HIGH THROUGHPUT PARAMETRIC STUDIES OF THE STRUCTURE OF
COMPLEX NANOMATERIALS
By
PENG TIAN

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics and Astronomy
2011

ABSTRACT
HIGH THROUGHPUT PARAMETRIC STUDIES OF THE STRUCTURE OF
COMPLEX NANOMATERIALS
By
Peng Tian

The structure of nanoscale materials is difficult to study because crystallography,
the gold-standard for structure studies, no longer works at the nanoscale. New tools
are needed to study nanostructure. Furthermore, it is important to study the evolution of nanostructure of complex nanostructured materials as a function of various
parameters such as temperature or other environmental variables. These are called
parametric studies because an environmental parameter is being varied. This means
that the new tools for studying nanostructure also need to be extended to work quickly
and on large numbers of datasets. This thesis describes the development of new tools
for high throughput studies of complex and nanostructured materials, and their application to study the structural evolution of bulk, and nanoparticles of, MnAs as a
function of temperature.
The tool for high throughput analysis of the bulk material was developed as part
of this PhD thesis work and is called SrRietveld. A large part of making a new
tool is to validate it and we did this for SrRietveld by carrying out a high-throughput
study of uncertainties coming from the program using different ways of estimating the
uncertainty. This tool was applied to study structural changes in MnAs as a function
of temperature. We were also interested in studying different MnAs nanoparticles
fabricated through different methods because of their applications in information
storage. PDFgui, an existing tool for analyzing nanoparticles using Pair distribution
function (PDF) refinement, was used in these cases. Comparing the results from the
analysis by SrRietveld and PDFgui, we got more comprehensive structure information

about MnAs.
The layout of the thesis is as follows. First, the background knowledge about material structures is given. The conventional crystallographic analysis is introduced in
both theoretical and practical ways. For high throughput study, the next-generation
Rietveld analysis program: SrRietveld, is coded in Python. The details of SrRietveld
are provided in the thesis. More importantly, two real applications of SrRietveld are
demonstrated to show its use cases. For the error analysis on Rietveld refinement, it
is found that the results from two popular Rietveld programs are very sensitive to the
input errors and the subset sampling method is particularly useful when the errors
on diffraction pattern are unknown. In order to show the power of SrRietveld, I did
the parametric study on MnAs bulk and nanoparticles. It is found that the magnetostructural property exists in bulk and one type of MnAs nanoparticles. However,
when I want to probe the other type of MnAs nanoparticles that has smaller size,
the conventional Rietveld method doesn’t work and I turn to PDF for help. Using
PDF approach, the structures of bulk and nanoparticle MnAs have been explored.
Finally the conclusion is that I can either retain or modify the bulk properties by
using different synthesis methods.

To my family

iii

ACKNOWLEDGMENTS
At the time when I finish my PhD thesis, I would like to thank those who have
helped me to accomplish this important goal in my life.
First of all, I want to give my deepest thanks to my advisor, Professor Simon J.
L. Billinge, for guiding me through my PhD study with his expertise, insights and
great patience. My future work will continuously benefit from what I have learned
from him.
I am grateful to all professors in my PhD guidance committee, Prof. Phillip
Duxbury, Scott Pratt, Subhendra D. Mahanti and James T. Linnemann, for all the
help and advice they have offered me.
I would also like to acknowledge all my officemates. I learned many things about
scientific research and computer techniques from working with Jiwu Liu, Wenduo
Zhou, Pavol Juhas, Chris Farrow and Emil Bozin. Chris Farrow and Yingrui Shang
gave me suggestions on my thesis writing. And Timur Dykhne, Xiaohao Yang, Hrishi
Tiwar and many other visiting scholars make the office an enjoyable place to work
in. Thanks to all my friends in graduate school, Jiwu Liu, Teng Yang, He Huang,
Wenduo Zhou, Chuan Lu, Jiping Li, Yingjie Li and many others, for standing by me
and giving me help whenever I need.
Most importantly, my great gratitude to my grandparents, my parents and my
fiancee. You have given me enormous support and love. This work is dedicated to
you.

iv

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

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

vii

1

Introduction to material structure
1.1
Basics of Crystal system . . . .
1.2
Bragg’s Law . . . . . . . . . . .
1.3
Diffraction Method . . . . . . .

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1
2
3
5

2

Crystallographic Analysis . . . . . . . . . . . . . . . . . . . . . . .
2.1
Theoretical Background . . . . . . . . . . . . . . . . . . . . . . .
2.2
Rietveld Method . . . . . . . . . . . . . . . . . . . . . . . . . . .

10
11
14

3

Next-generation Rietveld analysis: SrRietveld . . .
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . .
3.2
Design Principles . . . . . . . . . . . . . . . . . . .
3.3
Implementation . . . . . . . . . . . . . . . . . . . .
3.4
Application of SrRietveld I: NaCl . . . . . . . . . .
3.5
Application of SrRietveld II: Error estimations . . .

20
20
21
22
25
30

4

The total scattering and atomic pair
4.1
Theory of PDF . . . . . . . . . .
4.2
Corrections of PDF derivation . .
4.3
Cases of PDF calculation . . . . .

5

MnAs: Diverse structural and magnetic properties
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . .
5.2
Methods . . . . . . . . . . . . . . . . . . . . . . . .
5.3
Results and Discussion . . . . . . . . . . . . . . . .
5.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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55
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57
61
68

6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

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46
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53

List of Tables

3.1
3.2

5.1

5.2

Refined values, estimated standard deviations for LaMnO3 from the
GSAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Difference in the values of refined parameters between FullProf refinements on data converted from time-map format using the SE and NSE
methods, respectively. The numbers in the parethese are the standard
deviations on the differences, calculated as the square-root of the sum
of the variances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

The refined structural parameters for bulk and type-A nanoparticles
at 295 K and 335 K using α (P 63mc) and β (P nma) model. All models
are refined with isotropic thermal factor Uiso . . . . . . . . . . . . . .

62

The refined structural parameters for type-B nanoparticles at 295 K
and 335 K using the α (P 63mc) and β (P nma) model. All models are
refined with isotropic thermal factor Uiso . . . . . . . . . . . . . . . .

65

vi

List of Figures

1.1

Bragg diffraction [1, 2]. Two beams with the same wavelength and
phase approach the sample and are scattered by two different atoms
inside it. The lower one travels an length of 2d sin θ longer. . . . . . .

4

1.2

Ewald construction [3] . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.1

SrRietveld software architecture. The User Interface, SrRietveld, HDF5
Data Structure, PyGSAS, and PyFullprof are python packages in the
SrRietveld project. The GSAS and Fullprof represent the underlying
refinement engines. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

Temperature dependence of SrRietveld refinement results for NaCl using FullProf (blue dots) and GSAS (red squares) engines. (a) lattice
parameter, a, and (b) atomic displacement parameters, Uiso , at Na
(crosses for FullProf, plus markers for GSAS) and Cl (dots for FullProf, squares for GSAS) sites. Dashed lines mark fitted Debye model
curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

(a) TOF neutron powder diffraction pattern on LaMnO3 (blue circles)
with the best fit calculated pattern from FullProf (red line). The low
d-spacing, short TOF, region to the left of the dashed line has been
zoomed in 6 times for clarity. The green line offset below is the difference between the calculated and measured patterns. The black markers
indicate the peak positions. (b) As (a) but the red curve is calculated
using GSAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

The probability distribution functions of refined lattice parameters
from FullProf based on data converted using the SE and NSE methods.
(a), (b) and (c) show the distributions of lattice parameters a, b and c,
respectively. For each subfigure, red, green and black symbols are from
the NSE method; blue, orange and brown are from the SE method.
Diamonds are from Monte Carlo resampling, squares are from subset
sampling and circles are from the standard Hessian matrix method. .

39

3.2

3.3

3.4

vii

3.5

5.1
5.2

5.3

5.4

5.5

5.6

5.7

The probability distribution functions of thermal factors Uiso . (a) is
for La, (b) is for Mn, (c) is for O1 and (d) is for O2. All colors and
symbols have the same meanings as Fig. 3.4. . . . . . . . . . . . . . .

40

TEM images and particle size distributions of type-A (a) and type-B
(b) MnAs nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . .

70

PDF refinements on bulk (left) and type-A nanoparticle (right) MnAs
data. For each subfigure, the bulk circles represent the PDF from
the experimental data and the red solid line is the calculated PDF
after refinement. The green curve offset below is the difference curve
between data and model. . . . . . . . . . . . . . . . . . . . . . . . . .

71

(a) comparison of the difference (orange curve) between 295 (blue line)
and 335 K (red line) diffraction pattern for bulk MnAs. (b) same as
(a) but for type-A nanoparticles with violet-colored difference curve.
(c) comparison of the difference curves from (a) and (b). The orange
is the same as the difference curve in (a) and the violet is the result of
scaling the curve in (b) by a factor of 4 for comparison. . . . . . . .

72

(a) comparison of the difference (orange curve) between 295 (blue line)
and 335 K (red line) PDF for bulk MnAs. (b) same as (a) but for typeA nanoparticles with violet-colored difference curve. (c) comparison of
the difference curves from (a) and (b). The orange is the same as the
difference curve in (a) and the violet is the result of scaling the curve
in (b) by a factor of 4 for comparison. The simliar comparison based
on diffraction pattern is shown in Fig. 5.3. . . . . . . . . . . . . . . .

73

Lattice parameters ((a), (b)) and atomic displacement factors ((c),
(d)) from PDF refinements on bulk and type-A samples using α phase
structure model. (a) and (b) are for lattice parameter a and c in unit
of ˚. (c) and (d) are for Uiso of Mn and As in unit of ˚2 . For
A
A
each subfigure, the blue and red are for bulk MnAs on the cooling and
heating runs respectively. The cyan and magenta are for type-A MnAs
nanoparticles on cooling and heating runs. . . . . . . . . . . . . . . .

74

PDF refinements on data acquired on type-B nanoparticles at 295
((a), (b)) and 335 K ((c), (d)) using the α ((a), (c)) and β ((b), (d))
phase model, respectively. For each subfigure, the symbol and color
representations are the same as Fig. 5.2. . . . . . . . . . . . . . . . .

75

Same as 5.3 but for type-B nanoparticles. In (c) the difference curve
(violet) of type-B nanoparticles is scaled by a factor of 10 since its
amplitude is too small to compare with the bulk one. . . . . . . . . .

76

viii

5.8

5.9

Same as 5.4 but for type-B nanoparticles. In (c) the difference curve
(violet) of type-B nanoparticles is scaled by a factor of 10 since its
amplitude is too small to compare with the bulk one. The simliar
comparison based on diffraction pattern is shown in Fig. 5.7. . . . . .

77

Real part of the AC magnetization (a) and unit cell volume (b) in
bulk, type-A and type-B nanoparticles. All of the unit cell volumes
are based on the results of α-model refinements. The blue (cooling)
and red (heating) are for the bulk sample; cyan (cooling) and magenta
(heating) are for type-A nanoparticles; and the green (cooling) and
orange (heating) are for type-B. The dashed lines at 303 K and 317 K
show the temperature range of the structural transitions in bulk MnAs
as reported in the literature as we mention in the text. . . . . . . . .

78

ix

Chapter 1
Introduction to material structure
and diffraction method
All materials are made of atoms. There are about 100 kinds of atoms based on
the periodic table. The universe is composed of them. The properties of the materials
are determined by the atoms of which they consist and how the atoms are bonded
together [4].
The structures of the materials have to be studied so that the researchers can
understand the properties of useful materials [5]. Studies of material structures can
also help chemists in synthesis and modification of materials with desired properties [6, 7]. In general, solid materials are the most commonly studied by scientists
because of their applications [8, 9, 10], for example in the technology of transistors
and semiconductors.
Diffraction techniques, including X-ray diffraction, neutron diffraction and electron diffraction, have been developed to investigate the structures of different materials, especially for the crystalline solids where the atoms are composed in an ordered
fashion. More details about this topic can be found in many good books [11, 12, 13].
This thesis focuses on high throughput parametric study on the structure of complex

1

nanomaterials.
A brief introduction to the structure of materials is given in the following three
sections, with emphases on issues that are relevant to this thesis. Section 1.1 covers
some basic knowledge about crystal structure, such as it description and classification.
In section 1.2, the foundation of crystallography: Bragg’s law is introduced. The
last section describes the relationship between the diffraction pattern and material
structure.

1.1

Basics of Crystal system

The structures of the materials appear differently in nature. The simplest one of
these structures normally is the one with clearly repeated pattern of atomic positions.
The material with such structure is called crystal [8, 9, 10], e.g.NaCl and diamond.
Scientists who study them are focus on their inside geometry. The crystallography [14,
15] is the field that people study the crystal.
The structure of a crystal is made up of a number of points (atoms or molecules)
arranged in a ordered way. Inside a crystal, a lattice (a repeated pattern of atoms or
molecules) exhibits long-range order and symmetry. The so-called unit cell is formed
by the points. The lengths of the edges in the unit cell and the angles between them
are called the lattice parameters [8]. The space group [8] is used to represent the
symmetry property of the structure. It is a mixture of the translational symmetry
of a unit cell, the point group symmetry operations of reflection, rotation, the screw
axis and glide plane symmetry operations [16]. In summary, there are seven lattice
systems: triclinic, monoclinic, orthorhombic, rhombohedral, tetragonal, hexagonal,
and cubic, which includes 14 unique ones. Normally scientists use one of 230 unique
space groups to describe the crystal symmetry[17].
The Miller index is usually used to determine the atomic plane. If we choose

2

the three lattice vectors a1 , a2 and a3 to determine the unit cell, the corresponding
primitive reciprocal lattice vectors are given by [8]

b1 =

2πa2 × a3
a1 ⋆ (a2 × a3 )

(1.1)

b2 =

2πa3 × a1
a2 ⋆ (a3 × a1 )

(1.2)

b3 =

2πa1 × a2
a3 ⋆ (a1 × a2 )

(1.3)

Using the three Miller indices l, m and n, we can define the plane orthogonal to the
lattice vectors:
glmn = lb1 + mb2 + nb3

(1.4)

The perpendicular distance d between two closest lattice planes is determined by:

d=

2π
glmn

(1.5)

In order to understand the structures, it is necessary to know the arrangement
of atoms relative to each other, the number of the nearest neighbors, a list of interatomic distances and etc.. To detect the structure information in atom level, people
have been relying on diffraction technique since about 100 years ago. In the following
section, I will describe the basics of the diffraction theory.

1.2

Bragg’s Law

Bragg’s Law is the fundamental rule of modern diffraction theory. It is also the
milestone where people start to understand the diffraction in more profound way. It
was discovered by physicist Sir William L. Bragg in 1912 [1, 2], who later won the
Nobel Prize with his son because of their work in determining crystal structures [18].

3

Since then, the X-rays became a power tool to investigate the structure of crystals
and other forms of materials.

Figure 1.1: Bragg diffraction [1, 2]. Two beams with the same wavelength and phase
approach the sample and are scattered by two different atoms inside it. The lower
one travels an length of 2d sin θ longer.
X-rays can force the cloud of atom’s electrons to move when they are incident on
an atom. If the cloud movement results that the same-frequency waves are radiated,
this phenomenon is known as elastic scattering [19]. In such case, the new waves can
be scattered but it is often negligible for the high-order scattering. If the incident
beam is made up of neutrons, elastic scattering could also happen when it hit on
the nuclei or it flies through the electronic magnetic field around the nuclei. The big
difference between these two type of beams is that X-ray scattering is not sensitive
to the atom with few number of electrons, e.g., hydrogen, especially when there are
heavy atoms with many electrons while neutron scattering does. The fields produced
by the scattered waves can interfere with each other. Sometimes it produces stronger
signals while other times it could cancel out each other. Therefore, the diffraction
pattern are formed and can be observed through a detector or film, which is usually
the experimental result. It serves as the first step of diffraction analysis. In general,

4

the constructive interference happens when the following formula is satisfied:

nλ = 2d sin θ

(1.6)

where n is an integer representing the given order, λ is the wavelength of the incident
wave, d is the spacing between two planes, and θ is the angle between the incident
beam and the planes. It is called the Bragg’s law. As shown in Fig. 1.1, the diffraction
pattern is determined by the parameters mentioned in Eq. 1.6. When the equation is
satisfied, there will be strong peak appearing in the diffraction pattern. The strength
of those peaks relative to each other are determined by the order parameter n.

1.3

Diffraction Method

Diffraction method is one of the most popular scientific techniques to probe the
structures of materials. Normally the X-ray, neutron, or electron beam is used as
the probed beam. In particular, for the study in this thesis, I am focus on powder
diffraction, which was first used by Debye and Scherrer in 1916 [20]. It has advantage
over other diffraction techniques because it is relatively easy to prepare the powder
sample for common cases. Meanwhile it can still provide the comparable structure
information from analyzing the diffraction pattern.
The most common type of radiation used in powder work is X-radiation since the
technique was developed a century ago. For later study in this thesis, I also use the
X-ray powder diffraction method as the experimental tool. The X-ray we refer here is
a high energy electromagnetic radiation with corresponding wavelength in the order
of 1 ˚. In order to access high energy X-ray source, we usually go to synchrotron at
A
national laboratory (e.g., Advanced Photon Source at Argonne National Laboratory
or National Synchrotron Light Source at Brookhaven National Laboratory) to perform
our powder diffraction experiments.
5

As we know, the powder sample is made of large number of crystals with random
orientations. In order to understand the basis of X-ray powder diffraction, we have
to study the interaction of X-ray and crystal first. As I mentioned earlier in this
chapter, X-rays can be diffracted from crystals since their electric fields interact with
the electron clouds of atoms in the crystals. The X-rays scattered from adjacent
atoms interfere and a diffraction pattern is produced. In this section, I will introduce
the X-ray diffraction on crystal and later powder diffraction in general.

1.3.1

Reciprocal space and Ewald construction

Figure 1.2: Ewald construction [3]
Assume the incident X-ray beam targeting on the crystal has a momentum ki ,
and the diffracted beam has a momentum kf . Therefore the momentum difference
between these beams is defined as

Q = △ k = kf − ki
6

(1.7)

where Q is known as the scattering factor (the change of momentum). It defines
the so-called reciprocal space that the bright spots are shown on in the diffraction
experiments. There is direct relationship between reciprocal space and real space that
I will talk about later. It is another basis of diffraction theory.
If the amplitude of ki is 2π/λ, where λ is the wavelength, and the diffraction is
elastic, then we can calculate the amplitude of Q by the following equation:

|Q| = 2k sin θ = 4π sin θ/λ

(1.8)

where |ki | = |kf | = k = 2π/λ. The angle between |ki | and |kf | is 2θ, which is shown
on Fig. 1.2. In fact respective to ki , for the elastic diffraction, the kf will always fall
on the surface of the sphere in the figure. The sphere is called Ewald sphere.
When Bragg condition is satisfied, it means Eq. 1.6 is established. By substituting it into Eq. 1.5 and comparing it with Eq. 2.11, we can clearly see that the
scattering vector Q is equal to a reciprocal lattice vector glmn when the order is
1. More importantly, it implies that the diffraction signals can only appear on the
surface of the Ewald sphere as shown in Fig. 1.2.

1.3.2

Basic theory of crystal diffraction

For X-ray diffraction, we can use ki to define the momentum of the incident wave.
Based on classical wave theory [12], the incident wave can be totally represented by
Aeiki ·r , where r is the position with the sample. Notice here, we only consider the
static case, where the momentum vector doesn’t depend on the time. If we also
assume that the electron density as ρ(r) and ignore constant term here, then the
diffracted wave is determined by
amplitude of diffracted wave = Aeiki ·r ρ(r)dV

7

(1.9)

where dV is the volumn that is filled by electron cloud around the position r.
For simplicity, we always consider the elastic diffraction case first. It means
for both incident and diffracted wave, they have the same amplitude of momentum
vector, as |ki | = |kf |. Assume that the diffracted wave hits at the position rd on the
detector, the phase change of the wave is given by
ik ·(r −r)
e f d

(1.10)

Since there is no change on the amplitude of diffracted wave after the diffraction, the
total wave that arrive at detector is determined by

A

ik ·(r −r)
ik ·r
drρ(r)eiki ·r e f d
= Ae f d

i(ki −kf )·r

drρ(r)e

(1.11)

Plugging Eq. 1.7 into the above equation, we get
ik ·r
Ae f d

i(ki −kf )·r

drρ(r)e

ik ·r
= Ae f d

drρ(r)e−iQ·r

(1.12)

If we define
F (Q) =

drρ(r)e−iQ·r

(1.13)

which means the F (Q) is the Fourier transform of ρ(r). The Eq. 1.12 can be simplified
as
ik ·r
Ae f d

ik ·r
drρ(r)e−iQ·r = Ae f d F (Q)

(1.14)

Finally the intensity of the diffraction peak is determined by
A2 |F (Q)|2

(1.15)

where F (Q) is called structure factor. Furthermore, the structure factor is the one to
connect the real space (r) to reciprocal space (Q). Because of this one-to-one direct
8

relationship, the reciprocal and real space analysis potentially can give us the same
set of structure information.
For perfect crystal, its electron density ρ(r) is periodic. Then its counterpart in
reciprocal space, F (Q), has non-zero values only when the Bragg condition is satisfied.
It implied that there are only Bragg peaks appearing in the diffraction pattern on
the detector. However, in real experiment, all crystals are not perfect. They have
different defects and heterogeneity in their configuration. Therefore, Bragg peaks
have a finite width and there may be significant diffuse (inelastic) scattering that fall
between the Bragg peaks [21].

1.3.3

Powder diffraction

In our experiments, we use the powder samples, which usually contain large
number of randomly oriented crystals. By saying that, we assume that each possible
orientation is equally represented in the powder sample. For some crystals in the
sample, the Bragg condition will be satisfied and the Bragg peak will appear. Because
of homogeneity in crystalline orientation, the azimuthal angle of the incident beam
doesn’t matter in powder case. Therefore, in powder diffraction pattern, we can’t
directly obtain the phase information but we can get the magnitude from the intensity
of the peaks.
For instance, when Bragg’s condition is satisfied for any hkl plane, the reflections
from all crystals in the sample lie on a cone, which is called Debye-Sherrer cone. From
the perpendicular view of the detector area, we can see the smooth circles formed
on the powder pattern because many bright spots from crystals are continuously
connected. The obvious result is that the information from the three-dimensional
reciprocal space has been compressed into one dimension. Due to the complexity of
such powder patterns, it is usually very difficult to interpret them without further
information.

9

Chapter 2
Crystallographic Analysis
Crystallographic analysis [14] is a method based on diffraction techniques to
study the structure information inside crystals [22]. The method for studying crystal
structure has been developing since last century [23, 24]. It is focus on analyzing the
diffraction patterns coming from the certain sample. In particular, powder diffraction
pattern is one of such cases that many scientists are working on. Since most materials
can not form large crystals, the powder diffraction [25] overcomes the single crystal
technique and becomes one of the most popular techniques to characterize the crystallographic structure, crystallite size, and preferred orientation in polycrystalline or
powdered solid samples [26]. There are several databases [17, 23, 27] existing now
for scientists to index their diffraction pattern in order to find new phases in the
sample. However, it was hard to do when the databases didn’t contain many known
structures. So powder diffraction was not popular until Rietveld method [28, 29] was
developed in 1960s. Rietveld method is also one of the techniques I have been using
for the high throughput parametric study during my PhD research.
This chapter focuses on the Rietveld method only as one of major crystallographic
analysis methods. Firstly, section 2.1 provides an overview of powder diffraction
pattern, as the basis of the crystallographic analysis. Then in section 2.2, The detailed

10

introduction of Rietveld method is presented.

2.1

Theoretical Background

In this section, I will focus on introducing the background knowledge of powder diffraction data. Usually a diffraction histogram is used to present the powder
diffraction data. As we discuss in chapter 1, the intensity I in diffraction pattern
is a function of the structure factor, which can be the function of scattering vector
Q or scattering angle 2θ eventually. Normally people use the scattering vector Q
as the independent variable since it contains the effects of the scattering angle and
wavelength λ. Another advantage to use Q is that we can measure our diffraction
data under different experimental setup, e.g., different scattering angles and different
wavelengths. Finally we can compare them in the same space, reciprocal space [25].
As noted in many books [30] on powder diffraction, the total intensity detected
from diffracted X-ray or neutron beam, IT , is composed of the following parts:
IT = Ic + IIC + IM C + IBG

(2.1)

where Ic is the intensity of coherent scattering, IIC is the intensity of incoherent
scattering, IM C is the intensity of multiple-scattering and IBG is the intensity of the
background. Next I will explain them one by one in more details.
Normally the intensity of coherent scattering, Ic , is the largest term in Eq. 2.1.
Coherent Scattering refers to the elastic scattering. It usually results all the Bragg
peaks in the diffraction pattern as function of 2θ in X-ray diffraction or time of
flight in neutron diffraction. As the intensity is determined by structure factor as
we see in chapter 1, it contains the information about atom and lattice, e.g., atomic
coordinates, lattice parameters and etc. According to classical books [12, 31] about
the coherent scattering, its intensity is determined by scattering cross section, dσc ,
dω
11

the absorption factor A, and polarization factor P as following the equation below:

Ic = AP C

dσc
dω

(2.2)

where C is to used for normalizing the cross section term in the unit of intensity
per atom. The cross section term, dσc , can be computed directly given the charge,
dω
mass of the particle in the sample and scattering angle of the incident beam. The
analytical determination form for it can be found in many other books [12, 31].
Comparing with coherent scattering, the incoherent scattering is the inelastic
scattering. For X-rays, the incoherent scattering is also called Compton scattering [12]
while for neutrons it is from nuclear spin scattering [32]. Since it is inelastic, the
energy of scattered beam always is less than the energy of incident beam, which means
that there is energy transferred from incident beam to targeted atoms. It may lead
atoms to diffuse. So it is also part of diffuse scattering, which generally includes all
intensity that is outside Bragg reflections. It is usually difficult to directly separate the
intensity of incoherent scattering from multiple-scattering and background although
it can be calculated approximately in some cases, e.g., if the multiple-scattering is
ignored.
Multiple-scattering [33] is very complex process in general. Since it is the term
only giving greater contribution to the total intensity than the background, people
usually ignore it although it happen in most scattering process. However for some
application, e.g., X-ray absorption Spectroscopy data analysis, it becomes especially
important. Scientists develop different modeling such as diffusion process [34] to simulate it. Because it is computationally intensive, it also attracts computer scientists
to apply different algorithms to solve it numerically [35].
The intensity of the background in Eq. 2.1 refers to all scattering without the
sample, e.g., the scattering from the sample holder, air and etc. In the experiments,

12

people always try to get as less background as possible in order to make the data
analysis easier. Usually high background implies certain problem and we have to
change the condition to solve it. For example, it could be because of the strong
fluorescence and then we have to choose different wavelength to do the measurement.
If the sample is in amorphous phase, it is likely to produce strong background that
make the data analysis more difficult [36]. It is the case where the conventional
crystallographic analysis may not work. In common cases, we model the background
by using different type of polynomial functions.
From the powder diffraction data, we can extract the structure and phase information [37]. For example, For each diffraction peak, it represents a particular plane
denoted by Miller index as we discuss in the first chapter. The peak position contains
the information of the size and shape of the unit cell for the crystalline sample [25].
The peak width depends on more factors: the size of the crystallites, the strain in
the sample and instrumental parameters. Combining the information of diffraction
peaks including their relative intensities, we can identify the sample and determine
its structure. More importantly, for high throughput parametric study, we can control the certain parameter, e.g., doping or temperature, to do the powder diffraction
experiments in order to see how the sample system response. As one of the possible
results, we could discover a new phase appearing during this process. In this way,
the insitu measurement can help us understand new science in a complex material
system.
Overall, powder diffraction is very useful technique in characterizing the structure
information of the sample. However, it usually produces highly overlapped diffraction
peaks, which leads the analysis on the data becomes very challenging.

13

2.2
2.2.1

Rietveld Method
Introduction

Rietveld method is a technique to extract structure information based on the experimental diffraction data and given structure model. It was developed by Hugo Rietveld to analyze the experimental data from either X-ray or neutron powder diffraction [28, 29]. It accounts for the instrumental effect and structural factors into the
regression model, which make itself very complex. Theoretically, Rietveld method
is a non-linear least square optimization process and provides the values of physical
parameters by matching the experimental pattern with calculated pattern based on
the predetermined structure model [38]. By varying the values of refined parameters
during the process, the best fit can be obtained. In this section, I will introduce the
mathematical model of Rietveld method and how it is implemented in order to solve
real problems.

2.2.2

Mathematics

Within the weighted least square framework, Rietveld method is to minimize the
following function based on all the intensities in the experimental dataset [39].
N

My =
i=1

wi (yoi − yci )2

(2.3)

where yoi refers the ith data point in measured dataset and yci is the calculated
intensity for the point based on given structure model. The wi is equal to the inverse
of the variance on the ith data point and N is the number of data points in the
experimental dataset.
Since Rietveld method itself is a structure refinement method rather than structure solution [26], as part of the setup, it requires the structure model that make both

14

physical and chemical sense and a set of reasonable initial values for the parameters
inside the model.
Rietveld method models the intensity at ith point of the diffraction pattern by
the following equation, according to Young’s book [38]:

yci = As

g

Lg Pg |Fg |2 φ(2θi − 2θg ) + ybi

(2.4)

where A is an absorption factor, s is the scale factor. The g represents the Miller
indices, hkl. Lg is a function that contains the factors of Lorentz, polarization and
multiplicity. Pg is the preferred orientation function that depends on g.Fg is the
structure factor as we discuss in the first chapter. φ is the peak profile function that
can be selected from a set of available functions based on the sample and experimental
setup. As the last one, ybi is the intensity of the background at the ith point, which
is often fitted by one of several polynomial functions.
The absorption factor, A, is usually a constant for flat sample in X-ray diffraction.
In this configuration, it depends on the wavelength and the scattering angle. Since the
absorption factor varies for different instrument and sample geometry, many Rietveld
programs provides different functions to calculate it in different setting [40, 41]. For
example, normally a exponential polynomial function is used to model the absorption
factor for cylindrical sample in the Debye-Sherrer geometry, which I have been used
for all the experiments in my PhD study, while there are other nonlinear functions
for other instrumental setup in order to consider the surface effect.
Unlike the absorption factor, the Lorentz and polarization factors are rather
modelled differently for different type of radiation. The polarization factor only applies to X-rays and the Lorentz factor depends on it while for time-of-flight (TOF)
neutrons it depends on the d-spacing and scattering angle [40, 41].
Among all the parameters that are defined above in Eq. 2.4, the peak profile part

15

is the most popular topic that scientists have been discussing [42, 28, 29, 43, 44, 45]
even before Rietveld method was developed since it plays the most important role
in Rietveld refinement. Different peak profile functions are provided to deal with
time-of-flight neutron diffraction, constant wavelength neutron and X-rays diffraction,
respectively [38]. The simplest profile function is Gaussian function although it is
proper only when there are no particle size and inhomogeneous strain effects [43].
Specifically, the Gaussian peak function is a function of peak width and scattering
angle, which is widely used in the model of X-ray and neutron diffraction profiles. In
reality, the peak function is usually asymmetric because of the instrument geometry
and strain inside the sample. Lorentzian function is applied to fit the asymmetric
feature in the peaks. Combining Gaussian and Lorentzian function, Pseudo-Voigt
function is the most useful one among all peak profile functions [45]. It consistently fits
better than pure Gaussian or Lorentzian function for diffraction patterns up to now
although there may be better function for certain sample and instrument setup [43].
After introducing the structure model, now let us focus on the weighting factor
at data point i, wi , in Eq. 2.3. As we mention above, wi is the inverse of the
variance on observed data point i [39, 46]. For ideal case, we assume the observation
is independent of each other and for each point its statistical fluctuation has the
Poisson distribution since it is the counting number on certain position of detector
screen. As in Eq. 2.3, the probability that we have observed intensity yoi at point i
is
P (yoi) = λyoi exp(−λ)/yoi !

(2.5)

where λ is the expected value of the observed intensity at ith point. Furthermore, it
is also the variance of yoi . So to get better weighting factor now, we need to estimate
the λ. On the other hand, Eq. 2.5 can be view as the the probability distribution
of parameter λ given the known intensity yoi . In this case, we can get the unbiased
estimation of λ by calculating its mean, which is yoi +1 if we assume λ is a continuous
16

variable [47]. It is the application of Bayesian inference [48]. However, the estimates
from Bayesian approach heavily depends on the prior distribution. In the literature,
Rainwater et al. [49]use an unbiased estimate of λ based on the assumption of the
prior distribution is uniform over all possible values.It also leads to the same result
as yoi + 1. However, Box et al. [38] argue that the prior distribution as λ−0.5 is more
proper, which results that the estimate of λ is equal to yoi + 0.5. Overall, considering
the yoi is much larger positive number from experimental data, yoi is good enough to
use as the estimate and still widely used in conventional Rietveld softwares [40, 41].
With understanding the means of all parameters in Eq. 2.3, I am going to describe how Rietveld method works as the following. The purpose of the method is to
find the refined values of all variables by matching the calculated pattern with the
experimental pattern. In other words, we need to find the minimum of Eq. 2.3. In
order to do so, we take the derivative of it respect to parameter pk and set it to zero
if we assume the calculated model is a function of parameters pk , k = 1, ..., n:
∂M
= −2
∂pk

N
i=1

wi (yoi − yci )

∂yci
=0
∂pk

(2.6)

Since the model to calculate yci is non-linear, it has no analytic solution. So we can
apply Taylor series expansion about al for parameter pl to approximate it to the first
order:
yci (pl ) = yci (al ) +
l

∂yci
△pl
∂pl

(2.7)

If we assume that
△yi = yoi − yci (al )

(2.8)

Then Eq. 2.6
N

2
i=1

wi (△yi −

l

∂y
∂yci
△pl ) ci = 0
∂pl
∂pk

17

(2.9)

By rearranging the above equation, we get
N

wi
i=1



∂yci 
∂pk

l



∂yci
△pl  =
∂pl

N
i=1

wi △yi

∂yci
∂pk

(2.10)

Here we get the matrix form
x = A−1 b

(2.11)

where the element xl of vector x is equal to △pl . The element Akl of matrix A is equal
to

N w ∂yci ∂yci , and the element b of vector b is equal to
k
i=1 i ∂pk ∂pl

N w △y ∂yci .
i ∂p
i=1 i
k

The matrix A is so-called the Hessian matrix [26, 40, 41]. Given an initial values of
every refined parameters, Hessian matrix can be calculated explicitly. Based on final
equation 2.11, the increment of the parameter value can be obtained and then the
current value of parameter is recalculated. The whole process will run repeatedly until
it doesn’t decrease the residual M any more or it reaches the termination criterion.
This approach is called Gauss-Newton method [50]. Because of the probability of
divergence [51, 52], there is a shift factor, f , to control the absolute value of the
increment as
pm+1 = pm + f △pl
l
l

(2.12)

where m is the iteration number. In this way, it avoids some failed cases by decreasing
the increment systematically. Still if the initial values of parameters are far from the
right ones, it may still come to divergent point.
As one of more efficient ways, Levenberg-Marquardt method is widely used [26,
40, 41]. It has the same procedure as Gaussian-Newton method but instead uses the
shift factor it employs the damping factor based on Eq. 2.10. By using this factor,
it can force the incremental vector to move along the direction of steepest decent.
Although it is implemented in many existing Rietveld programs [53, 54, 40, 41, 55,
56, 57, 58], it can’t ensure that the found result is the global minimum of function

18

M [51, 52]. Usually people always want to plot experimental and calculated patterns
during Rietveld refinement to see whether the problems appear.
In practice, Rietveld refinement needs a lot of human effort to get as better
result as possible. First we want to check whether there are a lot of sharp peaks in
diffraction pattern in order to use the method [38]. Normally Rietveld can’t work
with the pattern with only broad background that can be obtained from amorphous
material. Next an initial structure model that is close to the right model and initial
values for parameters in the model are required . Furthermore, we need to select
proper peak profile and background functions as I mentioned earlier. During the
refinement, we have to consider how to refine so many parameters (in which order
or how to group them). Overall for high throughput parametric study, these become
especially problematic.

19

Chapter 3
Next-generation Rietveld analysis:
SrRietveld
3.1

Introduction

Currently, many computer programs have been developed for Rietveld refinement [40, 41, 55, 56, 57, 58]. However, most software programs of this type require
intensive user inputs and there is also a sharp learning curve for new users. The
high data throughput from new generation diffractometers [59, 60, 61, 62], such as
POWGEN at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory, is more than the conventional Rietveld refinement software can easily handle.
Poor convergence of current programs is another obstacle for accurate determination
of refined parameters and conventional refinement software often diverges, requiring
significant human intervention to find the optimal structure solution. SrRietveld is
designed to improve the user experience in computer aided Rietveld refinement.
The new software makes use of the existing refinement programs GSAS [40] and
FullProf [41], which are already widely used. These programs are used by SrRietveld
as refinement engines and SrRietveld provides an automation layer and a graphical

20

user interface. The new functionalities are implemented in Python packages that can
manipulate and communicate with the refinement engines. The refinement process
can be controlled from SrRietveld and the results can be displayed and analyzed. This
architecture separates the controlling scripts from the refinement engines and also
allows the implementation of other engines in the future. Using SrRietveld, average
users can set up high-throughput refinement quickly and save time in analyzing large
numbers of data-sets. Meanwhile, advanced users are still able to customize the
refinement routines, create their own Python scripts, or even develop new software
applications based on SrRietveld’s functionalities. The SrRietveld codes are open
source and delveloped by a team (in Alphabet order): Chris Farrow, Jiwu Liu, Pavol
Juhas, Peng Tian, Simon Billinge, Wenduo Zhou and Yingrui Shang.

3.2

Design Principles

SrRietveld has been developed as part of the NSF-funded distributed data analysis for neutron scattering measurements (DANSE) project in the Python language [63]
using object oriented programming (OOP) concepts. Python is extensively used in
scientific software development. It is cross-platform and suitable for fast development. Also there are many well designed packages for scientific programming and
visualization, such as Matplotlib [64], NumPy [65] and SciPy [66]. The OOP design
enables SrRietveld to be easily maintained and extended.
The philosophy of the SrRietveld design is to improve user efficiency during a
typical work flow by controlling existing Rietveld refinement engines. For example,
human intensive steps to guide a refinement to a local minimum can be automated,
and multiple (hundreds or thousands) of refinements can be carried out using the
same refinement template. Refinement information and results are stored in a flexible HDF5 data structure [67] allowing easy organization of a large number of refine-

21

ments. SrRietveld also provides powerful tools for visualizing refinements. Fits can
be plotted in real-time during refinement, as can selections of refined parameters plotted against refinement number or some other meta-parameter such as temperature.
Pre-written scripts are provided for automating common tasks, such as refining from
coarse guesses of initial parameters, sequential refinement, etc. It is also possible for
advanced users to develop their own macro scripts and contribute to the development
of SrRietveld. Users are supported through an online community where features can
be requested and bugs reported.

3.3

Implementation

The architecture of SrRietveld is shown in Figure 3.1. SrRietveld consists of
several programmatic units, indicated by boxes in Figure 3.1. The SrRietveld user
interface is a full-featured graphical user interface (GUI) that enables the user to
conveniently configure the refinement routines, modify inputs, and investigate and
analyze the results. The GUI interacts with a control layer that coordinates communication with the refinement engines, PyGSAS and PyFullProf. PyGSAS and
PyFullProf are Python libraries that give programmatic access to GSAS and FullProf. The implementation of PyFullProf, PyGSAS and the control layer ensures that
SrRietveld has consistent behavior even while using different engines, allowing refinements to be readily carried out on the same data using either engine. The HDF5
data structure has been designed to be relatively small in size and quick to load, even
when handling thousands of datasets.
As one of the developers, I lead the development of PyGSAS, which is comparable
with the earlier developed PyFullProf by Wenduo Zhou. As a team, Jiwu Liu, Yingrui
Shang and I have been involving the design and development of SrRietveld control
layer and the GUI. Specifically, Jiwu Liu developed the HDF5 data structure and

22

Figure 3.1: SrRietveld software architecture. The User Interface, SrRietveld, HDF5
Data Structure, PyGSAS, and PyFullprof are python packages in the SrRietveld
project. The GSAS and Fullprof represent the underlying refinement engines.

23

Yingrui Shang is more focus on the GUI while I coded most of the SrRietveld control
layer. SrRietveld incorporates most of the major functionality supported by the
current versions of FullProf and GSAS. However, some features, such as the refinement
of magnetic phases, are not implemented, though this is planned for a later release.
SrRietveld is easy to use. It uses automation processes and intelligent control of
the work flow whereby users are relieved of a large number of repetitive inputs. Tools
are also provided in SrRietveld for extending and interrogating refinement results.
Users are able to carry out a series of Rietveld refinements on one or more data sets
automatically. The program captures errors and attempts to recover automatically
when a refinement diverges, and thus increases the robustness of this sequential refinement process. In addition to the ease-of-use, SrRietveld provides flexibility and
extensibility to advanced users. Since SrRietveld is modular and designed with focus
on extensibility, the default behavior of the software can be redefined. Advanced
user scripts can be written in Python to give new functionalities. Such a script is
demonstrated below, which fits a Debye model for the lattice dynamics to the temperature dependence of the refined thermal parameters. In future releases we will
allow easy incorporation of user-defined scripts into the program for easy sharing.
Scientific programmers can also implement the application programming interfaces
provided in SrRietveld into their own applications so SrRietveld’s functionality can
be incorporated into other programs.
SrRietveld is open source software distributed under the BSD License [68]. It
is free to use, subject to the copyright restrictions and disclaimer, though we ask
that papers published from work done using SrRietveld cite our work, as well as
the paper describing the particular refinement engine used (FullProf or GSAS). More
information can be obtained from the project web pages [69] or by contacting professor
Simon Billinge (sb2896@columbia.edu).
SrRietveld is under active development and we encourage users to post bug re-

24

ports and feature requests on the SrRietveld online community group (details in the
documentation). The plans for future releases include support for parallel computation, magnetic diffraction, and features for engineering diffraction analysis.

3.4

Application of SrRietveld I: NaCl

To demonstrate a typical use case, sequential refinements were carried out on xray powder diffraction patterns measured from NaCl at a series of temperatures. The
data were collected at beamline 11-ID-B of the Advanced Photon Source (APS) at Argonne National Laboratory (ANL). The NaCl powder was obtained from Puratronic R
with 99.999%, 5N grade purity. To demonstrate the ability to handle data from a
high throughput instrument, many data-sets were collected using a low-resolution
high throughput mode [70]. The sample was contained in a 1 mm diameter kapton
tube mounted perpendicular to the synchrotron beam and cooled using an Oxford
cryostream liquid nitrogen cooler. Data were collected at 307 temperature points
between 155 K and 500 K on a 2D Perkin-Elmer amorphous silicon detector mounted
128 mm behind the sample and perpendicular to the beam. The beam energy was
58.26 keV giving an x-ray wavelength of 0.2128 ˚. The temperature was ramped
A
continuously at 3 K/min during data collection.
To improve the convergence during the sequential refinement, the refined results
from the previous temperature point were used as initial values in the refinement on
the data set at the following temperature point. The first point to be refined was
the lowest temperature point. A refinement scheme was used whereby parameters
were turned on in turn according to normal practice [38] beginning with a Lebail
refinement [71] of the lattice parameters and the background, followed by Rietveld
refinements where the scale parameter, zero shift parameters, lattice parameter, etc.,
were switched on in turn. This procedure is automated in SrRietveld and requires no

25

user input after initial setup. Although in principle in a sequential refinement it should
not be necessary to follow this scheme at every step since the initial parameters from
the previous refinement are used and are already close to the minimum in practice we
find that at some temperature points the refinement will still diverge if this scheme is
not followed. However, SrRietveld provides great flexibility. For example, if speed of
refinement is an issue the scheme could be changed so that all parameters are switched
on at once after the first refinement, and the step-by-step scheme only turned on if
a refinement diverges. Because SrRietveld controls the engines with scripts, there is
great flexibility for automating such a procedure.
In the NaCl refinements, the full-set of refinable parameters used was the scale
factor, the zero shift factor, the lattice parameter, the background parameters, the
peak profile parameters, and the isotropic thermal displacement parameters. The
starting model was taken from the literature [72]. Refinements were carried out
on the same data using both GSAS and FullProf engines within SrRietveld. Both
temperature series refinements used the same SrRietveld script, with the refinement
engine changed from FullProf to GSAS by simply loading a different type of of template in the GUI. The ease of refining using both GSAS and FullProf engines is one
of the great strengths of SrRietveld as we show here. Although GSAS and FullProf
models are highly similar, there are subtle differences in how they handle experimental effects. For example, the way they handle the background function is quite
different. In the FullProf example, user defined points are imported and a linear
interpolation is applied between them for background correction. In the GSAS case,
the background function is fitted to the data together with Rietveld profile. We have
used the background function 6, a power series with negative power terms [40]). Also,
the peak profiles are slightly different. The pseudo-Voigt peak profile function was
used in the FullProf case. The GSAS refinements used profile function 3, which is
also a pseudo-Voigt, but parametrized in a slightly different way [40]. These model

26

Figure 3.2: Temperature dependence of SrRietveld refinement results for NaCl using
FullProf (blue dots) and GSAS (red squares) engines. (a) lattice parameter, a, and
(b) atomic displacement parameters, Uiso , at Na (crosses for FullProf, plus markers
for GSAS) and Cl (dots for FullProf, squares for GSAS) sites. Dashed lines mark
fitted Debye model curves.

27

differences result in slight differences in the physical parameters refined from the data
from each engine, as evident in Fig. 3.2.
Another feature of SrRietveld is that the parameters from many refinements are
retained in the Python data-structure, allowing post-processing scripts of arbitrary
complexity to be written. Here this feature is illustrated by writing a script to fit the
Debye model to the refined Debye-Waller factors from Na and Cl from each of the
engines. It is possible for advanced users to make plug-ins for SrRietveld, which can
be saved and reused by the user, and also readily shared with other users.
The theoretical values of Uiso are calculated from [73]
1
Uiso (T ) = 2
8π

6h2
MkB θD

1
+
4

T 2 θD /T xdx
θD
ex − 1
0

+ σo

(3.1)

where the refinable parameters are θD and σo , the Debye temperature and offset,
respectively. Additionally, T is temperature in Kelvin, M is the mass of the ion in
question, kB and h are the Boltzmann and Planck constants, respectively. The fitting
to the refined Uiso data is carried out with the least square optimization algorithm
available in the SciPy Python scientific computing package [66]. The best-fit lines are
shown in Fig. 3.2 as dashed lines through the data.
The differences between the Debye temperatures and offsets refined from each
engine are much larger than the estimated standard deviations. The esds give a
measure of the precision of the fit, and not the accuracy of the Debye temperatures. To
understand this, note that the dashed lines fit well to the Uiso curves, but the slopes
and offsets of the curves are clearly different. The differences in slopes and offsets come
from differences in the mathematical model for the line-shapes and backgrounds used
in FullProf and GSAS, respectively. This illustrates the advantage of being able to
refine data using two refinement engines. If the FullProf and GSAS models are equally
valid and we cannot say one or the other is inferior, then the dispersion of results from

28

the two engines gives a realistic estimate of our uncertainty in the accuracy of the
refined parameters. In this case we may take into account the result from both engines
to get estimate of ΘD,Na = 305 ± 17 K and ΘD,Cl = 257 ± 9 K where the estimated
uncertainties now reflect accuracy rather than precision. The corresponding offset
parameters obtained by combining the Rietveld and GSAS results are 30 ± 50 ˚−1
A

and 20 ± 40 ˚−1 for Na and Cl, respectively. Therefore there is no evidence for a
A
static offset in the Debye model that would indicate a significant number of static
defects in the structure.
In other experiments, the Debye temperature of NaCl can be determined either
by measuring the elastic constant or by fitting the specific heat data. The ΘD values
measured with these two methods are 321.2 ± 1.6 K and 320.6 ± 1.5 K, respectively,
reported by Lewis et. al. [74]. Broadly speaking the Debye temperatures measured
here are in good agreement. Because of the simplicity of the Debye model, and
the different way that different measurements weight the phonon density of states,
the different approaches are not expected to yield the same Debye temperatures.
Furthermore, the thermodynamic measurements give some average of the full density
of states whereas the diffraction measurement differentiates the behavior of Na and
Cl. The agreement between the values determined from diffraction and those obtained
from other methods are therefore quite acceptable.
The refinement engines used in the current release of SrRietveld are FullProf
and GSAS. Along with many others in the community, we would like to thank Juan
Rodriguez-Carvajal and the other FullProf developers and Robert B. Von Dreele,
Brian H. Toby and the other GSAS developers, for their enormous efforts in developing FullProf and GSAS. We would also like to thank the other developers in the
DANSE project for useful advice and help. We appreciate the cooperation and valuable suggestions from Emil Boˇin, Jason Hodges, Ashfia Huq, Ke An, Paolo Radaelli
z
and Laurent Chapon. We thank Timur Dykhne for help with the documentation. In

29

addition to their suggestions, Aziz Daoud-Aladine, Jon Hanson, Vanessa Peterson,
and Andrew Studer shared data for testing which we are grateful for. The authors
also would thank Christos Malliakas for providing the NaCl sample and assistance on
the measurements on this sample. SrRietveld was developed as part of the DANSE
(distributed data analysis of neutron scattering experiments) project, which is funded
by the National Science Foundation under grant DMR-0520547. Use of the Advanced
Photon Source was supported by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences, under Contract No. W-31-109-Eng-38.

3.5

Application of SrRietveld II: Error estimations

In Rietveld refinement [75], a non-linear least squares method is applied to refine
model parameters in order to give a best fit to experimental data. It is also important
to get accurate estimations of the uncertainty on the corresponding parameter values, for example, to differentiate between competing structural models based on the
refined parameters. This section explores three different methods for estimating uncertainties, applied using two different popular refinement programs, GSAS [40] and
FullProf [76, 41]. To facilitate this process we use the SrRietveld program [77] that
automates GSAS and FullProf using Python [63] scripts and a graphical user interface. The bootstrap error estimation methods required many thousands of Rietveld
refinements to be carried out which was straightforward in SrRietveld.
Based on the standard least-square weighting schemes built into most of the
Rietveld refinement programs, the estimated standard deviations (esd) of refined parameters are calculated through the Hessian matrix [75]. Since the Rietveld method
forms a nonlinear multivariate model, it is not possible to get a closed form Hessian matrix and an iterative optimization procedure is used to refine parameters and
calculate their standard deviations.

30

Estimates of uncertainties from Rietveld refinements are widely thought to be
problematic. For example, the model is approximate and includes things such as
peak profile and details of how the background is calculated, which are often not
ideal. It is difficult to tell whether the refinement is converged at the global minimum point, which is a requirement for the estimated uncertainties to reflect the real
uncertainty [78, 27, 79]. Even at a global minimum, if the model is inadequate to
reproduce all the features in the diffraction pattern, the correct standard deviations
cannot be calculated by statistical methods [80].
In this section, in addition to the normal uncertainty estimation coming directly
from the least-squares minimization procedure we use a Monte Carlo resampling
and a subset sampling approach where we use a pseudo random number generator
through SrRietveld [77] to do uncertainty estimation for refined parameters. This
is done on LaMnO3 data from the neutron powder diffractometer, NPDF, at Los
Alamos National Laboratory. In the Monte Carlo resampling method, an ensemble
of equivalent data-sets are created by fluctuating the intensity on each data-point
using a random number generator consistent with the known measurement errors.
In subset sampling a subset of data-points is selected randomly and the fit is made
to that reduced data-set. This is done many times to determine the distribution of
refined parameter values from the many subsets. It works because the number of
independent measured points greatly exceeds the number of refined parameters. This
method has the advantage that it can estimate uncertainties on refined parameters
even when the measurement error is not known.
We find that all three methods yield uncertainty estimates that are comparable
to each other. For all parameters we find that GSAS and FullProf yield the same
refined values. Therefore, model errors coming from subtle differences in how GSAS
and FullProf parameterize peak profiles and backgrounds don’t result in larger errors
than those from the statistical fluctuations in the data. We also carry out a sensitivity

31

analysis of refined values when different statistical weights are used in the least-squares
equation. This is done by setting different, though reasonable, estimated errors on
the data-set before refining the data in FullProf. Applying different weights to the
same data-set resulted in a number of parameters refining values that sometimes
exceed five standard deviations from the values with the correct weights, and so we
note a high sensitivity of refined values to the statistical weights used in the least
squares equation. The use of both GSAS and FullProf within SrRietveld has yielded
a very complete picture of precision and accuracy of all refined parameter values. We
also summarize the subtle differences between how GSAS and FullProf calculate and
report standard deviations and reconcile the two approaches.

3.5.1

Methods

We carried out the refinements on previously published data [81, 82]. The data
are neutron powder diffraction data from LaMnO3 measured at 300 K on the NPDF
diffractometer at the Lujan Center of Los Alamos National Laboratory. The data
were refined using the GSAS [40] and FullProf [76, 41] programs controlled by the
SrRietveld program [77]. SrRietveld is a highly automated program for Rietveld
refinement. It not only can do the conventional refinement by using GSAS and
FullProf as the engines but also can automate the refinement of large numbers of
datasets with minimal human effort.
The original diffraction data are stored in GSAS time-map format [40]. These
files can be read by GSAS but not FullProf. In order to use both engines to refine the
data we therefore must transform the data so that it can be recognized by FullProf. To
convert the data we used functions in rawplot, part of the GSAS package, controlled
by SrRietveld. Rawplot does not propagate the errors on the data and so we estimated
the uncertainties using

Iobs /Ii , where Iobs is the observed intensity and Ii is the

incident spectrum intensity. During the conversion in rawplot the incident spectrum

32

is smoothed, and our approach assumes that after smoothing the incident spectrum
does not contribute error to the normalized intensities, but without having access to
details of how this was done we took this as being a reasonable starting point. We
refer to this as the no-spectrum-error (NSE) method.
The problem with this calculation is that it doesn’t consider the errors on the
incident intensity. On the other hand, GSAS uses a fitted function with a few parameters to describe this spectrum. When GSAS carries out a refinement, the variance and
covariances obtained from the fitting of the spectrum are used by GSAS to propagate
the estimated uncertainty for each point in the incident spectrum and this is then
included in the calculation of the statistical weight used in the least-squares equation
for each time of flight (TOF) channel. However, this is happening during a GSAS fit
but not in Rawplot, which suggests a two step approach is needed for converting timemap data to FullProf compatible data while propagating the errors correctly. First,
run rawplot to get the intensities. Then, carry out a GSAS refinement to generate
the .LST file [40] which includes the weights for each point in the diffraction pattern.
The standard deviation on each point in the .LST file of normalized intensities is the
square-root of the inverse weight. Finally, a data file in (x, y, σ) format is created
using these values. We refer to this as the spectrum-error (SE) method for converting
the time-map data.
The SE method gives more accurate estimates of the standard errors on the data,
but it requires a GSAS refinement to be carried out as part of the steps in generating
a FullProf input file from the Los Alamos time-map data. It would be preferable if
the NSE method works sufficiently well to be used. However, as we see below, this
is not the case. The results of Rietveld refinements are highly sensitive to the values
used as weights in the least-squares equation. Automated versions of the conversion
program for converting GSAS-time-map to FullProf compatible data will be included
in future releases of SrRietveld.

33

Before the refinement, instrumental parameters were calibrated using standard
nickel data. The standard orthorhombic structure [81, 83] was used for LaMnO3 in
the P bnm space-group. The refinable structural parameters were a, c, La’s X and Y
fractional coordinates, O1’s X and Y , O2’s X, Y and Z fractional coordinates and
all isotropic thermal factors. The profile used was time of flight convolution pseudoVoigts with back-to-back exponential function (TOF profile function 3 in GSAS and
Npr= 9 in FullProf) and the background function used was a polynomial function
for both engines. SrRietveld allows parameters to be turned on sequentially to improve convergence. The refinement strategy used was to turn on scale factor, then
background (polynomial function with 6 coefficients), zero shift, lattice parameters,
peak profile parameters (sigma1, sigma2), all allowed atom fractional coordinates and
finally isotropic thermal factors. Examples of Rietveld fits to the data from GSAS
and FullProf are shown in Fig. 3.3.
Standard Uncertainty Estimation in Rietveld refinement
The standard Rietveld method is to calculate the diffraction pattern ycalc,k (pi ),
on a grid of N points that match the data points, from a model with n parameters,
pi , and vary the parameters in such a way as to minimize the properly weighted sum
of squared differences,
N

M=
k=1

wk (yobs,k − ycalc,k )2 ,

(3.2)

between the calculated function ycalc,k and a measured one yobs,k . Here wk is a
weighting factor that depends on the statistical uncertainties on the measured data as
we discuss below. If we find the minimum value of the above function, we find the best
fit. As described in standard textbooks [75, 30], the minimum is found by taking the
first derivative of Eq. 3.2 with respect to the parameters, pi , and setting it to zero. The

34

15

X1

X6

(a)

Intensity (a.u.)

10
5
0
15

(b)
10
5
0
10000

20000

30000

TOF (ms)
Figure 3.3: (a) TOF neutron powder diffraction pattern on LaMnO3 (blue circles)
with the best fit calculated pattern from FullProf (red line). The low d-spacing, short
TOF, region to the left of the dashed line has been zoomed in 6 times for clarity.
The green line offset below is the difference between the calculated and measured
patterns. The black markers indicate the peak positions. (b) As (a) but the red
curve is calculated using GSAS.

35

resulting system of equations that must be solved can be written in the matrix form
∂y

∂y

calc calc . Inverting this
w ∂p
∂pj
i
−1 b. The estimated standard deviation for parameter
matrix equation, we get x = A

as Ax = b where A is the Hessian matrix and Aij =

pi , σi , can be determined from the diagonal elements of the variance-covariance matrix
that is the properly normalized form of A−1 , thus, σi ∝

(A−1 )ii [30].

The standard deviation on refined parameters pi are calculated using the same
formula in both FullProf and GSAS,

σi =

(A−1 )ii χ2 ,
ν

(3.3)

where χ2 = M/(Nobs − Nvar ) and Nobs is the number of observations in all hisν
tograms and Nvar is the number of variables in the least-squares refinement.
As the FullProf manual mentions, it reports two different χ2 values. One is
ν
computed by considering all points in the fit and the other is computed by summing
over only those points that are under Bragg peaks [76, 41]. This latter definition
results in slightly larger χ2 values since it gives greater weight to regions of the
ν
pattern where the fit is less good. For example in our fits this resulted in χ2 P = 1.77
ν,F
compared to 1.63 for the conventional definition that includes all points in the fitted
spectrum. It is this definition of χ2 P that FullProf uses in its estimation of the
ν,F
parameter uncertainties in Eq. 3.3, which increases the estimated errors reported by
FullProf compared to those from GSAS.
Monte Carlo Resampling of Data Probability Distribution
In a powder diffraction experiment, each observation is assumed to come from a
Poisson distribution due to the random nature of the scattering processes. If more
than around 20 counts have been measured at a particular point, the Poisson distribution is well approximated by a Gaussian and least-squares analysis is the appropriate

36

minimization procedure for fitting the diffraction pattern [84]. The Gaussian distribution of counts, d, at the ith point in the diffraction pattern, with mean µi and
standard deviation σi , is given by

pi (d) =

1
σi (2π)1/2

exp

−(d − µi )2
.
2
2σi

(3.4)

In the absence of a model, our best estimate for µi is the measured number of counts.
Given that we know σi by propagating errors from the raw counts [75], we can create duplicate data-sets that are consistent with the measured data-set by choosing a
different value of d from pi (d) using a pseudo-random number generator. This is not
strictly correct since the measured d on the ith point is not equal to µi . However,
when we apply this to a large number of independent points we expect that on average the approximation will hold quite well. An ensemble of 5000 new data-sets was
generated in this way. Each data-set in the ensemble was then refined using SrRietveld. A histogram was then made of the refined values for each parameter yielding
its probability distribution. The mean and standard deviation of the distribution was
then calculated for each parameter. The results are shown in Table 3.1, which show
the probability distributions for representative refined parameters.

The values in

the table for the refined parameters and the standard deviations from the MC resampled data are the values of the mean and standard deviation, respectively, of the
probability distributions. The reported profile weighted agreement factor, Rw , and
χ2 ’s were calculated by determining the diffraction pattern using the mean values
ν
of the parameters from the probability distributions (the ones reported in the table)
and comparing this to the original, non-fluctuated, data-set.

37

Table 3.1: Refined values, estimated standard deviations for LaMnO3 from the GSAS.
Parameter
a (˚)
A
b (˚)
A
c (˚)
A
La

O1

O2

Mn
Rw
χ2
ν

X
Y
Uiso
X
Y
Uiso
X
Y
Z
Uiso
Uiso

(˚2 )
A
(˚2 )
A

(˚2 )
A
(˚2 )
A

Standard
value
5.537938
5.741537
7.697324
-0.007697
0.047808
0.00595
0.073758
0.488184
0.00743
0.725351
0.306210
0.038524
0.00769
0.00422
4.86%
1.65

σH
1.17
1.28
1.76
2.15
1.80
2.1
2.42
2.45
3.2
1.86
1.80
1.27
2.1
3.4

′
σH

0.91
1.00
1.37
1.67
1.40
1.6
1.88
1.91
2.5
1.45
1.40
0.99
1.6
2.7

Monte Carlo
value
5.537974
5.741540
7.697319
-0.007699
0.047801
0.00594
0.073745
0.488180
0.00743
0.725352
0.306211
0.038526
0.00769
0.00422
4.89%
1.67

σM
1.18
1.34
1.76
2.21
1.85
2.1
2.38
2.52
3.2
1.82
1.78
1.27
2.2
3.5

′
σM

0.91
1.04
1.36
1.71
1.43
1.6
1.84
1.95
2.5
1.41
1.38
0.98
1.7
2.7

Subset
value
5.537972
5.741548
7.697400
-0.007665
0.047704
0.00609
0.073741
0.488044
0.00790
0.725243
0.306252
0.038540
0.00764
0.00445
4.43(2)%
1.49(2)

σS
2.24
2.09
2.82
2.64
2.44
2.3
2.88
3.46
3.3
1.98
2.45
1.65
2.5
3.6

′

σS
1.83
1.71
2.30
2.16
1.99
1.9
2.35
2.83
2.7
1.62
2.00
1.35
2.0
2.9

Table 3.2: Difference in the values of refined parameters between FullProf refinements
on data converted from time-map format using the SE and NSE methods, respectively. The numbers in the parethese are the standard deviations on the differences,
calculated as the square-root of the sum of the variances.
Parameter
Standard MonteCarlo
Subset
˚)
a (A
0.0014(2) 0.0014(2)
0.0014(3)
b (˚)
A
0.0014(2) 0.0014(2)
0.0014(3)
c (˚)
A
0.0019(2) 0.0019(2)
0.0019(4)
La
X
0.0001(3)
0.0001(2)
0.0001(4)
Y
0.0003(3)
0.0003(2)
0.0003(3)
2 ) 0.0015(3) 0.0017(2)
Uiso (˚
A
0.0016(4)
O1
X
0.0001(4)
0.0001(3)
0.0001(4)
Y
0.0002(4)
0.0001(3)
0.0001(5)
Uiso (˚2 ) 0.0011(5) 0.0012(4)
A
0.0012(5)
O2
X
-0.0001(3) -0.0001(2)
-0.0001(2)
Y
-0.0001(3) -0.0001(2)
-0.0001(3)
Z
0.0000(2)
0.0000(2)
0.0000(2)
2 ) 0.0011(3) 0.0012(3)
Uiso (˚
A
0.0011(4)
2 ) 0.0010(5) 0.0011(4)
Mn
Uiso (˚
A
0.0010(5)

38

0.2

(a)

Probability distribution

0.1
0

5.5356

5.5368

5.538

0.16

(b)

0.08
0

5.739

5.7405

5.742

0.12

(c)

0.06
0

7.695

7.6975
Lattice Parameter (Å)

Figure 3.4: The probability distribution functions of refined lattice parameters from
FullProf based on data converted using the SE and NSE methods. (a), (b) and
(c) show the distributions of lattice parameters a, b and c, respectively. For each
subfigure, red, green and black symbols are from the NSE method; blue, orange and
brown are from the SE method. Diamonds are from Monte Carlo resampling, squares
are from subset sampling and circles are from the standard Hessian matrix method.

39

Probability distribution

0.06

0.05

0

(c)

(a)

0.1

0.03

0.003

0
0.003

0.006

(b)

0.009

(d)

0.1

0.06

0.05

0.03

0

0.006

0.002

0.004

0.006

0

0.006

-2

0.008

Atomic displacement parameter (Å )
Figure 3.5: The probability distribution functions of thermal factors Uiso . (a) is for
La, (b) is for Mn, (c) is for O1 and (d) is for O2. All colors and symbols have the
same meanings as Fig. 3.4.

40

Subset Sampling
In subset sampling we remove some percentage of the measured data-points before carrying out the refinement. This is done by using a pseudo-random number generator to randomly select which data-points to discard. It is done in such a way that
sufficient data are retained to carry out a stable refinement. In our case we removed
50% of the data-points to create each sub-sampled data-set. We tested discarding
more data-points but it led to unstable refinements. In this way we created 5000
sub-sampled data-sets. We carried out a Rietveld refinement on each sub-sampled
data-set using GSAS and FullProf, respectively, running under SrRietveld. In the
same way as for the Monte Carlo method described above, the probability distribution for each refined parameter is shown in Figs. 3.4 and 3.5 and the means and
standard deviations of the resulting probability distributions are listed in Tables 3.1.
In this case we report in the table the average Rw and χ2 values from 5000 refineν
ments. We also report the standard deviations of the distributions, which are sharply
peaked around the mean value.

3.5.2

Results and Discussion

We first compare the three different methods for assessing uncertainties on the
refined parameters by considering the results of the GSAS refinements in Table 3.1.
No bias is introduced into the refined values themselves by using the resampling methods as evidenced by the fact that the refined values are the same for each method
within the standard deviations. None of the average values are more than one standard deviation from the values refined using a different method. Furthermore, we see
excellent agreement between the unweighted (primed) standard deviations obtained
from the Hessian matrix and the MC resampling method. The computationally efficient use of the variance-covariance matrix gives the same error estimates as the
′

computationally expensive MC resampling method. The unweighted σ s from the
41

sub-sampling method are significantly higher. This is expected because only half of
the data has been used in each of these refinements, so we expect that the uncertainty
√
on our results will be higher by roughly 2 since half the data have been removed (it
√
won’t be an exactly 2 change since not exactly half the counts have been removed
√
′
′
in the sub-sampling). The average value of σS /σH = 1.38, which is close to 2 as
expected.
The subset sampling method gives good unbiased estimates of the uncertainties
but at the expense of yielding lower precision estimates of the refined values themselves. The information content in the data is being used to estimate each number
(the mean and standard deviation) independently whereas in the Hessian and MC
resampling we are using additional information (the counts and the uncertainties on
the counts) to estimate the refined parameters and their standard deviations. If the
data uncertainties are known accurately then the standard Hessian matrix approach
is best. In principal, the sub-set sampling method is preferable in the case where the
data uncertainties are not well known. However, as we discuss later, incorrect weights
in the least squares equation result in biased estimates of the refined parameters and
in practice it is important to have good uncertainty estimates on the data, as well as
good data.
The same overall behavior is seen when we consider the FullProf refinements.
There is excellent agreement between the unweighted esd’s of the Hessian Matrix
and MC resampling methods, with enlarged values on the subset sampled esd’s. Of
greater interest is to compare the GSAS and FullProf refinement results to check for
consistency by comparing the corresponding values in Table 3.1. We see that there is
excellent agreement in all the values between the tables, including the refined values
and their esds. Despite using subtly different parameterizations for backgrounds and
peak parameters, the two refinement methods give excellent agreement with each
other.

42

Here we show the results of refinements to the data when reasonable, but incorrect, weights have been applied in the least-squares equation. In all other respects
the refinements are identical to those. The esd’s on the refined values from both the
Hessian and MC methods are smaller than the those on GSAS and FullProf refinements on SE converted data. This is because the esd’s on the data points are smaller
in the NSE converted data, where no error was propagated from the spectrum normalization. In this case the Hessian and MC methods under-estimate the true esd’s
on the refined parameters. However, interestingly the sub-sampling method returns
esd’s on the refined parameters that are in good agreement with those obtained from
from the GSAS and FullProf refinements on the SE converted data. This can be
′

seen by comparing the unweighted σS ’s with those in Tables 3.1. This is because the
sub-sampling method uses the refinements to sample the real fluctuations in the data
and does not use the propagated esd values on the data in its estimates of refined
parameter esd’s (although it does use them as weights in the least-squares equation). The subset sampling method can be relied upon to report unbiased estimates
of uncertainties in the data even when the data uncertainties have not been properly
propagated, as before at the expense of yielding less precise estimates of the refined
values themselves. However, we show below that the sub-sampling method does not
prevent biased estimates of the refined values being obtained if the data uncertainties
are incorrectly propagated.
The values that are refined using FullProf for the SE and NSE converted data can
be compared. For a number of parameters the refined values are significantly different
(by more than 5 standard deviations) even though the refinements were done with
the same model on the same data. The only difference was the values of the weights
used in the least squares equation. This is highlighted in Table 3.2 which shows these
differences, and the esd’s on the differences, for a number of structural parameters.
This effect can also be seen in Figs. 3.4 and 3.5 where the probability distributions

43

for refinements to the NSE and SE converted data are clearly offset.
This result shows a clear sensitivity of refined parameter values to using the
correct weights in the least-squares equation. Not only is it essential to have a good
model, but also to have accurate uncertainty estimates on the data, in order to refine
unbiased values for parameters. An interesting question is whether it is possible to test
for the situation that the esd’s quoted in a data file are accurate or not. This is possible
in principle by comparing the uncertainties obtained from the Hessian and sub-set
sampling methods. As we showed above, the ratio of the uncertainties estimated using
these two methods should be roughly

1/R where R = Nsubsampled /Ntotal . If this

criterion is significantly violated it may indicate the data esd’s are not accurate.

3.5.3

Conclusion

We have compared three ways of estimating uncertainties on refined parameters
in a Rietveld refinement using two different Rietveld programs, GSAS and FullProf:
the estimated standard deviations obtained from the programs themselves, and by
resampling the data in two different ways using a pseudo random number generator.
There is no apparent advantage of using the computationally expensive Monte Carlo
resampling method with respect to using the standard deviations obtained directly
from the variance-covarience matrix. However, in certain circumstances the use of the
sub-sampling method is warranted. These uncertainty estimation methods will be included as features in future versions of the SrRietveld Rietveld refinement automation
program.
We have demonstrated that there is a significant sensitivity of refined parameters
to the use of correct data-esd’s as weights in the least squares equation. Significantly
biased refined values result when incorrect weights are used even when the same model
is fit to the same data-set. This underscores the importance in Rietveld refinements
of having accurate data-esd’s propagated from the raw data or estimated in some

44

other way. The use of the sub-sampling method yields accurate estimates of standard
deviations on refined parameters even when the data-esd’s are incorrect or unknown.
However, it does not correct for the biased values of the refined parameters themselves
that result in this case. Comparing the esd’s from the sub-sampling method to the
values obtained from the variance-covariance matrix may be a useful diagnostic, indicating situations when there may be a problem with the reported data-esd’s. In this
case, the accuracy of the refined values is certainly lower than the precision which
should be reported. This is generally true because of model errors and parameter
correlations that are not included in these, but we note an additional contribution
due when data uncertainties are uncertain.

45

Chapter 4
The total scattering and atomic
pair distribution function method
Due to the emerging demand for the alternative energy and electronic applications, the more complex materials are synthesized and characterized. They are made
of multiple elements, have large unit cells and often low dimensional or incommensurate structures [30, 32]. Commonly, they have aperiodic disorder: some aspect of the
structure that is different from the average crystal structure. In the case of nanoparticles, the concept of crystal is invalid as the approximation of infinite periodicity is
no longer a good one. It is often critical important to know their structure details
in order to explain their special properties. Nevertheless, powder diffraction is an
important method for characterizing these materials, but we have to go beyond the
Bragg equation and crystallographic analysis.
In this chapter, we present an alternative approach which treats both the Bragg
and diffuse scattering on an equal basis, the so-called total scattering technique.
Data from throughout reciprocal space, over a wide range of Q-values, are utilized.
The technique is both straightforward and intuitively easy to comprehend. Fourier
transformation helps us translate the reciprocal space information into real space. For

46

total scattering, Fourier analysis of the data is known as pair distribution function
(PDF) analysis.
This chapter will give the background knowledge of PDF and the physical meaning of it. Finally I will introduce the interesting cases for calculating the PDFs that
will be used later in the thesis.

4.1

Theory of PDF

Recently, the work of Chris Farrow in Prof. Simon Billinge’s group resolves a
long-standing ambiguity in the PDF literature by deriving the PDF equations from
the beginning. Since the full derivation is not reproduced even in many textbooks
on the subject and it leads the precise relationship between the measured correlation
function in an X-ray or neutron total-scattering experiment and the underlying model.
So this section will briefly go through the procedure as the following to give the clear
ideas about the PDF and its application and for more details please refer to Chris’s
paper [85].
From classical wave theory [12], the scattering amplitude from a set of i atoms
at points ri is:
ψ(Q) =

ψi
i

fi (Q)eiQ·ri .

=

(4.1)

i

Assume there is no inter-atom contribution on the intensity, the incoherent intensity

47

is calculated as the following:
∗
ψi ψi

Iinc =
i

∗
fi (Q)fi (Q)

=

(4.2)

i

= N f2 ,
where ∗ means the complex conjugate. Also we can define the square of the average
scattering power as
1
f 2= 2
N

∗
fj fi .

(4.3)

ij

On the other hand, the coherent scattering intensity is given by
iQ·(ri −rj )

∗
fj fi e

Ic =
i

j
iQ·rij

∗
fi fi +

=
i

∗
fj fi e

(4.4)

i=j
iQ·rij

= N f2 +

∗
fj fi e
i=j

From the classical book [32] about PDF, the total scattering structure function
is defined as

S(Q) =

Ic
(f − f )2
−
.
N f 2
f 2

(4.5)

Based on Eq. 4.4, we get
S(Q) − 1 =

1
N f 2

48

iQ·rij

∗
fj fi e
i=j

.

(4.6)

For a powder sample, since there is equal probability for all direction, thus we
have
iQ·rij

e

iQrij cos θ 2
2π
π
rij sin θ
0 dφ 0 dθe
=
2π
π
2
0 dφ 0 dθrij sin θ
iQr cos θ π
2
−2πrij e ij
0
=
2
4πrij iQrij
iQrij
−iQrij

e

−e

=

=

(4.7)

2iQrij
sin(Qrij )
Qrij

.

Therefore, based on Eq. 4.6, the reduced total scattering structure function, F (Q) =
Q[S(Q) − 1], is
F (Q) =

1
N f 2

∗
fj fi
i=j

sin(Qrij )
rij

.

(4.8)

Since we have the following relationship according to Fourier transformation on
F (Q)

f (r) =

2 ∞
F (Q) sin(Qr)dQ.
π 0

(4.9)

This is the common definition of the PDF. Substituting F (Q) by Eq. 4.8 we get
2 ∞ 1
∗ sin(Qrij ) sin(Qr)dQ
fj fi
π 0 N f 2
rij
i=j
∗
fj fi ∞
2
=
sin(Qrij ) sin(Qr)dQ
rij 0
πN f 2

f (r) =

i=j

1
=
N f 2

∗
fj fi
i=j

rij

[δ(r − rij ) − δ(r + rij )]

49

(4.10)

Assume we have only positive axis, then it becomes
f (r) =

1
rN f 2

i=j

∗
fj fi δ(r − rij ).

(4.11)

According to Chris’s paper [85], we can interpret f (r) by the radial distribution
function (RDF), R(r). It means the number of atoms in the shell at a distance r when
the thickness of shell is one. So R(r) = 4πr 2 ρ(r). On the other hand, by integrating
f (r)r along r direction, we also can get the number of atoms as shown in the paper.
Therefore, we have
f (r) =

R(r)
r

(4.12)

= 4πrρ(r).
In our experiments, we can only measure Q down to certain value, Qmin due to
the experimental setup. So we should consider f (r; Qmin ) function as the following:
2 ∞
F (Q) sin(Qr)dQ
π Q
min
2 Qmin
= 4πrρ(r) −
F (Q) sin(Qr)dQ.
π 0

f (r; Qmin ) =

4.2

(4.13)

Corrections of PDF derivation

First we consider the simplest case: RDF of an infinite scatterer with uniform
density, ρ0 . In this system the intensity will be a delta function at Q = 0, which results
f (r) = 0 based on Eqs. 4.6 and 4.9. This shows the conflict with the assumption as
pointed out by the paper [85]. As mentioned in chapter 1, the scattering amplitude
can be expressed as ψ = f eiQ·r ρ0 dr. In this simple system, the intensity is given
by
′
eiQ·(r−r ) drdr ′ ,

Ic (Q) = ρ2 f 2
0

50

(4.14)

where the integrals are over all space.
Considering the shape of the scatter, we define a shape function s(r) such that
inside the shape s = 1 and outside the shape s = 0. Then,
′
s(r)s(r ′ )eiQ·(r−r ) drdr ′ .

Ic (Q) = ρ2 f 2
0

(4.15)

We redefine variables so that r ′′ = r − r ′ , and dr ′′ = dr, in which case we have
dr ′′ eiQ·r

Ic (Q) = ρ2 f 2
0

′′

s(r ′ )s(r ′ + r ′′ )dr ′ .

(4.16)

If we define
γ0 (r) =
where V =

1
V

s(r ′ )s(r ′ + r)dr ′ ,

(4.17)

s(r)dr is the volume. This γ0 (r) is the characteristic function of the

shape [86], and is also called the nanoparticle form factor in the PDF literatures [82,
87, 88]. With the above definition and Eq. 4.16, we have
Ic (Q) = ρ2 f 2 V
0

γ0 (r)eiQ·r dr.

Substituting the above equation into S(Q) − 1 =

S(Q) − 1 =

1
ρ2 f 2V
2 0
N f

= ρ0

The second term,

(4.18)

Ic − f 2 , we get
N f 2
f 2

γ0 (r)eiQ·r dr −

f2
γ0 (r)eiQ·r dr −
.
2

f2
f 2

(4.19)

f

f2
, is very small compared to the first term and can be ignored
f 2

normally.
Since we assume it is isotropic scatter, we can use the result from Eq. 4.7 as the

51

following

S(Q) − 1 = ρ0
= ρ0

∞
0
∞
0
∞

2π

dr
dr

0
2π
0
2π

π

dφ
dφ

0
π
0
π

dθγ0 (r)eiQ·r r 2 sin θ
dθγ0 (r)eiQ·r r 2 sin θ

sin(Qr) 2
dr
dφ
dθγ0 (r)
= ρ0
r sin θ
Qr
0
0
0
∞
sin(Qr)
γ0 (r)
= ρ0
4πr 2 dr.
Qr
0

(4.20)

This gives

F (Q) =

∞
0

4πρ0 rγ0 (r) sin(Qr)dr.

(4.21)

Then we take the inverse Fourier transform to get
fu (r) =

2 ∞
F (Q) sin(Qr)dQ
π 0

(4.22)

= 4πρ0 rγ0 (r),
where the subscript u indicates that this result is based on the assumption that it is
an uniform density system.
As Chris’s paper [85] shows, for a macroscopic crystal, in the region of Q below
the first Bragg peak, the distinct scattering is zero except at very low-Q where small
angle scattering region is reached. The small angle scattering intensity, Isas from the
sas
sas
crystal is identical to that from the solid with uniform density: Iu = Icrystal . The

small and wide angle scattering regions are well separated in Q and Isas decays to
zero before Qmin is reached in the crystal. Thus,
fsas (r) =

2 Qmin
F (Q) sin(Qr)dQ
π 0

= fu (r)

52

(4.23)

and therefore
2 Qmin
F (Q) sin(Qr)dQ = 4πρ0 rγ0 (r).
π 0

(4.24)

Finally, by substituting Eq. 4.24 into Eq. 4.13, we get

f (r; Qmin ) = 4πrρ(r) − 4πrρ0 γ0 (r).

4.3
4.3.1

(4.25)

Cases of PDF calculation
Calculation of PDF for bulk crystals

In the case of bulk crystals, the region of interest in the PDF is usually r ≪ D,
where D is the smallest dimension of the crystal. In this region, γ0 (r) ≈ 1. Thus,
f (r; Qmin ) = G(r)

(4.26)

= 4πr(ρbulk (r) − ρ0 ),
The pair density function, ρbulk (r), is calculated from a model with periodic boundary
conditions [89, 90], or from a box of atoms that is much larger in extent than the
range of r of interest [91]. The average number density ρ0 is given by the number of
atoms per unit volume, which in the case of crystals is the number of atoms in the
unit cell divided by the unit cell volume.

4.3.2

Calculation of PDF for nanoparticles modeled as attenuated bulk crystals

The pair density, ρ(r), in Eq. 4.25 is the function for the nanoparticle, which
can be approximated as γ0 (r)ρbulk (r) [92]. In this case ρbulk (r) is determined in the
53

same way as in the above section. Thus,

(4.27)

f (r; Qmin ) = 4πrγ0 (r)(ρbulk (r) − ρ0 ).

If there is a distribution of nanoparticle sizes and shapes, the characteristic function, γ0 (r) can be replaced with an appropriately averaged characteristic function
γ(r) =

γ0 (r; R1 , R2 , . . .)p(R1 , R2 , . . .)dR1 dR2 . . . .

(4.28)

Here, p(R1 , R2 , . . .) is the normalized distribution of nanoparticle shapes parameterized by R1 , R2 , . . .. For example, for spherical nanoparticles of radius R, p(R1 , R2 , . . .) =
p(R), the distribution of nanoparticle radii. Finally, we replace Eq. 4.27 with

(4.29)

f (r; Qmin ) = 4πrγ(r)(ρbulk (r) − ρ0 ).

4.3.3

Calculation of PDF from the normalized Debye Function

For this case, the F (Q) function is evaluated using Eq. 4.8 and then Fourier
transformed to obtain the desired real-space function. To account for thermal and
zero-point motion in reciprocal-space calculations, Eq. 4.8 is replaced using DebyeWaller effects,

F (Q) =

1
N f 2

∗
fj fi
i=j

1 2

− σ Q
e 2 ij

2

sin(Qrij )
rij

.

(4.30)

2
Here, σij is the correlated broadening factor for the atom pair [90, 93, 94]. There are

more details and other cases presented in Chris’s paper [85] that is a good guide on
this material.

54

Chapter 5
MnAs: Diverse structural and
magnetic properties
Discrete nanoparticles of MnAs with distinct magnetostructural properties have
been prepared by small modifications of solution-phase arrested precipitation reactions. Rietveld and X-ray atomic pair distribution function based approaches were
used to explore the evolution of the structure of the samples with temperature and
these data were compared to the magnetic response measured with AC susceptibility.
Relative to a bulk standard, one type of MnAs nanoparticles was found to demonstrate similar but smaller structural transitions and corresponding magnetic changes.
However, both magnetic and structural transitions in the second type of nanoparticles
are strongly suppressed.

5.1

Introduction

MnAs has been noted as an attractive candidate for information storage and energy applications due to its interesting magnetic and structural properties. [95, 96, 97]
Bulk MnAs has a magneto-structural phase transition in which a first-order magnetic
transition occurs from a high-spin ferromagnetic to a low-spin paramagnetic state
55

with a concommitant structural transition from the hexagonal α phase (NiAs-type)
to the orthorhombic β phase (MnP-type) [98]. The transition occurs at 315 K upon
warming. At a higher temperature 400 K, a reentrant second-order phase transition
takes place back to the α phase, but the high-spin state is now paramagnetic. This
sequence of transitions is shown in the following.
In studies on bulk MnAs, scientists observed that the structural transition can
be driven by the application of a magnetic field [95], which confirms that the coupling
between the magnetic transition and structural transition is strong. These properties
result in a tunable magnetocaloric effect in bulk MnAs which could be applied in
microelectronic circuit [99, 100] applications. A number of theoretical studies [101, 98]
have discussed the correlation between structure and magnetism though there is no
clear understanding of the details of the relationship.
Compared to bulk materials, nanoscale materials exhibit size and shape tunable
physical properties such as electronic, magnetic and catalytic properties. Research
on nanoscale MnAs has been largely focused on epitaxially grown MnAs particles or
disks on semiconducting thin films [102] or nanowires [103]. Such nanoscale MnAs is
a promising material for thermally assisted magnetic recording [102]. However, these
epitaxial structures are subject to external pressure due to lattice strain from the substrate, resulting in significant shifts in the magnetostructural transition temperatures.
Recently, a method for preparing MnAs as discrete unsupported nanoparticles [104]
enables the effect of size on the first order transition to be probed in the absence of
external stress. Particles of size 9-23 nm were prepared in solution by reaction of
dimanganesedecacarbonyl (Mn2 (CO)10 ) and triphenylarsine oxide ((C6 H5 )3 As=O)
in coordinating solvents at temperatures ranging from 523-603 K. Intriguingly, small
changes in the synthetic method led to two distinctly different products, as detected
by powder X-ray diffraction conducted at room temperature. Notably, in one case, the
metastable β structure, rather than the thermodynamic α structure, was observed.

56

Despite the presence of a magnetic transition consistent with the magnetostructural
transformation in bulk MnAs, no evidence was observed of a corresponding structural transition in temperature dependent powder X-ray diffraction measurements
for either material [104].
Here we perform a detailed structural evaluation of the two types of MnAs
nanoparticles using an advanced nanostructure determination method, the atomic
pair distribution function (PDF) technique [32, 105, 106, 88], as a function of temperature in the region of the expected magnetostructural transition. The obsesrvations from these data are correlated with AC magnetic susceptibility data acquired
over the same temperature range. We show that nanoparticles synthesized using one
approach behave like the bulk, albeit at a reduced level. Nanoparticles synthesized
by the other method appear to be kinetically trapped and do not undergo a magnetostructural transition. This is a dramatic example of the modification of a material’s
structural and magnetic properties at the nanoscale.

5.2

Methods

5.2.1

Synthesis

The bulk MnAs we used for this study was purchased from Pfaltz and Bauer
Chemicals. A chemical etching process was performed to eliminate impurities. Briefly,
1 g of ground bulk MnAs was combined with 20 ml concentrated HCl solution, slowly
heated to 80 ◦C, and kept at that temperature for 20 minutes. The sample was then
washed with 50 ml de-ionized water several times to remove the soluble impurities.
The synthesis of MnAs nanoparticles is done by our collaborators, Prof. Stephanie
Brock’s group at Wayne State University. It can be achieved by either a slow heating method, or a high temperature fast injection method. We call these approaches
method A and method B, respectively. The resulting nanoparticles are called type-A

57

and type-B. In method A, 0.256 mmol of Mn2 (CO)10 is combined with 0.528 mmol
Ph3 As=O, 8.0 - 10.0 ml of 1-octadecene (ODE) and 5 g of trioctylphosphine oxide
(TOPO) in a Schlenk flask under argon. The mixture is gradually heated up to
250 ◦ C over 2 hours, changing from yellow to orange to black, and then maintained
at this temperature for 18 h. The as-prepared nanoparticles are then isolated by
centrifugation after dispersing in chloroform and precipitating in absolute ethanol.
This dispersion/reprecipitation process is repeated several times to ensure complete
removal of TOPO and reaction byproducts.
For method B, the fast-injection method, the same amount of Mn2 (CO)10 ,
Ph3 As=O and 1-octadecene are combined together in a Schlenk flask under argon.
The mixture is heated slightly using a heatgun until the powder precursors are dissolved. This mixture is then cannulated under inert conditions into hot TOPO (5.0 g)
maintained at 330 ◦ C in a second Schlenk flask. The reaction is maintained at 330 ◦ C
for 18 hours. The nanoparticles are isolated using the procedure described for type-A
MnAs.
Particle size and morphology were assessed using transmission electron microscopy
studies performed on a JEOL 2010 HRTEM operating at 200 kV. Samples were prepared by dispersion of solid precipitates in chloroform and then a drop of the colloidal
solution was deposited on a carbon-coated copper grid and allowed to air-dry.

5.2.2

Diffraction experiments

Total scattering powder diffraction experiments were performed at the 11-ID-B
beam-line at the Advanced Photon Source at Argonne National Laboratory, with
58.26 keV x-rays using the rapid acquisition (RaPDF) mode [70]. A large area 2D
GE Revolution 41RT flat panel detector [107] was mounted orthogonal to the beam
path. Each sample was packed in a kapton capillary 1 mm in diameter and measured
in the temperature range of 295 K - 335 K during heating and cooling. In the heating

58

(cooling) procedure we continuously increase (decrease) the temperature at 1 K/min
rate and collect data frames every 30 s, i.e., every 0.5 K. In the measured range of
295-335 K, this results in ∼ 80 data sets for both the heating and cooling runs. The
raw 2D data were integrated and converted to intensity versus 2θ using the software
Fit2D [108], where 2θ is the diffraction angle.
Rietveld refinements were carried out on the data using the GSAS [40] and FullProf [41] programs controlled by the SrRietveld program [77]. SrRietveld is a highly
automated program for Rietveld refinement. It not only can do the conventional
refinement by using GSAS and FullProf as the engines but also can automate the
refinement of large numbers of datasets with minimal human effort. As the first step
of the Rietveld refinement, instrument parameters were calibrated using standard silicon data. There are two structural models for the MnAs samples: the α phase model
in the P 63mc spacegroup and the β phase model in the P nma spacegroup. The refinable structural parameters were a, b and c, Mn’s X and Z fractional coordinates,
As’s X and Z fractional coordinates and isotropic thermal factors on each symmetryindependent atomic site. The peak profile used was the Thompson-Cox-Hastings
pseudo-Voigt function (CW profile function 2 in GSAS and Npr=7 in FullProf). The
background function used was a Chebyschev polynomial in GSAS and linear interpolation in FullProf. The refinement strategy used was to turn on scale factor, then zero
shift, background coefficients, lattice parameters, peak profile parameters, all allowed
atom fractional coordinates and finally, isotropic thermal factors.
The PDF, G(r), was obtained by Fourier transformation of the reduced total
scattering structure F (Q) = Q(S(Q) − 1) according to the following equation [32].
G (r) =

2 Qmax
Q[S(Q) − 1] sin Qr dQ,
π Qmin

(5.1)

where S(Q) is obtained from a diffraction experiment. The maximum range of Q

59

used in the Fourier transform was Qmax = 23.0 ˚−1 . The value of Qmax chosen
A
to optimize the PDF was selected by choosing the largest Qmax possible without
introducing too much statistical noise into the data. The PDFs were modelled using
PDFgui [109]. The structural models and refinable parameters are the same as Rietveld refinement discussed above, but the PDF refinements yield information about
the local structure rather than the average crystal structure probed by Rietveld refinement. Because of the relatively large size [88] of the nanoparticles in this study,
the results from average and local structural refinements matched each other well for
all the MnAs samples.

5.2.3

Magnetization measurements

The magnetic measurement of MnAs is done by our collaborators, Prof. Gavin
Lawes’s group at Wayne State University. AC magnetic susceptibility measurements,
which are more sensitive to small changes in magnetization arising from possible impurity phases than dc measurements, were carried out at 10 kHz using a standard
option on a Quantum Design Physical Properties Measurement System. The magnetic susceptibility was measured on heating and cooling from 280 K to 340 K at an
AC excitation field of 10 Oe with zero DC field. Bulk, type-A, and type-B MnAs
nanoparticles were prepared by sealing loose powders in evacuated quartz tubes. The
data were normalized to total Mn quantity based on chemical analysis data acquired
by atomic absorption spectroscopy using a Perkin-Elmer Analyst 700 instrument.
Solid samples were dissolved in nitric acid, diluted, and compared to a calibration
curve created using a series of Mn standards (High-Purity Standards).

60

5.3

Results and Discussion

Discrete nanoparticles of MnAs were prepared from arrested precipitation reactions using either a slow-heating method (method A, yielding type-A particles)
or a high-temperature rapid-injection method (method B, yielding type-B particles) [104]. Electron micrographs of the particles are shown in Fig. 5.1, along with
a size-distribution histogram. Type-A nanoparticles are well-dispersed and faceted
in shape with a size of 22.3 ± 3.9 nm, measured along the small axis. In contrast,
type-B nanoparticles show a strong tendency to aggregate and are slightly smaller,
at 17.8 ± 3.1 nm. The aggregation of Type-B MnAs particles revealed in the TEM
image is not a universal phenomenon for nanoparticles of Type-B MnAs [104] and
may be a consequence of surfactant loss due to vigorous washing.
First, I used the Rietveld analysis using α model on bulk MnAs. To compare
the quality of the refinements, I plot the zoomed-in figures for both temperatures. It
is clearly shown that the 335 K fit is much worse than 295 K at least in the range of
8-16o as we know from Rw . It implies that bulk MnAs stays at α phase at 295 K but
not at 335 K.
In order to know what the transition is for bulk MnAs during the temperature of
295 to 335 K, I do the Rietveld sequential refinements using α model on all datasets
of bulk MnAs that we collected. The obvious result is that there is structural phase
transition in the temperature range based on the hysteresis shown in the figure. The
transition starts at 303 K and ends at 317 K for heating run, which matches the
observations in the literature [98]. As we know from Fig. 5.1, the type-A MnAs
nanoparticles have relative large particle size so we can still use Rietveld refinement
to analyze them. Similarly, we see the transition also happens in type-A nanoparticles
for the same temperature range. For the sake of brevity, I only present the results of
lattice parameter a together with the ones from bulk. Interestingly, we see the change
in type-A is smaller than bulk and dies out broader. However when I want to analyze
61

Table 5.1: The refined structural parameters for bulk and type-A nanoparticles at
295 K and 335 K using α (P 63 mc) and β (P nma) model. All models are refined with
isotropic thermal factor Uiso .
Parameter
bulk
Type-A nanoparticles
295 K α
335 K β
295 K α
335 K β
˚)
a (A
3.724(2)
5.734(3)
3.686(5)
5.728(4)
b (˚)
A
3.724(2)
3.684(3)
3.686(5)
3.686(5)
˚)
5.712(5)
6.378(4)
5.731(13)
6.379(9)
c (A
Mn
X
0.0021(2)
0.0063(5)
Z
-0.00014(9)
0.271(7)
0.009(2)
0.27(2)
2)
Uiso (˚
A
0.014(3)
0.015(5)
0.021(12)
0.019(9)
As
X
0.231(7)
0.23(2)
0.25(2)
0.92(9)
0.24(3)
0.9(2)
Z
2)
Uiso (˚
A
0.009(2)
0.009(3)
0.013(9)
0.009(7)
Rw
0.096
0.084
0.203
0.189
the type-B nanoparticles, Rietveld refinement doesn’t work well since there are less
sharp peaks in the pattern. Therefore, we turn to pair distribution function (PDF)
for help.
For comparison purpose later on, I repeat the analysis on bulk and type-A MnAs
as the first steps here by using PDF technique. Examples of typical total scattering
powder diffraction patterns in the form of PDFs are shown in Fig. 5.2. We first
verify that our results for the structure of the bulk material are in agreement with the
literature [98], i.e., just below room temperature the samples are in the α phase and at
just above room temperature in the β phase. In Fig. 5.2(a), we see a good agreement
between the best-fit α model and the bulk MnAs data at 295 K, as additionally
indicated by the low Rw = 0.096. Similarly, there is a good fit of the β model to the
bulk data at 335 K as shown in Fig. 5.2(c) and by an Rw = 0.084. For comparison,
the best fit of the α model to the data at 335 K gives Rw = 0.196 and is shown
in Fig. 5.2(b). The poor fit shows that the PDF is clearly capable of differentiating
between these phases. The refined values of the good fits are presented in Tab. 5.1.
Our PDF refinements of the bulk are in excellent agreement with crystallographic
results from the literature. [98]
62

We next consider type-A nanoparticles. In Fig. 5.2(d)-(f) we show the fits of
the bulk structural models to the PDFs of the type-A nanoparticles. The fits are
good overall indicating that these nanoparticles have similar structures to the bulk
material. For the 335 K data (Fig. 5.2(e), (f)) the fit of the β structural model is
significantly better than the α model, as in the bulk. The Rw = 0.189 for the β
model and Rw = 0.217 for the α model and the improved fit is evident by comparing
the difference curves of (e) and (f) in the figure. However, the overall fit of the β
model to the 335 K data is less good than for the bulk material. This is evident
from the larger Rw (0.189 vs. 0.084) and the clear features in the difference curves.
This shows that there are structural modifications from the purely β structure in the
nanoparticles, possibly from the surface region.
Fig. 5.2(d) shows the fit of the α model to the type-A nanoparticle data collected
at 295 K. There is a good overall fit, but the Rw = 0.203 is somewhat high, and there
is some residual in the difference curve in the Figure. However, the β model also fits
the 295 K data quite well (Rw = 0.187). This could be because the structure is still
in the β-form at this temperature and the sample has not transformed. However,
another possibility is that the structure has transformed to the α-phase but the β
structural model fits better because it has more refinable prameters (12 for β vs. 9
for α). This is possible because the α phase is similar to β and can be obtained in a
continuous manner from β by adding some symmetry elements and moving atoms onto
special positions. Thus, the observation of a lower Rw is not sufficient to establish
that the material is in the α phase and we need to look further. In order to establish
definitively whether a structural transition is taking place in the type-A nanoparticles
similar to that in MnAs bulk, we consider the change in the PDF of the material as
its temperature is reduced from 335 to 295 K, shown in Fig. 5.4.

The difference

curve in Fig. 5.4(a) shows the changes in the PDF of bulk MnAs due to the structural
transition and Fig. 5.4(b) shows the same for the type-A nanoparticles. It is clear

63

that the structural changes in the nanoparticles are smaller than those in the bulk.
However, close inspection of the difference curves in Fig. 5.4(a) and (b) shows that
they are highly correlated and have similar features. To test the degree of correlation
we plot the two difference curves on top of each other in Fig. 5.4(c), multiplying the
curve from the nanoparticles by a factor of 4 to account for the smaller amplitude of
those changes. Based on the data in Fig. 5.4(c), we calculate the Pearson correlation
coefficient [110, 111] to show the degree of correlation between two difference curves.
The value of the correlation can fall in the range from 1 (fully correlated) to -1
(anticorrelated), with the middle value zero meaning there is no correlation. For our
case, the Pearson correlation coefficient between the two difference curves is 0.959,
which clearly shows that the structural changes between 335 K and 295 K of the
bulk and type-A nanoparticles are the same. Thus, the type-A nanoparticles are
undergoing the same structural phase transition as the bulk, although the amplitude
of the changes is less.
Having established that the type-A nanoparticles are undergoing the same transition as the bulk, we next sought to compare the evolution of the transition with
temperature and measure the structural hysteresis curves of the two samples. The
changes in a and c lattice parameters, and in Uiso for Mn and As of bulk and type-A
samples are shown in Fig. 5.5. The hysteresis curves are very similar between the bulk
and the type-A nanoparticles. For example, on cooling, the onset temperature of the
structural transition is very similar. However, the hysteresis curves of the nanoparticles are less square than those of the bulk, and the size of the structural modifications
(for example, the change in lattice parameters) are smaller in the nanoparticles. We
conclude that the structural response of the type-A nanoparticles are very similar to
the bulk in most respects, but of a smaller magnitude, and the transition is smeared
out in temperature on the low-T side. This transition broadening may be expected
due to finite size effects of the nanoparticles.

64

Table 5.2: The refined structural parameters for type-B
and 335 K using the α (P 63mc) and β (P nma) model. All
isotropic thermal factor Uiso .
Parameter
295 K α
295 K β
335 K α
˚)
a (A
3.669(6)
5.702(3)
3.675(6)
˚)
b (A
3.669(6)
3.664(4)
3.675(6)
5.719(12)
6.344(9)
5.724(11)
c (˚)
A
Mn
X
-0.0023(3)
Z
0.008(2)
0.224(12)
0.008(7)
˚2 )
Uiso (A
0.03(2)
0.018(16)
0.03(2)
As
X
0.27(7)
Z
0.24(3)
0.91(2)
0.24(3)
˚2 ) 0.017(14)
0.009(7)
0.016(13)
Uiso (A
Rw
0.305
0.201
0.275

nanoparticles at 295 K
models are refined with
335 K β
5.713(3)
3.672(5)
6.356(9)
-0.0011(2)
0.227(13)
0.019(16)
0.27(7)
0.91(2)
0.009(5)
0.197

In our previous published MnAs paper [104], we have described the synthesis
of Type-B MnAs with different sizes and compared the structural properties. TypeB MnAs nanoparticles have been prepared in sizes up to 26 nm by this synthetic
method by adjusting the reactant concentrations and reaction times. The size in the
reported range does not appear to be a primary determiner for the structure being
probed. It is also true for this study as we observe that the behavior of the type-B
nanoparticles is very different. Structural results from representative temperature
points are presented in Tab. 5.2. Fig. 5.6(c) and (d) show the fits of the α and β
structural models to the type-B nanoparticle PDFs at 335 K. The β model clearly
provides a better fit to the data and a lower Rw (0.197 vs. 0.275) suggesting that
at high temperature the material is in the β phase. However, as is the case for
the type-A nanoparticles, the fit is far from ideal suggesting additional structural
relaxations. Fig. 5.6(a) and (b) show the fits of the two structure models to the
295 K data. The β model provides a significantly better fit than the α model (Rw =
0.201 and 0.305, for β and α-models, respectively), which suggests that the structural
transition may have been suppressed in these nanoparticles. As with the type-A
nanoparticles, we test this hypothesis by looking at the change in the PDF between

65

335 K and 295 K for the type-B nanoparticles and compare it to the bulk. The
result is shown in 5.8.

The first observation is that the overall changes in the

PDFs of the β nanoparticles are very small. The changes are comparable to the
noise, with some small features that might be explained by a small change in lattice
parameter due to thermal expansion. However, there is no indication of any structural
phase transition on cooling for these samples. To make sure there is not a smaller
magnitude phase transition as was observed in the type-A nanoparticles, we again
look for correlations in the difference curves between 335 K and 295 K compared
to the bulk, which are plotted on top of each other in Fig. 5.8(c), after scaling up
the type-B difference curve by a large factor of 10 to match the amplitude of the
fluctuations. Visually there is no evidence of correlation and the Pearson correlation
coefficient is -0.197 which indicates the difference curves are essentially uncorrelated.
This result indicates that the structural transition has been completely suppressed in
the type-B nanoparticles. Furthermore, plots of the temperature dependence of the
structural parameters from type-B nanoparticles (not shown) do not show hysteresis
similar to those of the bulk and type-A nanoparticles. Interestingly, the unit cell
volume of the type-B nanoparticles is significantly less than the bulk or the type-A
nanoparticles as shown in Fig. 5.9(b). For example, the cell volume is decreased by
0.89% at 335 K and by 2.71% at 295 K, compared to the bulk. This synthesis method
has clearly produced nanoparticles that, although similar in size and composition, are
structurally distinct.
Finally, we would like to establish the relationship between the magnetic and
structural transitions. This comparison is shown in Fig. 5.9. In agreement with the
literature [98] we see a strong correlation between the structural and magnetic transitions in the bulk as can be seen by comparing the red and blue curves in Fig. 5.9(a)
(magnetic) and 8(b) (structural), where the structural parameter we choose to plot
is the unit cell volume. The cyan and magenta curves show the same comparison

66

for the type-A nanoparticles. Again, the magnetic and structural transitions appear
to be coupled, although the hysteresis in the magnetic data is far smaller than that
observed in the structural data. The increase in the magnetization on cooling is
decreased in the type-A nanoparticles, relative to the bulk, and this correlates to a
smaller increase in cell volume. Most significantly, the width of the thermal hysteresis in the magnetization is much smaller for the type-A nanoparticles, on the order
of only 1 K, than for the bulk sample, close to 10 K. This is observed despite the
structural hysteresis being broader in the nanoparticles than in the bulk. Thus, the
structural and magnetic transitions are related, but not completely dependent on
each other.
The volume of the type-A nanoparticles in the low-spin β phase is the same as
for the bulk material, though the increase in the volume on entering the α phase
is less, consistent with the decreased magnetization in that phase. The small suppression in the magnetic signal can perhaps be attributed to surface effects in the
nanoparticle sample. This suggestion is supported by the observation that the high
temperature susceptibility in the type-A nanoparticles is smaller than that of the bulk
sample, so that the relative change in magnetic susceptibility, χ′ (T = 295K)−χ′ (T =
335K)/χ′ (T = 335K), is approximately equal. These measurements support the
conclusion that the magnetostructural transition in the type-A nanoparticles is qualitatively similar to that occurring in bulk MnAs, but involving a smaller amount of
distortion.
The temperature dependence of the magnetic susceptibility for the type-B nanoparticles, also plotted in Fig. 5.9, differs strikingly from the behavior of the type-A
nanoparticles and bulk MnAs. The type-B nanoparticles have a negligible magnetic
susceptibility at 335 K, consistent with a weakly paramagnetic low-spin state. There
is a small, but distinct increase in susceptibility at lower temperatures. This transition
can be seen most clearly in the data collected on warming, which suggest a transition

67

from a weakly ferromagnetic to paramagnetic state at a temperature similar to what
is observed in the type-A nanoparticles. In addition to the magnitude of the susceptibility being greatly suppressed from that of the type-A nanoparticles, the thermal
hysteresis in the susceptibility is also much broader for the type-B nanoparticles,
extending over at least 30 K. The fact that the susceptibility in the type-B nanoparticles is a significant fraction of that found in the type-A nanoparticles together with
the different temperature dependence of the two samples makes it unlikely that the
magnetic response in the type-B nanoparticles arises from some type-A impurity that
was not detected by our other techniques.
This magnetic transition in the type-B nanoparticles does not appear to be linked
to a structural transition but may be connected to an observed irreversibility in the
unit cell volume on cooling. This evidence for a magnetic transition in the typeB nanoparticles in the absence of any clear structural transition suggests that the
interplay between magnetism and crystal structure in MnAs may be more complex
than previously believed.

5.4

Conclusion

Structurally distinct samples of MnAs nanoparticles have been prepared using
either slow-heating (method A) or high-temperature rapid-injection (method B) arrested precipitation reactions, and their temperature-dependent structure and magnetic characteristics have been probed. Using Rietveld and PDF methods, we confirm
that the structure transition in bulk MnAs is highly correlated with the magnetic
transition, as previously reported. Type-A nanoparticles have a similar structural
transition that happens in the same temperature region as bulk, but the changes are
smaller in amplitude. Once again, a clear magnetic transition correlates with what is
happening structurally. This is in contrast to previous reports wherein no structural

68

transformation was observed for type-A nanoparticles by traditional temperaturedependent powder X-ray diffraction methods, despite the persistent observation of
a magnetic transition [104] and highlights the sensitivity of the PDF approach in
probing structural transformations in nanoscale materials. Type-B nanoparticles are
distinct from both the bulk and type-A nanoparticles adopting the β structure over
the entire temperature range. The fact that type-A particles predictably cycle between α and β, whereas type-B nanoparticles are kinetically trapped, suggests the
structure is pinned, possibly by impurity ion inclusion during the rapid nucleation
inherent in the method B synthesis. Previous studies have demonstrated a slow conversion over time to the thermodynamic α phase. Partial conversion during the data
acquisition could explain some of the irreversibility observed in both the structural
and magnetic data. A detailed study of the kinetic stability of type-B nanoparticles
as a function of particle size is underway.

69

Figure 5.1: TEM images and particle size distributions of type-A (a) and type-B (b)
MnAs nanoparticles.

70

Type-A nanoparticles

Bulk
(a) 295K, α

4

0

-4

-3

G (Å )

0
-2

(d) 295K, α

3

(b) 335K, α

4

(e) 335K, α

3

0

0

-4

-3
3

(c) 335K, β

4

0

0
-4
0

(f) 335K, β

10

20

-3
30
0

r (Å)

10

20

30

Figure 5.2: PDF refinements on bulk (left) and type-A nanoparticle (right) MnAs
data. For each subfigure, the bulk circles represent the PDF from the experimental
data and the red solid line is the calculated PDF after refinement. The green curve
offset below is the difference curve between data and model.

71

(a)

4

8×10

Intensity (a.u.)

4

4×10

0

(b)

4

6×10

4

3×10

0
4

(c)

2×10

0
4

-2×10

0

5

10
ο

2θ( )

15

20

Figure 5.3: (a) comparison of the difference (orange curve) between 295 (blue line)
and 335 K (red line) diffraction pattern for bulk MnAs. (b) same as (a) but for typeA nanoparticles with violet-colored difference curve. (c) comparison of the difference
curves from (a) and (b). The orange is the same as the difference curve in (a) and
the violet is the result of scaling the curve in (b) by a factor of 4 for comparison.

72

(a)

5
0
-5

(b)

-2

G (Å )

3
0
-3
2

(c)

0
-2
0

10

r (Å)

20

30

Figure 5.4: (a) comparison of the difference (orange curve) between 295 (blue line)
and 335 K (red line) PDF for bulk MnAs. (b) same as (a) but for type-A nanoparticles
with violet-colored difference curve. (c) comparison of the difference curves from (a)
and (b). The orange is the same as the difference curve in (a) and the violet is
the result of scaling the curve in (b) by a factor of 4 for comparison. The simliar
comparison based on diffraction pattern is shown in Fig. 5.3.

73

0.024

3.72

(c)

0.015

2

0.016

3.69

Uiso (Å )

Lattice (Å)

(a)

5.73

(b)

(d) 0.01

5.715
300 310 320 330

300 310 320 330

Temperature (K)
Figure 5.5: Lattice parameters ((a), (b)) and atomic displacement factors ((c), (d))
from PDF refinements on bulk and type-A samples using α phase structure model.
(a) and (b) are for lattice parameter a and c in unit of ˚. (c) and (d) are for Uiso of
A
˚2 . For each subfigure, the blue and red are for bulk MnAs
Mn and As in unit of A
on the cooling and heating runs respectively. The cyan and magenta are for type-A
MnAs nanoparticles on cooling and heating runs.

74

(a) 295K, α

(c) 335K, α

(b) 295K, β

3

(d) 335K, β

-2

G (Å )

0

-3
3

0

-3
8

16

24

r (Å)

8

16

24

Figure 5.6: PDF refinements on data acquired on type-B nanoparticles at 295 ((a),
(b)) and 335 K ((c), (d)) using the α ((a), (c)) and β ((b), (d)) phase model, respectively. For each subfigure, the symbol and color representations are the same as
Fig. 5.2.

75

(a)

4

8×10

4

4×10

Intensity (a.u.)

0
4

6×10

(b)

4

3×10

0
4

(c)

2×10

0
4

-2×10

0

5

10
ο

2θ( )

15

20

Figure 5.7: Same as 5.3 but for type-B nanoparticles. In (c) the difference curve
(violet) of type-B nanoparticles is scaled by a factor of 10 since its amplitude is too
small to compare with the bulk one.

76

(a)

5
0

-2

G (Å )

-5

(b)

2.5
0

-2.5

(c)

2
0
-2
0

10

r (Å)

20

30

Figure 5.8: Same as 5.4 but for type-B nanoparticles. In (c) the difference curve
(violet) of type-B nanoparticles is scaled by a factor of 10 since its amplitude is too
small to compare with the bulk one. The simliar comparison based on diffraction
pattern is shown in Fig. 5.7.

77

Unit Cell Vol (Å ) χ’(emu Oe-1 mol-1 Mn)

3

7.5

(a)

5
2.5

FM

PM

0

68.5
68

α

β

(b)

67.5
67

66.5
300

310
320
Temperature (K)

330

Figure 5.9: Real part of the AC magnetization (a) and unit cell volume (b) in
bulk, type-A and type-B nanoparticles. All of the unit cell volumes are based on
the results of α-model refinements. The blue (cooling) and red (heating) are for the
bulk sample; cyan (cooling) and magenta (heating) are for type-A nanoparticles; and
the green (cooling) and orange (heating) are for type-B. The dashed lines at 303 K
and 317 K show the temperature range of the structural transitions in bulk MnAs as
reported in the literature as we mention in the text.

78

Chapter 6
Conclusion
This thesis work has been motivated by the following quote that was publish
in Science magazine by Prof David Waltz and Bruce G. Bruchanan [112]: ”The
main goals of automation in science have been to increase productivity by increasing
efficiency (e.g., with rapid throughput), to improve quality (e.g., by reducing error),
and to cope with scale, allowing scientific treatment of topics that were previously
impossible to address.” By automating scientific activities and making them high
throughput, it is possible to understand new science about a system. In material
science, one such problem is the study of the structure of complex material, the topic
of this thesis.
New powerful X-ray and neutron source are now becoming available allowing
us for the first time to do high throughput parametric studies on the structure of
complex material. For example, the most intense pulsed neutron beams are available
at Spallation Neutron Source (SNS) at Oak Ridge, TN. Inside SNS, as one of worldclass diffractometers, POWGEN, has been built to target for such research studies
through powder diffraction method.
Powder diffraction method was first used nearly a century ago. However, until the
introduction in 1967 by Rietveld of a method for the refinement of crystal structures

79

from powder data, interest in powder methods increased dramatically. However,
the main problem of Rietveld method is that it is non-linear least squares and is
notoriously convergent. To find the minimum, you have to carry out a complex
set of refinement steps, manually guiding the refinement to the minimum. For high
throughput studies, this become especially problematic.
To overcome the above problem, a new Rietveld analysis program: SrRietveld,
was developed in Prof. Billinge’s group. SrRietveld is a python program that wraps
two legency Rietveld refinement programs: GSAS and FullProf. By using hierarchic
data structure, it allows more flexibility and it is easy to use. As one of the developers,
I lead the development of PyGSAS and involve the development of other components.
The detailed description of SrRietveld has been described in the chapter 3 of the thesis.
As part of this research, I use SrRietveld to do bootstrap error estimation in
Rietveld analysis in order to test the accuracy of the error estimation in Rietveld
method. I have compared three ways of estimating uncertainties on refined parameters
in a Rietveld refinement using two different Rietveld programs, GSAS and FullProf:
the estimated standard deviations obtained from the programs themselves, and by
resampling the data in two different ways using a pseudo random number generator.
There is no apparent advantage of using the computationally expensive Monte Carlo
resampling method with respect to using the standard deviations obtained directly
from the variance-covarience matrix. However, in certain circumstances the use of
the sub-sampling method is warranted. These uncertainty estimation methods will
be included as features in future versions of the SrRietveld.
Besides I have demonstrated that there is a significant sensitivity of refined parameters to the use of correct data-esd’s as weights in the least squares equation.
Significantly biased refined values result when incorrect weights are used even when
the same model is fit to the same data-set. This underscores the importance in Rietveld refinements of having accurate data-esd’s propagated from the raw data or

80

estimated in some other way. The use of the sub-sampling method yields accurate
estimates of standard deviations on refined parameters even when the data-esd’s are
incorrect or unknown. However, it does not correct for the biased values of the
refined parameters themselves that result in this case. Comparing the esd’s from
the sub-sampling method to the values obtained from the variance-covariance matrix
may be a useful diagnostic, indicating situations when there may be a problem with
the reported data-esd’s. In this case, the accuracy of the refined values is certainly
lower than the precision which should be reported. This is generally true because of
model errors and parameter correlations that are not included in these, but I note an
additional contribution due when data uncertainties are themselves uncertain.
Finally, applying next-generation Rietveld and PDF analysis softwares that I
mentioned above, the bulk and nanoscale MnAs have been studied. It is confirmed
that the structure transition in bulk MnAs is highly correlated with the magnetic
transition as reported in the literature [101, 98]. Type-A nanoparticles have a similar
structural transition that happens in the same temperature region as bulk, but the
changes are smaller in amplitude. Type-B nanoparticles are distinct from both the
bulk and type-A nanoparticles adopting the β structure over the entire temperature
range. The fact that type-A particles predictably cycle between α and β, whereas
type-B nanoparticles are kinetically trapped, suggests the structure is pinned, possibly by impurity ion inclusion during the rapid nucleation inherent in the method
B synthesis. A detailed study of the kinetic stability of type-B nanoparticles as a
function of particle size is underway.
I believe that the high throughput Rietveld tool: SrRietveld, can contribute
more to the community by providing a way to probe the phase transition and other
scientific problems. The error estimation methods that I develop here have potential
application in PDF or other methods. More importantly, combining these techniques,
more new structure information from complex materials can be discovered in the

81

future.

82

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83

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