Cl-Mn-Cl
95.925(2)
Cl-Mn-Cl
93.688(2)
Cl-Mn-N
85.987(2)
Cl-Mn-N
85.950(1)
Cl-Mn-Cl
93.331(2)
Cl-Mn-N
95.352(2)
Cl-Mn-N
175.325(0)
Cl-Mn-N
171.299(0)
Cl-Mn-N
90.816(2)
N-Mn-N
80.485(2)
Mn-Cl-Mn
135.551(1)
N-C1-C2
119.789(2)
N-C1-H1
118.964(0)
C2-C1-H1
121.073(2)
C1-C2-C3
120.983(2)
C1-C2-H2
117.105(2)
C3-C2-H2
121.696(0)
C2-C3-C4
119.081(0)
C2-C3-H3
122.350(2)
C4-C3-H3
118.259(2)
C3-C4-C5
121.238(2)
C3-C4-H4
120.017(0)
C5-C4-H4
118.488(2)

230
Bond
Angle (deg.)
N-C5-C4
119.272(2)
N-C5-C6
113.193(0)
C4-C5-C6
127.187(2)
C5-C6-C7
123.843(0)
C5-C6-N
118.893(2)
C7-C6-N
117.149(2)
C6-C7-C8
120.999(1)
C6-C7-H5
114.957(2)
C8-C7-H5
124.034(2)
C7-C8-C9
120.011(2)
C7-C8-H6
121.336(0)
C9-C8-H6
118.190(2)
C8-C9-C10
117.846(2)
C8-C9-H7
128.477(1)
C10-C9-H7
113.527(1)
C9-C10-N
122.701(1)
C9-C10-H8
122.426(2)
N-C10-H8
114.865(2)

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BIOGRAPHICAL SKETCH
Garrett Earl Granroth was born the youngest son of Earl G. and Barbra
J. Granroth on October 13, 1971 in Rockford, Illinois. He started elementary
school in the fall of 1976. During these years, his interest in the physical sciences
developed through several chemistry classes. Garrett graduated from Booker
High School in Sarasota, Florida in the spring of 1989. At Stetson University, in
DeLand, Florida, and under the direction of Professor Kevin Riggs, he conducted
research on magnetic thin films using the techniques of ferromagnetic resonance
and the magneto-optical Kerr effect. After four years, Garrett graduated Cum
Laude with a B.S. in physics on a hot summer day in 1993. The fall of 1993 saw
Garrett beginning his graduate studies at the University of Florida in physics. In
the spring of 1994, he joined the research group of Professor Mark W. Meisel to
begin the work of which this thesis is the culmination. Besides the work of this
thesis, he also took the opportunity to make several ultrasound measurements in
normal and superfluid 3He at T « 200 ¡jK.
240

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Mark W. Meisel, Chairman
Associate Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Selman P. Hershfield
Associate Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Gary G. Ihas
Professor of Physics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Fred Sharifi
Assistant Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Dan R. Talham
Associate Professor of Chemistry
This dissertation was submitted to the Graduate Faculty of the Department of
Physics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
May 1998
Dean, Graduate School

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UNIVERSITY OF FLORIDA
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