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This is to certify that the
dissertation entitled

LOCAL STRUCTURE STUDY OF DISORDERED
CRYSTALLINE MATERIALS WITH THE ATOMIC PAIR
DISTRIBUTION FUNCTION METHOD

presented by

XIANGYUN QIU

has been accepted towards fulfillment
of the requirements for the

 

 

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LOCAL STRUCTURE STUDY OF DISORDERED CRYSTALLINE MATERIALS
WITH THE ATOMIC PAIR DISTRIBUTION FUNCTION METHOD

By

Xiangyun Qiu

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2004

ABSTRACT

LOCAL STRUCTURE STUDY OF DISORDERED CRYSTALLINE MATERIALS
WITH THE ATOMIC PAIR DISTRIBUTION FUNCTION METHOD

By

Xiangyun Qiu

The employed experimental method in this Ph.D. dissertation research is the
atomic pair distribution function (PDF) technique specializing in high real space res-
olution local structure determination. The PDF is obtained via Fourier transform
from powder total scattering data including the important local structural informa-
tion in the diffuse scattering intensities underneath, and in-between, the Bragg peaks.
Having long been used to study liquids and amorphous materials, the PDF technique
has been recently successfully applied to highly crystalline materials owing to the
advances in modern X-ray and neutron sources and computing power. An integral
part of this thesis work has been to make the PDF technique accessible to a Wider
scientific community. We have recently developed the rapid acquisition PDF (RA-
PDF) method featuring high energy X-rays coupled with an image plate area detector,
allowing three to four orders of magnitude decrease of data collection time. Corre-
spondingly in software development, I have written a complete X-ray data correction
program PDFgetXZ (user friendly with GUI, 32,000+ lines). Those developments
sweep away many barriers to the wide-spread application of the PDF technique in
complex materials. The RA-PDF development also opens up new fields of research
such as time-resolved studies, pump-probe measurements and so on, where the PDF
analysis can provide unique insights. Two examples of the RA-PDF applications are
described: the distorted Ti2 square nets in the new binary antimonide Tigsb and
in-situ chemical reduction of CuO to Cu.

The most intellectually enriching has been the local structure studies of the colos-

sal magneto-resistive (CMR) manganites with intrinsic inhomogeneities. The strong

coupling between electron, spin, orbital, and lattice degrees of freedom result in ex-
tremely rich and interesting phase diagrams. We have carried out careful PDF analy-
sis of neutron powder diffraction data to study the local Mn06 octahedral distortions.
For example, in the updoped compound LaMnOg, the Jahn-Teller (JT) transition
around 750 K is characterized as distorted (JT active) to undistorted Mn06 octahedra
transition by the conventional crystallographic analysis. However, our PDF results
show local Mn06 octahedral distortions persist above the JT transition, and it is
their random orientations that make the existing local distortions invisible to average
structure analysis. The nature of the JT transition around 750 K is orbital order to
disorder. Our first local structure study of the high temperature rhombohedral phase
(T 2 1010 K) additionally discovered the existence of locally JT distorted Mn06
octahedra. More significantly, the range of the orbital order in the high temperature
dynamic-J T phases, estimated from the crossover from the local to the average struc-
ture, is around 16 A (~four Mn06 octahedra), suggesting strong nearest neighbor JT
anti-ferrodistortive coupling. In the bi-layered Lao_928r2,03MngO—,, we found the shape
of Mn06 octahedron changes from oblate (4 long, 2 short Mn-O bonds) to prolate
(2 long, 4 short Mn-O bonds) as the material goes from type A anti-ferromagnetic
to type CE charge ordered phase. This can be understood as the dx2_y2 to d(x,y)2_,.2
orbital occupancy transition of the Mn3+ eg electrons. Evidences for nano-scale in-
homogeneities in the Mn4+ rich region of La2_ngr1+2angO7 (0.54 S a: g 0.80) are

also discussed.

To my loving and beloved,

iv

Acknowledgments

It’s such a thrill to be in this moment. This moment wouldn’t have been possible
without all the loving and beloved people around me. This moment wouldn’t have
any meaning without them. Each one of them is so special on the way to this moment.
This moment belongs to them.

No words can express my greatest appreciation and respect to S. J. L. Billinge, my
Ph.D. adviser. The past four and a half years spent in his group has been invaluable
and extremely beneficial. I saw unlimited patience, unsparing guidance, and unfailing
trust. Above all, he always strives to help me succeed.

I am full of gratitude to my thesis committee members, P. M. Duxbury, S. Tessmer,
M. Thoennessen, and W. Repko for their devoted supervision. My highest respect
to each of them. Consistent encouragement from S. D. Manhanti during my stud-
ies is greatly appreciated. My Ph.D. research benefits greatly from collaborations
with P. J. Chupas, C. P. Grey, J. C. Hanson, H. Kleinke, C. Kmety, P. L. Lee, and
J. F. Mitchell. Special thanks to each of them. It has been a great pleasure work-
ing with them. I have fortunately received magnanimous help from D. S. Robinson,
S. Short, and D. Wermeille during experiments, to whom I am very very thankful. I
also like to thank C. Malliakas for his help with sample characterizations.

Deepest appreciation goes to Th. Proffen who has helped me a great deal in almost
all aspects of my research. V. Petkov and I-K. Jeong are cordially thanked for their
generous help with X-ray experiments and data analysis. My most sincere salute to
E. Bozin who taught me neutron experiments and gave his lavish encouragements
along with his amusing accompany. It has been great fun working with M. Gutmann,
P. F. Peterson, J. W. Thompson, A. Masadeh, and H. J. Kim. They are all great
teachers and friends.

This moment is especially for my family. My father, my mother, all my family, I

should have spent all the time with you all. This moment is especially for you.

Contents

1 Introduction 1
1.1 Importance of the local structure .................... 1
1.2 Experimental methods of local structure determination ........ 3
1.3 The atomic pair distribution function technique ............ 6
1.4 Recent advances in the PDF method .................. 9
1.5 Outline of this thesis ........................... 10

2 The Rapid Acquisition Pair Distribution Function Method 12
2.1 Introduction ................................ 12
2.2 Description of the modern PDF experiments .............. 13

2.2.1 Neutron powder diffraction .................... 14
2.2.2 X-ray powder diffraction ..................... 15
2.3 Use of the image plate as an X-ray detector .............. 16
2.3.1 Principle of X-ray image plate detection ............ 16
2.3.2 Characteristics of image plate detectors ............. 17
2.3.3 The IP and PDF method ..................... 21
2.4 RA-PDF experiment ........................... 22
2.5 The proof of principle study ....................... 23
2.5.1 Description of the experiments .................. 23
2.5.2 Early results ............................ 26
2.6 Comparison of the RA-PDF with conventional X-ray PDF method . . 32
2.6.1 Description of the experiments .................. 32
2.6.2 Results ............................... 33
2.7 New data analysis program: PDFgetX2 ................. 36
2.7.1 The crystallographic problem .................. 36
2.7.2 Method of solution ........................ 37
2.7.3 Software and hardware environment ............... 37
2.7.4 Program specification ....................... 38
2.7.5 Documentation and availability ................. 38
2.8 Conclusions ................................ 38

3 Orbital Correlations in the Pseudo-cubic 0 and Rhombohedral R
Phase of Undoped LaMnO3 40
3.1 Introduction to manganites ....................... 40

3.1.1 Earlier studies ........................... 41
3.2 The colossal magneto-resistive effect ................... 43

vi

3.3 Physics of manganites .......................... 46

3.3.1 Crystal field effect ........................ 46
3.3.2 J ahn-Teller distortion ....................... 48
3.3.3 Double exchange mechanism ................... 50
3.3.4 Semi-covalent exchange and the Goodenough-Kanamori rule . 52
3.3.5 Summary ............................. 55
3.4 Phase diagram of La1_$Ca$MnOg as an example ............ 55
3.5 Introduction to the question: high temperature phases of the undoped
LaMnO3 .................................. 57
3.6 Experimental methods .......................... 61
3.6.1 Sample preparation ........................ 61
3.6.2 Differential thermal analysis ................... 62
3.6.3 Neutron powder diffraction .................... 64
3.6.4 Obtaining experimental PDFs .................. 64
3.6.5 Real space PDF refinement ................... 66
3.6.6 Reciprocal space Rietveld refinement .............. 70
3.6.7 Model independent peak analysis ................ 74
3.7 Local structure from PDF analysis ................... 74
3.8 Average structure from the PDF and Rietveld analysis ........ 79
3.8.1 Crystallographic phase determination .............. 79
3.8.2 Lattice structure parameters ................... 82
3.8.3 Summary ............................. 88
3.9 Intermediate structure from PDF analysis ............... 89
3.10 Discussion and conclusions ........................ 91

Local Structure of the Bi-layered Manganite La2_2er1+23Mn207 in

the Mn“ Rich Region 97
4.1 The bi-layered manganite: La2_2ISr1+2an207 ............. 97
4.2 Orbital occupancy transition at :r = 0.54 ................ 100
4.2.1 Introduction to the question ................... 100
4.2.2 Experimental methods ...................... 105
4.2.3 Results ................. ' .............. 106
4.2.4 Discussion and conclusions .................... 113
4.3 Evidence for nano—scale inhomogeneities in the Mn4+ rich region: 0.54 S
a: S 0.80 .................................. 115
4.3.1 Introduction to the question ................... 115
4.3.2 Experimental methods ...................... 117
4.3.3 Model independent analysis ................... 120
4.3.4 Structural modeling ........................ 124
4.3.5 Discussion ............................. 126
4.3.6 Conclusions ............................ 128
Applications of the RA-PDF Method 130
5.1 Prelude .................................. 130
5.2 Locally distorted Ti square nets in TIQSb ................ 130

vii

 

 

 

5.2.1 Introduction to the question ................... 130

5.2.2 RA—PDF analysis of local Ti2 distortions ............ 134

5.2.3 Discussion and conclusions .................... 140

5.3 In—situ chemical reduction of CuO to Cu ................ 141
5.3.1 Experiment ............................ 142

5.3.2 Results ............................... 143

5.3.3 Conclusion ............................. 145

6 Concluding Remarks 147
6.1 Summary of the thesis .......................... 147
6.2 Future work ................................ 150
6.2.1 Measuring more PDFs ...................... 151

6.2.2 Obtaining high quality PDFs .................. 152

6.2.3 Combining complementary probes ................ 154

6.2.4 Applying the PDF technique in new areas ........... 154

6.3 Acknowledging funding agencies ..................... 155

A The rapid acquisition pair distribution function method experiment 157
A.1 Requirements of the experiment ..................... 157
A2 Setup of the RA-PDF experiment .................... 158
A3 Collecting RA-PDF data ......................... 162
AA Processing the data ............................ 165
A41 Obtain calibration parameters .................. 166

A.4.2 Integrate the IP raw data .................... 166

A43 Normalization by the monitor counts .............. 173

A44 Additional corrections to the RA-PDF data .......... 173
Bibliography 178

viii

 

 

 

 

 

List of Figures

2.1

2.2
2.3

2.4

2.5

2.6

2.7

2.8

3.1

3.2

3.3
3.4

3.5
3.6
3.7
3.8
3.9
3.10
3.11

3.12

Energy level scheme of detection media BaFBrzEu2+ used in the image

plate phosphor layer ........................... 18
The dependence of the IP response on the incident photon energy . . 20
Two dimensional contour plot of the Ni powder diffraction raw data
from the Mar345 image plate detector .................. 24
Experimental F (Q), G (r), and the crystallographic model fit of the Ni
powder ................................... 27
Experimental F (Q), C(r), and the crystallographic model fit of the
a-AlF 3 powder .............................. 29
Experimental F(Q) and C(T) Of BI4V2011 and BI4V1.7TIO.3010.85, and
their G (r) differences ........................... 30
Comparison of the F (Q) and G (r) of oz-Ang between the IP and SSD
data collection methods ......................... 34
Comparisons of the calculated PDFs from the crystal structures refined
from the IP and SSD G(r)s ....................... 35
Crystallographic representation of the perovskite structure with generic
chemical formula ABO3 .......................... 42
The MR ratio AR/ R as a function of temperature for thin films of
La1-zCaanO3 at 1:20.33 ........................ 44
Contour visualization of the five 3d atomic electron orbitals ...... 47
Schematic plot of the Q1, Q2, and Q3 Jahn-Teller distortion modes of
one Mn06 octahedron .......................... 49
The schematic configuration of the Mn-Mn magnetic exchange inter-
action in consideration of the theory of semi-covalence exchange . . . 53
Phase diagram of La1_$CaanO3 as a function of temperature and Ca
doping :1: .................................. 56
Cartoon view of the orbital ordering in one LaMnO3 ab plane . . . . 58
Crystal structure of LaMn03 ....................... 59

The DTA measurement data from sample LaMnO3(A) and LaMnOg(B) 63
One exemplar the experimental raw data from neutron powder diffraction 65
Example of experimental F (Q) and C(r) of LaMn03(A) and (B) at

300 K ................................... 66
Examples of PDF refinement results at 650 K, 730 K, 880 K, 1150 K
of sample LaMn03(A) .......................... 67

ix

 

 

 

 

3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23

3.24
3.25

3.26

3.27
3.28

4.1
4.2

4.3
4.4
4.5

4.6
4.7

4.8

4.9

Examples of PDF refinement results at 20 K, 300 K, 603 K, 805 K of
sample LaMnO3(B) ............................ 69
Demonstration of the consequence of missing Bragg peaks at low Q
region of the experimental F (Q) on later PDF refinement ....... 70
Examples of Rietveld refinement of sample LaMnO3(A) NPDF data . 71
Examples of Rietveld refinement of sample LaMnO3(B) SEPD data . 72
Comparison of the neutron powder diffraction pattern of sample L3MIIO3(A)
between the beginning and end of the experiment ........... 73
Low 1‘ PDF region of experimental data at all temperatures for both
sample LaMn03(A) and (B) ....................... 75
Extracted peak profile parameters from the double Gaussian curve fitting 77
Lattice constants and cell volumes of sample LaMnO3(A) and LaMnOg(B) 80
The three Mn-O bond lengths and Mn06 octahedral volume as func-

tions of the temperature ......................... 82
The isotropic atomic thermal displacement parameters of La, Mn, and

O atoms as functions of the temperature ................ 83
The lattice and MnOB octahedral distortion parameters, and the tol-
erance factor F as functions of the temperature ............ 85
Octahedral tilt angle and O-Mn-O angles as functions of the temperature 86
Q2 and 62;; JT modes and c1 and c2 coefficients as functions of the
temperature ................................ 88
Mn-O bond lengths from varying range PDF refinements and the fits

with PDF form factor of a solid sphere ................. 90
The inverse of the domain radius as a function of temperature . . . . 92
The thermal displacement parameters of oxygen atoms along the di-
rections parallel and perpendicular to Mn-O bonds .......... 93
Crystal structure of the bi—layered manganite La2-2ISr1+2angO7 . . 98
Crystallographic structural and magnetic phase diagram of the bilayer
manganite La2_2xSr1+gangO7 in the range 0.3 S :1: _<_ 1.0 ....... 100
Schematic illustrations of magnetic structures of La2_2xSr1+2angO-, . 101
Cartoon representation of type CE charge ordering in manganites . . 102
Illustration of the in-plane dx2_y2 orbital occupancy in half doped
La2_21,-SI'1+2$MI1207 ............................ 104
Experimental F (Q) and G (7‘) from LagggsrgggMngO'z at 4 K ..... 107
Rietveld refinement results of La2_2xSr1+21MngO7 at a: = 0.54 neutron
powder diffraction data at 4 and 300 K ................. 108
Low 1' region of experimental PDFs of La2_2xSr1+2angO7 at all tem-
peratures. Temperature dependence of the PDF peak heights from
Mn-O and O-O correlations ....................... 109
The thermal displacement parameters of all oxygen atoms along the di-

rection parallel and perpendicular to the Mn-O bonding directions, as
a function of temperature. The in-plane a = b and apical c lattice con-
stants of L82_21-SI'1+2$M112O7 at :r = 0.54 as a function of temperature,
along with the extracted fraction of d322_,.2 type orbital occupancy . . 112

 

4.10

4.11
4.12
4.13
4.14
4.15

4.16
4.17

5.1
5.2
5.3
5.4

5.5
5.6

5.7

5.8
5.9

A.1

A2

The three crystallographically distinct Mn-O bond lengths within one
Mn06 octahedron, along with the octahedral and unit cell volume, as
functions of the temperature .......................
Experimental F(Q) and 0(7) of La2_2xSr1+2an207 at :1: = 0.80 at
4 K, with a fit of a structure model ...................
Comparison between experimental PDFs of La2_2xSr1+2$MngO7 at 1: =
0.54, 0.60 and the model PDF assuming 46% prolate octahedra . . .
Comparison between experimental PDFs of La2_2$Sr1+2$MngO-, at :r =
0.78, 0.80 and the model PDF assuming 20% prolate octahedra . . .
PDF peak heights of the first Mn-O peak around 1.94 A as a function
of Sr-doping :2: ...............................
Developments of in-plane lattice constants with Sr—doping :r .....
Two in-plane Mn-O bond lengths dependence on the Sr-doping :1:

Thermal displacement factor of the in-plane O atoms along the direc-
tion of Mn-O bonds ............................

Crystal structure of T ing in the ball and stick model .........
Planar net of Ti2 atoms in Tigsb, with thermal ellipsoids shown. . . .
The hypothetical x/2 x x/2 x 1 superstructure of the Ti2 square nets.
The experimental reduced structure function F (Q) from X-ray and
neutron measurements ..........................
Experiment and calculated PDFs from the distorted model of Tigsb .
PDFs of TiQSb and the PDFFIT results of the undistorted and dis-
torted models, for both X—ray and neutron data ............
Schematic diagram of the flow cell used in the RA-PDF experiment .
Powder diffraction pattern during the chemical reduction of CuO

The experimental G (r)s at the beginning and end of the chemical re-
duction of CuO to Cu ..........................

Schematic diagram of the rapid acquisition PDF (RA-PDF) experi-
ment layout ................................
Estimated relative standard deviations of the IP response as a function
of photon number absorbed in the area .................

xi

114
119
121
122
123
125
126

127

132
133
134

136
138

139
143
144

145

159

175

List of Tables

2.1 Lattice parameters of a-A1F3 from PDF and Rietveld refinements of
both IP and SSD data. The estimated standard deviations are shown
in the parenthesis immediately following the values ..........

5.1 List of expected inter-atomic distance splits/changes from the undis-
torted to distorted Ti2Sb structure model ................
5.2 Comparison of refined parameters of the distorted Ti2Sb model be-
tween the (RA-)PDF and single crystal diffraction methods ......

xii

36

133

140

 

 

 

 

Chapter 1

Introduction

1.1 Importance of the local structure

Materials are made up of atoms with charge and spins, and all carry some proper-
ties/ functions. How atoms are assembled together refers to the atomic and magnetic
structure of materials. The material structure and property/function relationship
proves to be very important, as well as very interesting. For example, graphite (dark,
soft, cheap) and diamond (transparent, hard, precious) are dramatically different
materials. However, they are both made up of the same kind of carbon atoms and
the only difference is their atomic crystal structures. What is more, the buckyball
(C60) and nanotube are made up of the same carbon atoms too. Certainly, there
are many other cases than carbon, such as Si02 (quartz, sand, agate, onyx, opal),
A1203 (sapphire, alumina), and so on. In a word, the structure property relationship
is fundamental and its in-depth understanding is vital to manipulate the material
function. Thus, precise knowledge of the structure is of paramount importance.

The importance of structure is further argued here by its most relevant energy
scale in modern functional materials. Among the millions of ways that materials
are used nowadays, with the exception of nuclear reactions in nuclear reactors and

possibly a few others, the atoms in materials always keep their original elemental

 

 

 

species. For example, gasoline is burned to propel automobiles, but the number of
carbon, oxygen, and hydrogen atoms stays the same before and after the chemical
reaction; the magnetic moment of each storage grain in the hard disk is flipped, but the
atoms comprising the disk never change their types. It is the structure that is closely
related to their practical function. The bonding interaction between atoms gives the
relevant energy scale (order of eV per atom) to our daily energy use. The practical
functions of materials are often directly expressed material’s macroscopic properties,
such as the mechanical characteristics, electric conductivity, thermal conductivity,
etc.. Though seemingly irrelevant, the microscopic structure is the underlying origin
of many macroscopic properties.

The local structure in this context refers to the atomic neighboring environment at
the length scale of nanometers or less. When the long range overall structure can be
simply constructed from the local structure by periodic assembly, this results in crys-
talline materials with perfect long range order; for example, diamond, quartz, and
sapphire. Preparation of perfect crystals has received significant amount of efforts
as they often possess astonishing properties such as mechanical strength. However,
one growing modern trend is that, in many interesting functional materials, the 10-
cal structure does not order perfectly long range as in crystalline materials. We are
observing that modern functional materials are getting more and more structurally
complex and disordered. Their properties of both scientific and technological interest
are often exploited and tuned by either intrinsic disorder or extrinsic disorder induced
by chemical doping, mesostructure patterning, etc.. For example, the high tempera-
ture superconducting cuprates reach the highest transition temperature with optimal
levels of defects induced by chemical doping. Therefore, detailed knowledge of local
structure (getting to know the neighbors) is crucial in modern times to understanding

the properties of complex and disordered materials.

 

 

1.2 Experimental methods of local structure de-

termination

Let us first look at the structure determination of crystalline materials, where the
local structure is the same as the average (overall) structure. There exists in crystals
a periodic box called the unit cell of the lattice. The long range ordered stacking of
the unit cells makes it very convenient and effective to look at the structure’s Fourier
counterpart in reciprocal space. In fact, a reciprocal lattice uniquely corresponds to
a specific lattice structure. Measuring the reciprocal lattice with diffraction experi-
ments [1] will lead to solution of the crystal structure in real space. Father and son
Bragg first demonstrated crystal structure solution by the means of X-ray diffrac-
tion [2, 3, 4] and won the Nobel prize in 1915 for their pioneering work. Ever since
then, the analysis of Bragg peak intensities (the reciprocal lattice reflections) for
Crystal structure determination has become routine and the almost exclusive experi-
lTlental method for this purpose, giving rise to conventional crystallographic analysis.
This mature technique is still enjoying great success in areas such as the protein
c ry st allography.

Probing the local structure in non—perfect crystalline materials proves very chal-
1enging to conventional crystallography. The assumption that Bragg peaks contain
all structural information does not hold in the presence of disorder and/ or local dis-
tOrtions. This is because the local deviations from perfect long range order result
in diffuse scattering in-between and underneath the Bragg peaks in reciprocal space
Other than the reciprocal lattice. Neglecting those diffuse scattering intensities re-
Sults in loss of the local structure information. Other experimental approaches have
been invented to probe the local structure (neighboring surrounding environment, for
eXample, the atomic pair distribution function (PDF) method [5] employed in this

thesis. Let us first briefly review a few other approaches in the following.

 

 

 

Extended X-ray Absorption Fine Structure (EXAFS) The measured quantity
in one EXAFS experiment is the sample absorption coefficient as a function of
the incident X-ray energy usually ranging between 20 and 800 eV above the
K or L absorption edge of one specific constituent element [6, 7]. Either the
transmitted photons or the fluorescence intensities can be measured for this
purpose. How the local structure information is contained in the absorption co-
efficient can be understood qualitatively as the following. The incident photons
excite either the K or L shell core electrons into the continuum as a conse-
quence of the photoelectric effect when the photon energy is closely above the
absorption edge. The excited electron will carry some kinetic energy which is
the difference between the photon energy and the absorption threshold of the
probed atomic shell. As the photoelectron energy (2 20 eV) is large compared
with its interaction energy with the surrounding atoms (~ 3 eV), simple plane
wave approximations can be made on those photoelectrons with the surround-

ing atoms taken as perturbations. The cross section of the photoelectric effect
is then proportional to the wave function overlap between the photoelectron
plane wave and the atomic K or L core level states. While the atomic core
levels are little affected by the neighboring atomic environments, the photoelec-
trons are scattered by the surrounding atoms. Therefore, the final state of the
photoelectron is modified by the backscattered electrons, and the backscattered
Electron waves can either add or subtract depending on their relative phases.
As the wavelength of the photoelectron simply follows the de Broglie relation
/\ = %, their relative phases directly depend on the distance between the probed
atom and its neighbors, i.e. half of the traveling distance of the backscattered
electron. Oscillations in the sample absorption with incident photon energy
a-re then expected, as the photoelectron energy, thus the wavelength A, and the

I‘elative phases, are varied. To properly extract the local structure information

from EXAFS data, many other complications have to be taken into considera-
tions, such as backscattering phase shift, finite life time of electron states, and

multiple electron scattering events.

It is worthwhile to mention its surface version SEXAF S. Instead of measur-
ing either transmitted or fluorescent photons, the photoelectron yield from the
surface is measured. This also requires the experiments to be performed in

ultra-high vacuum (10‘10 Torr).

X-ray Absorption Near Edge Structure (XANES) The experimental aspects
of XANES are close to EXAFS except that the covered absorption spectrum
range is between the threshold and the point at which EXAFS begins [7]. This
range is poorly defined, and taken usually as up to 50 eV above the threshold.
The XANES method is very sensitive to local electronic state changes. As the
photoelectrons in XANES measurements have low energy, the theoretical con-
siderations are rather different. The focus has been the electronic structure of
the low—lying extended excited states. The interaction between the photoelec-
trons and atoms is also significantly stronger, thus multiple scattering events
become significant. Though the oscillating amplitude in the XANES region is
generally significantly larger than in the EXAFS region, extensive theoretical

efforts are still required for detailed quantitative analysis [7].

NHCIear Magnetic Resonance (NMR) Within the present context, the NMR
technique probes the chemical environment of the specific nucleus under res—
Onance. During an NMR measurement, the energy of different nuclear spin
States are shifted by an amount proportional to their net spin by the external
Static magnetic field. Tiansitions between the different spin states can be initi-
ated by a secondary oscillating magnetic field by either changing its frequency

01' the amplitude of the applied static magnetic field. The zero static field N MR

 

 

 

method is called nuclear quadrupole resonance (N QR). The important quantity
coming from NMR measurements is called chemical shift, which is defined as the
difference between the resonance frequency of the nucleus and a standard [8],
relative to the standard. One origin of the chemical shift is from the electrons
around the nucleus. The applied magnetic field also causes the electrons to cir-
culate about the direction of the applied magnetic field. This circulation causes
a small magnetic field at the nucleus which opposes the externally applied field
(in some cases, it enhances the external field). This shifts the nuclear resonance
frequency, which is proportional to the magnetic field strength at the nucleus.
Therefore, the magnitude of the chemical shift carries information about the
surrounding electrons, thus the local chemical environment, such as valence,
bonding, geometry, etc.. As the probed nucleus is required to have net spin,

isotope substitution is sometimes necessary.

1.3 The atomic pair distribution function technique

As suggested by its name, the atomic pair distribution function (PDF), G (r), tells the
Probability of finding atomic pairs separated by the real space distance r. It should
be stressed that the PDF of interest is radially averaged and is a one dimensional
function. To obtain the G (r) for a given a structure, we first sit on one atom i, then
look out for neighbors. A peak is assigned for every atom j found at the position
corresponding to the inter-atomic distance rij, and zero everywhere else. Then we
repeat the same procedure over all atoms, and average all obtained curves to get the
tOtal G (r) of the structure. Another way to get the G' (r) is to first obtain the atomic

radial density function n(f') with an arbitrary origin. The convolution of n(r') with

 

itself results in the PDF. The mathematical definition of the PDF is

com = $2: 2]; [$5a — up] - 47rrp0

= 47T7‘lp(7‘) - pol, (M)

where r,,- is the distance between atoms i and j; the value b, is the scattering length
for atom i; (b) is the average scattering length over all atoms; and p0 is the number
density. The p(r) here is equivalent to the spherical average of f n03) - n(F' — Pym.
Though less intuitive than the radial density function n(F), the PDF, C(r), is the
directly measurable quantity from powder diffraction experiments.

The path from a powder diffraction experiment to the G (r) can be conceptually
understood in the following way. We will first concentrate on one single grain, then
do the spherical averaging. Given the radial density function n(f') of the scatterer,
from the kinematic scattering theory in the Born approximation, the scattered wave

at momentum transfer Q takes the form of F (Q) = f n(7")eiQ"Fdr', which is actually
the Fourier transform of n(F). Well known as the phase problem in crystallography,
the measurable quantity is the scattered intensity I (Q) = |F(Q)|2, where the phase
information is lost. We cannot directly and uniquely recover the atomic density
function n(f'). Now, we can rewrite the HQ) as F(Q) - F163), where F*(Q) is in
fact the Fourier transform of the real function n(—r'). Then based on the convolution
t1190113111, the Fourier transform of I (Q) is the convolution of the Fourier transforms of
FAQ”) and F*(Q). This is f n(73)-n(13—r')d73. Then we do the spherical averaging due
to the random grain orientation in powders. This will result in the one dimensional
form 0f G(r). Therefore, the PDF can be obtained by Fourier transforming the

measuer scattering intensities. The measured scattering intensities can be expressed

+°° bibj ' Qt” .. 1 oobz'b' ' Qi'
S(Q) =/_oo Wp(rij)fllch;—zldr,j = 5/0 (7)12-47rrijp(r,j)%T—J)drij (1.2)

from the Debye scattering equation [9]. The PDF can be directly obtained via a sine
Fourier transform: G(r) = :3me Q[S(Q) — 1] sin Qr dQ.

The key difference between the PDF method and conventional average structure
analysis of powder diffraction data is how to treat the diffuse scattering intensities in-
between and underneath the Bragg peaks. The Bragg peaks come from the long range
ordered structure, while the diffuse scattering intensities come from local/ temporal
deviations from the long range ordered structure. The PDF technique takes into
account both, as evident from the equations above, while conventional average struc-

ture analysis only looks at the Bragg peaks. Therefore, the PDF method is capable of
probing both the local and average structure. This combination of local and average
Studies from the same scattering data represents a clear advantage for a wide range of
applications. For example, with the presence of Bragg peaks in the case of crystalline
materials, the PDF technique and conventional analysis are complementary while the
former provides additional information about the local structure. However, in the
case Of nano—crystalline and non-crystalline materials, with few or no Bragg peaks,
only the PDF technique is applicable.

Local structural information is intuitively contained in the low-r PDF peaks.
Thus, the real space resolution is an important experimental factor to resolve closely
n(Eighbonng peaks. PDF peaks are first broadened by the atomic thermal motions.
Second, the finite measurement range also broadens the peaks coming from the ter-
minatiOn effects [10]. High real space resolution requires usually extended Q range (2
25-0 A\l) data to be collected, and low temperatures if possible. Other than identi-
fying lOw—r peak positions (existing atomic pair distances), extracting the peak width

an . . . .
d area reveals further details about the local distortions and number of neighbor-

 

ing atoms as well. The average structural information is usually recovered from the
PDF structure model refinements using the non-linear least square regression fitting
program PDFFIT [11]. The agreement between the average structure and experi-
mental G(r) provides a good check for the possible existence of local disorder, and
hypothetical distorted structure models can then be fit to improve the agreement.
Knowledge of the detailed local and average structures often serves as the basis of
many theoretical calculations of material properties. For example, the bond valence
sum can be computed to investigate the strength of chemical bondings. Electronic '
band structure calculations can be carried out if necessary to check whether the
obtained structure gives macroscopic properties consistent with other types of mea-
surements. For more details about the PDF technique, please turn to a recent book

by Egami and Billinge [5].

1.4 Recent advances in the PDF method

Having long been used to study liquids and amorphous materials [12, 13, 14, 15],
the PDF technique has been recently successfully applied on nano-crystalline and
Crystalline materials [16, 17], owing to advances in modern x—ray and neutron sources
and much improved computing power [5]. Why does the PDF technique require those?
The first is the wide Q range necessary for highly crystalline materials whose coherent
scattering extends to the high Q region. The high energy high flux synchrotron
sources and intense spallation sources made quantitative PDF measurements possible
in a reasonable amount of data collection time. Secondly, the intensive data post-
proceSSing requires high speed computers, which have become widely available over
the last decades.
PDF analysis of perfect crystalline materials, such as Si, Ni, not only obtain
the average structures in quantitative agreement with conventional crystallographic

In . . .
ethOCis [18], but also give additional information about the correlated motions be-

 

tween neighboring atoms [19, 20]. Proffen et al. have successfully applied the PDF
method to probe the chemical short range order in Cu3Au [21]. Complementary to
the average structure analysis, Billinge et al. and Bozin et al. have applied the
PDF technique extensively to study the local disorder and inhomogeneities in the
highly crystalline colossal magneto—resistive manganites and high temperature super-
conducting cuprates [22, 23, 24, 25, 26]. When the long range crystallinity gradually
fades such as in nano-crystalline and non-crystalline materials, the reciprocal space
approach taken by the conventional crystallographic methods becomes less and less
effective. However, this decrease of structural coherent length has little effect on the
PDF technique where no long range periodicity is assumed. For example, Petkov et
al. have shown that local and intermediate structure can be obtained from PDF data
when a certain level of local atomic order is preserved [27, 28, 29, 30]. As for non-
crystalline materials such as liquids, glass, and amorphous materials, the very local
Structure only persists and the Angstrom scale becomes the only meaningful length
scale- Here, the PDF technique has been the method of choice for structure investi-
gations [31, .1, 32]. At present, much higher real space resolution is being achieved
[33, 34 , 35]. Sophisticated and intensive modeling methods, such reverse Monte Carlo
(RMC) and potential based regression algorithms, have also become popular to gain

more insight [36, 37, 38, 39: 40l-

1.5 Outline of this thesis

Investigation of the local structure employing the PDF technique is the the central
theme of this thesis work, covering both technical developments and scientific appli-
Chalpter 2 will describe a recent development of a new way to collect PDF data:
rapid acquisition pair distribution function (RA-PDF) method, where the coupling of

hi .
gh energy X—ray with an area detector leads to three orders of magnitude decrease

10

of data collection time.

Chapters 3 and 4 focus on the high resolution PDF analysis of neutron pow-
der diffraction data on two manganite systems: undoped LaMn03 and bi-layered
La2_2$Sr1+2angO7, respectively. The significance of local Jahn-Teller distortions
will be emphasized, and their strong influence on the electronic, and magnetic de-
grees of freedom will be discussed.

Chapter 5 will demonstrate a few applications of the RA—PDF method, e.g., re-
solving the distorted square Ti nets in Tigsb, and the in-situ chemical reduction of
CuO to Cu.

Chapter 6 summarizes the thesis and discusses some future work.

Appendix A describes the the experimental aspects and post-data processing of

the RA-PDF method in detail.

 

11

Chapter 2

The Rapid Acquisition Pair

Distribution Function Method

2. 1 Introduction

In recent times, new materials are emerging at unprecedented rates, and one growing
trend is that their structures become more and more disordered and complex. It
is of great current interest to characterize those nanocrystalline material structures
E3fi'eCtively and routinely. However, the long range structural coherence length at nano-
meter scales challenges conventional crystallographic analysis. As reviewed by Billinge
and Kanatzidis [16], the PDF technique is a promising candidate as evidenced by its
SUCCessful application to solve structures with different levels of complexities [5, 16].
However, conventional PDF measurements are very slow and generally take more
than eight hours, even at a third generation synchrotron or intense spallation neutron
souI‘Ces. This, coupled with the novelty of the approach and the somewhat intensive
data analysis requirements, has prevented widespread application of the technique
in 3$883 such as nano—materials. This chapter will introduce a recent development
T the rapid acquisition PDF (RA-PDF) method where the data collection time is

reduCed by three to four orders of magnitude. A new program PDFgetX2 [41] will

12

 

 

also be briefly described as a user friendly program to obtain the PDF from X-
ray powder diffraction. I will show that high-quality medium-high resolution PDFs
(Qmax S 35.5 A”) can be obtained in a few seconds of data collection time using a
two-dimensional (2D) image plate (IP) detector with easy to use programs. These
developments open the way for more widespread application of the PDF technique to
study the structure of nanocrystalline materials. They also open up the possibility
for experimentally demanding experiments to be carried out such as time-resolved

studies of local structure.

2.2 Description of the modern PDF experiments

Let us first briefly review modern PDF experiments. As we are trying to resolve
inter—atomic distances to one tenth of an Angstrom resolution, this sets the relevant
wavelength of the probing wave to use. From the De-Broglie wavelength equation
/\ = {3‘ with A of 0.2 A, the corresponding energy scales are around 60.0 keV for X-
l'aYS (E = Lifi—WkeVD, 2.04 eV for neutrons (E = fi%(meV)), and 3.76 keV for
electrons (E = %(6V)). Electrons have seldom been the choice for bulk studies due
to their short penetration depth (strong scattering). High energy X-rays and neutrons
have much better penetrating power suitable for bulk studies, which also means their
interactions with matter are rather weak. Thus most quantitative experiments need
to be performed at either high flux synchrotron source or intense neutron spallation
SOUrces, and they are slow because of the weak interactions. X—rays and neutrons are
complementary probes for PDF studies, providing different contrasts for the different

elemental species in the structure. For example, X-rays are scattered by the electrons,

t311118 heavy atoms with more electrons are better “seen” by X-rays [5]

13

2.2.1 Neutron powder diffraction

In spallation neutron sources, neutrons are first generated by bombing the target
materials (mercury, tungsten, etc.) with high energy charged particles (H+). The
hot neutrons are cooled down by a constant temperature moderator (chilled water,
liquid methane, etc.). The neutrons delivered to the sample consist of two major com-
ponents, the thermal neutrons in equilibrium with the moderator, and the “hotter”
epithermal neutrons not completely equilibrated. Therefore, you can imagine a train
of neutrons with different velocities hit the sample and get scattered in all directions.
The scattered neutrons are recorded by the installed detector banks covering wide
solid angles. Those neutrons are also timed at the beginning and end of the flight,
so that their speeds can be computed given the known traveling distances, and thus
the wavelength /\ via the De—Broglie wavelength equation. The momentum transfer
Q simply follows from Q = 47"sin 6, where 26 is the scattering angle. This gives
the method name of neutron time of flight (T OF) method. Neutrons in spallation
sources are pulsed with a repetition rate from 20 to 50 Hz. It is noteworthy that it is
mostly those epithermal neutrons that give the important high Q data. The program
PDFgetN [42] provides a user friendly graphic user interface (GUI) to obtain the PDF
from neutron powder diffraction data.

Even with multiple detector banks, neutron experiments still take around 6 hours,
and this is for a large amount of sample, more than 6 grams. Compared with X—rays,
the neutron incident flux normally achieved is three orders of magnitude lower. One
Way t0 improve the data collection is to increase the covered solid angle of the detector
banks, as in the upgrade of the neutron powder diffractometer (NPD) to NPDF at
the Lujan center at Los Alamos National Laboratory [43]. The newly upgraded
NPDF gives the state of the art PDF data collection time of around 2 to 3 hours
available in the USA. We can also boost the flux of the neutron source for the same

p urpOse. The new spallation neutron source (SNS) project, still under construction,

14

is expected to significantly boost the data throughput rate. However, the reduction

of data collection time will be partially canceled by the smaller sample size used in

future neutron studies.

2.2.2 X-ray powder diffraction

In synchrotron sources, electromagnetic radiation due to the acceleration of cycling
electrons traveling at relativistic speeds give a wide energy spectrum of X-rays. Wig-
glers and insertion devices are used in situations to enhance the brightness at certain
photon energies. Monochromaters are used to select the desired wavelength X—rays.
High energy X-rays, e.g. Z 100 keV, can be routinely achieved at third generation
synchrotron sources such as the Advanced Photon Source (APS) at Argonne National
Laboratory. Multiple slits are used to confine the beam to dimensions in consideration
of either sample sizes or other reasons such as experimental resolution, background,
etc.. With monochromatic X-rays, the scattering angle 26 is scanned step by step
by moving a point solid state detector (SSD). Again from the formula Q = ~47" sin 6,
the scattering intensity I versus Q data are collected. Similar to neutron PDF data
Processing, many steps have to be performed to obtain the PDF from X-ray powder
diffraction data. For this purpose, the menu driven program PDFgetX [44] has been
used,

Conventional PDF X-ray measurements with a SSD normally take more than 8
hours, though the data collection time at a single 26 point usually takes less than 10
seconds. As the wide Q range required by PDF analysis needs easily more than 2000
p OintS, it is the step by step scanning that significantly changes the experimental data
collection time scale. Considerable flux increase will be expected when the fourth
generation free electron laser X-ray source becomes available. However, the data
collection time then will possibly be limited by the time taken to move and stabilize

t . . . .
he detector. One alternative approach 18 to use multiple pomt detectors, or area

15

".1

 

h‘

detectors. Use of multiple detectors has been tested in our group and others. First, the
experiment becomes rather difficult technically, e.g. even the accurate measurement
of sample to detector distance is different for a point detector. And also different
detector elements are found hard to work in a consistent way due to various electronic
issues. Attempts with an area detector such as an image plate have also been realized
in our group with the Debye-Scherrer geometry [45]. The counting rate is greatly
improved. However, the limiting factor becomes the time cost to take out the image
plate, scan it, and then put it back in. Also, signal-background ratio issues were never

statistically resolved. The RA—PDF technique developed here makes use of a planar

2D image plate used in a transmission mode.

2.3 Use of the image plate as an X—ray detector

The detection media of an image plate (IP) is one layer of very small crystalline grains
0f photo-stimulable phosphor mixed with organic binders. The grain size is about
5 Hm; the layer thickness is usually around 150 pm. There is usually a protective
layer (about 10 pm) on the top and one support layer on the bottom (about 250 um).
A POIyester backing plate is often used to make it reusable and easy to handle as a
flexible plastic plate. The IP’s sensitive surface can be easily identified by its all-blue-

White or all-white appearance, while the backing side is usually black or gray.

2-3- 1 Principle of X-ray image plate detection

The photo-stimulable phosphor used nowadays is BaF(Br,I):Eu2+ (previously it was
BaFBrzEu2+) [46]. This material is capable of storing a fraction of the absorbed X-ray
energy, and emitting photo—stimulated luminescence (PSL) later when stimulated by
visible laser light. The mechanism of the PSL is illustrated by the energy level scheme

of 13a-IfBrzEu” shown in Fig. 2.1. The energy level of interest is the Eu2+ ground

16

 

 

 

state, 6.5 eV below the conduction band, and the vacant F +(Br+, 1+) centers 2.0 eV
below the conduction band. The ground state has Eu2+ and vacant F+ centers. The
incident X—ray will pump electrons from the valence band to the conduction band.
Despite being a complicated process, the X-ray irradiation results in a certain num-
ber of Eu3+ and F pairs proportional to the absorbed X-ray energy. The F centers
are caused by the absence of halogen anions from their designated positions in the
lattice. The energy trapping state of F centers is meta-stable with a long lifetime.
However, the trapped electrons can be easily excited by visible light (2 2 eV) to
return to the conduction band. One of the occurring processes is the recombination
of Eu“+ with one election to Eu“, along with the emission of a photon of 3.2 eV
(blue light). A red He—Ne laser is usually used for read-out. Its wavelength (632.8
nm) is considerably separated from the PSL wavelength (390 nm). A conventional
high-quantum—efficiency photo-multiplier tube (PMT) is used to collect the photo-
stimulated photons. The signal is then amplified and digitalized to be processed by
computers. The remaining F centers in the phosphor after read-out can be further

erased by exposing to visible light. In practice, it is more efficient to bleach it with a

Powerful halogen lamp (500W).

2.3.2 Characteristics of image plate detectors

The performance of the image plate as an X-ray area detector has been reviewed by
Amemiya and others [48, 46, 47, 49, 50, 51, 52, 53] in great detail. Here we briefly

summarize those characteristics for completeness.

Detective quantum efficiency The definition of detective quantum efficiency (DQE)
is DQE = (.S‘(,/N,,)2/(.S”,/N,-)2 where S is signal and N is noise (the standard
deviation of the signal), and subscripts o and i refer to the output and input,
respectively. As scattering events are random following the Poisson distribu-

tion, the (Si/M)2 term is just the 5,. The background noise level of the IP

17

 

..i

conduction band

 

26V

.-_-__----.__?

 

 

b-----4I

 

 

F+:——-‘-F

6.5 eV

 

 

4.6 CV 8.3 CV

3.2 eV

 

 

 

 

‘--------q-----

 

 

 

Eu2+ _—“‘< - -- Eu3+

 

__ l

Valence band
\— Excitation
‘ - - -- PSL

 

Figul‘e 2.1: Energy level scheme of image plate phosphor material BaFBrzEu2+ pro—
posed in Ref. [47]. Arrows indicate the excitation and PSL process as suggested by
Takshashi et al. [48].

18

 

is usually less than 3 X-ray photons/ (100 um)2, which in practice largely de-
pends on the noise level of the IP readout system. The relative uncertainty
of the IP deviates from an ideal detector at high exposure levels (2 100 X-
ray photons/ (100 um)2). The “system fluctuation noise” saturates the relative
uncertainty of the IP results around the 1% level. The origin of the system
fluctuation noise includes non-uniformity of absorption, non-uniformity of the
color-center density, fluctuation of the laser intensity, non-uniformity of PSL

collection, and fluctuation of the high-voltage supply to the PMT.

Dynamic range and linearity of the IP response The dynamic range refers to
the detectable weakest and strongest signal with acceptable distortion. One

dimensionless definition is taken as the ratio between the maximum counts in

 

the linear regime and the lowest detectable signal (determined by the intrinsic
noise of the detector system). The IP has rather good dynamic range of 105:1,
with linear response range from 8x101 to 4x 104 photons/ (100 pm)? The error
rate is less than 5%. Sometimes, two sets of PMTs are necessary to cover the

entire dynamic range of the IP.

Spatial resolution and active area size The determining factor of the spatial res-
olution of the IP is the laser-light scattering in the phosphor during the readout.
The laser-light scattering originates from a mismatching of the refractive indices
at the boundaries of phosphor crystal grains. When the IP is read with a 100 um
Square laser size, the spatial resolution is limited by the linear spread function
with 170 um full width at half maximum (FWHM). The point spread function
of IP systems has been studied by Bourgeois et al. [54] in detail. The active
area size of the image plate is rather important for PDF studies which need
large IPs. Commercially available IPs have various standard sizes ranging from
127x127 mm2, 201x252 mm2, to 201x400 mm2. The largest automated IP

System currently contains the 345 mm diameter IP disk (MAR345).

19

 

 

 

:15

J i

3

215

 

IP response (D.S.U)
1.5. 2

1

 

“ .

(i) per incident photon .

 

 

 

'0. ..

c:

o L I A I a I A I n I A I n I
O 8 1 6 24 32 4O 48 56

Energy (keV)

Figure 2.2: The IP response as a function the the energy of an X—ray photon. (i) is
the IP response per incident X-ray photon. (a) is the IP response per absorbed X-ray
PhOton. The unit of the ordinate corresponds toe the background noise level of the IP
SCanner (the data were digitized from the the corresponding figure in reference [55]).

Energy dependence The measured intensity on the IP is proportional to the de-
posited energy in the phosphor layer. This depends on the absorption efficiency
per incident photon and the deposited energy of one absorbed photon. Un-
fortunately, the IP response is energy dependent, and its energy dependence
is shown in Fig. 2.2 from 4.0 to 60.0 keV. We are only concerned with the IP
response per incident photon (lower curve). Except for the discontinuity at the
Ba K edge, the detection efficiency decreases with photon energy in the high
energy regime of our interest. This necessitates an energy dependent detection

efficiency correction discussed later in this chapter.

20

 

Fading and other factors The IP stores the absorbed energy in the meta-stable F
centers and thus will fade with time after exposure. Exposing the IP to visible
light or high temperature will expedite the fading. At room temperature, the
IP is usually stable within one hour. The fading lifetime does not depend on the
X-ray energy during exposure, as it is the F centers that store the energy. The
surface roughness and the non-uniformity of the IP is about 1-2%. A calibration
image is usually obtained by exposing the IP to a flood field from a radioactive
source, and then used to correct the recored images. In principle, the IP, being
an integrating-type counter, is not counting rate limited. However, extremely
intense X-rays seem to cause permanent damage to the phosphor layer. With
the exercise of common precautions, the IP has proved to be a very reproducible

and reusable X-ray area detector.

To summarize, the IP detector system meets various requirements of X-ray area
detectors. They are, high detective quantum efficiency, a wide dynamic range, a
linearity of response, a high spatial resolution, a large active area size, and a high

counting rate capability.

2.3.3 The IP and PDF method

Recent developments have shown the utility of 2D detector technology in scattering
studies of liquids. A recent report by Crichton et al. [56], has made use of integrated
two—dimensional IP data for in-situ studies of scattering from liquid GeSeg. This

Stud)’, and others more recently, demonstrate the feasibility of using IPs for diffuse
scattering measurements though the measurements have been limited in real-space
resolution, with Qmax g 13.0 A‘1 [57, 58], making them less suitable for the study
Of Crystalline and nanocrystalline materials. Image plates have also been successfully
used to study diffuse scattering from single crystals [59]. A Debye-Scherrer camera

utlliZing IPs has also been tested in our group and shows promise for lower energy

21

X-ray sources such as laboratory and second generation synchrotron sources [45].
Successful application of IP technology to the measurement of quantitatively re-
liable high real-space resolution PDFs requires that a number of issues be resolved.
For example, it is necessary to correct for contamination of the signal from Comp-
ton and fluorescence intensities and for angle and energy dependencies of the IP
detection efficiency [55, 60]. Here we show that high quality medium-high real-space
resolution PDFs can be obtained by applying relatively straightforward corrections.
As expected, the quality of the PDFs is lower in samples comprising predominantly
low atomic-number elements. Nonetheless, even the PDFs of these samples prove
adequate. Further quality studies were also carried out to investigate the RA-PDF
capabilities. We also compared the IP data with the conventional solid state detector
data to verify that quantitatively reliable structure information can be obtained from

the RA-PDF method.

2.4 RA-PDF experiment

Our first PDF experiment with an IP detector took place in October, 2002. Two
essential components were key to the success of this first experiment. One was the
experience with the online image plate detector, mainly from Dr. Peter Chupas in
Prof. Clare Grey’s group at Stony Brook University, Dr. Jon Hanson from Brookhaven
National Lab, and Dr. Peter Lee from Argonne National Lab. The other was the
expertise with the PDF technique from our group. Part of the results have been
published [61]. Technical descriptions, together with the experimental setup diagram,

can be found in Appendix A.

22

 

2.5 The proof of principle study

2.5.1 Description of the experiments

All diffraction experiments for our first study were performed at the l-ID beam line
at the Advanced Photon Source (APS) located at Argonne National Laboratory, Ar-
gonne, IL (USA). High energy X—rays were delivered to the experimental hutch using a
double bent Laue monochromator capable of providing a flux of 1012 photons/second
and operating with X-rays in the energy range of 80—100 keV [62]. Two energies
of X-rays were used for the experiments, 80.725 keV (0.15359 A) and 97.572 keV
(0.12707 A), with the 80.725 keV experiments being performed first. In the optics
hutch, a gold foil was installed after the monochromator between two ion chambers,
attenuating the flux of the beam by approximately 30%. Calibration of energy at

80.725 keV was achieved using the gold absorption edge as the reference.

A Mar345 image plate camera, a round disk with a usable diameter of 345 mm,
was mounted orthogonal to the beam path, with the beam centered on the IP. When
operating with 80.725 keV X—rays, a LaB6 standard was used to calibrate the sample to
detector distance and the tilt of the IP relative to the beam path, using the software
Fit2D [63, 64]. For calibration of the IP, the wavelength was fixed to represent
the gold absorption edge, 0.15359 A, at 80.725 keV. When the X-ray energy was
increased from 80.725 to 97.572 keV, the sample to detector distance was first fixed
at the value determined at the Au edge and the wavelength was calibrated using the
standard LaBs. The IP camera was then moved closer to the sample and the new
sample to detector distance was obtained from refinement by fixing the wavelength
at 0.12707 A. Sample to detector distances of 317.28 and 242.12 mm were used for
collection of data with measured Qmax of 21.0 A‘1 and 30.0 A’l, respectively. In
practice, Qma,c of 18.5 A‘1 and 28.5 A”1 were used due to the corrupted data near

the IP edges. The beam stop is a solid tantalum cylinder (diameter 3.1 mm), with an

23

 

 

800 1000 1200 1400 1600 1800 2000 2200 2400 2600

2600 2600
2400 2400
2200 2200
2000 2000
I 800 1 800

g

o

C!
1600 1 600
1400 1400
1200 1200
1000 1000

800 800

 

800 1000 1200 1400 1600 1800 2000 2200 2400 2600

 

 

Columns
‘~ . i I I l l l I I I 1 -. .» . ‘ . I
I I T I I 7* I I I 'IT I I I l l I
300 1000 3000 10000 30000
Intensity

Figure 2.3: (color) Two dimensional contour plot from the Mar345 Image Plate Detec-
tor. The data are from nickel powder measured at room temperature with 97.572 keV
incident X—rays. The concentric circles are where Debye—Scherrer cones intersect the
area detector.

24

indentation machined to a depth of approximately 2 mm to accept the beam. With a
beam stop to sample distance of approximately 150 mm, the scattering angle blocked
by the beam stop is less than 1 degree, dependent upon sample to detector distance.
This gives a Qmin limited to approximately 1 A"1 . With our beam stop to sample
distance of approximately 150 mm, the scattering angle blocked by the beam stop is
less than 1 degree. This gives the energy dependent Qmin limited to approximately
0.6 A"1 with 80.725 keV X-rays. In many crystalline materials with small unit cells
this is not a problem. When Bragg—peaks are lost at low-Q due to this limit, a weak,
long-wavelength oscillation results in G(r), which is not fatal but, ideally, is to be
avoided.

The methods used to synthesize a-A1F3, Bi4V2011 and Bi4V1.7Tio_3Olo.35 have
been reported elsewhere [65, 66]. Ni was purchased from Alfa Aesar (99.9%, 300
mesh) and was used as received. Fine powders of all the samples were measured in
flat plate transmission geometry, with thickness of 1.3 mm packed between kapton
foils. The beam size on the sample as defined by the final slits before the goniometer
was 0.4 mm x 0.4 mm. Lead shielding before the goniometer, with a small opening
for the incident beam, was used to reduce background. All raw data were integrated
using the software Fit2D and converted to intensity versus 26 (the angle between
incident and scattered X-rays). An example of the data from nickel measured at
97.572 keV is shown in Fig. 2.3. The integrated data were then transferred to a
home-written program, PDFgetX2 [41], to obtain the PDF.

Distortions in intensities when using flat IPs have been addressed in single crystal
crystallography, and arise from the fact that IPs are of a finite thickness, often between
100-200 microns [60, 67]. At the high energies (Z 60 keV) needed to measure S (Q) to
a high value of Q with commercial IP cameras, absorption of the X-ray photons by the
phosphor is very small and most of the X-rays travel straight through the phosphor.

The absorption, and thus the measured intensity, is then dependent on the path

25

- ‘ T.nI' -mq

 

length of the beam through the IP phosphor at all incident angles and can easily be
corrected [60]. This oblique incidence correction assumes that the correction is equal
in all directions, thus, care was taken to ensure the IP was mounted orthogonal to the
beam. The attenuation of scattered photons after the sample, due to air absorption, is
angle dependent. This effect is negligible due to the very small absorption of the high
energy X-rays between the sample and the detector and is ignored. The count time is
always adjusted to ensure that there are no saturated pixels in the detector. Standard
corrections for multiple scattering, polarization, absorption, Compton scattering, and
Laue diffuse scattering were also applied to the integrated data to obtain the reduced
structure function F (Q) (Fig. 2.4(a)). Direct Fourier transformation gives the pair
distribution function G(r) (Fig. 2.4(b)). The average structure models were refined
using the profile fitting least-squares regression program, PDFFIT [11]. Rietveld

refinements of the data were performed with GSAS [68].

2.5.2 Early results

All the experimental data from samples Ni, a-AlF 3, Bi4V2011 and Bi4V1.7Tio.3Olo,85
show excellent counting statistics over the entire Q range, reflecting the significant
advantage obtained by extracting 1D data sets by integration of a 2D area detector.
Image plate exposure time, ranging from one second (Ni) to 10 seconds (a-A1F3), was
chosen carefully for each sample to maximize the dynamic range of the IP without
saturating the phosphor on the Bragg peaks. Data collection was repeated a number
of times until acceptable statistical errors were observed at high-Q in the reduced
structure functions, as shown in Figures 2.4(a), 2.5(a), and 2.6(a). All the PDFs,
G (r), show either superior or acceptable qualities.

The first example, shown in Fig. 2.4, is from standard Ni powder with Qmax of
28.5 A“. The Ni PDF, G (r), in Fig. 2.4 (b) appears to have minimal systematic errors

which appear as the small ripples before the first PDF peak at r = 2.4 A. These result

26

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3‘03)
Gel.
I N
a _
E37 21 24 27 _
cr - l .
I l . l _ l ..
O I'luu'ulrllflyvtll'.
0 8 16 24
6:103")
m I T I ' I #-
H
5 .i . '
’3: _ ‘lll ll 1:
23° my H
m
I’- I . L .. I
0 5 10 15

Figure 2.4: (a) The experimental reduced structure function F(Q) = Q * (S (Q) - 1)
of Ni powder. The inset is a zoom in of high Q region showing the excellent signal to
noise. (b) The experimental G (r) (solid dots) and the calculated PDF from refined

r (A)

structural model (solid line). The difference curve is shown offset below.

27

from imperfect data corrections and their small amplitude is a good indication of the
high quality of the data. A structural model (space group Fm3m) was readily refined,
and gave excellent agreement with the data as shown in Fig. 2.4(b). The lattice
parameter (3.5346(2) A) and an isotropic thermal displacement parameter (U =
0.005184(6) A2) were refined. The lattice parameters reproduce the expected values
given in previously published data [18]. In spite of the simplicity of the Ni crystal
structure, the exceptional quality of both the experimental Ni PDF and refinement
indicates that the necessary data corrections of image plate data with high Q range
(28.5 A‘1 in this case) can be carried out properly and with an acceptable level of
accuracy.

The weakly scattering a-AIF3 compound, in addition to a slightly more complex
structure than Ni, presents a greater challenge to proper data correction due to the
majority contribution of Compton scattering in the high Q region. The reduced
structure function and resulting PDF are shown in Fig. 2.5. The data were successfully
refined using a previously reported model from the literature (space group R3c) [69].
The fit is shown in Fig. 2.5(b). The overall quality of the fit is good with a weighted
profile agreement factor [11] of 3 percent. The refined lattice parameters are a =
b = 4.9420(3) A, c = 12.4365(2) A. The fractional coordinate of the F atom is
:1: 2 0.4267(4). These numbers are consistent with the Rietveld refinement of the
same data, which produced refined lattice parameters of a = b 2 4.9383(4) A and
c = 12.4271(1) A, and a fractional coordinate, :1: = 0.4287(4), for the F atom. Both
refinements are consistent with the published data of 4.9381(5) A, 12.4240(3) A, and
0.4309(4), respectively [65]. The successful application of PDF analysis of oz-Ang
data indicates this technique is capable of handling weakly scattering materials and
still gives reliable information.

In addition to the highly crystalline and rather simple nickel and a-A1F3 struc-

tures, the more complex and disordered layered Aurivillius type oxide anion conduc-

28

 

 

 

Q(S(Q)-1)

 

 

 

 

 

 

 
  
  

 
     

.0... .0...”

 

G(r) (ii—2)
s? 0

 

-4

 

 

o I 5 .1OI1I5
1‘03)

Figure 2.5: (a) Experimental reduced structure function F(Q) = Q * (S(Q) — I) of
a-AlF3, (b) G(r) and modeled PDF of a-AIF3, notations as in Fig. 2.4.

29

 

 

 

 

Q(S(Q)-1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N

H l

..- I

g : (b)

(5

Cl

0;»:

«I, V '_ i I "i ‘I _l‘
O 5 10 15 20

r (A)

Flgure 2.6: (a) Reduced structure function F(Q) of Bi4V2011, (b) Experimentally
‘f’btained G(r)’s of Bin/20,, (solid dots) and Bi4V1,7Tio_3010,85 (solid line). The dif-
.erences between them are plotted below with an offset, noting the high reproducibility
In the low 7‘ region (also shown in the inset).

30

tors Bi4V2011 and Bi4V1,7Ti0_3010.35 were examined. The structure is derived from
the ordered Bi4M02012 structure, which contains alternating BigOg+ layers spaced
by corner sharing perovskite M006 layers [66, 70]. The vanadium analogues contain
significant disorder in the vanadium layers due to oxygen vacancies. Vanadium can
exist in 4, 5, and 6 coordinate environments, inducing significant static disorder in
these layers, which is further complicated by dynamic disorder resulting from the
high anionic conductivity of these materials. Titanium substitution on the vanadium
sites leads to the stabilization of the high temperature y-phase. The experimen-
tally obtained reduced structure function, F(Q), of BI4V2011 is shown in Fig. 2.6(a).
A combined PDF and Rietveld analysis was carried out on the same data set and
detailed results will be reported elsewhere. To examine the reproducibility of the sys-
tematic errors in the IP data analysis, the measured data for the compounds Bi4V2011
and Bi4V1_7Ti0,3010,85 were compared (Fig. 2.6(b)). Clearly, the ripples in the low r
regions are rather small and highly reproducible, implying the high reliability of the
PDF data especially for comparative studies. For example, very small differences exist
around the first V-O bond length of 1.80 A, as evident in the inset of Fig. 2.6(b). The
Significant differences beyond the first peak reveal the somewhat drastic structural
Changes on longer length scales. This also shows that RA-PDF is capable of capturing
Structural changes where they exist, reproducing local structures if unchanged.
Medium-high real-space resolution PDF data analysis from crystalline materials
was performed by using image plate data and shows promising results. Comparable
or even better statistics than from conventional X-ray measurements can be achieved
With Significantly shorter counting times. The new combination of a real space probe
and fast counting time opens up a broad field for future applications to a wide variety
of ITlaterials of both scientific and technological interest. For example, PDF methods
could be used to study structural changes under in situ conditions [71] and the time

dev'5310pment of chemical reactions and biological systems over short time scales of

31

seconds may be studied.

2.6 Comparison of the RA-PDF with conventional

X-ray PDF method

The use of an image plate area detector for quantitative PDF analysis needs to be
properly validated due to the experimental uncertainties described in early sections.
For example, the lack of energy resolution intrinsic to the IP, and the energy de-
pendence of the IP response. Therefore, we quantitatively compare the X—ray Data

collected with the IP with that from energy resolved solid state detector (SSD).

2.6.1 Description of the experiments

A powdered sample a-A1F3 was measured with the IP and SSD detectors under
the same experimental conditions at the l-ID beam line at the Advanced Photon
Source (APS) located at Argonne National Laboratory, Argonne, IL (USA). The X-
ray energy used is 99.8 keV, with beam size of 0.4 by 0.4 mm. The sample was
loaded into a quartz capillary with diameter of 1 mm. The beam stop is a solid
tantalum cylinder (diameter 3.1 mm), with an indentation machined to a depth of
approximately 2 mm to accept the beam. Data collection with the IP detector was
Performed first. A Mar345 image plate camera was mounted orthogonal to the beam
path , with the beam centered on the IP. Experimental calibration measurement such
as tllle sample to detector distance is the same as described elsewhere [61]. A single
e’“I>()sure of 20 seconds was collected. Then, an SSD was used by moving the IP
0111; and keeping all other setups. The Ge SSD detector (Canberra GL 01108) was
mounted on the 26 arm for an angle dispersive scan over the same 26 range as in the
1P measurement (z 35 degrees). The energy window of the single channel analyzer

Was set to include only elastic and Compton intensities. Two 26 scans were made

32

with the same step size of 002°. The first is from 0.8 to 15.0 for 4 seconds at each
point; the second is from 15.0 to 35.0 for 6 seconds. The total measurement time
was around 10 hours. Scattering from the an empty capillary was also collected with
both detectors for background subtraction. The program PDFgetX2 [41] was used
to process the X-ray powder diffraction data with details described earlier. Structure
model refinements with experimental PDFs were carried out with PDFFIT [11], a
profile fitting least-squares regression program. The powder diffraction data are also

analyzed by reciprocal space Rietveld refinements with program GSAS [68].

2.6.2 Results

Fig. 2.7(a—1) and (a-2) show the respective reduced structure factor function F (Q)
with Qmax of 21.0 A'l. The smaller scale of the IP data in (a-l) comes from its
lower Q space resolution than the SSD data. A more significant difference is the
counting statistics in the high Q region. The IP data with 20 seconds collection time
appear to show much better signal to noise ratio than the SSD data (10 hours) in
the high Q region. The corresponding G(r)s are shown in Fig. 2.7(b-1) and (b—2) as
filled circles. The Al-F bond length around 1.80 A gives the first physical PDF peak.
The high frequency spurious ripples below 1.80 A serve as a good indicator of the
level of experimental uncertainties coming from systematic errors. Both PDFs show
comparably high qualities with those ripples being rather small. It is noteworthy
that the somewhat expected high ripple levels in SSD data from the significantly
Worse counting statistics is absent in the G (r) As a direct comparison of the two
exPerimentaI PDFs subject to different experimental Q space resolutions, the crystal
Strlleture (space group R30 [69]) of a-A1F3 is then fitted to both PDFs to see whether
the Same structural information can be obtained. The calculated PDFs from refined
crystal structures are shown in Fig. 2.7(b—1) and (b—2) as solid lines accordingly with

the difference curves plotted below. Both fits show excellent agreement with the

33

 

 

 

 

 

 

 

 

 

 

In
{‘2 ‘1?
5 ' S

In
6‘ . 3
T: o

o

m

l

o
9* f
l I
53 ‘5
E? 3
v
Lo 0

0

(7| 1 . I .

0 5 15 20

 

10
Q (11")
Figure 2.7: (a—l) and (a—2) show the experimental reduced structure function F (Q) =
Q(S(Q) —- 1) of a-A1F3 from IP and SSD, respectively. (b—l) and (b-2) show the
corresponding C(r) obtained via a sine Fourier transform of (a-l) and (a—2) in filled
circles. The solid curve on top of each experimental C(r) is the calculated PDF
frorn the refined crystal structure, with the difference curve show below offset. The
horizontal dashed lines indicate the G(7‘) uncertainties at l a level.
Weighted residual factor pr of 15% and 26% for IP and SSD data, respectively. The
origin of both difference curves (random noise like) appears to be determined by the
exp erimental statistical noise, and the fact of higher statistical noise level in SSD data
bec<DIhes rather apparent. To see how the refined crystal structures differ from each
Other, we calculated PDFs from them with the same experimental factors, and show
the tWo calculated PDFs in Fig. 2.8. Their minimal differences evidenced from their
difi‘el‘ence curve shown below indicates the two refined models contain quantitatively

the Same structure information. Rietveld refinement of both the IP and SSD data

were also carried out. Table 2.1 shows the obtained lattice parameters from PDF and

34

 

 

 

 

.ng

A -.,. 1.. -;.~ ., r f
“aw-«J WWW7:i/‘tw'v‘llfxwi‘w'v'fiy'"-"‘L-'\I’B'U"“”""‘~I"Lv-"'i“! \‘Ifwwllvw VWWVWM¢¥
' 1
. I .

G(r) (3‘2)
0
a
q
< .
i
<3“

 

 

-5

 

 

5 1o 15 20
r (K)

Figure 2.8: Calculated PDFs from the refined crystal structures with the IP and SSD
G(r)s. Their difference is shown offset below.

 

0

Rietveld refinements of both IP and SSD data. The IP and SSD data refinements
result in somewhat large deviations on the lattice constants. One possible reason
COUld be the large uncertainties during our experimental calibrations. In addition,
the different instrumental resolution functions of the IP and SSD also contribute to
the deviations in lattice constants, which is discussed in detail in [10]. We would like
to point out that other lattice parameters from PDF refinements of both data sets are
in Very good agreement, while the Rietveld refinements give rather different thermal

factOrs.

35

 

 

IP (PDF) 1P (Rietveld) SSD (PDF) SSD (Rietveld)
a=b(A) 4.9468(3) 4.9381(3) 4.9221(5) 4.9216(2)

c(A) 12.4610(17) 12.444304) 12.3911(28) 12.4000(10)

Fa: 0.4257(3) 0.4281(2) 0.4267(5) 0.4277(3)

A1U...(A2) 0.0034(1) 0.0010(1) 0.0027(1) 0.0060(6)
( ( ( (

FU,-,0(A2) 0.01001) 0.00101) 0.00971) 0.0059 6)
R... 0.15 0.04 0.26 0.10

 

 

 

 

 

Table 2.1: Lattice parameters of a-A1F3 from PDF and Rietveld refinements of both
IP and SSD data. The estimated standard deviations are shown in the parenthesis

immediately following the values.

2.7 New data analysis program: PDFgetX2

PDFgetX2 is a GUI driven program to obtain the pair distribution function from X-
ray powder diffraction data. All codes used now are written by me under the guidance

of Prof. Simon Billinge. All source codes amount to more than 40,000 lines.

2.7. 1 The crystallographic problem

The pair distribution function (PDF) reveals directly in real space the inter-atomic
distances in a material. Recent applications have proved the PDF technique as a
Powerful local structural probe of nanostructured materials [5, 16], as well as its
tr adi tional use to study liquids and glasses [31, 32]. The experimental PDF is obtained
by a. Sine Fourier transformation of the total-scattering structure function S (Q), where
Q is the magnitude of the scattering vector. To obtain the S (Q) from raw scattering
intellSities many corrections have to be made to account for various instrument and
Sample effects. Adding to this complexity, most existing X-ray data PDF analysis
80ft"’Vaares are menu driven and less user-friendly. Reproducing an earlier data analysis
has been difficult because data processing parameters are usually kept in the notebook

0t, . . . . .
her than With the data. Cross-platform compatiblllty has also been an issue.

36

 

 

2.7.2 Method of solution

In program PDFgetX2, a user—friendly graphical user interface (GUI) has been built
to facilitate user interactions with data. Standard corrections [5] due to background
subtraction, sample absorption, polarization, and Compton intensities are available.
Particularly, for the recent RA-PDF development [61], oblique incident angle cor-
rection and and empirical energy dependence of the detection efficiency are also im-
plemented. Standard uncertainties due to finite counting statistics are estimated
and propagated in all steps. The final S (Q) and G (7‘) data files are multiple-column
ASCII files with the processing parameters in the header. The S (Q) data also contain
the Faber-Ziman coefficients for all partial structure factors as additional columns.
The G (7') file format is compatible with the PDF modeling programs PDFFIT and
DIS CUS [11, 72]. The interactive data language (IDLI) is chosen as the software
environment, ensuring cross-platform compatibility. In comparison with our menu
driven program PDFgetX [44] (no longer supported by us), the PDFgetX2 offers

numerous new features and expanded capabilities.

2.7 .3 Software and hardware environment

The Commercial IDL licensed distribution (version 6.0 or higher), or the freely down-
I(Jada—IDIe IDL Virtual Machine (IDLVM), is the only prerequisite to run PDFgetX2.
The IDLVM is freely available from the download section of the IDL website at
http: / / Www.rsinc.com / download / . Platforms supported by IDL include Linux/ UNIX,
WINDOWS, and MACINTOSH. No specific hardware is used by PDFgetX2. Instal-
lation of IDL may use up to 200 MB hard disk space, while the current PDFgetX2

dlStril>11tion takes about 17 MB disk space.

1
5011371va is a registered trademark of Research Systems, Inc. for their Interactive Data Language
are

 

37

 

 

 

2.7 .4 Program specification

Program PDFgetX2 should run in the same way on all supported platforms as the
same source is used. The look and feel of the GUI may vary slightly. The program
offers flexible choices of data corrections. Essentially all processing parameters are ac-
cessible from the GUI. Most intermediate data during correction steps can be directly
visualized for quick problem diagnosis. The S (Q) and G(r) data are automatically

saved by default.

2.7.5 Documentation and availability

A short tutorial with example data sets can be found in the PDFgetX2 manual
that is distributed with the main program, which also includes a reference guide.
Program executable and related help files are downloadable from PDFgetX2 web

page at http: / / www.pa.msu.edu / cmp / billinge-group / programs / PDFgetX2 / .

2.8 Conclusions

The RA-PDF method features high energy X—rays coupled with an area detector. Use
Of high energy X-rays provides enough Q space range for quantitative PDF analysis.
Use of an area detector offers three to four orders of magnitude reduction of data
collection time. The new data analysis program PDFgetX2 implements necessary
corrections to account for the characteristics of the IP detector used. We have shown
With Convincing examples that high quality medium-high resolution PDFs can be
Obtailled from wide range of materials. The RA-PDF method and easy to use program
PDB‘getX2 expect to significantly lower the barrier for wide-spread PDF applications
to a. broad scope of crystalline and nano-crystalline materials. For example, more
Samples can be measured, more detailed phase diagrams can be mapped out with

(“11¢er data collection.

38

 

 

The PDF technique can now be introduced to new types of experiments. Time
resolution of the RA-PDF method is around 2 minutes, which is limited by the read-
out and erasing time. Attempts with the use of solid state area detectors (18 frames
per second) have also been made in our collaborations. A large fraction of time
resolved experiments can benefit from high real space resolution PDF local structure
analysis simply by moving the detector closer to the sample and using high X-ray
energies. The overall experimental setup is also significantly simplified with no moving
parts during data collection. Systems at transient states can be probed with RA-PDF
method as well. This puts much less stringent requirements on sample conditions.
Materials under extreme conditions such as high pressure, high temperature become

possible for PDF studies. Chapter 5 describes some recent RA-PDF applications.

39

 

Chapter 3

Orbital Correlations in the
Pseudo-cubic 0 and Rhombohedral
R Phase of Undoped LaMn03

3.1 Introduction to manganites

Manganites here refer to a family of manganese oxides known to show the colossal
magneto-resistive (CMR) effect, where the electric resistivity changes drastically un-
der relatively weak magnetic fields. Partially driven by their potential technological
applications in magnetic field sensors and magnetic read-write heads, the CMR man-
ganites have received extensive studies both theoretically and experimentally over
the past decades. Maybe, even greater interest in manganites is propelled by their
challenge to our current knowledge of condensed matter physics. Of the family of
manganites, the very rich phase diagram and fascinating properties provide excellent
playground for scientific exploration. The manganite compounds represent one of the
best examples of strongly correlated electron systems, a dominant theme of current

times.

40

 

. .
’ \ .— ._

 

3.1.1 Earlier studies

The name “manganites” was first given by J onker and Van Santen in 1950 [73], whose
work is the first on the family of manganites. The starting material was LaMn03
which crystallizes in the cubic structure of the mineral perovskitel CaTi03. The per-
ovskite structure for a generic compound ABO3 is shown in Fig. 3.1. The important
structural motif is the corner shared 806 octahedron, i.e., Mn06 in the mangan-
ites. In their pioneering experiments [73, 74], the trivalent La atoms were substituted
by divalent Ca, Sr, or Ba atoms to obtain polycrystalline samples of (La,Ca)Mn03,
(L3,SI‘)MDO3, or (La,Ba)MnO3. One of the most intriguing observations was the
striking correlation between the ferromagnetism (FM) and electrical conductivity:
concomitants of paramagnetism to PM and insulator to metal (IM) transition driven
by either doping or temperature.

The origin of the ferromagnetic phase was first attributed to a positive indirect-
exchange interaction. This view was soon replaced by the so called “double ex-
change” mechanism proposed by Zener in 1951, which still remains at the core of our
understanding of these magnetic oxides. In a series of seminal papers [75, 76, 77],
Zener “interpreted the ferro-magnetism as arising from the indirect coupling between
incomplete d-shells via the conducting electrons”. A quantitative relationship be-
tween electrical conductivity 0 and the Curie temperature (Tc) was also developed:
a ’5 (are2 / ah) (Tc / T), where a: is the fraction of Mn4+ among all Mn ions and a is the
lattice constant. Excellent agreement with the data of Jonker and Van Santan was
found within a factor of 2. Another corner stone of double exchange theory came
from Anderson and Hasegawa [78]. Their perturbative approach deduced that the
effective hopping amplitude between neighboring Mn ions is proportional to cos(6 / 2),
where 6 is the classical angle between the core spins of neighboring Mn ions.

In 1955, the structure of magnetic ordering over the entire composition range of

 

1Perovskite is named for a Russian mineralogist, Count Lev Aleksevich von Perovski

41

 

 

 

 

Figure 3.1: Crystallographic representation of the perovskite structure with chemical
formula ABOa. A is the cation on the corner, B is the cation in center of the Os
octahedron, O atoms lie on the face centers.

42

La1_xCaanOg was discovered in a remarkable early neutron diffraction studies by
Wollan and Koehler [79] from Oak Ridge National Laboratory. The early magnetic
phase diagram obtained by the neutron scattering technique closely matches the mod-
ern version coming 50 years later [80]. As a function of Ca doping, many interesting
ferromagnetic and anti ferromagnetic phases were found. It is worth noting that
evidence of charge ordering was also reported in the anti ferromagnetic phases. Lat-
tice distortion was also studied with X-ray diffraction measurements in these early
experiments.

Following the paper by Wollan and Koehler, Goodenough presented a theory of
semi-covalent exchange [81], and successfully explained the observed rich magnetic
and structural phase diagram. Kanamori revisited this problem, and compiled the
still widely used Goodenough—Kanamori rules [82, 83]. More details will be discussed
later.

In the following years, not much attention was given to manganites. Only to men-
tion a few, Morrish et al. in 1969 [84] found a way to grow high quality millimeter
long single crystal manganites with composition (La,Pb)MnO3. The results of pre-
vious studies on polycrystalline materials were confirmed. Searle and Wang [85, 86]
proposed a phenomenological model based on a strongly spin-polarized conduction
band to explain the ferromagnetism. Nonetheless, the family of manganites enjoyed a
greatly renewed interest since the discovery of the colossal magneto-resistive (CMR)

effects.

3.2 The colossal magneto-resistive effect

The earliest account of magneto-resistance (MR) data was reported by Volger in as
early as 1954 [87]. The observed decrease of resistivity in magnetic fields in the
ferromagnetic phase was puzzling, but the true size of this effect was not appreciated.

About 45 years later, notably large MR effects were found in Nd0,5Pb0,5MnO3 bulk

43

 

 

-1soo
L30.67“0.33”"°x f“
l
; [ ~300
-1000 .. AR”? 1' l . ’5
<§ 9
e.
n: 200 g
a .2
-500“ 100 ‘E' E
e O
o. 200 3
- 100 E
0 o lo :5

 

 

 

 

 

o 100 200 300
T(K)

Figure 3.2: The MR ratio AR/ R (dashed line) as a function of temperature for thin
films La1_xCaan03 at $20.33 after 750°C treatment for 0.5 hour. The electric
resistivity p and magnetic susceptibility M are also shown. Taken from Ref. [92].

samples [88] (Kusters et al. , 1989) and Lag/3Ba1/3MnO3 thin films [89] (von Helmolt et
al. , 1993). Similar observations were also published by Chahara et al. [90] and Ju et
al. [91]. However, the name “colossal” was not coined till the truly colossal MR
ratios reported by Jin et al. [92] in 1994 on films of Lag/3Ca1/3Mn03. In Jin et al. ’3
paper, the MR ratio is defined as AR/ R 2 (RH — R0)/RH, where R0 is the resistance
with zero magnetic field strength, and RH is the resistance in a magnetic field of
6 T. Fig. 3.2 shows the temperature dependence of the MR ratio AR/R directly
taken from the original paper. The largest MR ratio in Fig. 3.2 occurs at 200 K,
reaching close to 1500%, which is much larger than any previously reported value in
any system. They also succeeded in accomplishing the largest MR ratio over 100 000%
after some optimization treatment of the films. A rather exhaustive literature search

found that the record value of MR ratio is approximately 100 000 000% observed on

44

 

La0.5Cao,5MnOg films at 57 K by Gong et al. [93].

The enormous magnitude of the MR found in manganites should be distinguished
from the previously discovered giant magneto-resistance (GMR). GMR materials are
multi-layer metallic films showing a relatively large sensitivity to magnetic fields.
The mechanism in these films is largely due to what is known as the spin-valve effect
between spin polarized metals. If an electron in a regular metal is forced to move
across a spin-polarized metallic layer (or between spin-polarized layers) it will suffer
spin-dependent scattering. If the electron was initially polarized parallel to that of the
layer the scattering rate is relatively low; if initially polarized anti-parallel to that of
the layer the scattering is high. The effect of an external field is to increase the ratio
of the former events to the latter by aligning the polarization of the magnetic-layer
along the direction of the external field. This effect has the very important advantage
of not being limited to low temperatures. Another important difference from CMR is
that usually only several tenths of a Tesla magnetic field is necessary, while the CMR
materials may require a field of over 5 T.

The CMR effect in manganites can be thought of as a magnetic field driven insu-
lator to metal (IM) transition. This has great similarity to the temperature driven
IM transition concomitant with the PM to FM magnetic phase transition. Quali-
tatively, the observed CMR effect can be explained within the context of theory of
double exchange. The external magnetic field will drive the material from PM to PM
phase when sufficiently large. This would induce the insulator to metal transition.
However, the enormous size of the CMR effect cannot be explained quantitatively by
double exchange only [94]. Significant theoretical developments over the past decade

have greatly improved our understanding of the physics of manganites.

45

 

3.3 Physics of manganites

In the chemical formula AMIIO3, oxygen has valence of -2. Thus manganese is triva-
lent when the ion on A site is +3 valenced, e.g. La3+. Substitution of ions on A sites
by divalent ions (Ca2+, Sr“, etc) results in mixed valenced manganese ions: Mn3+
and Mn“. The trivalent and tetravalent Mn ions have very different properties in the
lattice. The electronic configuration of a Mn3+ ion is 1322822p63s23p63d4. Therefore,
the 3d shell, being the only partially filled shell, is of interest. When placing the 4
electrons in the five 3d (l=2, m)=—2, -—1, O, 1, 2) orbitals, a few relevant interactions,
within current context, need to be considered. The first is the on-site Coulomb repul-
sion energy U between electrons on the same site and same orbital (same ml). U was
estimated to be 5.2i0.3 eV and 3.5:]:03 eV for CaMnO3 and LaMn03 respectively,
by Park et al. [95], using photo-emission techniques. The second is the Hund (fer-
romagnetic) coupling between electrons on different orbitals. The exchange constant
J H was suggested to be larger than 1 eV from both experimental and theoretical
studies [96, 97, 98]. Thus, as a result of the strong on-site coulomb repulsion, the four
3d electrons will have their spins lined up in parallel, and occupy 4 of the 5 orbitals,
resulting in a total spin of 822. In the case of Mn“, we can simply remove one of

the four electrons, resulting a total spin of 823/ 2.

3.3.1 Crystal field effect

Mn ions sit in the center of the octahedral cage formed by six 02’ ions. Let us first
assume that the octahedral cage has normal shape, meaning six equi-distant Mn-O
bond lengths. As a consequence, the spherical symmetry of the Mn ions is broken;
the degeneracy of the five I = 2 3d orbitals is also broken. This is known as the
crystal field effect, though most times only the influence of neighboring 02‘ ions is

considered. Considering the cubic symmetry of the surrounding 02‘ ions, the split

46

 

 

dxy dx2_y2 dz2_r2

Figure 3.3: Contour visualization of the five 3d atomic electron orbitals. Shown are
the electron equi-density surfaces.

of the five d-shell orbitals results in two groups, the 7r bonding orbitals dig, dyz, d”.
with symmetry class tgg, and the a bonding orbitals d312_,2, dz2_y2 with symmetry
class eg. The five 3d orbitals are visualized in Fig. 3.3. e9 orbitals have higher energy
than tzg orbitals. This can be understood qualitatively as electrons in eg orbitals
feel more repulsion from the 02" ions than in tzg orbitals (eg orbitals project toward
neighboring 02‘ ions, while L29 orbitals points into the empty space in between 02‘
ions). The actual energy split can be calculated to be A = $04), where Zq is the
charge of the oxygen ion, a is the Mn-O distance, r is the distance of the electron to
the centering Mn ion [99]. In practice, the gap A typically varies between 1 and 2
eV.

47

 

 

3.3.2 J ahn-Teller distortion

Let us now review how the four 3d electrons of an Mn3+ ion would occupy those five
orbitals. Since the largest energy scale is still the on-site coulombic repulsion, no dou-
ble occupancy of a single orbital would be allowed. Therefore, the triplet tgg orbitals
have one electron in each orbital. The fourth electron becomes the single eg electron.
We need to recall that the results of eg doublet and tzg triplet are obtained assuming
a cubic symmetry, i.e. in a normal shape Mn06 octahedron. Though all tgg orbitals
are singularly occupied, the two e9 orbitals have only one electron. Further energy
level splitting due to lattice distortion will be favored by the one electron occupied e9
orbitals, as the single eg electron can occupy the lower energy orbital (thus lower the
total energy). This energy splitting is found to be linear to the lattice distortion, while
the energy compensation of lattice distortion grows quadratically with the distortion
(when the distortion is small). Thus, a symmetry lowering lattice distortion becomes
spontaneously energetically favorable. This is known as the Jahn-Teller (JT) effect
resulting in the lifting of degeneracy due to the orbital-lattice interaction.

Even when only one Mn06 octahedron is considered, the possible JT distortions
become non-trivial, as there are 21 degrees of freedom (seven atoms: one Mn and
six 0). The following simplifications will be made. First, translational degrees of
fI‘eedom of the octahedron as a whole can be thrown away (-3). Second, the rotation
of the octahedron is ignored since it does not contribute to the energy splitting within
one octahedron, and the orthogonality of the octahedron is maintained (—12). Then
only 6 degrees of freedom are left. At last the Mn ion is always constrained to be

in the center of the octahedron (-3). Now we have 3 degrees of freedom. Borrowing
the notation used by Kanamori [82], the normal modes of our interest are Q1, Q2,
Q3- They are depicted in Fig. 3.4 for one Mn06 octahedron. Under the influence
of mode Q1, the octahedral shape is kept normal, it only expands or shrinks as a

Whole, mimicking the movement of breathing. Q1 mode is thus sometimes called the

48

 

         

l 1

Figure 3.4: Schematic plot of the Q1, Q2, and Q3 (from left to right) Jahn-Teller
distortion modes of one Mn06 octahedron. Arrows point to the moving direction of
oxygen atoms relative to a normal octahedron.

breathing mode. Mode Q2 will distort the octahedron, resulting in two long, two
short, and two medium Mn—O bonds. The Q3 mode also distorts the octahedron,
resulting in two long, four short (or four long, two short) Mn-O bonds. The observed
J T distortion would presumably be a linear combination of the normal modes Q1,
Q2, Q3. In practice, Q3 is the dominant mode, usually together with some degree of
the Q2 mode.

The magnitude of the J T distortion is determined by the balance of the degeneracy
lifting of eg orbitals and the lattice strain energy compensation. The corresponding
Eigenenergy E JT of this electron-lattice coupling is estimated to be around 0.25 eV
by Dessau and Shen [100]. Relevant to this study, this static J ahn—Teller energy E JT
cOrresponds to a temperature of 2,901 K, much higher than normally probed temper-
ature range. As Mn06 octahedra share their corner 02’ atoms, the JT distortions
betWeen the neighboring octahedra are directly correlated, giving rise to a strong
anti‘ferrodistorsive coupling. We would like to point out that the ignored rotation
0f MnOs octahedron leads to the buckling mode, which may play a significant role
in releasing the lattice strain. However, most theoretical calculations have simply
ignol‘ed this lattice stiffness effect.

One very important result that should be kept in mind is that the Mn4+06 octahe-

49

 

 

 

 

dron does not subject to the JT distortion. The reason is the absence of eg electrons,
and its tgg orbitals all have single occupancy. Thus, its total energy is not sensitive

to the lifting of degeneracy of eg or 529 orbitals induced by JT distortions.

3.3.3 Double exchange mechanism

The version of the theory of double exchange introduced by Zener [75, 76, 77] of-
fered a qualitative and descriptive understanding of the origin of ferro-magnetism.
It states that the conducting electrons will remember their spin orientations when
wandering around. In transition metal oxides, those electrons come from the incom-
plete d—shells. Suppose we have one Mn“ and one Mn4+ bridged by an O”- ion, the
eg electron on Mn3+ is contributing to the electric conductivity. The spin of this eg
electron will first be parallel to the spins of core electrons on the tzg orbitals. When
this electron moves to the Mn4+ via the bridging 02‘, the total energy of the system
would increase due to the Hund coupling if Mn4+ has a different spin orientation.
Thus, high conductivity is expected when the neighboring Mn ions have their spins
aligned in parallel. On the other hand, increasing the mobility of the eg electron helps
to lower its kinetic energy. One way to understand this is the following: the more
mobile an electron is, the broader its wave function is, the smoother its wave function
becomes. The kinetic energy of the electron is related to the second derivative of the
Wave function, thus the kinetic energy is reduced with increased mobility. The re-
81111; of this interaction is that the conducting electrons favor ferromagnetism to lower
their kinetic energy. Thus, the anomaly of electrical conductivity in the ferromagnetic
phase [73] is successfully explained.
The name “double exchange” originated from the proposed scenario of charge
transfer by Zener [75]. The eg electron hopping from Mn3+ to Mn4+ was first consid-
ered to be a one-step process. Direct hopping is not very likely due to the presence

of bridging 02‘ atoms. Zener then suggested that the process can be completed

50

 

 

 

by simultaneous charge transfer from the 02‘ to Mn“, and from Mn3+ to 02‘. It
is a double exchange of electrons with the same spin, though it is hard to imagine
how it actually occurs in real systems. A more quantitative study by Anderson and
Hasawaga [78] took a perturbative approach with the introduction of the hopping
amplitude t reflecting the electron mobility. They proposed that the transfer of elec-
trons from Mn3+ to Mn4+ occurs one by one, not simultaneously as believed by Zener.
One fundamental result is the dependence of effective hopping amplitude on the an-
gle 6 between the classic spins of core electrons (t2g electrons) of the neighboring Mn
ions: teff = t - cos(6/ 2). The value of the hopping amplitude t is not well estimated
from either experiments or theories. Dessau and Shen [100] have reported a value
as large as 1 eV, while a magnitude between 0.2 and 0.5 eV is suggested by Arima
et al. [101, 102]. Nevertheless, the fairly widely accepted value is a fraction of 1
eV [103, 104].

The 21) orbitals of oxygen atoms are the intermediate medium of the electron
transfers from Mn to Mn. Thus the h0pping amplitude should also depend on the
Mn-O-Mn bond angle ¢, as the efficiency of electron transfer relies on the wave
function overlap between the involved orbitals. The overlap is maximized when 05 is
180 degrees due to the linearity of the 2p orbitals. T he significance of angle 0’) leads to
the important geometrical factor, tolerance factor, defined as I‘ = dA_o/(\/2dMn_0).
Here dA_0 is the nearest neighbor distance between ions on the perovskite A site

and oxygens; dMn_0 is the Mn-O bond length. Generally, localization (insulating)
and delocalization (conducting) of the eg electrons affects the tolerance factor by
changing the average Mn—O bond length. More effectively, chemical substitution
IIlodifies the lattice due to different ionic sizes. The order of some commonly used
ions with increasing size is Mn4+ (0.52 A), Mn3+ (0.70 A), Ca2+ (1.06 A), La3+ (1.22
A), Sr2+ (1.27 A), 02‘ (1.32 A), and Ba2+ (1.43 A). A perfect cubic system results

in a tolerance factor of 1, and <6 of 180 degree. When the A site ion is too small and

51

 

 

 

demands a small d A_0, the dominant effect is the decrease of angle d) (octahedral tilt),
along with a relatively small decrease of d Mn—O- As discussed before, this would cause

the hopping amplitude to decrease, as observed experimentally [92, 80, 105, 106].

3.3.4 Semi-covalent exchange and the Goodenough-Kanamori

rule

Semi-covalence originates from the strong overlap between the empty cation (Mn3+
or Mn“) orbitals and the filled orbitals of neighboring anions (02‘). Then electrons
will be shared by the empty cation orbitals and the anion orbitals (oxygen 2p). Those
empty cation orbitals are lattice-orbitals that are nearly degenerate with the atomic
3d orbitals. It is assumed that the energy difference between the lattice-orbitals
and the atomic d shells is smaller than the exchange energy between the shared
electron and the cation d shell electrons. Under the above conditions, due to strong
exchange forces, the electron whose spin is parallel to the net cation spin will be
more preferably shared than the one whose spin is anti-parallel. In this scenario, only
one electron contributes to the “covalent” bond, giving the name “semi-covalence”.
Though the semi-covalence only directly acts on neighboring cations (magnetic) and
anions (non-magnetic), magnetic interactions between neighboring magnetic cations
Can be initiated through the semi-covalent exchange via the bridging anions. In the
Case of manganites, Fig. 3.5 shows the schematic electron-spin configuration of Mn-
Mn magnetic exchange. In manganites, the Mn-O-Mn bond is approximately linear,
thus the same oxygen 2p orbital (linear) mediates the exchange interaction with the
two bridged Mn atoms. The two electrons in one 2p orbital have opposite spins.
Thus, anti ferromagnetic Mn-Mn coupling will result if each 2p electron forms a semi-
COValent bond with each Mn atom. This gives the configuration No.1 of Fig. 3.5. If
Only one of the electron initiates semi-covalence, the O atom will have a net spin

a1lti-parallel to this election. The direct exchange between the O atom and the other

52

 

scuemnc ELECTRON- Mn-Mn T‘RANsfion
sem conneuanious SEPARATION TEMPS. RES'STWITY CASE

ORDERED LATTICES

 

 

ANTIFERROMAGNETIC

1W] SMALLEST To >Tc HIGH l

4+ OR 3+ 4+ OR 3+

FERROMAGNETIC

] om 0] LARGE To ) Tc HIGH 2

3+-OR4+ 3+

PARAMAGNETIC

\0 an 7 LARGEST Tc: 0 men 3
3+ 3+

 

DISORDERED LATTI CES

 

FERROMAGNETIC

1.701205?

+ SMALL 1'0 a 1° Low 4
how-w
3+ 4+

 

 

 

 

 

 

 

Figure 3.5: The schematic configuration of the Mn-Mn magnetic exchange interaction
in consideration of the theory of semi-covalence exchange. Taken from Ref. [81].

Mn atom (ionic bond) causes their spins to be anti parallel. Therefore, the Mn-Mn
magnetic coupling becomes parallel (case No.2 in Fig. 3.5). There is no semi-covalence
in the paramagnetic state. The most interesting case takes place in the disordered
lattice where a metallic-like bond forms if an 02’ is between a Mn3+ and Mn4+ ion
(the last two cases in Fig. 3.5).

Though theoretical treatment of manganites is elegant and powerful, the resulting
physical pictures can not be obtained without extensive scientific computation. Here
‘Ve introduce an intuitive phenomenological rule for qualitatively understanding the
experimental phase diagram. The Goodenough-Kanamori (GK) rules make a corre-

Spondence between the magnetic structure and charge and orbital arrangements of

53

.u—x

 

 

 

 

the Mn ions. It is based on the semi-covalence exchange mechanism first proposed
by Goodenough [81, 82] to describe the non ionic bonding nature between Mn and O
atoms. In 1961, Kanamori [82] compiled the GK rules after revisiting the relation be-
tween electronic, magnetic, and lattice structures. It states that, for neighboring Mn
ions bridged by an oxygen atom, if one and only one of the anti bonding eg orbitals
with respect to the oxygen atom is filled, ferromagnetic coupling between Mn ions will
result. Anti-ferromagnetic coupling is resulted in all other cases. Together with the
consideration of lattice elastic energy, the GK rules have successfully predicted the
magnetic and structural phase diagram of the cubic perovskite manganites [79, 81].
Though simple, even now, the GK rules are a useful intuitive guide to manganite
prOperties.

It might be useful to distinguish the semi-covalence exchange from the superex-
change commonly occurring also via the non-magnetic anions. In the superexchange
model based on the non-ionic character of the lattice as well, it is assumed that the
exchange energy is smaller than the energy difference between the lattice-orbitals and
the cation d orbitals. Thus the interaction is through virtual hopping of the anion
electrons. Another assumption is that only one electron can be excited from the anion
orbital to neighboring cation (1 orbitals at one time, and the hopping probability de-
Pends on the electron spin due to its coupling with the cation d orbital electron spins

f0110wing the Hund rules. As a result of both virtual hoping and anti ferromagnetic
direct exchange of the uncompensated anion spin with the other cation, the magnetic
COlipling between the cations is ferromagnetic when the cation d orbitals are less than
hfitlf filled, and anti-ferromagnetic when half, or more than half, filled. Therefore,
in the case of manganites, the superexchange model will always lead to ferromag-

r1etism, which apparently can not explain the anti ferromagnetic phase observed in

tnanganites.

54

 

 

3.3.5 Summary

The heart of the physics of manganites lies in the intricate competition between elec-
tron, spin, and lattice degrees of freedom, resulting in a very rich and intriguing
magnetic, structural, and electronic phase diagram. A realistic physical model of
manganites needs to include the on—site coulombic repulsion, strong Hund’s coupling,
electron hopping, and electron-phonon coupling, and maybe more. However, even
with some high level of approximation, the Hamiltonian still usually proves too hard
to be solved analytically. One important tool employed rather extensively is the large
scale physical model computation, such as Monte Carlo simulations and dynamic
mean field calculations [107, 108, 99]. Nevertheless, extensive theoretical and exper-
imental efl'orts to understand the physics of manganites have greatly improved our

understanding of strongly correlated electrons systems.

3.4 Phase diagram of La1_xCaan03 as an exam-

ple

The richness of the manganite phase diagram will be briefly illustrated by the exam-
ple of La1_xCaanO3, one of the most extensively studied, due to its robust CMR
effect and full range of possible doping levels [80, 109, 110, 99, 111, 112]. The phase
diagram of La1_ICaanOg is indeed very rich and interesting as shown in Fig. 3.6
as a function of Ca concentration :1: and temperature. The metallic phase only exists
Over a narrow range of 0.17 S :1: g 0.50, where the important CMR effect is observed.
At low dOping levels, the lattice structure in ground state is orthorhombic, and the
magnetic structure is type-A anti ferromagnetic [80]. This magnetic structure can
be considered as ferromagnetic planes coupled anti ferromagnetically. The proposed
Cranted anti ferromagnetism (CAF) is nowadays considered as coming from phase

Separation at nano—meter scales [99]. It is very interesting to note the coexistence

55

 

 

 

 

 

 

 

 

1000 Ssisésis;3523;Ezizisisisisisisésésisi' """"""""
900 "5353533h9m99h99'5'55535'
:2 800
j; 700 Orthorhombic
5 I octahedra
‘9'; 600 .I rotated
(D l
0- III
«E» 5‘” :15. 0'
+— 400 ,!.:,I 3/3 48 5f
300 533:. l 7/
200 .;.:_1/ F co )8
I ' Fl .-
100 .u- _
0 -__.5 CO \\ =AFE9A
oo 02 0.4C 0.6 0.8 1.0
ax

Figure 3.6: Phase diagram of La1_ICaanOg as a function of temperature and Ca
doping :10, after Cheong et al. [115].
of the ferromagnetic and insulating phases (F 1) between 0.1 and 0.17. At rational
doping of a: = 3/8, the highest curie temperature (~ 260 K) is observed. The 50%
doping presents one particularly interesting case where the type CE charge ordering
(CO) [113, 114] is the stable ground state (discussed in more detail in Chapter 4). The
highly doped La1_zCazMnOg (a: > 0.5) shows various types of anti ferromagnetism.
The end member CaMnO3 has a perfect cubic lattice structure and type G anti fer-
1‘ Ornagnetic (AF) structure where every Mn“ is coupled anti ferromagnetically with
its six neighboring Mn4+ ions [80]. There is no JT distortion in CaMn03 because all
0Ctahedra are Mn4+05 octahedra.

Other than changing doping or temperature, external pressure and isotope sub-
Stitution also considerably modify the phase diagram. Most studies have focused on
the effects on the resistivity and Curie temperature [116, 117, 106, 118, 119, 120,
121, 122, 123, 124, 125]. Interestingly, the electric resistivity drastically decreases

56

 

and the TC increases with pressure, which is still not understood [99]. Within cur-
rent understanding, increasing the pressure will reduce the lattice constants, thus
bend more the Mn-O-Mn angle. The hopping amplitude t is reduced by bending the
Mn-O-Mn angle. Therefore, we would expect the electric resistivity to increase with
pressure. One suggestion by this author, though this remains to be verified, is that
the pressure may modify the density of states near the Fermi surface as well. Isotope
substitution of 1”O with 180 results in change of the T0 of 20 K, which is much larger
than observed than in other oxides. These experimental findings clearly indicate the

significant involvement of lattice phonons in the important CMR effect.

3.5 Introduction to the question: high tempera-
ture phases of the undoped LaMn03

The perovskite manganites related to LaMn03 continue to yield puzzling and sur-
prising results despite intensive study since the 1950’s [110, 111, 99]. The pervading
interest comes from the delicate balance between electronic, spin and lattice degrees
of freedom coupled with strong electron correlations. Remarkably, controversy still
exists about the nature of the undoped endmember material, LaMnOg, where every
manganese ion has a nominal charge of 3+ and no hole doping exists. The ground-
state is well understood as an A-type antiferromagnet with long-range ordered, J ahn-
Teller (JT) distorted, Mn06 octahedra [79, 126] that have four shorter and two longer
bonds [127]. The elongated occupied eg orbitals lie pointing down in the xy-plane and
alternate between along :13 and y directions, the so-called 0’ structural phase [79, 126].
Fig. 3.7 shows the orbital ordering (00) of the occupied eg orbitals in the ab plane.

Here the MnOe octahedra are tilted, giving the crystal structure as shown in Fig. 3.8.

At TJT ~ 750 K the sample has a first-order structural phase transition to the

57

 

   

CO
6"" 6"

Figure 3.7: Cartoon view of the orbital ordering (00) in one LaMnO3 plane. Only
the occupied eg orbitals are shown.

 

       

 
 

   

   

   

0 phase that formally retains the same symmetry but is pseudo-cubic with almost
regular Mn06 octahedra (six almost equal bond-lengths). In this phase the c00p-
erative JT distortion has essentially disappeared. It is the nature of this 0 phase
that is unclear. The O-phase has special importance since it is the phase from which
ferromagnetism and colossal magnetoresistance appears at low temperature at Ca, Sr
dopings > 0.2 [79].

The additional complication of disorder due to the presence of alkali-earth dopant

ions, and the doped-holes on manganese sites, is absent in LaMnOg. However, there

58

 

 

Figure 3.8: Crystal structure of LaMn03. The Mn06 octahedron is shown as one
unit, noting its tilting.

59

is still disagreement about the precise nature of the O-phase in this simple case. In
1996 Millis [128], based on fits to data of a classical model that included JT and
lattice terms, estimated the energy of the JT splitting to be 2, 0.4 eV and possibly
as high as 2.4 eV. This high energy scale clearly implies that the JT distortions
are not destroyed by thermal excitation at 750 K and the O’- 0 transition is an
order-disorder transition where local JT octahedra survive but lose their long-range
spatial correlations. The barrier that is overcome at this transition is the lattice
stiffness that serves to orientationally order the distorted octahedra. This picture is
supported by some probes sensitive to local structure, for example XAFS [129, 130]
and Raman scattering [131], each of which present evidence that structural distor-
tions consistent with local JT effects survive above TJT. However while compelling,
this picture seems hard to reconcile with the electronic and magnetic properties. In
the O-phase the material’s conductivity increases (despite being at higher tempera-
ture and more disordered), and becomes rather temperature independent [132], and
the ferromagnetic correlations become stronger compared to the O’-phase [132, 133].
These results clearly imply greater electronic mobility in the O-phase, which appears
at odds with the persistence of JT distortions above TJT that are becoming orien-
tationally disordered. The disordering should result in more carrier scattering and
higher resistivity, contrary to the observation [132]. Furthermore, a sharp decrease
in the unit cell volume at TJT has been reported [134], which mimics the behavior
observed when electrons delocalize and become mobile at TC in the doped materi-
als [135]. The authors of Ref. [134] reconcile this observation with the order-disorder
picture of the phase transition by analogy with the volume drop on melting of ice,

though the connection seems somewhat tenuous.

60

 

 

3.6 Experimental methods

Two LaMn03 samples with slightly diflerent oxygen stoichiometry were measured.
The first sample, referred to as LaMnO3(A), is nearly perfect stoichiometric with
negligible excess oxygen content, while the other sample, refereed as LaMnO3(B), has
small but non-negligible amount of excess oxygen content. We will show that the two

samples give qualitatively the some results.

3.6.1 Sample preparation

Powder LaMnO3(A) sample of ~ 6 g was prepared from high purity Mn02 and
La203; the latter was pre-fired at 1000 °C to remove moisture and carbon dioxide.
Final firing conditions were chosen to optimize the oxygen stoichiometry at 3.00. The
crystallographic behavior was confirmed by a Rietveld refinement of the same data
used in the PDF, using program GSAS [68]. Both the difl'erential thermal analysis
(DTA) and Rietveld measurements estimated the same phase transition temperature
of TJTN 735 K and TR ~ 1010 K (discussed in detail later). These values indicate
that the sample is highly stoichiometric, e.g. T JT:750 K for perfectly stoichiometric
sample and around T ”W 600 K reported for 0.005 excess oxygen per chemical formula
unit [136, 126].

The LaMnOg(B) sample was prepared using standard solid state reaction methods.
Stoichiometric amounts of La203 (Alfa Aesar Reacton 99.99%) and Mn02 (Alfa Aesar
Puratronic 99.999%) were ground in an A1203 mortar and pestle under acetone until
well mixed. The powder sample was loaded into a 3/ 4” diameter die and uniaxially
pressed at 1000 lbs. The pellet was placed into an A1203 boat and fired under pure
oxygen for 12 hours at 1200—1250 °C. The sample was cooled to 800 °C and removed,
reground, repelletized, and refired at 1200-1250 °C for an additional 24 hours. This

process was repeated until a single phase, rhombohedral, X-ray diffraction pattern

61

 

was obtained. Total reaction time was approximately 5 days. Thermogravimetric
analysis (TGA) indicated that the as-prepared sample had an oxygen stoichiometry
of about LaMnO3,10. The LaMnOuo sample was ground, left in powder form, and
placed into an A1203 boat. The sample was post-annealed in ultra high purity Ar at
1000 °C for 24 hours then quenched to room temperature. The oxygen stoichiometry
was again determined using TGA under forming gas. The final oxygen stoichiometry

was 3.006.

3.6.2 Differential thermal analysis

Differential Thermal Analysis (D TA ) data collection: The main goal of the DTA mea-
surement was to find the phase transition temperatures T JT and TR, which would be
used to guide the later neutron powder diffraction experiments. Sample LaMn03(A)
of 0.1033 gram was sealed under vacuum in one quartz tube end of spherical shape.
The standard reference material used is A1203 of about 0.0500 gram. The differential
thermal measurement was performed on a Shimadzu DMA-5O from 283 to 1373 K,
with temperature ramping rate of ten degree per minute and holding time of one
minute. Two heating and cooling cycles were run to check reproducibility. About
0.0583 gram of LaMnO3(B) sample was also measured with similar settings.
Qualitative results: Fig. 3.9 shows the raw data from DTA measurements on both
samples. Sample LaMnOg(A) shows rather pronounced thermal peaks across the JT
transition (z 735 K), and nonetheless distinguishable peaks across the orthorhom-
bic to rhombohedral (OR) phase transition around TR~ 1010 K. In contrast, the
LaMnO3(B) response from the JT transition appears around 660 K, and the peaks
are rather weak and broad. No identifiable feature is visible across the expected
second OR phase transition around 800 K. The phase transition temperature was

estimated as the mean of the thermal peak positions during heating and colling 2. By

 

2noting only qualitative analysis was carried out while proper DTA data analysis will require
considerations of experimental factors such as the relative weight and specific heat of the reference

62

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C C a
‘ i
O
O
r “I"
a E
V v
E ............ ‘ E
o%?_ .1 a??? '''''''' rI'I:
( .
._ 15 5500 .‘l
O O ’.— q ‘
‘ F 1
h d
8 ’~ 4.
8. ' 7 1
I '- a
1" 4
..............
#4

 

 

1 000 400 600 800 1 000 1 200
Temperature (K) Temperature (K)

Figure 3.9: The DTA measurement data from sample LaMnO3(A) (left panel) and
LaMnO3(B) (right panel).

doing this, we found the T JT of LaMnO3(A) sample is around 735 K, which is very
close to the 750 K for the “perfect” stoichiometric sample. The T R is around 1010 K,
about the same as reported in the literature for the “perfect” stoichiometric sample.
The quite weak signals from LaMnO3(B) sample are somewhat surprising (as the ex-
cess 0 content is rather small, 3 0.006), suggesting the Mn4+ contents (due to excess
oxygen) drastically affect both phase transitions. Those transition temperatures were
further confirmed by our neutron powder diffraction experiments discussed later.

To summarize, the DTA measurements successfully located the two phase tran-
sition temperatures. The LaMnO3(A) sample is shown to be very close to be stoi-
chiometric, and the presence of small amount of Mn4+ should not compromise our
studies. The LaMnOg(B) sample appears to be less stoichiometric as expected. Still,
it is very interesting to see how our microscopic structure results depend on the slight

excess oxygen content.

 

material, the cooling rate, holding time, etc.

63

 

3.6.3 Neutron powder diffraction

To reliably determine the oxygen atom positions, i.e. Mn-O bond lengths, neutron
powder diffraction data were collected, because of the much improved oxygen atom
contrast compared to X-rays [5].

Sample LaMnO3(A) was measured on the newly upgraded Neutron Powder Diffrac-
tometer (NPDF) [43] at Los Alamos Neutron Science Center (LANSCE) at Los
Alamos National Laboratory. Powder sample of about 6 gram was sealed in a vana-
dium can suitable for high temperature measurements. A high temperature furnace
was used to collect data at temperatures in the order of 300, 650, 700, 720, 730,
740, 750, 800, 880, 980, 1050, 1100, 1150, and 550 K. An additional data set at
300 K after the high temperature measurement was also collected in the displex to
check for possible change of the sample after reaching as high as 1150 K. Each tem-
perature point was measured for 3 hours which is found sufficient to provide good
counting statistics. Sample LaMnOg(B) was measured on the Special Environment
Powder Diffractometer (SEPD) at the Intense Pulsed Neutron Source (IPNS) at Ar-
gonne National Laboratory (ANL). Powdered sample of about 10 gram was sealed
in a cylindrical vanadium tube with helium exchange gas. Data were collected from
20 K to room temperature in a closed cycle helium refrigerator, and from room tem-
perature to 805 K in a Howel furnance. Each temperature point was measured for
6 hours. In both experiments, calibrations runs such as the empty diffractometer,
empty sample container can, and a vanadium rod standard were also measured for

proper corrections of sample data.

3.6.4 Obtaining experimental PDFs

Program PDFgetN [42] was used to carry out many correction steps, such as the
detector deadtime and efficiency, scattering background, sample absorption, multiple

scattering, in-elasticity eflects, and normalization by the incident flux and the total

64

 

 

- 1200

Raw Counts
800

 

 

.. . 1..

5000 10" 1.5x10‘ 2x104
Time of Flight (as)

 

Figure 3.10: One example of experimental raw data from neutron powder diffraction.
The 9: axis is the neutron time of flight (TOF), and y axis denotes the measured
intensity.

sample scattering cross-section. to yield the total scattering structure function, S (Q),

where Q is the magnitude of the scattering vector. The PDF, G (r), is then obtained

by a sine Fourier transform according to
2 °° ,
C(r) = ;/ Q[S(Q) — 1] Sin Qr dQ. (3.1)
0

Fig. 3.10 shows one example of the raw diffraction data directly coming from the
experiments. Examples of F(Q) = Q(S(Q) — 1) and 0(7) from both neutron exper-
iments are shown in Fig. 3.11. Qmm of 32.0 A‘1 was used for LaMnO3(A) NPDF
data, while QmaI of 26.0 A—1 for LaMnO3(B) SEPD data. The 980 K NPDF data
appeared to subject to fluctuating incident beam profiles during collection, and will
be ignored in our data interpretations. All other PDFs show high quality. Extensive

structural modeling immediately followed.

65

 

 

 

141
102

  

A‘. A
T as 'I- 8
S S
A A
3'; 81;
u. Li.

 

 

 

 

 

 

 

 

60) (3'2)
0

—4

 

 

 

 

 

 

 

 

8
r (A) r (A)

Figure 3.11: Left half shows the NPDF data from sample LaMnOg(A) at 300 K and
right half shows the SEPD data from sample LaMnOg(B) at 300 K. The annotations
are, (a) The experimental reduced structure function F(Q) = Q(S(Q) — 1). (b) The
experimental C(r) obtained by Fourier transforming the data in (a).

3.6.5 Real space PDF refinement

PD F FIT [11], a real space non-linear least square regression refinement program, was
used to fit structural models to all experimental PDFs. The real space distance r
range of the experimental PDF used is 1.5 to 20.0 A 3. The initial structure model
for the orthorhombic phase has space group anm, with asymmetric cell made up
Of La (-0.0078, 0.049, 0.25), Mn (0.00, 0.50, 0.00), 01 (0.075, 0.49, 0.25), 02 (0.73,
0-31 , 0.04). The high temperature rhombohedral R phase has space group R30 with
its asymmetric cell comprising of La (0.0, 0.0, 0.25), Mn (0.00, 0.00, 0.00), O (0.44,

0-0, 0.25). Necessary constraints are properly set up manually to maintain the anm

 

3The rm," of 20.0 A PDF refinements already obtain the crystallographic structures in excellent
aEmernent with Rietveld analysis

66

 

 

Fitting plot of 650K

”El

 

 

fi’ 1 v y f' i 1—v

(7), ‘1 AM- All A
jWAVWW ”i

1': j - it '3
EA M. ”FAA/ill int/tnl AA

Fitting plot of 730K

4
‘ w

 

'7
4

 

 

 

g?

 

 

 

 

 

 

 

gé:
cm at")

 

 

 

 

6(r) (1")

:6 :4 v :2 fl -
(I q-
é
<

-a -e -4 -2 o

 

 

 

 

   

 

 

4

2

I?

 

 

 

 

 

 

 

 

 

 

 

GM (1")

-a'-a'-4_-2 n

G(r) (r’)
-4 -2
:1

2’
<
S

 

 

 

 

 

 

 

'10 4":5' ' _'

 

-LAL;1-1-IA14141‘

 

2 4 6 81012141618
I'(l)

 

Figure 3.12: (upleft) The PDFFIT refinement results of sample LaMn03(A) data
at 650 K, (upright) 730 K, (downleft) 880 K, (downright) 1150 K. Denotations are
the following, the experimental G (r) is shown as solid dots, and the calculated PDF
from refined structural model is shown as solid line. The difference curve is shown
Offset below. The dashed lines show the estimated standard deviations due to finite
collIlting statistics at 20 level.

67

 

or R§c space group symmetry. This also means that special atomic positions are not
refined. It is noteworthy that constraints can be set based on the atomic environ-
ments regardless of the space group, as PDFFIT assumes no inherent symmetry of the
refined structure. Refined structure parameters include lattice constants, non-special
atomic fractional coordinates, and isotropic thermal displacement parameters. Addi-
tionally, finite instrument resolution factor is modeled as the Qsig, the full width at
half maximum (FWHM) of the instrumental resolution function; the renormalization
factor (scale factor) is also refined to account for small but non-negligible systematic
errors during data analysis. All refinement agreements are rather good with RM, val-
ues less than 0.10. Examples of the PDF refinement are shown in Fig. 3.12 and 3.13
for sample LaMn03(A) and (B), respectively.

In the difference curves shown in Fig. 3.12, sine wave like fluctuations are somehow
apparent. A clearer plot is shown in Fig. 3.14(a). This is found to originate from the
missing low Q Bragg peaks in F(Q), which was directly sine Fourier transformed to
obtain the G (1‘) From Fig. 3.14(b), three peaks (first one is a doublet) are missing

from the experimental F (Q) with positions at 1.57, 1.61, and 1.77 151-1. With the
known peak positions and the relative peak intensities estimated from the structure
factors, the analytical formula of s * expim'owl)2 ~(200 sin( 1.571") ~ 1.57 + 83 sin(1.61r) -
1-6 1 + 38 sin(1.77r) - 1.77) was fit to the difference curve. 7‘ is the real space distance.
3 is the scale factor as the only fitting parameter. The fit is shown in Fig. 3.14(c).
The excellent agreement of the fit strongly indicates the missing Bragg peaks are
the origin of the fluctuations in the difference curves. Subtraction of the fitted sine

Waves resulted in a much improved refinement as expected, which can be seen in

Fig- 3.14(d).

68

 

 

 

 

 

 

 

 

 

 

 

 

 

2f 1
20K
on- 'n'
? :6“
I I
5° ' So
3 3
on o
I |O_
I
53
I —
9
4 a 12 1s '
'0‘)
3K
10L "”

 

AAAAAAHAO
v11) V“ w

6" 97‘

1 = l I o
:5 5
1': 1:
5’ b”

".1- “.1

 

 

 

 

 

 

r (l) r (K)

Figure 3.13: Examples of PDF refinement results at 20 K, 300 K, 603 K, 805 K of
saLl'nple LaMnO3(B). Denotations are the following, the experimental 0(7) is shown
as SOlid dots, and the calculated PDF from refined structural model is shown as solid
line- The difference curve is shown offset below. The dashed lines show the estimated
Sta-ndard deviations due to finite counting statistics at 20 level.

69

 

 

0.5

G(r) (AW)
0
G(r) (r2)
0

—O.5

 

 

 

 

 

A A”.
T ‘3‘
S '50
A A
0 b
v

u. (D

—4

   

 

 

 

O 0.5 1 1.5 2 2.5 3 4 8 12 16
oar-1) r(fi)

 

—8

Figure 3.14: Demonstration of the consequence of missing Bragg peaks at low Q
region of the experimental F (Q) on later PDF refinement (see text for details).

3.6.6 Reciprocal space Rietveld refinement

PrOgram GSAS [68], the general structure analysis system package, was used to carry
Ollt Rietveld analysis of Bragg peaks in reciprocal space. The Q range used is from
1-0 to 12.5 .451. Only the two highest angle banks are refined. Initial structure
mOdels are the same as the aforementioned PDFFIT refinements, i.e. anm and R§c.
FOI‘ both N PDF and SEPD data sets, the lowest temperature data were first refined.
Instrument and sample dependent parameters, such as diffractometer zero constants

and peak profiles, are only refined at this initial step. Then, in the order of low to

high temperatures, starting from the refined parameters of previous temperature, the

following are refined: lattice constants, non special atomic fractional coordinates, and

70

 

 

1600

Intensity (a.u.)
800
Intensity (a.u.)

 

 

 

2000

3200
Intensity (a.u.)
1000

Intensity (0.1.1.)
1600

O

 

0

 

 

 

 

 

A n “A
‘ v w w v—W‘

 

2 4 6 10 2 3 4 5 6
o (r‘) o (r‘)

Figure 3.15: (upleft) The Rietveld refinement plot of bank 5 of 650 K LaMnO3(A)
data, (upright) 730 K, (downleft) 880 K, (downright) 1150 K. The annotations are:
experimental observed intensities are shown as crosses (x); calculated pattern from
crystallographic model is shown as solid line; their difference is shown offset below;
Bragg peak positions are indicated by the short vertical bars.

isotropic thermal displacement factors. Background for each bank is approximated by
a shifted Chebyschev polynomial with 7—-12 terms (type 1 in program GSAS). Along
the way, addition of the second phase or the change of space group was necessary
near phase transition temperatures. Rather satisfactory agreements were achieved at
all temperatures with X2 ranging from 2.0 to 5.0. Examples of Rietveld analysis are
ShOWn in Fig. 3.15 and 3.16 for sample LaMn03(A) and (B), respectively.
The sample LaMn03(A) after the high T measurement was also measured at 300 K
at a. later time in displex, to check whether the sample stoichiometry has changed
during the high temperature measurement. Rietveld refinements were carried out on
both 300 K data sets, and shown in Fig. 3.17. Between these two measurements, the
CTYStal structures are identical, and the only different parameters are scale factors

and Scattering background. Both refinements have rather small X2 values, 2.46 (be-

71

 

 

 

 

1080

2800

Intensity (a.u.)
540
Intensity (a.u.)
1400

  

III I-IIl-II o
I I I'll—II-

 

 

 

 

 

 

 

 

2800

"tensity (a.u.)
400
Intensity (a.u.)

 

 

 

 

 

 

 

6 8
o (11")

FigLIIe 3.16: (upleft) The Rietveld refinement plot of bank 5 of 20 K LaMn03(B)
data, (upright) 300 K, (downleft) 603 K, (downright) 805 K. The annotations are:
experimental observed intensities are shown as crosses (x); calculated pattern from
crystallographic model is shown as solid line; their difference is shown offset below;
Bragg peak positions are indicated by the short vertical bars.

72

 

 

 

 

 

 

 

 

 

 
    

 

 

 

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1 .. u 0 ‘ I O In 1.1 n,l. ‘ O
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swk.cm....llu..c n.»
A I
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'
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P D I D b I l

 

 

 

 

 

8.8 8w: 1 0 8m. own o
Ajdv 36:35 Ajdv £235

(left) Rietveld re-

of 300 K data before the high T measurement, (right) 300 K data after the

igh T measurement.

parison of the neutron powder diffraction pattern of sample
73

(A) between the beginning and end of the experiment.

FiEUre 3.17: Com
LaMnO3
f1Ilement

h

 

 

 

 

 

 

L

fore), 1.87 (after). Also in consideration that the measurement was all carried out in
vacuum, we reached the conclusion that the sample stoichiometry had not changed

during our experiment.

3.6.7 Model independent peak analysis

The PDF tells directly the distribution of inter-atomic distances regardless of the
inherent lattice symmetry. Thus, even without a structure model, we are able to
extract quantitative information about one or more specific bonds, such as the bond
length and coordination numbers. To our advantage, usually only a few distinct
peaks exist at the low 1' PDF region, corresponding to the few bonding lengths in
the structure. It is generally more difficult to directly analyze the high 1‘ PDF peaks,
which requires serious effort to decompose the contributions from many inter-atomic
distances. With rather good accuracy [137, 138], PDF peaks are of Gaussian curve
shape, along with the consideration of termination effects [18]. Therefore, we can
simply fit the low T PDF peaks with a few Gaussian peaks, whose number corresponds
to the number of distinct (and resolved) bond lengths. The obtained Gaussian peak
profiles reveals the local structure details quantitatively. PWID and XYFIT (home

written programs) are used for this purpose.

3 - 7 Local structure from PDF analysis

From now on, our discussion will focus on the experimental results from the sample
L3-1\/11103(A). The corresponding data from sample LaMnO3(B) will be also shown side
by Side for comparison. Qualitatively, the two samples show behaviors in excellent
8agreement.
Local structure information is contained in low 7' PDF regions. Thus we show

in Fig. 3.18(a) the low 1" region of the PDF, which directly reflects the Mn-O bond

74

 

 

 

 

 

 

 

 

In
{$1 a a |
l ' o
. i ° T
*1 ..- 1'
O '- 3' o
:5 - S
S 3 '
(.9 (.9 “'3

 

 

C(r) (3’2)
Gm (r2)
6 12

 

 

 

 

 

 

 

 

 

 

 

 

   

. . . . 1.6 A 2 i 2.4
r (A) r (A) r (A) r (A)

Figure 3.18: Left half (four panels) show the low T experimental PDF data from sam-
ple LaMnO3(A). (a) Low r region of experimental PDFs at all temperatures shown
with offsets. Positive y axis points to the increase of temperature. Two dashed lines
indicate the Mn-O long and short bonds at 1.94 and 2.16 A, respectively, (b) shown
are the PDFs at 650, 720, and 880 K without offset. The differences between 650
and 720 K, 720 and 880 K are shown below offset. (0) fitting result of PDF data at
880 K with two Gaussian curves. Solid dots denote the experimental data. Solid line
denotes fitted curve. Plotted below is the difference curve. (d) fitting result of PDF
data at 1150 K with annotations as in (c).

Right half (four panels) show the low 7‘ experimental PDF data from sample
LaMn03(B). (a) low 7' region of the PDF at all temperatures shown with offsets.
Positive y axis points to the increase of temperature. (b) low 1" region of the PDFs
at 300 K, 517 K, and 770 K are shown. The difl'erences between 300 K and 517 K,
517 K and 770 K are shown below. (c) fitting result of PDF data at 770 K with two
Gaussian curves. Solid dots denote the experimental data. Solid line denotes fitted
curve. Plotted below is the difference curve. ((1) fitting result of PDF data at 20 K
with annotations as in (c).

75

length distributions. It should be first noted that Mn—O peaks are upside-down due
to the negative Mn neutron scattering length [5]. In the low temperature 0’ phase,
the JT distorted MnOG octahedra are ordered in the staggered pattern as illustrated
by Fig. 3.7. This perfect long range ordering results in identical local and average
structure. Consequently, we should observe the four shorter and two longer Mn-
0 bonds in experimental PDFs (the two short and two medium bonds can not be
resolved directly). The low temperature PDFs in Fig. 3.18(a) clearly show the Mn-O
doublet at 1.94 and 2.16 A, e.g. the 300 K data at the bottom of Fig. 3.18(a).

Within the low temperature 0’ phase, increasing the temperature will enhance
the atomic thermal motions, which in turn will broaden PDF peaks [5]. This is what
we see by comparing the bottom five PDFs in Fig. 3.18(a). Across the JT transition
at T JTN 735 K, the average crystallographic structure indicates nearly regular MnOs
octahedra. If this is also true for the local structure, the first Mn-O PDF doublet
peak should turn into a single peak around 2.0 A. However, the experimental PDFs
show the opposite: the Mn-O doublet persists above JT transition. The long Mn-O
peak does not separate itself from the more pronounced short Mn-O peak, however,
is clearly evident as the high 7' shoulder. The splitting remains larger than the ther-
mal broadening of the peaks. This behavior extends into the highest temperature
measured. As our local structure data covered the high temperature R phase for
the first time, the local JT distortion is then discovered to exist in R phase as well.
The PDFs at low 7‘ region show qualitatively and intuitively that the JT distortion
survives at all temperatures. The peaks do broaden at higher temperature due to in-
creased thermal motion [5] and the two contributions to the doublet are not resolved
at high temperature. However, it is clear that the peak is not a broad, single-valued,
Gaussian at high temperature as predicted from the crystallographic models.

We would like to see if there is any change in the peak profile on crossing the

T JT beyond normal thermal broadening. To test this we plot the change in this

76

 

2.2

2

0.12

0.09

 

peak width (A) peak position (A)

 

 

 

co

0.

o 400 600 800 1000

T00

* ,
‘1?" - ' '1 il'(o')
v
.504 000 no
gm
” II
o
o.
x
80“...- I-
0.9, I
go
£8 I" _I
2 - I.
.x Q.
8 ..
Q01”. I 0' II? I

 

 

0 200 400 600 800

T (K)

0

 

 

 

c3110

20‘

0|

‘6

on“!

Q'-

I

10

O.

o

.- I | .
a
ma " R
Q- -
o IF
OAIAJAlLJAq

 

 

400 600 800 1000
T (K)

Y

 

iI '<c'>

peak area

 

 

 

 

 

 

N
gr:
(I O
(D
O. 0' 0
o l A l n l . II A l
0 200 400 600 800
T (K)

Figure 3.19: Extracted peak profile parameters from the double Gaussian curve fit-
ting. Upper half is from sample LaMnO3(A); lower half is from LaMn03(B). The low
1‘ peak is shown as filled squares, while the high 7' peak is shown as filled circles: (a)
peak positions, (b) peak widths, (c) peak area, ((1) RM.

nn Mn-O peak between T=720 K and T2880 K, as it crosses TJT, and compare
this to the change in the peak on going from 650 K to 720 K. The latter case has
approximately the same temperature differential (thermal broadening will be the
same) but there is no structural transition in that range. The difference curves from
these two situations, shown in Fig. 3.18(b), are almost identical. We conclude that
there is virtually no change to the Mn06 octahedra as they go from the 0’ to the
0 phase. It is noteworthy that the shape of the difference curves mimics the typical
differences between two Gaussian with different widths.

Finally, to quantify the nature of the local JT distortions, we have fit two Gaussian
peaks to the nearest-neighbor MnOG doublet in the experimental data at all temper-
atures. The quality of the fits at “low” (880 K) and high (1150 K) temperature are
rather good, as can be seen in Fig. 3.18(c) and (d). The position and width of each
peak, and the total integrated intensity of the doublet, were allowed to vary (the
intensities of the long and short bond distributions were constrained to be 1:2 in the
fits). The weighted residual factor pr varies from 1% at high temperature to 5% at
low-temperature indicating excellent agreement over all temperatures. Attempts to
fit a single Gaussian in the 0 phase resulted in significantly worse agreements. The
results of the temperature-dependent fits are shown in Fig. 3.19.

No anomaly of any kind can be identified across the T JT:735 K. The integrated
intensities and peak positions do not change significantly with temperature from 300 K
to 1150 K indicating that the average bond-lengths of the short- and long-bonds in
the JT distorted octahedra are rather temperature independent. The peak widths
increase slightly with temperature due to increased thermal motion (Fig. 3.19(b));
however, there is no clear discontinuity in the peak broadening associated with the
phase transition. The broader low 7' peak (larger peak width) seems to come from
the fact that two Mn-O bonds (two short and two medium) contribute to it, while

the high 7' peak only comes from the two long Mn-O bonds. The results confirm that

78

the local JT distortions persist, virtually unstrained, into the 0 and R phases.

To summarize here, we first confirmed the earlier XAFS results [129, 130] and
establish that in the 0 phase the full JT distortion persists in the local structure.
Second, we show for the first time that the high temperature R phase (T2 1010 K)
is also locally fully JT distorted. Both qualitative and quantitative analysis has been

carried out.

3.8 Average structure from the PDF and Rietveld

analysis

3.8.1 Crystallographic phase determination

Both PDF refinements over a wide r range and Rietveld refinements result in a crys-
tallographic model to represent the average structure. The average structure here
means that it is obtained by spatially averaging over all cells, which can be all the
same as in perfect crystals, or different in the presence of disorder. As the detailed av-
erage structures have been reported in the literature [136, 126, 139], only brief results
will be shown. We first show the lattice constants in Fig. 3.20. According to conven-
tion, c/\/2 is shown in the orthorhombic phase for easy comparison with a and b. The
“equivalent” egg 2 c/\/2 for the rhombohedral phase is obtained from the relations:
a” 2 b0, by = %(a0 — b0 - CO), CH 2 2a0 +c0. So that 1802,, 2 Ci, + Hail, where the
subscripts H and 0 denote the orthorhombic and rhombohedral phase, respectively.
The experiment data clearly show the JT transition temperature T JT around 735 K
for LaMn03(A), and 610 K for LaMnO3(B), indicated by vertical dashed lines in
Fig. 3.20, where the sample goes from the 0’ into the O (pseudo-cubic) phase. It is
worth mentioning that the transition temperatures are in good agreement with earlier

DTA results. Refined values from PDF refinements are plotted as filled symbols, while

79

 

 

 

 

 

 

 

 

 

 

to TT'
:1- 1% O! O
1 1'8‘ " 11 I! s
6" 16‘
5'13- I IE'A A5 !: "nA
0'- .m ”.5 On I 5
E I I WA 5"” I " A
2* E C | o 2 I. H .‘D'g
[~1- . I ‘ v f ' V
i" I a! 1 Thu '5 11
0 b I 0h I -1
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= : 5. Jr: 1]: :1 if- L -44
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5: )H 1' ‘ 5'“ 151111
. O . 0' O 0
a o ’ ' 2,000... L
3112. AR | . 310 A A II
.A 0 I0 lR ] *AA II 4
L . - . -1. s 1- «5‘. . . - °- 11°.
400 soo 300 1000 o 200 400 500 300
T00 T00

Figure 3.20: Results from LaMnO3(A) are shown in the left panel; while LaMnO3 (B)
in the right. Both PDF (filled symbols) and Rietveld (Open symbols) refined values
are shown. (up) The unit cell volume (0) and the average of three lattice constants (I)
as a function of temperature. (down) The lattice constants a(l), MO), and c/x/2(A).

the Rietveld refined values are shown as open symbols. They appear to be in excellent
agreement. It is interesting that the three lattice constants a, b, and c/\/2 become a
little more separated with increasing T after T JT. The cell volume is shown in the
upper panel of Fig. 3.20. One striking feature is the collapse of cell volume across
the JT transition, which is believed to be from the orbital disorder above JT transi-
tion. The same reason is believed to account for the decrease of the average of the
three lattice constants (to be discussed later). For sample LaMnOg(B), the seemingly
rather large JT transition temperature difference of 150 K from only 0.006 (nominal
value) excess 0 atoms is in fact somewhat expected. For example, a T JT=600 K was
observed by Norby et al. in a slightly non-stoichiometric LaMnO3,005 [136].
Another signature of this JT transition is the disappearance of the MnOs octa-

hedral JT distortion upon warming. Fig. 3.21 shows how the three “orthogonal”

80

Mn—O bond lengths within one Mn05 octahedron, their average, and the Mn06 oc-
tahedral volume evolve with temperature. Clearly, in the low T 0’ phase, the MnOs
octahedron is fully JT distorted with two long, two medium, and two short bonds.
This is not contradicting the local structure results where only two Mn-O bonds were
observed. The reason is that the short and medium Mn-O bonds are not directly
resolved in the experimental PDFs, as the Mn-O peak is shown as a doublet (not a
triplet) in Fig. 3.18. It is the same reason why one Gaussian curve was used to fit
these two shorter bonds (in Fig. 3.19). It is also self-consistent that the low 1' Gaus-
sian curve appears to have an area twice as large as the high 1' one. In the average
structure, upon the JT transition, the diflerence between those long and short bonds
reduces drastically, which is what we expect from this 0’ to O (pseudo-cubic) phase
transition. The JT transition is confirmed to be orbital order-disorder type as the
local JT distortion persists at all temperatures. The anomalous change of the Mn-O
bond lengths in the average structure comes from the orientational disorder of the
distorted Mn06 octahedra in the 0 phase, while the Mn06 octahedra always lay their
long Mn-O bonds in the ab plane in the 0’ phase. In both Fig. 3.20 and 3.21, the
OR transition of the sample LaMnO3(A) is also identified to occur around 1010 K in
agreement with early DTA results. In the R phase, Mn06 octahedra are strictly nor-
mal with six equi-distant Mn—O bonds, and the lattice symmetry becomes hexagonal.
No anomaly is observed on the cell volume, however, the volume of the Mn06 octa-
hedron collapses again. This most probably indicates the structure experiences two
relaxation steps: JT and OR transition. The lattice relaxation introduces more dis-

order, and thus results in “seemingly” reduced volumes across both phase transitions.
Therefore, both the JT and OR phase transitions have been confirmed on our

samples, and are consistent with the published literatures. Average structure analysis

with combined PDF and Rietveld methods has been shown to have quantitative

81

 

 

 

 

 

 

 

 

 

 

l I 'l N ' 1 ll
0
g_ (a) -01 I: (o) 2
I
'5: ’ "i I ‘n 15.: m 0' I]
v0. FAV . A
..~- I ~25.” MI --111 s.
E" A E... m 0 A
2 ’ j 0| o 261— ' 3 8 LI Arno
° 0 ° m ._I 0n 0|
>'- m In Q1: > o-c
or,” '] "N: o l a N:
o 0 0 l v o . v
§ ' l I '5'“, § I W!-
q 'l g .— ' o
u- I ll - . d- . || -
" °' 1° 1“ N "’ ° 11°
1‘1. 1 : .1 ., N”: 1 H N
" (b) N (b)
I I .4 I . . . .. 8 ll
5:3, I 59” t
a h l 56:“ II
5 5' ll 5 I‘iofl I
" . fig III. " .8
0“ 0| 0“ I
| A ‘ “’8 ' AAAAL 4 ‘ “mm
= | c In”
2 z
02.. 00 I . mil-ll i'fl
.° 1? 1R . . .° 11°
400 600 500 1000 o 200 400 600 800
T00 1'00

Figure 3.21: Results from LaMnO3(A) are shown in the left panel; and LaMnOg(B)
in the right. Both PDF (filled symbols) and Rietveld (open symbols) refined values
are shown. (up) The Mn06 octahedral volume, and the average of three Mn-O bond
lengths as a function of temperature. (down) The long, medium, and short Mn-O
bonds.

agreement. LaMnO3 in high temperature 0 and R phases shows no JT distortion on
the average structure. In the context of the existence of local JT distortions at all

temperatures, the O and R phases are orbital disordered.

3.8.2 Lattice structure parameters

In the following, we will show other extracted structural data coming from the aver-
age structural analysis. The isotropic atomic displacement parameters are shown in
Fig. 3.22. Basically, La shows rather normal temperature dependence, while Mn ex-
periences a small jump only across the J T transition. Only 0 atoms show significant
increase of U,” across both the JT and OR transitions. Rietveld results indicate this

anomaly on both 0 atoms, while PDF results only suggest the in-plane O atom. In

82

 

 

 

 

 

 

 

 

 

 

'0 I ' I v u 1 i I r I f f j 1
3‘ I La IL Lo ”0 I
_ (L Mn F 0‘ N Mn H I
AN 00 o"- 0000‘
ago.- I o 0| . £0 c|)|o
O
" ° ol'afi V II .I
O ’ DD 0 O
.9 '_ o O a I .L’.’ o_ 3630‘
DO..- 8 . a. III-d Do 0 a I...
°I i a l . 38 I - II .
i '3
l .I 9 II
o : : : :l : + A. o: a : : 4 llfi ‘f
3 01 lo 3_ 01 o
O, l 02 Faun- o 02 n I DD
‘33, I cl) 9 ell-I B “8 0
W i 'l ow l}? n-
.2 °- 0 ‘ - .22 o” D '0"
3° I I D a g
i _ I
-. I I . 358 - II
Ii
o J A l ‘1 1 A J A 01 A l A l L II A l
400 600 800 1000 o 200 400 600 800
T00 T00

Figure 3.22: Results from LaMnO3(A) NPDF data are shown in the left panel; while
LaMn03(B) SEPD data in right. Both PDF (filled symbols) and Rietveld (Open
symbols) refined values are shown. Data represent the isotropic atomic displacement
parameters for all atoms in the asymmetric cell (La, Mn, 01, 02).

both refinements, the Um of both 0 atoms are released/refined at the same time,
not one by one. This should rule out the side effect of parameter fitting orders. The
discrepancy between the quantitative values between PDF and Rietveld refinements
has been discussed in Ref. [41]. The excess increase of O atom thermal displace-
ment parameters is consistent with the scenario of increasing disorders across both
transitions.

The lattice and MnOs octahedron appear to be more metrically normal upon
increasing temperature. We can define some geometric parameters to characterize

the distortions in the following. The lattice orthorhombicity factor D is defined as

 

, (3.2)

where a and b are the in-plane lattice constants. The distortion parameter A of a

coordination polyhedron MON is

A = 523(1).- — <b>>/<b>}2, (3.3)

where b.- is the ith M-O bond length and (b) is their average. One important parameter

frequently quoted in the literature is the tolerance factor defined as

bA-O

= 755.: ‘34)

where bA_0 is the distance between the A site and the nearest oxygen, and an_0 the
average Mn-O bond length. In a perfect cubic system, the tolerance factor I‘ is equal
to 1.0. We show the above defined parameters in Fig. 3.23. The main result is the
reduction of lattice distortions in the 0 and R phases. Anomalies are evident across
the JT transition, while OR transition appears to be a rather smooth crossover. The
tolerance factor I‘ is approaching the “perfect system” value from below (noting that
I‘ > 1.0 is not possible), but does not reach 1.0 even at the R phase.

The smaller than unity tolerance factor F also means that the Mn06 octahedra
have to tilt for self accommodating in the lattice matrix. In the orthorhombic phase,
there are two Mn-O-Mn angles, while only one in the R phase. Those Mn-O-Mn angles
are not 180 degrees due to the tilt of Mn06 octahedra around the pseudo-cubic [111]

direction. The average tilt angle (cp) can be computed from the two Mn-O-Mn angles

01 and 62 by

 

 

2 - 5cos 9012
9 = 3.5
C08 1 2+cos<p12 ’ ( )
1 — 4 2
cos 02 = :08 ('02 , (3.6)

and (90) = ($1 + cp2)/2. Another set of angles of interest is between the O-Mn-

84

 

 

 

 

 

 

 

 

 

 

n llaétige l : ,,, lattice 1' 0
°.- n . 0' MnO I '
° L001: IA 1 ° L001; H
c A c -AAAA
9 A . ”A“ .9 ‘ A A
go] I. Ball-"gojllll ll
U - -‘o -
3- o '4 I . 3. 3 ' a II
$0 $0.00
N : : I = II :7 l : l :7 I II T l
m.- I ll .- I I I II I
0
~ I I“ .. II '
a . s .. pa
2 a- l o °| _ .2 2' l '
8° ' ol 8 . @0000.
s - 0 ° - s
53 I 5a 9.“ _
8 o' . 9| I ' .9 °‘ 6 II
I J I '8888 J? ‘
400 600 800 1 000 0 200 400 600 800
T(K) T (K)

Figure 3.23: Results from LaMnO3(A) NPDF data are shown in the left panel; while
LaMnO3(B) SEPD data in the right. Both PDF (filled symbols) and Rietveld (open
symbols) refined values are shown. (up) The distortion parameters for lattice, MnOs
octahedron, and La012 polyhedron. (down) The tolerance factor F.

O triplets within one MnOs octahedron. Fig. 3.24 shows the development of these
angles with temperature. We need to mention that for these results, the anm model
was used in PDF refinements of the R phase, which actually has a higher symmetry
R30. If space group R30 is used in PDF refinements, the same results were obtained
as from the Rietveld refinements. The use of lower symmetry group lem for the R
phase aimed to look for local structure motifs, which would otherwise be constraint by
the R30 symmetry. The trend is that Mn-O-Mn angles are getting more “straight”,
i.e. approaching 180 degrees. However, change of the average tilt angle (,9 is not
drastic, and does not show anomalous decrease upon either phase transition. We

thus can not simply attribute the change of lattice constants and Mn-O bond lengths

85

 

I ' I

 

 

 

 

 

8 I .

.- Mn7-020-Mn8 O .
%, Mn5-09—Mn7 | I l I I g”
C ' C
o I I a o
c I on c
z 8- | ' | . 2
c') F I o 2000 CI)
I . I B B Ell I
g a g | g
8': I '

” : I I : I . 4|—-o—

" I

l I I | .

*_ I _
:7; I Q“ RI... g
on I 2 9' on
c 39" I ‘ c
z | I|IBI 2

2|— | 'III-I

o A_ 1 l A] l

m I I V r 'l I +—'_
2 a;- I III.“ 2
2’ . 2’
a N | I I I a
o a" I I1 I ‘ o
2 °_Cl D O O O _ 2
I °’ A g I I
O '. A A Al°°°l o

.. A ‘ ..
8L I In .1 . .‘JAA.‘
400 600 800 1000

T (K)

Figure 3.24: Results from LaMnOg(A) N PDF data are shown in the left panel; while
Both PDF (filled symbols) and Rietveld (open
symbols) refined values are shown. (up) the two Mn-O-Mn angles 91 and 02, (middle)
the Mn05 octahedral tilt angles cpl, (p2 (I, o), and their average (cp) (A) (down) the

LaMnO3(B) SEPD data in right.

O-Mn-O angles within one Mn06 octahedron.

86

160

16 155

14

12

94

90 92
0'.» 'El ' "

88

 

I f

Mn7-020-Mn8 ll 0

Mn6-OQ-Mn7 I : I

 

 

CIDE-

 

OO) DI
O. IIIIII

0.13.

L A

4A

0
08:23:

 

 

O

200 400 600

T (K)

800

without careful thoughts. The exact picture can not be obtained without knowing the
local structure in disordered systems (discussed later). The O-Mn-O angles are rather
intriguing. The Mn-O bonds are not orthogonal even in the 0’ phase, probably due to
the lattice strains. However, this non-orthogonality seems to increase with increasing
temperature, and shows a jumping at the JT transition. We think this is an indication
of increasing lattice strain in the high temperature phases.

The Mn06 octahedral JT distortion has been characterized by the normal JT
modes such as Q2 = 2(l — s)/\/§ and Q3 2 2(2m — l - s)/\/fi, where the l, s, m are
the long, short, and medium Mn—O bond lengths. The ground-state wave functions

\119 and We in the eg state are given [140, 141, 82] by

\I’g = 01¢12—y2 + C2¢322—r29 (3'7)

\I’Ie = C2¢:r:"’——y2 "' Cl¢3z2—r27 (38)

where c1 and c2 are coefficients obtained from relations: tan cp = Q2 / Q3 and tan cp/ 2 =
c1 /c2. The self-normalization condition C? + c3 = 1 should also be satisfied. In the
absence of octahedral distortions, we should obtain c1 2 c2 = 0.7071, i.e. orbitals
¢$2_y2 and ¢322_,2 are degenerate. Those parameters are shown in Fig. 3.25. Both
Q2 and Q3 values approach zero as the Mn06 octahedral JT distortion diminishes,
which is as expected. As the Q2 mode is more pronounced at low T, the 4532242
orbital is more preferably occupied, as is evident from the large 02 values. Upon
increasing T and reducing the average Mn06 JT distortion, c1 and 02 merge to the
equal probability value of 0.7071. Above T JT, c1 and c2 values are not physically

meaningful, as the distortion is essentially gone as from the average structure.

87

 

 

"'le2'3' ""razlla',

    

 

 

 

 

 

 

 

 

* *..
Aa" . . 03 p no...0. 803 “A
a . I ‘5 ' '
N “E.
=5 30 . I-‘a . l
‘5' °‘ .1?” 3"" 3“ 1
LA ‘ ‘ 1 ’AAAA‘ ‘ 4
3 I I I J I I J = gI I I I I = H = I
' ' C1 1‘ ' c1 ”0
I CZ A} - 02 ”A
AAA
D no
80" ‘A4 AAI "86AAAA‘ AA '
«a . .§ I ‘. .3
02a 0“ “cl-.03..... 0. "I
h I .|.OO . l'fib
l A PAL 1 A J A l A l A l A II A l
400 600 800 1000 o 200 400 600 800
T(K) T(K)

Figure 3.25: Results from LaMnO3(A) NPDF data are shown in the left panel; while
LaMnO3(B) SEPD data in right. Both PDF (filled symbols) and Rietveld (open
symbols) refined values are shown. (up) The Q2 and Q3 values, (down) the c1 and 02
values.

3.8.3 Summary

In summary, our average structure analysis has established the crystallographic JT
and OR phase transition of our samples. Obtained transition temperatures from our
neutron powder diffraction and DTA data are in good agreement. PDF wide 1' range
refinements and Rietveld analysis give quantitatively consistent results. Below TJT,
Mn06 octahedra are fully JT distorted with two long, two short, and two medium
Mn-O bonds. The e9 orbitals order in a staggered pattern, giving the significantly
orthorhombic 0’ phase. Across the JT transition, the lattice enters the 0 (pseudo-
cubic) phase where the MnOG distortion is drastically reduced, though not completely

removed. The OR transition completes the removal of Mn06 J T distortions, and leads

88

to the more symmetric hexagonal phase (Rgc).

3.9 Intermediate structure from PDF analysis

Fits to the nearest neighbor Mn-O peaks show a large JT distortion in the 0 phase,
whereas crystallographically the Mn06 octahedra are almost regular (six equal bonds).
This implies that the average structure results from a loss of coherence of the ordering
of the JT distorted octahedra and that the JT transition is an orbital order-disorder
transition. The locally JT distorted Mn06 octahedra orient the long bonds randomly
along three main axis. Thus, on the average, the lattice appears to be pseudo cubic,
and the Mn06 octahedron appears to be almost regular. However, the placements of
the long bonds of neighboring Mn06 octahedra are not independent, because of the
shared oxygen atom. It would cost too much lattice elastic energy (strain) to have
two long bonds aligned back to back. An anti-ferrodistorsive first-neighbor interaction
has to be considered, maybe a weak second neighbor interaction as well. Thus, we
expect locally JT distorted Mn06 octahedra to form small short-range ordered clus-
ters (“polarons”), and the cluster size (“correlation length”) depends on the relative
strength of the JT distortion and the lattice stiffness.

In principle we can test the extent of any orbital short-range-order in the PDF
by refining models to the data over different ranges of 7‘. Fits confined to the low-
I" region will yield the local (JT distorted) structure whereas fits over wider ranges
of 1' will gradually cross over to the average crystallographic structure. To extract
the size of the short—range ordered clusters, we have fit the PDF from rm,“ = 1.5 to
rum, where 5 < rm“. < 20 A 4. This was done for all data-sets. The model used
at all temperatures was the low-temperature structure in the anm space group.
Representative results are shown in Fig. 3.26 as open circles The amplitude of the

refined JT distortion is constant as a function of 7'me at 300 K (Fig. 3.26(a)) reflecting

 

413mm of 20 A is found to already give the crystallographic results.

89

 

 

2.2
I

.1

 

   

   

0.10.12:

 

2

' "’f.“nounn'fl‘fl'fl'l'“
u "I" ' ., u

dMn-O (A)

 

 

2.2 1 .9

.1

 

 

 

 

dMn—O (A)
2

1.9

 

 

 

 

4 8 I 12 16 20
rmax (A) rmax (A)

Figure 3.26: Mn—O bond lengths from the refined structure model as a function of
rum are shown as open circles, and the corresponding fits with the PDF form factor
of a solid sphere is shown as filled circles. The horizontal lines indicate the Mn—O
bond lengths from Rietveld refinements. (a) 300 K, (b) 740 K, (c) 800 K, (d) 1100 K
(see text for details).

 

the fact that the orbital order is perfectly long-range. Regardless of the range fit
over, the full JT distortion is recovered. At higher temperature, and especially in
the 0’ phase the amplitude of the refined distortion falls off smoothly as the fit
range is extended to higher-r, until it asymptotically approaches the much smaller
crystallographically refined value (Fig. 3.26(b,c,d)). We understand this behavior
in the following way. Domains of local orbital order exist in the 0 phase. These
may resemble the pattern of orbital order at low temperature, and for convenience
this is how we have modeled them; however, we cannot rule out the possibility that
another type of orbital order exists as we have not done an exhaustive search and

compared different models. These domains do not propagate over long range and are

90

 

orientationally disordered in such a way that, the observed pseudo-cubic structure is
recovered on average. We can estimate the domain size by assuming that the orbitals
are ordered inside the domain but uncorrelated from one domain to the neighboring
domain. This results in a fall-off in the amplitude of the refined distortion with
increasing fit range with a well defined PDF form-factor [142], assuming spherical
domains, that depends only on the diameter of the domain. The three curves of
refined-bond-length vs. rm” from the short, medium and long bonds of the MnOs
octahedron could be fit at each temperature with the diameter of the domain as the
only one parameter. Representative fits are shown in Fig. 3.26 as the filled circles,
evidencing rather good agreements.

The temperature dependence of this domain size is shown in Fig. 3.27. We find
that in the pseudo-cubic 0 phase these clusters have a diameter of ~ 16 A independent
of temperature except close to T JT where the size grows. Below T JT the correlation
length of the order is much greater, although some precursor effects are evident in
the data just below T JT. The refined domain size of orbital order is similar in the
R-phase though the quality of the fits becomes worse in this region. This may be
because the nature of the short-range orbital correlations changes, i.e. different from
the 0’ phase. Domain size of ~ 16 A spanning over four Mn06 octahedra suggests
the JT distortion has strong first nearest neighbor coupling, and weak second nearest

neighbor coupling.

3.10 Discussion and conclusions

We now address the issue of the unit-cell volume collapse observed crystallograph-
ically [134]. This is most readily explained if charges are delocalizing resulting in
more regular Mn06 octahedra and a smaller unit cell, as observed at higher doping
in the La1_zCaan03 system for example [23]. This appears to be borne out by our

detailed Rietveld results on the same data which indicate that the volume contraction

91

 

 

 

 

 

 

I ' 1 I ll ' I ' '—
In
C5 1/r=a(T-Tfl)'+b I O
p= 0.48 l
b= 0.09
5‘ |
I v-_ .
SO
(0
.2 .
U
2
>3. -
O
or - : 3.4—04$ - g . g .

 

300 450 600 750 900 1050
T (K)

Figure 3.27: The inverse of the domain radius as a function of temperature is shown
as the filled circles. The dashed line indicates the JT transition temperature. The
solid line is a fit with the formula shown in the upper left corner.

occurs as a result of a shortening of the average Mn—O bond length, (TMn—o>- This
is shown in Fig. 3.21(a)s where the results of both Rietveld and wide-r-range PDF
fits are in excellent agreement and show the rapid shortening of (rMn_0). Detailed
calculations show that the volume reduction of Mn06 octahedra accounts for the full
volume reduction of the unit cell, i.e. La012 polyhedron does not show significant
volume reduction at all. However, we have clearly shown that locally the Mn06 oc-
tahedra do not change their shape and (rMn_0) is independent of temperature. The
volume collapse must therefore have another physical origin. The most likely scenario
is an increase in octahedral tilting amplitude, although none is seen in the average
octahedral tilt angle. To test this, we made a x/i x W x 1 supercell so as to de-

compose the O atom’s thermal displacement parameters along directions parallel and

92

 

 

 

 

 

 

 

 

n ‘ I ' fi ' I
Q.
0
Ag_
as c5
v
j.
o.-
O
01
l A

 

 

 

C ' ,
250 500 750
T (K)

Figure 3.28: The thermal displacement parameters of oxygen atoms parallel and
perpendicular to Mn-O bonding directions. Filled symbols (0, I) are the parallel;

Open symbols are the perpendicular.

perpendicular to the Mn-O bonds. Fig. 3.28 shows the refinement results. It is clear
that a jump in the oxygen displacement parameters, refined over a wide range in the
PDF, is seen at T JT consistent with the appearance of increased, but disordered, tilt
amplitudes. The change in the average tilt angle that is necessary to accommodate
the collapse in unit cell volume is just 0.4 degree (cell volume decreases by 0.38%
across T JT)- Note that a sharp increase is seen in the components of the oxygen
displacement parameter both parallel and perpendicular to the Mn-O bonds. This
is expected due to the disordered distribution of the Mn—O long bonds as well as
the increased inhomogeneous tilting that is required to accommodate the short-range

ordered orbitals.

It may become necessary to discuss the somewhat counter-intuitive decrease of the

93

(TMn_0) upon warming in the average structure, while the local structure analysis
shows it to actually slightly increase as expected from the normal thermal expansion.
The local structure PDF analysis gives the physical distance between atomic pairs
(i.e., the Mn and O atoms in this case) without any assumption about the long range
ordered structure. The average structure analysis first assumes one crystallographic
unit cell with some symmetry, then obtains the the average position of each atom in
the unit cell. Therefore, the position of each atom is averaged before the atomic pair
distance is calculated. The average structure analysis will give the physical atomic
pair distances in the perfect crystalline materials. However, these distances may not
represent the real physical values in the presence of structural disorder. This is in fact
what occurs in the high temperature LaMn03 phases. Locally JT distorted Mn06
octahedra lay their long bonds randomly along each crystallographic axis. The three
Mn-O bond lengths thus approach their averaged value across the JT transition.
Still, this does not explain the decrease of the (rMn_0), as none of the three Mn-O
bonds shorten across the JT transition. In the mean time, the average of the lattice
constants also decreases. The angle between the Mn-O bonds and the corresponding
crystallographic axis has to increase for this to happen, i.e., an increase of the Mn06
octahedral tilting angle. As mentioned in the previous paragraph, an increase of 04°
Of the tilting angle is sufficient to account for the decrease of the decrease of (rMn_0)
as well.

The sudden decrease of resistivity across the JT transition is still somewhat puz-
Zling. As the nature of the JT transition is orbital order to disorder, we would expect
the more disordered state has lower conductivity due to the increased scattering by
the “defects”. Apparently, this is not the case, and there has to be some other phys-
iCal origin. Zhou and Goodenough [132, 143, 144, 145] have pr0posed a vibronic
l’nOdeI where small fractions of Mn3+ disproportionate into Mn2+ and Mn“. While

this Charge disproportionation improves the conductivity, the JT distortions of those

94

involved Mn06 octahedra also disappears. Though our PDF studies strongly suggest
the fully JT distorted MnO6 octahedra at all temperatures, the error bar of a few
percent in our analysis may be incapable of resolving such small fractions. This au-
thor thinks the dynamic JT nature of the high T phases may offer an explanation to
the sudden increase of conductivity. In the scenario of dynamic JT distortions, the
Mn06 octahedron is constantly changing its shape, which means the eg electron is
moving around at least inside one octahedron. This eg electron jumping is supposed
to be a slow process as our neutron powder diffraction study (energy resolution ~ 20
meV) sees the fully distorted state. Nevertheless, due to the non-ionic nature of the
Mn-O bonding, we would expect the e9 electron to possibly hop to its neighboring
octahedron, thus increasing the conductivity. Particularly in the case of short range
JT distortion correlations in the 0 and R phases, this gives the “large polaronic
transport” picture.

To conclude, we have used a diffraction probe of the local atomic structure, neu-
tron pair distribution function (PDF) analysis, to see if the local and average behav-
iors can be reconciled and understood. This method allows quantitative structural
refinements to be carried out on intermediate length-scales in the nanometer range.
We confirm that the JT distortions persist locally at all temperatures, and quantify
the amplitude of these local JT distortions. No local structural study exists of the
high-temperature rhombohedral R-phase that exists above TR = 1010 K. We show for

the first time that even in this phase where the octahedra are constrained by symme-
try to be undistorted, the local J T splitting is constant. Fits of the PDF over different
7‘-ranges indicate that locally distorted domains have a diameter of ~ 16 A with four
M 1106 octahedra spanning the locally ordered cluster. The unit cell volume collapse
seen crystallographically comes from a reduction in the average Mn-O bond-length;
however, this quantity is constant with temperature in the local structure. We show

that the observed volume reduction is consistent with increased octahedral rotational

95

degrees of freedom. We speculate that local clusters of the low-T structure form on
cooling at TR but are disordered and align themselves along the three crystal axes.
These domains grow, and the orbital ordering pattern becomes long-range ordered

b8lOW T JT .

96

 

 

Chapter 4

Local Structure of the Bi—layered
Manganite La2_2er1+2an207 in

the Mn4+ Rich Region

4.1 The bi-layered manganitezLa2_23Sr1+23Mn207

The bi-layered manganite La2_2xSr1+2an207 is the n = 2 member of the so-called

Ruddlesden-Popper series An+1Mnn03n+1, while the “cubic” compound LaMn03 is

the n = 00 end member. The crystal structure of the bi-layered manganite is shown

in Fig. 4.1. This system appears to be a prospective candidate for future technolog-
ical application by showing a large colossal magneto-resistance (CMR) effect [146],
and has been widely studied [147]. This compound consists of double layers of cor-
ner shared Mn06 octahedra (perovskite structure type) separated by one rock—salt
layer of (La,Sr)-O. Common to many transition-metal oxides with perovskite-related
Structures, the interplay between spin, charge, and lattice degrees of freedom is of
Critical importance [148]. Their delicate interactions result in a rich and interesting,
bUt somewhat poorly understood, phase diagram [149, 150]. The bi—layered man-

ganite system La2_ngr1+2$Mn207 shows behavior qualitatively similar to the cubic

97

Figure 4.1: Crystal structure of the bi-
layered manganite La2_2ISr1+2Il\/lnzO7. It
consists of double layers of cornered shared
MnOG octahedra, which are separated by
one rock-salt layer of (La,Sr)-O.

 

 

 

CM R manganites and is particularly interesting to study because of the extra factor
0f its reduced dimensionality. This material is also thought to be a good candidate for
applications because its low dimensionality increases the ferromagnetic critical fluctu-
atiOIlS close to the paramagnetic-ferromagnetic transition that are important for the
CMR phenomenon. The reduced dimensionality obtained by constraining the lattice
degree of freedom in this bi—layered system also facilitates the investigation of strong

correlations between electron-lattice coupling and magnetic properties, proving it to

98

 

 

be one of the most interesting CMR manganites [147].

The crystallographic structural and magnetic phase diagram of La2_2xSr1+23 Mn207
is shown in Fig. 4.2. The tetragonal unit cell with space group I 4/mmm is found
in all Sr doping levels, except between :1: = 0.76 and a: = 0.94 where the tetragonal
and orthorhombic (space group I / mmm) structural phases coexist. The most heavily
studied region is between :1: = 0.30 and 0.42, where the important CMR phenomenon
and the temperature driven Insulator-Metal (IM) transition are observed [146]. The
pioneering work of Moritomo et al. [146] discovered a Curie temperature of 126 K
for this bi-layered compound at a: = 0.40. The observed magneto-resistance (MR)
ratio at the Curie temperature reached as high as 200% under 0.3 Telsa magnetic
field, significantly higher than in the equivalent 3D Sr-based compound. The large
MR effect was also reported for the :L' = 0.3 compound [151]. Ling et al. [150] have
used temperature dependent neutron powder diffraction to systematically map out
the magnetic and structure phase diagram in wide Sr—doping range of 0.3 S a: S 1.0,
which serves as the basis to the more modern version shown in Fig. 4.2. While
the paramagnetic state appears at high temperatures at all doping levels, the ground
states of this bi-layered compound show very interesting magnetic structures. Fig. 4.3
Shows the schematic diagrams of the magnetic phases. The magnetic coupling be-
tween the double layers of Mn06 octahedra (inter-layer) is significantly weaker than
intra-layer, giving quasi-2D magnetic interactions. We will get into more details of
the different structural and magnetic phases in the regions of interest in the sections
followed. Please refer to Ref. [150] for others. Most studies have focused on the Mn3+
rich portion (:1: < 0.5) of the La2_2xSr1+23Mn207 phase diagram, particularly because
of the observed CMR phenomenon. However, the phase diagram of the less studied

Mn4+ rich region shows many novel and intriguing properties, and is the subject of

this Chapter.

99

Irv-rifiv

400 fi..,v.........

l
l
350 ‘*
Paramagnetic Insulator T if [[1
I l
i l

300 Tetragonal

T (K) Charge Order —>

Type A AFI

AFM—’

 

Figure 4.2: Magnetic and crystallographic phase diagram of La2_2er1+2IMn207.
Solid data points represent magnetic transitions determined from neutron powder-
diffraction data (antiferromagnetic Neel temperature TN, ferromagnetic Curie tem—
perature TC, charge-ordering temperature T00 , and orbital ordering temperature
To). Open data points represent crystallographic transitions. Lines are guides to the

eye.

4.2 Orbital occupancy transition at a: = 0.54

4.2- 1 Introduction to the question

In manganite systems close to 50% doping (Mn3'5+) charge and orbital ordering is very
commOH [152, 153, 154, 155]. This is understood as an interplay between coulombic
and lattice strain degrees of freedom [99]. Charges disproportionate towards Mn3+ and
Mn4+ and form a lattice on every second site. The occupied orbitals also order so as to
loWer the energy of the System. In the cubic manganites the occupied orbital results in

a longer Mn-O bond as a result of the Jahn-Teller distortion. These elongated d3a2_,2

100

 

(b) FM (2) (0) FM (XY) ((1) Type-A (X7)

Ti" ’ 1 (f 7’

fit} ’17 {El—1 VI A

' polytypism I ; tilting :4

i C—-—-—-—O

G ’“I'I 1}} WI

0") TYPE-G (Z)

(8) AFM (Z)

(6) TYPe C 0’) (f) TYPO-0‘00 (9) Type-G (Xv-Z)

Figure 4. 3: Schematic illustrations of magnetic structures of L32 QISngangOh

SHOWS representing spin orientations on Mn siteS: (a) AFM, (b) FM at 17 — 0 32 (C)
M at a:— — 0. 40, (d) type-A AFI, (e) type-C AFI, (f) type-C" AFI, (g) type-G AFI

at a: 2 o 92 and (h) type-G AFI at :1: = 1.00. Taken from Ref. [150]

101

Figure 4.4: Cartoon representation of type CE charge ordering in manganites. The
solid dots (o) are the Mn4+ ions. Shown orbitals are the occupied d3(z,y)2_,2 orbitals
of Mn3+ ions. Dashed lines indicate the ferromagnetic zigzag chains coupled anti-
ferromagnetically.

orbitals, (a = 11:, y, z), lay down in a crystallographic plane (shown in Fig. 4.4) and

form a zig—zag pattern resulting in the CE magnetic ordering arrangement [152, 156,

l 154]. The simple, semi-empirical, understanding for this behavior comes from the
Goodenough-Kanemori rules [81, 82].

In common with the cubic manganites, CE charge and orbital-ordering is ob-

35“ charge, in

served in this bi-layered compound close to 50% doping (nominal Mn

this case composition LaSrgMn207) [153, 154], appearing at 210 K on cooling. How-

! ever, in contrast to the cubic manganites the ordering disappears again around 170K
and the sample has an A-type antiferromagnetic insulating ground-state. A-type
magnetic ordering is observed in undoped and lightly doped cubic manganites and
ConSists of ferromagnetic planes that are antiferromagnetically coupled along the c-

aXiS- In the undoped manganites this state is easily understood in the framework of

the Goodenough-Kanemori (G-K) rules [81, 157]. However, at the high doping levels

102

of this sample it is inconsistent with these rules, as we discuss below.

The G-K rules give a simple road-map for understanding the diverse magnetic
ground-states observed with changing doping in the manganite systems. It is assumed
that charges reside in particular d-orbitals (and their associated covalent bonds with
neighboring oxygen atoms). A particular orbital can then be considered as “filled” or
“empty” (though in reality this disproportionation of charge will not be complete).
The network of Mn ions is then considered. The Mn eg d-orbitals point towards
neighboring Mn ions through the nearest neighbor oxygen ions. When two filled, or
two empty, orbitals point towards each other the coupling is locally antiferromagnetic
due to superexchange through a (nearly) 180° Mn-O-Mn bond. When a filled orbital
points towards an empty one, double-exchange results in a local ferromagnetic inter-
action. The orbitals can then order to produce long-range magnetic order of different
types according to these local interactions. For nominal Mn charge of 3.5, the G-K
rules straightforwardly lead to the CE structure. The elongated, occupied, d3$2_,2
and d3y2_,2 orbitals form a zigzag chain with ferromagnetic correlations along the
chain. Each zigzag chain is then coupled antiferromagnetically with its neighboring
chain, and also with the chain residing in the layer above (refer to the OO cartoon
representation shown in Fig. 4.4). In contrast, in the undoped cubic manganites the
occupied d3$2_,.2 or d3y2_,2 exist on every site and form into a checkerboard pattern.

In this case, in the plane, a filled orbital always points towards an empty one resulting
in 2-D ferromagnetic correlations in the plane (see Fig. 3.7). The orbitals pointing
Perpendicular to the plane are all empty d(x,y)2..22 resulting in antiferromagnetic cor-

relations between neighboring planes and therefore A-type antiferromagnetism.

The problem comes when one tries to explain A-type antiferromagnetism at 50%

doping. Here we have to take these orbitals and tile a plane when half of the sites

are IVIn4+ in such a way as to have alternating filled and empty orbitals. In fact, it

IS possible if the electrons change their usual d($,y)2_22 orbital occupancy to d32_y2.

103

Figure 4.5: Illustration of the in-plane dx2-y2 orbital occupancy in half doped
La2_ngr1+gangO—,. The solid dots (o) are the Mn4+ ions. Shown orbitals are the
occupied d$2_y2 orbitals of Mn3+ ions.

The Mn06 octahedron distorts due to the J T effect that breaks the degeneracy of the
d3z2_,2 and the d32_y2 e9 orbitals (where :r, y and 2 can be permuted depending on
the orientation of the octahedron). In general, in the manganites the d3z2_,.2 orbital is
stabilized and becomes occupied and elongated. However, it is possible given differ-
ent external perturbations, such as crystal field effects, that the dxz_y2 could become
stabilized and the occupied orbital. This orbital forms bonds with four neighboring
oxygens instead of two. When this is the occupied orbital we expect that there will
be four long bonds and two short bonds and the octahedral distortion will be oblate
instead of prolate. It also changes the relative number of filled and empty orbital
pairs at constant charge. A-type antiferromagnetism at 50% doping is then predicted
by the G-K rules if we tile the plane alternately with Mn3+ and Mn4+ ions and the

filled d32_y2 orbitals of the Mn3+ ions lie in the plane. Every site in the plane is then

joined by a filled-to—empty orbital pair and therefore ferromagnetic correlations, as

ill‘JStrated in Fig. 4.5. All the out-of-plane orbitals are empty resulting in AF cor-

104

relations between the planes. This would seem a natural explanation for the A-type
magnetism at 50% doping in the layered manganites. That it is energetically plausible
has also been verified theoretically [158, 159], due to the reduced quasi—2D geometry
stabilizing the dx2_y2 orbitals. Note, it has been shown by resonant X-ray scatter-
ing [160] that this orbital occupancy explains the A-type antiferromagnetism [161]
in Nd0,45Sr0.55Mn03, a 50% dOped cubic manganite. What makes the current case
particularly interesting is that CE-type charge ordering is observed over a narrow
temperature range on cooling, before the sample passes into the A-type ground-state.
Since the CE style orbital ordering requires d322_,2 occupancy and the A-type re-
quires d$2_y2 occupancy, this implies a transition in orbital occupancy happening as
a function of temperature in this material.

We have carried out local structural measurements using the atomic pair distri-
bution function (PDF) analysis of neutron powder diffraction data on a sample of
La2-QxSr1+2angO-, with :1: = 0.54 as a function of temperature to investigate this
orbital occupancy transition. The corresponding local structure response from oblate
to prolate Mn06 octahedra transition is evidenced consistently. The PDF technique
has proved to be capable of directly measuring the local JT distortion effects in both

cubic and layered manganites [127, 162].

4.2.2 Experimental methods

A powdered sample of LaflggsfzggMIlQO'z was synthesized using the method described
1'11 Ref. [163]. The samples were characterized using X-ray diffraction and suscep-
ti bility measurements. The oxygen content were verified by measuring the c—axis

Parameter that was found to fall on the expected curve for stoichiometric samples.
Neutron powder diffraction measurements were carried out on the Special Envi-
r on IIlent Powder Diffractometer (SEPD) at the Intense Pulsed Neutron Source (IPNS)

at Argonne National Lab (ANL). The sample of about 7.0 g was sealed in a cylin-

105

drical vanadium can with helium exchange gas. Temperature dependence data were
collected from 4 K to room temperature using a closed cycle helium refrigerator.

Experimental PDFs were obtained the program PDFgetN [42]. The resulting
total-scattering structure function, S (Q), and the PDF, G (r), from Lao,ggSr2,08Mn207
(a: = 0.54) at 4 K is shown in Fig. 4.6. Superimposed on the PDF is a fit to the
data of the average structure model using the profile fitting least-squares regression
program, PDF F IT [11]. The S (Q) data were terminated at Qmaz = 30 A’1 before the
Fourier transform. This is a reasonable value for Qmax in typical PDF measurements
on SEPD. Rietveld refinements of the same diffraction data were also carried out
using program GSAS [68] with the conventional tetragonal crystallographic cell: space
group I4/mmm, a = b = 3.8666 A, c = 19.8713 A, La(Sr)1 (0.00, 0.00, 0.50), La(Sr)2
(0.00, 0.00, 0.32), Mn (0.00, 0.00, 0.10), 01 (0.00, 0.00, 0.00), 02 (0.00, 0.00, 0.19),
03 (0.00, 0.50, 0.09). The type A magnetic phase was added as the second phase to
account for the magnetic diffraction peaks. Excellent fits were obtained for data at
all temperatures. Fig. 4.7 shows the exemplary Rietveld fit plot of diffraction data at
4 and 300 K.

4.2.3 Results

The PDF, G (r), tells the probability of finding pairs of atoms separated by the real
Space distance 7'. Thus every existing inter-atomic distance results in a Gaussian
Shaped peak, the width of which is determined by the atomic thermal motions [5]. In
the case of one eg electron complete d3zz_,2 occupancy, a long Mn-O bond around 2.15

A becomes possible [127] in CE ordering phase. However, there is no direct evidence
for a r = 2.15 A peak at any temperature (see Fig. 4.8). Note that Mn-O peak is
uIDSide down due to the negative Mn neutron scattering length [5]. The existence

0f Such long bonds has been claimed in earlier PDF measurements at a much lower

doping (:6 = 0.32) [162], but are not apparent in our data. In addition, the integrated

106

 

100

Q(S(Q)—1)
50

 

0

 

 

 

O
H
O
N
O

 

; WU“! W" 1 3.1

_
_A_. A. AA A “A AA A. ALLA-AA._
vv Vr'vv vv v-r ‘v vvv'v ‘1va
l A n l A

G(I‘) (3-2)
—12—6 0 6 12 18

 

 

 

l

4 8 12
r (A)

Figure 4.6: (a) Reduced total scattering structure function, F(Q) = Q(S(Q)—1) from
La0_9BSr2,ogl\1n207 at 4 K. (b) The obtained experimental PDF, G(r) (open circles).
The solid line is a fit to the data of the crystallographic model with the difference

curve shown below offset.

 

area between 1.6 and 2.1 A (i.e. the number of short Mn-O bonds) shows no sign of
Changes over the whole measured temperature range. We attribute the absence of 2.15
A Mn-O bond to mixed dx2_y2 (minority) and d3z2_,2 (majority) orbital occupancies
in CE phase. The long Mn-O bonds in both the type A AFM and CE CO phases
a-1313ear to be shorter than 2.1 A. This scenario has also been suggested by theoretical

CalCulations [164], as a result of the reduced quasi-2D geometry.
The proposed dx2_y2 to d3,2_,2 orbital occupancy transition also changes the shape

of Mnog octahedra from oblate to prolate. The Mn-O bonds change from four long

107

 

 

 

 

 

 

V I ' l V I V I
0
CL .
o X
I? [n
3. . '- ‘
0 :
v 8
.4? '0 '
m N
C .
0
4" :7!
E -‘ _. « .
o . - .§
I I II "III IIII.IIIIII-I- l-.—-I--
I I I I IIII-IIIlulll-I
. 1 4 I . I . n
O 3 6 9 12
o (11")
I V I V l V 1 V f V T
p x 1
O
0
IO
N

Intensity (a.u.)
1300

O

 

 

 

 

 

0 (r1)

Figure 4.7: Rietveld refinement results of La2_2$Sr1+2an207 at x = 0.54 neutron

DOWder diffraction data at 4 (upper panel) and 300 K (lower panel). Crosses (x) are
experimental data, solid lines are the calculated values from refined structure model,
the difference curve is shown offset below. The short bars indicate the Bragg reflection
pOSitions from nuclear (upper) and magnetic structures, respectively.

108

 

 

50
13

 
    

 

 

 

 

 

 

 

 

 

 

4.. I
I O
.9 _ .
.v . §.
‘ :3. : z“ ‘ -.
e % O I
0 ' ' I
Q I a
<.>=- ' I - -
O c _ _ __. I
= fifi
o. I I .
5 '— : : 1*. § : i: 4. :
’co : :
29’" ,, Lifi‘f ; (c) .
Hg, \.
‘6 In- IE I\ _
o _c In . i . \ \
§ . E if .
Q ~:\ \:I
o ("’8' I , ‘
o I I
L L l . l A ll . | I A l A: l . :l n l
1.8 2.1 2.4 2.7 3 O 75 150 225 300
r01) T(K)

Figure 4.8: (a) The low 1‘ region of all experimental G(r)s from 4 K to 300 K of
La2_2er1+2angO7 at :c = 0.54, with offsets along y axis for clarity. The two dashed
lines indicate the Mn-O and 0-0 peaks at 1.94 and 2.72 A, respectively. (b) Temper-
ature dependence of the Mn-O absolute peak height around 1.94 A. Vertical dashed
lines indicate the transition temperatures around 110 and 210 K. Debye behaviors of
the PDF peak height are indicated by solid lines, with extrapolated dashed lines. (c)
The 0-0 peak height around 2.72 A. Annotations as in (b).

109

and two short to two long and four short. Though with the same four and two combi-
nations, the prolate octahedron actually has a broader Mn-O bond length distribution
than the oblate, because of the larger separation between the long and short bonds.
The underlying reasoning is that one eg electron will lengthen two Mn-O bonds more
than the case of four bonds. A broader distribution directly leads to a broader PDF
peak with the area conserved. Therefore, the height of the PDF Mn-O peak, as a
directly obtainable experimental quantity, is expected to drop excessively across the
oblate to prolate octahedra transition. Fig. 4.8(b) shows the temperature dependence
of the Mn-O (z 1.94 A) peak height, with dashed vertical lines indicating the transi-
tion temperatures. The PDF peak height normally decreases with temperature due to
thermal broadening, following the Debye behavior. One guiding-to-the-eye FDF peak
height Debye curve was then calculated and plotted on top of the data in Fig. 4.8(b)
with different offsets to match the high and low T data. The required offset between
the low and high T behaviors evidences the excess reduction of peak height, as ex-
pected from the above reasonings. This trend from the somehow noisy Mn-O peak
height data is further substantiated by the peak height temperature dependence of
the O-O pairs (z 2.72 A) edging the Mn06 octahedron. The 0-0 peak height data
shown in Fig. 4.8(c) indicate similar excess peak height falloff, though less sensitive
to Mn06 octahedral distortions. These observations directly support the scenario of
the oblate to prolate Mn06 octahedra transition.

The long Mn-O bonds in the oblate and prolate Mn3+06 octahedra all lay down
in the MnOz plane implied by the (132.12 to d322_,.2 orbital occupancy transition.
However, the orientations of the JT distorted octahedra are in-extractable from the

PDF peak height data, which only tell the overall octahedral shape. We then fit
the conventional crystallographic model (space group I4/mmm, a = b = 3.8742 A,
C 2 19.9844 A)) where all Mn06 octahedra have the same shape with four equal in-

plane Mn-O bonds. Though with the presence of local structure disorders, excellent

110

fits to all experimental PDFs were obtained, suggesting the JT distortions in this
bi-layered system are indeed rather small. The the refined thermal displacement
parameters (TDP) of the oxygen atoms are examined for signs of disorders. As no
symmetry is assumed in PDFFIT, the TDPs are conveniently decomposed into two
components, parallel and perpendicular to the Mn-O bond directions. One direct
consequence of the oblate to prolate octahedra transition is the increased Mn-O bond
length mismatch in the Mn02 plane (as the long bonds in the prolate octahedra are
significantly longer). This effect should show up on the parallel TDP components in
Fig. 4.9(a)), where only the in-plane oxygen atom shows anomalous increase across
the transition (deviating from the low T Debye behavior at high T). This is a strong
indication that the long Mn-O bonds always stay in the Mn02 plane. Note that the
TDP of Mn atoms show no sign of anomaly. To accommodate the long bonds in the
prolate octahedra, we also expect the Mn06 octahedra to increase tilting amplitude.
As no octahedral tilting is present in the structure model, sudden increase of the
perpendicular TDP components is expected. Fig. 4.9(b) evidences the anomalous
increases on both the in-plane and out-of-plane oxygen atoms. These results suggest
the oblate to prolate octahedra transition to be the d32_y2 to d322_,.2 orbital occupancy
transition type.

Consistent behaviors are also observed on the lattice constants from the Rietveld
analysis of the same data (shown in Fig. 4.9(c,d)). Clearly, the intermediate range
bEtWeen the two transition temperatures appears to be a cross-over region between the
low and high T regions, which can be well described with the same Debye model (solid
lines) offset-ed differently. The in-plane (a = b) and apical (c) constants show similar
trerIds, though with opposite signs. We argue that both behaviors can be qualitatively
explained by the dx2_y2 to (132242 orbital occupancy transition as follows. At low T, eg
electrons occupy the dz2_y2 orbitals. Around 115 K, the orbital occupancy transition

to C13,2_,.2 sets in. This process in fact distributes more electron densities along the

111

 

 

 

2r “‘I'°2 . <°> « s
(__03 i a:

 

U x 103 (32)
19.92

9.88

1

 

 

 

12
I
1—9—1
3869

 

3: 34143;? :33
3"“ [$11,119

 

 

 

 

 

F I : : ‘ '5
. 1 .-1 .-1 . 1 m1 . n .-/.-1 . 1‘
n

o 75 150 225 300 0 75 150 225 300
T (K) T (K)

F ignre 4.9: Dotted lines in all panels indicate the transition temperature. (a) and (b)
Show the thermal displacement parameters of the 01 (O), 02(A), and 03 (I) oxygen
a-‘3()Ins along the direction parallel and perpendicular to Mn-O bonds, respectively. (c)
and ((1) show the in-plane a&b and apical c lattice constant, respectively. The solid
_a-Ild dashed lines are the fitted and extrapolated Debye curves, noting that each pair
in each panel are the same curve with different offsets. The inset in (c ) shows the
phase” fraction of the prolate MnOa octahedron as a function of the temperature.

112

c axis, due to the donut shaped ring in the d322_,2 type orbitals. This will lengthen
the c axis. To the opposite, a & b axis would decrease correspondingly, e.g. bond
valence and/or volume considerations. The coexistence of deyz and d322_,.2 orbital
occupancy is expected in the cross-over region. Using the extrapolated values (dashed
lines) from both the low and high T Debye curves, we then extract the fraction of
occupied d3zz_,2 orbitals by assuming linear combinations. The inset in Fig. 4.9(c)
shows thus obtained fraction of the d322_,2 orbital occupancy (type CE phase), noting
the good agreements between values calculated from the in-plane and apical lattice
constants.
Finally, we show the three crystallographically distinct Mn-O bond lengths from
Rietveld refinements within one Mn06 octahedron in Fig. 4.10(a). The in—plane Mn-
03 distance behaves in the same way as the a = b lattice constants, though not
apparent under the shown scale. The significant observation is that the three Mn—O
bonds are rather close in length, i.e., the difference is less than 0.04 A at any tem-
perature. The absence of long Mn-O bonds around 2.15 A is consistent with our
PDF results, supporting the mixed dxz_y2 and d322_,.2 type orbital occupancy pic-
ture. As the eg electron stays always inside the MnOG octahedra, the octahedral
VOlume is not expected to show significant anomalies. Fig. 4.10(a) shows both the
0Ctahedral and unit cell (scaled) volume increases normally with temperature as ex-
pected. Note that drastic octahedral and unit cell volume changes are found near the
iIlslllator-metal transitions where the eg electron mobility changes [165]. The absence
of volume anomalies suggest the residence of the eg electron within the octahedron

at, all temperatures, and only its orbital occupancy changes.

1 ~ 2.4 Discussion and conclusions

3 LII results are consistent with the widely believed scenario of nano-scale phase sepa-

a-t 1on and compet1t10n, by show1ng the oblate to prolate octahedra transmon to be a

113

 

 

 

 

 

 

 

 

A (a) C I Q . (b) i 2 .

58F : O . . J 6‘10 I I I

g... ...0. .2. 1‘5:- ' I .-

C . ' q) '- o . l

.9 ' - E _ - - ‘

3. . I . 3 I I

u . ' ' B In I .1

c '-

0 A ...... - .3, ..... to. > it- . ' . __

‘3 - 2 : 00" ‘I :

o - -I I . - .

Hz- i-I-3 I - £10. ".i i

2" II': : 5 . : _
o 100 200 300 F o 100 200 300

T (K) T (K)

Figure 4.10: (21) Temperature dependence of the three crystallographically distinct
Mn-O bond lengths within one Mn06 octahedron: Mn-Ol (I), Mn-02 (0), Mn-O3
(A). (b) The Mn06 octahedral volume (I) and scaled unit cell volume (0). Note
that both volumes increase smoothly with temperature, showing no anomalies.

smooth cross-over spanning the phase coexisting temperature range between 115 and
210 K. The structural and magnetic states of the ~50% doped layered manganites
have been well characterized by diffraction [155, 149. 153, 154]. There is some dis-
agreement about the details of the transitions (e.g., compare Refs. [153] and [154]);
however, qualitatively the progression is as follows. On cooling, charge—order (CO)
sets in around 210 K. Below this temperature both A-type antiferromagnetism and
CE-type antiferromagnetism appear and coexist in nano-scales over some range of
temperature. At low temperature the A-type ferromagnetism prevails and the CE is
strongly suppressed or destroyed. High resolution x-ray diffraction [154] suggest that
in the coexistence region the sample is phase separated into distinct regions contain—
ing the A and CE phases. A hysteresis to the destruction of the CE phase [153] also
suggests that this transition is first order. Conductivity goes down at the CO tran-
sition. It recovers as the CE phase is destroyed around 115 K bearing a remarkable
similarity to the MI transition in the CMR manganites. However, at low tempera-

ture the resistivity again rises indicating that the A-type magnetic state is insulating,

114

albeit with better conductivity than the CE-phase. This behavior reflects a compe-
tition between the CE and A phases, and this delicate balance of power between the
phases presumably explains the variability in quantitative results between different
investigations [153, 154]. We also note that CE—type magnetism is observed in the
higher-doped region of the phase diagram [166]. By the same arguments we expect
that this requires d3z2_,.2 occupancy. Therefore, if this argument is correct the region
of stability of the d12_y2 orbital occupancy is rather narrow and bounded on the low-
and high— doped side, and at high-temperature, by d322_,.2 orbital occupancy.

To conclude, our PDF analysis of high resolution neutron powder diffraction data
consistently evidence the oblate to prolate JT distorted Mn05 octahedra transition
when the material goes from type A AFI phase into type CE CO phase. This can also
be understood as an unusual charge occupancy transition from (132-3,2 to d3(x,y)2_,2
orbitals. Our experimental results provide consistent evidences for theoretical calcu-

lations.

4.3 Evidence for nano—scale inhomogeneities in the

Mn4+ rich region: 0.54 S a: S 0.80

4.3.1 Introduction to the question

In a wide range 0.46 S :c S 0.66, the type-A anti-ferromagnetic insulator (AFI) phase
is the ground state while it is more commonly found in cubic manganites at the lowest
Sr2+ dOpings, though with some exceptions [167, 161]. Also, a spin disordered “gap”
region takes over for doping 3: from 0.66 to 0.74; then a tetragonal to orthorhombic
crystallographic phase transition occurs sharply at $20.74, followed by the type C / C“
AFI phase up to 0.90 doping. There is also a wide charge ordered (CO) region

(0.48 g :1: g 0.66). These observations challenge the simple Goodenough-Kanemori

115

(GK) rules [81] that successfully correlate the magnetic and structural properties of
cubic perovskite manganites and are not well understood.

An A-type AF I phase, of ferromagnetic sheets antiferromagnetically coupled, at
50% doping can be explained within the framework of the GK rules if the Mn 3d32_y2
orbitals are occupied by the (3g electrons rather than the more commonly observed
3d3y2_,2 occupancy. However, to explain the C/C* AFI phase (linear FM coupled
chains that are antiferromagnetically coupled to their neighbors) at high doping re-
quires 3d3y2_,.2 occupancy. Because of the symmetry of these different orbitals the
former, 3dx2_y2 occupancy, will result in oblate (two short and four long bonds) and
the latter, 3d3y2_r2 occupancy, in prolate (two long, four short bonds) JT-distorted
octahedra. Alignment of the long-bonds in the plane along the b—axis naturally ex-
plains the observed orthorhombic symmetry [150]. In addition, a theoretical model
based on the two 69 orbitals by Okamoto et al. [168] also suggests the stabilization of
3d12_y2 and 3d3y2-,2 orbitals in type A and type C/C" magnetic phases respectively.

One outstanding question is the origin of the wide spin disordered region (0.66 S
a: g 0.74). Because of the change in symmetry of the occupied orbitals a transition
from A- to C/C*-type order must be first order. Presumably in the spin disordered
region a competition exists between these two magnetic orders that frustrates the sys-
tem preventing magnetic long-range order from forming. It would be interesting to
study the local spin correlations to verify this, but lack of single crystals has, thus far,
prevented such studies. However, the local structure can be studied straightforwardly
using the atomic pair distribution function (PDF) analysis of neutron powder diffrac-
tion data [5] and the local magnetism can be inferred from this through application
of the GK rules.

In the spin-disordered region the crystallographic structure is metrically tetrago-
nal. This would be observed both if the local structure is tetragonal and also if it is

locally orthorhombic but the locally orthorhombic domains are disordered along the

116

a and b directions. For example, this would occur if the long bonds of JT-distorted
octahedra are randomly arranged along a and b. The PDF method could distinguish
these two cases. Similarly, if Mn3+ and Mn4+ ions are spatially disordered their
presence will be more apparent from a local structural study [23].

We applied Atomic Pair Distribution Function (PDF) analysis of neutron powder
diffraction data to search for the presence of JT distorted Mn06 octahedra in the
doping range 0.54 S x S 0.80 at low temperature. Being a high resolution local
structure probe, PDF technique has proved to be capable of resolving different levels
of MnOs octahedral JT distortions in cubic perovskite manganites [23]. The advan-
tage of this technique is that both Bragg and diffuse scattering intensities are used,
reflecting both long and short range structural correlations. This enables us to study
local structures contained in diffuse scattering found underneath and between the
Bragg peaks [5].

The results indicate a gradual change of the local structure with doping rather than
an abrupt phase transition as seen in the average structure. Local orthorhombicity
is evident as early as :1: = 0.60 where the average structure is clearly tetragonal. This
supports the idea that the sample is inhomogeneous on the nano-scale with 3d,,2/y2_,.2
symmetry JT distorted Mn3+ octahedra coexisting with undistorted Mn3+ and Mn4+

octahedra. The number of JT distorted octahedra varies smoothly with doping.

4.3.2 Experimental methods

Finely powdered samples of La2_2xSr1+2angO7 (:L‘ = 0.54, 0.60, 0.64, 0.66, 0.68, 0.70,
0.72, 0.76, 0.78, 0.80) were synthesized at Argonne National Laboratory (ANL). The
synthesis method is described elsewhere [163]. All samples were characterized using
X—ray diffraction and susceptibility measurements. The oxygen content was verified
by measuring the c—axis parameter and was found to fall on the expected curve for

stoichiometric samples.

117

Neutron powder diffraction measurements were carried out on the Special Environ-
ment Powder Diffractometer (SEPD) at the Intense Pulsed Neutron Source (IPNS) at
ANL. The samples of about 7.0 g were sealed in cylindrical vanadium cans with helium
exchange gas. Data were collected at 4 K for all the compounds using a closed cycle
helium refrigerator. The :1: = 0.64 and x = 0.68 samples were measured four months
after the others. Standard corrections were made to the raw data to account for exper-
imental effects such as detector dead time and efficiency, background, sample absorp-
tion, multiple scattering to obtain the normalized total scattering structure function,
S (Q), where Q is the magnitude of the scattering vector. These procedures are de-
scribed in detail elsewhere [5]. All corrections were carried out using the program
PDFgetN [42]. The PDF, C(r), is obtained by a Fourier transformation according to
G (r) The PDF gives the probability of finding an atom at a distance r away from
another atom. An example of the PDF from La2_2xSr1+2angO7 (:1: = 0.80) at 4 K is
shown in Fig. 4.11(b) with the diffraction data in the form of F (Q) = Q[S (Q) — 1] in
Fig. 4.11(a). Superimposed on the PDF is a fit to the data of the average structure
model using the profile fitting least-squares regression program, PDFFIT [11]. The
S (Q) data were terminated at QmaJD = 33.0 A“. This is a reasonable value for Qmax
in typical PDF measurements on SEPD. Uncertainties at the level of 0‘ are drawn as
dashed lines on the difference curves.

Peaks in G (r) represent the probability of finding pairs of atoms separated by
the distance—r, weighted by the product of the corresponding atom pair’s scattering
lengths. In a perfect crystalline La2_2$Sr1+2an207 structure, the nearest neighbor
distance comes from the 6 equidistant Mn-O bond lengths in one Mn06 octahedron,
corresponding to the first peak in PDF at about 1.94 A. This is negative due to
Mn atom’s negative neutron scattering length. Peaks at higher-r generally contain
contributions from more than one unresolved pair. The peak at 2.72 A is dominated

by high multiplicity O-O correlations, though it also contains a contribution from

118

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r I ‘U ' I ' r.
g_ (a) _
A 1- .
10.. _
A0
0 . .
We
a»: j
0 "'dl'l'r'll"
O 8 16 24 32
0(1“)
0.
A .
‘7' °°- »
5 . ,5.
0° v
m-
I

 

 

 

-16_

O 5 1 0 1 5

I' (A)
Figure 4.11: (a) The experimental reduced structure function F (Q) = Q =1: (5' (Q) — 1)
of L32_238f1+2an207 at a: 20.80. (b) The experimental G(r) obtained by Fourier

transforming the data in (a) (solid dots) and the calculated PDF from refined struc-
tural model (solid line). The difference curve is shown offset below.

119

La/Sr-O correlations. The decomposition of multiple contributions is handled by a
real space Rietveld refinement program: PDFFIT [11] with which a structural model
can be obtained without the constraints posed by space group symmetries. Therefore,
both local structural and average structure analysis can be performed on the same

data set.

4.3.3 Model independent analysis

Low~r PDF peaks directly reflect the local structural details, and disorder in the local
structure can cause excess peak broadening [23], extra shoulders [162] and even split
peaks [127]. The prolate JT distorted octahedra of Mn3+ ions result in four short
Mn-O bonds in the range 1.92—1.97 A and two long bonds at 2.16 A [169, 126, 127].
In the cubic manganites, in the absence of disorder, these are easily resolved in the
PDF [127]. With dOping the loss of orientational order of the orbitals quickly sup-
presses the coherent J T distortion and the average structure changes from orthorhom-
bic to rhombohedral. However, the presence of fully JT distorted octahedra is evident
in the local structure, though the peak in the PDF from the long-bonds is not resolved
and is evident only as a broad shoulder [23, 170, 171].

We first investigated the PDFs from these layered, doped, manganites to search
for qualitative evidence for the existence of long r = 2.16 A bonds. This is shown in
Figures 4.12 and 4.13. Fig. 4.12 shows the experimental PDFs at :1: =0.54 and 0.60
with a calculated PDF from a model assuming all Mn3+06 octahedra are prolate.
In this case the number of Mn3+ octahedra is determined from the doping and of
these, two out of six Mn-O bonds are set to 2.16 A. The clear discrepancies between
experiments and model rule out the existence of fully JT distorted prolate Mn06
octahedra in the type A AFI phase. As we discuss elsewhere, this is probably due to
the fact that electrons are largely delocalized in the planes (though not perpendicular

to them) in this region of the phase diagram at low-T [172], in analogy with the

120

 

 

    
    
 

2.16 A

 

(D

I“ /o.54+——c'
/7 0.60 .._._.

no g, model : =

 

 

| ' 1 . 1 . 1 . 1 . 1

1.8 1.9 2 2.1 2.2
r (3)

Figure 4.12: Two dashed lines show the experimental PDFs of doping 2: at 0.54 and
0.60. While the model PDF shown as the solid line is calculated assuming 46% prolate
octahedra (two long Mn-O bonds at 2.16 A, four short ones at 1.935 A) mixed with
54% normal octahedra (6 Mn-O bonds at 1.935 A).

 

situation in the CMR region of the cubic manganites [23, 170, 171].

In the type C/C“ orthorhombic phase, we expect eg electrons to stay in 3d3y2_,.2
orbitals, and therefore the 2.16 A long bonds are expect to be present. In this
case the number of Mn3+ sites, and therefore the number of long-bonds, is rather
small. Nonetheless, there is rather good agreement between the prediction of the
simple model and the data in the region around 7‘ = 2.16 A. Figure 4.13 shows
the experimental PDFs at .7: = 0.78 and 0.80 with the model-PDF. Because of the
small number of long bonds (6.7% at a: = 0.8) this result is not conclusive evidence
supporting the existence of these long bonds, though the data are consistent with
their presence.

The PDF peaks represent the bond length distributions in the material. The

121

 

 

 

 

 

 

 

r (A)

Figure 4.13: Two dashed lines show the experimental PDFs of doping 2: at 0.78 and
0.80. While the model PDF shown as solid line is calculated assuming 20% prolate
octahedra (2 long Mn-O bonds at 2.16 A, 4 short ones at 1.935 A) mixed with 80%
normal octahedra (6 Mn-O bonds at 1.935 A).

proposed Mn06 octahedral shape change of the Mn3+ octahedra, from normal to
prolate JT distorted with increasing doping, induces more local structural distortion
and thus would cause the low-r PDF peak to broaden. An increase in disorder in the
local structure will result in this peak broadening, and therefore lowering, with doping
as observed. In Fig. 4.14 we show the change of the peak height, inversely related to
the peak width, of the first PDF peak around 1.935 A with doping. The a: = 0.64
and 0.68 samples lie above the others. These samples were measured at a later date
and evidently the systematic errors have not been perfectly reproduced. Note that
the data were all collected at 4 K so no temperature broadening efl'ects are expected.

Also, increasing the doping in this highly doped region moves the composition towards

the pure stoichiometric end-member and so dopant ion induced disorder coming from

122

 

 

13
]

. 12.5
l-I-l

12

11.5

l-I-l

Peak height (a.u.)
1'1 .
l-I-l
l-I-l
I-I-l
..|_-_|...-..........-

I-I-l
l-I-l

 

d

 

 

1 0.5

l 1 l 1 l l I n I

0.55 0.6 0.65 0.7 0.75 0.8
doping x

 

Figure 4.14: Solid squares show the magnitudes of the heights of the first Mn-O
peak around 1.94 A. The vertical dotted lines indicate positions of magnetic phase
transitions from type-A to spin disordered to type C/C*.

the alloying is decreasing with increasing doping. The observation of an increase in
disorder with increasing dOping, in this peak that is highly sensitive to the Mn-O
bonds, is therefore strong evidence that an electronically driven change is occurring
in the Mn—O octahedral shape.

The smooth evolution of the peak heights (Fig. 4.14) suggests that the local struc-
tural changes occur continuously with changes of doping concentration ac, in contrast
to what is observed in the average structure. The change in global symmetry from
tetragonal to orthorhombic at a: = 0.76 is presumably related to a transition of the

JT long-bonds from being randomly oriented to having a net orientation along b.

123

4.3.4 Structural modeling

Structural modeling gives a more quantitative picture of the local structure than
the qualitative analysis described above. Two structural models were fit based on
the tetragonal and orthorhombic crystallographic models [150]. It is worth noting
here that any constraint by the space group and symmetry during average structure
analysis can be relaxed in our real space full profile modeling. Additionally, we can
add any kind of constraints based on physical reasons. It was found that the best
agreement was found for doping :5 Z 0.60 when the tetragonal symmetry is relaxed
to orthorhombic. In the case of a: = 0.60 the improvement in fit of the orthorhombic
model is barely significant and in this case we cannot unambiguously assign the
local symmetry as orthorhombic. In the following, only results from the relaxed
orthorhombic symmetry are reported.

The two in-plane lattice constants a and b are shown in Fig. 4.15 together with
the lattice constants obtained from Rietveld refinement using GSAS [68]. The two
dashed vertical lines show the phase transition from type-A AFI to spin disordered and
tetragonal to orthorhombic (which is almost coincident with the magnetic transition
from spin disordered to type C/C“ AF I), respectively. The PDF refinements suggest
the structure is already locally orthorhombic as early as a: = 0.60, while the sharp
crystallographic phase transition occurs around a: = 0.76. This could be explained if
JT distorted MnO6 octahedra are beginning to appear on Mn3+ sites around a: = 0.60,
but the long bonds lie along the a and b axes randomly.

It is important to determine whether the JT long-bonds that appear at a: Z 0.60 lie
in the plane, perpendicular to the plane, or are distributed between these possibilities.
This can be studied by looking at the refined values of the Mn-O bond lengths.
The four different Mn-O bond lengths within one Mn06 octahedron, determined
by PDFFIT, are shown in Fig. 4.16. What is clear from this Figure is that over

this doping range there is no clear trend in the apical (perpendicular) bonds. This

124

 

0 & b (A)
3.84

I

66

¢ -

3.81

 

 

0.55 0.6 0.65 0.7 0.75 0.8
Doping x

 

Figure 4.15: Solid circles are the in-plane lattice constants a and b from PDF refine-
ments with orthorhombic models, while open circles are from Rietveld refinements
on the same data with tetragonal and orthorhombic models in 0.54 S x S 0.76 and
0.78 S a: S 0.80, respectively. Vertical dotted lines indicate positions of magnetic
phase transitions from type-A to spin disordered to type C / C‘.

implies that the observed increase in the Mn-O bond-length distribution with doping
is coming primarily from the in-plane bonds. The J T distorted Mn3+ ions that appear
with doping are predominantly locating their long-bonds in the plane. The electronic
states along c remain largely unchanged with doping, and little or no charge transfer
occurs between in-plane and out of plane.

We have observed evidence for JT long bonds lying in the plane from the PDF
refinements. If this picture is correct we would expect to see a response in the refined
in-plane Mn and O displacement factors. These should be small for a: < 0.60 because
there is little disorder in the structure and the models that we are using, based on
the average structure, should work well also for the local structure. We might expect

them also to be small and largely thermal in origin for z > 0.76 where the J T distorted

125

 

     
   

¢

q. d
a?"
v 3: .E% I
5 . \ . [if
m 8
cu -
ge-qj @-
'o'- i
g 1
m I\I'I .

02. ' \Hi-

 

 

 

0.55 0.6 0.65 0.7 0.75 0.8
doping x
Figure 4.16: Solid squares are the two in-plane (ab) Mn-O bond lengths. The bond
length between Mn and the out of plane 0 atom is shown as open squares, while
the bond length between Mn and the intra plane 0 atoms is shown as open circle.

Vertical dotted lines indicate positions of magnetic phase transitions from type-A to
spin disordered to type C/C".

orbitals are ordered along the b axis. In the spin disordered region we see evidence in
the local structure for significant numbers of JT distorted an"+ ions that are not fully
ordered. By allowing the local structure to be orthorhombic much of this disorder
will not show up in PDF derived displacement factors. However, it is interesting to
note that there is a peak in the value of the planar Mn-O displacement parameters

in this region, as shown in Fig. 4.17.

4.3.5 Discussion

In the earlier section, we present the evidence supporting the fact that, at low tem-

perature at a: = 0.54, the 6g electrons associated with Mn3+ ions may be delocalized

126

 

1.2

l-I-l

1 .05

U (x10"2 AZ)
0.9

A

H

1 l

0.75

b

 

 

l—I—l

1.5

I L I 1 I 1 I

I

 

0.55 0.6 0.6 0.7 0.75 0.8
doping x
Figure 4.17: Thermal displacement factor of the in-plane O atoms along the direction

of Mn-O bonds. Vertical dotted lines indicate positions of magnetic phase transitions
from type-A to spin disordered to type C/C’.

in the plane. At this point there are no JT distorted Mn06 octahedra and the local
structure agrees with the average structure. When the C / C*-type antiferromagnetism
appears around a: = 0.76, coincident with a global orthorhombic distortion, it seems
clear that JT distorted Mn06 octahedra have appeared with the 3d3y2_,2 orbitals
occupied. The main result from the current work is the observation that, in the 10-
cal structure, the crossover between these two behaviors happens continuously with
doping and is not abrupt as it is in the average structure. The structure is locally
orthorhombic as early as a: = 0.60 suggesting the presence of JT distorted Mn06
octahedra that are orientationally disordered within the plane. From this work we
cannot tell if this disorder is static or dynamic.

This picture could qualitatively explain the spin-disordered region since, from the

GK rules, local magnetic correlations will randomly fluctuate between ferromagnetic

127

and antiferromagnetic from site to site.

More detailed consideration of this model suggests that the sample is likely to
be nano-phase segregated in this region since we believe the low-doping end-member
(the :1: = 0.54 sample) has its cg electrons delocalized in the plane [173]. For this
to make sense, delocalized clusters with locally A-type magnetic correlations must
persist in this spin disordered region. Since there is no evidence of macroscopic phase-
separation these clusters are likely to be nano—scale. Presumably they coexist with
nanoscale regions of the sample where the 69 electrons are localized as Mn3+ ions with
a local JT distortion. With increasing doping the number of these localized Mn3+
sites first increases as the proportion of the sample in the localized state increases
at the expense of the delocalized state. Once the entire sample has transformed to
the localized state, with increasing doping the number of Mn3+ sites will decrease as
(1 — :r) in the normal way. This is apparent from the decreasing orthorhombicity that

is evident for a: > 0.80 [150].

4.3.6 Conclusions

Based on PDF results, we suggest that the local structure of La2_2xSr1+2angOy
evolves smoothly as a function of doping at low temperature in the region of the
phase diagram 0.54 _<_ a: g 0.80. The material evolves smoothly from being locally
tetragonal at a; = 0.54 to having a well established orthorhombicity at a: = 0.80. The
local and global structures agree well at these end-points. However, in between, and
associated with the spin disordered region of the phase diagram, the local structure
appears orthorhombic even though the material is metrically tetragonal. These re-
sults can be reconciled if JT distorted Mn06 octahedra exist with their long-bonds
lying in the plane but disordered along the a and b axes. We have discussed that
these results are consistent with the presence of inhomogeneities resulting in a coex-

istence of delocalized and localized electronic states, possibly due to nano-scale phase

128

separation, in this intermediate region of the phase diagram, into regions that have
the characteristics of the two end-members at :1: = 0.54 and a: = 0.80 respectively. We
have argued that such a nano phase separation into disordered and possibly fluctuat-
ing A-type and C / C*-type magnetic domains may explain the frustrated magnetism
in this region. Making certain assumptions we have quantified the evolution of the

phase separation with doping.

129

Chapter 5

Applications of the RA-PDF
Method

5. 1 Prelude

In chapter 2, we described in great detail the recent development of the RA-PDF
method. Some exemplary data were also shown to justify the high reliability and
quality of RA-PDF data analysis. As the underlying drive is to apply it to answer
significant scientific questions, several recent applications will be shown in this chap-

ter.

5.2 Locally distorted Ti square nets in Ti2Sb

5.2.1 Introduction to the question

The new binary antimonide Ting was discovered in Prof. Kleinke group [174] during
attempts to further verify the usefulness of the recently published Kleinke-Harbrecht
structure map of the MgQ pnictides and chalcogenides [175]. This structure map was
previously utilized to correctly predict the hitherto uncovered arsenides ZrTiAs and

ZrVAs that exhibit undistorted Ti and V square nets, respectively [176, 177]. The

130

TiQSb sample synthesis procedures is described elsewhere [174].

The crystal structure has been solved through conventional crystallographic anal-
ysis with both X-ray powder diffraction and single crystal diffraction refinements.
Starting with a tetragonal unit cell (space group I4/mm, a = b = 3.95 A, c = 14.61 A,
asymmetric unit: Til (0.0, 0.0, 0.33), Ti2 (0.0, 0.5, 0.0), and Sb (0.0, 0.0, 0.14)), both
diffraction data were fitted reasonably well. Fig. 5.1 shows the ball and stick model
of Ti28b crystal structure. However, the thermal displacement factor U11 (0.125) for
the Ti2 atom is anomalously large compared to U22 (0.006) and U33 (0.003) (all U
values given in A2). All other atoms show neither strong anisotropy nor abnormal
U values. Fig. 5.2 shows the thermal ellipsoid of the Ti2 square nets. As such U11
anomaly is usually a good indication of static disorder along the anomalous direction,
a distorted model with split Ti2 sites was then introduced. In this model, Ti2 atoms
are shifted along the U11 direction in the asymmetric cell to have none zero :1: values,
but with 50% occupancy to maintain the sample stoichiometry. Refinement with this
distorted model improved the agreement significantly, resulting the T12 :1: value of
0.0701(6). Due to the absence of super-lattice peaks, increase of the unit cell was not
considered at the first place. Two samples with different synthesis conditions were
studied and gave the same distortion of the Ti2 atoms.

This simple distorted model with split T12 sites needs to be carefully examined.
First, the introduced split Ti2 sites resulted in a unphysical Ti2-Ti2 bond length
of 0.56 A. This can be resolved if the two split sites originating from the same Ti2
site can not be occupied at the same time, though both with 50% occupancy. One
hypothetical superstructure of the Ti2 net (shown in Fig. 5.3) was proposed, but can
not be fully justified without the observation of super-lattice reflections. While the Ti2
square net distortion is nevertheless strongly suggested, the Ting structure appears
to be a case of local distortions without long range ordering. If the Ti2 splitted sites

scenario correctly describes the local structure, some specific set of local inter-atomic

131

 

Figure 5.1: Crystal structure of Tigsb in ball and stick model. The connected Ti2
square nets lie at the bottom and in the middle.

132

 

Figure 5.2: Planar net of Ti2 atoms in Ting, with thermal ellipsoids shown.

distance splits/ changes is expected as shown in table 5.1. To establish beyond doubt
about the local structure, a high resolution local structure probe is necessary. We
employed the RA-PDF method to verify the local inter-atomic distances coming from

the distorted model.

 

 

atomic pair undistorted distorted

1 Ti2-Sb 2.84 2.86

2 Ti2-Sb 4.87 4.65 and 5.10

3 Ti2-Ti2 2.80 2.82

4 Ti2-Ti2 3.95 3.40 and 4.51

5 Ti2-Ti2 5.59 3.99, 5.22, 5.59, and 6.09
6 Ti2-Til 3.17 3.00 and 3.35

7 Ti2-Til 5.07 4.97 and 5.18

 

 

 

 

 

 

Table 5.1: List of expected inter-atomic distance splits/ changes from the undistorted
to distorted model. All lengths are given in unit of A.

133

 

 

 

 

 

 

 

 

 

 

n'

 

 

Figure 5.3: The hypothetical \/2 x \/2 x 1 superstructure of the T i2 square nets.
5.2.2 RA-PDF analysis of local Ti2 distortions

The distorted model with split Ti2 sites ultimately yields different inter-atomic dis-
tances, in particular involving Ti—Ti bonds (Table 5.1), but also Ti-Sb distances in the
second coordination sphere. Experimental proof for the presence of the bonds / distances
stemming from the split of Ti2 sites was directly obtained utilizing the real-space
pair distribution function (PDF) technique [5]. The PDF, C(r), gives the probabil-
ity of finding pairs of atoms separated by distance r, and thereby comprises peaks
corresponding to all discrete inter-atomic distances. The experimental PDF is a di-
rect Fourier transform of the total scattering structure function 5' (Q), the corrected,
normalized intensity, from powder scattering data given by C(r) = % f0°° Q[S (Q) —
1] sin Qr dQ, where Q is the magnitude of the scattering vector. Unlike crystallo-
graphic techniques, the PDF incorporates both Bragg and diffuse scattering inten-

sities resulting in local structural information [5]. Its high real-space resolution is

134

ensured by measurement of scattering intensities over an extended Q range (35.0
A“) using short wavelength X-rays or neutrons.

Both X—ray and neutron powder diffraction experiments were carried out. These
give complementary data-sets due to the different relative scattering lengths of Ti
and Sb to X-rays and neutrons: fT,(0)/f5b(0) = 0.43 for X-rays whereas for neutrons
bT,(0)/b5b(0) = —0.61. Thus, the scattering from Ti, and therefore the PDF peaks
originating from Ti correlations, are somewhat more apparent in the neutron data.
The X-ray experiment was performed at the 6-ID beam line at the Advanced Photon
Source (APS) at Argonne National Laboratory. A powdered Ting sample of disk
shape (thickness of 1.0 mm, diameter of 1.0 cm) was loaded into a hollow flat metal
plate, and then sealed between thin kapton films. Data acquisition at 300 K employed
the recently developed rapid acquisition PDF (RA-PDF) technique [61] with the X—
ray energy of 98.0 keV. A single exposure of the image plate detector was limited to
2 seconds to avoid detector saturation, and was repeated 20 times to achieve better
counting statistics in the high-Q region.

The neutron experiment was performed at the newly upgraded neutron powder
diffractometer (NPDF) at the Lujan center at Los Alamos National Laboratory [21].
A powdered TiQSb sample of 5.92 g was loaded into a standard 3/8” vanadium can
under helium exchange gas. The sample height was measured to be 2.92 cm. Low
temperature is preferred to sharpen the peaks in the PDF and so data acquisition was
carried out at 15 K in a displex closed cycle refrigerator. Data were collected for 10.0
hours. The long collection time was necessitated because the large amount of incoher-
ent scattering from this sample. Calibration measurements of the background, empty
containers, etc, were also performed. Standard corrections were applied to obtain the
total scattering structure functions, 5 (Q), using the programs PDFgetX2 [41] and
PDFgetN [42] for X-ray and neutron data, respectively. The experimental reduced
structure function, F (Q) = Q(S(Q) — 1), is shown in Fig. 5.4 (a) (X-ray) and (b)

135

 

 

 

 

 

F(Q) (11")

 

 

 

 

 

 

 

 

 

 

 

I 1 I L I

0 I 10 20 30
Q (11“)

 

Figure 5.4: The experimental reduced structure function F (Q) = Q(S(Q) — 1) of
Tigsb (a) at room temperature from the X-ray measurement and (b) at 15 K from

the neutron data.

136

(neutron). The resulting PDFs, C(r), are shown in Fig. 5.6 as the solid dots. Panels
(a) and (b) show the X-ray data and (c) and (d) have the neutron data.

The data-PDFs were examined to search for direct evidence of the short and long
Ti2-Ti2 distances implicit in the distorted model. A large thermal factor with no
underlying splitting results in broad peaks in the PDF centered on the average position
whereas a split position results in peaks in the PDF splitting into two components
that retain their sharpness. Thus, the distorted and undistorted models should be
directly distinguishable in the PDF. The low—r region of the PDF is shown in Fig. 5.5
for X-rays (upper curve) and neutrons (lower curve). Features in the PDF can be
compared to expected bond distances from Table 5.1. The strength of the PDF peak
depends on the scattering power of the two atoms contributing to the peak, and the
multiplicity of the pair. For both X-rays and neutrons Ti is a weaker scatterer than
Sb and so the low multiplicity T i-Ti bonds that would show the distortion directly
are hard to see. In particular, the Til-Ti2 peak centered at 3.17 A and the Ti2-Ti2
distance at 3.95 A are barely evident. There is some suggestion from the neutron data
that the 3.17 A is split (apparent as a highlighted “M”-shaped feature in both the
x-ray and neutron difference curves), but by itself this is hardly convincing. However,
the relatively strong Ti2-Sb peak centered at 4.87 A gives a clear and unequivocal
indication of this splitting. The peak from the undistorted model centered at this
distance is absent in the data which, instead, has relatively sharp features at 4.65
A and 5.10 A. This results in a clear feature in the difference curves below the data
in Fig. 5.5. In the X-ray data it is M-shaped and in the neutron data W-shaped;
the difference due to the change in sign of the Ti-Sb peak in the PDF due to the
negative neutron scattering length of Ti. These M /W features clearly show that the
peak centered at 4.87 A has split into a shorter and a longer peak.

The data were then modeled quantitatively using a real-space profile fitting pro-

gram, PDFFIT [11]. This program can refine the crystallographic model to the data,

137

 

3

 

4.65 5.10

G(r) (a.u.)
0
5"
8
5"
0‘
4.

—3

  
 

' Neutron

-

-6

 

 

 

 

 

2 3.4.5 6
r(A)

Figure 5.5: Experimental PDF (solid dots) and the calculated PDF from the distorted
model (solid line). Upper curves: X-ray data; lower curves: neutron data. The
difference curves based on the undistorted model are shown below.

138

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

m A
‘i‘
r ..
'1' N
‘5, o
. ‘—
1‘ V
5’0. 1*
: v
1......“ 1.....11.....-...1- 0
--P'I""VVI-rv-w .‘vr‘ "TW-
w A
‘i‘
0'7" N: 0'
‘1' N
‘5 o
.__.
3 V0
0 O; 3 I
:.H:‘.‘f.'1hw“kw‘v~‘m- 'fi‘“ O -=- - ' 'W‘I'r "iv v ' ‘-- - '17—"
1_ I 1 I 1 I 1 I 1 1 I 1 I 1 l 1 I
O 3 6 9 12 0 3 6 9 12
r (R) r01)

Figure 5.6: PDFs, G(r), from the data shown in Figure 5.4 (a), (b): X-ray data;
(c), (d): neutron data (shown as the blue dots in both cases). Solid lines are the
fits to the model data, with (a) and (c) from the undistorted model and (b) and (d)
from the split site model. Difference curves are shown below the data. The dotted
lines around the difference curves indicate the standard uncertainties due to random
counting statistics on the data at the 1 a level.

in analogy with Rietveld refinement. However, local structural distortions can be
introduced into the models even when they are not long-range ordered and no crys-
tallographic super-lattice reflections are observed as in the current case. We first fitted
the data with the undistorted model, and show the results in Fig. 5.6 (a) (X—ray) and
(b) (neutron). Clearly, the undistorted model proves to be insufficient, as evident
from the difference curves below the data. Fluctuations significantly larger than the
experimental uncertainties (the estimated errors at the 1 a level are shown as dotted
lines associated with the difference curves) are observed over the whole fitting range.
One of the strengths of the PDF technique is that it yields quantitatively reliable
intermediate range information on nanometer length-scales. The persistence of the

fluctuations in the difference curve of Fig. 5.6 (c) is significant and a clear indication

139

that the undistorted model fit to the data is lacking. Next, the distorted model with
split Ti2 sites was tried, with the fitting results shown in Fig. 5.6 (c) (X-ray) and (d)
(neutron). The agreements to both X-ray and neutron PDFs improved significantly
over the entire range, with the addition of only one additional refinement parameter,
and gave rather satisfactory fits given that the distorted model is an average of the
unknown superstructure. The superstructure model, in the Cmca space group, was
also tried and gave comparable fits. The structure data from the PDF refined models
are in excellent agreement with those refined from the single crystal study as shown
in Table 5.2. The full profile PDF refinements confirm that the distorted model de-
scribes both the local and average structure to great accuracy and is far superior
to the undistorted model. The superstructure model in the Cmca space-group gives
comparable agreement to the split site model which shows that this model is also

consistent with the data as well as making better physical sense.

 

 

 

 

 

 

 

Crystal data X-ray 300K Neutron 15K Neutron 300K
3 (A) 3.9546(8) 3.9611(10) 3.9472(2) 3.9575(3)
c (A) 14.611(3) 14.655(6) 14.565(1) 14.576(2)
Til 2 0.3305(1) 0.3307(3) 0.33069(8) 0.3310(1)
Ti2 x 0.0701(6) 0.0708(20) 0.0711(9) 0.067(1)

Sb z 0.1398(1) 0.1398(1) 0.13977(6) 0.1400(1)
Til Um 0.0074(4) 0.0073(2) 0.00273(5) 0.0062(1)
Ti2 Um 0.0067(2) 0.0069(2) 0.00359(6) 0.0079(1)

Sb Um 0.0063(2) 0.0035(1) 0.00242(3) 0.00573(7)

Rm, 0.057 0.213 0.201 0.227

 

 

Table 5.2: Comparison of refined parameters of the distorted Tigsb model between
the (RA-)PDF and single crystal diffraction methods.

5.2.3 Discussion and conclusions

Structure characterization of the new metallic binary antimonide Tigsb was carried
out with single crystal diffraction, powder diffraction, and the RA-PDF method. Both
the average structure and local structure were successfully obtained with those com-

plementary studies. More importantly, this study proves again the Kleinke-Harbrecht

140

structure map capable of predicting the previously unknown Tigsb structure. One
very interesting structural feature for this compound is a new type of distortion of
one metal atom layer that is a perfect square planar net in its aristotype Lang. The
corresponding distorted Ti2 net of TIQSb comprises squares and rhombs in a 1:1 ra-
tio, as shown in Fig. 5.3. It is worth noting that combined X-ray and neutron PDF
refinement with the same model was also performed.

High real space resolution RA-PDF analysis of Tigsb powder diffraction data is
exemplified in this study. In the mean time, high resolution PDF analysis of neutron
powder diffraction was carried out and gave structure details in excellent consistency
with the RA-PDF method. It is noteworthy that their reciprocal space resolutions are
rather different from the F (Q) peak widths in Fig. 5.4. However, the local structure
in the low 1' PDF region is not affected at all, while the high 1‘ PDF data from the
RA-PDF method quickly loses intensity compared to the neutron PDF (Fig. 5.6).
This stands as a limitation of the RA-PDF method where high real space resolution
compromises the reciprocal space resolution due to the finite IP size [10]. Nevertheless,
the RA-PDF method shows to be a very powerful and fast way to probe the local
structure. The author also would like to note that the X-ray sample was received on
the last morning of the experiment. The successful data collection of a complete data

set would not have been possible without the fast RA-PDF method.

5.3 In-sz'tu chemical reduction of CuO to Cu

This is an example of in-sz’tu time resolved powder diffraction studies in heterogeneous
catalysis by coupling the study of long range and local structural changes. Most of
the work described here was done by our collaborators (see reference [178]). The
application of in-situ time resolved powder diffraction to study structural changes
that occur during chemical reactions has become a standard method at synchrotron

sources. With respect to heterogeneous catalysis, the understanding of the structural

141

changes, both on the local and long range length scales, that are the consequence
of reaction conditions is paramount to fully understanding the functioning of many
catalysts. Time resolved diffraction experiments which combine high-energy X—rays
(>80 keV) with area detectors open the possibility of coupling both reciprocal and
real space methods to probe both long range and local structural changes simulta-
neously [61]. This is particularly advantageous in the study of chemically induced
structural changes, commonly encountered in heterogeneous catalysis. Here we illus-
trate the application of the approach to monitor the reduction of CuO, a material that
is ubiquitous in compositions of methanol synthesis and water-gas shift catalysts [179].
Activation of the catalysts requires reduction of the CuO, producing small metallic Cu
particles which are the active centers for catalysis. We have therefore been studying
the reduction of CuO under controlled conditions using H2 to examine the formation

of centers active for catalysis [180].

5.3.1 Experiment

All diffraction experiments were performed at the 1-ID beamline at the Advanced
Photon Source (APS) located at Argonne National Laboratory, Argonne, IL (USA).
High energy X-rays were selected using a double bent Laue monochromator capable
of providing a flux of 1012 photons/second and operating with X—rays in the energy
range of 50-200 keV.

In-sz'tu diffraction experiments were performed with a specially designed flow cell
(Fig. 5.7), which allows for flow of reactant gas over the sample during acquisition
of powder diffraction data. The temperature was controlled in steps as follows; 2
hour ramp from room temperature to 350°C C. During the experiment 4% H2 in He
was passed over the sample at a rate of 8 cc/min. A MAR345 image plate camera
was mounted orthogonal to the beam path, with the beam centered on the IP. When

using the 99 keV X-rays, a LaB6 standard was used to calibrate the sample to detector

142

Gas Flow Catalyst

Sample Thermocouple

PyrexIQuartz
Cappilary
- l

- I "—_I TI

 

 

 

To Gonlometer Head To Mass Spec'l

Figure 5.7: Schematic diagram of the flow cell used in the RA-PDF experiment.

distance and tilt of the IP relative to the beam path, using the software Fit2D [64].
All raw two-dimensional image data were integrated using the software Fit2D and
files output as intensity versus 20. Ten second exposures yielded data that provided

very good statistics for the refinement techniques described below.

5.3.2 Results

The in-situ reduction of CuO was followed using high energy X-rays coupled with a
two dimensional image plate (IP) detector. The results are shown in Fig. 5.8. The
reduction of CuO to Cu metal takes place at approximately 280°C C. The reduction
to Cu is quite fast and occurs on the order of minutes. During this period both CuO
and Cu are evident, and we are currently investigating the structural changes that
occur.

High energy X-rays provide several experimental advantages in the structural
analysis of time resolved data. Utilization of high-energy X—rays allow for larger
values of momentum transfer to be probed (> 18 A‘l), opening up the possibility for

the application of real space methods (pair distribution function, PDF, analysis) to

143

 

 

 

 

 

 

 

 

 

 

  

 

oil
1] e
W”) L

2 i 4

 

 

 

 

 

Figure 5.8: Powder diffraction pattern during the chemical reduction of CuO to Cu.

probe changes to local structure during reaction conditions. The G (r)s obtained from
the diffraction patterns (IP data) at the beginning of the in-situ experiment and at
the end of the in-situ experiment are shown in Fig. 5.9. For example, the first Cu-O
peak drops dramatically and a new peak at 2.6 A (Cu-Cu) is observed. Traditional
Bragg refinement of the same diffraction data used for PDF analysis provides for an
improved parameter to data ratio. Exceptional Bragg profile refinements of beginning
and final phases can be obtained, but disorder in mixed phase results in poorer
fits and significant features in the difference electron density maps, which we are
attempting to interpret. Following the reduction using real space methods may open
up the further understanding of the reduction process, as the changes to the Cu-O

coordination during the final stages of reduction may not be evident in the Bragg

144

 

 

 

 

 

 

 

 

‘

m’——i——Ih—-_———-'

 

A
C'
v“
I 1

 

 

 

 

 

 

 

 

 

Figure 5.9: G (r) for (a) CuO at the beginning of the time resolved experiment and
G(r) for (b) Cu formed at the end of the time resolved experiment. The guideline
shows the Cu-Cu first shell distance.

profile refinements.

5.3.3 Conclusion

High energy X—rays open up the possibility for qualitatively new time-resolved exper-

iments which couple both reciprocal and real space approaches to structural analysis.

145

The application of X-rays with energies between 80—120 keV allows for collection of
powder diffraction data to high momentum transfer (>18 A‘l) using commercial im-
age plate cameras at sampling rates which will allow monitoring relevant structure
changes under process conditions. This allows for both Rietveld and PDF analysis
to be carried out on the same data sets probing both long range and local structural

changes simultaneously.

146

Chapter 6

Concluding Remarks

6.1 Summary of the thesis

We set out to determine the local structure of disordered crystalline materials, em-
ploying the atomic pair distribution function (PDF) method [5]. As the demand of
atomic local structure characterizations is growing (see Chapter 2), the PDF tech-
nique is emerging as a promising tool for this purpose. Neutron and X-ray powder
diffraction experiments have been exclusively used to obtain the experimental PDFs.
The novelty of the PDF technique is to take into account both the Bragg peaks and
diffuse scattering intensities, as a total scattering probe. Obtained via the Fourier
transform from the reciprocal space, the PDF method is a unique local structural
probe that presents information in the more intuitive real space [22], and contains
structural information at all length scales. The ultimate goal is to understand the
fundamental structure-property relationship driven by our sheer scientific curiosity,
and then be able to manipulate the structures and/or prOperties according to our
technological needs.

The recently developed rapid acquisition PDF (RA-PDF) method [61] made a
leap forward by reducing the data collection time by three to four orders magnitude.

The key components of the RA-PDF method are high flux high energy X-rays and an

147

automated image plate area detector. A new program PDFgetX2 [41] implemented
necessary additional corrections to the RA-PDF data, and significantly simplifies the
intensive data analysis with the built-in user-friendly graphic user interface (GUI).
The proof-of-principle and systematic quality studies established that high quality
experimental PDFs can be routinely obtained from materials with wide ranges of
atomic Z numbers. The measurable extended reciprocal space range provides the
high real space resolution essential to local structure approaches [10]. RA-PDF data
have low Q-resolution and this method is not suitable for measurements requiring
high reciprocal space resolution. The RA-PDF method has resulted in numerous
publications [61, 178, 174, 181], and opens up future possibilities for wide-spread
PDF applications.

Relevant to this thesis work, the RA-PDF method has been applied to study
the distorted T 12 square nets in the new binary antimonide Ting, in combination
with neutron PDF data analysis. The Ti2 square net is confirmed convincingly to
distort locally. This example shows one strength of the PDF technique, probing
the local distortions inaccessible (or inconclusive) to conventional crystallographic
analysis. The other example was the combined PDF and Rietveld study of the in-
situ chemical reduction of CuO to Cu. The fast RA-PDF measurement makes it
possible for following the in-situ chemical reactions in real time, while still giving
high quality PDFs. In fact, the PDF analysis appeared to be more advantageous
than the Rietveld analysis due to the presence of significant diffuse scattering and
non-crystalline forms during the reduction. The PDF directly probes the local inter-
atomic distance distributions regardless of the long range structural coherence. The
resulting coordination numbers (Cu-O, Cu—Cu) reveal important information such
as the “phase” fractions. Within its current capabilities, the time deve10pment of
chemical reactions and biological systems over time scales of minutes can be studied,

as well as, in the future, the pump-probe experiments.

148

The colossal magneto-resistive (CMR) manganites are fascinating strongly corre-
lated electrons systems challenging our current understanding of condensed matter
physics. The atomic local structure plays important roles via the strong interplay
between electron, spin, and lattice degrees of freedom. The Mn06 octahedral Jahn-
Teller (JT) distortion is a direct result of the strong coupling between the electrons
and the lattice. The undOped parent compound LaMnO3 shows a very interesting, but
not fully understood, phase diagram, even without the additional complications such
as the admixture of Mn3+ and Mn“ ions and pinning effects induced by doped cations
in the doped compounds. Our PDF analysis of the J T transition around 750 K contin-
ued and extended the same local structural results as earlier XAFS studies [129, 130].
The nature of the JT transition is found to be orbital order to disorder. Our high
temperature data in the rhombohedral phase (T > 1010 K) discovered the locally JT
distorted Mn06 octahedra while strictly regular Mn06 octahedra are indicated by an
average structure analysis. We have also shown that the PDF analysis can obtain the
average structure in quantitative agreement with the Rietveld refinements. More sig-
nificantly, the intermediate range structure information could be extracted from our
PDF analysis by following how the refined structure evolves with the change of the
PDF r range that is fit over. In the orbital disordered states at high temperatures, we
found the existence of short range ordered domains. The domain size was estimated
by the crossover from local to average structure, both of which are obtained from the
PDF analysis. This domain size is found to be around 16 A spanning over four Mn06
octahedra, suggesting strong nearest neighbor anti-ferrodistortive JT coupling and
weak second nearest neighbor coupling. The importance and delicacy of the interplay
between electron and lattice degrees of freedom is again borne out of our study, which
also illustrates that the PDF gives structural information at all length scales.

The bi—layered manganite La2_2xSr1+2zMngO7 shows great similarities to the equiv-

alent “cubic” manganite, as well as intriguing differences. In particular, we studied

149

the Mn“ rich region (:1: > 0.5). One example is to detect the local Mn06 octahedral
shape change across the temperature driven orbital occupancy transition from d$2_y2
to d322_,2 at :1: = 0.54. This orbital occupancy transition brings the Mn06 octahedron
from oblate (two short, four long Mn-O bonds) to prolate (four short, two long Mn-O
bonds) upon warming. This Mn-O bond length distribution transition is evidenced
by the‘excess decreases of the Mn-O and O-O PDF peak heights across the transition.
Behaviors consistent with this proposed oblate to prolate octahedral transition are
also observed in the thermal displacement parameters of the in-plane oxygen atoms,
and the lattice constants. Coexistence of the two orbital occupancies is invoked to
explain the observed smooth crossover. The other example concerns the Sr-doping :1:
driven phase separation at nano-scales within the range 0.54 S :1: S 0.80 [182]. We can
also understand this as the increasing stability of the d3zz_,.2 orbitals with increased
Sr-doping. The d3z2_,.2 orbital occupancy at low Mn3+ concentrations (large 51:) re-
sults in the orthorhombic phase. We show evidence supporting the smooth growth of
the local orthorhombic phase with Sr-doping, instead of the sharp abrupt tetragonal
to orthorhombic crystallographic phase transition at x = 0.74 observed on the aver-
age structure [150]. It can also be understood that the sample La2_2xSr1+2an207
in the Mn4+ rich region 0.54 g :1: S 0.80 shows intrinsic inhomogeneities (“phase”

separation) at nano-scales.

6.2 Future work

In this section, some future possibilities are discussed. The discussion will be taking

the view of the PDF technique, instead of each individual research project.

150

6.2.1 Measuring more PDFs

The RA-PDF method considerably simplifies X-ray PDF measurements, while being
three orders or more faster than the conventional X-ray PDF data collection [61].
With high energy X—rays, thicker samples can be measured due to the weak sample
self-absorption effect. The third generation synchrotron source provides X-ray beams
with significantly boosted brilliance [62]. As a consequence, the necessary total sample
exposure time is usually in the order of tens of seconds, which already provides high
counting statistics. The limiting factor to data collection time becomes the image
plate read-out and erase-off time, which is about three minutes for a single exposure.
Thus, we can further improve our data rate by another factor of three if the data
operation time of the area detector can be significantly reduced. One solid state area
detector (detection media is amorphous silicon) has been tested in our collaborations,
which gives 18 frames per second data throughput. The other alternative is the charge
coupled device (CCD). The active area of commercially available CCD is close to the
image plate size we are currently using (345x345 mmz), e.g., 315 mm of Quantum 315
model. It will be great to test those two types of area detectors for quantitative PDF
analysis in the near future. Various technical issues / difficulties are expected, however,
these detectors present a very interesting prospect. For time resolved measurements,
a few seconds of exposure time usually suffices to obtain PDFs with acceptable quality
for time dependent comparisons. Even in this case, the time resolution of the RA-
PDF can be improved by at least two order of magnitude, i.e. from around three
minutes to one second level. This would make another leap forward in the PDF
applications especially on time-resolved studies. Just to mention about neutron PDF
measurements, we currently are using large sample size (6-10 gram), large beam size
(5 x1 cm2), and multiple banks of detectors. Thus, neutron data collection is to a large
extent beam flux limited. Though slow, the neutron measurements are indispensable,

being complementary to X-ray experiments.

151

A significant amount of additional work would be very much welcome on the
bi-layered manganite compounds. One very interesting experimental observation is
the re-entrant charge ordering behavior of the La2_2xSr1+2$MngO-/ near :1: = 0.50
around 50 K [183, 155, 184, 185]. We would expect the PDF peaks heights to be
lower in the charge ordering temperature region. As the deviations from normal
Debye behaviors is better characterized with denser data points, the RA-PDF method
should be useful. Anomalies in electric transport data have also been reported in the
higher Sr-doped regions [185], temperature dependent measurements at high doping
levels would be very promising. In addition, I am particularly interested in the
local structure of La2_2xSr1+2angO-; at :1: = 0.4. Though the local structure of
the Sr-doping level at :1: = 0.3 has been studied with the PDF method [162], the
x = 0.4 compound is still very interesting due to their rather different magneto-

resistive transport behaviors [151].

6.2.2 Obtaining high quality PDFS

The accessible reciprocal space range has been significantly extended with the avail-
able high energy X-rays and faster epi-thermal neutrons [5]. The real space resolution
of the PDF has been shown to reach the order of one tenth of an Angstrom [16, 186],
which becomes comparable to the intrinsic PDF peak width broadening set by the
atomic thermal motions. Though with high real space resolution, the obtainable qual-
ity of the PDF is sample dependent. As high quality PDFS can be straightforwardly
obtained from the “good” materials, improved data analysis methodologies are neces-
sary to handle the “bad” compounds. For neutron PDF data analysis, samples with
high absorption cross section or significant incoherent scattering cross section belong
to the “bad” category, e.g. highly absorbing samples necessitate very accurate ab-
sorption and multiple scattering corrections. For X-ray PDF data analysis, very low

Z (S 10) materials are troublesome because of the dominating Compton scattering

152

at high Q region. It is worth noting that the low Z materials represent an extremely
interesting and important group such as organic and biological samples.

The origins are the imperfections in the data corrections. In some cases, the
assumptions made in the theoretical / empirical corrections simply fail, e.g., multiple
scattering is usually calculated up to secondary scattering only [187, 188, 189, 190],
and absorption of elastic and inelastic scattering intensities are taken as the same [44,
41]. Other cases are that some compiled theoretical profiles used in the data correc-
tions have large uncertainties [191, 192, 193] in the high Q region in real material
systems. For example, the tabulated atomic form factors are for neutral atoms and
some common ions. While this is a reasonable approximation for high Z elements,
the change of outer-shell electron densities of low Z elements in the material modifies
the scattering form factors rather considerably. The Compton theoretical profiles
only have high accuracy up to Qmax of 16 A‘1 [192, 194], while the experimental
data commonly reach as high as 35.0 A“. Therefore, quantitative analysis of the
scattering intensities at high Q calls for more accurate theoretical calculations. On
the other hand, to extend the capability of the PDF method, careful experimental
studies of those various effects can be carried out and corresponding solutions can be
implemented. More specifically, neutron samples with wide ranges of absorption or
incoherent cross section can be studied to investigate which corrections are limiting
our data analysis. For RA-PDF experiments, the energy dependence of the image
plate phosphor layer transmission coefficients are best measured over a wide X-ray
energy range, e.g. 20 to 120 keV; the calibration data on the energy dependence of
the image plate response needs to be extended to higher X-ray energies (~ 120 keV)
as well; the angular dependence of the fluorescence intensities due to self-absorption

can also be verified with a multiple channel analyzer (MCA).

153

6.2.3 Combining complementary probes

The strength of the PDF technique lies in the local and intermediate range struc-
ture information. Thus the conventional crystallographic methods probing the long
range ordered structure can be naturally combined with the PDF analysis. This has
been routinely exercised, providing complementary and unique insights into the real
structure of materials. The PDF measurement takes the average over all powder
grains, all atoms, and the solid angle. This makes it less sensitive to longer range
structure motifs, such as phase separation in the range of tens of nano-meters. In this
case, the joint probes with microscopy measurements are very appealing Such as scan-
ning tunneling microscopy (ST M), high resolution transmission electron microscopy
(HR-TEM), and atomic force microscopy (AFM).

The neighboring environment of atoms influences quite diverse material proper-
ties, such as the electronic structure, magnetic exchange coupling, phonon dispersions.
Knowledge of the atomic structure will be greatly complemented by the knowledge
of the lattice thermal dynamics, macroscopic electronic and thermal transport prop-
erties. In view of this, it will be very beneficial to collect first-hand data on the
same sample with complementary techniques such as thermal analysis measurement,
electric conductivity measurement, neutron and X-ray inelastic scattering measure-
ment. Detailed atomic structure also often serves as the basis for many theoretical

calculations such as the band structure computations.

6.2.4 Applying the PDF technique in new areas

The purpose of the PDF technique is to answer significant scientific and/or techno-
logical questions suitable to this method. One very effective way is to bring the PDF
technique to a wider user community. Widespread use of the PDF method could
bring in a large number of new users and have significant impact in various scien-

tific disciplines. On the other hand, the capability and application scope of the PDF

154

technique needs also to be expanded. Here we will discuss a few possibilities.

Single crystal diffuse scattering (SCDS) measurements also look at the same diffuse
scattering intensities as the PDF powder diffraction measurements. Though more
time consuming, the SCDS method gives more information due to the absence of
powder averaging. Borrowing from the PDF method the idea of a real space probe
via Fourier transform, we can obtain the three dimensional PDF from SCDS data.
It will be also interesting to obtain the two dimensional PDF from two dimensional
structures such as films, surfaces, and interfaces. Another fermenting idea is the
magnetic PDFs with polarized neutron scattering experiments, which can separate
the magnetic and nuclear scattering components through polarization analysis.

Conventional time resolved diffraction experiments should benefit from the PDF
analysis by adapting the RA-PDF setup, i.e., using high energy X-rays and shortening
the sample to detector distance. In particular, the potential PDF applications on
the time development of biological systems are very appealing. While the shape
and conformational changes are probed by the global structure approaches such as
optical microscopy and small angle scattering, the change of local environments, e.g.,
breaking or forming of some specific bonds, can be studied with the PDF method,
which is rather sensitive to small local structural changes. Much improved RA-PDF
time resolution is also expected with either the solid state area detector or CCD. I am
looking forward to the future exciting developments. As the amount of data coming
from those experiments will be huge, the efficiency of current data analysis methods

needs to be improved correspondingly.

6.3 Acknowledging funding agencies

The work at MSU benefited from support from the following agencies, National Sci-
ence Foundation through grants, DMR-0075149, DMR-0211353, CHE-9903706 and
CHE-0211029, Center for Fundamental Materials Research at Michigan State Uni-

155

versity, US. Department of Energy through grant DE-FG02-97ER45651.

This work has benefited from the use of NPDF at the Lujan Center at Los Alamos
Neutron Science Center, funded by DOE Office of Basic Energy Sciences and Los
Alamos National Laboratory funded by Department of Energy under contract W—
7405-ENG-36. The upgrade of NPDF has been funded by NSF through grant DMR
00-76488.

The work at ANL and beam time at IPNS and APS is supported by the US
Department of Energy through grant No. W—31-109-ENG-38.

156

Appendix A

The rapid acquisition pair
distribution function method

experiment

The rapid acquisition pair distribution function (RA-PDF) experiment will be de-

scribed in detail as follows.

A.1 Requirements of the experiment

To collect powder diffraction data suitable for PDF analysis, a few requirements have

to be met.

Extended Q range The finite Q space measurement range directly determines the
experimental real space resolution of the PDFS [10]. As the physical size of the
IP is fixed, the desired Q range defines the relation between X-ray energy and
the sample to detector distance. In practice, the probed Qmaz is determined by
the particular question to be answered by the experiment. However, to reduce
the effect of the termination ripples due to the cutoff in Q space, QmaxZ 18.0

A‘1 is generally necessary. Much higher Qmax, e.g. 2 30.0 A“, is required for

157

highly crystalline materials.

Good counting statistics To obtain quantitatively reliable structural information,
a certain level of counting statistics is necessary. As the scattering is a random
event, the measured counts follow the Poisson distribution. When the number
of measured scattered photons is large, its uncertainty is simply the square root
of the measured counts. An empirical rule of thumb is to collect around one

million counts per inverse Angstrom.

Good signal to background ratio This can be part of the second requirement.
However, this requirement is of special importance, and tricky as well, mostly
because more than one scattering process contributes to the measured total
counts and only elastic coherent scattering is used in the PDF analysis. To
achieve a good signal to background ratio, it is necessary to have a good esti-
mate of what fraction of the total counts is the elastic signal sought after. For
example, at Q just as high as 25.0 A‘1 for Si atoms, the Compton scattering
intensity is 8 times the elastic intensity. If 1% is the desired relative uncertainty,
you will need as many as 640,000 counts. Still chances are you are looking for
even smaller changes in the elastic scattering intensities, which calls for more
counts. In principle, it is difficult to know the expected signal a priori. The
best approach is to analyze the data during the experiment, then decide whether

more collection time is necessary.

A.2 Setup of the RA-PDF experiment

For practical reasons, the procedure to set up the experiment will follow the flight
path of the X-rays, which means from upstream to downstream. The first thing
is to choose the right beam line with proper X-ray characteristics for your specific

experiment. Experimental setup in the early stages is similar to our conventional

158

3193 detector sample monochromator >60.0 keV

‘\

’4‘

        

RA-PDF experimental setup slits storage ring

Figure A.1: Schematic diagram of the rapid acquisition PDF (RA-PDF) experiment
layout.

PDF X-ray experiments. The schematic diagram of the RA-PDF experimental setup

is shown in Fig. A.1.

1. Set the energy. This step doesn’t need to be the first, and commonly it’s
necessary to change the X-ray energy during the experiment. Nonetheless, a
few ways to calibrate the energy are briefly explained here. In principle, we
can compute the X-ray energy given the monochromator tilting angle and the
reflection plane used. However, those calculated values are usually only good
approximations due to the various uncertainties in the tilting angle, etc.. One
commonly used approach is to scan across the absorption edge of a standard
material, such as Pb, or Au K edges, and use their well known absorption edge
energy values as references. The other way is to use multiple channel analyzer
(MCA) to locate the incident beam energy. One complication of this method is
the MCA needs to be calibrated first, usually with a set of standard radiative
materials which have characteristic emission lines. A common step after this
is to align the center of the slits so that the incident beam passes through the
centers. This can be done automatically by monitoring the downstream beam

intensities when moving the slits.

2. Set the slit openings to give you the proper beam size. We have used 0.5x0.5

159

mm at both llD-D and 61D-D at APS. The primary considerations for beam size
are usually the Q space resolution and incident X-ray flux. With some approx-
imations, starting from Q = 47" sin 6, the resolution function can be expressed

35,

 

2
égzéififiiijféfitwf %
where d is the sample to detector distance, r is the distance of the pixel from the
center of the image plate. In our typical RA-PDF experiments, the maximum
20 angle is about 30.0 degrees. We can show that the first term % is the
smallest, usually in the order of 10‘4 determined by the characteristics of the
monochromator(s). The prefactor of the Q dependent second term varies from

1.0 to 0.65 as 20 goes from 0.0 to 30.0 degrees. The typical value of 9;“ is

1.0

m = 0.005 with 1 mm thick sample and 200 mm sample to detector distance.

Ar can be estimated as the sum of beam dimension and image plate pixel
size (0.1x0.1 mm), while 7" value ranges from 0.0 to 172.5 mm. If we simply
take 100.0 mm as the mean value of r, the % becomes comparable to % with
beam dimension of 0.5x0.5 mm. A much larger beam size than this would
make it the limiting factor to the instrument resolution. There are also some
other considerations as well, such as the intrinsic beam size, the incident beam
intensity, scattering powder of the sample and so on. Around 0.5x0.5 mm beam

size is used in our RA-PDF experiments.

. Align the center of the diffractometer with the beam center. This involves two
steps. We first locate the center of rotation of the diffractometer usually with
help from a telescope. A sharp metal pin is mounted on the goniometer, and
the goal is to get the tip of the pin in the same place when any circle of the
diffractometer is rotated. Once you have done this, use the motor to adjust

position of the diffractometer to set the tip of the pin on the beam center. You

160

can use a monitor or burn paper or fluorescent screen to do it. The easiest
way is to place a Silicon diode detector after the pin and do x-y scanning of
the diffractometer. Once this is done, remember the current position of the pin
tip in the telescope, or move the telescope so that the pin tip sits in an easy
to remember position, e.g. the center cross. This is the beam center and the

center of the diffractometer after the alignment.

. Put the image plate detector to its approximate position. You will need to
adjust it later. In most cases, the only variable here is the sample to detector
distance. As the dimension of the image plate (e.g. 345 mm in diameter for
Mar345 detector) is fixed, given the energy you will be working at, you can
compute the necessary maximum 20, and then the sample to detector distance
d(mm) = 172.5/ tan(26). For example, if you want to achieve Qmam of 35.0 A“1
at 100 keV with the Mar345 detector, the sample to detector distance needs to

be 202.5 mm.

. Install the beam stop and align it. Before you open the beam shutter, make
sure to have Pb shielding right in the front of the image plate detector. Direct
beam WILL damage the IP permanently! The best way is to put a Silicon
diode monitor after the beam stop installed on a translation stage. Through
scanning the beam stOp positions, you can quickly position the beam stop to

fully intercept the incident beam.

. Align the image plate detector within a reasonable amount of time. One goal
is to set the beam center close to the detector center. The other is to achieve a
good orthogonality of the plate plane to the beam direction. This requires the
use of a calibrant, for example, fine Si powder. Program FIT2D [64] provides
an easy-to-use interface to do this job. One important consideration for this

purpose is to minimize the uncertainties in later corrections. For example, the

161

oblique incident angle dependence correction assumes that the same Q values

around the ring have the same incident angle to the IP.

7. Take efforts to reduce the background. The RA-PDF experiment subjects to
rather high background levels. Most background intensities come from air scat-
tering. Thus the first step here is to shield out the air scattering before the
sample by placing a thick Pb plate with a hole at the center for the incident
beam. If there is a considerable gap between the Pb shield and the sample, a
collimator right in front of the sample is preferred. During experiments, usually
more than 50% incident beam is allowed to pass through the sample. The rather
long sample to beam stop distance (2 150 mm) creates a significant amount of
air scattering. The collimation after the sample is in principle hard to imple-
ment. One way is to shorten the sample to beam stop distance, however, this
would increase the blocked scattering angle by the beam stop. A very effective
way would be to use an evacuated second flight path, however, more difficult
in practice. The background scattering in practice only becomes an issue for

weakly scattering samples.

8. Install additional equipment required for your experiment. For example, the
displex with cryostat needs to be set up for low temperature measurements. In

the case of high temperature measurements, the furnace should be installed.

Once you are satisfied with the position and tilt of the image plate, write down
the values you get, such as the sample to detector distance, beam center, tilt angle,

etc. We are ready now to do some real measurements on samples.

A.3 Collecting RA-PDF data

This briefly describes some common actions during RA-PDF measurements. The

RA-PDF data collection mostly involves changing samples and controlling the data

162

acquisition system. Operation of data acquisition system works varies at different
beamlines. However, changing samples is essentially the same. As the sample to
detector distance is obtained from the calibration runs (with Si or LaBs), we need
to put the sample at the same position as the calibrant. This is usually done with
the help of the telescope indicating both the center of the beam and center of the
diffractometer. However, the RA-PDF setup requires a very close sample-to-detector
distance, e.g. 202 mm. It might happen that the telescope view of the sample is
blocked by the image plate detector. If this is the case, alternative ways of ensuring
the same sample positions should be implemented, such as a monitor screen, identical
sample holders etc.. If possible, a small telescope can be mounted on the diffractome-
ter for sample alignment. When changing the sample, be sure not to bump the beam
stop. A significant shift of the beam stop requires realignment of the beam stop, and
another set of measurements of the background scattering.

If the X-ray energy needs to be changed and a new sample to detector distance
becomes necessary, extra attention needs to be paid in order not to lose the calibration.
This is because the sample to detector distance and the X—ray energy are strongly
correlated in the calibration. ONLY one of them can be unknown to get any sensible
calibration. You can either change the X-ray energy or move the detector first, then
you do the calibration, and then you change the other one. If only room temperature
measurements are performed, moving the sample is much simpler than moving the
detector because the later almost always requires some aligning adjustments of the
detector.

Beam exposure time depends on the scattering power of the sample and the inci-
dent beam intensity. The rule of thumb is to keep all the counts in the linear response
regime of the image plate. In our case, the maximum count of in each pixel is 64,000
set by the electronic readout device. A reasonable maximum intensity for our data

is advised not to exceed 60,000 (though a maximum of 131,000 of 17 bits is possible

163

 

with two sets of PMTs). Grainy samples may cause sparse pixel saturations even
with very short exposure times because of those crystals with considerable sizes. Cer-
tainly the best solution is to have fine powdered samples. As grinding samples may
be not possible during the experiments, we may try to do more sample rocking, or
try another section of the sample. However, we should not worry about the small
number of saturated pixels in our data. First, they can be removed during the data
processing. Second, the relative error would be rather small compared to the total
number of pixels, e.g. ~ 10,000,000 pixels in the Mar345 detector.

To properly analyze the RA-PDF data, we need to also measure the background
and other calibration data if necessary. Here the background refers to the sum of
the container scattering and the empty instrument scattering (mostly air scattering).
The background intensities will be subtracted from the sample total intensities during
data analysis. When the background level is rather high, the absorption correction
due to the existence of the sample in the beam (when collecting sample data) be-
comes necessary. This correction is usually rather tricky since it depends on the
detailed experimental setup. Strictly speaking, different absorption corrections also
become necessary when the container itself scatters (absorbs) considerably. For the
most commonly used kapton tape in our lab, the scattering from the container itself
is small compared to the air scattering, and there is no need to distinguish between
the container and empty instrument scattering. If a glass capillary is used, the scat-
tering from it may be very strong. We may need to measure the empty instrument
separately. As the background scattering is usually slowly varying intensities, they
can be smoothed during analysis without compromising the analysis. Thus, the issue
of counting statistics is less critical for background measurements.

In the end, you should have the data for the sample, data for the background, and

data for the calibration.

164

A.4 Processing the data

Here the data processing refers to the procedure to obtain PDFS from the RA-PDF
data (for more detail see Chapter 5 of [5]. The conventional way to obtain PDFS
starts from the SPEC file format data collected with a solid solid detector (SSD), and
is described in detail in the tutorial of program PDFgetX2 [41, 195]. A simplified
version is described in the following. The raw data from our SSD detector are in
SPEC data file format. The program PDFgetX2 [41] was used to preprocess the
raw data to obtain the intensity versus 20 two column data. In the preprocessing,
the measured scattering intensities were first corrected for detector dead time (e.g.,
0.84 ,us), then normalized by the corresponding incident beam intensities (monitor
counts collected by one ion chamber right before the sample). Then the two scans
were merged together to yield the whole range. The same procedure was repeated
on the background scattering data. After the preprocessing, the background data
is subtracted from the sample data to obtain the sample only scattering intensity
versus 26. Following applied corrections are the sample absorption, polarization,
and Compton scattering to obtain the elastic scattering only. The last step is to
normalize the elastic scattering by the Q dependent average scattering power of the
material to get the total structure factor S (Q), which is then transformed to obtain
the experimental G(r) with the formula: G (r) = g- f0°° Q[S (Q) —- 1] sin Qr dQ.
However, the data coming from the image plate are two dimensional images, and
thus need to be integrated first to give the multiple column ASCII data. One column
should be the intensity, while there should also be one column being either 26 or Q.
The integrated data can then be fed to program PDFgetX2 to obtain the PDF as

described above.

165

 

 

A.4.1 Obtain calibration parameters

The image plate data are simply a two dimensional matrix of the intensities. Thus
we first need to know the 26 or Q value of each matrix element, which requires the
following parameters: the pixel size, the beam center (51:, y), the sample to detector
distance, the X-ray energy, and the tilt of the image plate relative to the incident
beam. In practice, we know before hand the pixel size that depends on the laser
scanning mode during read-out, and either the X-ray energy or the sample to detector
distance. The other parameters are obtained through the calibrations with sample
standards. The program FIT2D [64] has been our choice of software for obtaining
those parameters. One additional parameter, the X-ray polarization factor, will be
used during data integration. You can get this number from the instrument scientist.
All those parameters are critical to the accuracy of our data, and thus should be
kept well. This also means to repeat the calibration whenever the experimental setup

changes.

A.4.2 Integrate the IP raw data

The goal here is to get a two column data set equivalent to the conventional angle
dispersive scan, where every data point represents the sum of scattering intensities
over the same solid angle. In the image plate data, different Q values have different
number of corresponding pixels. This necessitates an additional geometric correction
to properly normalize the data. The X-ray polarization correction should also be
done at this step, because it depends not only on the distance of the pixel from
the beam center, but also on the azimuthal angle. Data integration can be done
straightforwardly with FIT2D with the choices of above corrections. The integrated
data should be saved in “CHI” file format. It’s also recommended to save it to
GSAS [68] format for later Rietveld analysis.

For each sample, we usually take multiple exposures to achieve better counting

166

 

statistics. Since the multiple data sets are measured under identical conditions, we
directly sum them together to get the total counts. To avoid too large values, we
divide the total counts by the number of exposures .

Batch integration is possible with the FIT2D GUI macro functionality. This
becomes necessary when there are a large number of data files to integrate, as is the
case for our RA-PDF experiments. A more efficient way is to use the keyboard mode
in which program FIT2D can be driven by command line inputs. In this case, we
can put all the commands in a macro file, and give the macro file name to FIT2D to
achieve the same result. Certainly the macro file should contain information such as
the needed processing parameters and the related image plate data file names. Many
samples are usually measured with the same experiment setup, thus only the data file
names need to be changed in the macro file. In consideration of this, the macro file
is designed to take parameter values from variables, and the variable values can be
passed when calling FIT2D. The current version of FIT2D macro file is the following

for your reference.

%!*\ BEGINNING OF mar_integrate.mac MACRO FILE
%!*\ This is a comment line

Z!*\ 1) Deduce the file sequence and calculate 2theta step size
DEDUCE

#START_FILE

#END_FILE

CALC

#SCAN_BINS

#MAX_ANGLE

/

VARIABLE

#STEP_2THETA

167

 

 

EXIT

%!*\ 4) Initialize data and set up experiments
CREATE DATA
3450

3450
EXCHANGE
CREATE DATA
3450

3450
GEDMETRY
KEYBOARD

#X_BEAMCENTER

 

n-.

#Y_BEAMCENTER

#WAVELENGTH

#SAM2DET

#X_PIXELSIZE

#Y_PIXELSIZE

#TILT_ROTATION

#TILT-ANGLE

0.0

%!*\ 5) Read in all data and add them, then divide by number of files
D0 #COUNT = ##START, ##END, ##STEP

%!*\ Get MAR_FILE by patching different variables
I2C

#COUNT

N0

##NUM_CHARS

168

 

#CVALUE

CDNCATENATIDN

##PREFIX

#CVALUE

#MAR_FILE

CUNCATENATION

#MAR_FILE

##EXTENSION

#MAR_FILE

%!*\ Read in the data file
INPUT DATA

MAR RESEARCH FORMAT
#MAR_FILE

%!*\ Add it to the memory
ADD

EXCHANGE

END DU

%!*\ Make the memory to be the current data
EXCHANGE

%!*\ Calculate number of files
CALC

##START

##END

##STEP

/

1/X

169

 

 

l

+

VARIABLE

#NUM_FILES

EXIT

CDIVIDE

#NUM_FILES

%!*\ 6) Using powder diffraction to integrate the data
FIT

%!*\ 7) Threshold masking of the data, greater than mask
THRESHOLD MASKING

NO

64000.00

POWDER

YES

#X_BEAMCENTER

#Y_BEAMCENTER

#X_PIXELSIZE

#Y_PIXELSIZE

#SAMZDET

#TILT_ROTATION

#TILT_ANGLE

%!*\ Input polarization factor?
YES

#POLARIZATION_FACTOR

%!*\ Lorentzian correction?

PARTIAL POWDER

170

 

%!*\ NONE

%!*\ YES: Euqal angle pixel scan, N0: Equal radial distance
YES

#STEP_2THETA

%!*\ Correct spatial distortion?
NO

%!*\ No to "save powder diffraction data standard file"
NO

%!*\ Exit fit submenu

EXIT

EXCHANGE

OUTPUT

CHI.

#CHI_FILE

%!*\ Output rows?

YES

1

OUTPUT

GSAS

#GSAS_FILE

YES

1

EXIT

YES

Instead of passing the variable values from the keyboard, a setup file can be used,
and is shown below. The meaning of each field is noted by the comments on the same

line. Basically, this file contains all the information necessary to integrate the image

171

 

 

plate data. We can simply add more and more data to the list of data to the end.

# Setup for automation of MAR3450 file integration

# Format:

# 1) You can insert comment lines at any place, starting with "#"

# 2) NO space should be left before the parameter values

# 3) When commenting on the same line, at least ONE space should

# be left between parameter value and its comment

# Written by Xiangyun on 07/23/2003

3450

3450

0.126514

0.94

179.4899

100.0

100.0

1720.870

1705.101

7.866515

0.0234912

44.0

3450

#
CsIFR6_426_annealed3d_001.mar3450
CsIFR6_426_annealed3d_007.mar3450
CsIFR6_426_annealed3d.gsa

CsIFR6_key_p-90.chi

31:43:33:

*1:

x dimension

y dimension

X-ray wavelength
Polarization factor
Sample to detector distance
X pixel size

Y pixel size

X beam center

Y beam center

Tilt rotation angle
Tilt angle

Max angle

Number of points

Start file to read
End file to read
GSAS file to save

CHI file to save

The batch FIT2D data integration is initiated by command manintegratesh <

172

 

setup-file-name under the directory containing all the raw data. The SHELL script
mar-integrate.sh reads the setup file to extract the parameter values, then calls FIT2D
with the macro file and proper variable values. One additional operation in the script
is to comment out the first four lines in the integrated CHI files, in order to make it

directly readable by the data visualization program KUPLOT [196].

A.4.3 Normalization by the monitor counts

The last step of this preprocessing before using PDFgetX2 is to normalize the data.
This is very important because sample and background are usually measured with
different exposure time. Even with the same exposure time, the incident beam in-
tensities most likely are still different during the self-decay and re-filling process of
the synchrotron storage ring, as well as the unpredictable slight fluctuations. The
normalization constant should be the incident beam monitor closest to the sample.
There should be no beam size defining slits/collimation’s after this monitor, other-
wise its reading doesn’t reflect the real incident beam intensity on the sample. In the
case of multiple image plate data being averaged, the corresponding total monitor
counts should also be averaged, i.e. divided by the number of data files (exposures).

Finishing this, we are ready to use PDFgetX2 to obtain the PDFS.

A.4.4 Additional corrections to the RA-PDF data

The RA-PDF data collection is much quicker (three to four orders of magnitude
faster) and requires a simpler experimental setup. However, the data analysis becomes
more difficult, in addition to the extra steps to integrate the image plate data. The
characteristics of the image plate detector requires some additional corrections, as
well as the high background levels. We will focus on two aspects, the lack of energy

resolution and the energy dependent response.

Estimation of standard deviations Proper estimation of the standard deviations

173

 

 

(BSD) and their propagation along the data corrections are rather important, as
the experimental PDFs are commonly used in least square regression programs
for structure refinement [197, 198]. For X-ray photon counting detectors, the
ESD of the observed count (after dead time correction) is simply its square
root when the count is large (more than a few hundreds). However, the IP
detector is not a photon counter. Quite a few experimental factors contribute
to the relative uncertainties of IP response, such as the read-out system, and
the non-uniformity of the IP, etc. [49, 52]. To complicate the situation, in the
published literatures, the ESD is calculated with respect to the noise level of
the IP system under study [54, 55]. Fig. A.2 shows the ESD of the IP response
as a function of exposure level. It was found that the relative ESD of the IP
response is rather close to an ideal photon counter at low exposure levels, while
it saturates to a little less than 1% at high exposure levels. The IP response
relative to the noise is found to be close to 1—2 per incident photon at the high
energies of our interest. Therefore, based on the specifications of the IP system
in our experiment (Mar345, sensitivity of 1 X-ray photon per ADC-unit at 8
keV, intrinsic noise 1-2 photon equivalents), we approximate the ESD of each

pixel count to be the larger value between its square root and its one hundredth.

Once the ESD of each pixel is determined, we can propagate the ESD through
the integration from the two dimensional raw image and geometric corrections.
However, program F IT2D does not generate/ propagate ESDs, and only gives
the two column data: intensity versus the scattering angle 26. Thus we need
to correctly retrace the procedure done in FIT2D to obtain proper ESDs. The
steps are the following. The integrated data from FIT2D are the averaged
intensities among the pixels within each bin grid corrected by the geometric

factor. As each pixel has the same area, the geometric factor is simply cos3(26).

174

-1. a; 41m

 

 

 

 

 

 

 

N

8 1 10 102

“r l— I I
§ 1,.
‘5 \\
'; 2 l- X \‘
3 \3\
E \N\
g l . I“. \8\
2;; \\ \Omx— ——x
SimsL OT 4
s
g 103 105 10’

Photon Number Absorbed In NxN pixels Area

 

Figure A.2: Estimated relative standard deviations of the IP response among areas
of N x N pixels as a function of photon number absorbed in the area. The scale of
N is shown in the upper part. Open circles are for the drum scanner and crosses
for BAS2000. Dashed lines represent the relative standard deviation of the photon
number absorbed in the area. Taken from [55].

So the first step is to multiply the intensities by cos3(26) to get the averaged
total counts. Then, we need to compute the number of pixels in each bin grid.

For the scattering angle bin width of 626, the area for this bin on the IP is

 

approximately 27rd2 sin(26)/ cos3(26) . 626, where d is the sample to detector
distance. Dividing the area by each pixel size gives the number of pixels. Given

the integrated count c, pixel area Apixela we obtain

c - cos3(26)

\/27rd2 sin(20)/ 0053(29) ' 529/Apz'ze1/ 0083(20), (A.2)

ESD =

 

 

noting that ‘/c - cos3(26) should be replaced by 0.01 ' c ' cos3(26) if the latter is

larger due to the 1% saturation of relative uncertainties of the IP.

175

Oblique incident angle This correction accounts for the angular dependence of
the scattered photon effective path length in the IP phosphor layer [60], as
a direct result of its incomplete absorption of the scattered photons. This
correction becomes very significant at high X—ray energies and large incident
angles (both present in the RA-PDF experiments). Two parameters are used
here. The scattered photon absorption coefficient of the IP phosphor layer and
the incident angle (equal to 26). The X—ray energy used in our experiments is

highly penetrating with the absorption coefficient less than 1%.

Compton intensities at high Q and fluorescence The image plate counts only
give the detected X-ray intensities, with no knowledge of the energy of the X-
ray. The lack of energy resolution makes it impossible to distinguish between
elastic, Compton, and fluorescent photons. This poses rather serious problems
to PDF analysis. Early use of image plates to collect diffraction data reached
Qmaz less than 13.0 A‘1 [199, 57, 58]. In the low Q region, elastic scattering
is usually the strongest signal, thus removal of the parasitic Compton and flu-
orescence intensities doesn’t introduce much error into the analysis. However,
elastic scattering decays exponentially with Q; Compton intensity increases
with Q; and fluorescent intensity is constant. As a consequence, in the high Q
region (QmazZ 30.0 A”), the situation is quite different. For example, even
the case of Ni (Z228) atoms, the Compton scattering is about 4 times as large
as the elastic scattering at Q = 30 A“. This ratio increases with decreasing
Z values. The fluorescent intensities can be effectively suppressed by going to
higher incident X-ray energies for low Z elements or just below the edge for
high Z elements. However, for the medium to high Z elements, the currently
routinely achievable highest energy X-rays still give rather significant fluores-
cent intensities. Therefore, the extraction of elastic signal in the high Q region

is not a trivial task. Small errors on the estimation of Compton or fluorescent

176

 

 

intensities would result in significant deviations to the elastic signal. Another
complication is the statistical uncertainty of the extracted elastic intensities, as
the uncertainty of both Compton and fluorescent intensities will be added to
it. To achieve the same counting statistics as when measuring elastic scattering
separately, the total counts from the image plate would have to be ~ n2 times

as many (if elastic scattering accounts for 1/n of the total intensities).

Energy dependence of the IP response Due to the detection mechanism of the
image plate, X-rays with the same number of photons but different energies will
result in different counts. Though elastic scattering has the same energy at all
Q values, the energy of Compton scattering decreases with increasing Q and the
fluorescence intensities usually comprise of several characteristic energy lines.

The high Q region data again are sensitive to this effect due to the dominat-

 

ing Compton and/or fluorescent intensities. This energy dependent detection
efficiency is not a simple linear function, and in fact is difficult to measure ac-
curately. In program PDFgetX2, either a linear or quadratic empirical formula
is used as the energy dependence of the image plate detection efficiency while

ensuring the detection efficiency decreases with increasing X-ray energy.

The obtain PDF, G(r), is in a compatible file format for the PDF refinement
program PDF FIT [11]. Please refer to the user’s guide of the program PDFgetX2 [195]

for details.

177

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