A TIME-RESOLVED X-RAY SCATTERING STUDY
OF THE ORDERING KINETICS IN Cu3Au
By
ROBERT FRANCIS SHANNON, JR.
A DISSERTATION PRESENTED
TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1990
ACKNOWLEDGMENTS
Stephen E. Nagler's, patience and understanding, as well as guidance, have
been instrumental in my progress as a graduate student. Without him this work
could not have been done.
I thank Curtis R. Harkless who spent many a late night watching the
experiment so that I could get some sleep. He also was very helpful in the tuning
and operation of the temperature control system. I could list many ways in which
Curt contributed to this work, but most of all I found discussions with him to be of
the most use.
Our engineer Ward Ruby, who is overqualified for his position, lent technical
expertise that was of course very valuable, but his sense of humor and good nature
was probably even more helpful by keeping my spirits up.
Marsha E. Singh spent two years as a post doc in our lab. Although she was
working on a different project, she found time to help with the convolution fitting
routines, and discussed problems that I encountered.
Thanks go to Gary Venn and Lee Smith for their knowledge with computers,
Gary for writing the scan program used to control the diffractometer and Lee for
configuring and linking our computer system.
Also, I would like to thank P. H. Holloway and J. K. Truman for their help
in preparing the sputtered films, R. M. Nicklow for the bulk sample, as well as
P. Flynn and J. Dura for the M.B.E. films.
I thank my family, Bob and Kay my parents, and Kathy and Kevin, my
siblings. Their support, both moral and financial, cannot be overlooked. Bob and
Kathy, having both received a PhD., Kay presently working on one, and Kevin
having received an MD, all understand life as a graduate student. Their advice
comes from knowledge and experience and has been helpful.
Finally, I wish to thank the U.S. D.O.E. for their support of this research.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ............................................... ii
LIST OF FIGURES ................................................... vi
LIST OF TABLES .................................................ix
ABSTRACT ............................................................ x
CHAPTERS
1 BACKGROUND ......................................... 1
Introduction ............................................ 1
Previous Work ........................................... 5
2 THE GROWTH OF ORDER...............................10
3 X-RAY DIFFRACTION BY Cu3Au .......................... 20
Basic X-ray Scattering ..................................20
Diffraction with Cu3Au and its
Relation to the Order Parameter ....................... 25
Anisotropy in Cu3Au ........................................29
The Effect of Type 1 Walls on Scattering .......................34
4 EXPERIMENTS ..........................................37
The Bulk Single Crystal .................. ..................37
Experimental Details ................................37
Analysis of the Bulk Data ...............................45
Results from the Bulk Data .................. ........ 54
The Sputtered Films and the Effect
of Stoichiometry ...........................83
Experimental Details ..................................83
Analysis for the Data on the
Sputtered Films ....................85
Results from the Sputtered Films ......................87
The M.B.E. Films ..........................................93
Experimental Details ...................................93
Analysis on the M.B.E. Film Data ..................... 97
Results from the M.B.E. Films ..........................98
Scaling and System Dimension ........................107
A Brief Look at the Early Time ...............................119
5 CONCLUSIONS ........................................... 127
Current Results ............................................ 127
Future Directions ........................................ 129
APPENDIXES
THE EFFECT OF TYPE 1 WALLS ON SCATTERING ...... 130
FUNCTIONAL DEFINITIONS USED IN FITTING ..................136
HOW TO CALIBRATE THE BRAUN PSD .... .............. 138
SPECTROMETER ALIGNMENT .........140
DETECTOR DEAD TIME CORRECTION ........................ 143
REFERENCES ........................................................ 144
BIOGRAPHICAL SKETCH ......................... .............. 151
LIST OF FIGURES
Figure page
1. Schematic of (fcc) structure for ordered Cu3Au 3
2. Coexistence and classical spinodal curves for a
first order transition in a typical binary alloy. 11
3 Ordering and coarsening after a quench; 13
4. Difference in path length for scattering 23
5. Domains of different phase separated by a type-1 wall 30
5. Domains of different phase separated by a type-2 wall 30
6. Domains of different phase separated by a type-2 wall 31
7. Domains of 3 phases separated by curved walls 33
8. Anisotropic scattering disks profiles 35
9. First version of the heating element/sample mount 38
10. Lay out of equipment: a) Radial b) Transverse 43
11. Ge [1,1,1] or radial instrumental resolution 47
12. Ge [1,1,1] or radialnsverse instrumental resolution 48
12. Ge [1,1,1] or transverse instrumental resolution 48
13. Mosaic spread at the [0,1,0] .............. 50
14. Effective radial resolution 52
15. Effective transverse resolution 53
16. Change in line shape after a quench 55
17. Raw data on the [0,1,0] bulk superlattice peak. 68
18. Data shown in fig. 17 scaled by amplitude and width. 69
19. Intensity vs. time ... 71
20. Bulk Cu3Au sample history .....................................73
21. Scaled intensity vs. scaled time. 74
22. Amplitude vs. time 75
23. Logl0(amplitude) vs. Logl0(time) for the bulk 77
24. Integrated intensity, amplitude, and background 78
25. Log, (Length) vs. Logl0(time). L a 2r/Ir 80
26. Typical [1,-1,0] profiles from sputtered film B 88
27. Logl0L (2r/u in X) vs. Log(time) ..............89
28. L vs. Logl0(t) ................................................ .91
29. Logl0(L) vs. Log0[Log0(t) ....................................... 92
30. New heating element/sample mount 96
.................................
31. L=2T/r vs. time for the 45001 M.B.E. film 100
32. Log10[L] vs. Logl0[time] for the 4500a M.B.E. film 101
33. L=2r/r vs. Logl0[time] ..............102
34. Size of a domain L=2w/r vs. time for the 7101 104
35. Logl0[L] vs. Logl0[time] for the 710A M.B.E. film ... ......105
36. L=2w/r vs. Logl0[time] for the 710a M.B.E. film .........106
37. L=2r/r vs. time for the 260R M.B.E. film 108
38. Logl0[L] vs. Logl0[time] for the 260A M.B.E. film ..................109
39. L=24/r vs. Log10[time] for the 2601 M.B.E. film .......110
40. 3D scaling plot for 7109 M.B.E. film 112
41. 2D scaling plot for 710R M.B.E. film 113
42. 3D scaling plot for 2601 M.B.E. film ............... 114
43. 2D scaling plot for 260, M.B.E. film ................................ 115
44. Raw scans of the 260a M.B.E. film ......116
45. Weak scaling and incorrect scaling in the 710 film .........117
46. Weak scaling and incorrect scaling in the 260A film .................118
47. Early time scans of the [0,1,0] of Cu Au....... ................ 120
48. Amplitude of the [0,1,0] of bulk Cu3Au vs. time. ..........121
49. Logl0width] vs. Logl0[time] ...................... .......... 122
50. Integrated intensity vs. time ................. .............. 124
51. Radial scan showing the satellites ................................ 125
LIST OF TABLES
Table page
1. Chi-squared values for selected scans as a function of tine after quench
on the bulk Cu3Au sample. Arrangement is by decreasing temperature and
direction in reciprocal space .. ......58
2. Exponents for power law growth from the bulk data 81
3. Quenches and fitted exponents for the sputtered films 87
4. M.B.E. film resolution corrections 98
Abstract of Dissertation Presented
to the Graduate School of the University of Florida
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
TIME-RESOLVED X-RAY SCATTERING STUDY OF ORDERING KINETICS
IN Cu3Au
by
Robert Francis Shannon, Jr
August 1990
Chairman: Stephen E. Nagler
Major Department: Physics
Time- resolved x- ray scattering has been used to study ordering kinetics in
single crystal bulk Cu3Au, as well as in sputtered and molecular beam epitaxy
grown films. After annealing at high temperatures the sample is rapidly quenched
to fixed temperatures below the order- disorder transition temperature. The
development of order is monitored in real time using scattering techniques.
The bulk sample clearly showed three regimes: nucleation, ordering, and
coarsening. The anisotropic superlattice peaks that reflect the domains structure
are investigated in connection with the ordering kinetics. The line shape of the
scattering function exhibits a crossover from gaussian to lorentzian- squared as the
system goes from the ordering regime to the coarsening regime. Coarsening in
Cu3Au is consistent with curvature driven growth.
Domain coarsening in stoichiometric sputtered films is also consistent with
curvature driven growth. However, coarsening in copper rich films proceeds much
more slowly. The results suggest the extra copper affects the ordering kinetics in
the same way diffusive impurities would, resulting in a logarithmic like time
dependence.
The M.B.E. films show a slowing of the growth at late times. The 4500X
film starts out with curvature driven growth but then continuously slows down as
the domains grow. The 710 film shows an interesting temperature dependence for
the growth, in such a way that at temperatures close to the transition, the domain
growth almost freezes at late times. The dominate factor is probably strain, all of
the trends for slower growth are consistent with greater strain. The dimensionality
in the M.B.E. film systems is considered. The scaling in the 4500 and 7101 films
is clearly three dimensional. However, the dimension of the scaling in the 260A film
is unclear.
CHAPTER 1
BACKGROUND
Introduction
The kinetics of phase transitions in systems that exhibit ordering after a
quench from above a critical ordering temperature T has received much recent
interest.(1-6) One focus of this interest has been on whether universality classes for
the kinetics of these systems exist as they do for ordinary static critical
phenomena(6). Conservation laws are known to greatly affect the growth kinetics.
However, the effects of symmetry, structure and sharpness of domain walls, order of
the degeneracy of the ground state, impurities, and stoichiometry are not well
understood. Examples of kinetic behavior can be found in many systems including
magnetic materials, copolymers, and alloys.
This dissertation focuses on a careful study of the ordering kinetics in the
alloy Cu3Au. Cu3Au is a prototypical system with a first order transition, and a
nontrivial asymmetry. The elements Cu and Au have considerably different atomic
numbers (Z), making x-ray measurements of superlattice peaks easier. Moreover,
the equilibrium properties of Cu3Au are well studied!7) Two important properties
of Cu3Au are its four-fold degenerate ground state and the existence of two types of
domain walls that separate the four ordered phases. These two differences make
Cu3Au a good choice for furthering our understanding of the non-equilibrium
kinetics of first order phase transitions. The non-equilibrium kinetics are less well
understood, particularly in the context of the ideas of modern statistical mechanics.
Interesting effects in the non-equilibrium kinetics are introduced by the existence of
different domain wall structures and a corresponding anisotropy of the correlations.
Alloys like Cu3Au exhibit a change in their atomic arrangement as a
function of temperature. Such alloys have markedly different equilibrium properties
as a function of temperature. However, the present interest is focused on the
approach to equilibrium.
The crystalline form of Cu3Au is face centered cubic (fcc), and below the
critical temperature Tc (Tc 3900C) has an equilibrium state where the Au atoms
preferentially reside at cube corner sites and the Cu at face center sites, (see
figure-1). Above Tc the atoms are randomly distributed, but short range order still
persists. The transition from the ordered state to the disordered state is first order,
and like so many things in nature the change from order to disorder takes place
much faster than the change back to order.
The experiments discussed here are all concerned with the change from
disordered to ordered states. In a typical experiment the sample is held at a
temperature well above Tc in order to insure that the sample is in a disordered
state, from which the temperature is rapidly lowered to a point below Tc and held
there for several days. X-ray scattering scans are collected as a function of time.
The peak grows with time, becoming narrower and more intense. Moreover, the
shape of this x-ray peak is important. Many theories predict this shape and hence,
a careful investigation is important to understanding the underlying physics.
Surprisingly, there has been little experimental work in this direction.
Unfortunately, it is usually difficult to obtain the exact shape due to effects from
instrumental resolution, finite size, strain, and mosaic spread.
Experiments have been performed on a bulk single crystal, sputtered films,
including one Cu rich film, M.B.E. (molecular beam epitaxy) grown films from 260
Au- 0
Cu- 0
Cu3Au
ordered
cube
FIG. 1. Schematic of (FCC) structure for ordered Cu3Au.
of a domain increases. Larger domains grow at the
expense of their smaller neighbors.
to 4500 angstroms thick. For each sample the experiment was'repeated for a series
of temperatures.
From the experiments on the bulk single crystal it is evident that three
distinct mechanisms are involved, nucleation, ordering, and coarsening as defined
below. Nucleation, as the name suggests, refers to a delay between the quench and
the start of the development of long range order. Although in several quenches a
nucleation period is observed, the time resolution in these experiments was such
that this regime was not well characterized. Others using a different experimental
set up and a more powerful x-ray source have probed this aspect of the transition in
more detail 8) Nucleation refers to a delay between the quench and the start of the
development of long range order. Ordering (or growth) is the process by which the
disordered state condenses onto the various nucleated droplets of the ordered phase.
One of our results is that the ordering process can be characterized by a change in
the functional form of the scattering associated with the structure factor. The
movement of domain boundaries that separate the different ground states is known
as coarsening. This name refers to the fact that as time progresses the average size
of the domains grow as the larger domains convert their smaller neighbors to the
same phase.
Another sample is an off-stoichiometric sputtered film of Cu3Au. It is
interesting that an off-stoichiometric sample exhibits a different type of growth
than its stoichiometric counterpart. Moreover, the off-stoichiometric film is
probably in a different class altogether. The extra Cu in the sputtered film acts as a
diffusive impurity. However, the stoichiometric sputtered films (all over 7000
angstroms) exhibited curvature driven domain coarsening, where the size of an
ordered domain is proportional to the square root of time. The differences in the
time dependence usually involve a lowering of the power of the time dependence
from 1/2 (square root) to a value less than 1/2. However, the growth in the
off-stoichiometric film is also consistent with a logarithmic time dependence.
The last samples are single crystal films grown by molecular beam epitaxy
(M.B.E.), which are reviewed in a contexts similar to the off-stoichiometric
sputtered film. Both the off-stoichiometric sputtered film and the M.B.E. films
display a slower type of time dependence for growth. However, the cause of this
change from the bulk behavior at critical composition arises from slightly different
causes. Consequently, similarities exist, but so do differences.
Previous Work
The earliest studies of the order disorder transition in Cu3Au date at least as
far back as the early 1900's.(9) However, the interest then was how the level of
order affects bulk properties. The present interest is in the ordering itself and the
non-equilibrium properties associated with it. At the time of the earlier work the
intensity of available x-ray sources did not permit the in-situ type of measurement
made in this work. The early experiments measured only bulk effects like resistance
as a function of temperature, time, or order. The only x-ray work that could be
done was static measurements. An extensive review of order-disorder transitions in
binary alloys prior to the late 1930s can be found in an article by F. C. Nix and W.
Shockely.(10) Two experimental papers both from 1936 are worth noting, one by
Sykes and Evans(11) and the second by Sykes and Jones.(12) These two works
describe the electrical resistance, specific heat, and the crystal structure as related
to the order-disorder transition. A later paper by Jones and Sykes(13) along with
their earlier work are the earliest references relating the width of the superlattice
peak to domain size.
Bragg and Williams(14-16) in 1934 and 35 put forth the first equilibrium
theory from which the qualitative behavior of bulk properties such as the specific
heat, the level of long range order, and resistance could be derived. Formally this
theory is analogous to the Weiss theory for ferromagnetism.(4) This theory worked
only below the transition temperature T The Bragg-Williams theory is a static
mean field approximation that does not consider fluctuations. The Bragg-Williams
theory considers the long range interactions between atoms and therefore, does not
explain short-range order and only qualitatively describes the equilibrium
properties near T The theory of Bethe(17) considers nearest neighbors. As in the
Bragg-Williams case the energy just below Tc is the same as just above T .
However, in Bethe's theory additional energy is required to destroy short range
order above T This gives rise to an anomaly in the specific heat in agreement
with experiment. Due to the complicated nearest neighbor configuration and
asymmetry in composition found in A3B systems like Cu3Au, many of Bethe's
assumptions regarding the structure do not apply to our case. Fortunately
Peierls(18) has applied Bethe's theory to this case. A more formal approach to the
theory of order-disorder kinetics appears in an article from 1956 by G.
Vineyard(19). His theory starts with a set of multi particle distribution functions
and equations of motion. The simplest approximation for these equations yield the
Bragg-Williams result. The Bethe-Peierls approximation can also be derived from
this theory if only nearest neighbor interactions are considered.
In the 70s interest in the kinetics of Cu3Au was renewed by the development
of theories that considered the movement of domain walls(1,20-22) as well as the
availability of more intense x-ray sources. Allen and Cahn(20) derived the time
dependence of the size of an ordered domain. Their theory is often referred to as
curvature driven (or controlled) growth because the driving force goes as the
curvature of the wall. The final result is that the size of an ordered domain L(t) is
proportional to t1/2 (the square root of time). Earlier theories(1) predicted that the
velocity of the wall would go as the product of the mean curvature and the interface
free energy. Therefore, the earlier versions predicted critical slowing down of the
wall as T is approached from below. In the Cahn-Allen case the diffusional
dissipation of free energy of a domain wall exactly equals the available driving force.
Hence, the velocity of a domain wall is not a function of the driving force,(20) and
the power law that describes the time dependence is not a function of temperature.
The temperature dependence is only in a proportionality term that precedes the
power law time dependence. This renewed interest is actually part of a growing
interest in universality classes and non-equilibrium phenomena which occur in such
transitions.
The kinetics of transitions have been of great interest in the past decade.
Theoretical,(1,2,23-39) experimental,(40-57) and computer simulation(3,39,58-76)
studies on first order phase transitions abound. A variety of systems,(30) binary
alloys, binary fluids, magnetic materials,block copolymers, glasses, ferroelectric
crystals, and others can be used to study this phenomenon. The problem is often
discussed in terms of an Ising model,(78) partly because the order parameter is often
simply the magnetization, an easily measured quantity, and partly because Ising
like systems are easily classified. Ferromagnets with spin exchange (Kawasaki)(3)
dynamics are systems with a conserved order parameter (COP), ferromagnets with
spin flip (Glauber)(79) dynamics have a non-conserved order parameter (NCOP),
and antiferromagnets with spin exchange dynamics(28) have a NCOP (sub lattice
magnetization) even though the net magnetization is conserved. The reason being
easily classified is so important is that the nature of universality classes(30) for
growth is not fully understood.
Binary alloys are also useful systems for kinetic studies for similar reasons.
One obvious division in classes is whether the order parameter is conserved or
non-conserved. A recent computer simulation(60) on a three-dimensional binary
alloy model (of the AB type) with a NCOP shows qualitatively different ordering
kinetics than the equivalent spinodal decomposition found in binary alloys with a
COP. This simulation possesses a characteristic length that increases as tl/2 and a
corresponding scaling of the structure factor of Ld, where d is the dimension.
Phase separation is an example of a process with a COP that has been
extensively studied experimentally(1029,47,50). Phase separation exhibits diffusion
controlled grouth(31) where L(t) is proportional to t1/3 as compared to the NCOP
process of curvature controlled grouth(18'19) where L(t) is proportional to t1/2
There are also systems that exhibit interface controlled growth(4) where L(t) is
proportional to time. Hence, the power of the time dependence seems to be an
important difference between classes(29). Another class may be found in systems
with impurities(80-84) where L(t) is proportional to ln(t)m. It has been suggested
this class can be further divided by the value of the power m(30), possibly
indicating whether the impurities are quenched(80-83) or diffusive(84,85). This
type of growth might also be found for critical quenches (close to Tc) involving
spinodal decomposition(38)
One purpose of this work is to distinguish different universality classes,
specifically the effect of dimensionality, composition, the order of the ground state
degeneracy, and the nature of the scaling form for the structure
factor(1,2,23-29,39). There is evidence(86) that the mechanism of domain growth
will change for thin films. Also, there was some question as to whether the order of
the degeneracy lowers the power of the time dependence(56'73'74'77'87- 89
Another reason to look at Cu3Au is that it has two types of domain walls(21,90,91)
The first of these walls conserves the nearest neighbor configuration and due to the
energies involved(89) is easily formed compared to the second. A comparison of the
growth of these two very different walls should provide further insight into the
study of order disorder kinetics.(92) Moreover, less work has been done on systems
involving a NCOP(7,43'93'94) than COP. After a thermal quench both types of
systems exhibit three distinct regimes;(8,94,95) nucleation, ordering, and
coarsening. The functional form of the structure factor can be derived from most
applicable theories,(8'35,39,94,96) however, instrumental resolution, finite size, and
strain often have sizable contributions to the shape of the scattering, making it
difficult to discern the exact shape of the structure factor experimentally.
Determining this shape is another purpose of this work.
The rest of this dissertation is arranged as follows: Chapter 2 discusses the
theories of domain coarsening and nonequilibrium scaling. The first part of chapter
3 is on basic x-ray diffraction and how the level of order is related to the diffraction
pattern. A description of the different types of domain walls in Cu3Au can be found
at the end of chapter 3. The experimental analysis and results are discussed in
chapter 4. Conclusions are presented in chapter 5.
CHAPTER 2
THE GROWTH OF ORDER
Cu3Au is a prototypical system for order-disorder transitions(90). The
transition is driven by minimizing the free energy
F=U-TS. (2.1)
At low temperatures the free energy is dominated by the internal energy, U, causing
the minimum in free energy to correspond to the ordered state. However, at high
temperature the minimum for the free energy corresponds to a disordered state
because it is dominated by the entropy term, (-TS). The critical temperature is a
function of the concentration of the different atoms. Figure 2 is a generic phase
diagram for alloy systems with order disorder transitions. Below the solid
coexistence curve the equilibrium state corresponds to order and above to disorder.
The dashed curve in figure 2 represents the classical spinodal curve separating
regions of metastability and instability of the disordered state after a thermal
quench to below T For phase separation these two curves meet at the critical
composition. However, for the process of superlattice formation 8) this is not the
case, producing an extended nucleation period even for the stoichiometric
composition.
For instantaneous quenches to temperatures just below Tc the short-range
order fluctuations associated with temperatures above Tc are metastable. This
metastable state then decays during a finite incubation time towards a new
equilibrium value(8). During the incubation time the fluctuations are such that the
--
CD
3
--
C-)
(D
CD
CD
Concentration
FIG. 2. Coexistence curve (solid) and classical spinodal curve (dashed) for a
first order transition in a typical binary alloy. Points M and U label
regions as metastable or unstable.
m U
critical size for the stability of an ordered droplet is not realized. The nucleation
stage starts when the fluctuations have relaxed sufficiently to allow a stable ordered
droplet to form. As quenches become deeper the fluctuations associated with the
disordered state become unstable and nucleated ordering changes to continuous (or
spinodal) ordering. There is some indication that this change occurs well below the
classical spinodal temperature.(8) In fact there is no abrupt change as nucleated
ordering changes to spinodal. After nucleation droplets of the ordered phase grow in
a matrix of the disordered phase (see fig 3a). The individual droplets may be in any
of the p allowed ground states. Eventually the domain boundaries meet, and the
system is composed of domains in different ground states separated by domain walls
(see fig. 3b,c). The coarsening process then begins as the larger domains grow at the
expense of their smaller neighbors.
In the coarsening regime the system can be characterized by L(t), the
average size of an ordered domain at time t. During coarsening it is expected that
the system will be self-similar under a rescaling of both space and time. The
growth of the domains is characterized by the time dependence of L(t). If all the
excess energy is at the domain walls the velocity of the wall is proportional to its
curvature, as discussed below. This leads to an equation of motion;
dr K (2.2)
Separating variables gives
r dr a dt (2.3)
Integration shows
12
1r at (2.4)
which implies
L(t) t1/2 (2.5)
(a)
FIG. 3 Ordering and coarsening after a quench.
a)Ordering shortly after the on set of critical nucleation.
b)Early stages of domain coarsening. c)Late stages of domain coarsening.
This is Cahn-Allen (C-A) curvature driven growth. For a NCOP system with p=2
and no random impurities the C-A result has been found to be correct.(20,22)
Another growth mechanism is that of Lifshitz and Slyozov, where the rate of
change in the volume of a domain is constant. This type of growth is expected in
systems with a COP that exhibit phase separation.
dV
d = constant (2.6)
To convert 2.6 to a velocity is a simple matter. The volume is proportional to r3 so,
dV r r2dr. (2.7)
Substitution into 2.6 shows,
dr 1r2. (2.8)
The velocity of a domain wall is proportional to the inverse of the area of the wall.
Integration as done above leads to,
L(t) a t1/3 (2.9)
a slower power law.
An even slower power law of t1/4 has been recently proposed 97) When this
new class was first proposed it was explained by the failure of the assumptions made
in the C-A theory and hence, rightfully came under attack. It is only recently that
the correct reasons for this classes existence have been revealed. The t1/4 class is
associated with the case of zero curvature in the Cahn-Allen theory. This new class
exhibits competition between sharp and broad domain walls. It has also been found
that this class displays a cross over from t1/4 to t1/2 as the temperature is
increased.
A fourth type of time dependence may be found in systems with energy
barriers to the growth which depend on the characteristic length L,(3
E(L) = L1/m/A (2.10)
where A is weakly temperature dependent. If the time necessary to over come such
a barrier is of the form,
t = r exp(E(L)/T) (2.11)
then inverting, and substituting for E(L) gives,
L(t) = AT [ln(t/r)]m (2.12)
a logarithmic dependence on time, where m is a constant, that depends on the type
of barrier, and r is a rate constant.
Obtaining the time dependence from the velocity of a domain wall
(anti-phase domain boundary) as done above is simple. It is considerably more
difficult to show that the velocity either goes as the curvature, or as the inverse of
the surface area. The velocity in general is proportional to the thermodynamic
driving force(14) and associated constant of proportionality is called the mobility.
One way to consider thermodynamic nonequilibrium is through a mean field
approach. Mean field theories to nonequilibrium dynamics of metastable states are
an alternative to cluster dynamic theories, and are of practical importance in
determining the thermodynamic driving forces involved in domain coarsening.(1-4)
Attention is focused on a small set of semi-macroscopic variables that define an
order parameter like the ones discussed at the end of chapter three. These variables
are considered to have slow dynamic evolution as compared to the microscopic
degrees of freedom, that are averaged, or enter only as random forces. The slowness
is important because the derivatives with respect to time are frequently
approximated as zero.
The typical attack is to use a Ginzburg-Landau Hamiltonian(1) where the
free energy density has a double well structure below some critical temperature.
The double well corresponds to the two different phases ordered or disordered. In
the COP process of phase separation the order parameter is the local concentration
of one species of atoms, where for the NCOP process of superlattice formation the
order parameter is the occupation of a species of atoms on a superlattice. This
difference results in very different behavior. For the COP process the structure
factor or scattering not only grows in amplitude as the width decreases, but the
position moves from high to low q, as phase separation increases. Moreover,
because the order parameter is conserved the integrated intensity must be a
constant. For the NCOP case intensity at a superlattice peak is a function of the
order parameter as shown earlier.
The Cahn-Allen result starts by assuming the free energy per unit volume of
a homogeneous phase fo is a function of the order parameter 9. The important
quantity is Afo, the difference between a state of arbitrary order and an ordered
state. The equation of motion is then,
f=- Af 2 (2.13)
'F = -a + MV 19
where a is a positive kinetic coefficient and M=2ax is a diffusion coefficient (m2/s).
Rewriting 2.13 to correspond to a direction g normal to a surface of constant I
gives,
f89 Af 2 -9K+K 1 al (2.14)
= -a o-M 2 -(K+K2)
where (K1+K2) is the mean curvature which is equal to the negative of the
divergence of a unit vector normal to the surface. In the original paper by Cahn
and Allen g was defined such that for a spherical domain g was directed towards the
interior. Although many reverse this convention it will be followed here. Now
consider a gently curved domain wall. A domain wall is a surface of constant 1.
The velocity of the wall is just og/Ot of a constant # surface.
V = = / (2.15)
but from 2.13 for this surface,
Ir = -M(K1+K2) (2.16)
L'OTJ 1P 2) 0 J
therefore, from 2.14 and 2.15,
V = M(K1+K2). (2.17)
As previously discussed this result leads to the time dependence given in equation
2.5. Also this result was derived in the contexts of a second order transition which
is being applied to the late time development of a first order transition.
The degree of order and microscopic structure of a system can be
characterized by the order parameter correlation function. The fourier transform of
the correlation function, S(Q), is called the structure factor. As discussed in
chapter 3, S(Q) is directly measured in x-ray experiments. In a nonequilibrium
situation, such as that following a quench into a metastable or unstable state, the
structure factor may be generalized and written as S(Q,t), where t is the time after
a quench.
Self similarity of the ordering system under a rescaling of space and time for
a COP process is exhibited by the scaling form of the structure factor.
S(Q,t)= [q ( t )]S[q/q ( t)] (2.18)
max max
For a NCOP system,
S(Q,t) = [(t)]Se[(q)/o(t)] (2.19)
where q=Q-G, G is the ordering wave vector, d is the spatial dimensionality, and
e=q/q. In the COP case qmax is a function of time and in the NCOP case it is a
constant. Although the two types of scaling are different they can be made to look
similar. In the COP case the scattering arises from the domains forming an
imperfect lattice. Hence, the position [qmax(t) a 1/L(t)] is a function of the
distance between domains, which on average is the size of the domains. In contrast
the length scale in the NOCP case is inversely proportional to the peak width, a(t).
Substitution of L(t) into 2.18 and 2.19 gives the identical result.
S(Q,t) = [L(t)dS[L(t) (2.20)
Scattering is defined by the structure factor, therefore, this type of scaling
implies that the functional form of the scattering remains constant. If scaling is
valid, only the amplitude and width change for the NCOP system. Equations
2.18-2.20 also describes how the dimensionality of a system can be detected. In
general the scaling function S is anisotropic, reflecting the structure of both the
domains and their walls.(27)
A recent proposal(30) divides growth kinetics into four classes characterized
by different low temperature behaviors. For class 1 systems the temperature is not
important, such as a simple Ising ferromagnet with Glauber dynamics. Class 1
systems obey power law domain growth at all temperatures. Cu3Au probably
belongs to class two. Class two differs from class one only in that members have
local defects that cause freezing of the domain growth at absolute zero. Such local
defects are expected to have activation energies that are independent of domain size
L. Classes three and four, on the other hand, have defects with L(t) dependent
activation energies, leading to logarithmic growth which is discussed in connection
to equation 2.12. Type Three and four systems are differentiated by the power m of
the logarithmic growth. Class three has m=1, and is likely to contain the random
field Ising model. Class four has ml#, and may include dilute ferromagnets and spin
glasses.
The integrated intensity is independent of the existence of anti-phase
domains.(90) Experimentally, however, if the domains are very small, it is possible
to lose some intensity in the tails of the reflection. The width of a superlattice peak
is of course greatly affected by the existence of domains. Broadening from the finite
size of domains, in the Cu3Au case, has a dependence on (h,k,l)(90) as will be shown
in the next chapter. An effective size can be defined but it is purely formal since
there are corrections.
19
Leff = 2r/r (2.21)
where o is the width in units of inverse distance.
CHAPTER 3
X-RAY DIFFRACTION BY Cu3Au
Basic X-ray Scattering
The diffraction of x-rays depends on the crystal structure and the
wavelength. X-rays are scattered by the electric field created by the electrons.
Hence, the diffraction of x-ray photons contains information concerning electron
density.
The position of the diffraction peak can be easily derived by considering the
difference in path length of two beams reflected off two parallel crystalline planes a
distance d apart.(98) If the difference in path length is a multiple of the wavelength
of the radiation used, then constructive interference forms a peak. This logic leads
to Bragg's Law,
2dsin( )=nA (3.1)
where 0 is measured from the plane, and n is an integer. Bragg's Law is valid, if
A<2d, which is why x-rays are used to probe crystals. Bragg's Law considers the
periodicity of the lattice. However, it does not consider the structure or material
that make up the lattice. Consequently, Bragg's Law does not contain any
information about the intensity of the diffraction.
Before considering the electron density, which truly defines the diffraction, it
is convenient to define a reciprocal lattice. Given three primitive lattice vectors
(al,a2,a3), the primitive reciprocal lattice vectors are defined as;
a2" a3 a3" al al" a2
b =2r. 3 b2=2r. ; b3=2r- al a2 (3.2)
1 al*a2= a3 2 ala2 a3' 3 alba2x a 3
The primitive reciprocal lattice vectors are constructed so that each is
orthogonal to two of the three crystal primitive lattice vectors. Thus,
bi.a = 2r6ij (3.3)
where 6ij is the Kronecker delta function. The factor of 2r is omitted by
crystallographers but are used for convenience in physics. The reciprocal lattice is
the set of points defined by the reciprocal lattice vector,
G = v1b1+v2b2+v3b3 (3.4)
where vi is an integer. All crystal structures have two interdependent lattices, the
crystal lattice and the reciprocal lattice. For example, the crystal lattice for Cu3Au
is face centered cubic (fcc), it is a simple mathematical exercise to show that the
reciprocal lattice associated with an fcc crystal lattice is a body centered cubic (bcc)
lattice.(90,98)
The properties of a crystal are invariant under a transformation between
equivalent lattice sites. The electron number density n(r) must, therefore, be
periodic in r with periods al,a2,a3 in the directions of the crystal axes.(98 This is
expressed mathematically as,
n(r+T) = n(r) (3.5)
where T is the crystal lattice vector,
T = n1al+n2a2+n3a3 (3.6)
where ni is an integer.
Applying fourier analysis to the periodic function n(r) is equivalent to finding
a set of vectors J such that n(r) is invariant under translations of the crystal lattice
vector;
n(r) = Enj.exp(iJ.r) = n(r+T). (3.7)
By Bloch's theorem G exactly satisfies the condition expressed in 3.7 for
By Bloch's theorem G exactly satisfies the condition expressed in 3.7 for J.
The fourier coefficients nG determine the intensity of the scattered radiation,
and that the Bragg condition is equivalent to the difference between the incident
and scattered wave vectors being G. As seen in figure 4 the difference in phase
factor between the incident (k) and scattered (k') wave vectors is exp[i(k-k') r].
The scattering amplitude is an integral over space of the electron concentration
times this phase factor;
F = fdV n(r) exp[i(k-k').r]. (3.8)
Repeatedly using the invariance of n(r) by translations T, the scattering amplitude
F can be written as,
F = E fdV nG exp[i(G-Ak).r] (3.9)
G
where k+Ak = k'. For (G = Ak) F is a maximum and F = V E nG showing that
G
the scattering amplitude is a function of the fourier coefficients nG. The width of a
peak for a one dimensional lattice goes as the inverse of the number of atoms,(94)
< = 2r/M (3.10)
where M is the number of atoms.
It is convenient to rewrite 3.9 at G=Ak as the integral over one cell
multiplied by the number of cells. The scattering amplitude for one cell is known as
the structure factor FG(98)
FG =f dV n(r) exp(-iG.r) (3.11)
cell
The electron concentration function can also be written as the superposition of
electron concentrations nj of each atom of a cell.( )
n(r) = E n.(r-r.) (3.12)
j= 1J
where s is the number of atoms in a cell. The scattering amplitude then becomes,
FG = E fdV nj(r-rj)exp(-iG.r) (3.13a)
J
= S exp(-iG-r.) dV nj(p)exp(-iG.p) (3.13b)
J
,,-Crystal
Incident
Outgoing
ik1 r
Ik'o r
FIG. 4. Difference in path length between two wave vectors scattered from
points 0 and r. The total difference in phase angle is (k-k') r.
The corresponding difference in phase between wave vectors scattered
from dV at r and 0 is exp[i(k-k')-rj.
where a = r-r.. Defining the atomic form factor fj
f = dV nj(s)exp(-iG.s) (3.14)
it is apparent that f. contains the same information as nj and the structure factor
J
becomes,
FG = .exp(-iG-r). (3.15)
J
The scattering amplitude is usually written in terms of the integers v1,V2,V3,
replaced by h,k,l and rj in units of the crystal lattice {rj = xjal+yja2+z.a3},
FG = fjexp[-2ri(hxj+kyj+lzj)]. (3.16)
The intensity is proportional to F F, where F denotes the complex conjugate of F.
Therefore, FG does not need to be real, and hence, is not directly observable. In
addition, FG is defined in terms of a cell or basis, where there is freedom in the
choice of a cell.
The scattered intensity is more naturally described in terms of a correlation
function. The time independent pair correlation function is defined in real space as,
G(r)= f
d r (3.17)
where p(r') is the charge density at position r', and < > represent a
thermodynamic average. G(r) is an equilibrium quantity. The fourier transform of
G(r) is commonly called the structure factor, S(Q).
S(Q)= f exp(iQ-r)G(r)d3r (3.18)
If the system of interest is in a non-equilibrium state the structure factor becomes a
function of both position and time, S(Q,t).
Some confusion may arise between structure factor S(Q,t) and the time
dependent equilibrium structure factor normally written S(Q,t) which refers to
fluctuations about equilibrium as opposed to the approach to equilibrium. Here
S(Q,t) is a non-equilibrium S(Q) at a time t, the time is added to emphasize the
fact that we are measuring a non-equilibrium function. Another confusion in
definitions comes from the scattering amplitude F historically being called a
structure factor. In older text books on x-ray scattering F is commonly called a
*
structure factor. Although F F and S(Q) are interchangeable, physicists generally
use S. The newer theories use S and the older theories often use F. Since 1954,
following a paper published by Leon Van Hove, physicists have preferred to use
S(Q). In Van Hove's paper equation 3.18 is derived in the Born approximation.99)
In the above discussion and throughout this work S(Q,t) is a non-equilibrium
quantity, and F will be call the scattering amplitude.
Diffraction with Cu3Au and its Relation to the Order Parameter
For an fcc lattice the usual choice of a basis is one corner atom at [0,0,0], and
three face centers at [0 ], [,0,], and [,]. For this choice,
F(h,k,l) = fatom{1+exp[-ri(k+l)]+exp[-ri(h+l)]+exp[-ri(h+k)]}. (3.19)
If the indices (h,k,l) are unmixed (all even or all odd) FG=4fatom. If the indices
are mixed (2 even 1 odd, or 1 even 2 odd) two of the exponents will be odd multiples
of-ri and one even, so FG will vanish.
The above assumes that only one type of atom makes up the crystal. This is
not the case for Cu3Au. The values of fAu and fCu can be found in the
international tables for x-ray crystallography Vol. 3(100) Since a real sample of
Cu3Au is never fully ordered, a partially ordered sample should be considered. The
following discussion is a long-range order theory only, and is equivalent to the
Bragg-Williams theory!4) Starting with the fraction of Cu and Au atoms located
at corner and face center sites a definition of long range order [L] can be
introduced.(90) This long range order variable I is not the same as a long range
order parameter used in statistical mechanics even though it was originally named
as such. Here long range order L is defined in terms of the occupancy of lattice sites.
r = fraction of Au atoms on a cube corner
w = fraction of Cu atoms on a cube corner
r = fraction of Cu atoms on a face center
f = fraction of Au atoms on a face center
where r stands for right type of atom, w for wrong type, c for corner, and f for face
center. Of these four variables three are dependent, hence, a few results are
immediately obvious,
rc+c = 1; rf+w = 1 (3.20)
1 3
rc+wf = zAu the fraction of (3.21)
gold atoms
3 1
frtf+w = zCu the fraction of (3.22)
copper atoms
1 3
where the fractions and I refer to the concentration of corner and face center sites
respectively. Long range order can now be defined,
L = rc+rl- = rc-wf = rfwc (3.23)
The order variable defined this way is one, if the sample is fully ordered, and zero, if
the sample is fully disordered. Using 3.19 or 3.20 the last dependent variable can be
eliminated,
3 1
S= (rc- ZCu)/ = (rf ZAu)/I (3.24)
The scattering amplitude for crystalline Cu3Au is a function of the occupation of
sites and the atomic form factors for Cu and Au as defined in equation 3.19.
Therefore, the relation between the scattering amplitude and the order variable is
easily derived.(90) Dividing the sum in 3.18 up into sums over cube corners and
face centers gives,
F = E ( r cfAu+WcfCu)exp[2i(hx c+kyc +lzc)]
corners
+ ( rfCu+wffAu)exp[2i(hxf+kyf+lzf)]. (3.25)
f aces
The only corner is at [0,0,0]. Substituting the corner position for [h,k,1] as well as
the face center positions as done to get 3.16 it is found that for unmixed (h,k,l),
F = (rcfAu+WcfCu) + 3(rffCu+wffAu) (3.26a)
= (rc+3wf)fAu+ (wc+3rf)fCu (3.26b)
1 3 1 3
= 4( rc+wf)fAu+ 4( wc+ rf)fCu (3.26c)
= 4(zAufAu+Z CufCu) (3.26d)
and for mixed (h,k,l),
F = (rcfAu+cfCu) (rffCu+ffAu) (3.27a)
= (rc-wf)fAu- (rfwc)fCu (3.27b)
= L(fAu- Cu) (3.27c)
where fN is the atomic form factor for N, and zN is the atomic fraction of itom N.
For stoichiometric Cu3Au, zAu = 1/4, and zCu = 3/4. The unmixed indices or
fundamental peaks are just those for an fcc crystal with an amplitude corresponding
to scattering from an average atom [ (3fCu+fAu)/4 for stoichiometric Cu3Au]. The
scattered intensity is the structure factor multiplied by its complex conjugate.
Therefore, the intensity of a mixed indices or superlattice reflection is proportional
to the square of the difference between the atomic form factors for Cu and Au,
multiplied by the square of the order variable. Hence, from a comparison of the
intensity of a superlattice peak relative to a fundamental, the long range order
variable (L) can be theoretically determined.
In real applications there are several effects which can produce sizable
errors.(90) Extinction effects from different grains can reduce the intensities of
fundamentals, hence, high-order peaks should be used in the calculation.
Nevertheless, the intensity of a superlattice reflection is proportional to the square
of the long range order variable.
In the modern theories a proper order parameter must be used. Often the
systems investigated are equivalent to Ising models, and therefore, have a one
component order parameter. For the COP process of phase separation the order
parameter is the local concentration of one of the two types of atoms. This is
equivalent to an Ising model with Kawasaki (spin exchange) dynamics with the
order parameter defined as the total magnetization. For order-disorder transitions
the analogous Ising model also uses Kawasaki dynamics only now the order
parameter is the magnetization on a sub-lattice, which is not conserved. Cu3Au
can be described by a three component order parameter.(92101) In a detailed
theory of Lai(92) the three component order parameter is defined in terms of the
occupation of sites on one of four interpenetrating simple cubic sublattices described
by Ising like occupation numbers si'
+1 if occupied by Cu (3.28)
i -1 if occupied by Au
where i = 1, 2, 3, and 4 corresponding to the four equivalent ways of defining a
cube corner in the FCC lattice or equivalently the four cubic sublattices. On the
basis of symmetry Lai constructs the order parameter according to the Landau
Lifshitz rules,
S1(R)=o[il(R)--2(R)-43(R)+s4(R)] (3.29a)
f2(R)= [ls,(R)-42(R)+s3 (R)-4(R)] (3.29b)
93(R)=o[sl(R)+s2(R)-s3(R)-44(R)]. (3.29c)
Using conservation of atoms a conserved concentration I can be written,
I = o[s (R)+s2(R)+s3(R)+s4(R)-2]. (3.30)
From here an equation of motion is found using a Ginzburg-Landau potential. The
square modulus of the order parameter used in modern theories is proportional to
the intensity of a superlattice peak.
When, as is the case for binary alloys, the order parameter is proportional to
the charge density (atomic concentration), the charge density correlation function is
directly proportional to the order parameter correlation function. Therefore, the
structure factor measured in the scattering experiment is linearly proportional to
the order parameter correlation function which is normally modeled theoretically.
Anisotropy in CuAu
The solid Cu3Au solution forms a fee crystal and below Tc 3900C orders in
the L12 structure (Au on the corners, Cu on the face centers as in figure 1) with a
ground-state degeneracy of p=4. The four different phases correspond to redefining
the cube corners as one of its three nearest face centers, and of course the original
phase one started with. Mathematically one shifts the position of the ordered state
by 0, (al+a2)/2, (al+a3)/2 or (a2+a3)/2. The presence of domain walls leads to
anisotropic line shapes in superlattice peaks as discussed below.
Two types of domain walls exist in Cu3Au. Type-1 walls(21) are easily
formed due to their low energy. These walls can be formed by displacing the atoms
on one side of a plane perpendicular to the a3 axis by (al+a2)/2 as in figure 5.
Type-1 walls can also be formed by any cyclic permutation of (al,a2,a3) in the
above method. A Type-2 wall can be formed by displacing the atoms on one side of
a plane perpendicular to a3 by (al+a3)/2, as in figure 6. The important difference
between type-1 and type-2 walls is that type-1 walls conserve the nearest neighbor
configuration where type-2 walls do not. The type-2 wall in figure 3 has gold
atoms as nearest neighbors, one could form a slightly different type-2 wall say
type-2b by redefining all the Au atoms to be Cu atoms and all the in plane Cu
atoms to be Au. Type-2b walls have Cu atoms as nearest neighbors in the wall
Type-1 Domain Wall
-Au -
\2, 1) *r -9~Au -Cu
-Cu be lw aro
(1 ,1 ) above Theclane
FIG. 5. Domains of different phase separated by a type- 1 domain wall as
described in the text. The line marks the position of the wall.
Type-2 Domain Wall
0 vN
( ", -Cu atoms beeow nd-
above the plne
FIG. 6. Domains of different phase separated by a type- 2 domain wall as
described in the text.
instead of Au atoms as in type-2a walls. Moreover, if one looks in a direction
perpendicular to a type-2 wall one finds planes of all Cu alternating with planes of
half Au and half Cu, the difference between type-2a and type-2b walls is just which
of the two types of planes is repeated to create the wall. In order to move a type-2
wall there must be some diffusion of atoms due to the off stoichiometry of the wall.
On the other hand a type-1 wall can be moved by shifting the wall up or down by
(al+a2)/2, hence, it is often called a half-diagonal glide domain wall. This
difference posed the question of whether the'two types of walls grow at the same
rate or not. If the two walls do not grow at the same rate then scaling would not be
valid.
One way to measure the anisotropic profile is to look in both the radial and
transverse directions during coarsening. This corresponds to observing the motion
of both types of walls. At the [1,0,0], the radial direction probes type 1 walls and
the transverse type 2. At the [1,-1,0], the situation is almost reversed, the radial
direction probes type 2 walls and a specific transverse direction probes type-1. So
another way to consider both types of walls is to observe the [1,0,0] and the [1,-1,0]
in the radial (or transverse) direction. In this study the validity of scaling during
coarsening in an anisotropic system is established. The fact that both walls behave
in the same way suggest it is the same mechanism driving the growth in all
directions. In figures 5 and 6 straight domain walls have been shown, however, the
growth is curvature driven suggesting that a curved wall should be considered. A
curved wall is shown in figure 7, in it one can find elements for all types of walls.
Although it is possible to envision this wall growing with different velocities in
different directions, that has been found not to be the case.
Curved Domain Walls
0 0
0 0 **Q
,-Cu atoms celow ana
atove The plane
( 1, 1, )
FIG. 7. Domains of three different phases separated by curved domain walls
constructed from a combination of both type- 1 and type- 2 walls.
constructed from a combination of both type-~ 1 and type- 2 walls.
The Effect of Type 1 Walls on Scattering.
The width of a superlattice peak is of course greatly affected by the existence
of domain walls. In the following consideration of the scattering a number of
assumptions are made: 1. type 2 walls are not present, 2. the half diagonal glide
domain boundaries perpendicular to each of the axes are completely independent
from each other, and 3. the intensity drops off sharply away from the peak (a small
q approximation). For convenience the convention that all superlattice peaks be
written such that h and 1 have the same parity has been adopted. The shape of the
peak, as shown in appendix A, can then be written as (in reduced units),
2 N1a sin (N2h2) N3a
I(hh2h) = F2 Na 2(N2h2) 3 (3.31)
3 a+(rh) 2 (rh2)2 a+(rh)
where h1, h and h3 represent the departure of the diffraction vector from the
[h,k,l] peak position [ie.,q =(2r/a).(hl,h2,h3), Q =G+Q, G =(2r/a).(h,k,l)], and a
is the probability of crossing an anti-phase domain boundary in a distance a. The
intensity for a superlattice peak as given above is roughly a thin disk lying in the h1
h3 plane with a thickness determined by the crystal dimension N2a2. type 2 walls
would, of course thicken the disk. Figure 8 shows the orientations of theses disks for
one plane of reciprocal space. Also this equation shows that the radius of the disk
increases with the probability a of anti-phase boundaries.
Normally, the condition I(hlh2h3) equal to (1/2)I(hlh2h3) would be used to
define the width of the peak in terms of L. However, the approximation of hlh2h3
being small lead to problems. Therefore, one starts with a more general equation
for the width.(90)
SI dq
FWHM max (3.32)
max
[ 1,0,0]
[2,0,0]
FIG. 8. Shapes of the reflections in reciprocal space for type 1 walls
normal to the three axes. The fundamental reflections are spheres,
and the superlattice reflections are disk-shaped.
Also in appendix A, the relation between domain size and superlattice peak width is
shown from 3.32 to be,(90)
FWHM a(2r/L) ( Ih + k ) (3.33)
Sh2+k2+l2
This result is for type 1 walls, however, type 2 walls also produce broadening with a
2r/L dependence. The correction of 2.21 for the [0,1,0] type peak is one, and for the
[1,0,1] is V4. For a given functional form equation 3.32 can be used to relate the
corresponding width to L. For a gaussian with width o, defined as,
F(x)=A. exp [[(x-xo)/2g]2] (3.34)
equation 3.33 gives,
= (2i/L) ( hl+kl ) (3.35)
Sh2+k 2+12
where as for a lorentzian squared with width r, defined as,
r 2122
F(x)= [1+ j[(x-x )/r] (3.35)
equation 3.33 gives,
r /L) (I h + k ) (3.36)
/ h2+k2+12
The reader should be aware of the above corrections for type 1 walls. However,
throughout this work it is assumed that 2.21 is correct for both types of walls.
CHAPTER 4
EXPERIMENTS
The Bulk Single Crystal
Experimental Details
The crystalline sample of Cu3Au used in these experiments was cut at Oak
Ridge National Laboratories to expose the [h,0,0] face. In order to remove any
oxidation the surface was etched in nitric acid and then annealed for 12 hours at
7000C under an Ar atmosphere.
The furnace used in these experiments is composed of an evacuable chamber
and a heating element. Heating was performed by running AC current through a
0.003mm diameter tungsten coil wrapped around a 9mm diameter copper rod that
had been covered with a 1.5mm thick layer of electrically insulating boron nitride.
The tungsten coil consisted of 20 to 25 loops covering a length of 35mm. The sample
was mounted to the rod as shown in figure 9. This heating rod is placed in the
chamber which is equipped with a 3600 Beryllium window. Beryllium is a light
metal that is almost transparent to 8 KeV x-rays. The tantalum shown in figure 9
was used to keep Cu from diffusing from the mount into the sample. Tantalum foil
was also placed above the copper rod for the same reason.
A proportional relay (Douglas Randall model RD04A) was used to control
the current to the heating coil, and hence, the temperature. The proportional relay
was connected to a model 828D Micristar temperature controller which used a
K-thermocouple placed at the top of the copper rod to get a reference temperature.
copper plate
0o tantalum plate
29m tantalum clip
29m-- sample
16mm
copper rod
35mm tungsten coil
FIG. 9. Heating element and sample mount used in connection with the bulk
sample, the sputtered films, and the 4500 X (M.B.E.) film.
In the initial experiments quenches were done by putting the Micristar on manual
which fixed the current, then the Micristar's temperature set point was changed to 2
to 50C above the final post temperature desired. Next the current was turned off at
the source allowing the sample to cool. As the post temperature reached the
Micristar's new set point the current source was turned back on and the Micristar
put on automatic. The system would soon reach an almost constant temperature a
few degrees below the set point. If left alone the system temperature would slowly
be raised till it reached the set point. Therefore, the Micristar was temporarily put
on manual and the set point lowered to the post temperature. The Quenches took
place in 30 to 90 seconds depending on depth. Quench times were measured from
when the temperature crossed Tc to when the temperature was within one degree
celsius of the final temperature. The quenches did not over shoot but smoothly
went to the final temperature. The quenches were significantly improved in later
experiments as will be disused in connection with the M.B.E. films.
In addition to the thermocouple used for temperature control there also was
a K-thermocouple held directly onto the face of the sample. The temperature
difference between the sample and the top of the heating rod was about 90C. The
vast majority of this gradient existed in the tantalum plate protecting the sample
from the Cu plate. The gradient across the sample is believed to be small due to the
geometry of the heater. Moreover, an estimate of the gradient suggests that the
temperature gradient across the sample was 0.01C/mm. This estimate was based on
the thermal conductivity of Cu and Au and the Stefan-Boltzmann law, weighted by
the emissivity of Cu and Au. The x-ray spot size on the sample was roughly
1x6mm2, consequently the temperature difference within the area of interest was
about 0.060C. The temperature control was good to *0.3C over several hours and
*0.5C over several days. Therefore,the gradient was negligible relative to the
temperature stability.
In order to determine the transition temperature the intensity of the [0,1,0]
peak was recorded as a function of time for several temperatures after raising the
temperature from below Tc to near T The intensity at each temperature was then
plotted as a function of inverse time and the curve was extrapolated to 1/time = 0.
At 385.20C the peak was observed for 3.5 hours and the extrapolation suggested
that the sample would almost disorder but not completely. On the other hand at
385.70C the peak was nearly gone after 1.5 hours and the extrapolation clearly
showed that the sample was above T The transition temperature was then taken
to be the value of a weighted average based on the extrapolations, and found to be
385.30C. This extrapolation is not very important because the temperatures used
are only half a degree apart, but it was clear that the transition was closer to
385.2*0.50C than 385.7*0.50C.
The wavelength of the x-rays used is named from the core electron levels of
the. metal target used in the Rigaku RU300. The doublet line of Cu is referred to as
CuKa (A=1.541781) or just Ka. The two components of this doublet line are Ka
(A=1.540511) and K2 (A=1.54433A).
The x-rays were produced by a rotating anode. The major components of a
rotating anode are, the anode and cathode or filament. To produce x-rays, electrons
are accelerated by a potential difference from the filament to the anode or target.
In so doing x-rays are produced in several ways. The first is from bremsstrahlung,
the sudden deceleration of the electrons. The second is through the process of
exciting core energy levels of the metal target, which then decay producing x-rays
of specific wavelengths. Almost all of the work done here used a Cu target. The
most intense core energy levels are labeled Ka and KO. This method of x-ray
production is very inefficient, only about one percent of the energy used goes into
the production of CuKa radiation. The majority of the energy goes into heating the
target. Therefore, the target must be cooled. Tube sources have a fixed target that
is typically cooled by running water. Tube sources at best are run at around 3kW.
In order to increase the intensity the anode (or target) is rotated increasing the area
that is heated. In such cases the anode is in the shape of a cylindrical bowl that is
rotated about the axis of the cylinder with the electrons bombarding the side.
For geometrical reasons the resolution or spot size is best at smaller angles.
An angle of six degrees as measured from the surface of the target was used here.
Six degrees gives a 1 to 10 reduction of the spot size. Other commonly used angles
are 3, 9, and 12 degrees.
The RU300 can be operated in two different geometries (point or line focus).
If the filament is mounted perpendicular to the scattering plane associated with the
Huber diffractometer, then the anode must spin in the scattering plane. The
electrons must then hit the anode in a line perpendicular to the scattering plane.
This produces a bar or line of x-rays perpendicular to the scattering plane. If the
situation is reversed then the bar lies in the plane at an angle of 6 degrees to the
direction of the incident beam. The projection of this bar onto a cross section of
beam perpendicular to its motion is a point. For a PSD a line focus has better
instrumental resolution than a point focus, but the point focus generally gives higher
counting rates, because more of the x-rays are focused onto the sample.
The optics following the x-ray source differ according to whether the
direction scanned in reciprocal space was parallel to the ordering vector Q (radial
scan) or perpendicular to Q (transverse scan). Radial scans probe reciprocal space
along a line that passes through the origin and the peak in question. Hence, in a
radial scan the magnitude of Q is directly varied. Mathematically any point in a
radial scan is definable as (c.h,c.k,c.l) where c is a real number. For the [0,1,01 a
radial scan is along the [0,k,0] direction. Transverse scans in contrast probe a
direction perpendicular to the radial direction. For example, at the [0,1,0] a scan
that varied h, 1, or, a linear combination of the two would be a transverse scan. In
this study the magnitude of Q is almost constant in a transverse scan. Hence, a
transverse scan is almost identical to a mosaic scan in which the sample is rocked
while Q is held fixed.
Recalling the disk shaped scattering profiles for a superlattice peak
previously shown in figure 8 a transverse scan of the [0,1,0] is along the diameter of
the disk and a radial scan is through the disk: Equation 3.31 implies that type 1
walls do not affect the width in the radial direction at the [0,1,0]. Therefore, the
radial direction probes type 2 walls only. The transverse direction probes type 1
and 2 walls. However, the broadening in this direction is dominated by type 1
walls.
The sample was placed at the center of rotation of a Huber Eulerian cradle
by adjusting an x-y translation at the base of the furnace. Figure 10a and 10b
shows schematically the apparatus in the two geometries used; 10a was used to
collect the radial data and 10b the transverse data. Set up 10a uses a line focus and
a Braun model OEO-50M linear platinum wire based position sensitive detector
(PSD). The Braun detector's signal is binned into 1024 channels by a multi channel
analyzer. For the sample to detector distance used (23.5cm) each channel
corresponds to 0.0125 degrees in 20. The window on the PSD was 5cm long which
corresponds to a solid angle of 10.3 degrees. The transverse set up (figure 10b)
employed a point focus and a Bicron model 1XMP040B scintillation detector. The
Bicron's signal was fed into an E.G.&G. Ortec model 590A single channel analyzer
which is connected to an E.G.&G. model 974 quad counter/timer. The
counter/timer is in turn connected to an IBM XT PC through an I.E.E.E.-488
interface bus. The PC also directs the Huber through a Klinger Scientific MC4
Slits
A
Four Circle
DIffractomete
Monitor
PO (0,8,2)
Monochromater
FIG. 10. Lay out of equipment: a) Radial setup b) Transverse setup.
stepping motor controller. Both set ups use a second Bicron detector, interfaced
with the PC as an incident intensity monitor, so that fluctuations in intensity can
be corrected for.
For the radial set-up over 90% of the K2 (A=1.54433R) intensity was slit
out leaving mostly Ka (A=1.540511) where the subscripts define the two
components of the doublet line. The slitting out of K2 was done on the [0,2,0] peak
with the pre-sample slits shown in figure 10a. At the [0,2,0] Ka and K' where well
separated as opposed to at the [0,1,0] where the two components were not
completely resolved. It was important to slit out K2 due to the complications it
would have added to the resolution correction. In the transverse set-up it was not
necessary to do so because of the analyzer. The resulting instrumental resolution for
both set-ups, in the corresponding directions of the scans in reciprocal space was
0.003V-1(HWHM).
In the above discussion, PSD scans have been called radial scans. A radial
scan in real space involves varying the detector angle 20 while half angling the
sample (0=n). The major difference in a PSD scan relative to a radial scan is the
lack of half angling of the sample. The two are approximately the same for small
distances about a peak. For the sample to detector distance used the 1024 PSD
channels covered *6.4 degrees in 20. The maximum error in the sample angle is
then 3.2 degrees. Typically a range less than 400 channels (or 2.5 degrees) was used
in the analysis. At the [0,1,0] the radial direction probes the thin part of S(Q).
Therefore it is reasonable to assume that S(Q) is constant (or flat) for small
deviations from the radial direction. Under this assumption, a PSD scan differs
from a radial scan in reciprocal space by a displacement of 90 degrees from the
radial direction of (Q-q)-tan(O-n). For the range used in the analysis, the
difference in the distances from Q between the two types of scans is less than 7.7
percent at the end points and 3.4 percent at the mid-points. The mosaic spread of
the sample (0.25 degrees) would further lessen the difference between the these two
types of scans. This small difference between a PSD scan and a radial scan only
affects the results in that the width of a PSD scan is slightly dependent on type 1
walls where a true radial scan probes only type 2 walls.
Analysis of the Bulk Data
One of the purposes of this work was to consider the growth kinetics in an
anisotropic system. Noda, Nishimura, and Takeuchi(43) measured the [1,1,0]
superlattice peak in Cu3Au by time resolved x-ray scattering techniques. They
found evidence for scaling of S(Q,t), a Lorentzian-squared line shape, and
curvature-driven growth. However, they incorrectly assumed isotropic scattering
profiles, casting some doubt on the validity of their conclusions. It was postulated
that type 2 walls, which were not considered in this earlier work, would have a
slower growth due to the need to diffuse one type of atom relative to the other in
moving a type 2 wall. The off stoichiometric nature of type 2 walls that might have
caused the growth to become diffusion limited is discussed in chapter 3 section C.
In these experiments the intrinsic x-ray peak width and shape is of interest
for several reasons as discussed earlier. Measuring the intrinsic line shape of S(Q,t)
is complicated by several extraneous contributions to the line width, arising from
the effects of, instrumental resolution, finite grain size, strain, and the mosaic
spread. The correction for instrumental resolution is the most important. The
accuracy of all results depends heavily on a careful handling of the resolution. The
radial data was corrected for instrumental resolution as well as finite size and strain
as discussed below. The correction for the transverse direction was further
complicated by the mosaic of the sample. In both set ups the instrumental
resolution was measured by placing a perfect Ge single crystal at the sample
position and observing the [1,1,1] reflection, which is at an angle near that of the
[0,1,0] of our sample. The intrinsic width of a Ge single crystal peak is negligible
compared to the instrumental resolution. Estimates of broadening from sample
characteristics other than the ordered domain size are made from the broadening of
fundamental Cu3Au bragg peaks. Graphs of the measured instrumental resolution
functions are shown in figures 11 and 12 for the radial and transverse set ups
respectively. A detailed study, involving both theoretical and experimental
determinations of the instrumental resolution function show the complex nature of
this function!102,103) In this work the instrumental resolution was carefully
measured. The validity of the procedure discussed here is supported by the detailed
study. The contribution to the width of the [0,1,0] arising from finite size and strain
was estimated from radial scans of the [0,2,0] and [0,4,0]. These scans were fit to
gaussian profiles, and the resulting widths in degrees were corrected for instrumental
resolution assuming the widths add in quadrature ( cor meased res
Ultimately the effects of finite grain size and strain can be included in an
effective resolution function. Finite grain size leads to an identical contribution to
the reciprocal space line width of each bragg peak in the sample. Strain can be
viewed as providing a distribution of lattice constants, leading to an additional
width at each peak that is linearly proportional to Q. To obtain the contribution to
the width at the [0,1,0] arising from finite size and strain the [0,2,0] and [0,4,0]
corrected widths were converted into units of inverse angstroms and fit to [a + bQ],
where a and b are fit parameters and Q is the magnitude of the corresponding
scattering vector. The widths due to finite size and strain correspond to a and bq
respectively. The result of this fit suggested that the correction at the [0,1,0] was
Radial Resolution
1.91
Position
1.92
(A-)
FIG. 11. Scan of Ge [1,1,1] used to measure instrumental resolution in the
radial direction. The graph shows only the central portion of the scan,
many more points were taken in the tails.
1/3
0
U
1.90
cOc
cOc
.00X09c'$<
1.93
r r T 1 I
1~11111111111
1
O
I 00000000
Transverse Resolution
c0
0
Q
-15 -10 -5 0 5 10 15
Position (x10-3 A-')
FIG. 12. Scan of Ge [1,1,1] used to measure instrumental resolution in the
transverse direction.
4cP
I I I I I II
O
ICb~C1G~lrrindri~YiY~
... .. T I f i I I I i I I I I I I I I I
"" ""
l
""
l
"
I. I
. I ,
0.002a-1 (FWHM), one third that of the instrumental resolution. One could argue
that the widths should be added as a+bQ2, not linearly as was done. If the fits are
done this way the correction for finite size and strain becomes ten percent larger and
the overall correction 1.4 percent larger. The simplest correction is what was done,
this method is equivalent to assuming the widths add like lorentzian widths. The
case for using a+bQ2 assumes the widths add like gaussian widths and that the
width from strain is small compared to that for finite size, both of which are poor
assumptions. In any case the final result is not sensitive to the choice. A high
resolution set-up similar to 10b but with a germanium monochromator and
analyzer was used to measure the mosaic of the Cu3Au sample at the [0,2,0].
Mosaic spread is a measure of the distribution of orientations of the grains in the
sample. The mosaic spread contributes to the each peak a constant angular width
in real space, and hence, in reciprocal space is proportional to Q. The transverse
scans were taken as a function of Q so the positions in the high resolution scan were
converted to Q at the [0,1,0] after a small correction for finite size and strain. The
mosaic at the [0,1,0], shown in figure 13, has a FWHM of 0.0044 radians
(0.007X-1), a value that is a little larger than instrumental resolution.
Initially the data from the bulk sample was fit to a lorentzian squared form
and then the width were corrected by subtracting the effective resolution(104). That
method, although useful for preliminary analysis, introduces a systematic error.
Unlike a lorentzian and a gaussian the convolution of two lorentzian squared
functions is not a similarly shaped function, hence, there is not a simple analytical
way to remove the resolution from a lorentzian squared fit. In order to correctly
correctly account for the resolution the data was fit to the convolution of the
resolution function and a model functional form. Although this procedure was a
better method, attempts to assign a functional form to the resolution failed to model
Mosaic of Bulk Sample
C4-
U
/
i-
I I I
4oac' 1O II
-15 -
10 -5 0 5 10
Position (x10-3 A-)
FIG. 13. Scan of the [0,2,0] of Cu3Au with the position linearly
corrected to correspond to the mosaic spread at the [0,1,0].
15
I I I
I
I
I ...................
Ir i t 11~ ll1lr
O
~1
o
o
v\,
the data well. The reason for this is simple: The various contributions to the
resolution were in general equally poorly fit by a gaussian and/or a lorentzian
squared. Other forms fit even worse. The instrumental resolution had a slight
asymmetry from slitting, and the mosaic had a little structure in it (see figures
11-12). Therefore, fitting was done by convolving the instrumental resolution data
itself point by point with a narrow gaussian representing finite size and stain and
the model line shape. This method worked very well for the radial data. The radial
data was collected with a PSD as described earlier which took many points in in the
radial direction minimizing any error from the uncertainty in the shape of the peak.
Figure 14 is the result of convolving the instrumental resolution (figure 11) with a
gaussian representing finite size and strain. Moreover, the correction in the radial
direction did not involve a mosaic spread. In the transverse direction this method
also worked well. However, the uncertainty in the line width, which is dominated
by errors in the resolution correction, is larger in the transverse data than the radial
data. The convolution of the transverse instrumental resolution (figure 12) with a
gaussian representing finite size and strain as well as the mosaic (figure 13) is shown
in figure 15. Despite the larger absolute resolution correction, the raw transverse
data shows changes in the shape of the peak, and hence, S(Q,t) more clearly than
the radial data. In fact the resolution is not very important to the change in S(Q,t)
because the change occurs early when the resolution correction is small compared to
the peak width.
Nucleation is associated with very early times and the equilibration of short
range order. The short range order peak intensity is very low, consequently only its
existence could be seen by the lack of a long range order peak. However, changes
during the growth stage that are revealed through changes in the shape of the
scattered intensity are seen. This change is analyzed by fitting the data to both
Radial Correction
1.91
Position
C%3
*i-l
tf
^11
c,
c^
S=
3
O
o
[.92
(A-')
FIG. 14. Convolution of the instrumental resolution (fig. 11.) with a gaussian
of width 0.001 A-1 (.002k-1FWHM) (representing finite size and strain.
The convolution represents the correction in the radial data.
I I I
I I I I
I I I i I
[9
1.90
1.93
111111
I
Transverse Resolution Correction
tOO
U
-30 -20 -10 0 10
Position (x10-3
20 30
A)
A-')
FIG. 15. Convolution of the instrumental resolution (fig. 12.), mosaic spread
(fig. 13.), and a gaussian of width 0.001 A-1 (.002X-1 FWHM)
representing finite size and strain. The convolution represents the
correction in the transverse direction.
theoretical and fundamental functions. Another method for seeing changes in line
shape is to scale peaks from different times by their amplitude and width. In such
scaling plots, differences in shape are apparent. However, they do not define or
systematize the change as do the fits.
The fits to the data are used to extract a reciprocal space line width which is
inversely proportional to an ordered domain size L(t). L(t) is compared to various
functional forms, specifically a power law (equation 2.5) and/or a logarithmic law
(equation-2.12). The fits also provide an amplitude and integrated intensity (width
multiplied by the amplitude) as functions of time and temperature. Scaling the
intensity and time gives information relating to changes, or lack thereof, in the
mechanisms for growth.
Results from the Bulk Data:
The shape of our peak at early times ( first half hour ) is gaussian (GS).
During this time the superlattice peak arises from isolated ordered clusters
embedded in a disordered matrix. A GS shape at early times is not surprising, after
all it is associated with finite size broadening. At later times the line shape crosses
over to a Lorentzian-squared (LSQ) form. The LSQ shape is characteristic of the
coarsening process, and results from a random distribution of ordered domains
separated by sharp domain walls.(96) Figure 16 compares the shape of the peak at
early and late times.
The simplest model for superlattice peak line shapes in Cu3Au is a
Hendrichs-Teller(105) (HT) approach, which assumes independent domain walls as
discussed in chapter 3 section D. The predicted scattering function(8) in this
approximation is an anisotropic lorentzian in the small q limit. The function
X2 -- -2 10000
/ 365.8 C
Co 750o 2250
I- time (sec)
S......z GS
D LSQ
S 5000 -T- 5000
A 450 sec 2600 sec
0 -0
-0.125 0.0 0.125 -0.125 0.0 0.125
77( A-')
Fig. 16. Change in the line shape after a quench. Transverse scans of the
[0,1,0] with a*=1.666A-1. Solid (dotted) line is a GS (LSQ)
least-squares fit. The FWHM of the instrumental resolution is
equal to the width of the tic marks on the I7 axis.
Inset: 2 (as later defined) of the fit vs. time for the GS(diamonds) and
LSQ(circles).
definition is presented in appendix B. This HT approach, was used by Ludwig et
al8) to parameterize early-time kinetic experiments on Cu3Au. The small q limit
is a modified Lorentzian where the line width depends on the direction in k-space.
Although this approach effectively parameterized their data it would not do so here.
However, this is not a contradiction because all of their scans were taken in the time
it took to take our first scan, the resolution they used was very broad compared to
the present work, and their sample was a poly-crystalline wire. The modified
Lorentzian produced poorer fits to our data than a straight Lorentzian which did
not fit well at any time. However, if the shape was convolved with the resolution
function the fits were poor but acceptable. It is worth noting that the background
for these fits was constrained to be positive, if negative backgrounds were permitted
considerably better fits were found in the transverse scans which contained
considerably few points in the tails.
Several theoretical predictions on coarsening suggest(35'39) that the tails of
S(q,t) should decay as q consistent with a LSQ peak. The theory of Ohta,
Jasnow, and Kawasaki(35) (OJK) derives an equation for the line shape that
involves an integration for each q. Using numerical integration non-convolution fits
to this line shape have been done on selected peaks from quenches to various
temperatures and times such that the resolution is small compared to the overall
width. However, the resulting fits during coarsening were significantly worse than
the LSQ fits for the transverse scans but only slightly worse for the radial scans.
Definitions for this theory and the one that follow are summarized in appendix B.
Recently Z. W. Lai has developed a coarse-grained model of the phase
transition kinetics for a Cu3Au type system(92). The model is based on a
Ginzburg-Landau hamiltonian with the symmetry of Cu3Au. This model exhibits
distinct regimes from early to late time. Moreover, the line shape of the structure
factor starts off gaussian and then crosses over to approximately lorentzian-squared
in agreement with our observations especially in the transverse direction. At late
times this model produces a modified lorenztian squared of the form,
Fl(k)= 1 2 (4.1)
S(+bk2)2+2b2k
Fits of this form to our data do not favor this form over that of a pure
lorenztian-squared (the b=0 case). However, Z. Lai points out that this variance
could be due to the size and time used in the computations for this model.
Moreover, due to the need for a larger system, and hence, more computer time a
smaller anisotropy in the shape of the domains was used in the numerical
calculations than is found in Cu3Au. Z. Lai also suggested that this model might
not be "completely settled into the longest time asymptotic regime". Recalling that
during the crossover there is a period where the peak is neither gaussian or
lorenztian-squared, our fits were checked to see if b2 could be related to the
crossover. However, there was no evidence of b2 contributing in any systematic
way. In fact even at the crossover this modification has very little effect on the
shape of the fits. Nevertheless this model exhibits the qualitative behavior that we
have observed.
In table 1 are listed the reduced chi-squared (X2) values for non-convolution
and some resolution convolution fits to gaussian, lorentzian, lorentzian-squared,
lorentzian to a variable power, and the two shapes based on the discussed theories
(see appendix B for functional definitions). In the fits, up to a second order
polynomial background [a(x-x) 2+b(x-x )+c] was added to the model function.
Chi-squared is the goodness of fit parameter used in the fitting routine
[X2=((Ifit -meas 2/(degrees of freedom)] All but the latest times given in table 1
are such that the resolution width is small compared to the over all width.
Therefore, the resolution is not important for the first four times listed in table 1.
Table 1.
Chi- squared values for selected scans as a
function of time after a quench on the bulk Cu3Au sample. Tables are
arranged by decreasing quench temperature and direction in reciprocal space.
Transverse
\order
line \ of
shape \bg
0
Gaussian 1
2
Lorent- 0
zian 1
2
0
LSQ 1I
2
Lor. to m 0
(m) chi 1
2
HT(8) 0
Theory 1
2
O.J.K(35) 0
Theory 1
2
(non convolution fits)
8 2
II minutes min
I I
4.6)
4.8
6.2
2.5
2.0
1.5
8.3
8.3
8.2
2.9
2.5
2.1
2.5
2.2
2.2!
10.0i
10.1
9.8!
2.8
2.5
2.3:
3.3
3.3)
3.6
T=3820C [X2=(Ifit-meas)2/(npnt-npar
3
utes
16.2'
16.3
11.1I
135.6
138:1
122.2
9.2
9.3
7.0
1.9
1.8
1.6!
151.8'
155.1
131.21
31.4'
32.0
29.7'
48
minutes
2.1
2.1
2.1
72.3'
73.81
56.8
229.51
234.41
188.3
2.2
2.3
2.2
2.0
2.0
2.0!
264.8'
244.41
189.0!
77.9'
79.9
76.31
123
minutes
1.8
1.8
1.7
24.8'
25.0
22.0
48.8
48.4
41.3
2.61
2.5
2.4
1.91
1.81
1.8,
67.41
64.4
62.11
17.01
17.0
14.81
493
minutes
1.8)
1.8)
2.0
48.3
49.2
34.8
82.6
84.1
70.2
6.6
6.7
4.3
5.4
5.5
4.21
85.1'
86.7
71.4!
119.4'
331.2
90.5
convolutionn fits)
Gaussian
Lorent-
zian
LSQ
Lor. to m
(m)chi
HT
Theory
2.7
2.1
1.5
8.5
7.9
8.0
2.81
2.5
2.1
1.9
1.4
1.4
8.71
8.3
8.4!
16.9'
16.6
11.1
134.6
134.1
115.3
8.2
8.3
7.4
1.9
1.8
1.7
136.51
139.51
118.11
4.6)
4.8
9.0
75.5'
75.4
56.8
217.9
217.8
167.7
2.0
2.0
2.0
1.9
1.9
1.9
213.0'
214.6'
172.8!
3.2
3.2)
3.5
25.9'
25.5
22.0
42.3
41.0
33.2
3.4
3.3
3.0
1.51
1.4
1.4
41.81
43.01
29.81
2.0
2.0
2.0
50.71
50.6
35.1
65.7
65.3
18.41
11.5
11.7
6.5
1.71
1.71
1.6
43.91
44.8
28.4!
1.3
1.3
1.4
: :
Table 1. (cont.)
Values of chi-squared
Transverse
(non convolution fits)
T=375.6C
\order
line \ of
shape \bg
0
Gaussian 1
2
Lorent- 0
zian 1
2
0
LSQ 1
2
Lor.to(m) 0
(m) chi 1
2
HT 0
Theory 1
2
O.J.K. 0
Theory 1
2
convolutionn fits)
Gaussian
Lorent-
zian
LSQ
Lor.to(m)
(m) chi
O.J.K.
Theory
8
minutes
23
minutes
48
minutes
123
minutes
494
minutes
3.9
3.5
2.2
101.8
102.6
92.0
21.3
21.6
16.3
5.8
5.6
4.5
155.0
138.0
100.1
5.31
5.1
5.01
18.5)
14.5
93.4
31.0'
30.8
24.5
191.9
194.8
174.2
8.3
8.3
6.4
3.3
3.1
3.2
200.4
207.8
174.9
19.3
19.2
16.91
2.78
2.8
2.8
81.8
83.0
64.3
283.4
288.6
245.4
5.1
5.1
5.2
4.7
4.7
4.6
295.5
300.3
239.2
101.7
104.1
38.6!
2.1
2.1
2.2
133.7'
136.0
117.8
325.4
332.0
307:71
11.71
11.8
11.1
10.4
10.4
10.4
342.3
349.7
307.7
92.5
94.2
80.4'
1.9
1.9
1.9
55.6'
56.8
42.0
93.3
95.2
79.2
7.6
7.7
4.81
5.6
5.7
4.5
133.2
121.1
95.5
39.1
40.8
28.1
1.7
1.7
1.9
4.11
3.5
2.2
103.5
102.1
89.1
20.5
20.8
20.7
3.7
3.3
2.2
111.11
127.01
90.81
II
I I
I I
1 I
I I
I I
I I
!! (20.3
(23.4)
11(139.6)
SI
I I
I1
32.4'
31.5
24.5
186.9
185.9
161.1
6.9
6.9
5.9
2.8
2.5
2.6
207.91
212.41
159.3'
85.5'
84.9
64.3
260.4
259.4
210.9
4.0
4.0
4.0
3.9
3.9
3.9
314.01
317.9
190.4!
139.91
139.2
118.0
273.4
272.9
187.9
14.3
14.4
12.4
6.7
6.7
6.8
265.91
271.71
182.1
2.0
2.0
2.0
58.6'
58.5
42.4
24.7
24.7
16.7
15.1
15.4
9.4
1.5
1.6
1.6
23.1
23.4
20.1
1.3
1.3)
1.3
Table 1. (cont.)
Values of chi-squared
Transverse
(non convolution fits)
\order
\ of
\ bg
I I
I I
miI
II
II
II
II
II
II
I I
II
(19.3
III 151
1|(59.7
I I
I I
I I
I I
1'1
8
nutes
3.5
2.8
2.1
127.6
128.4
114.4
25.81
26.1'
17.9
6.5
6.2
5.6
145.7,
132.91
126.6
6.91
6.71
6.41
T=370.80C
23
minutes
2.9)
3.0
3.1
29.8'
29.4
22.2
224.7
227.2
204.0
10.9
11.0
8.5
3.8
3.4
3.3
279.4'
257.71
213.2
19.91
20.21
17.11
2.2
2.2)
2.5
convolutionn fits)
Gaussian
Lorent-
zian
LSQ
Lor.to(m)
(m) chi
HT
Theory
line
shape
48
minutes
123
minutes
494
minutes
Gaussian
Lorent-
zian
LSQ
Lor.to(m)
(m) chi
O.J.K.
Theory
HT
Theory
47.91
45.1
31.5
199.51
190.1
159.4
5.3
4.7
4.8
4.8
3.9
3.0
208.9
203.8
169.4
27.0
24.9
17.8
2.0
2.0
2.0
24.9
24.9
22.0
85.6
85.0
76.8
3.3
3.1
3.0
3.2
3.0
2.9
93.7
92.0
81.6
17.3
17.3
14.61
55.0
56.21
41.7
122.7
125.1
107.2
7.7
7.8
5.8
6.9
7.1
5.6
161.11
158.6'
119.21
38.4
40.2
26.81
1.8
1.8
2.0
I
II
II
II
I I
I I
I I
I I
I I
11(23.7)
I1(28.4)
(209.4)
I I
I I
3.7
2.9
2.1
130.1
128.1
111.1
24.9
25.3
25.8
3.3
2.6
2.1
143.91
129.51
125.5,
31.2
30.0
22.2
219.5
217.1
189.2
9.1
9.2
8.3
2.5
2.1
2.0
216.31
218.1'
196.2!
50.0
46.1
31.6
184.7
170.5
136.0
4.2
3.5
3.4
4.1
3.3
2.6
209.31
163.61
127.5!
2.8
2.8)
3.0
26.0'
25.5
22.1
73.9
71.4
48.4
3.2
3.0
2.7
2.3
2.1
2.1
58.9'
57.5
51.8!
2.0)
2.1
2.3
57.9
57.9
42.21
37.1
37.0
25.9
13.2
13.4
8.3
2.5
2.5
2.5
34.8
35.6
29.9
1.4)
1.3
1.4)
Table 1. (cont.)
Values of chi-squared
Transverse
(non convolution fits)
T=365.80C
0
Gaussian 1
2
Lorentz- 0
zian 1
2
0
LSQ I
Lor.to(m) 0
(m) chi I
2
HT 0
Theory 1
2
O.J.K. 0
Theory 1
2_
II
I I
II mil
II
II
I I
I
I
II
II
20.0)
14.7
I I
I I
I I
I I
I
8
nutes
2.8
2.4
1.8
129.31
131.0
112.6
24.61
25.11
16.41
5.5
5.5
5.01
172.6
161.6
123.0
5.9
5.9
5.61
convolutionn fits)
I
I I
II
II
SI
I I
I I
I I
1I(24.1)
1 27.4'
S89.0
I I
I I
I I
2.9'
2.5
1.8
132.1
130.9
109.5
24.1
24.5
25.7
2.6
2.3
1.8
148.1
142.5
109.7!
25.7'
24.7
17.2
213.2
211.2
184.4
11.51
11.6'
10.7
2.8 2.3
2.4 2.3'
2.1 2.55
222.71
221.61
178.0'
line
shape
\order
\ of
\ bg
23
minutes
48
minutes
123
minutes
514
minutes
3.2
3.2
3.5
35.7'
36.2
26.8
187.91
191.61
68.21
5.3'
5.4
5.2
3.1
3.1
2.7
207.9!
163.7'
19.31
19.81
15.1!
25.61
25.4
23.0
97.31
96.9!
88.6'
3.0
2.9
2.9
3.0
2.8
2.9
116.3
103.5
94.1
18.1
17.7
15.2'
2.0
2.0
2.0
60.5
61.2
40.2
142.0
143.2
118.3
8.5
8.51
6.8
8.3
8.3
6.2
158.9
168.61
120.6
42.6
43.1
27.9!
1.9)
1.9)
2.2
Gaussian
Lorent-
zian
LSQ
Lor.to(m)
(m) chi
HT
Theory
37.3'
37.0
26.8
174.8
174.3
147.3
3.5
3.6
3.6
2.3
2.3
2.0
193.9'
198.31
148.8!
26.8'
26.0
23.1
84.6
81.9
71.1
2.9
2.7
2.7
2.3
2.2
2.2
110.71
81.41
61.31
62.9
62.8
41.1
45.5
45.0
27.7
13.0
13.1
7.0
3.0
2.9
2.8
40.2
40.6
29.21
1.8
1.8)
1.8
1.4)
1.4)
1.4
Table 1. (cont.)
Values of chi-squared
Radial
(non convolution fits)
\order
line \ of
shape \ bg
0
Gaussian 1
2
Lorent- 0
zian 1
2
0
LSQ 1
2
Lor.to(m) 0
(m) chi 1
2
HT 0
Theory 1
O.J.K
O.J.K 0
8
minutes
1.1)
1.4
1.0)
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
8.7
8.6
2.5
1.6!
T=382.00C
22
minutes mi
2.5 13.
2.5
1.3
2.4
2.4
1.3
1.0
1.0
1.0
2.4 0.9 2.4
2.4 0.9 2.4
2.9 0.9 3.0
3.4
3.4
3.2
3.11
47
nutes
13.4
6.5
23.0
23.1
11.7
2.3
2.3
2.3
1.7
1.7
1.3
43.3
45.21
34.11
7.61
convolutionn fits)
I I
1 I
I I
I I
I I
I I
II 1o)
I I
I I
I I
II
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2 2.2
1. 2.2
1.2 2.6
6.6
5.3
1.6'
123
minutes
485
minutes
38.6'
38.7
23.7
103.1
103.7
63.5
6.3
6.3
6.4
3.9
3.9
2.2
178.5
168.31
149.2
19.2'
2.4
2.43
2.8
23.0
23.1
16.6
87.9
88.4
62.4
6.4
6.5
6.5
4.1
4.2
2.7
138.8
139.8
122.8
11.41
2.5
2.5)
2.9
Gaussian
Lorent-
zian
LSQ
Lor.to(m)
(m) chi
HT
Theory
O.J.K.
2.5
2.5
1.3
2.4
2.4
2.3
0.9
0.9
0.9
0.9
0.9
0.9
3.1
3.1'
2.6'
11.9'
11.9
5.6
20.0
20.0
16.7
1.4
1.4
1.3
1.4
1.4
1.2
31.8
31.6
19.8
28.91
28.7
17.1
38.4
38.2
36.5
2.8
2.7
1.4
1.9
1.8
1.3
133.6
132.7
79.01
I
2.0
2.0)
2.4)
13.3
13.3
9.4
8.8
8.8
8.8
4.5
4.6
3.1
2.4
2.41
2.21
208.91
208.9
66.3
9.31
1.4
1.4)
1.5
i!
Table 1. (cont.)
Values of chi-squared
Radial
(non convolution fits)
T=375.60C
\order
line \of
shape \bg
0
Gaussian 1
2
Lorent- 0
zian 1
__ 2
0
I D
LSQ 1
2
Lor.to(m) 0
(m) chi 1
2
HT 0
Theory 1
O.J.K.
O.J.K. 0O
8
minutes
I I
3.6)
3.7
5.1)
1.9
1.8
1.2
2.9
2.9
1.4
1.4
1.4
1.2
1.1
1.1
1.1
9.0
8.71
4.4
2.41
23
minutes
8.1
8.1
3.5
19.5
19.6
8.4
3.3
3.3
2.8
.0 1.7
.0 1.7
.0 1.4
48.7
47.31
31.2
4.01
49
minutes
20.3'
20.3
10.5
51.3
51.6
25.3
4.5
4.5
4.2
2.7 2.2
2.7) 2.1
3.2 1.5
112.8
111.9
79.2
10.21
convolutionn fits)
Gaussian
Lorent-
zian
LSQ
Lor.to(m)
(m) chi
HT
Theory
O.J.K.
7.5
7.5
4.7
31.2
31.0
21.1
2.31
2.3
2.11
1.7 2.2
1.6 2.2
1.4 2.5
30.7
30.41
22.21
i
126
minutes
491
minutes
42.8
43.0
24.0
150.7
151.7
95.8
11.4
11.5
11.5
5.5
5.5
2.2
244.2
245.71
209.3
19.2'
50.1'
50.41
36.0
215.9
217.4
153.3
15.1
15.2
15.3
7.9
7.9
4.71
347.21
349.2'
279.8
26.01
2.6
2.6)
3.2
2.6)
2.6)
3.0)
0 "
1
2 II
0 II
1 II
2
0
1
2 I
0
1
2 II
0
1 II
2
0 I
1.9
1.8
1.2
7.2
7.0
3.8
1.3
1.3
1.2
1.1
1.1
1.1
7.2
6.9
4.01
3.4
3.4)
4.7
17.41
17.3
16.4
51.8
51.6
38.9
1.8
1.8
1.8
1.7
1.6
1.3
61.0
59.31
47.21
31.1'
31.1
63.4
129.6
129.4
58.5
3.0
3.0
1.6
2.8
2.8
1.3
149.9
150.71
80.31
I
27.5
27.4
149.8
21.6
21.6
33.5
7.4
7.4
4.4
3.4
3.4
3.0
463.8
466.8
270.6
20.9!
1.9
1.9)
2.2
1.5
1.5)
1.6
Table 1. (cont.)
Values of chi-squared
Radial
(non convolution fits)
T=370.80C
\order
line \ of
shape \ bg
0
Gaussian 1
S2
Lorent- 0
zian 1
2
0
-----~I ,
LSQ 1
2
Lor.to(m) 0
(m) chi 1
2
HT 0
Theory 1
O.J.K.
O.J.K. 0
II
II mi
I I
I
II
'I
II
II
II
I I
I I
I I
I I
I I
I I
I I
8
nutes
1.9
1.9
1.0
3.6
3.7
1.5
1.5
1.5
1.1
1.0
1.0
0.9
17.0
16.6'
6.3
2.51
23
minutes
3.2
3.2
4.5
8.8'
8.7
3.1
23.6
23.7
8.9
3.8'
3.8
2.8
1.5
1.4
1.1
69.7
68.4
40.2
4.01
49
minutes
3.0
3.0
3.8
126
minutes
19.4
19.5
9.1
65.4
65.8
30.5
7.5
7.6
6.3
2.5 2.7
2.5 2.7
1.7 3.2)
144.6
152.11
101.61
8.91
487
minutes
49.1'
49.4
28.7
170.0
171.1
100.3
12.6
12.6
12.4
5.4 2.7
5.5 2.7)
2.8 3.1)
317.3
320.21
249.2'
21.71
convolutionn fits)
Gaussian
Lorent-
zian
LSQ
Lor.to(m)
(m) chi
HT
Theory
O.J.K.
51.7
52.0
37.2
237.7
239.3
164.9
17.3
17.4
17.4
7.9
7.9
4.9
362.8
366.5
304.8
22.71
I
I I
II
II
II
I I
I I
I I
II
I I
I I
I I
I I
1.9'
1.9
1.0
14.0
14.0
5.5
1.4
1.4
1.4
1.0
1.0
0.9
13.8
13.6
5.7J
1
8.3
8.1
4.2
47.3
47.0
28.9
2.6
2.6
2.4
1.4
1.3
1.1
47.0
46.81
29.51
1
16.91
16.8
14.7
81.2
81.1
58.7
3.51
3.51
3.3
2.1 2.0
2.1 2.0
1.5 2.3
88.7
86.31
66.01
I
37.0'
37.0
20.6
101.5
101.4
87.2
2.8
2.8
2.1
2.8
2.8
1.6
165.3
165.51
109.51
I
2.5
2.5)
3.2
29.3
29.3
20.4
68.7
68.7
41.4
5.6
5.6
3.4
3.1
3.11
2.71
442.71
449.21
64.31
17.11
1.6
1.67
1.7
Table 1. (cont.)
Values of chi-squared
Radial
(non convolution fits)
T=365.80C
\order
line \ of
shape \ bg
0
Gaussian 1
2
Lorent- 0
zian 1
2
LSQ 1
2
Lor.to(m) 0
(m) chi 1
2
HT 0
Theory 1
O.J.K 0
convolutionn fits)
Gaussian
Lorent-
zian
LSQ
Lor.to(m)
(m) chi
HT
Theory
O.J.K.
16
minutes
27
minutes
52
minutes
128
minutes
496
minutes
5.0
4.9
1.8
12.2
12.2
3.6
2.6
2.5
1.6
1.1
1.0
0.9
45.4
43.31
21.6
3.71
10.5'
10.5
3.8
26.4
26.6
9.6
4.1
4.1
3.0
1.6
1.5
1.2
80.0
79.9
45.1
5.1!
3.5
3.5
4.6
24.11
24.3
10.9
67.4
67.9
31.1
7.5
7.6
6.5
3.0
3.0
2.1
167.0
168.11
110.9
11.7!
51.61
51.91
29.71
170.01
171.21
96.11
12.6
12.7
12.2
5.3
5.3
2.9
322.0
323.11
252.5'
23.2
2.9
2.9)
3.6
55.0'
55.4
37.91
237.7
239.3
161.9
16.3
16.5
16.5
7.7
7.8
4.3
364.0
366.5
302.7
23.9
2.7
2.7
3.2
2.7
2.7)
3.0
4 8'
4.8
4.7
1.7
32.8
32.4
17.1
2.0
2.0
1.9
1.0 2.8
00. 2.8
0.9 3.8'
32.7
32.1
17.4
9.9
9.8
3.5
55.0
54.7
32.8
2.8
2.8
2.6
1.5
1.4
1.2
54.5
54.41
34.11
1
3.2
3.2)
4.2
21.4'
21.4
9.3
97.2
97.1
64.9
3.8'
3.8
3.71
2.5
2.6
1.9
100.1
100.8
72.71
40.2'
40.1
22.2
115.8
115.7
93.5
3.0
3.0
2.6
3.0
2.9
1.8
171.5
172.61
157.41
1
2.5
2.5
3.0
32.7'
32.7
21.5
42.1
42.1
42.1
5.2
5.21
2.7
2.8
2.8
2.1
425.4
428.3
62.9
20.11
2.0
2.0)
2.3
1.6
1.6)
1.8
: :
One would expect for late times the convolution of the resolution function with a
LSQ would fit better than a straight LSQ, as is the case in the radial data.
However, this is so for the transverse data, implying that the uncertainty in the
transverse resolution function is too great to adequately define its shape, which is
the case due to the uncertainty in the shape of the mosaic.
The difference in the chi squared values between the LSQ and lorentzian to
the mth power during coarsening is never so great as to make one believe that the
peak is not a LSQ shape. In fact, the dependence of chi squared on m is relatively
flat between about 1.3 to over 3 but rises quickly near 1.0.
Fits to the convolution of the resolution function and the first theoretically
based shape discussed have not been done in the transverse direction. It is unlikely
that such a fit would be an improvement over the non-convolution fit, because this
shape is similar to a LSQ in the tails and the convolution fitting with a LSQ
function did not improve the fits in the transverse direction. However, despite the
computer time needed for such a convolution fit a few have been done for the radial
data. Improvement in the chi squared values in the convolution fits relative to the
non-convolution fits for this first theoretical shape were similar to what was found
in the LSQ shape. The chi squared values for the LSQ function at late times are
approximately half those of the OJK theory. Keeping in mind that the justification
for assuming a LSQ peak has always been that the tails decay as q-4 as OJK theory
predicts, it is some what surprising that a LSQ shape actually fits better than the
full theory. However, several approximations were made in the theory that may
explain this. The theory assumes an isotropic system, and a gaussian distribution of
fluctuations of the domain wall positions. The authors point out the dependence of
their results on the choice of this distribution!35)
Another way to see the change in the line shape, and hence, S(q,t) is to scale
the peaks. If the peaks were the same shape, plots of counts divided by the fitted
amplitude vs. position divided by the width, would lie on the same curve. Figure 17
shows some of the raw data for the transverse 365.80C quench with LSQ fits to all
but the earliest one which is fit to a GS. Figure 18 shows the data after being scaled,
the two lines are a scaled LSQ fit and a scaled GS fit. From the scaled data it is
apparent that the earliest peak has a different shape than all the rest. This
demonstrates that S(q,t) can not be scaled from the coarsening regime to the growth
regime, but does scale during coarsening. Moreover, S(q,t) should not scale in this
way during growth when the ratio of the volume of the sample in one of the ordered
phases to the volume in a disordered phase is constantly changing. In order for
scaling to be valid the physical picture after rescaling must be identical to an earlier
time. This is not the case during the growth stage, if the length scale is reduced
during growth the percentage of the sample that is in a disordered state will be
greater than it was at all earlier times.
The crossover in the line shape has been observed in over a dozen quenches.
Most of these quenches were to 3750C (Tc-100C). In one quench to 378C (Tc-70C)
the crossover is present, however, the greatest difference in their respective
chi-squared values while the peak favored a GS line shape was 1.37 to 2.21 where
for deeper quenches ratios of 1 to 9 where found. Moreover, in the 382.00C quench
the first peak mildly favored a GS line shape with chi-squared values of 1.9 and 2.2
for GS and LSQ fits respectively. In the deeper quenches to below 3760C where the
crossover is pronounced the peak does not fit well to either a GS or LSQ shape at
the point the chi-squared values for the GS and LSQ fits cross. In fact, the peak
starts off GS and then continuously changes until it is LSQ, but during the change
8
x 7.60 min.
375.60C o 22.75 min.
03 47.95 min.
S122.72 min.
m 45 A 514.15 min.
2-
O
Position (x 102 A')
FIG. 17. Raw data on the [0,1,0] bulk superlattice diffraction peak. Solid
lines are fits to the data. For the 7.60 min. scan a gaussian fit is
shown, the rest are lorentzian- squared fits.
1.1
x 7.60 min.
375.6oC 0 22.75 min.
37 .0D 47.95 min.
So 122.72 min.
O0.7 A 514.15 min.
O
-0. -
Transverse
-0. 1 1 1 11
-5 -3 -1 1 3 5
Scaled Position
FIG. 18. Data shown in fig. 17 scaled by amplitude and width. The solid
lines describe a gaussian and a lorentzian- squared shape. The 7.60 min.
peak is gaussian, the rest are lorentzian- squared.
it is neither. The rate of change seems to be slower with lower temperature, in fact
for quenches above 3760C it seems as if the change to LSQ was almost complete by
the end of the first scan. In a typical quench 300 second scans were taken at the
start of the experiment, attempts with shorter scanning times only made the effect
less dear, due to the shape of the peak being statistically less well defined.
Moreover, for quenches close to Tc, the intensity of the peak is weaker than it is for
deeper quenches, making it more difficult to see this change at higher temperatures
for the same reason as above.
The change in line shape also occurred earlier with each quench. This
suggests that the coarsening stage is reached sooner if the sample has been ordered
for long times and disordered at temperatures within (Tc+400C) prior to the
quench. If the sample is heated to 7000C for 10 or more hours the change in line
shape is again found to take place over the original longer time period. Coarsening
being reached sooner suggest that nucleation and or growth is faster in samples that
have not been annealed at high temperatures than in ones that have. This effect
may involve the migration of vacancies to domain wall and grain boundaries during
coarsening.
The change in line shape although present in the radial data is not as clear as
in the transverse data. There are two reasons for this. The first is that the radial
data was taken after the transverse data and the history dependence discussed above
would lessen the effect. The second is that the width in the radial direction is
narrower being due to only type 2 walls. Therefore, in the radial direction
resolution is more important. Type 2 walls being farther apart may also have an
effect on the peak shape.
Figure 19 displays the integrated intensities (fitted amplitude multiplied by
fitted width) versus time after a thermal quench for the transverse directions at
0 2 4 6 8
time (x104
sec.
FIG. 19. Intensity (width of convolution fits multiplied by the amplitude of
the fit) vs. time. The 380.0C data shows a delay of approximately 1900
seconds that corresponds to a period of nucleation. The data shown was
taken in the transverse direction at the [0,1,0].
10
several temperatures. The 380.0OC transverse data shows a delay immediately after
the quench, similar delays have also been observed in other quenches to
temperatures closer to T This delay is a result of an incubation time for
nucleation. Such delays have been clearly observed in single grains of Mg3In by
Konishi and Noda(106). The delay time depends on the sample history which for
the bulk sample is shown in figure 20. In light of the history dependence discussed
above it is not too surprising that the deepest quench in which a delay is seen is the
first one after etching and annealing at 7000C. After the delay a period of rapid
growth in intensity is observed corresponding to the linear rise with time. At late
times the intensity starts to saturate, asymptotically approaching some maximum
(I (T)) that increases with decreasing temperature. The intensity after the delay
fits well to
I(t) = I[1-exp[-(t-t)/T]]1. (4.2)
Figure 21 shows the data from 19 scaled by plotting I/I vs. t/b where b is the time
for the intensity to reach I /2. As expected,(43) the data falls on a universal curve.
In theory, one could use r from the fits to equation 4.2, however, that proved to be
less accurate. The reason for this is simply that the r in the fits corresponds to a
crossover time, which is early and our time resolution, being at best 1/2 the time to
take a spectrum, is a sizable percentage of r.
The delay seen in the integrated intensity is also in the amplitude. A plot of
amplitude vs. time is shown in figure 22. The amplitude used in the plot is the
convolution fitted amplitude and as such is corrected for resolution effects in the
direction of the scan. The amplitude is proportional to the square of the order
parameter and should scale as L(t)d.. Therefore, the amplitude should have a time
dependence td/2. However, the data has only been corrected for resolution effects in
one direction and the measurement approximately integrates over the other two
Sample History
8.0
6.0
4.0
2.0
U
0
C'M
c3
0
'I
S
(D3
0.3 0.7 1.1 1.5
time (x103 hours)
FIG. 20. Bulk Cu3Au sample history starting from etching and annealing in an
Ar atmosphere at 7000C to end of radial quenches. An etching and
annealing followed by quenches similar to above was done prior to
what is shown.
-0.1 !
.0
0.8
8
0.6
0.4
0.2
0 10 20 30 40 50 60 70
time/time(L/2)
FIG. 21. Scaled intensity vs. scaled time. The nucleation period for the 380.OC
data was subtracted from the time before scaling. The unsealed data is
shown in fig. 19.
Transverse Amplitude
0.3 0.6 0.9
1.2
time (x105
sec.)
FIG. 22. Convolution fitted amplitude vs. time for the temperatures
shown. The delay seen in the plot of the integrated intensity
is also seen here.
3
<
<^
0.0
dimensions, therefore, the measured amplitude should roughly scale as L(t)d-2 and
have a time dependence of t1/2. A plot of Logl0(amplitude) vs. Logl0(time) is
shown in figure 23. The late time slope for the deeper quenches is 0.46, a value close
to the value found for the time dependence of L as will shortly be discussed. Trying
to extract the time dependence of L from the amplitude would not be a good
method due to the uncertainty of the resolution effect. However, it is reassuring
that a value close to the expected value of 1/2 is found, indicating that the scaling
involved is as expected from theory. The curvature in the Log-Log plot is at least
in part due to the resolution function.
From a simplistic Ising model calculation it might be argued that the
integrated intensity should be constant and that in the previous graphs the rise in
integrated intensity is due to the resolution function integrating over more of the
peak as the peak becomes narrower. This is not the case. First of all the integrated
intensity is calculated from the product of the fitted width and the fitted amplitude,
not from the sum of all the counts. Hence, the intensity in the tails is accounted
for(at least in the direction scanned). Figure 24 shows the integrated intensity,
amplitude and constant background as a function of time. In this multiple plot the
background has been multiplied by 1000 to plot it on the same scale as the
amplitude. The fact the fitted background is 1000 times or more smaller than the
fitted amplitude implies that the missing intensity would have to be spread out in a
volume 103 times larger than what was effectively covered. It is obvious that some
of the intensity is being falsely counted as the background due to the rise in the
background with time. However, this change in fitted background is in the wrong
direction because the argument says intensity is missing at early times not late.
This is a different problem that is discussed in connection with power law growth.
I 4
375.60C
So l 365.8C
2.0 2.8 3.5 4.3
Logo[time (sec.)]
FIG. 23. Loglo(amplitude) vs. Loglo0(time) for the
same data at in figure 22.
0.0 0.2 0.4 0.6 0.8
time (x105 sec.)
1.0
FIG. 24. Plots of integrated intensity, amplitude and fitted constant
background. The background has been multiplied by 103 in order
to plot it on the same scale as the amplitude. The integrated
intensity has also be scaled to fit on the graph.
The background should probably be quadratic due to a diffuse peak arising from
thermal fluctuations. Moreover, the fitted background for the second and third
points is slightly negative, once again indicating that if any thing the change in
intensity is greater than suggested by the plots.
The above argument shows that any missing intensity can not be coming
from the direction that was scanned. The resolution function in the vertical
direction is large which strongly implies that any missing intensity is lost in the
radial direction. As it happens several quenches with radial scans using the graphite
monochromator and analyzer (what has been called the transverse set up) were
tried. The resolution in that direction turned out to be too large; within an hour
after a quench the instrument was resolution limited. The width after six hours
suggests that the resolution was close to .02R-I(FWHM). Remember that at the
[0,1,0] the radial direction probes the thin part of the scattering disk. Based on this
value for the radial resolution the intrinsic peak width is equal to the instrumental
resolution width 16 minutes after the quench. At this point the peak should be
almost completely integrated over (for the deeper quenches). Some of the intensity
could still be lost in the tails due to differences in line shapes. The intrinsic line
shape at this time is approaching that of a LSQ and the instrumental resolution
(figure 11) is some thing between a gaussian and a LSQ. A time of 16 minutes
corresponds to the fourth point in figure 24, which has an intensity of 35 percent of
the final intensity. Hence, about 65 percent of the change in intensity is real. An
even stronger argument can be made for the radial data. A change in integrated
intensity has also been seen in other systems by N. Wakabayashi(107)
Figure 25 shows a plot of logl0(L) vs. log10(t) whose slope is "a" in;
L(t) a ta (4.3)
In table 2 are listed the exponents "a" for curvature driven growth, as a function of
Bulk Cu3Au
3.75
3.50
3.25
3.00
2.75
2.50
2.
6 3.2 3.8 4.4
Logo[time (sec)]
FIG. 25. Log10(Length) vs. Logl0(time after the quench). L x 2r/r where F is
the LSQ fitted width. Solid lines are linear fits whose slope corresponds
to the exponent "a" for power law growth.
I I
I I
temperature and type of background. For the radial data the order of the
polynomial background was not important because only the constant term gave any
sizable contribution. The exponent for the radial data was found to be .50*.03 and
temperature independent. Here, the error in "a" was estimated from the effect of
changing the resolution correction, as well as the standard deviation as calculated
from values at different temperatures, which was 0.014. An error in the resolution
correction would contribute a systematic error, however, this correction as discussed
earlier is dominated by the instrumental resolution, which is well known. This is
not the case in the transverse direction, where the resolution correction is less well
defined and the order of the polynomial background seems to affect the result.
Table 2. Exponents "a" for power law growth (equation 4.3).
direction I"' transverse I'' radial
background constant linear quadratic ,constant
temperature "a" "a" "a" "a"
382.00C 0.45 0.47 0.49 0.49
380.00C 0.54 0.56 0.58
375.60C 0.40 0.40 0.45 0.51
370.80C 0.42 0.43 0.46 0.51
365.80C 0.40 0.43 0.46 0.49
average 0.44 0.46 0.49 0.50
std. dev. 0.06 0.06 0.05 0.01
It should be noted that the 3800C transverse data is inconsistent with all the
other data, yielding a significantly larger exponent than the others. The reason for
this is unknown. However, this set of data was from the first quench after etching
and annealing of the sample, and hence, can be separated temporally from the rest
of the data. During the experiment the filament in the x-ray source burnt out and
was replaced, however, this happened late in the series of quenches rather than after
the first one. However, changes in the filament before failing may be important.
Evidence for the change in "a" being due to the filament can be found in the fact
the exponents "a" can be correlated to the order that the data was collected. The
value obtained for "a" in the transverse direction decreases with time up till the
filament was replaced then goes up. This could be a result of the filament sagging
with time, and hence, changing the effective resolution. Movement of the x-ray
beam would have at least two effects, one it would very slightly change the area of
the sample considered, perhaps having an effect through the mosaic, two it would
misalign the sample. However, the most likely way that the resolution would be
changed is simply by increasing the spot size on the anode that the x-rays are
emitted from. A lower value for "a" would be consistent with an under correction
for resolution. Furthermore, this change in "a" cannot be correlated with
temperature. The filament was replaced before the 3820C quench and after all the
others listed in table 1. Consequently, the instrumental resolution, which was
measured after the series is better known for the 3820C data. As can be seen in
table one the exponents "a" in the transverse direction found from assuming a
constant or linear background are smaller than those from a quadratic background.
All values of "a" in the transverse direction are less than 0.5 except for those
connected with the 380 C quench, which are all above 0.5. In any case, the average
values for each type of background are within their respective standard deviations to
each other and 0.5 in spite of the systematic errors in determining "a" from the
transverse scans. More importantly it was type 2 domain walls that were suspected
of not following curvature driven growth. Type 2 walls are probed by radial scans,
and hence, "a" is known to a high degree of accuracy for the domain walls in
question. This implies that curvature-driven growth applies to different types of
walls and confirms scaling in an anisotropic system.(27)
The possibility of a lower exponent for the transverse data might be
explained as being due to impurities or incorrect stoichiometry. Random
imperfections lead to slower growth, possible a log(t) growth law.(81,83,84,108)
However, a systematic error in the resolution correction is the more likely
explanation, especially in light of the fact type 2 walls are not affected.
The Experiments on Sputtered Films and the Effect of Stoichiometry
Experimental Details
A single target of Cu3Au was used to produce the fine grained sputtered
films on substrates of 0.13 mm thick HN kapton or 0.13 mm thick A1203. A glass
slide was placed in the chamber with the other substrates so that film thickness
could be determined with a stylus profilometer. Electron-microprobe techniques
verified the stoichiometry and Rutherford back scattering showed the composition
to be uniform as a function of depth. The film characteristics are independent of
which substrates was used. The stoichiometric film (film A) was deposited on
kapton to a thickness of 7500 300 angstroms, with a composition
Cu0.75 .01Au.25.01 as determined by electron microprobe techniques.
The nonstoichiometric film (film B) was deposited to an initial thickness of
10,000+\-400 angstroms on an A1203 substrate. Subsequently the film was placed
in contact with a clean Cu block and annealed under vacuum for 2.5 hours at 5000C.
The clean Cu block was the heating rod previously discussed in connection with the
furnace. The tantalum foil used in later experiments to shield the sample was added
due to experience with this film. After the data was collected and the transition
temperature was found to be 2990C the film was recharacterized by
electron-microprobe techniques, the results showed the sample to have a
homogeneous concentration profile of Cu0.79.01 Au0.21 .01. This stoichiometry is
consistent with the transition temperature.
Preliminary x-ray characterization of the films found the [1,1,1] direction to
be normal to the substrate to within a mosaic spread of approximately 0.2 A-1 or
30 half width at half maximum (HWHM). The in plane orientations were random.
The lattice constant at room temperature was 3.74 A. Prior to the first quenches
both films were annealed at 3500C for over 12 hours in a vacuum. Earlier
measurements on other films showed changes in the relative intensities of the [1,1,1]
and in plane fundamentals as a result of the first annealing, implying an
improvement in the orientation of the grains along the [1,1,1] direction.
Initially kapton substrates were used. Kapton is transparent to x-rays, easily
cut to fit any sample mount, and inexpensive. However, it bows when heated to
high temperatures. Heating the kapton prior to film deposition may have helped,
but it must be heated in a vacuum. Even in a vacuum of a few millitorr the kapton
would deteriorate over several weeks if held above 350C, as was done during the
experiments. Another problem with kapton substrates is its low thermal
conductivity, most of the thermal conduction was in the film itself. Substrates of
Al203 have better thermal properties than kapton for these experiments.
Kapton was also used for the x-ray windows on the first version of the
furnace which was used in connection with all the sputtered films. Although the
vacuum obtained with kapton windows is a little poorer than with beryllium
windows, they are considerably less costly and worked very well in the testing of the
design of the furnace, prior to manufacturing the present version. Moreover, kapton
unlike beryllium is reasonably transparent to visible light allowing visual
observations of the interior of the furnace which helped pinpoint weakness in the
design. The only other relevant difference between the furnace described for the
bulk sample and this one is that the sample mount was a frame not a plate, allowing
the diffraction peaks to be observed in a transmission geometry. Improvements in
the heating element are discussed in connection with the experiments on the M.B.E.
films where the changes where implemented.
The data was collected in the radial direction with a PSD as described earlier
and shown schematically in figure 10a. The instrumental resolution was .060
(HWHM in 20) or 0.0042X-1 at the [1,0,0], and 0.0041A-1 at the [1,-1,0].
Quenches on both the [1,0,0] and [1,-1,0] were done, in order to observe both types
of walls. Radial scans widths of the [1,0,0] are broadened by type 2 walls. The
[1,0,-1] peak width is affected by both types of walls but is dominated by type 1
walls. The notation [1,-1,0] instead of [1,1,0] is used to indicate that it is
perpendicular to the [1,1,1], or in the plane of the film. The transition temperature
Tc for each sample was determined from the disappearance of superlattice peaks
upon heating. The samples were heated well above Tc to ensure complete disorder,
then cooled to an initial temperature Ti > Tc. After establishing equilibrium the
films were quenched rapidly to Tf < Tc. X-ray scattering scans were collected and
stored sequentially at fixed Tf.
Analysis for Data on the Sputtered Films
In analyzing the data a gaussian shape was assumed for all the fitted peaks.
Superlattice peaks from a bulk sample are lorentzian squared like, not gaussian.
The shape of the peak due to the existence of anti-phase domains is difficult to see
experimentally in films due to the contribution from finite size. A gaussian shape,
however, can be supported. Each grain in the film that contributes to the
scattering, will in general be out of phase with the rest, producing the superposition
of (possibly LSQ shaped) peaks with a random distribution in phase. The resulting
peak should be well approximated by a gaussian. Furthermore, the peaks as
measured where gaussian. Therefore, the data was fit to gaussians, and then
instrumental resolution, finite size and strain were subtracted in quadrature,
(4.4)
where c is the correction from instrumental resolution, finite size and strain. The
values of rc are listed with the results. Instrumental resolution was measured by
the width of a Si [1,1,1] as was done with Ge for the bulk data. The contribution
from finite size and strain was estimated from measurements of fundamental Bragg
peaks in the same directions in reciprocal space as the superlattice peak. The
measurements from the fundamental peaks were not sufficient to consistently
determine the corrections for these samples. The actual correction used was a
probable upper bound, which as discussed below, may cause a cause a systematic
error in analyzing the data.
The time dependence is analyzed by fitting to functional forms as discussed
in chapter four, specifically power law (equations 2.5 and 4.3) and logarithmic
(equation 2.12) time dependence. The comparisons made between samples are such
that the quench depths are the same. The quench depth being the same means that
the absolute temperatures are 1870C apart. In order to rule out the possibility of
any effect being due to the difference in absolute temperature one quench to 2780C
on the bulk sample was done. The results indicated curvature driven growth as is
found at higher temperatures.
Results from the Sputtered Films
Some typical raw data from a time resolved run on the Cu rich film are
shown in figure 26. The data on the stoichiometric film is similar but with better
signal to background. Figure 27 shows log-log plots of L vs t for the [1,0,0] and
[1,-1,0] superlattice peaks in both film A and B. The quench depth (T -Tf) is
roughly the same for each run shown. In both films the [1,0,0] peaks are much
narrower than the [1,-1,0] peaks, due to the anisotropy of the domains. These
results support, although not as strongly, the isotropic growth found in the bulk
sample.
Comparing the results from the different samples, it is clear that domain
growth proceeds much faster in the stoichiometric film. Fitted growth exponents, a
in equation 4.3, are listed in Table 3. The quoted errors are dominated by estimates
of the systematic error arising from uncertainty in the resolution and finite size
corrections. The stoichiometric film has a 0.4, roughly consistent (within
experimental uncertainties) with curvature driven growth (a = 1/2). On the other
hand, values of a 0.2 are obtained for the film with extra Cu. This cannot be
reconciled with curvature driven growth.
Table 3. Quenches and fitted exponents.
film peak Tf(oC) Ti (C) Tc (C) c (channels) a [Eq. 4.3] m [Eq. 2.12]
A 1,-1,0) 365 393 386 17.1 0.42*0.08 4.9
A 1,0,0) 364 396 386 14.8 0.41*0.11 4.2
B 1,-1,0) 278 320 299 13.5 0.23*0.04 2.2
B 1,0,0) 283 320 299 19.0 0.18*0.06 1.9
At the (1,-1,0) peak position, 1 channel corresponds to AQ=8.51 10 X-1.
At the (1,0,0) peak position, 1 channel corresponds to AQ=8.71x104 x-1
88
Cu.79Au21 T-T=210C
-- *2000
(1,--I, 0)
1500
() -- 1000
()-- 1000
C-3
(a)- 1000
I I -- 1000
1000
0 250 500
20 (channels)
FIG. 26. Typical [1,-1,0] profiles from film B quenched to 2780C. For clarity
of presentation, the data points are averaged in groups of ten, and
spectra are offset in increments of 500 counts. Counting times are
1000 sec. for each 1024 channel scan. Only the central 500 channels
are shown. Each channel corresponds to AQ=8.51x10 A -1. The solid
lines are gaussian fits. Times after the quench: (a) 1.50x103 sec.,
(b) 3.48x103 sec., (c) 6.48x103 sec., (d) 3.65x104 sec., (e) 1.70x105 sec.
S2.8 -
+ o
0
0
CD
0000
C_ 2.0 o0
2 +(1, 0, 0)
0o(1- 0) -
1.6
3.0 4.0 5.0
1o got (sec
FIG. 27. Logl0L (2r/1 in X) vs. Log(t) (time after quench in seconds) for
film A (upper curves) and film B (lower curves). The solid lines are
power law fits to the late- time data. The discontinuity in the film B
1,0,0] data is a known experimental artifact.