TIMERESOLVED MEASUREMENTS OF THE UNDERPOTENTIAL DEPOSITION OF COPPER
ONTO PLATINUM(111) IN THE PRESENCE OF CHLORIDE
Adam Craig Finnefrock
Abstract
We have studied
in situ the ordering of a twodimensional CuCl
crystal electrodeposited on a Pt(111) surface. We simultaneously measured
highresolution timeresolved xray scattering and chronoamperometric
(current vs. time)
transients. Both measurements were synchronized with the leading edge of an
applied potential step that stimulated the desorption of Cu and subsequent
ordering of the CuCl crystal. In all cases, the current transient occurs on
a shorter timescale than the development of crystalline order. The
timedependent xray intensity data (
$2\times {10}^{4}$ data points) are well fit
by an Avramilike function with only three parameters. By performing a series
of voltagestep experiments, we demonstrate that the ordering time diverges
with applied potential
$\phi $ as
$\tau ~\mathrm{exp}[1/(\phi {\phi}_{0})]$,
consistent with the nucleation and growth of twodimensional islands.
Monitoring the timedependent widths of the xray peak, we see a narrowing
corresponding to the growing islands.
Adam Craig Finnefrock was born in Long Beach, California in 1970. Before the
age of 18, his family had moved over a dozen times. In spite of this
emotional trauma (or perhaps, because of it) the young preppie left New
England to study in Houston, Texas in 1987. He graduated from Rice University
with bachelor degrees in Physics and Mathematical Sciences in May 1992. He
matriculated to Cornell University and joined Professor Joel Brock's research
group in late 1992. He will be taking a postdoctoral position with Professor
Sol Gruner, where he will attempt to overcome his crushing ignorance of all
things biological. The culpability for his questionable choice of career
resided on his parents' bookshelves, teeming with optimistic science fiction
from the '60s and early '70s.
The results presented in this dissertation would not be possible
without the assistance of many people.
First, I have worked in the laboratories of Joel Brock for the past six
years. Alternately serving as teacher, taskmaster, advisor, and friend, he
has provided the motivation and means for all of the work described herein.
He has been extremely generous with his time, and has taught me most of my
experimental skills, and what I know of xray scattering.
Professor Abruña has been my unofficial mentor. A primary collaborator, he
has also given me excellent advice throughout and helped me to find the
"larger picture". He has also provided courage (and food, occasionally) in
the face of hardship, on and off of the beamline. His advice on the entire
academic process was invaluable.
Lisa Buller has been my counterpart in the research group of Professor
Abruña. Her assistance was crucial, especially during the earliest stages
of this project, where hours were long and data were few. She has provided
for the practical aspects of the electrochemical results described herein.
Kristin Ringland has been an enormous help throughout this project. She has
been present for virtually all of the data acquisition and has assisted in
many other ways too numerous to detail. Throughout the rigors of grim and
intense synchrotron runs, she has uttered not even one complaint. As has been
remarked, she is still probably "the perfect graduate student". Arthur Woll
has been a constant friend in the lab, and I have appreciated his good humor
throughout. He and I have had many discussions on xray scattering from
surfaces, and have discussed the need for a comprehensive treatment that we
can understand. I hope to see more about this in his dissertation. Emma
Sweetland, the first student to graduate, helped me through the earliest
years. She is also responsible for much of the laboratory infrastructure that
we often take for granted.
Samantha Glazier was the newest member of this collaboration. Her dauntless
enthusiasm and relentless curiosity have been refreshing for us veterans.
I would also like to thank two other xray electrochemists. Mike Toney of IBM
gave me some early encouragement and practical advice on cell design and xray
measurements. Ben Ocko of NSLS has also given me lots of advice, and acted as
the most steadfast critic.
Jean JordanSweet at IBM has been responsible for the beamline where this
xray data was taken. Supervising the steady stream of users, ensuring that
the beamline is ready for use, and repairing the broken/altered equipment
afterwards is a thankless job. I would like to thank her for it.
I would like to thank the members of my special committee, who have taken the
time to read this dissertation, and are likely to be the ones to ever do so.
(Though if you are reading this, who knows?) I really appreciated their
comments and careful readings; they have made this dissertation far better and
more readable that I could have done alone.
I would particularly like to thank the Chair, Carl Franck, for his
encouragement, support, and advice (particularly on the choice of postdoctoral
appointments) throughout most of my graduate studies. I also would like to
thank him for hosting the Easy Physics seminars, which have been an
interesting staple of my physics diet.
This would not be possible without my family, who got me to this stage in the
first place and gave me the tools to continue. Jennifer Mass, my fianceé,
has helped me through this dissertation from the other side of a Ph.D. I
could not have made it without her patience and support throughout the entire
process. I would simply be lost without her.
The U.S. taxpayer has provided generously, if somewhat unwittingly,
for this research. This work was supported by Cornell's Materials Science
Center (NSF Grant No. DMR9632275). Additional support was provided by the
NSF (Grant Nos. DMR9257466 and CHE9407008) and the Office of Naval
Research. The xray data were collected at the Cornell High Energy
Synchrotron Source (CHESS), which is supported by the National Science
Foundation (Grant No. DMR9311772), and at the IBMMIT beam line X20A at the
National Synchrotron Light Source (NSLS), Brookhaven National Laboratory.
NSLS is supported by the U.S. Department of Energy, Division of Materials
Sciences and Division of Chemical Sciences. (Contract
No. DEAC0276CH00016).
Chapter 1
Introduction
The electrodeposition of a metal adsorbate onto a solid surface is a key
aspect of important technological processes such as electroplating and
corrosion inhibition. In a number of cases, metal overlayers can be
electrodeposited onto a dissimilar metal substrate at a potential that is less
negative than the Nernst potential (that required for bulk deposition).
Experimentally, this "underpotential deposition" (UPD) provides a precise
means for quantitatively and reproducibly controlling coverage in the
submonolayer to monolayer (and in some cases multilayer) regime
[
89,
7,
128].
The initial stages of adsorption/deposition, along with the
growth mechanism, dictate the structure and properties of the deposit. UPD is
an important experimental technique for investigating the early stages of
deposition, and the diverse factors that influence it, for several reasons.
First, in contrast to vacuumsurface experiments, the electrochemical
interface provides direct control over the chemical potential of adsorbed
species. This has been recently exploited by Ocko and coworkers [
105]
to study twodimensional Ising lattice dynamics. Second, the charged
doublelayer (section
2.3) produces
enormous (up to 10
${}^{7}$ V/cm) electric fields, capable of driving surface
rearrangements [
106]. Third, UPD is generally reversible. Thus, it
is possible to perform repeated measurements of a deposition/desorption
transition using the same sample and systematically varying the control
parameters.
The strongest interaction in a UPD process is between the metal to be
deposited and the substrate
[
89,
6,
51].
Thus, UPD is usually restricted to the deposition of one monolayer prior to
the onset of bulk deposition; in some systems, however, up to three atomic
layers can be deposited. Although the metalsubstrate interaction usually
dominates, other interactions can also be important. For example, strongly
adsorbing anions in the electrolyte are particularly important as both
anionmetal and anionsubstrate interactions significantly affect UPD
processes. Furthermore, the adsorbed species rarely loses its charge
completely during the early stages of deposition
[
119,
120,
149,
129,
94,
155,
156].
Rather, it becomes completely reduced only when the applied potential is close
to the Nernst potential. This variable charge state alters the electrostatic
interaction between the deposit and the anions. At more positive potentials,
there is a strong attractive electrostatic interaction that disappears as the
metal is discharged. This attractive interaction can produce a metalanion
bilayer on the electrode surface at intermediate
potentials [
156,
123,
91,
133,
131].
In addition to the surface coverage, both the presence of other adsorbates,
especially anions, and the surface structure of the substrate can profoundly
affect the structural and electronic characteristics of the deposit
[
90,
149,
94,
150,
71].
Although there is a great deal of existing work on UPD lattice formation, the
early stages of deposition are not wellunderstood
[
121,
33]. In much of this earlier work, the structure of
a UPD overlayer was determined by transferring the electrode into an
ultrahigh vacuum (UHV) chamber and employing established surface science
techniques such as lowenergy electron diffraction (LEED). However, such
measurements are inherently
ex situ and cannot provide information on
the kinetics of deposition.
Recently,
in situ probes such as scanning tunneling microscopy (STM)
[
92,
64,
76],
atomic force microscopy (AFM) [
93], and surface xray scattering
(SXS) [
101,
132,
133,
139,
140,
107] have been applied to UPD
systems. In addition to eliminating the ambiguity of
ex situ
measurements, they offer the possibility of studying the kinetics of
deposition. Kinetic studies are crucial for identifying the ratelimiting
steps in the electrochemical growth of not only metals but also of
technologically relevant materials such as GaAs [
136] and CdTe
[
135]. Such studies can also provide important tests of the
large body of theoretical work on nonequilibrium statistical physics,
especially on the kinetics of growing surfaces and
interfaces [
17].
The UPD of metal overlayers onto single crystal electrodes provides an
excellent family of experimental systems for studying fundamental aspects of
materials growth. In particular, Cu UPD on Pt(111) has been extensively
studied by a variety of techniques. The process is very sensitive to the
presence of anions and appears to be kinetically controlled. The exact
structure and nature of the overlayer, particularly at intermediate coverages,
has been the subject of some controversy. Based on LEED studies, Michaelis
et al. [
102] identified
the intermediate overlayer as a
$4\times 4$ structure.
However, more recent
in situ anomalous xray
diffraction measurements of the overlayer structure as a function of potential
by Tidswell
et al. [
131] indicate
that the intermediate overlayer structure is a more complicated incommensurate
CuCl bilayer.
Surface xray scattering techniques have been previously applied to UPD. For
example, Toney and coworkers have studied Pb, Tl, and Cu UPD on Ag and Au
surfaces [
101,
132,
133]. In addition, Ocko and coworkers have
studied a variety of equilibrium surface structures as a function of both the
solution concentration of the adsorbate (especially anions) and the surface
charge, with emphasis on gold substrates [
139,
140,
107]. However,
all of these studies have been static in nature and have not addressed the
kinetics of adlayer formation. This is due, in part, to the severe
experimental challenges that such measurements present.
Timeresolved surface xray scattering represents a nearly ideal probe for
studying the time evolution of the overlayer structure during UPD. X rays
in the
$0.5$ Å to
$1.5$ Å region are not significantly absorbed by
aqueous solutions allowing for the
in situ study of the
electrode/solution interface. In addition, the line shape of the scattered x
rays can be interpreted simply in terms of wellknown correlation functions,
allowing direct tests of theory. Using signal averaging techniques, transient
structures with lifetimes as short as a few microseconds can be studied
[
127].
In this dissertation, I report the first timeresolved surface xray
scattering measurements of metal electrodeposition. The specific system
chosen system is the UPD of Cu
${}^{2+}$ onto Pt(111) in the presence of Cl
${}^{}$
anions. Some of the results have been already published
[
65,
78,
4,
66];
inclusion of these results in this dissertation is with the written permission
of these journals.
To my knowledge, these are the only timeresolved xray measurements of any
UPD process. This is not surprising, because these measurements are extremely
difficult to perform. UPD is extremely sensitive to contaminants, requiring
special protocols and rigorous cleanliness throughout the preparations of the
sample, the solutions, and the electrochemical cell. To observe the
scattering from only a single monolayer, a synchrotron xray source is
necessary. Additional scattering from the solution and the film that
contains it can easily overwhelm the signal of interest. These considerations
imply that static xray scattering measurements from the UPD layer are quite
difficult. Compounding this by performing timeresolved measurements of the
nonequilibrium UPD system adds another challenge. The signal to noise ratio
must be sufficiently high in each time bin to obtain useful data. This ratio
can be improved by depositing the UPD layer under voltage control, and then
pulling out most of the solution. This is the traditional method for studying
UPD structures
in situ. However, this configuration completely
prevents further manipulation of the UPD layer; the contact between the sample
face and the other electrochemical electrodes is diminished. The kinetics of
the UPD formation or dissolution are then completely inhibited.
Because of all these difficulties, the conventional wisdom was that
timeresolved xray scattering measurements of UPD were not possible. To
resolve these challenges, we had to develop and successively improve several
aspects of our experiment. The first and most dramatic improvement came in
the observation that annealing the sample at high temperature for up to an
hour dramatically improved the crystal mosaic (an indication of the size and
relative alignment of domains within the crystal). Similar behavior has been
observed in single noble metal crystals in UHV
[
69]. The next limitation was
the misorientation of our samples' faces with respect to the crystal axis.
Ultimately, this imposed a constraint on the maximum terrace size. Following
Joel Brock's suggestion, I designed a crystal polishing apparatus that could
orient the sample in any arbitrary direction with a precision of
0.001
${}^{\circ}$. The sample could be repolished along this axis, producing a
crystal face with the desired orientation. (Other experimental groups in
Clark Hall have adopted our design and procedures to obtain dramatic
improvements in sample quality.) Lisa Buller had a electrochemical cell
built, based upon Mike Toney's original design. The final challenge was to
interface the potentiostat to the rest of our timing hardware and software.
This involved months of programming and testing.
After overcoming these challenges, we were able to obtain very useful and
interesting data. We have studied
in situ the ordering kinetics of the
twodimensional CuCl crystal electrodeposited on a Pt(111) surface. We
simultaneously measured highresolution timeresolved xray scattering and
chronoamperometric (current vs. time) transients. Both measurements were
synchronized with the leading edge of an applied potential step that
stimulated the desorption of Cu and subsequent ordering of the CuCl crystal.
In all cases, the current transient occurred on a shorter timescale than the
development of crystalline order. The timedependent xray intensity data (
$2\times {10}^{4}$ data points) were well fit by an Avramilike function with only
three parameters. By performing a series of voltagestep experiments, we
demonstrated that the ordering time diverged with applied potential
$\phi $ as
$\tau ~\mathrm{exp}[1/(\phi {\phi}_{0})]$, consistent with the nucleation and
growth of twodimensional islands. Monitoring the timedependent widths of
the xray peak, we observed a narrowing corresponding to the growing islands.
This dissertation is organized into chapters as follows.
Chapters
2
and
3 are introductory in nature.
Chapter
2 is an introduction to
electrochemistry, specifically oriented to the phenomenon of UPD. It is aimed
at a physicist who may be unfamiliar with electrochemical phenomena, and the
presentation is from a fundamental perspective. Wherever possible, I have
made analogies to examples familiar to most physicists.
Chapter
3 is a derivation of some xray
phenomena, starting with the classical xray scattering from an electron.
The experimental apparatus and procedures are documented in
chapter
4. These include sample preparation
and data acquisition procedures. Static xray measurements and their
subsequent analysis are in chapter
5. Kinetic
(timeresolved) xray and chronoamperometric (current vs. time) measurements
are found in chapter
6. These data are analyzed
in terms of a nucleation and growth model. Finally, conclusions are presented
in chapter
7. Long derivations and discussions are
relegated to the appendices.
Chapter 2
Introduction to Electrochemistry
1 Introduction
In this chapter, I will introduce and discuss some of the rudiments of
electrochemistry from a physics perspective. The first section introduces the
electrochemical potential. The second section concerns the nature of the
electrode  solution interface, and discusses several models for the electric
charged doublelayer. Then, the behavior and transport of ions in solution is
discussed. The following sections describe bulk deposition and underpotential
deposition. Finally, specific adsorption is explained and adsorption
isotherms are examined.
What is electrochemistry? I like the definition with which Schmickler
[
114] begins his recent book:
Electrochemistry is the study of structures and processes at the interface
between an electronic conductor (the electrode) and an ionic conductor (the
electrolyte) or at the interface between two electrolytes.
The first electrochemical experiments were also some of the first biophysical
experiments. These are the famous studies of electrified frog legs, performed
by Luigi Galvani [
68,
67].
Since then, experimental science has fragmented into a multitude of
disciplines and spawned many industries.
Presently, electrochemical processes are crucial to a wide variety of
commercial processes. These include batteries, which are of great importance
in the quest for lowemission electric vehicles. Corrosion is an
electrochemical process under active study, especially in industry.
Electroplating, for either the prevention of oxidation, or coating one metal
with a more precious one (such as in jewelry) is another process of
importance. Recently, specific multilayer semiconductor structures have been
electrochemically synthesized.
Two more examples illustrate the importance of electrochemistry.
Electroanalytic processes alone account for $68.2 billion worldwide
[
134]. The primary products include
chlorine, aluminum, copper, and sodium hydroxide. Electrolysis of water is
still used in Europe to produce high purity hydrogen and oxygen. The global
production of aluminum consumes the same amount of electricity as 10% of the
United States electricity sales.
Electrochemistry is also crucial to
electrochemical biosensors, which are now used in medical settings. They
monitor the concentrations of various gases dissolved in
the blood (such as carbon dioxide, oxygen, pH) or
electrolyte levels (sodium, potassium, calcium, chloride). These sensors are
portable and give continuous realtime results at the patients' bedside.
Previously, the alternative was to periodically take blood samples and send
them to the hospital lab for analysis.
2 Electrochemical Potential
In analogy to the chemical potential
${\mu}_{i}$, let us define the
electrochemical potential of species
$i$ with charge
${q}_{i}$ in an electric
potential
$\phi $
[
22]:
${\tilde{\mu}}_{i}={\mu}_{i}+{q}_{i}\phi \mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(1)$

Obviously, for a neutral species, the electrochemical potential and the
chemical potential are identical.
For convenience, chemists often separate the chemical potential into a
concentrationdependent term and a concentrationindependent term.
${\mu}_{i}(T,{c}_{i})={\mu}_{0,i}(T)+\mathrm{kT}\mathrm{ln}{a}_{i}(T,{c}_{i})$ 
$(2)$

where
${a}_{i}$ is called the
chemical activity of species
$i$.
Values for the standard chemical potentials
${\mu}_{0,i}$ are tabulated in the chemical reference literature.
Although only
$T$ is listed here, the chemical potential may depend upon
numerous parameters (pressure, pH, etc.), depending upon the nature of the
experiment.
This separation in (
2.2) is reminiscent of the
isolation of the pressuredependent terms in gas mixtures:
${\mu}_{i}(T,p)={\mu}_{0,i}(T)+\mathrm{kT}\mathrm{ln}({p}_{i}/{p}_{0,i})\mathrm{\hspace{0.5em}\hspace{0.5em}},$ 
$(3)$

where the pressures
${p}_{i}$ take the place of the concentrations
${c}_{i}$ and the
pressure ratios
${p}_{i}/{p}_{0,i}$ take the place of the activities
${a}_{i}$.
For ideal gases,
${p}_{0,i}=1$, and the chemical potential of an ideal gas
is recovered [
96],
${\mu}_{i}(T,p)={\mu}_{0,i}(T)+\mathrm{kT}\mathrm{ln}{p}_{i}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(4)$

Likewise, we define the
activity coefficients
${\gamma}_{i}$,
${a}_{i}={\gamma}_{i}{c}_{i}\mathrm{\hspace{0.5em}\hspace{0.5em}},$ 
$(5)$

and require that
${\gamma}_{i}=1$ for an "ideal" solution (like an ideal gas,
there is no interaction among ions of species
$i$), such that the
analogous equation obtains:
${\mu}_{i}(T,{c}_{i})={\mu}_{0,i}(T)+\mathrm{kT}\mathrm{ln}{c}_{i}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(6)$

This approximation is expected to hold in the limit
of very dilute solutions (
${c}_{i}\to 0$),
or when the ions are wellscreened.
Returning to the electrochemical
potential (
2.1), the definition of chemical
activity (
2.2) can be incorporated as
${\tilde{\mu}}_{i}={\mu}_{i,0}+\mathrm{kT}\mathrm{ln}{a}_{i}+{q}_{i}\phi$ 
$(7)$

For a pure phase (for instance, a solid metal electrode), the activity
is unity and
${\tilde{\mu}}_{i}={\mu}_{i,0}$. The electrons within that
metal have an electrochemical potential
${\tilde{\mu}}_{{e}^{}}={\mu}_{{e}^{},0}\phi $. Their concentration never
varies appreciably, so we can ignore the effects of activity within the metal.
3 Electrode  Solution Interface
A complete model of the electric double layer
[
23,
39,
115]
was given by
Bockris, Devanathan, and Müller [
34].
This is illustrated in figure
2.1. It contains positively
charged species adsorbed onto the electrode, polar solvent (water) molecules,
and solvated species both near and far from the electrode surface.
We will take these components in turn, and gradually build up to this complex
arrangement.
Figure 1: BDM model of charged double layer. Adapted from
[
38,
18]. The polar solvent molecules are shown
as ellipses, with the arrows pointing toward the positive end. The
specifically adsorbed (section
2.7) ions of
indeterminate charge are labeled as "q". Note that some of the ions are
solvated, which limit the closest approach distance.
If we apply a negative charge onto our electrode surface, then it will attract
positive ions from solution. This will have the effect of making the
charge of the electrode appear to be less negative to a test charge deep in
the bulk solution. This "charge screening" is exactly analogous to the
screening of point particles described by Debye, which is present in a broad
range of physical contexts. Charge screening is discussed again
in section
2.4.
Throughout this section, the potential far from the electrode (in
the "bulk solution") is set to zero. Relative to this potential, the
electrode is at an electric potential
${\phi}_{0}$.
The earliest model of the double layer was proposed by Helmholtz
[
137,
138]
in 1879.
He considered just the a layer of positively charged ions, tightly bound to
the negatively charged electrode surface. The centers of these ions were
postulated to lie on a
single "Helmholtz" plane at a distance
${z}_{H}$ from the electrode surface.
The resulting potential is identical to that within a capacitor, and is a
linear interpolation between the electrode and bulk potentials, as shown in
2.2.
$\phi (z)=\{\begin{array}{cc}{\phi}_{0}(1\frac{z}{{z}_{H}})\hfill & z\le {z}_{H}\hfill \\ 0\hfill & z\ge {z}_{H}\hfill \end{array}$ 
$(8)$

In analogy with a parallelplate capacitor, the capacitance is a constant
independent of voltage. This is in conflict with experimental observations,
indicating that this model is incomplete.
Figure 2: (a) Diagram of the Helmholtz model. The negative ions adsorbed
onto the surface are shown with solid lines. The positive "image"
charges are shown with dotted lines. (b) Potential
$\phi $ vs. distance
$z$ from the electrode.
An entirely opposite approach was undertaken by Gouy in 1910 [
73] and
Chapman in 1913 [
54]. They proposed that none of the ions were
tightly bound to the surface. Each ion is not constrained to lie in a tight
double layer, but is sensitive to the electric potential formed by the other
ions. In this way, the positions of the ions are not predetermined, but
are the
result of a statistical equilibrium with respect to this potential. This is
only a meanfield model; individual ions are expected to react only to the
overall field produced by all the other ions.
This meanfield model can be analyzed with some rigor.
First, the electric potential
$\phi $ depends on the charge density
$\rho $
as stated by Poisson's equation (in Gaussian units),
$\frac{{d}^{2}{\phi}^{2}}{d{z}^{2}}=\frac{4\pi \rho}{\epsilon}$ 
$(9)$

where the
$\epsilon $ is the dielectric constant of the solution.
The charge density is the sum of the charges from species
$i$
$\rho (z)=\sum _{i}{q}_{i}{c}_{i}(z)\hspace{0.5em}.$ 
$(10)$

In equilibrium,
each charge density follows a Boltzmann distribution, determined by the mean
field
$\phi (z)$,
$c}_{i}(z)={c}_{i}(z=\hspace{0.5em}\infty )\mathrm{exp}\frac{{q}_{i}\phi (z)}{kT$ 
$(11)$

Combining these two equations leads to the PoissonBoltzmann equation
$\frac{{d}^{2}{\phi}^{2}}{d{z}^{2}}=\frac{4\pi}{\epsilon}\sum _{i}{q}_{i}{c}_{i}(z=\infty )\mathrm{exp}\frac{{q}_{i}\phi (z)}{kT}$ 
$(12)$

Using the relation
$\frac{{d}^{2}{\phi}^{2}}{d{z}^{2}}=\frac{1}{2}\frac{d}{d\phi}{\left(\frac{d\phi}{\mathrm{dz}}\right)}^{2}$ 
$(13)$

we can rewrite (
2.12) and integrate to find
$\left(\frac{d\phi}{\mathrm{dz}}\right)}^{2}=\frac{8\pi \mathrm{kT}}{\epsilon}\sum _{i}{c}_{i}(z=\infty )\mathrm{exp}\frac{{q}_{i}\phi (z)}{kT}+\text{constant$ 
$(14)$

To solve for the constant, we apply the boundary conditions that far from the
electrode the potential and its derivative
fall to their bulk value (zero). That is,
$\begin{array}{cccc}\multicolumn{1}{c}{\underset{z\to \infty}{lim}\phi (z)}& =\hfill & 0\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(15)\\ \multicolumn{1}{c}{\underset{z\to \infty}{lim}d\phi (z)/\mathrm{dz}}& =\hfill & 0\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(16)\\ \multicolumn{1}{c}{}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(17)\end{array}$

Then, (
2.14) becomes
${\left(\frac{d\phi}{\mathrm{dz}}\right)}^{2}=\frac{8\pi \mathrm{kT}}{\epsilon}\sum _{i}{c}_{i}(z=\infty )(\mathrm{exp}\frac{{q}_{i}\phi (z)}{kT}1)$ 
$(18)$

In order to continue and keep the equations manageable, it is helpful to
restrict the discussion to a "symmetrical" electrolyte. These are also
called
$z:z$ electrolytes, because they consist of only one cationic and one
anionic species, of equal charge magnitude (often denoted
$z$). The
electrolyte used in these experiments, HClO
${}_{4}$ (which dissociates into H
${}^{+}$
and (ClO
${}_{4}$)
${}^{}$), is an example of a symmetrical electrolyte.
Then (
2.18) becomes
${\left(\frac{d\phi}{\mathrm{dz}}\right)}^{2}=\frac{16\pi \mathrm{kT}{c}_{\infty}}{\epsilon}(\mathrm{cosh}\frac{q\phi (z)}{kT}1)$ 
$(19)$

where
${c}_{\infty}$ denotes the common
${c}_{i}(z=\infty )$, and
$q$ the
magnitude of the charges. Taking the square root,
$\frac{d\phi}{\mathrm{dz}}={\left(\frac{32\pi \mathrm{kT}{c}_{\infty}}{\epsilon}\right)}^{1/2}\mathrm{sinh}\frac{q\phi (z)}{2kT}$ 
$(20)$

We take the negative square root by our assumption that the electrode sits at
a positive potential with respect to the bulk solution. Substitute
$\psi =\frac{q\phi (z)}{2kT}$ 
$(21)$

to rewrite (
2.20) as
$\frac{d\psi}{\mathrm{dz}}=\kappa \mathrm{sinh}\psi$ 
$(22)$

where we have also substituted for the inverse of the "Debye length"
$\frac{1}{L\_\text{Debye}}=\kappa ={\left(\frac{8\pi {q}^{2}{c}_{\infty}}{\epsilon kT}\right)}^{1/2}.$ 
$(23)$

The significance of this length will become apparent below.
Rewriting as
$\int \frac{d\psi}{\mathrm{sinh}\psi}=\int \mathrm{dz}\hspace{0.5em}(\kappa )$ 
$(24)$

and then integrating,
$\mathrm{ln}\mathrm{tanh}\psi /2=\kappa z+\text{constant}$ 
$(25)$

The constant can be determined by using
$\psi (z=0)={\psi}_{0}$, and then
$\frac{\mathrm{tanh}(q\phi /4kT)}{\mathrm{tanh}(q{\phi}_{0}/4kT)}=\mathrm{exp}(\kappa z)$ 
$(26)$

In the limit that
${\phi}_{0}\to 0$, we obtain
$\phi ={\phi}_{0}\mathrm{exp}(\kappa z)$ 
$(27)$

For aqueous solutions,
$\epsilon =78.49$ [
24] at 25
${}^{\circ}$C, and
then
$\kappa =3.29\times {10}^{7}(q/e){c}_{\infty}$, where
$\kappa $ is given
in cm
${}^{1}$ and
${c}_{\infty}$ in M (moles/liter). Our electrolyte is 0.1 M HClO
${}_{4}$,
so
$\kappa =3.3\times {10}^{6}$ cm
${}^{1}$ or
$3.3\times {10}^{2}$ Å
${}^{1}$.
This constitutes an enormous voltage gradient, on the order of 10
${}^{6}$ to
10
${}^{7}$ V/m.
Figure 3:
Normalized potential profiles
$\phi /{\phi}_{0}$ vs.
$z$ for the
GouyChapman model at
${\phi}_{0}$ = 1000 mV (solid), 100 mV (dashed), 10 mV
(dotted). The
${\phi}_{0}$ = 10 mV case is indistinguishable from the
${\phi}_{0}\to 0$ limiting form (
2.27) .
Note that larger values of
${\phi}_{0}$ have steeper descents. I have used
the case of 0.1 M HClO
${}_{4}$, just as in the experiments described later.
For this case, the Debye length is
$L\_\text{Debye}$ = 9.6 Å. Adapted
from a similar figure in [
19].
The GouyChapman model is an improvement over the Helmholtz model, but it does
not take into account the finite size of ions. There must be a plane of
closest approach, just as predicted by Helmholtz. The minimum distance of
this plane from the electrode surface is the ionic radii. If the ions are
solvated, then they will not even be able to approach that closely. Stern
realized this in 1924 [
125] and proposed a model to incorporate
this. Essentially, it is a combination of the two previous models.
Call the distance of closest approach
${z}_{\mathrm{OHP}}$.
(OHP stands for "Outer Helmholtz Plane". The "Inner" plane will be
defined shortly.)
The Helmholtz description applies for
$z\le {z}_{\mathrm{OHP}}$,
and the GouyChapman description applies for
$z\ge {z}_{\mathrm{OHP}}$.
At this boundary, we require continuity in the potential
${\phi}_{\mathrm{OHP}}$ and its derivative.
In the
$z\le {z}_{\mathrm{OHP}}$ region, we have the Helmholtz linear potential drop
$\phi (z)={\phi}_{0}{\frac{d\phi}{\mathrm{dz}}}_{\mathrm{OHP}}z$ 
$(28)$

In the
$z\ge {z}_{\mathrm{OHP}}$ region, the potential follows the previous
PoissonBoltzmann result, but displaced by
${z}_{\mathrm{OHP}}$,
$\frac{\mathrm{tanh}(q\phi /4kT)}{\mathrm{tanh}(q{\phi}_{\mathrm{OHP}}/4kT)}=\mathrm{exp}[\kappa (z{z}_{\mathrm{OHP}})]$ 
$(29)$

If we choose
${z}_{\mathrm{OHP}}$, then we can find
${\phi}_{\mathrm{OHP}}$ by
applying (
2.20) and (
2.28)
at
$z={z}_{\mathrm{OHP}}$ and requiring the continuity of
$d\phi /\mathrm{dz}$ there.
Numerically solving these selfconsistent equations produces a value for
${\phi}_{\mathrm{OHP}}$. The Stern model adds only this one additional parameter,
${z}_{\mathrm{OHP}}$.
Figure 4: Potential profile
$\phi $ vs.
$z$ for the Stern model
at
${\phi}_{0}$ = 130 mV. The transition point is set at
${z}_{\mathrm{OHP}}$ =
5 Å (dashed line), which then corresponds to a transition voltage of
${\phi}_{\mathrm{OHP}}$ = 75 mV. I have used the case of 0.1 M HClO
${}_{4}$, as
in the experiments described later. Adapted from a similar figure in
[
20].
We will discuss the following models only qualitatively. As we add more
components to the model, additional variables are added that are difficult
to measure. But it is important to keep the additional components in mind, if
only to appreciate the difficulty of predicting exact quantitative behavior.
The Grahame model [
74] (1947) includes the possibility of ions
specifically adsorbed (section
2.7) on the electrode
surface. This specific adsorption is chemical in nature, and cannot be
explained simply by electrostatic arguments. These ions can have either
positive, negative, or no charge. We expect, however, that their ionization
state and degree of attraction to the electrode will be influenced by the
electrode's potential. The closest approach of these adsorbates defines the
"Inner Helmholtz Plane".
The BockrisDevanathanMüller model includes (polar) solvent molecules that
bring us to our complete picture shown earlier
2.1.
In retrospect, all of this modelbuilding may seem somewhat
ad hoc. A
contrasting approach has been put forth recently by
Borukhov
et al. [
36]. The
authors start with the
PoissonBoltzmann equation but include the contributions to the free energy
from the finite size of the ions. This remediates some of the defects of the
GouyChapman model and is in agreement with experiments they cite where large
multivalent ions are adsorbed onto a charged Langmuir monolayer.
4 Charged Ions in Solution
Having discussed the electrode surface and its ionic neighborhood, we turn to
the charged ions in bulk solution.
First, the various modes of transport
of charged ions to the electrode surface are discussed and compared.
These play an important role in
the kinetics of deposition at that interface.
Second, the importance of a supporting electrolyte in
electrochemical experiments is described.
4.1 Transport of Ions
In any deposition/growth system, the transport of particles to the surface is
an important consideration. Often, the evolution of the surface morphology is
determined by the relative rates of transport to the surface and reactions at
the surface. Two familiar limiting cases are
diffusionlimited aggregation (DLA)
[
153,
154]
and
kineticlimited
growth (such as the KPZ model
[
86]).
All electrochemical reactions take place only at the electrode surface, so
transport of ions to that interface is of paramount importance.
We expect the current of species
$i$ to be proportional to its respective
electrochemical potential gradient,
${J}_{i}=\frac{{c}_{i}{D}_{i}}{kT}\nabla {\tilde{\mu}}_{i}+{c}_{i}\nu$ 
$(30)$

with the usual diffusion constant
$D$.
$J$ is the flux of ions and
has units of concentration times velocity.
We consider the possibility that the solution itself is in motion, with
velocity
$\nu $.
If the concentrations are small, the ideal gas approximation can be
used (
2.6):
${\tilde{\mu}}_{i}={\mu}_{0,i}+kT\mathrm{ln}{c}_{i}+{q}_{i}\phi .$ 
$(31)$

The electrochemical potentials, which are not directly measurable, have been
replaced by concentrations and the electrostatic potential energy.
Substituting (
2.31) into (
2.30),
we obtain the NernstPlanck equation
${J}_{i}(r)={D}_{i}\nabla {c}_{i}(r)\frac{{D}_{i}{c}_{i}(r){q}_{i}}{kT}\nabla \phi (r)+{c}_{i}(r)\nu (r)\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(32)$

Using the continuity equation
$\frac{\partial c}{\partial t}+\nabla \xb7J=0$ 
$(33)$

we obtain an equation of motion for the concentration profile
$\frac{\partial {c}_{i}(r)}{\partial t}={D}_{i}{\nabla}^{2}{c}_{i}(r)+\frac{{D}_{i}{c}_{i}(r){q}_{i}}{kT}{\nabla}^{2}\phi (r)\nabla \xb7({c}_{i}(r)\nu (r))\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(34)$

These three terms respectively represent diffusion, electromigration, and
convection.
Diffusion refers to the random, Brownian motion of particles that follow
Fick's first and second laws of diffusion:
$\begin{array}{cccc}\multicolumn{1}{c}{J(r)}& =\hfill & D\nabla c(r)\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(35)\\ \multicolumn{1}{c}{\frac{\partial c(r)}{\partial t}}& =\hfill & D{\nabla}^{2}c(r)\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(36)\end{array}$

Fick's first law can be derived by considering the Brownian motion of
particles. A simple treatment can be found in
[
25],
which considers a random walk of fixed step size.
More sophisticated treatments make use of the Langevin equation
[
97].
Fick's second law of diffusion (
2.36) follows from the
first (
2.35) by the continuity
equation (
2.33).
The diffusion constant can be solved for by using a Fourier transform
(appendix
B)
$\u27e8{r}^{2}(t)\u27e9=2dDt$ 
$(37)$

where
$d$ is the spatial dimensionality (
$d=3$ for a typical solution
vessel, although
$d=1$ is probably more appropriate for the thinlayer cell
described in section
4.3.4).
The second mode,
electromigration, refers to the motion of charged ions under the influence of a
The final mode of ionic transport is due to the transport of the solution
itself. Formally, this is termed
hydrodynamic transport, though it
is usually known as convection.
Because of the extremely small solution volume in our xray cell, hydrodynamic
transport is not a consideration. However, it is an important consideration
in larger electrochemical cells. When the solution is stirred or the
electrode rotated, this will give rise to hydrodynamic effects. Rotating
ringdisk
techniques [
50]
make use of this effect, for instance.
Convection must also be taken into account for sensitive measurements that take
place over several minutes and where significant depletion occurs near the
electrode.
4.2 Supporting Electrolyte and Charge Screening
In addition to the species of interest, most electrochemical experiments
incorporate a
supporting electrolyte. This is either the solution or
is dissolved in the solution at a high concentration with respect to the
species of interest. For instance, in our experiments the primary solution
was H
${}_{2}$O, and the supporting electrolyte was 0.1M HClO
${}_{4}$. A brief and
practical discussion of supporting electrolytes can be found in Brett and
Brett [
40].
There are several advantages to using a supporting electrolyte. First,
the
doublelayer does not extend far into the solution; the majority of the
potential drop is very close to the electrode
(section
2.3).
Second, ions are wellscreened. As described by DebyeHückel theory
(see, for example, McQuarrie [
98]), a charge in
solution tends to attracts charges of opposite sign. The gives rise to an
effective "ionic atmosphere" that diminishes the effective net charge
felt by a test charge some distance away.
Third,
because there are far more charged ions in solution, the overall resistance of
the solution is much diminished.
Fourth, most of the current is carried by the electrolyte, not the dilute ions.
This has implications on the dominant
mode of transport.
In any electrochemical system, it is important to determine what
fraction of the measured current is derived from diffusion as opposed
to electromigration.
When both effects are present, the analysis becomes complicated.
However, electrochemical experiments are generally carried out with a
large concentration of a
supporting electrolyte
relative to the concentrations of active species.
The supporting electrolyte does not take part in the
reaction at the electrode, but does carry the majority of the current through
the solution.
The electrolyte serves to screen the ions,
making the "ideal gas" approximation more realistic.
Hence, the deposited ions arrive mostly via diffusion, and
electromigration effects can be neglected [
26].
5 Bulk Deposition
In this section I will discuss the deposition of "bulk" amounts of material
onto an electrode surface. In the next chapter I will turn to
"underpotential" deposition, which occurs at voltages closer to the rest
potential, and is sometimes the precursor to bulk deposition.
As a prelude to understanding underpotential deposition, it is necessary to
understand something about bulk deposition. First I will discuss the Nernst
equation, which determines the onset of bulk deposition. Then I will
introduce the Cottrell equation, which is a simple realization of bulk
deposition. Both of these are covered in the more analytical electrochemistry
texts.
5.1 Nernst Equation
In discussing chemical activities
(section
2.2) and
and electrochemical potentials (section
2.2),
we have already developed the necessary machinery to write down the Nernst
equation. The following treatment parallels Bard and Faulkner
[
27].
Using the relation between chemical potentials and chemical
activities (
2.2), the Gibbs free energy is
$\Delta G=\Delta {G}_{0}+kT\sum _{i}{\nu}_{i}\mathrm{ln}{a}_{i}$ 
$(38)$

In an electrochemical reaction, the change in Gibbs free energy is equal in
magnitude to the electrical energy dissipated or produced. Conventionally,
the sign of the voltage difference
$\Delta V$ is taken to be
$\Delta V>0$
when the reaction is spontaneous (
$\Delta G<0$). Then
$\Delta G=ne\Delta V$ 
$(39)$

The "standard" (when all activities are unity) potential of the reaction
$\Delta {V}_{0}$ is just
$\Delta {G}_{0}=ne{V}_{0}$. Then,
$\Delta V=\Delta {V}_{0}\frac{kT}{ne}\sum _{i}{\nu}_{i}\mathrm{ln}{a}_{i}$ 
$(40)$

This equation, known as the
Nernst equation is of great importance to
electrochemistry. Knowing the stoichiometry of a reaction
${\nu}_{i}$ and the
activities
${a}_{i}$, one can predict the electric potential necessary to drive
the reaction forward (or spontaneously generated). The voltage
$\Delta V$ is
often termed the
Nernst potential. In electrochemistry textbooks,
$\Delta V$ is often written as
$E$, and
$\Delta {V}_{0}$ as
${E}^{0}$. I am using my
notation to emphasize that this voltage is the potential difference across the
electrodes, and to avoid confusion with the electric field.
It is also common to write the Nernst equation in terms of concentrations,
which are easily measured. (The activities are often not known.)
Using activity coefficients (
2.5)
${a}_{i}={\gamma}_{i}{c}_{i}$,
$\begin{array}{cccc}\multicolumn{1}{c}{\Delta V}& =\hfill & \Delta {V}_{0}\frac{kT}{ne}\sum _{i}{\nu}_{i}\mathrm{ln}{\gamma}_{i}{c}_{i}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(41)\\ \multicolumn{1}{c}{}& =\hfill & \Delta {V}_{0}\text{'}\frac{kT}{ne}\sum _{i}{\nu}_{i}\mathrm{ln}{c}_{i}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(42)\end{array}$

where the
formal potential
$\Delta {V}_{0}\text{'}$ incorporates the activity coefficients. The
formal potentials for various reactions can be easily measured and are
tabulated in the literature.
Bulk deposition is just an example of these reversible reactions we have been
discussing. Consider the deposition of copper ions from solution onto an
inert electrode. This is written as
$\mathrm{Cu}}^{0}(\mathrm{ads})\leftrightarrow {\mathrm{Cu}}^{2+}(\mathrm{sol})+2{e}^{$ 
$(43)$

For bulk deposition, we must have
$\Delta V<0$ for this reaction (as
written). For potentials above the Nernst potential, we expect to see no
deposition. As the potential of the working electrode is lowered below the
Nernst potential, deposition is possible. The more negative the applied
potential, the more favored the reaction becomes, and the faster it goes.
Measuring the deposition by the current flow, we find a response as
in figure
2.5.
Figure 5: Cartoon of Bulk Deposition
5.2 Cottrell Equation
Consider a deposition experiment where the potential is abruptly shifted
from above
the Nernst potential (no deposition) to below the Nernst (bulk deposition).
Before
$t=0$, the system is in equilibrium with a mean
concentration of
${c}_{\infty}$ everywhere;
$c(z,t\le 0)={c}_{\infty}$.
At
$t=0$, the voltage is altered such
that deposition occurs. The electrode surface is assumed to be perfectly
adsorbing so that
it is a perfect sink for the adsorbing ions;
$c(z={0}^{+},t>0)=0$.
Each of these adsorbed ions transfers
$n$ electrons to/from the electrode.
Furthermore; the solution container is semiinfinite and hence inexhaustible.
Far from the electrode, the bulk solution concentration will be maintained;
$c(z\to \infty ,t)={c}_{\infty}$.
We must solve the
linear diffusion equation
$\frac{\partial c(z,t)}{\partial t}=D\frac{{\partial}^{2}c(z,t)}{\partial {z}^{2}}$ 
$(44)$

subject to the initial condition
$c(z,t\le 0)={c}_{\infty}$
and the boundary conditions
$\begin{array}{cccc}\multicolumn{1}{c}{c(z={0}^{+},t>0)}& =\hfill & 0\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(45)\\ \multicolumn{1}{c}{c(z\to \infty ,t)}& =\hfill & {c}_{\infty}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(46)\end{array}$

The standard method [
28,
41]
is to Laplace transform (
2.44)
and the initial condition:
$s\hspace{0.5em}c(z,s){c}_{\infty}=D\frac{{d}^{2}c(z,s)}{{\mathrm{dz}}^{2}}$ 
$(47)$

This ordinary differential equation has the solution
$c(z,s)=A(s)\mathrm{exp}[(s/D{)}^{1/2}z]+B(s)\mathrm{exp}[(s/D{)}^{1/2}z]+{c}_{\infty}/s$ 
$(48)$

Applying (
2.46) requires
$A(s)=0$ and (
2.45)
then requires
$B(s)={c}_{\infty}/s$. The solution to the transformed
equation is
$c(z,s)=\frac{{c}_{\infty}}{s}\{1\mathrm{exp}[(s/D{)}^{1/2}z]\}$ 
$(49)$

Taking the inverse Laplace transform yields the concentration profile
$c(z,t)={c}_{\infty}\{1\text{erfc}\left[\frac{z}{2(Dt{)}^{1/2}}\right]\}={c}_{\infty}\text{erf}\left[\frac{z}{2(Dt{)}^{1/2}}\right]$ 
$(50)$

Using the continuity equation (
2.33) and assuming each
ion transfers a charge of
$ne$, then the current density
$j(t)$ measured at
the electrode (normalized to electrode area) is
$j(t)=\underset{z\to 0}{lim}=neD\frac{\partial c(z,t)}{\partial z}$ 
$(51)$

The derivative
$\partial c/\partial z$ can be evaluated by applying
the Fundamental Theorem of calculus to the definition of the error function,
$\text{erf}(x)=\frac{2}{\sqrt{\pi}}{\int}_{0}^{x}\mathrm{dy}\hspace{0.5em}{y}^{2}$ 
$(52)$

and then substituting back into (
2.51) to arrive at
$j(t)=ne{c}_{\infty}{\left(\frac{D}{\pi t}\right)}^{1/2}$ 
$(53)$

The concentration profile and current density are plotted in
figure
2.6 for
$D=9\times {10}^{6}{\mathrm{cm}}^{2}/s$,
typical of aqueous solutions. Note that the current is arbitrarily large at
early times (limited by extrinsic factors), and decays away as a powerlaw.
The concentration profile begins as a steep distribution, but broadens at
later times as diffusion makes more ions available.
Figure 6: (a) Current density at the electrode surface from (
2.53).
(b) Concentration profiles from (
2.53), for
$t=0.001$s (solid),
$0.01$s (dotted),
$0.1$s (dotdashed), and
$1$s (dashed).
6 Underpotential Deposition
Imagine that we start at the rest potential and slowly sweep the potential in
a negative direction. In contrast to the cartoon of bulk deposition
(figure
2.5), small peaks in the current
response can be observed. This phenomenon is not ubiquitous. These
peaks are only observed for particular species deposited onto particular
electrodes. In addition, this phenomena is very surfacesensitive. For
instance, the peak positions and heights when Cu is deposited onto Pt vary
depending upon the particular Pt crystal face: (111), (100), or (110). This
process is termed
underpotential deposition, because the deposition
takes place at potentials "under" the Nernst potential (closer to the rest
potential). A potential applied beyond the Nernst potential is often termed
the
overpotential, as in (
6.2).
Figure 7: Indication of underpotential deposition during a voltage sweep.
Although underpotential deposition (UPD) is a complex process, we can
qualitatively justify this behavior. Presuming that a Cu ion has a greater
affinity to bond to a Pt atom than it does to another Cu atom, then we can
imagine that the underpotential deposition situation shown in the top panel
of figure
2.8 would be favored at some potentials for which
bulk deposition, shown in the bottom panel, would not. In the top panel, each
deposited Cu is directly in contact with the Pt surface. In the bottom panel,
the subsequent Cu layers are only in contact with the prior Cu layers.
Figure 8: Difference between bulk deposition and underpotential deposition
In the next section, I present simple models of
"specific adsorption",
of which underpotential deposition is a particular example.
7 Specific Adsorption
In the discussion of the double layer
(section
2.3) we considered the
electrostatic interaction of charged ion species with the charged electrode.
In our most sophisticated model, we assumed that no ion could approach closer
than the radius of its solvation sphere. Ions that do lose their solvation
spheres and penetrate within the outer Helmholtz plane are said to be
specifically adsorbed. Their interaction is more than electrostatic, and is
comparable to a chemical bond. Bard and Faulkner [
29] make
the analogy:
The difference between nonspecific and specific adsorption is analogous to
the difference between the presence of an ion in the ions atmosphere of
another oppositely charged ion in solution (e.g., as modeled by the
DebyeHückel theory) and the formation of a bond between the two solution
species (as in a complexation reaction).
Needless to say, these interactions will be very complex. Models of these
processes need to combine the ionization of charges near surfaces, solvation,
chemical bonding, chargescreening, and surface phenomena such as work
functions.
8 Adsorption Isotherms
Even in the absence of any adsorption, there would be some concentration
(equal to the bulk concentration) of
ions of
species
$i$ in a region near the electrode surface.
The surface excess concentration
${\Gamma}_{i}$ [
116,
30]
is defined to be the concentration of species
$i$ in excess of
the bulk concentration, normalized by the
area of the electrode.
The "coverage"
$\theta $ is defined to be the surface excess
normalized by its saturation value,
$\theta =\Gamma /\Gamma \_\text{sat}$,
so that
$0<\theta <1$.
The definition of this region "near" the electrode surface is somewhat
arbitrary. In principle, it can include the diffuse doublelayer. However,
the
${\Gamma}_{i}$ in the diffuse double layer will have little effect if we have
a supporting electrolyte. First, most of the the ions drawn into the diffuse
double layer will be from the supporting electrolyte, not the species
$i$.
Second, the width of the diffuse doublelayer narrows exponentially as the
concentration of ions increases.
In the limit that the adsorbates on the surface to do not interact, we can
write down a simple model for the adsorption onto a surface. This is very
similar to standard the simplest latticegas models of adsorption discussed in
introductory statistical mechanics courses. I also assume that there is
no interaction between the species and the solution itself, and that there is
no speciesspecies interaction in the solution. If concentrations are not too
high, then the supporting electrolyte screens these charges.
The rate of adsorption is proportional to the concentration of ions in
solution
$c$, the number of sites available for adsorption
$(1\theta )$, and
a Boltzmann factor involving the Gibbs free energy of the activated complex
$G\text{'}$ (see figure
2.9) and the Gibbs free energy of the ion
in solution
${G}_{\mathrm{sol}}$. Isotherms, by definition, are equilibrium
measurements. Hence we usually assume that the system has come to equilibrium
such that the concentration near the electrode is equal to that in the bulk
solution,
${c}_{\infty}$. We are also implicitly neglecting the diffuse double
layer (which is described by the PoissonBoltzmann distribution).
$\text{adsorption:}\hspace{0.5em}\hspace{0.5em}\hspace{1em}\frac{d\theta}{dt}=Kc(1\theta )\mathrm{exp}(\frac{G\text{'}{G}_{\mathrm{sol}}}{\mathrm{kT}})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(54)$

Figure 9: Diagram of energy levels for adsorption process.
In this case, adsorption is energetically favored by an amount
$\Delta G$, but the system must first overcome an energy barrier
$G\text{'}$.
Similarly, the rate of desorption is proportional to the concentration on
the surface
$\theta $ and another Boltzmann factor involving
$G\text{'}$ and the
Gibbs free energy of the adsorbed ion
${G}_{\mathrm{ad}}$.
$\text{desorption:}\hspace{0.5em}\hspace{0.5em}\hspace{1em}\frac{d\theta}{dt}=K\theta \mathrm{exp}(\frac{G\text{'}{G}_{\mathrm{ad}}}{\mathrm{kT}})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(55)$

In equilibrium, these
$\theta $ are constant, and these two rates are equal in
magnitude. Then,
$\frac{\theta}{1\theta}=c\mathrm{exp}(\frac{\Delta {G}_{0}}{\mathrm{kT}})$ 
$(56)$

where
$\Delta {G}_{0}={G}_{\mathrm{sol}}{G}_{\mathrm{ad}}$. This describes the
Langmuir isotherm.
Figure 10: Langmuir isotherms for various values of
$\Delta {G}_{0}/\mathrm{kT}$.
To make a slightly more realistic model, we assume that there is some
interaction between the adsorbates. Using a meanfield approach,
let
$\Delta G=\Delta {G}_{0}+\gamma \theta $. If the adsorbates attract one
another, then
$\gamma >0$. If they repel, then
$\gamma <0$. In the case
that
$\gamma =0$, we recover the Langmuir result. Typically this isotherm is
written as
$\frac{\theta}{1\theta}=c\mathrm{exp}(\frac{\Delta {G}_{0}}{\mathrm{kT}})\mathrm{exp}(g\theta )$ 
$(57)$

where
$g=\gamma /\mathrm{kT}$. This is known as the
Frumkin isotherm.
(Note that I am defining
$g$ as following Schmickler [
117], other
sources [
42,
24] define
$2g=\gamma $. My
$g$ is equal to
$g\text{'}$ of Bard and Faulkner [
24].)
Also frequently used is the
Temkin isotherm, which is
just (
2.57) in the
$\theta \to 1/2$ limit and is
often rearranged like
$\theta =\frac{1}{g}\mathrm{ln}\left[c\mathrm{exp}(\frac{\Delta {G}_{0}}{\mathrm{kT}})\right]\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(58)$

Figure 11: Frumkin isotherms for various values of
$g$.
The dotted line corresponds to
$g=0$, which is identical to the Langmuir
isotherm.
The choice of which isotherm to use depends upon experimental conditions. The
Langmuir isotherm is an accurate description for small coverages (
$\theta $),
or equivalently, small concentrations. In this regime the adsorbates are
sufficiently sparse that they do not interact. Because of the approximation
used to derive it, the Temkin isotherm is only used for
$0.2<\theta <0.8$
and
$g$ not approaching zero. The Langmuir and Frumkin isotherms are
virtually indistinguishable as
$\theta \to 0$, and both are linear in
that regime. These two points become apparent upon
expanding (
2.57) in this limit and keeping terms to
second order,
$\theta +(1+g){\theta}^{2}\approx c\mathrm{exp}(\frac{\Delta {G}_{0}}{\mathrm{kT}})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(59)$

They are also evident from inspection of figure
2.11.
Often, only the terms linear in
$\theta $ is kept, and the resulting equation
is called the "linearized (Langmuir) isotherm".
Chapter 3
Introduction to Xray Scattering
1 Introduction
This chapter provides an introduction to xray scattering, suitable for a
firstyear graduate student studying physics, chemistry, or a related field.
It covers a broader range of material than is necessary simply to interpret
the results in chapters
5
and
6. The reader may safely decide to skip
ahead and return back to the specific sections that are referenced in those
chapters.
2 Generation of X Rays
Early xray experiments were performed with xray tubes, and today most still
are. Presently there are at least two other available sources, both superior
to tubes.
Rotating anode sources, while expensive, can easily fit within a room.
Synchrotron sources are extremely large multiuser facilities, of which
only a handful exist in the world. The benefits include an enormous gain in
flux and angular collimation. Both of these sources were used for
this dissertation, and I will discuss them in turn.
3 Conventional Xray Sources
"Conventional" xray sources [
57] work on the same principle
as Röntgen's original apparatus; electrons are accelerated into an block of
material (the anode), generating xrays. While the xray tube of Röntgen
used electrons ionized from gas, today the electrons are produced by a
highcurrent filament. These electrons are then accelerated by an electric
field into the anode. When struck by the electrons, the anode produces a
broad, continuous spectrum of xrays, due to the electron deceleration within
the anode. This is typically called
bremsstrahlung from the German
"braking radiation".
The more useful spectral components are the "characteristic" radiation
lines, which arise from electronic transitions within the anodic atoms. If an
electron kicks out an electron from an atom, the atom will be in an excited,
ionized state. Subsequently, one of the remaining atomic electrons will fall
into the unoccupied state, releasing an xray photon and conserving energy.
Due to the quantized energy levels, the resultant xray spectrum is also
discrete, and characteristic of the atomic element. The wavelengths are
labeled according to the energy transition. For instance, the
$K\alpha $ lines
correspond to transitions from
$L$ (
$n=2$) to
$K$ (
$n=1$), the
$K\beta $ from
$M$ (
$n=3$) to
$K$, and
$L\alpha $ from
$M$ to
$L$. The principal quantum
number here is denoted by
$n$. These characteristic lines can be exceedingly
narrow (
$<$ 0.001 Å), so it is possible to have nearly monochromatic
radiation for an xray experiment. Sometimes it is even possible to resolve
the lines even further. The
$K\alpha $, for instance, can split into the
$K{\alpha}_{1}$ and
$K{\alpha}_{2}$. These correspond to transitions from
$L$ states
with slightly different energies (the fine structure).
The intensity (per
$d\lambda $) in the characteristic lines is higher than the
bremsstrahlung by a few orders of magnitude. Nevertheless, the flux
per solid angle is low because the radiation is spread isotropically into all
directions. The overall intensity can be boosted by using the highest
electron beam current possible. In practice, this requires both watercooling
the anode and rotating it to prevent a single focus spot from overheating.
These
rotating anodes provide the highest flux presently available in a
"bench top" laboratory setting.
For the work presented in this dissertation, a Rigaku (Model RU200) rotating
anode was used primarily for orientation of samples, training, and as a
testing bed for the experiments. This instrument has a tungsten filament
that can support up to 200 mA current. The electrons are accelerated over
potentials as large as 60 kV into a rotating, watercooled copper anode.
4 Synchrotron Xray Sources
A synchrotron xray source begins with an ultrahigh vacuum (10
${}^{9}$ Torr)
storage ring. Within the ring are electrons circulating at nearlight speeds.
Whenever a charge is accelerated (for instance, if constrained to a circular
path) it emits radiation. In doing so, it loses energy. To keep the
electrons moving in stable orbits, energy in the radio frequency range is
added at intervals synchronized with the electron "bunches". While
electromagnetic radiation is produced for any acceleration, it is advantageous
to place additional accelerating devices at specific locations. In our
experiments, these were simple "bending magnets" that sharply steer the
electron beam. More sophisticated devices, such as "wigglers" and
"undulators" cause the electron beam to be accelerated up and down several
times within a narrow spatial region. This leads to a corresponding increase
in the intensity of the overall xray beam delivered.
A brief, if dated (1979), review of synchrotron radiation can be found in
[
56]. An even shorter overview is presented in
[
87]. A thorough and very
recent (not yet in print) account of synchrotron radiation and related devices
is [
77]. The remainder of this
section will use a few results derived in
[
130].
The most striking feature of synchrotron sources is the high degree of
collimation (unlike conventional sources, which radiate into all
$4\pi $ solid
angle). The "opening angle" for the radiation is peaked sharply forward and
determined by the speed of the electrons. The fullwidth at halfmaximum is
[
130]
${\theta}_{\mathrm{FWHM}}\approx 1/\gamma \mathrm{\hspace{0.5em}\hspace{0.5em}},$ 
$(1)$

where
$\gamma =\frac{1}{\sqrt{1{v}^{2}/{c}^{2}}}$ 
$(2)$

is the usual relativistic Lorentz factor [
81]. For a 2 GeV
beam of electrons (whose rest mass is 511 eV),
$\gamma \approx 4000$, leading
to an opening angle of
${\theta}_{\mathrm{FWHM}}=0.{015}^{\circ}$.
For a comparable energy resolution, synchrotrons deliver
${10}^{6}$ 
${10}^{7}$
times the
flux of rotating anodes.
A second feature is that synchrotron radiation has different polarization
characteristics from the unpolarized conventional sources. The ratio of
power in the parallel polarization (in the plane of the synchrotron ring)
to that in the perpendicular polarization
is
$\frac{{P}_{\parallel}}{{P}_{\perp}}=\frac{6+(v/c{)}^{2}}{2(v/c{)}^{2}}$ 
$(3)$

where
$v$ is the electronic speed. For a highly relativistic beam, the
electric field is mostly (by a factor of seven) polarized in the plane of
the electrons' motion.
The third feature is that synchrotron xray radiation has a broad, continuous
energy spectrum. The upper value is limited by the electronic velocity, but
the range is also dependent upon the characteristics of the beamline
acceleration device and "optics" within the beamline itself. Some
experiments make use of the entire multifrequency ("white") beam. The
majority (including the ones described herein) use a monochromator to select a
comparatively narrow range of frequencies. A typical monochromator consists
of one or more Bragg diffractions (
3.67) from
single crystals or specially engineered multilayer structures.
5 SingleElectron Scattering
Although xray scattering is inherently a quantum phenomenon, many important
features can be correctly derived from a classical treatment
[
141]. A quantum mechanical treatment can be found in
[
46,
112]. We will also
take the nonrelativistic limit and use Gaussian units.
Following the general treatment in
[
82] assume
a "free" charge of magnitude
$q$ and mass
$m$. This is subject to an
incident electromagnetic plane wave of frequency
$\omega $, wavevector
${k}_{1}=k{n}_{1}$, electric field amplitude
${E}_{1}$,
and polarization
${\text{\epsilon}}_{1}$
$\begin{array}{cc}\multicolumn{1}{c}{{E}_{1}(t)={E}_{1}{\text{\epsilon}}_{1}{e}^{i({k}_{1}\xb7x\omega t)}}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(4)\\ \multicolumn{1}{c}{{B}_{1}(t)={n}_{1}\times {E}_{1}}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(5)\end{array}$

As usual for electromagnetic radiation,
${E}_{1}$,
${B}_{1}$, and
${n}_{1}$ are mutually orthogonal.
By the Lorentz force law and Newton's second law, a free point charge
$q$ will
then accelerate as
$a(t)=\frac{q}{m}{E}_{1}{e}^{i({k}_{1}\xb7x\omega t)}.$ 
$(6)$

We know that accelerating charges emit radiation.
Following (14.18) from
[
83], the radiation field observed from a distance
$R$
along a unit vector
${n}_{2}$ is
$\begin{array}{cccc}\multicolumn{1}{c}{{E}_{2}(t)}& =\hfill & \frac{q}{{c}^{2}}{\left[\frac{{n}_{2}\times ({n}_{2}\times a)}{R}\right]}_{\mathrm{ret}}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(7)\\ \multicolumn{1}{c}{{B}_{2}(t)}& =\hfill & {[{n}_{2}\times {E}_{2}]}_{\mathrm{ret}}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(8)\end{array}$

so the radiation will be an electromagnetic spherical wave, of the same
frequency.
The "ret" refers to the fact that the quantities within the
square brackets must be calculated at "retarded" time
$t\text{'}=tR/c$.
Writing the electric field more definitely,
${E}_{2}(t)=\frac{{q}^{2}}{m{c}^{2}}\frac{{E}_{1}}{R}\mathrm{\hspace{0.5em}\hspace{0.5em}}{e}^{i({k}_{2}\xb7x\omega t)}\mathrm{\hspace{0.5em}\hspace{0.5em}}{n}_{2}\times ({n}_{2}\times {\text{\epsilon}}_{1})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(9)$

The instantaneous energy flux is described by the Poynting vector,
$S=\frac{c}{4\pi}{E}_{2}\times {B}_{2}=\frac{c}{4\pi}{E}_{2}{}^{2}{n}_{2}.$ 
$(10)$

and the power radiated per solid angle is (
3.10) multiplied by
${R}^{2}$:
$\frac{dP}{d\Omega}=\frac{c}{4\pi}{R}^{2}{E}_{2}{}^{2}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(11)$

If we define intensity as the average power per solid angle
and recall that
the timeaverage of a sinusoidally varying function over a period is
$\u27e8\mathrm{sin}\omega t\u27e9=\frac{1}{2}$,
then
$I=\u27e8\frac{dP}{d\Omega}\u27e9=\frac{c}{8\pi}{R}^{2}{E}_{2}{}^{2}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(12)$

Now think of this as a
scattering process, and so define a
scattering crosssection.
For the usual particlescattering
situation, this is
$\frac{d\sigma}{d\Omega}=\frac{\text{number of particles scattered into a unit solid angle per unit time}}{\text{number of incident particles crossing unit area per unit time}}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(13)$

In analogy to this, we can define the relevant crosssection as
$\frac{d\sigma}{d\Omega}=\frac{\text{energy radiated into a unit solid angle per unit time}}{\text{incident energy per unit area per unit time}}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(14)$

The denominator in (
3.14) is the timeaveraged incident
energy flux
$(c/8\pi ){E}_{1}{}^{2}$, so
$\frac{d\sigma}{d\Omega}={R}^{2}{\left\frac{{E}_{2}}{{E}_{1}}\right}^{2}={\left(\frac{{q}^{2}}{{\mathrm{mc}}^{2}}\right)}^{2}P({\text{\epsilon}}_{1},{n}_{2})$ 
$(15)$

where
$P({\text{\epsilon}}_{1},{n}_{2})={{n}_{2}\times ({n}_{2}\times {\text{\epsilon}}_{1})}^{2}$ 
$(16)$

is a factor dependent upon the incoming polarization and the observation point.
Using the vector relation
$a\times (b\times c)=(a\xb7c)b(a\xb7b)c\mathrm{\hspace{0.5em}\hspace{0.5em}},$ 
$(17)$

we have
$P({\text{\epsilon}}_{1},{n}_{2})={{n}_{2}\times ({n}_{2}\times {\text{\epsilon}}_{1})}^{2}={({n}_{2}\xb7{\text{\epsilon}}_{1}){n}_{2}({n}_{2}\xb7{n}_{2}){\text{\epsilon}}_{1}}^{2}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(18)$

Figure 1: Diagram for classical xray scattering
Take
${n}_{1}$ and
${n}_{2}$ to define the "scattering plane",
and define
the angle
$2\theta $ to lie between them:
${n}_{1}\xb7{n}_{2}=\mathrm{cos}2\theta \mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(19)$

The initial polarization
${\text{\epsilon}}_{1}$ is orthogonal to
${n}_{1}$;
separate it into components
${\text{\epsilon}}_{\parallel}$ and
${\text{\epsilon}}_{\perp}$
that lie parallel and perpendicular to the scattering plane. Noting that
${n}_{2}\xb7{\text{\epsilon}}_{1}={a}_{\parallel}\mathrm{sin}(2\theta )$ for an
incoming plane wave polarization
${\text{\epsilon}}_{1}={a}_{\parallel}{\text{\epsilon}}_{\parallel}+{a}_{\perp}{\text{\epsilon}}_{\perp}$, we have
$P(2\theta )=1{a}_{\parallel}^{2}{\mathrm{sin}}^{2}(2\theta ).$ 
$(20)$

For an incoming wave with the electric field polarized perpendicularly to the
scattering plane (
${a}_{\parallel}=0$), this polarization factor is unity.
This
case is applicable (
3.3)
to synchrotron radiation
(section
3.4), where the scattering plane is usually
perpendicular to the synchrotron ring. This takes advantage of the high
resolution along that
direction, determined by the opening angle (
3.1).
For a parallel polarization
(
${a}_{\parallel}=1$), this factor becomes
${\mathrm{cos}}^{2}(2\theta )$. Conventional
sources (section
3.3), such as rotating anodes,
produce randomly polarized radiation that has the factor
$\frac{1}{2}(1+{\mathrm{cos}}^{2}(2\theta ))$.
The intensity of scattered peaks also is proportional to
$\mathrm{sin}(2\theta )$.
This factor is a consequence of the Jacobian between angle space and
reciprocal space; a volume element in reciprocal space is smaller than its
generating volume in angle space by a factor
$\mathrm{sin}(2\theta )$. These
polarization factors
$P(2\theta )$ are sometimes bundled into the
"Lorentzpolarization" factor, which is just
$P(2\theta )/\mathrm{sin}(2\theta )$.
6 Scattering from Multiple Objects
In this section, I will discuss scattering from multiple objects. By
considering the phase difference between scattered waves from spatially
separated objects, I introduce the structure factor. Then I discuss the
connection between the structure factor and the correlation
function.
6.1 Fourier Transforms
I assume the reader is familiar with the use of Fourier transforms. This
section merely defines the Fourier transform as I will use it, as there is
some variety in the normalization and sign conventions in the literature.
Throughout this dissertation,
the Fourier transform of a function
$f(r)$ in
$d$ dimensions is
defined to be
$f(q)=\int dr\hspace{1em}{e}^{iq\xb7r}f(r)\mathrm{\hspace{0.5em}\hspace{0.5em}},$ 
$(21)$

and the corresponding inverse Fourier transform to be
$f(r)={\left(\frac{1}{2\pi}\right)}^{d}\int dq\hspace{1em}{e}^{+iq\xb7r}f(q)\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(22)$

Without explicit limits, the integration should be read as taking place over
all space. Some authors place a
$1/(2\pi {)}^{d/2}$ normalization on each
transform, and my choice of assigning the
${e}^{iq\xb7r}$ to the
direct Fourier transform (as opposed to
${e}^{+iq\xb7r}$) is
arbitrary.
This is the common choice in xray scattering texts. In quantum mechanics and
solidstate texts, the opposite sign convention is more common.
From the definition of a planewave
$\psi (q,t)={A}_{q}{e}^{i(q\xb7r\omega t)}$ 
$(23)$

I will likewise define the Fourier transform in the time domain to be
$f(\omega )={\int}_{\infty}^{+\infty}\mathrm{dt}\hspace{1em}{e}^{+i\omega t}f(t)$ 
$(24)$

and the inverse Fourier transform to be
$f(t)=\frac{1}{2\pi}{\int}_{\infty}^{+\infty}d\omega \hspace{1em}{e}^{i\omega t}f(\omega )\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(25)$

6.2 Structure Factors
Take the cross section derived in (
3.15) and
let the charge
$q$ equal the charge of an electron,
$e$. Defining the classical electron radius,
${r}_{0}={e}^{2}/{\mathrm{mc}}^{2}$ = 2.818
$\times $ 10
${}^{5}$ Å, we have
$\frac{d\sigma}{d\Omega}={r}_{0}^{2}P(2\theta )\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(26)$

We see that electrons scatter xrays only weakly (the cross section is very
small). Were it not for the large number of electrons in a macroscopic sample,
and the constructive interference under specific conditions, there would be
little observable signal. The lack of multiple scattering simplifies xray
analysis, in contrast to some electrondiffraction probes.
If we have a collection of
$N$ identical scatterers at various positions
${r}_{i}$, then the phase factors in the electric field amplitudes
(
3.4) and (
3.9) become important. The ratio
${E}_{2}/{E}_{1}$ contains the phase factor
${e}^{i({k}_{2}{k}_{1})\xb7r}$.
For the collection,
$\frac{d\sigma}{d\Omega}={r}_{0}^{2}P(2\theta ){\left\sum _{i}{e}^{i({k}_{2}{k}_{1})\xb7{r}_{i}}\right}^{2}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(27)$

The differential crosssection
$d\sigma /d\Omega $ has units of area, as all
cross sections do. The cross section, and hence the intensity, scales with
${N}^{2}$.
The square of the sum of the phase factors is called the
structure
factor, and is defined as
$S({k}_{2}{k}_{1})=\frac{1}{N}{\left\sum _{i}{e}^{i({k}_{2}{k}_{1})\xb7{r}_{i}}\right}^{2}=\frac{1}{N}\sum _{i}\sum _{i\text{'}}{e}^{i({k}_{2}{k}_{1})\xb7({r}_{i}{r}_{i}\text{'})}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(28)$

It is common to define the "momentum transfer"
$q={k}_{2}{k}_{1}$, in regards to the momentum that
is transferred to the scattered charge. Keep in mind that
$q$
has dimensions of inverse length (true momentum would require a factor of
$\hslash $).
For elastic scattering, the wavevector of the incident and scattered waves
have an identical magnitude
$k=\frac{2\pi}{\lambda}$.
From the simple vector addition in
figure
3.2,
$q=2k\mathrm{sin}\theta \mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(29)$

So we acquire the most common definition of the momentum transfer vector
$q=\frac{4\pi}{\lambda}\mathrm{sin}\theta \mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(30)$

Figure 2: Diagram illustrating that
$q=2k\mathrm{sin}\theta $.
For elastic scattering, the incoming wavevector
${k}_{1}$ and
outgoing wavevector
${k}_{2}$ both have length
$k$. The momentum
transfer
$q={k}_{2}{k}_{1}$. From
[
9].
If the scatterers form a continuous body rather than discrete point particles,
we can rewrite (
3.28) as an integral
$S(q)=\frac{1}{N}{\int}_{V}{d}^{3}{r}_{1}{\int}_{V}{d}^{3}{r}_{2}\hspace{1em}{e}^{iq\xb7({r}_{1}{r}_{2})}\hspace{1em}\rho ({r}_{1})\rho ({r}_{2})$ 
$(31)$

with
$\rho (r)$ as the number density of electrons at position
$r$ and
$V$ as the
scattering volume.
We can separate the integrals in the structure factor so that
$S(q)=\frac{1}{N}A(q)\hspace{0.5em}{A}^{*}(q)$ 
$(32)$

where the "scattering amplitude" is
$A(q)={\int}_{V}{d}^{3}r\hspace{0.5em}{e}^{iq\xb7r}\hspace{0.5em}\rho (r)\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(33)$

If we assume that the product
$\rho ({r}_{1})\rho ({r}_{2})$ is translationally invariant, then
from (
3.31) only the
relative separation
of
${r}_{1}$ and
${r}_{2}$ is important, and we can then shift both
vectors by
${r}_{2}$:
${r}_{1}\leftarrow ({r}_{1}{r}_{2})$ and
${r}_{2}\leftarrow 0$
$S(q)=\frac{V}{N}{\int}_{V}{d}^{3}({r}_{1}{r}_{2})\hspace{1em}{e}^{iq\xb7({r}_{1}{r}_{2})}\hspace{1em}\rho ({r}_{1}{r}_{2})\rho (0)$ 
$(34)$

and by defining the relative separation vector
$r={r}_{1}{r}_{2}$ and using
$\rho =N/V$, then
$S(q)=\frac{1}{\rho}{\int}_{V}{d}^{3}r\hspace{1em}{e}^{iq\xb7r}\hspace{1em}\u27e8\rho (r)\rho (0)\u27e9\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(35)$

The angle brackets denote a timeaverage.
In general, the system will have many fast modes of oscillation (due to
thermal motion, for example). Each xray scattering event is instantaneous,
so in principle, a snapshot of the system could be obtained. However, any
signal large enough to be measurable will have to be integrated over an
extremely long time period in comparison to these modes. So what is observed
is the timeaverage of these snapshots.
We can also define a modified structure factor with the forward scattering
(
$q=0$)
removed.
$\tilde{S}(q)=\frac{1}{\rho}{\int}_{V}{d}^{3}r\hspace{1em}{e}^{iq\xb7r}\hspace{1em}\u27e8\rho (r)\rho (0)\u27e9N\delta (q)$ 
$(36)$

For all practical cases, this will be identical to the standard
structure factor, because the
$q=0$ scattering
cannot be experimentally isolated from the primary, unscattered beam.
In the xray literature, the
$N\delta (q)$ term is often neglected.
In this treatment of scattering from multiple objects, we have taken the first
Born approximation. In our language of classical continuum fields, the
scattered field does not interfere with itself. In the quantum description,
this is equivalent to each xray photon scattering from, at most, one
electron.
The assumption of negligible "multiple scattering"
is usually valid because the numerical value of the classical
electron radius (2.818
$\times $ 10
${}^{5}$ Å) is so small. This
"kinematic" approximation fails when the observed scattering becomes large.
One instance of this is smallangle scattering (
$q\to 0$).
Another is when there is a coherent superposition of fields (Bragg
diffraction).
6.3 Correlation Functions
I will define the twopoint correlation function (also known as the pair
correlation function) as
$g({r}_{1},{r}_{2})=\frac{1}{{\rho}^{2}}[\u27e8\rho ({r}_{1})\rho ({r}_{2})\u27e9\u27e8\rho ({r}_{1})\u27e9\u27e8\rho ({r}_{2})\u27e9]$ 
$(37)$

This characterizes the probability of finding a particle at
${r}_{2}$
given one at
${r}_{1}$, relative to the probability of finding a particle
at
${r}_{2}$ without any conditions.
This expression gives the joint expectation value of the density at a spatial
position
${r}_{2}$ and at
${r}_{1}$.
If we think of these density elements as particles, then
$g(r)$ is
the probability of finding a particle at
${r}_{2}$ given a particle at
${r}_{1}$.
The angle brackets denote timeaveraging, as in the previous section. The
time average is assumed to be equivalent to an average over the entire
ensemble of microstates available to the system.
For an
translationally invariant system,
$\u27e8\rho ({r}_{1},{r}_{2})\u27e9=\u27e8\rho ({r}_{1}{r}_{2})\u27e9$, implying
$g({r}_{1},{r}_{2})=g({r}_{1}{r}_{2})$. For a
homogeneous system, the
oneparticle densities are equivalent:
$\rho ({r}_{1})=\rho ({r}_{2})=\rho $. In general, for large enough
$r={r}_{1}{r}_{2}$, we
expect that the particles will be uncorrelated:
$\underset{r\to \infty}{lim}g(r)=0$ 
$(38)$

Unfortunately, there is no standard definition for
the twopoint correlation function.
You will also find
$g({r}_{1},{r}_{2})=\u27e8\rho ({r}_{1})\rho ({r}_{2})\u27e9$,
$g({r}_{1},{r}_{2})=\u27e8\rho ({r}_{1})\rho ({r}_{2})\u27e9\u27e8\rho ({r}_{1})\u27e9\u27e8\rho ({r}_{2})\u27e9$
in the literature.
I prefer the definition (
3.37)
because it is normalized and the function falls to
zero at large distances.
From (
3.36), we see that the correlation
function is just proportional to the Fourier transform of the modified
structure factor. We can shift the coordinate system to place
${r}_{2}$
at the origin. Then the Fourier transform of
$\rho g(r)$ is
$\begin{array}{cccc}\multicolumn{1}{c}{{\int}_{V}dr\hspace{0.5em}{e}^{iq\xb7r}\rho g(r)}& =\hfill & \frac{1}{\rho}{\int}_{V}dr\hspace{0.5em}{e}^{iq\xb7r}[\u27e8\rho (r)\rho (0)\u27e9\u27e8\rho (r)\u27e9\u27e8\rho (0)\u27e9]\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(39)\\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{\rho}{\int}_{V}dr\hspace{0.5em}{e}^{iq\xb7r}\u27e8\rho (r)\rho (0)\u27e9\rho {\int}_{V}dr\hspace{0.5em}{e}^{iq\xb7r}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(40)\\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{\rho}{\int}_{V}dr\hspace{0.5em}{e}^{iq\xb7r}\u27e8\rho (r)\rho (0)\u27e9\rho V\delta (q)\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(41)\\ \multicolumn{1}{c}{}& =\hfill & \u27e8\tilde{S}(q)\u27e9\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(42)\end{array}$

The significance of this result is that if we could measure
$S(q)$
for all
$q$, we could completely determine the correlation function
$g(r)$. In practice, unfortunately, we can only measure
$q$
for a limited range of angles and lengths of
$q$. This limits our knowledge
of
$g(r)$.
Because it is a convenient abbreviation that I will use in
section
3.10, let me mention one more relation.
$\begin{array}{cccc}\multicolumn{1}{c}{\u27e8\rho (q)\rho (q)\u27e9}& =\hfill & \u27e8\int d{r}_{1}{e}^{iq\xb7{r}_{1}}\rho ({r}_{1})\int d{r}_{2}{e}^{+iq\xb7{r}_{2}}\rho ({r}_{2})\u27e9\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(43)\\ \multicolumn{1}{c}{}& =\hfill & \int d{r}_{1}d{r}_{2}{e}^{iq\xb7({r}_{1}{r}_{2})}\u27e8\rho ({r}_{1})\rho ({r}_{2})\u27e9\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(44)\\ \multicolumn{1}{c}{}& =\hfill & S(q)\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(45)\end{array}$

7 Scattering from Atoms
Previously (section
3.5), we discussed the
scattering from a free electron.
In this section, we consider the xray scattering from
an atom, based upon references
[
2,
58,
59,
60,
142,
143].
As shown in
(
3.15), a particle's cross section depends on its mass
$m$
as
$1/{m}^{2}$. Since the proton/electron mass ratio is over 1800, the xray
scattering from the nucleus is miniscule, and is usually neglected.
Ultimately, we would like to have a function, the
atomic form factor,
which accounts for all the subatomic structure, and tells us how the scattered
amplitude is modified (compared with the case of a single free electron).
Because this is a common desire, these functions are tabulated in
standard references [
152,
1,
2].
If we consider just the electron probability density cloud
$\rho (r)$
surrounding the nucleus and approximate each of the
$Z$ electrons as being
free, then we arrive at the standard form factor
$f}_{0}(q,Z)=\int dr\hspace{1em}{e}^{iq\xb7r}\rho (r)={\int}_{0}^{\infty}4\pi {r}^{2}\mathrm{dr}\hspace{1em}\rho (r)\hspace{0.5em}\frac{\mathrm{sin}qr}{qr$ 
$(46)$

(The latter equality holds in the case that
$\rho (r)$ is spherically
symmetric.) By substitution, it is clear that
${f}_{0}=Z$ for
$q=0$.
The standard atomic form factors
for the elements Pt, Cu, and Cl are plotted
in figure
3.3.
This
${f}_{0}$ term is the primary component of the atomic form factor
$f$, because for
energies far above 100 eV (and all of the data in this dissertation were taken
with xrays in the keV range), we can basically approximate most atomic
electrons as being free; Xrays are far too energetic to efficiently excite
intraatomic energy transitions.
Figure 3: Standard atomic form factors
${f}_{0}(q)$, normalized by atomic number
$Z$, for Pt (solid), Cu (dotted), and Cl (dashed). Note that
${f}_{0}(0)=Z$. From [
151].
However, the freeelectron assumption is only an approximation, and fails most
noticeably near "adsorption edges". When the xray energy is tuned close to
an absorption edge, it can eject a corelevel electron from the atom. (This is
related to the electroninduced ionization discussed in
section
3.3).
Deviations of the measured form factor
$f$ from
${f}_{0}$ are known as as
"anomalous dispersion". Typically, this modification to the amplitude is
separated into a real term
$f\text{'}(E,Z)$ and an imaginary term
$f"(E,Z)$. The
latter term allows for a change of phase in the scattered beam and is
manifested as absorption. Physically, the anomalous dispersion arises from
the resonance of the incident xray with differences between atomic energy
levels.
Summarizing the various terms that comprise the atomic form factor
$f(q,E,Z)$:
$f(q,E,Z)={f}_{0}(q,Z)+f\text{\'}(E,Z)+f"(E,Z)+{f}_{\mathrm{NT}}$ 
$(47)$

where
${f}_{\mathrm{NT}}$ is the nuclear Thomson scattering, usually neglected.
The scattering amplitude from an atom is the amplitude from a single free
electron, multiplied by this
atomic form factor,
$f(q,E,Z)$. Here
$E$ is the energy of the incident xray photon,
$Z$ is the atomic number, and
$q=\frac{4\pi}{\lambda}\mathrm{sin}\theta $
(equation (
3.30)) is the
momentum transfer (discussed in section
3.6.2).
7.1 Absorption
Absorption of xrays is caused by an incident xray striking an atom and
causing a core level electron to be ejected [
61]. This
is simply the photoelectric effect, which is usually presented in the context
of ultraviolet photons incident upon the outer shells. Historically, this was
one of the most dramatic experiments leading to the quantum paradigm. After
the photoelectron is ejected, the atom is in an excited state. Just as
described in section
3.3, an electron will fall into
the vacated state and emit a characteristic xray; this process is called
"fluorescence". Because the electron falls from an atomic level and not
from the vacuum, the fluorescence energy is always less than the energy of the
absorbed xray.
Empirically, the absorption of xrays by matter is observed to be
${I}_{z}={I}_{0}\mathrm{exp}(\mu z)$ 
$(48)$

where
$\mu $ is proportional to the electronic density. At these large xray
energies, the phase (solid, liquid, gas) of the material is unimportant,
except in its effect upon the mean density. The more fundamental quantity
(and the one most frequently tabulated) is the mass absorption coefficient
$\mu /{\rho}_{m}$, where
${\rho}_{m}$ is a mass density. Since xrays scatter from
atoms, it is sufficient to add up the elemental contributions to determine a
molecular compound's mass absorption coefficient, as
$\frac{\mu}{{\rho}_{m}}=\sum _{i}{w}_{i}{\left(\frac{\mu}{{\rho}_{m}}\right)}_{i}$ 
$(49)$

where
${w}_{i}$ is the weight fraction of element
$i$.
Figure 4: An illustration of a mass absorption coefficient vs. wavelength.
This is intended only for illustration, so it corresponds to no element.
The sharp drop in intensity is known as the absorption edge.
The absorption coefficient
${\mu}_{i}$ peaks dramatically near energies that
correspond to the absorption edges. Following each edge is a branch of the
absorption curve following the form [
62]
$\mu /{\rho}_{m}=k{\lambda}^{3}{Z}^{3}$ 
$(50)$

where
$k$ is a constant that has a different value for each branch.
An illustration is provided in figure
3.4.
For this fictitious element, the edge is at
$\lambda =2$Å.
In order to minimize absorption, we used an incident xray energy of 8800 eV,
near the bottom of the absorption branch, but sufficiently far from the Cu K
(8979 eV) edge. This choice of energy also reduced fluorescence from the Cu
atoms that occurs at or above the absorption edge.
8 Crystals
A Bravais lattice is the set of points that can be reached from a single point
by applying translation vectors. These translation vectors are called "basis
vectors" and are equal to the number of dimensions of the space of the
lattice. For real crystals described by Bravais lattices, there are three
basis vectors. Calling these basis vectors
${a}_{x}$,
${a}_{y}$,
${a}_{z}$, the Bravais lattice consists of the set of vectors
$\{R\}$,
$R={n}_{x}{a}_{x}+{n}_{y}{a}_{y}+{n}_{z}{a}_{z}$ 
$(51)$

where
${n}_{a}$,
${n}_{b}$,
${n}_{c}$ span all the integers.
Any Bravais lattice also has a reciprocal lattice [
10],
which is the set of plane wave vectors
$\{q\}$ that have the same
periodicity as the Bravais lattice. That is,
${e}^{iq\xb7r}={e}^{iq\xb7(R+r)}$, or
As the symmetric nature of (
3.52) makes clear, the
reciprocal lattice to the reciprocal lattice is just the original direct
lattice. In other words, the (direct) Bravais lattice in real space and the
reciprocal lattice are dual.
9 Diffraction from Crystals
9.1 Infinite Crystal
Consider an infinite array of point charges
$\rho (r)=\sum _{R}\delta (rR)$ 
$(53)$

where
$R$ are the Bravais lattice vectors.
But
for simplicity, let's begin with the case of an infinite onedimensional
crystal. The lattice vectors are
$R=ja$, where
$j$ runs over all
integers, so the scattering
amplitude is
$A(q)=\sum _{j}{e}^{iqja}=\sum _{j=\infty}^{\infty}{\left({e}^{iqa}\right)}^{j}=\sum _{j=1}^{\infty}{\left({e}^{iqa}\right)}^{j}+\sum _{j=0}^{\infty}{\left({e}^{iqa}\right)}^{j}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(54)$

Ordinarily, we would use
$\sum _{n=0}^{\infty}{x}^{n}=\frac{1}{1x}\mathrm{\hspace{0.5em}\hspace{0.5em}},$ 
$(55)$

but this fails the requirement
$x<1$ for convergence.
However, we can multiply
${e}^{iqa}$
by a small correction factor
${e}^{\mu}$
where
$\mu >0$ and take the limit
$\mu \to {0}^{+}$ after the sum is
taken. Then the sums in (
3.54) converge to
$A(q)=\delta (qja2n\pi )=\delta (qR2n\pi )$ 
$(56)$

where
$n$ is any integer.
This mathematical device has a basis in physical reality. As discussed in
section
3.7.1, all crystals absorb
some fraction of the xrays that diffract through them. Other processes
that cause a
finite scattering intensity are grouped under the term
extinction [
144].
The calculation for a threedimensional crystal is just a trivial extension of
the onedimensional case. Take a Bravais lattice as
$\rho (r)=\sum _{R}\delta (rR)=\sum _{R}\delta ({r}_{x}{j}_{x}{a}_{x})\hspace{0.5em}\delta ({r}_{y}{j}_{y}{a}_{y})\hspace{0.5em}\delta ({r}_{z}{j}_{z}{a}_{z})$ 
$(57)$

where
$\{R\}$ corresponds to
$\{{j}_{x},{j}_{y},{j}_{z}\}$.
$\begin{array}{cccc}\multicolumn{1}{c}{A(q)}& =\hfill & \sum _{R}{e}^{iq\xb7R}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(58)\\ \multicolumn{1}{c}{}& =\hfill & \sum _{\{{j}_{x},{j}_{y},{j}_{z}\}}{e}^{i{q}_{x}{j}_{x}{a}_{x}}\hspace{0.5em}{e}^{i{q}_{y}{j}_{y}{a}_{y}}\hspace{0.5em}{e}^{i{q}_{z}{j}_{z}{a}_{z}}\hfill \\ \multicolumn{1}{c}{}& =\hfill & \sum _{{j}_{x}}{e}^{i{q}_{x}{j}_{x}{a}_{x}}\sum _{{j}_{y}}{e}^{i{q}_{y}{j}_{y}{a}_{y}}\sum _{{j}_{z}}{e}^{i{q}_{z}{j}_{z}{a}_{z}}\hfill \\ \multicolumn{1}{c}{}& =\hfill & \delta ({q}_{x}{a}_{x}2H\pi )\hspace{0.5em}\delta ({q}_{y}{a}_{y}2K\pi )\hspace{0.5em}\delta ({q}_{z}{a}_{z}2L\pi )\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(59)\end{array}$

For nonzero
$A(q)$,
we thus require the Laue conditions
$\begin{array}{cccc}\multicolumn{1}{c}{{q}_{x}{a}_{x}}& =\hfill & 2\pi H\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(60)\\ \multicolumn{1}{c}{{q}_{y}{a}_{y}}& =\hfill & 2\pi K\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(61)\\ \multicolumn{1}{c}{{q}_{z}{a}_{z}}& =\hfill & 2\pi L\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(62)\end{array}$

where the indices
$H$,
$K$,
$L$ span all of the integers.
These conditions define a lattice of points in reciprocal space.
These points are just the reciprocal lattice described
by (
3.52), as we can show with a simple example.
The vector
$q$ is just
$q={q}_{x}{\hat{q}}_{x}+{q}_{y}{\hat{q}}_{y}+{q}_{z}{\hat{q}}_{z}$ 
$(63)$

where the
${\hat{q}}_{i}$ represent arbitrary unit vectors that form a basis
in reciprocal space. For simplicity, take
${\hat{a}}_{i}\xb7{\hat{q}}_{i}={\delta}_{\mathrm{ij}}\hspace{0.5em},$ 
$(64)$

although we could make some other choice. Then
$\begin{array}{cccc}\multicolumn{1}{c}{q\xb7R}& =\hfill & {n}_{x}{a}_{x}{q}_{x}+{n}_{y}{a}_{y}{q}_{y}+{n}_{z}{a}_{z}{q}_{z}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(65)\\ \multicolumn{1}{c}{}& =\hfill & 2\pi H{n}_{x}+2\pi K{n}_{y}+2\pi L{n}_{z}\mathrm{\hspace{0.5em}\hspace{0.5em}}.\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(66)\end{array}$

This is equivalent to
${e}^{iq\xb7R}=1$, the definition of the
reciprocal lattice (
3.52). We might choose another
basis for
${\hat{q}}_{i}$, instead of (
3.64), but
$q\xb7R$
will remain the same. So, the diffraction pattern from a Bravais lattice is
its dual reciprocalspace Bravais lattice.
Take
$d$ as the projection of a given
$R$ along
$q$. This
$d$ is the distance between two planes of the lattice. From
$q\xb7R=qd=2n\pi $ and the definition of
$q$ (
3.30), we recover Bragg's law:
$n\lambda =2d\mathrm{sin}\theta \mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(67)$

9.2 Lattices with a Basis
In this dissertation, we will be most concerned with the structure of
platinum. This is a facecentered cubic crystal, which can be visualized as a
cubic crystal with an extra atom at the center of each of the cubic faces.
Some crystal structures are not Bravais
lattices
${}^{1}$,
for example, silicon and diamond. They can, however, be described as a
facecentered cubic crystal with another facecentered cubic crystal
superimposed upon it. The second lattice has a relative displacement of
$1/4$
along the cubic body diagonal. The "diamond structure" is described by
vectors that run over the Bravais lattice, but also include the basis vector
for this displacement.
Although the facecentered cubic structure is a Bravais lattice, it is
convenient to describe it as a simple cubic lattice with a basis. The basis
vectors in this case describe the atoms on the faces.
For a simple cubic crystal with basis vectors
$a\hat{x}$,
$a\hat{y}$,
and
$a\hat{z}$,
the displacement basis vectors that generate the facecentered cubic
crystal are
$\begin{array}{cccc}\multicolumn{1}{c}{{d}_{1}}& =\hfill & 0\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(68)\\ \multicolumn{1}{c}{{d}_{2}}& =\hfill & \frac{a}{2}(\hat{x}+\hat{y})\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(69)\\ \multicolumn{1}{c}{{d}_{3}}& =\hfill & \frac{a}{2}(\hat{x}+\hat{z})\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(70)\\ \multicolumn{1}{c}{{d}_{4}}& =\hfill & \frac{a}{2}(\hat{y}+\hat{z})\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(71)\end{array}$

For a crystalline sample, we can divide the space into
unit cells that are repeated over all Bravais lattice vectors.
Then we can consider the scattering from one unit cell and replicate it
over the Bravais lattice.
For the facecentered cubic crystal, the scattering amplitude due to
one cube is
$\begin{array}{cccc}\multicolumn{1}{c}{A(q)=\sum _{i=1}^{4}{e}^{iq\xb7d}}& =\hfill & 1+{e}^{i\pi (H+K)}+{e}^{i\pi (H+L)}+{e}^{i\pi (K+L)}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(72)\\ \multicolumn{1}{c}{}& =\hfill & 1+(1{)}^{H+K}+(1{)}^{H+K}+(1{)}^{K+L}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(73)\\ \multicolumn{1}{c}{}& =\hfill & 4\hspace{0.5em}\hspace{0.5em}\text{if}\hspace{0.5em}\hspace{0.5em}H,K,L\hspace{0.5em}\text{all odd or all even}\hfill \\ \multicolumn{1}{c}{}& \hfill & 0\hspace{0.5em}\hspace{0.5em}\text{if}\hspace{0.5em}\hspace{0.5em}H,K,L\hspace{0.5em}\text{mixed}\hfill \end{array}$

So the diffraction pattern will be identical that of a simple cubic lattice,
except that intensity at
$H,K,L$ values which are all odd or all even
will be enhanced, and the intensity at other
$H,K,L$ values
will be extinguished.
10 Thermal Effects and Inelastic Scattering
This section is based upon
[
11,
63,
145,
146,
47].
Consider an ideal crystal as in (
3.53),
$\rho (r)=\sum _{R}\delta (rR)$ 
$(74)$

The structure factor from a Bravais lattice is found by applying
the definition of the structure
factor (
3.32) to the
scattering amplitude from a Bravais lattice (
3.58),
$S(q)=\sum _{R,R\text{'}}{e}^{iq\xb7(RR\text{'})}$ 
$(75)$

To consider an inelastic scattering process, take the
dynamical structure factor [
8]
$\begin{array}{ccc}\multicolumn{1}{c}{S(q,\omega )}& =\hfill & \frac{1}{N}{\int}_{\infty}^{\infty}\mathrm{dt}\hspace{0.5em}{\int}_{V}{d}^{3}{r}_{1}{\int}_{V}{d}^{3}{r}_{2}\hspace{1em}{e}^{iq\xb7({r}_{1}{r}_{2})+i\omega t}\hspace{1em}\rho ({r}_{1},t=0)\rho ({r}_{2},t)\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{N}{\int}_{\infty}^{\infty}\mathrm{dt}\hspace{0.5em}{e}^{i\omega t}\hspace{1em}\u27e8\rho (q)\rho (q,t)\u27e9\mathrm{\hspace{0.5em}\hspace{0.5em}},\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(76)\end{array}$

and allow the atoms to oscillate about their respective mean positions
such that their instantaneous positions are
$R+u(R)$.
Then,
$S(q,\omega )=\frac{1}{N}\sum _{RR\text{'}}{e}^{iq\xb7(RR\text{'})}\int \mathrm{dt}\hspace{0.5em}{e}^{i\omega t}\u27e8{e}^{iq\xb7u(R\text{'})}{e}^{iq\xb7u(R,t)}\u27e9\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(77)$

Take the harmonic approximation, which is that
$u(R)$ has
a Gaussian distribution (appendix
A).
Specifically, use
the result (
A.13) to evaluate the product in the angle brackets:
$\u27e8{e}^{iq\xb7u(R\text{'})}{e}^{iq\xb7u(R,t)}\u27e9={e}^{\frac{1}{2}\u27e8[q\xb7u(R\text{'}){]}^{2}\u27e9}{e}^{\frac{1}{2}\u27e8[q\xb7u(R,t){]}^{2}\u27e9}{e}^{\u27e8[q\xb7u(R\text{'})][q\xb7u(R,t)]\u27e9}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(78)$

The DebyeWaller factor
$W(q)$ is defined as
$W(q)\equiv \frac{1}{2}\u27e8[q\xb7u(R\text{'}){]}^{2}\u27e9=\frac{1}{2}\u27e8[q\xb7u(R,t){]}^{2}\u27e9=\frac{1}{2}\u27e8[q\xb7u(0){]}^{2}\u27e9\mathrm{\hspace{0.5em}\hspace{0.5em}},$ 
$(79)$

and then the structure factor is
$S(q,\omega )={e}^{2W}\hspace{1em}\int \mathrm{dt}\hspace{0.5em}{e}^{i\omega t}\sum _{R}{e}^{iq\xb7R}{e}^{\u27e8[q\xb7u(0)][q\xb7u(R,t)]\u27e9}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(80)$

Expanding the exponential
${e}^{\u27e8[q\xb7u(0)][q\xb7u(R,t)]\u27e9}$, the zerothorder term (replacing the
exponential with unity) yields
$S(q,\omega )={e}^{2W}\hspace{1em}\int \mathrm{dt}\hspace{0.5em}{e}^{i\omega t}\sum _{R}{e}^{iq\xb7R}$ 
$(81)$

which evaluates to
$S(q,\omega )={e}^{2W}\hspace{1em}\delta (\omega )\sum _{G}\delta (qG)$ 
$(82)$

where
$\{G\}$ is the reciprocal lattice to
$\{R\}$.
This is just the elastic scattering from Bragg peaks derived in previous
sections. So the net effect of the small thermal motions is to lower the
Bragg peak intensities by a factor of
${e}^{2W}$. The widths, however, are
unchanged.
The extra intensity goes into the other modes, which are the higherorder
terms in the expansion of the exponential. The scattering from these modes is
called "thermal diffuse scattering". The successive terms in the expansion
consists of zero, onephonon, twophonon, etc. processes. While the
zerothorder term is elastic, all the others are not. The firstorder thermal
diffuse scattering has a discrete energy spectrum, while the secondorder and
higher terms are smooth functions of scattered xray energy
[
12]. The secondorder thermal diffuse scattering has the
same intensity as Compton scattering,
and is often ignored.
It is often difficult to separate out the
elastic from the inelastic terms, because of the poor relative energy
resolution in xray detection.
The incident
xray
energy is on the order of keV, while the thermal excitations are on the order
of
${k}_{B}T$, 1/40 eV at room temperature. In practice, we are accepting such a
broad range of energies that
all energy transfers are selected.
This is equivalent to performing an integration
of (
3.76)
over all energy transfers
$\omega $
$\begin{array}{cccc}\multicolumn{1}{c}{\int d\omega \hspace{0.5em}S(q,\omega )}& =\hfill & \int d\omega \hspace{0.5em}\frac{1}{N}{\int}_{\infty}^{\infty}\mathrm{dt}\hspace{0.5em}{e}^{i\omega t}\hspace{1em}\u27e8\rho (q)\rho (q,t)\u27e9\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(83)\\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{N}{\int}_{\infty}^{\infty}\mathrm{dt}\hspace{0.5em}\delta (t)\u27e8\rho (q)\rho (q,t)\u27e9=\u27e8\rho (q)\rho (q)\u27e9\hfill \end{array}$

Thus, xray diffraction measures the static structure factor automatically.
Although sometimes called elastic scattering (because there is no
energy dependence), this is a misnomer. In fact, we are integrating over all
energy transfers
$\omega $, not selecting out the elastic (
$\omega =0$) term.
11 Xray Scattering from Surfaces
Xray scattering from surfaces is usually presented in either of two ways. In
section
3.11.1, I present the method of
truncating an infinite crystal, popularized by Robinson [
111].
The following section (
3.11.2) describes the more
traditional approach from classical electrodynamics. The final
section (
3.11.3) establishes the connection between
these two approaches.
11.1 Truncating the Infinite Crystal
This section builds upon the treatment of the infinite crystal in
section
3.9, and extends it to semiinfinite
and finite crystals.
11.1.1 SemiInfinite Crystal
In analogy with (
3.54), consider
an ideal semiinfinite crystal.
One face of the crystal is
presumed to be truncated at
$x=0$ while the opposite "end" stretches to
infinity. As before, we begin with the onedimensional case, which
illustrates the relevant behavior.
$A(q)=\sum _{j=0}^{\infty}{e}^{iqja}=\frac{1}{1{e}^{iqa}}$ 
$(84)$

Note that the sum over
$j$ now runs from
$0\dots \infty $. Like the
infinite crystal (
3.56), this function is infinitevalued for
$qa=2n\pi $. Unlike the infinite crystal,
(
3.84) has nonzero
values everywhere else. The structure factor has a minimum value of
$1/4$:
$S(q)={A(q)}^{2}=\frac{1}{4}\mathrm{\hspace{0.5em}\hspace{0.5em}}\frac{1}{{\mathrm{sin}}^{2}\frac{qa}{2}}$ 
$(85)$

11.1.2 Finite Crystal
Now, we truncate the crystal at both ends, so that it is a onedimensional
crystal containing
$N$ scatterers.
Using the relation
$\sum _{n=0}^{N1}{x}^{n}=\frac{1{x}^{N}}{1x}$ 
$(86)$

then
$A(q)=\sum _{j=0}^{N1}{e}^{iqja}=\frac{1{e}^{iqNa}}{1{e}^{iqa}}$ 
$(87)$

and
$S(q)=\frac{{\mathrm{sin}}^{2}\frac{1}{2}Nqa}{{\mathrm{sin}}^{2}\frac{1}{2}qa}$ 
$(88)$

Figure 5: Graph of
$\frac{{\mathrm{sin}}^{2}Nx}{{\mathrm{sin}}^{2}x}$ vs.
$x$, for
$N=10$.
The structure factor here is just like the diffraction intensity from
a diffraction grating with
$N$ slits (shown in figure
3.5).
Near the Bragg diffraction peaks
(
$qa\to 2\pi n$), the limit
$\underset{x\to 0}{lim}\frac{\mathrm{sin}Nx}{\mathrm{sin}x}=N$.
Because the crystal is finite, the structure factor maxima are now
${N}^{2}$, not
infinite. The minimum value is zero, which is also in contrast with the
semiinfinite crystal, but identical to the infinite crystal.
The extension to three dimensions is straightforward [
111].
Consider a threedimensional crystal with
${N}_{x}$,
${N}_{y}$,
${N}_{z}$ scatterers in
the
$x$,
$y$,
$z$ directions.
The scattering amplitude is then
$A(q)=\left(\sum _{{j}_{x}=0}^{{N}_{x}1}{e}^{i{j}_{x}q\xb7{a}_{x}}\right)\left(\sum _{{j}_{y}=0}^{{N}_{y}1}{e}^{i{j}_{y}q\xb7{a}_{y}}\right)\left(\sum _{{j}_{z}=0}^{{N}_{z}1}{e}^{i{j}_{z}q\xb7{a}_{z}}\right)$ 
$(89)$

Each factor is just a geometric series, which may be summed to yield
$A(q)=\left(\frac{{e}^{i{N}_{x}q\xb7{a}_{x}}1}{{e}^{iq\xb7{a}_{x}}1}\right)\left(\frac{{e}^{i{N}_{y}q\xb7{a}_{y}}1}{{e}^{iq\xb7{a}_{y}}1}\right)\left(\frac{{e}^{i{N}_{z}q\xb7{a}_{z}}1}{{e}^{iq\xb7{a}_{z}}1}\right)$ 
$(90)$

The structure factor
is just
$S(q)=A(q){A}^{*}(q)$, so
$S(q)=\left(\frac{{\mathrm{sin}}^{2}\frac{1}{2}{N}_{x}q\xb7{a}_{x}}{{\mathrm{sin}}^{2}\frac{1}{2}q\xb7{a}_{x}}\right)\left(\frac{{\mathrm{sin}}^{2}\frac{1}{2}{N}_{y}q\xb7{a}_{y}}{{\mathrm{sin}}^{2}\frac{1}{2}q\xb7{a}_{y}}\right)\left(\frac{{\mathrm{sin}}^{2}\frac{1}{2}{N}_{z}q\xb7{a}_{z}}{{\mathrm{sin}}^{2}\frac{1}{2}q\xb7{a}_{z}}\right)$ 
$(91)$

Taking the
${N}_{x}\to \infty $,
${N}_{y}\to \infty $ limit but
holding
${N}_{z}$ finite, we obtain a physically reasonable depiction of a
crystal surface.
$S(q)=\delta ({q}_{x}{a}_{x}2H\pi )\hspace{0.5em}\delta ({q}_{y}{a}_{y}2K\pi )\hspace{0.5em}\left(\frac{{\mathrm{sin}}^{2}\frac{1}{2}{N}_{z}q\xb7{a}_{z}}{{\mathrm{sin}}^{2}\frac{1}{2}q\xb7{a}_{z}}\right)$ 
$(92)$

In comparison with the infinite crystal,
the two Laue conditions (
3.60) on
${q}_{x}$ and
${q}_{y}$ are
maintained, but the condition on
${q}_{z}$ has
been relaxed. Where the final Laue condition
${q}_{z}{a}_{z}=2\pi L$ holds,
a Bragg peak will exist. The difference is that there is still residual
scattering at other
${q}_{z}$. The resultant intensity is sharp
in
${q}_{x}$ and
${q}_{y}$, but diffuse along
${q}_{z}$. These rods of scattering that
arise from the truncation of the crystal lattice are thus known as "crystal
truncation rods" (CTR).
As mentioned in section
3.9.1, absorption and extinction
limit the xray scattering intensity. Because of this finite penetration
depth, even a truly semiinfinite crystal will scatter like a finite crystal.
A reasonable orderofmagnitude for
${N}_{z}$ is 1000, given a penetration depth
$\approx 1\mu $m [
111]. Figure
3.6
plots (
3.92), proportional to the intensity along
a CTR.
Figure 6: A logarithmic plot of
$S(q)$ vs.
${q}_{z}$ as given in
(
3.92) for
${N}_{z}=1000$. This is the same
function shown in figure
3.5, but with a larger
${N}_{z}$. Also,
a finite
resolution function has been convolved through the data. This eliminates
the numerous minima seen in figure
3.5, and is
reasonable from an experimental standpoint. The minimum value is near
$1/2$, because
$\u27e8{\mathrm{sin}}^{2}x\u27e9=1/2$. Without the convolution,
the minimum value is zero.
11.2 Reflectivity from Smooth Surfaces
An alternative treatment is to consider xray scattering from a surface as an
example of the more general problem of reflection and refraction at a boundary
between two dielectric media. Since xrays are just electromagnetic
radiation, this should be perfectly valid. For the moment, however, we
neglect the atomistic nature of the sample and assume it to be a smooth,
continuous structure. The atomic periodicity can be added in later
(section
3.11.3).
11.2.1 Fresnel Equations
Consider a smooth interface between air (
$n=1$) and a block of amorphous
material (
$n<1$ for xrays). This ties into the classical electrodynamic
treatment described by Jackson [
84], with
$n\text{'}\to n$,
$n\to 1$, and
$\mu =\mu \text{'}$.
Jackson uses
${\theta}_{i}$, the angle between the incident beam and the surface
normal. For consistency with the rest of the dissertation,
I write results in terms of its complementary angle
$\alpha $ between the incident beam and the surface.
Figure 7: Diagram for the Fresnel equations.
Then the component of the electric field perpendicular to the
plane of reflection are
$\begin{array}{cc}\multicolumn{1}{c}{\frac{{E}_{t}^{\perp}}{{E}_{0}}=\frac{2\mathrm{sin}\alpha}{\mathrm{sin}\alpha +\sqrt{{n}^{2}{\mathrm{cos}}^{2}\alpha}}}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(93)\\ \multicolumn{1}{c}{\frac{{E}_{r}^{\perp}}{{E}_{0}}=\frac{\mathrm{sin}\alpha \sqrt{{n}^{2}{\mathrm{cos}}^{2}\alpha}}{\mathrm{sin}\alpha +\sqrt{{n}^{2}{\mathrm{cos}}^{2}\alpha}}}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(94)\end{array}$

and the parallel components are
$\begin{array}{cc}\multicolumn{1}{c}{\frac{{E}_{t}^{\parallel}}{{E}_{0}}=\frac{2n\mathrm{sin}\alpha}{{n}^{2}\mathrm{sin}\alpha +\sqrt{{n}^{2}{\mathrm{cos}}^{2}\alpha}}}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(95)\\ \multicolumn{1}{c}{\frac{{E}_{r}^{\parallel}}{{E}_{0}}=\frac{{n}^{2}\mathrm{sin}\alpha \sqrt{{n}^{2}{\mathrm{cos}}^{2}\alpha}}{{n}^{2}\mathrm{sin}\alpha +\sqrt{{n}^{2}{\mathrm{cos}}^{2}\alpha}}}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(96)\end{array}$

The Fresnel reflection and transmission coefficients are plotted in
figure
3.8.
If
$n<1$, as we will show is true for xrays incident on most materials,
then there exists
a critical angle
${\alpha}_{c}$ such that
$n=\mathrm{cos}{\alpha}_{c}$.
For
$\alpha <{\alpha}_{c}$,
$\mathrm{cos}\alpha >n$, so
the square roots become imaginary and the magnitude of
${E}_{r}/{E}_{0}$ is unity for both polarizations. This is termed total
external reflection.
Figure 8: Fresnel reflection and transmission coefficients.
In this example,
${\alpha}_{c}={1}^{\circ}$.
The transmission is sharply peaked at the critical angle, then
quickly falls to unity.
11.2.2 Index of Refraction
While a long derivation of the index of refraction
can be found in Warren [
147], a
simpler and more illuminating treatment can be found in Jackson
[
85]. I will not repeat the entire model here, but
just connect the results to our discussion.
For large enough photon energies (
$\hslash \omega $), the dielectric constant
approaches the "plasma limit" and
$\epsilon (\omega )\approx 1{\omega}_{p}^{2}/{\omega}^{2}$ 
$(97)$

where the "plasma frequency" is
${\omega}_{p}^{2}=4\pi NZ{e}^{2}/m$ 
$(98)$

Here,
$N$ is the density of atoms per unit volume and
$Z$ is the atomic
number.
The index of refraction is defined as
$n=\sqrt{\mu \epsilon}$ and we can
assume that
$\mu \approx 1$. Since the refractive index is close to unity, we
can expand the exponential
$n=\sqrt{1({\omega}_{p}^{2}/{\omega}^{2})}\approx 1\frac{1}{2}\frac{{\omega}_{p}^{2}}{{\omega}^{2}}=1\frac{2\pi NZ{e}^{2}}{m{\omega}^{2}}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(99)$

Using
$\omega =2\pi c/\lambda $, the classical electron radius
${r}_{0}={e}^{2}/{\mathrm{mc}}^{2}$, and the electronic density
$\rho =NZ$, we find that the index
of refraction is
$n=1{r}_{0}\frac{\rho {\lambda}^{2}}{2\pi}\equiv 1\delta$ 
$(100)$

Our assertion that
$\delta <<1$ will hold true in all cases, as demonstrated
for a practical example in section
3.11.2.
The critical angle is defined by
$n=\mathrm{cos}{\alpha}_{c}$.
By expanding the cosine to second order in the critical angle (which
should be small), we obtain
$\alpha}_{c}=(2\delta {)}^{1/2}={\left({r}_{0}\frac{\rho {\lambda}^{2}}{\pi}\right)}^{1/2$ 
$(101)$

11.2.3 Critical Angle Calculations for Platinum
As an example, the critical angle for platinum is calculated in this section.
The atomic weight of platinum,
$W$, is 195.078 g/mol, its atomic number
$Z$ =
78, and the mass density
${\rho}_{m}$ = 21.090 g/cm
${}^{3}$. Hence, the density of
electrons in platinum is
$\rho =\frac{{\rho}_{m}Z{N}_{A}}{W}=5.078\times {10}^{24}{\text{cm}}^{3}=5.078{\text{\xc5}}^{3}$ 
$(102)$

where
${N}_{A}$ is Avogadro's number (
$6.022\times {10}^{23}$).
For a Cu
$K\alpha $ emission line,
$\lambda $ = 1.54 Å, so
$\delta =5.401\times {10}^{5}$ by (
3.100) and
${\alpha}_{c}=1.04\times {10}^{2}$ (radians) by (
3.101).
Note that
$\delta <<1$ as claimed previously.
This is true even for this extreme example of a high
$Z$, high massdensity
material.
11.3 Scattering and Reflectivity
There are some apparent disparities between the results of the CTR theory
(section
3.11.1) and the classical Fresnel
reflectivity (section
3.11.2). The former
predicts Bragg peaks connected by crystal truncation rods. Given no
adsorption and a truncated infinite crystal, these Bragg peaks are predicted
to have infinite intensity. The Fresnel formulae have no Bragg peaks or
truncation rods, and the intensity maximum saturates at unity below the
critical angle.
The Fresnel treatment cannot predict Bragg peaks, because the scattering media
is assumed to be a solid block of constant density. The CTR treatment assumes
a perfect crystalline lattice. If we extend the CTR treatment to consider
continuous, homogeneous media, then integrals will take the place of
summations. The scattering amplitude is
$A(q)=\int dr\hspace{1em}{e}^{iq\xb7r}\rho (r)\hspace{0.5em}.$ 
$(103)$

A semiinfinite block of material, truncated at
$z=0$, will have a density
profile
$\rho (z)={\rho}_{0}\Theta (z)$ 
$(104)$

where
$\Theta (z)$ is the Heaviside step function.
Integrating by parts will not work, because
${e}^{\pm iq\infty}$
cannot be defined. However, following [
37], we
consider the sign function
$S(x)\equiv \{\begin{array}{cc}1\hfill & x<0\hfill \\ +1\hfill & x>0\hfill \end{array}$ 
$(105)$

The Fourier transform of
$S(x)$ can be found by considering
the function
${e}^{txS(x)}$ and then taking the
$t\to 0$ limit.
$\begin{array}{cccc}\multicolumn{1}{c}{{\int}_{\infty}^{\infty}\mathrm{dx}\hspace{0.5em}{e}^{iqx}{e}^{txS(x)}}& =\hfill & {\int}_{\infty}^{0}\mathrm{dx}\hspace{0.5em}{e}^{(tiq)x}+{\int}_{0}^{\infty}\mathrm{dx}\hspace{0.5em}{e}^{(t+iq)x}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(106)\\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{tiq}+\frac{1}{t+iq}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(107)\\ \multicolumn{1}{c}{}& {\to}_{t\to 0}\hfill & \frac{2}{iq}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(108)\end{array}$

Since
$\Theta (x)=\frac{1}{2}(S(x)+1)$,
and the Fourier transform of unity is a delta function,
then Fourier transform of the step function
$\Theta (x)$ is finally
$\int \mathrm{dz}\hspace{0.5em}{e}^{i{q}_{z}z}\Theta (z)=\frac{1}{i{q}_{z}}+\frac{1}{2}\delta ({q}_{z})\hspace{0.5em}.$ 
$(109)$

Then the structure factor (and the intensity) will be proportional to
$1/{q}_{z}^{2}$. The
$\frac{1}{2}\delta ({q}_{z})$ may seem unimportant. We can show
that it is significant by proving a familiar result
[
126]. Noting that
$\Theta (z)+\Theta (z)=1$ 
$(110)$

then
$\begin{array}{ccc}\multicolumn{1}{c}{{\int}_{\infty}^{\infty}\mathrm{dz}\hspace{0.5em}{e}^{i{q}_{z}z}}& =\hfill & {\int}_{\infty}^{\infty}\mathrm{dz}\hspace{0.5em}{e}^{i{q}_{z}z}\Theta (z)+{\int}_{\infty}^{\infty}\mathrm{dz}\hspace{0.5em}{e}^{i{q}_{z}z}\Theta (z)\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{i{q}_{z}}+\frac{1}{2}\delta ({q}_{z})+\frac{1}{i{q}_{z}}+\frac{1}{2}\delta ({q}_{z})\hfill \\ \multicolumn{1}{c}{}& =\hfill & \delta ({q}_{z})\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(111)\end{array}$

In fact, even without knowing (
3.109), one can show just
from (
3.110) that if
$F({q}_{z})\equiv {\int}_{\infty}^{\infty}\mathrm{dz}\hspace{0.5em}{e}^{i{q}_{z}z}\Theta (z)$, then
$F({q}_{z})+F({q}_{z})=\delta ({q}_{z})$.
The result
$S({q}_{z})\propto 1/{q}_{z}^{2}$ can be obtained in two other ways. Taking
the continuum limit
$a\to 0$ of the semiinfinite structure
factor (
3.85) yields this directly. The second method
is to
follow [
124] and consider the
structure factor (
3.31) again:
$S(q)=\frac{1}{N}{\int}_{V}{d}^{3}{r}_{1}{\int}_{V}{d}^{3}{r}_{2}\hspace{1em}{e}^{iq\xb7({r}_{1}{r}_{2})}\hspace{1em}\u27e8\rho ({r}_{1})\rho ({r}_{2})\u27e9$ 
$(112)$

Transforming these volume integrals to surface integrals,
$S(q)=\frac{1}{N}\hspace{1em}\frac{1}{(q\xb7\hat{n}{)}^{2}}\hspace{1em}{\int}_{S}(d{S}_{1}\xb7\hat{n})\hspace{1em}{\int}_{S}(d{S}_{2}\xb7\hat{n})\hspace{1em}{e}^{iq\xb7({r}_{1}{r}_{2})}\hspace{1em}\u27e8\rho ({r}_{1})\rho ({r}_{2})\u27e9$ 
$(113)$

where
$d{S}_{1}$,
$d{S}_{2}$ are differential surface vectors
constrained to lie along the surface. The choice of unit vector
$\hat{n}$
is arbitrary. If we take
$\hat{n}=\hat{z}$, then
$S(q)=\frac{1}{N}\hspace{1em}\frac{1}{{q}_{z}^{2}}\hspace{1em}{\int}_{S}{\mathrm{dx}}_{1}\hspace{0.5em}{\mathrm{dy}}_{1}\hspace{1em}{\int}_{S}{\mathrm{dx}}_{2}\hspace{0.5em}{\mathrm{dy}}_{2}\hspace{1em}{e}^{iq\xb7({r}_{1}{r}_{2})}\hspace{1em}\u27e8\rho ({r}_{1})\rho ({r}_{2})\u27e9$ 
$(114)$

and
$S$ is a constant
$z$ plane.
Throughout this chapter, I have implicitly used the "kinematic" theory,
which assumes that the first Born approximation holds. Because xrays
interact very weakly with matter, the approximation is valid for most points
in reciprocal space. However, at Bragg points (where Bragg's Law is
satisfied) and near the (000) reciprocal space point, the Born approximation
breaks down. That is why the CTR treatment fails to predict the existence of
the critical angle or the finite intensity for a semiinfinite crystal.
However, we can show the equivalence of the two approaches far from (000), in
the large
$\alpha $ limit. Starting with (
3.94), rewrite
${\mathrm{cos}}^{2}\alpha =1{\mathrm{sin}}^{2}\alpha $. Since
${n}^{2}1$ is very small, we can
approximate the terms within the square roots by
$[{\mathrm{sin}}^{2}\alpha +({n}^{2}1)]}^{1/2}=\mathrm{sin}\alpha +\frac{{n}^{2}1}{2\mathrm{sin}\alpha$ 
$(115)$

Then (
3.94) becomes
$\frac{{E}_{r}^{\perp}}{{E}_{0}}=\frac{{n}^{2}1}{4{\mathrm{sin}}^{2}\alpha}$ 
$(116)$

The reflectivity coefficient
$R={{E}_{r}^{\perp}/{E}_{0}}^{2}$ is proportional to
$1/{\mathrm{sin}}^{4}\alpha $.
Since
${q}_{z}=2k\mathrm{sin}\alpha $ from (
3.29),
$R\propto 1/{q}_{z}^{4}$.
The CTR intensity falls with
$1/{q}_{z}^{2}$, however. This apparent contradiction
can
be resolved by comparing the definitions of the
reflectivity coefficient
$R$
and the differential crosssection
$d\sigma /d\Omega $.
The reflectivity coefficient considers the total "reflected" energy and the
total incident flux. In contrast, the differential cross section is
normalized by the incident flux (proportional to
$1/\mathrm{sin}\alpha $) scattered
into a unit solid angle (also proportional to
$1/\mathrm{sin}\alpha $).
When these factors are taken into account,
$d\sigma /d\Omega =R{\mathrm{sin}}^{2}\alpha $.
To incorporate the Fresnel coefficients into simple kinematic xray
scattering, multiply each amplitude, both incoming and outgoing, by
${E}_{t}/{E}_{0}$
as in (
3.95) or (
3.93). Because
$n$ is so close to
unity, it hardly matters which polarization is assumed.
From (
3.29),
${q}_{z}=2k\mathrm{sin}\alpha $ and then defining
$({q}_{z}{)}_{c}=2k\mathrm{sin}{\alpha}_{c}$, we can simplify (
3.93) to be
$\frac{{E}_{t}^{\perp}}{{E}_{0}}=\frac{2}{1+\sqrt{1(({q}_{z}{)}_{c}/{q}_{z}{)}^{2}}}$ 
$(117)$

For
$\alpha <{\alpha}_{c}$, in addition to total external reflection, we
also have the simplification that
$\frac{{E}_{t}^{\perp}}{{E}_{0}}=2\mathrm{sin}\alpha /\mathrm{sin}{\alpha}_{c}$.
So the scattering amplitude will be multiplied by
$\frac{{E}_{t}^{\perp}}{{E}_{0}}(\alpha )$ and
$\frac{{E}_{t}^{\perp}}{{E}_{0}}(\beta )$.
These transmission factors are often neglected except near the
critical angle
${\alpha}_{c}$, because they factor approach unity quickly as
$\alpha >{\alpha}_{c}$ increases (see figure
3.8a).
Chapter 4
Experimental Procedures and Apparatus
1 Introduction
In this chapter, the specific procedures and apparatus used in these
experiments are documented. First, the sample preparation protocol is
described. The next section details the electrochemical apparatus and
procedures. The following section describes the xray scattering apparatus,
and the timing apparatus is discussed thereafter. Finally, suggestions for
future improvements are made.
2 Sample Preparation
2.1 Procurement
Samples (nominally Pt(111)) were obtained from the Materials Science Center
growth facility in Bard Hall.
These samples were oriented through Laue back reflection, and then
cut to the desired orientation by electrical discharge. Then, they were
polished with SiC paper and Al
${}_{2}$O
${}_{3}$ powder down to a grit size of
0.25
$\mu $m until a mirrorlike surface was obtained.
In principle, this procedure should produce crystals with welloriented faces.
To allow large terraces to form on metal crystals, it is desirable to reduce
the
miscut angle between the crystallographic axis (
e.g., (111))
and the surface normal. However, miscut angles as large as 2
${}^{\circ}$ were
measured in our lab by a combination of laser reflection and highresolution
Bragg diffraction. These can be traced to the Materials Science Center crystal
mounting apparatus, which was insufficiently rigid to ensure a miscut smaller
than a
few degrees.
2.2 Miscut Calculation
In this section, the angle
$\theta =(2\theta )/2$ is the Bragg diffraction
angle, while
$\phi $ is a rotation angle about the surface normal.
On a
miscut crystal, the Bragg diffraction peak will not be
coincident with the surface normal. The angle between them is defined to be
$\gamma $.
The surface normal was aligned with the
$\phi $ axis as follows. By reflecting
a laser beam from the mirrorlike face of the crystal, a tight spot was cast
onto a far wall or ceiling. Rotating the crystal about
$\phi $ caused the
laser beam to trace out a cone, causing the spot to trace out a corresponding
ellipse on the wall. By adjusting the tilt stages on the
sample goniometer, the ellipse could be narrowed until the spot did not move
with
$\phi $. Then, the surface normal was wellaligned with the
$\phi $ axis.
Moving the
$\phi $ angle to some fiduciary value, such as 0
${}^{\circ}$, the Bragg
diffraction angle
$\theta $ was recorded. Then,
$\phi $ was set to 180
${}^{\circ}$
and a different
$\theta $ was found. These two angles differ because
the diffraction peak traces out a cone, similarly to the laser beam. The
projection of the miscut along this one axis is (see figure
4.1)
$2{\gamma}_{x}=\theta (\phi ={0}^{\circ})\theta (\phi ={180}^{\circ})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(1)$

Figure 1: Determination of Miscut Angle
Likewise, by taking measurements at
$\phi ={90}^{\circ}$ and
$\phi ={270}^{\circ}$,
the orthogonal projection of
$\gamma $ is measured,
$2{\gamma}_{y}=\theta (\phi ={90}^{\circ})\theta (\phi ={270}^{\circ})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(2)$

By inspection of
${q}_{x}$ in figure
4.1,
${q}_{x}=q\mathrm{sin}{\gamma}_{x}$
(and
${q}_{y}=q\mathrm{sin}{\gamma}_{y}$), so the
true miscut angle is found from
$\mathrm{sin}{\gamma}^{2}=\mathrm{sin}{\gamma}_{x}^{2}+\mathrm{sin}{\gamma}_{y}^{2}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(3)$

2.3 Sample Preparation
After the miscut of each platinum crystal was measured, it was mounted onto
the orienting/polishing apparatus shown in figure
4.2.
The apparatus consists of a cylindrical barrel (E), with three dowels mounted
on it (C). (Only two dowels are shown in the figure.) These form one half
of the "kinematic mount"; the other half is a thick disk (B), into which the
dowels press. Opposite the first dowel is a circular depression (G), opposite
the second is a groove (H), and opposite the third is just the flat surface of
the disk. One of the dowels is fixed; the other two can be raised and lowered
my means of small adjustment screws running through the barrel. These permit
the disk to be oriented by a few degrees in any direction with respect to the
barrel axis. A long screw (D) attached to a spring and knurled knob (F) runs
through the barrel axis and passes through the clearance hole (I). When
tightened, the orientation is securely fixed. Finally, a mushroomshaped tip
(A) fastens to the kinematic disk (B) with three screws. The sample fits on
to the end of this tip.
Figure 2: Diagram of orienting/polishing apparatus. (a) Side view. (b)
Bottom view of the stage (B), showing the kinematic mount. Labels are
described in the text.
To prepare a sample, the tip of the apparatus was detached from the main body,
and then placed on a hot plate. A crystalbonding compound, liquid at high
temperatures, was dabbed onto the tip before the platinum crystal was added.
After cooling, the compound solidified to form a rigid, yet reversible, bond.
The tip was secured to the apparatus, and placed at the center of a rotation
of a fourcircle diffractometer. The orientation screws were adjusted until
the (111) Bragg diffraction peak was constant in
$\theta $ for any rotation
$\phi $ about the barrel axis. When this was attained, the locking thumb screw
was secured and the entire barrel was placed within the polishing sleeve.
The great advantage of this apparatus is that it can orient the sample face to
high precision, lock in that orientation, and then polish without loss of
precision.
Polishing took place on a
polishing wheel (Ecomet 4) run at the slowest speed (50 revolutions per
minute). Each polishing step took 15 minutes, and the sample was thoroughly
cleaned with water between steps.
The polish began with sandpaper
(Buehler 600 grit Carbimet paper discs #305112600)
and then successively finer (6
$\mu $m,
3
$\mu $m,
1
$\mu $m,
0.25
$\mu $m)
diamond powder (Struers DPSpray, P)
on a
nylon cloth (Buehler #407072)
with some lubricant (Struers DP Lubricant Blue, HQ or
Struers DP Lubricant Red, HQ).
At the end of this process, a mirror finish was invariably
obtained.
Thereafter, the tip was unscrewed from the apparatus and warmed on the hot
plate to remove the sample. The sample was then immersed in hot nitric acid
for at least four hours. This was done to remove any remaining contaminants,
particularly polishing powder. Also, even the smallest powder size
(0.25
$\mu $m) is extremely large on the length scales that xrays probe. To
remove strain and small grooves in the crystal surface, it was annealed in a
gas flame (available on tap in Clark Hall) for at least one hour. Finally,
the sample was characterized and the miscut calculated as described above.
Figure
4.3 demonstrates the dramatic improvement that
can take place after a sample is annealed. This sample was annealed for one
hour with a propane torch. For comparison, both intensities have been
normalized to yield a peak value of unity. Without this, the postannealing
peak would dwarf the preannealing peak. The true peak intensities differ by
a factor of 23. The fullwidth at halfmaximum was 0.60
${}^{\circ}$ before
annealing and 0.032
${}^{\circ}$ after annealing, a factor of 19.
Figure 3: Mosaic scans, before and after annealing, normalized to unit peak
height.
After several iterations of this orientingpolishingannealing
procedure, the bulk mosaic of the platinum
crystal was
$\approx 0.{018}^{\circ}$ (fullwidth at halfmaximum) and the
surface normal was oriented to within
$0.{027}^{\circ}$ of the (111) direction.
Empirically, we have found that both the mosaic and the miscut must be small
in order to observe the incommensurate overlayer. Furthermore, a high quality
substrate enhances the quality of voltammetric profiles.
The development of this procedure was crucial to the success of this
experiment. It has also propagated to other
groups (Cooper, Ho) in Clark Hall, and has enabled them to improve surface
quality and signaltonoise ratios in their own data.
3 Electrochemical Apparatus and Procedures
3.1 Solutions
Most of the solutions were prepared by Lisa Buller, and the following
paragraph is paraphrased from her dissertation [
52].
All solutions were prepared using water purified by a Hydro purification train
and a Millipore MilliQ system. The ionic salts were used as received and
always the purest available.
Perchloric acid solutions were prepared by dissolving either
CuO (99.999%, Aldrich) or CuCl
${}_{2}$ (99.999%, Aldrich) in Ultrex perchloric
acid. The addition of chloride anions was achieved through the addition of
CuCl
${}_{2}$ or NaCl (99.999%, Aldrich).
All solutions were bubbled for at least
15 minutes with prepurified nitrogen, which was further
purified by passage through oxygenabsorbing (MG Industries Oxisorb) and
hydrocarbon (Fisher Scientific Activated Carbon 614 Mesh)
traps to remove all traces
of oxygen.
3.2 Threeelectrode Electrochemical Cells
A typical welldesigned electrochemical cell has three electrodes
[
118,
43,
44]. The guiding
principle is to have all the interesting behavior occur at the
working
electrode. The other electrodes should be relatively inert and not complicate
the analysis of the processes that occur at the working electrode. All
potentials must be measured relative to some other reference value, which is
provided by the
reference electrode. The perfect reference electrode
would be "ideally nonpolarizable". That is, its potential remains constant,
regardless of the amount of current passing through it. Another purpose of
the reference electrode is to ensure that an applied potential change does
what we expect. Suppose we change the voltage of the potentiostat by
$\Delta V$. How do we know that this causes a
$\Delta V$ at the working electrode and
that part of the
$\Delta V$ does not go into the reference electrode? If the
reference electrode is ideally nonpolarizable, it maintains the same
potential value, and the full
$\Delta V$ is effected at the working electrode
interface.
The
counter (or
auxiliary) electrode assists in this process.
If the reference electrode is passing a significant amount of current, then
the assumption of ideal nonpolariziability is sorely tested. It is preferable
to have an alternate, lowresistance, pathway through which most of the current
flows. Counter electrodes are often composed of inert metals and have large
surface areas to minimize their overall resistance.
The various electrodes used in our experiments are shown in
figure
4.4. The large (10 mm) electrodes used for the
simultaneous
in situ xray and electrochemical measurements are
labeled by (a). The smaller (12 mm) "ball" electrodes, labeled by (b),
were produced by members of the Abruña group. These were of excellent
quality, and produced good electrochemical signals. However, they were too
small and too difficult to orient to be of use in our xray measurements. A
Ag/AgCl reference electrode is labeled by (c). These were constructed by Lisa
Buller [
52].
Figure 4: Electrodes used for the electrochemical measurements. (A) 10 mm
diameter electrode. (B) 12 mm diameter electrode. (C) Ag/AgCl
saturatedNaCl reference electrode.
3.3 Hanging Meniscus Cell
For electrochemical experiments on single crystals, a hangingmeniscus cell is
ideal.
A wire
is spotwelded to the sides of the crystal face, as in
figure
4.4(a,b).
The face of the crystal is then dipped into the solution compartment
(see figure
4.5), and
pulled upwards so that only a meniscus connects the sample with the bulk of
the solution. The reference and counter electrodes are placed in another
compartment, connected by a frit (partially fused glass).
Figure 5: Drawing of hanging meniscus cell.
The advantage of this arrangement is that only the crystal face of interest
in contact with the solution. Also, it is easy to use small (a few mm
diameter) crystals, which are often better quality than large (10 mm
diameter) crystals. The disadvantage is that the cell must remain in a
fixed vertical configuration; this requirement is incompatible with most xray
diffractometers. We used this cell only for voltammetric and current
transient measurements.
3.4 In Situ Xray cell
To perform simultaneous electrochemical and xray measurements, we constructed
a cell similar to the one developed by Toney and coworkers [
113].
This is a reflectiongeometry cell, as shown in figure
4.6.
The entire sample is immersed in solution, unlike the hangingmeniscus cell.
The solution is contained by 6
$\mu m$ polypropylene film, held in place by an
Oring.
Figure 6: Cartoon of in situ electrochemical xray cell.
A detailed illustration of the cell is provided by
figure
4.7. The majority of the cell is Teflon
(KelF is an alternative material with greater strength). The sample is
placed in the center and held in place by two noncircular KelF screws that
squeeze the sample laterally. The sample and screws are raised with respect
to a trough, where most of the solution resides. The reference electrode is
inserted from the side. The counter electrode is a platinum wire that
circumnavigates the trough several times.
Figure 7: Detailed plans for the
in situ electrochemical xray cell,
prepared by Lisa Buller [
52].
3.4.1 Absorption
The xray reflection geometry places a limit on the
in situ xray cell.
The polypropylene film is extremely thin, and while contributing to the
diffuse xray scattering background, does not
We must incorporate the absorption of xrays due to the layer of
solution that is covering the sample.
Figure 8: Absorption through a layer of thickness
$l$, given incident angle
$\alpha $ and reflected angle
$\beta $.
Consider an adsorbing layer of thickness
$l$ and attenuation per
unit length
$\mu $. From figure
4.8,
the total path length of xrays through the solution layer
will be
$x=l\mathrm{sin}\alpha +l\mathrm{sin}\beta $, where
$\alpha $ is the angle of
incidence, and
$\beta $ is the angle of reflection.
For grazing incidence (small
$\alpha $),
$l$ is limited by the horizontal
dimensions of the sample.
In the specular (
$\alpha =\beta $) case,
we have
$x=2l\mathrm{sin}\alpha $.
From the relation
${q}_{z}=\frac{4\pi}{\lambda}\mathrm{sin}\alpha $ (
3.30) and the absorption relation for
the intensity
$I={I}_{0}{e}^{\mu x}$ (
3.48),
then
$I={I}_{0}\mathrm{exp}(\frac{l\lambda}{2\pi}\mu {q}_{z})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(4)$

For aqueous solutions, the absorption coefficient can be calculated
from (
3.49):
$\mu =9.848\hspace{0.5em}{\text{cm}}^{1}$ for Cu K
$\alpha $ radiation (
$\lambda =1.542$Å)
and
$\mu =1.061\hspace{0.5em}{\text{cm}}^{1}$ for Mo K
$\alpha $ radiation
(
$\lambda =0.711$Å). The
$1/e$ absorption length is 1.0 mm for
K
$\alpha $ and 0.942 cm for Mo K
$\alpha $.
Clearly, using high energy xrays greatly reduces the
problem of absorption.
For an typical
$L=1.5$,
${q}_{z}=1.387$Å and
$\lambda =1.542$Å,
even
$l=1$mm of solution
causes an attenuation of 40%.
It is therefore important to remove as much solution from the cell as
possible, while still leaving enough to maintain good electrical contact
between the face of the working electrode and the other two electrodes.
3.5 Potentiostat
A potentiostat is an instrument to keep the sample under potential (voltage)
control and monitors the current. (A galvanostat, in contrast, keeps the
sample under current control and monitors the voltage.)
The simplest possible potentiostat circuit for a threeelectrode configuration
is shown in
figure
4.9.
The operational amplifier will supply sufficient current to keep the reference
electrode at a potential
$V$ with respect to ground (or the working
electrode). Significant current will pass from the counter into the working
electrode, but very little will pass through the reference electrode.
This is in accordance with section
4.3.2.
Figure 9: Simple potentiostat circuit for a threeelectrode electrochemical
cell.
Adapted from [
21].
The PAR 283 (Princeton Applied Research, Model 283) is a versatile instrument,
which can be run as either a potentiostat or a galvanostat. It accepts
commands over a GPIB (IEEE488) interface, and has its own sophisticated, if
unique, command language.
During most experimental runs, we used the PAR 283 to
acquire either cyclic voltammograms (described in
section
5.2)
or chronoamperometric transients.
Cyclic voltammograms were taken of each sample while the solution layer was
extended, when the solution was pulled out, and at various points during the
experimental run on a given sample.
3.6 Safety
In a dilute (0.1 M) form, perchloric acid poses a minor health hazard.
Contact with skin is mildly irritating, and should be rinsed off as soon
as possible. Contact with the eye is more serious. For this reason, splash
goggles should be worn at all times. In case of a large spill, sodium
bicarbonate should be available for neutralization.
The platinum sample glows yellowwhite during annealing. There is a
significant ultraviolet spectral component, and the sample needs to be kept
under continual supervision. To prevent permanent retinal damage,
ultravioletresistant goggles must be worn during this process.
3.7 Sample Treatment
Before a sample is inserted into the xray cell, a careful protocol must be
observed. UPD is extremely sensitive to chemical contaminants, especially
metallic and organic ones.
 Spotweld a clean platinum to the side of the sample, if not already present.

Clean the top of the cell with solution; it should bead over everything.
Then drain it away.

Rinse the cooling cell with solution at least three times,
draining with forced nitrogen.

Have nitrogen flowing into the cooling cell.

Flow solution into the cell, allowing a bubble to form on top of the cell.

Put the hood (which should have nitrogen flowing through it)
over the cell.

Wear goggles to prevent retina burn.

Anneal for 8 minutes the first time, 5 minutes each subsequent time.

Cool in cooling cell for 4 minutes (under nitrogen overpressure).

Flow solution into the cooling cell; fill to the level of the input port.
Immerse the sample for at least one minute; a longer period is
acceptable.
 The platinum surface will oxidize very quickly. The next step must be
done very quickly!

Remove the sample from the cooling cell and transfer it to the xray cell. To
buy time, it is often helpful to squirt some solution or (deoxygenated) water
on the face.

Tighten KelF screw to fix sample in place.
 Cover with polypropylene film, which should be prerinsed. Cover with
the Oring and metal sleeve. Screw down the four jeweler screws evenly,
for even pressure along the Oring.

The rest potential (no external potential applied) should be near 650 mV.

With solution layer extended, run a cyclic voltammogram from 650 mV to 200 mV
at 5 mV/s.

With a syringe, pull out most of the solution and run an identical cyclic
voltammogram.
The cooling cell used in the previous procedure is shown in
figure
4.10. The Pt(111) sample is shown hanging from its
hook. During operation, nitrogen is kept flowing through the cell. Solution
is drained from the bottom outlet, and introduced from either of the upper
inlets.
Figure 10: Drawing of cooling cell.
4 Xray Apparatus
Xray scattering is a nearly ideal probe of the ordering kinetics of the
twodimensional overlayers found in UPD systems. Unlike electrons or
neutrons, Xrays can penetrate through a
thin solution layer, allowing the experiments
to be performed
in situ. Xrays provide structural information on
atomic length scales without perturbing the system with mechanical probes or
large fields, as scanning probe microscopes may. Finally, the extremely high
flux from a modern synchrotron xray source, such as the National Synchrotron
Light Source (NSLS), permits the weak diffraction signal from a single CuCl
bilayer to be studied at high resolution.
In our experiments, the white beam produced by a bend magnet on the NSLS
electron storage ring was focused in both transverse directions by a total
external reflection mirror.
A monochromator consisting of two Ge(111) crystals was configured
to select 8.80 keV xrays.
The substrate was placed in a thin film geometry xray cell
similar to those used by Toney and coworkers [
113].
The cell was placed at the center of rotation of an Eulerian
cradle and two pairs of XYslits between the sample and the detector
determined the resolution of the scattered xrays.
The resolution is discussed in detail in section
5.5.
In the lab, xrays were produced by a Rigaku (Model RU200) rotating
Cu anode source. The Cu K
${\alpha}_{1}$ was selected by means of either a single
or triplebounce Si(111) monochromator. Although the instrument can provide
a 60 kV accelerating voltage and 200 mA filament current, the lowest power
setting (20 kV, 10 mA) was usually sufficient for sample orientation.
To detect xrays we used an integrated NaI scintillation crystal,
photomultiplier, and preamplifier (Bicron 1XMP 040BX). The resulting
electrical signal was sent through a combined amplifier and pulseheight
analyzer (Canberra Model 1718)
for broad energy discrimination.
The TTL pulses were then sent to a simple adding memory module (Kinetic
Systems 3610 Hex Counter) that also received timing pulses from another
module (Kinetic Systems 3655 Timing Generator).
In the lab, the signals were then acquired by a data acquisition card
(DSP 6001). At the NSLS, data acquisition was handled by a CAMAC to SCSI
interface module. In both places, the fourcircle diffractometer (Huber) was
under the control of a sophisticated software package ("spec", by Certified
Scientific Software) running on an Intel 486based computer.
5 TimeResolved Measurements
Timeresolved xray measurements can be accomplished in several ways.
For instance, Bergmann
et al.
[
32] used the timing of the
electron bunches around the synchrotron ring for Mössbauer experiments.
This is ideal for extremely short time ranges.
Very recently, Knight
et al.
[
88]
have demonstrated a prototype device etched onto a silicon wafer to
study protein folding. This works by mixing two jets together (for instance,
folded protein and a denaturing agent) and squirting the product through a
long channel. Because the flow is lamellar, the mixing occurs by diffusion.
Because the fluid volumes are extremely low (nanoliters), the diffusive length
scale is extremely short, and the mixing time is on the order of
microseconds. By moving the device along the xray beam, different times
after the mixing event are examined. In this way, position and time are
coupled.
In contrast, our method relies upon timing electronics to separate the xray
signal into various time bins. This "stroboscopic" method was first used by
our group to
study chargedensity wave kinetics
[
127].
Although the PAR 283 claims to have a trigger, it does not operate in the
standard sense of the term.
Normally, when an instrument (an oscilloscope, for example) is waiting for an
electronic trigger, operation ceases until the trigger is detected. Then, the
other operations are begun or resumed. Instead, the PAR 283 performs a
variety of operations, periodically polling the input to see if the trigger
signal has arrived. Only then is the specified series of actions initiated.
This can lead to an unpredictable delay between the trigger input and the
initiation of commands by the PAR 283. For this reason, it was decided to
have the PAR 283 be the master controller and send trigger signals to the
other instruments.
The control diagram is shown in figure
4.11. The
potentiostat applies a voltage to the sample and continuously reads current
from it. At the beginning of a voltage cycle, it sends a trigger pulse to
the waveform generator (Keithley 3940 multifunction synthesizer). This sends a
series of pulses to the multichannel scaling averager
(DSP 2190), which consisted of a multichannel scaling module (DSP 2090) and a
signal averaging memory (DSP 4101). These bin pulses both initiated the
averaging memory and incremented the current memory location (time bin).
These timing modules also received xray intensity data, which was added to
the time bin. At the end of a series of voltage cycles, the memory was dumped
to the computer for display and analysis.
The chronoamperometric traces (current vs. time)
were digitized into 5000 time bins, and collected by the potentiostat. At the
end of the voltage cycle, these were also sent to the computer.
Figure 11: Instrumentation for timing experiments.
6 Future Improvements
6.1 New Cell Design
With the advent of highenergy synchrotron sources, xray cell geometries with
a thick solution layer have become feasible. Brossard
et al.
[
49]
describe a cell very similar to ours, but without the thin solution layer
constraints. The cyclic voltammetry measurements they present are not high
quality; presumably, this is a function of sample preparation, and not the
cell itself.
6.2 Improved Sample Quality
As discussed in section
4.2.1, these crystals were not ideal.
The simplest course would be to procure samples from a reliable external
source. If annealing is still necessary, a new method should be found.
Heating with a torch sometimes produced
cloudy spots in the center of the sample, where the flame was hottest.
A more even annealing could be done in ultrahigh vacuum and by attaching it
to a heating stage.
6.3 Area (or Linear) Detectors
In our experiment, the highest resolution
(section
5.5)
was obtained
by rotating the sample about the surface
normal. In this case, an area or linear detector would not be helpful.
However, there may be cases in which the resolution is sufficient to simply
have an area detector mounted on the end of the detector arm.
A CCD (chargecoupled device) could be run in a mode such that each line
is shifted down. In this case, all of the timeresolved data could be
recorded on the device, which would speed up the data acquisition time by
the number of
$q$points.
6.4 Improved Electronics
In retrospect, the PAR 283 potentiostat was difficult to program, and not
as flexible as anticipated. A superior solution would be to purchase the
best possible analog potentiostat (BAS is an good choice) that accepts
an external line voltage. Then buy a good programmable digitaltoanalog card
that can be programmed easily and has sufficient time resolution.
It may also be advantageous to replace CAMAC modules with cards within the
computers. At the time of these experiments, we
needed to maintain compatibility with equipment at CHESS and NSLS X20A.
Now, the DSP timing modules could be replaced with a multichannel
scalar card (Oxford MCS, for instance). The counter/timer modules could be
replaced with an integrated counter/timer card (Keithley CTM010). These
particular upgrades are already underway
for the new spectrometer being set up in the Brock group laboratory.
Chapter 5
Cyclic Voltammetry and Static Xray Measurements
1 Introduction
This chapter begins the presentation of our data on the underpotential
deposition (UPD) of Cu onto Pt(111) in the presence of Cl.
The first section presents our cyclic voltammetry on this UPD system.
Subsequent sections discuss the hexagonal coordinate system and the structure
of the incommensurate UPD overlayer. Finally, our static xray measurements
of this overlayer are presented.
2 Cyclic Voltammetry
Cyclic voltammetry, as the name suggests, is a measurement of the current
while the voltage is being swept (usually linearly with time). The cyclic
adjective refers to the fact that the voltage is swept in both directions. As
a function of time, the applied voltage traces out a triangular wave
(figure
5.1a).
Figure 1:
(a) Applied voltage waveform.
(b) Current response for an ideal system.
Cyclic voltammetry is a commonly used technique in electrochemistry, with many
different applications. The next section illustrates a simple example: the
cyclic voltammogram from an adsorption / reduction reaction, where the
adsorption follows a Langmuir isotherm. In our experiments, this technique
provided information on the equilibrium phase diagram.
2.1 Cyclic Voltammetry for an Ideal System
This section follows the theory presented in
section
2.8. It may be helpful to review that
section before continuing.
As suggested by Bard and Faulkner
[
31],
consider the
reduction of species O at the electrode to form species R.
$O+n{e}^{}\to R\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(1)$

We begin at a sufficiently positive potential such that all of the adsorbates
are in the oxidized state (O). At
$t=0$, we sweep the potential negatively
and monitor the current generated at the electrode. We want an expression for
the current density
$j(t)$ in terms of the voltage
$\Delta V(t)$ and the
(constant) sweep rate
$v$. For reduction, we must sweep in the negative
direction, so
$d(\Delta V)/\mathrm{dt}=v$.
The current density comes from the reaction (
5.1), so
$j(t)=ne\frac{d{\Gamma}_{O}}{\mathrm{dt}}=nev\frac{d{\Gamma}_{O}}{d(\Delta V)}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(2)$

To find an expression for
${\Gamma}_{O}(t)$,
assume the reaction is completely reversible, so that the Nernst
equation (
2.41) applies:
$\Delta V=\Delta {V}_{0}\text{'}\frac{kT}{ne}\sum _{i}{\nu}_{i}\mathrm{ln}{c}_{i}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(3)$

In practice, this assumption means that the
$v$ must be very small in
comparison with the reaction rate.
Rewriting (
5.3) to find the ratio of concentrations at the electrode surface,
$\frac{{c}_{O}(z=0,t)}{{c}_{R}(z=0,t)}=\mathrm{exp}[\frac{ne}{kT}(\Delta V\Delta {V}_{0}\text{'})]\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(4)$

Assume there are no adsorbateadsorbate interactions, except for the
O
$\to $ R reaction and the competitive filling of adsorption sites.
If there were only one species on the electrode, then we would use the
Langmuir isotherm (
2.56)
$\theta =\Gamma /\Gamma \_\text{sat}=\frac{c\mathrm{exp}(\frac{\Delta G}{\mathrm{kT}})}{1+c\mathrm{exp}(\frac{\Delta G}{\mathrm{kT}})}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(5)$

When there are two species, O and R, competing for adsorption, then this
becomes
${\theta}_{O}={\Gamma}_{O}/{\Gamma}_{O,\text{sat}}=\frac{{c}_{O}\mathrm{exp}(\frac{\Delta {G}_{O}}{\mathrm{kT}})}{1+{c}_{O}\mathrm{exp}(\frac{\Delta {G}_{O}}{\mathrm{kT}})+{c}_{R}\mathrm{exp}(\frac{\Delta {G}_{R}}{\mathrm{kT}})}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(6)$

So the ratio of surface excesses is
$\frac{{\Gamma}_{O}}{{\Gamma}_{R}}=\frac{{c}_{O}{\Gamma}_{O,\text{sat}}\mathrm{exp}(\frac{\Delta {G}_{O}}{\mathrm{kT}})}{{c}_{R}{\Gamma}_{R,\text{sat}}\mathrm{exp}(\frac{\Delta {G}_{R}}{\mathrm{kT}})}=\frac{{b}_{O}{c}_{O}}{{b}_{R}{c}_{R}}$ 
$(7)$

where the abbreviations
${b}_{O}$ and
${b}_{R}$ are introduced for simplicity.
Combining (
5.4)
with (
5.7) the ratio
$x$ is
$x\equiv \frac{{\Gamma}_{O}}{{\Gamma}_{R}}=\frac{{b}_{O}}{{b}_{R}}\mathrm{exp}[\frac{ne}{kT}(\Delta V\Delta {V}_{0}\text{'})]\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(8)$

Since we assumed that
${\Gamma}_{R}(t=0)=0$, then
${\Gamma}_{O}(t)+{\Gamma}_{R}(t)={\Gamma}_{O}(t=0)$ for all
$t$.
Then
$\Gamma}_{O}={\Gamma}_{O}(t=0)\frac{x}{1+x$ 
$(9)$

and
now taking the derivative
$\frac{\partial {\Gamma}_{O}}{\partial (\Delta V)}={\Gamma}_{O}(t=0)\frac{\partial x}{\partial (\Delta V)}\frac{x}{(1+x{)}^{2}}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(10)$

Substituting (
5.10) into (
5.2)
and replacing
$x$,
$j=\frac{{n}^{2}{e}^{2}}{kT}v\Gamma (t=0)\frac{\frac{{b}_{O}}{{b}_{R}}\mathrm{exp}[\frac{ne}{kT}(\Delta V\Delta {V}_{0}\text{'})]}{{\{1+\frac{{b}_{O}}{{b}_{R}}\mathrm{exp}[\frac{ne}{kT}(\Delta V\Delta {V}_{0}\text{'})]\}}^{2}}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(11)$

Finally, we want to find the
fullwidth at halfmaximum. Start with the simplified function
$y(x)=\frac{{e}^{x}}{{(1+{e}^{x})}^{2}}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(12)$

We want to find
$x$ such that
$y(x)=y(0)/2=1/8$. Substituting
$z={e}^{x}$
and solving the resulting quadratic equation
${z}^{2}6z+1=0$ yields
$z=3\pm \sqrt{8}$, or
$x\approx \pm 1.7627$ (which is symmetric about
$x=0$, as
expected). Thus the full width at halfmaximum of (
5.11) is
approximately
$3.5255\times \mathrm{kT}/(ne)$, or about
$90.6/n$ mV at
25
${}^{\circ}$C.
2.2 Cyclic Voltammetry for Cu/Cl/Pt(111) UPD
The voltammetric profile is shown in figure
5.2.
These data were collected in our xray scattering cell at
$5$ mV/s with
$0.1$ M HClO
${}_{4}$ as a supporting electrolyte,
$1$ mM Cu
${}^{2+}$ and
$10$ mM
Cl
${}^{}$.
The current response exhibits two sharp and welldefined voltammetric
deposition peaks centered at about
$+0.47$ and
$+0.32$ V (vs. a Ag/AgCl
reference electrode).
Upon reversing the potential sweep,
the current response then exhibits two sharp stripping peaks corresponding to
the reverse reactions.
Figure 2: A cyclic voltammogram taken in the xray scattering cell at a sweep
rate of
5 mV/s with
$1$ mM
Cu
${}^{2+}$ and
$10$ mM Cl
${}^{}$, and
$0.1$ M HClO
${}_{4}$ as a supporting
electrolyte.
A schematic of the deposition process is depicted in
figure
5.3.
The labels A, B, C in this figure also correspond to the potential regions in
figure
5.2.
At the rest potential (region A),
chloride anions are adsorbed on the platinum surface
in a nonordered fashion [
158,
108].
As the potential is swept negatively,
copper is electrodeposited onto the platinum surface at a
welldefined potential
[
90,
149,
94].
The electrodeposited copper and chloride ions together form an ordered
CuCl bilayer structure
incommensurate
[
131]
with the platinum surface
(region B).
If the potential is then moved further in the negative direction,
there is further
copper deposition,
creating a full,
commensurate copper monolayer
(region C) [
94,
90,
156,
157,
95,
72].
The copper
monolayer is, in turn, believed to be covered by a disordered layer of
chloride anions.
On the reverse (positivegoing) sweep the reverse processes take place;
that is, some copper desorbs,
forming the CuCl lattice structure (region B) and at
more positive potentials the copper is completely stripped from the surface,
leaving the disordered chloride anions adsorbed on the surface
and returning the system to region A.
The sharp voltammetric features seen in Figure
5.2
are the electrochemical signature of a
clean and wellordered surface.
Figure 3: Cartoon of phases in the UPD of Cu on Pt(111) in the presence of
Cl.
In our experiments, we used cyclic voltammetry for three purposes. Most
importantly, it served as a qualitative "fingerprint" of the UPD process
itself. The cyclic voltammograms are extremely sensitive to contamination of
the solution, poor quality of the singlecrystal electrode surface, and
dissolved oxygen in solution. Empirically, we found that obtaining a good
cyclic voltammogram was a
necessary, but not
sufficient,
condition to finding a wellordered UPD layer with xray scattering.
Secondly, the width of the peaks tells us an important fact about this UPD
process. As derived in section
5.2.1 and
plotted in figure
5.1b, the fullwidth at halfmaximum of the
current peak should be close to
$90.6/n$ mV at room temperature.
The
fact that our peaks are significantly smaller than this value implies that
there is significant interaction among the adsorbed ions in the UPD layer. In
particular, once some ions are adsorbed/desorbed, this tends to enhance the
probability that other ions will follow.
Thirdly, unlike
figure
5.1b, the peak positions for the negative voltage
sweep are displaced from their partners on the positive voltage sweep.
This hysteresis, which is present even for
very slow sweep rates (1 mV/s), is an indication that the system is
kinetically limited.
The reason for this, which had been unclear, is explained by our timeresolved
data in chapter
6 in terms of a nucleation and
growth model.
3 Hexagonal Coordinates
The remainder of this chapter concerns xray scattering from the platinum
surface and the incommensurate CuCl overlayer.
For cubic crystals, the basis vectors are usually defined to be mutually
perpendicular and of equal length (like the
$x$,
$y$, and
$z$ Cartesian axes).
When dealing with the (111) surface of a facecentered cubic lattice, however,
it is convenient to redefine the basis vectors. The
$c$ axis is defined to be
along the (111) surface normal. Because of the ABCABC... stacking, the
reciprocal space (111) is mapped onto (003). The
$a$ and
$b$
realspace basis vectors, which lie in the plane of the surface,
are shown in figure
5.4a.
Because these basis vectors subtend 120
${}^{\circ}$, these are often
called "hexagonal surface units" [
75].
The circles in the figure represent
platinum atoms on the (111) surface.
These realspace lattice sites are indexed in
figure
5.5.
(a)
(b)
Figure 4:
(a) The Pt(111) surface with surface lattice vectors
$a$,
$b$, which are perpendicular to
$c$ =
(111).
(b) Reciprocal lattice vectors corresponding to the unit cell chosen
in (a);
${a}^{*}$ and
${b}^{*}$ subtend 60
${}^{\circ}$ and are
perpendicular to
${c}^{*}$.
From the convention (
3.64) that
${\hat{a}}_{i}\xb7{\hat{q}}_{i}={\delta}_{\mathrm{ij}}$,
${b}^{*}$ is orthogonal to
$a$ and
$c$,
and
${a}^{*}$ is orthogonal to
$b$ and
$c$. So
${a}^{*}$ and
${b}^{*}$ must point in the directions indicated in
figure
5.4b. These vectors subtend 60
${}^{\circ}$
and generate a triangular lattice. The reciprocalspace lattice sites are
indexed in figure
5.6. Although this figure
appears superficially identical to figure
5.5, the
indexing is different due to the different angles subtended by the
basis vectors. From this point on, Bragg peaks are indexed using these
hexagonal units.
The conversion from cubic to hexagonal units is easily accomplished.
Writing both
$q$vectors as column vectors, then the matrix product
${q}_{\mathrm{cubic}}={J}_{h\to c}{q}_{\mathrm{hexagonal}}$ and
${q}_{\mathrm{hexagonal}}={J}_{c\to h}{q}_{\mathrm{hexagonal}}$.
These transformation matrices are
${J}_{h\to c}=\frac{1}{3}\left(\begin{array}{ccc}\hfill 4& \hfill 2& \hfill 1\\ \hfill 2& \hfill 2& \hfill 1\\ \hfill 2& \hfill 4& \hfill 1\end{array}\right)$ 
$(13)$

and
${J}_{c\to h}=\frac{1}{2}\left(\begin{array}{ccc}\hfill 1& \hfill 1& \hfill 0\\ \hfill 0& \hfill 1& \hfill 1\\ \hfill 2& \hfill 2& \hfill 2\end{array}\right)$ 
$(14)$

where, of course,
${J}_{h\to c}{J}_{c\to h}=1$.
These matrices can be generated from any two noncollinear vector
transformations, such as
$(111)\to (003)$ and
$(\stackrel{\u203e}{1}11)\to (101)$.
Figure 5: Indexing of surface units in real space.
Figure 6: Indexing of surface units in reciprocal space.
4 Discussion of Incommensurate Structure
Tidswell and coworkers
[
131] have
characterized the incommensurate bilayer that is present for intermediate
potentials. They found a triangular array of xray scattering rods, sharp in
$H$ and
$K$ but diffuse in
$L$. The inplane spacing was approximately 0.765
that of the truncation rods from the underlying platinum crystal. This
corresponds to an inplane bilayer lattice spacing 30% greater than Pt(111).
Based upon their measurements of the positions and intensities of these
scattering rods, they propose the model shown in
figure
5.7. The Cu and Cl form a bilayer wherein the
Cu atoms (small gray circles) are close to the Pt surface (large gray
circles), and the Cl atoms (large empty circles) rest above the Cu,
coordinated in the threefold hollow sites of the hexagonal Cu lattice. If
the Cl atoms are partially ionized toward Cl
${}^{}$ (making them larger), it is
reasonable to assume that they are in close proximity to one another and
determine the incommensurate lattice spacing.
Tidswell
et al. claim that the spacing is near to that of
closepacked spheres with the Cl
${}^{}$ ionic radius.
Figure 7:
Realspace map of the incommensurate overlayer, looking down on the
Pt (111) surface (gray). The Cu atoms (black) lie above the Pt substrate
and
are incommensurate with it. The Cl ions (hollow) lie in threefold
hollow sites above the Cu layer.
It is surprising that the bilayer structure fails to follow the commensurate
lattice spacing, yet preserves the orientation of the underlying Pt lattice.
However, this scenario has been predicted by Novaco and McTague
[
103]. They hypothesize staticdistortion waves (analogous to
chargedensity waves) in cases where the adlayer is weakly adsorbed to the
substrate. Minimizing the energy leads to a preferred orientation of the
adlayer with respect to the substrate. The relative orientation angle need
not be zero. Shaw, Fain, and Chinn [
122] experimentally observed Ar
monolayers adsorbed onto graphite substrates at a range of low temperatures
(32  52 K). Their LEED (lowenergy electron diffraction) measurements
demonstrated the relative orientation angle was inversely related to the Ar
monolayer lattice spacing. Their previous measurements with other adsorbed
noble gases find cases where the relative orientation is zero, as in our case.
A complete review on this subject can be found in the Pokrovsky and Talapov
[
110].
Ben Ocko and coworkers
[
104]
have studied a commensurateincommensurate phase
transition in Br UPD on Au(100). Their results are compared with theoretical
predictions by Pokrovsky and Talapov
[
109].
Our interest is primarily in the kinetics of this system, rather than
performing more detailed crystallography on the static phases. Therefore, it
is necessary to find a useful parameter to monitor the emergence of order
during formation of the bilayer. We chose the (0.765 0
$L$) rod, because it
is the lowest index peak. Our choice of
$L=1.5$ depends upon two factors:
the minimization of background scattering from solution and polypropylene
film, and the
$L$dependent scattering from the bilayer itself. Absorption
effects (section
4.3.4) are most prominent for low
$L$, as is the diffuse background scattering. The bilayer scattering
oscillates with
$L$, as can be seen from examining the structure factor.
As mentioned above, the scattering is diffuse along the
${q}_{z}$ axis. This is a
consequence of the nearly twodimensional nature of the adsorbed layer.
However, the spacing between the Cu and Cl layers causes an interference
effect that is manifested in the oscillating intensity along
${q}_{z}$. A single
twodimensional layer of hexagonally arranged atoms has a sixfold rotation
axis about the surface normal. Add a second commensurate layer with the same
number of atoms, by putting a chloride atom at position
$R+a$ for every
copper lattice position
$R$. Assuming that these chlorides are attracted to
the copper layer, they will probably sit in the threefold hollow sites
between copper atoms.
At this point, the symmetry is broken and the bilayer is only threefold
symmetric.
To derive the structure factor, we need to find the coordinates of the
threefold hollow site. Consider the location of the threefold hollow site
between (0,0), (1,0), and (1,1). Referring to
figure
5.8, and recalling that the center of an
equilateral triangle is
$1/3$ of the distance from a side to the opposite
vertex, the position is
$\frac{1}{2}(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0)+\frac{1}{6}(1\mathrm{\hspace{0.5em}\hspace{0.5em}}2)=\left(\frac{2}{3}\mathrm{\hspace{0.5em}\hspace{0.5em}}\frac{1}{3}\right)$ 
$(15)$

Figure 8: Real space image of bilayer. The open circles represent the
positions of the copper atoms, and the closed circles represent the
positions of the chloride atoms. In reality, the chloride atoms occupy
considerably more space than the copper atoms, as shown in
figure
5.7.
Now we turn our attention to the phase factors that influence the structure
factor of the bilayer. We can consider the twodimensional CuCl bilayer as
though it were in isolation. The underlying Pt lattice has no fixed
periodicity with respect to it, and so will not change any of the structure
factors except at the specular condition
$H=K=0$. As shown in
section
3.9.2, the structure factor due to the
addition of another atom is
$S(q)={\left\sum _{j=1}^{n}{e}^{iq\xb7{a}_{j}}\right}^{2}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(16)$

The Cl atom sits at the threefold hollow site,
and using
$q=2\pi /a(H,\mathrm{\hspace{0.5em}\hspace{0.5em}}K)$ from (
5.15),
the structure factor at
$L=0$ is
$S(H,K,L=0)={1+\mathrm{exp}\frac{2\pi i}{3}(2H+K)}^{2}=\{\begin{array}{ccc}4\hfill & 2H+K=3n\hfill & \hfill n\mathrm{\hspace{0.5em}\hspace{0.5em}}\text{any integer}\\ 1\hfill & \text{otherwise}\hfill \end{array}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(17)$

This leads to a hexagonal lattice expanded by a factor of
$\sqrt{3}$, and
rotated by 30
${}^{\circ}$, as shown in figure
5.9.
Figure 9: Reciprocal space map of monolayer (circles) and bilayer (crosses).
The monolayer points correspond to figure
5.6.
The bilayer points incorporate the
$L=0$ structure factor
from (
5.17).
Now consider the
$L>0$ scattering.
Due to the threefold symmetry,
the points
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$,
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$, and
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}1)$ illustrate all of the
possible cases. (The
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}1)$ is equivalent to
the
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$.)
These structure factors are plotted in figure
5.10.
As expected, they follow a simple sinusoidal form, but the initial phase at
$L=0$ is determined by (
5.17).
Figure 10: Structure factors for various the
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ (solid),
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}1)$ (dotted), and
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ (dashed) rods as a function of
${q}_{z}$.
The careful reader will note that I have neglected the atomic form factors of
Cu and Cl (that depend upon the ionization state, which is not known), and
the "DebyeWaller" factors due to disorder within the CuCl bilayer.
However, in my opinion, there is a larger uncertainty that makes these
considerations moot. The illustration in figure
5.9 is only
one of two possibilities. The sixfold symmetry was broken by the assumption
that the chloride atoms fall into the upwardpointing triangles of
figure
5.5. We can equally well imagine that the
chloride atoms fall into the downwardpointing triangles. This is equivalent
to just a 60
${}^{\circ}$ rotation and changes the structure factors accordingly.
The threefold hollow site immediately above (along the yaxis) from the
origin in figure
5.9 is
$\frac{1}{3}(1\mathrm{\hspace{0.5em}\hspace{0.5em}}2)$. (This is not
shown in the figure, because we previously took the other choice.) The
resulting structure factor is
$S(H,K,L=0)={1+\mathrm{exp}\frac{2\pi i}{3}(H+2K)}^{2}=\{\begin{array}{ccc}4\hfill & H+2K=3n\hfill & \hfill n\mathrm{\hspace{0.5em}\hspace{0.5em}}\text{any integer}\\ 1\hfill & \text{otherwise}\hfill \end{array}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(18)$

We see that the roles of
$H$ and
$K$ are reversed
from (
5.17).
Even more likely is that there will be some combination of these two
possibilities. As the incommensurate bilayer forms, different domains
nucleate and grow on the surface (see chapter
6).
There will be domains with both possible orientations. These domains
are likely to be incoherent; that is, there will be no definite phase
relationship between them. The overall intensity will be the sum of the
intensities from all the domains.
This intensity is shown in figure
5.11,
which assumes an equal coverage for the two domain types.
As can expected from the sum of two sinusoidal functions, the
period of the oscillation is changed.
The
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}1)$ and
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ structure factors are identical, since we are
taking equal numbers from the two domain type, so
$H$ and
$K$ are identical.
Of course, due to the stochastic
nature of the nucleationgrowth process, the distribution may be skewed
toward one orientation over the other, instead of the
$1:1$ ratio
depicted here.
Figure 11: Structure factors for various the
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ (solid),
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}1)$ (dotted), and
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ (dashed) rods as a function of
${q}_{z}$.
The
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}1)$ and
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ coincide.
5 Static Xray Data
Because of the conflicting reports of the CuCl overlayer structure at
intermediate voltages, our first task was to make some static xray
measurements. We have observed the overlayer structure numerous times during
several experimental runs. Despite many attempts, we have never found
scattering at the
$(0.25m\mathrm{\hspace{0.5em}\hspace{0.5em}}0.25n\mathrm{\hspace{0.5em}\hspace{0.5em}}L)$ rods that can be attributed to the
incommensurate overlayer, where
$m$ and
$n$ are integers up to 6. On the
other hand, we have found scattering at the
$(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0\mathrm{\hspace{0.5em}\hspace{0.5em}}L)$ and
$(0\mathrm{\hspace{0.5em}\hspace{0.5em}}0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}L)$ rods that was voltagedependent. This indicates that the
measurements of Tidswell
et al.
[
131] are
correct, while the LEED measurements of Kolb
[
102]
cannot be confirmed. Assuming that the
$(0.75\mathrm{\hspace{0.5em}\hspace{0.5em}}0\mathrm{\hspace{0.5em}\hspace{0.5em}}L)$ seen in the LEED
corresponds to the
$(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0\mathrm{\hspace{0.5em}\hspace{0.5em}}L)$ peak, then the additional peaks may be
the result of multiple electron scattering. Alternatively, the
ex situ
experiment may change the ordering of the CuCl overlayer. This would not be
surprising, as the vacuum and solution environments are very different.
Figure
5.12 shows the
$(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0\mathrm{\hspace{0.5em}\hspace{0.5em}}1.5)$ overlayer Bragg
peak at two different values of the applied potential. These data clearly
demonstrate the presence of the incommensurate overlayer at 350 mV and its
absence at 250 mV. The potentialindependent background is due to
scattering from the solution layer and the polypropylene film that contains
it. By integrating for several seconds per
$q$point, the signal can
be easily resolved above this background. In the timeresolved measurements
(chapter
6), where the xray signal is split into
many time bins, this poses a considerable experimental challenge.
Figure 12:
Scattered intensity at
$q=(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0\mathrm{\hspace{0.5em}\hspace{0.5em}}1.5)+{q}_{\perp}$
at 350 mV (hollow) and 200 mV (filled) vs. Ag/AgCl.
The solid line is the best fit to a Lorentzian line shape.
The shape of the diffraction peak is wellfit by a
Lorentzian line shape (the solid line in figure
5.12),
and the halfwidth at halfmaximum
$\Delta $ corresponds to a correlation
length
$\xi =1/\Delta \approx 280$Å.
A Lorentzian is appropriate for systems with only
short range positional order.
The inset indicates the location of the Bragg rods of the twodimensional
incommensurate overlayer (hollow) and the crystal truncation rods
(section
3.11.1) of the Pt substrate (filled).
The arrow represents the transverse scan shown in
the main figure. The transverse direction is denoted by
${q}_{\perp}$ and is orthogonal to
$(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$
and at constant
$L$.
The corresponding xray scans through the overlayer Bragg peak, but along the
radial direction, are shown in
figure
5.13.
The radial direction is denoted by
${q}_{\parallel}$ and holds
$K$ and
$L$ constant.
In this case, the peak is broader than
scans through the
${q}_{\perp}$ direction.
At first glance, this appears to indicate that the correlation function is
strongly asymmetric, with
${\xi}_{\perp}>{\xi}_{\parallel}$.
This would be a surprising result.
However, as the next paragraphs will show, this can be accounted for by the
asymmetry of the resolution function.
Figure 13:
Scattered intensity at
$q=(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0\mathrm{\hspace{0.5em}\hspace{0.5em}}1.5)+{q}_{\parallel}$
at 350 mV (hollow) and 200 mV (solid) vs. Ag/AgCl.
The solid line is the best fit to a Lorentzian line shape.
The asymmetry of the resolution is due to the differing longitudinal and
transverse resolutions.
Figure
5.14 depicts the elements of the scattering
geometry that determine the resolution.
A variation in
$\theta =\frac{1}{2}(2\theta )$, the scattering angle between
${k}_{f}$ and
${k}_{i}$, causes
$q$ to trace out the
major axis of the resolution ellipse. A variation in the magnitudes
${k}_{i}$ and
${k}_{f}$, due to a variation in the
${\theta}_{\mathrm{beam}}$ striking the
monochromator, causes
$q$ to trace out the
minor axis of the resolution ellipse.
As shown in the figure, the
longitudinal
$q$ and transverse
${q}_{\perp}$
directions are not exactly coincident with the major and minor axes of the
resolution ellipse, but are rotated by
$\theta $ with respect to it.
A careful consideration of the resolution function is required for extremely
highresolution experiments
[
45].
For this relatively lowresolution experiment, the relative
rotation is neglected.
Figure 14: Cartoon of resolution function. (Created by Joel Brock.)
As already used, "perpendicular" (
${q}_{\perp}$) and "parallel"
(
${q}_{\parallel}$) refer to vectors in the
${a}^{*}$,
${b}^{*}$ plane (constant
$L$). The term "longitudinal" refers a
direction along the scattering vector
$q$, while "transverse" is
the direction orthogonal to this, but still in the scattering plane.
At this point in reciprocal space, the
${q}_{\perp}$ direction
corresponds to the transverse direction.
The
${q}_{\parallel}$ direction corresponds closely to the longitudinal
direction, but is slightly different because of the constraint that
$L$ remain
constant in
${q}_{\parallel}$.
The longitudinal resolution is found by
differentiating the definition of
$q$ (
3.30),
$\delta q=\frac{4\pi}{\lambda}\mathrm{cos}\theta \hspace{0.5em}\delta \theta \mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(19)$

In order to maximize the empirical signal to noise ratio, we used longitudinal
(corresponding to
$2\theta $) slits 2.0 mm wide .
This produced a resolution of
$\Delta (2\theta )=0.{20}^{\circ}$ (fullwidth at
halfmaximum).
The transverse resolution is found to be
$\delta {q}_{\perp}=q\hspace{0.5em}{\theta}_{\mathrm{beam}}$ 
$(20)$

where
$\Delta (2{\theta}_{\mathrm{beam}})=0.{012}^{\circ}$ (fullwidth at
halfmaximum) is determined by the opening angle of the
synchrotron (
3.1).
To confirm these calculations, we also measured xray intensities through the
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0\mathrm{\hspace{0.5em}\hspace{0.5em}}L)$ crystal truncation rod (CTR) at the same
$L=1.5$ and
${q}_{\perp}$,
${q}_{\parallel}$ directions as in
figures
5.12 and
5.13.
Xray scans through the CTR and the overlayer rod are shown
in figures
5.15
and
5.16. The top panel of each figure
illustrates the intensity data, normalized to the beam monitor.
The scattering from the overlayer is barely observable in comparison with the
CTR scattering. This is not surprising, because the
CTR intensity falls as
$~1/({q}_{z}{q}_{0}{)}^{2}$, where
${q}_{0}$ is
the Bragg peak position, and
$L=1$ for this rod.
The bottom panels illustrate the same data, but normalized to unity so that the
widths may be compared. The overlayer rod is broader than the CTR in each
figure.
Figure 15:
Comparison of the
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ crystal truncation rod (filled) and the
$(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ overlayer Bragg scattering rod (hollow) at
$L=1.5$.
Both are measured in the
${q}_{\perp}$ direction and at
350 mV.
The overlayer peak is barely visible near
$\phi =317.6$.
Figure 16:
Comparison of the
$(1\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ crystal truncation rod (filled) and the
$(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0)$ overlayer Bragg scattering rod (hollow) at
$L=1.5$.
Both are measured in the
${q}_{\parallel}$ direction and at
350 mV.
The calculated resolutions and measured peak widths are summarized in
table
5.1.
Each quoted value is a fullwidth at halfmaximum.
The first and third columns are calculated longitudinal (
5.19) and transverse (
5.20)
resolutions for the CTR and overlayer rod. The second and fourth columns are
measured widths of peaks in the shown in the previous figures
for the
$\phi $,
$\Delta {q}_{\perp}$
and
$H$,
$\Delta {q}_{\parallel}$
directions.
It may seem surprising that the measured width
$\delta {q}_{H}$ for the CTR is
less than the calculated resolution
$\delta q$. However, this is just an
indication that
$H$ is not collinear with
the longitudinal direction.
Referring to figure
5.14, a cut through the
resolution ellipse along a direction other than the major axis (the
longitudinal direction) will always produce a more narrow profile.
Table 1:
Summary of measured peak widths (
$\delta {q}_{H}$,
$\delta {q}_{\phi}$)
and calculated resolutions (
$\delta q$,
$\delta {q}_{\perp}$).
All values are fullwidth at halfmaxima.
peak 
$\delta q$ (Å
${}^{1}$) 
$\delta {q}_{H}$ (Å
${}^{1}$) 
$\delta {q}_{\perp}$ (Å
${}^{1}$) 
$\delta {q}_{\phi}$ (Å
${}^{1}$)

Pt(111) CTR 
$14.7\times {10}^{3}$ 
$7.1\times {10}^{3}$ 
$0.31\times {10}^{3}$ 
$2.5\times {10}^{3}$

CuCl overlayer 
$15.0\times {10}^{3}$ 
$18\times {10}^{3}$ 
$0.25\times {10}^{3}$ 
$7.1\times {10}^{3}$

As indicated, the resolution function is extremely asymmetric, with
$\delta q>>\delta {q}_{\perp}$.
By comparing
$\delta {q}_{H}$ and
$\delta q$, the
overlayer rod is seen to be resolutionlimited when measured along
${q}_{\parallel}$.
However, the overlayer rod is
found to be very wellresolved along
${q}_{\perp}$
by comparing
$\delta {q}_{\phi}$ and
$\delta {q}_{\perp}$.
From this consideration,
widths from scans of the overlayer along
${q}_{\perp}$
(figure
5.12) can be considered intrinsic to the CuCl
overlayer itself, and a correlation length of
$\xi \approx 280$Å
can be quoted
without resort to deconvolution.
The width of a CTR is related to the terrace size, but in a complicated way.
For a selfaffine surface (where there is no intrinsic length scale parallel
to the surface), then the CTR width at the antiBragg position (midway between
two Bragg peaks) is inversely proportional to the mean terrace size. When
there is a characteristic surface length scale, the CTR width is in general a
function of that length scale as well. In this case, the relationship to
terrace size is specific to the correlation function which generates that
length scale. As expected, the overlayer rods in our experiment are always
significantly broader than the CTRs. This suggests that the terrace size is
not the primary limitation on the mean island size. Without a separate and
very careful surface crystallography experiment, however, this cannot be
definitively proven.
Chapter 6
Kinetic Measurements
1 Introduction
In this chapter, simultaneous electrochemical and xray scattering
measurements of the ordering kinetics of the CuCl bilayer during the
transition from the commensurate copper overlayer to the incommensurate
bilayer are reported. First, the timeresolved data are presented. Then, a
simple theory for the nucleation of the incommensurate phase is
described. The subsequent section presents data to support this model. Next,
the entire
$q$
$t$ data set is presented, followed by a theory to describe
it. The data is analyzed in the context of this theory, and excellent
agreement is found. The final sections concern alternate models and further
theoretical explanations.
2 TimeResolved Data
To observe the ordering kinetics during stripping, we employed a simple signal
averaging technique. An example of the squarewave potential cycle that we
applied is shown in figure
6.1a. At
$t=0$, the potential
begins at 200 mV. The voltage is stepped to 350 mV at
$t=10$ seconds. At
$t=30$ seconds, the voltage is stepped back to 200 mV. This cycle repeats
with a period of 40 seconds. Throughout this cycle, we simultaneously monitor
both the current (figure
6.1b) and the
intensity of the scattered xrays at
${q}_{\perp}$ corresponding to the peak of
figure
5.12. As expected, the incommensurate scattering peak is
present only for values of the potential within the incommensurate phase.
Note that the rise in the intensity of the scattered xrays in
figure
6.1c is much slower than the corresponding current
transient in figure
6.1b. In contrast, the scattered intensity
falls on a time scale similar to that of the current transient.
Figure 1:
(a) Applied potential steps.
(b) Current transients.
(c) Time dependence of the integrated intensity of
the
$(0.765\mathrm{\hspace{0.5em}\hspace{0.5em}}0\mathrm{\hspace{0.5em}\hspace{0.5em}}1.5)$
overlayer diffraction peak.
The current transients describe the
charge transfer at the electrode interface. These are due to two
contributing processes: the capacitive charging of the doublelayer, and the
Faradaic charge transfer due to desorption/adsorption of ions.
Some previous chronoamperometric studies of closely related systems
[
79,
80]
exhibit distinct features in the current response that have been interpreted
as evidence of nucleation. These characteristic features are not present in
our data. We suspect that the geometry of the thin solution layer xray cell
may be responsible for this difference. The capacitive effect is greater for
our larger samples. This strong signal tends to mask other early features in
the current response. Also, in our apparatus, as compared with hanging
meniscus cells used in the other experiments, diffusion is comparatively
insignificant. First, diffusion from the "bulk" solution is not a
consideration for us, because the solution layer is so thin. Second, any
diffusion that does take place will occur in one dimension, rather than three.
Above the planar electrode face, the solution layer forms a very short
cylinder. The ions are in close proximity to the surface and conditions are
probably relatively uniform across the face, so diffusion is primarily along
the surface normal.
After numerous attempts, we can definitely say that there are no features in
the current response at the same time as the xray response. So
the measured current response is ascribed to desorption/deposition into a
disordered state (accompanied by charge transfer) which then gives rise to the
nucleation and growth of the equilibrium ordered phase. The rise in xray
intensity corresponds to an increased population in the incommensurate ordered
phase. The desorption process can be separated from the ordering process due
to the widely disparate time scales involved. Based on these data, we
hypothesize a scenario wherein the abrupt positive voltage step causes a
expulsion of some of the adsorbed copper ions. The remaining disordered ions
gradually reorganize into a twodimensional crystalline state with a larger
lattice constant, incommensurate with the platinum substrate.
3 Stochastically Nucleated Islands
Figure 2: Cartoon of a nucleation process.
Consider the nucleation of (ordered) islands from a disordered phase,
as depicted in figure
6.2.
As usual, we define the Gibbs free energy of an island to be proportional to
the (electro)chemical potential and to the number of particles that comprise
the island. An island in contact with another phase will give rise to an
excess free energy term proportional to the surface area of contact. So the
existence of an island with
$N$ particles will lead to an excess of free
energy
$\Delta G(N)=N(\stackrel{~}{\mu}\_\text{ordered}\stackrel{~}{\mu}\_\text{disordered})+\gamma S$ 
$(1)$

where
$\mu \_\text{ordered}$ and
$\mu \_\text{disordered}$ refer to
the electrochemical potentials of the respective phases. The surface free
energy generated by the boundary between phases is taken as the product of
a constant coefficient
$\gamma $ and the surface area.
Defining the overpotential as
$\eta =\Delta V\Delta {V}_{0}\text{'}$ 
$(2)$

then the Gibbs free energy is
$\Delta G(N)=Nze\eta +\gamma S\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(3)$

This linear dependence of
$G(N)$ upon
$\eta $ can arise
from least three different scenarios. Analogously to
the example described by Schmickler [
117], the desorption
may
be accompanied by a shift in the electric potential. For instance, if
the ordering process involves the desorption of ions, then those ions would
lose contact with the electrode surface and no longer be at the applied
potential
$\Delta V$.
Another possibility involves a change in the ionization state such that the
ionic charge changes by
$ze$.
The lack of any observed
charge transfer across the electrode interface at times
corresponding to the ordering process (see
section
6.3)
argues against these
scenarios.
In the third scenario, the ordering process is
driven by the change in adsorbate density after deposition. This is discussed
in detail in section
6.11.
Figure 3: Example of Gibbs free energy
$\Delta G$ as a function of particle
number
$N$.
For an arbitrary (perhaps fractal) island of dimensionality
$d$, then
$S=a{N}^{(d1)/d}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(4)$

The loss in
$\Delta G$ from the
$\eta $dependent first term
competes with the gain in
$\Delta G$ from the surface free energy. The loss
scales with the interior volume of the island (as
$N$), and dominates for
large
$N$. The gain scales more weakly as the surface area and dominates for
small
$N$. The consequence of this are functional forms for
$\Delta G$
exemplified in figure
6.3. There will be a critical
${N}_{\mathrm{crit}}$ such that islands with
$N<{N}_{\mathrm{crit}}$ will shrink to
vanishing and islands with
$N>{N}_{\mathrm{crit}}$ will grow arbitrarily
large.
This critical island number can be determined by setting
$\partial G/\partial N=0$,
$N}_{\mathrm{crit}}={\left(\frac{d}{d1}\frac{ze\eta}{\gamma a}\right)}^{d$ 
$(5)$

and the energy barrier that must be overcome to nucleate an island of this
size is
$\Delta {G}_{\mathrm{crit}}=\Delta G({N}_{\mathrm{crit}})=\frac{(d1{)}^{d1}}{{d}^{d}}\frac{(\gamma a{)}^{d}}{(ze\eta {)}^{d1}}$ 
$(6)$

For later discussion, the most important result of this derivation is that
$\Delta {G}_{\mathrm{crit}}\propto {\eta}^{1d}$, which is just the
$d$dimensional
generalization of an expression found in [
117].
Assume that there is some stochastic attempt frequency
${k}_{\mathrm{att}}$ to
make islands, only some of which are able to surmount the energy barrier.
Also assume a Boltzmann distribution of energies in the attempt profile. Then
the rate of islands successfully nucleated is
${k}_{N}={k}_{\mathrm{att}}\mathrm{exp}(\Delta {G}_{\mathrm{crit}}/kT)$ 
$(7)$

and a characteristic time is
$\tau \propto \frac{1}{{k}_{N}}=\frac{1}{{k}_{\mathrm{att}}}\mathrm{exp}\left(\frac{\Delta {G}_{\mathrm{crit}}}{kT}\right)~\mathrm{exp}({\eta}^{1d})$ 
$(8)$

4 Instantaneous vs. Progressive Nucleation
Let
$N(t)$ be the number of nuclei at time
$t$.
Following Schmickler, we assume "firstorder kinetics" as follows:
$N(t)={N}_{\infty}[1\mathrm{exp}({k}_{N}t)]$ 
$(9)$

This defines
${k}_{N}$, the nucleation rate. There is no strong reason to believe
that the nucleation rate follows some sort of firstorder restoring force as
$\mathrm{dN}/\mathrm{dt}=k({N}_{\infty}N)$, upon
which (
6.9) depends.
But this form allows an interpolation between two
limiting cases, often described in the literature. "Instantaneous"
nucleation refers to situations where the all possible nuclei have formed
before the time of observation of the system. This corresponds to
${k}_{N}t>>1$ in equation (
6.9) and yields
$N(t)={N}_{\infty}$. "Progressive
nucleation" refers to the opposite limiting case, where the nucleation
process is at its early phase throughout the time of observation. This
corresponds to
${k}_{N}t<<1$ in equation (
6.9) and yields a
linearized
$N(t)={N}_{\infty}{k}_{N}t$.
Of course, these are only limiting cases. Although many experiments in the
literature attempt to distinguish between instantaneous and progressive
nucleation, we expect in general to find systems that exhibit both types of
behavior, depending upon the rate of nucleation
${k}_{N}$ (which may be
controllable) and the time scale of
measurement.
In figure
6.4, these two cases are compared pictorially.
For each column, the time axis runs downwards. In the instantaneous case
(left side), all of the nucleation occurs before the first slide. As time
advances, each island grows larger, but no new ones are nucleated.
In the progressive case (right side), some islands have already nucleated
before the first slide. However, islands continue to be nucleated even as
their older siblings grow larger.
Figure 4: Cartoon contrasting progressive and instantaneous nucleation.
5 Characteristic Nucleation Time
Thus, we expect
that as we quench deeper and deeper into the incommensurate phase,
the transition will occur ever more rapidly.
To test this hypothesis,
we performed a series of voltage step measurements
in which the applied potential was stepped from 200 mV
(within the commensurate phase)
to varying potentials within the incommensurate phase.
As in the previous measurement,
we measured the scattered
intensity at the incommensurate overlayer peak position
as a function of time.
This corresponds to a series of experiments similar to
the one shown in figure
6.1, but varying the value of
the more positive voltage.
This is shown in figure
6.5.
Figure 5: Diagram of voltagequench experiments.
We characterized the resulting transition time by fitting the xray intensity
profiles (which resemble figure
6.1c) to a trapezoidal
functional form, as shown in figure
6.6. While this model
describes the data quite well, we ascribe no profound significance to it.
Rather, we use it simply to define a characteristic time,
$\tau $, which should
be inversely proportional to
${k}_{N}$. The inset to figure
6.7
plots the resulting
$\tau $ values. The characteristic time scale describing
the ordering of the bilayer ranges varies from 50 seconds for shallow quenches
to 0.7 seconds for deep ones. Figure
6.7 illustrates the
exponential dependence of
$\tau $ on
$1/\eta $. This fits Eq. (
6.6)
with
$d=2$, over the entire phase region. The linear slope demonstrates that
the growing islands are intrinsically twodimensional (rather than
threedimensional mounds or pits on the surface) and that these islands are
compact rather than fractal. The broad range of
$\tau $ also implies that our
twodimensional cell geometry has not inhibited the nucleation processes, but
is only limited by the accessible range of voltage values. Furthermore, in
all cases,
$\tau $ is longer than the current transient indicating that
capacitive charging effects are not dominating our results.
Figure 6: Typical trapezoidal fit
Figure 7: Characteristic rise time
$\tau $ vs. applied voltage.
Solid points represent xray transition times, while hollow points
represent the time scale for the desorption current transient to fall to
5% of its peak value. The straight line is a fit to the nucleation
model (
6.8) with
$d=2$. Inset:
$\tau $ on linear scale.
6
$q$
$t$ Data
Now we turn our attention to the development of order in the incommensurate
structure formed after desorption. In order to understand the kinetics of
this ordering process, we need to access the full
$q$
$t$ dependent
xray scattering. We repeat the timeresolved measurement of
figure
6.1 for a series of
$q$points linearly
spaced along the same
${q}_{\perp}$ direction as shown in
figure
5.12. An example of such a measurement is shown in
figure
6.8. The first thing to note is that the peak
remains centered at a constant value of
$q$ , ruling out the
possibility that the overlayer simply shifts its periodicity in response to
the change in potential.
Figure 8:
Scattered intensity as a function of time
$t$ and transverse
scattering vector,
${q}_{\perp}$, with a false
grayscale indicating intensity. Time bins have been merged for clarity.
From the discussion in section
5.5, the correlation
length is obtained from the width of the
diffraction peak and is believed to be determined by the finite size of
growing islands of CuCl. The diffraction peak narrows with time, indicating
that these islands are growing. The total coverage is proportional to the
integrated intensity.
Ideally, we would collect a
$q$
$t$ data set for each
$\phi $ voltage
transition. However, the single data set shown in
figure
6.8 consumed 29 hours of synchrotron beam time.
With our constraints, it was not feasible to consider collecting many data
sets during that beam time allocation.
Because xray intensities obey Poisson counting statistics, for a
measured signal intensity
$N$, the standard
deviation is
$\sqrt{N}$. As seen in figure
6.8, the
maximum signal in 1300 counts/second, while the
background is 1000 counts/second.
Because of this high background, we must count for long periods of time
to resolve the signal.
7 Growth of TwoDimensional Islands
Up to now, we have discussed the number of islands, but not
the total volume comprising this phase. As shown in
section
6.5, the islands are intrinsically
twodimensional. This is not intuitively pleasing, since the bilayer
itself is twodimensional.
Anticipating this result, in this section we will limit ourselves to the
case of twodimensional islands.
Assume that each island grows by the incorporation of atoms into its
boundary, and that this is the ratelimiting step for island growth. Then,
for an island of
$N$ atoms and radius
$r$,
$\mathrm{dN}/\mathrm{dt}={k}_{g}2\pi r$. This
defines
${k}_{g}$, the rate constant for individual island growth, which has
units of [length
$\times $ time]
${}^{1}$. The area of the island is simply
$A=N/\rho $. Since (for a circular island) we have
$A=\pi {r}^{2}$, we have
two expressions for
$\mathrm{dA}/\mathrm{dt}$:
$\frac{\mathrm{dA}}{\mathrm{dt}}=2\pi ({k}_{g}/\rho )r=2\pi r\frac{\mathrm{dr}}{\mathrm{dt}}$ 
$(10)$

which implies that
$r(t)=({k}_{g}/\rho )t$ and the area is
$A(t)=\pi ({k}_{g}/\rho {)}^{2}{t}^{2}$ 
$(11)$

Note that we are assuming that
${N}_{\mathrm{crit}}$ is much smaller than the mean island size. Otherwise
$A$ should
have a nonzero value at
$t=0$.
8 Avrami Theorem
Following Avrami
[
14,
15,
16]
consider a brief example of an area
$A$ with
$N$ circular islands, each of
area
$a$. The extended coverage is
${\theta}_{\mathrm{ext}}=Na/A$. Of
course, if the circles are placed randomly, then they will overlap somewhat
and the true coverage
$\theta $ will be less than the extended coverage
${\theta}_{\mathrm{ext}}$. While the true coverage is bounded by the limits
$\theta =0$ (no
coverage) and
$\theta =1$ (complete coverage), the extended coverage can
be infinite.
The probability that a particular point on the surface is
not covered by
a particular circle
$(1a/A)$. So the probability that it is not covered
by any of
$N$ circles is
$(1a/A{)}^{N}=(1(Na/A)/N{)}^{N}$.
Assume that
$a<<A$. Then in the limit
$N\to \infty $ and using
$\underset{N\to \infty}{lim}(1x/N{)}^{N}={e}^{x}$, this probability becomes
$\mathrm{exp}(\mathrm{Na}/A)=\mathrm{exp}(\theta )$.
Finally, the probability that a point is covered (which is just the
"coverage"
$\theta $) is
$\theta =1\mathrm{exp}({\theta}_{\mathrm{ext}})$ 
$(12)$

Using this assumption, we can find the total surface coverage from the
"extended" coverage (that is, the coverage if there were no overlap of
islands.) As
${\theta}_{\mathrm{ext}}\to \infty $,
$\theta \to 1$.
9 Extended Coverage
In this section, I combine results from
sections
6.7 and
6.8 to
derive some simple expressions for the extended coverage.
In general,
${\theta}_{\mathrm{ext}}=\sum _{i}^{N(t)}A(tt{\text{'}}_{i})$ 
$(13)$

where
$i$ is an index running over all the nuclei, and each island
$i$ is
nucleated and starts growing at
$t{\text{'}}_{i}$.
We can also change the sum from
$i$ to
$t\text{'}$ and write this as
${\theta}_{\mathrm{ext}}={\int}_{0}^{t}\mathrm{dt}\text{'}X(t\text{'})A(tt\text{'})$ 
$(14)$

where the multiplicity
$X(t\text{'})$ is simply
$\mathrm{dN}(t\text{'})/\mathrm{dt}\text{'}$. For our special
cases, this is
$2{N}_{\infty}\delta (t\text{'})$ (instantaneous) and
${N}_{\infty}{k}_{N}$
(progressive).
${}^{2}$
In the instantaneous limit, we have
${N}_{\infty}$ nuclei that have all nucleated
at
$t=0$, so the extended coverage is
$\theta}_{\mathrm{ext}}^{\mathrm{instantaneous}}=\pi {N}_{\infty}{\left(\frac{{k}_{g}}{\rho}\right)}^{2}{t}^{2$ 
$(15)$

In the progressive limit, we have a constant
$\mathrm{dN}(t\text{'})/\mathrm{dt}\text{'}$ that we can
substitute into (
6.14) to obtain
${\theta}_{\mathrm{ext}}^{\mathrm{progressive}}=\frac{\pi}{3}{N}_{\infty}{k}_{N}{\left(\frac{{k}_{g}}{\rho}\right)}^{2}{t}^{3}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(16)$

From the previous form of
$N(t)$ (
6.9)
we can compute the exact extended coverage
${\theta}_{\mathrm{ext}}=\pi {N}_{\infty}{k}_{N}({k}_{g}/\rho {)}^{2}{\int}_{0}^{t}\mathrm{dt}\text{'}{e}^{{k}_{N}t\text{'}}(tt\text{'}{)}^{2}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(17)$

Solving the integral, this becomes
${\theta}_{\mathrm{ext}}=\pi {N}_{\infty}{\left(\frac{{k}_{g}}{{k}_{N}\rho}\right)}^{2}[{k}_{N}^{2}{t}^{2}2{k}_{N}t+22{e}^{{k}_{N}t}]\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(18)$

In the two limits, this correctly reduces to the progressive and
instantaneous cases.
From the coverage alone, it is not possible to determine all of these
parameters individually. At best, in the intermediate cases, we can find the
variables
${k}_{N}$ and the ratio
$\alpha \equiv {N}_{\infty}({k}_{g}/\rho {)}^{2}$. In
the progressive case, we cannot even determine these two variables
independently, but only their product.
It is important to note that
${\theta}_{\mathrm{ext}}^{\mathrm{instantaneous}}$
is insensitive to
${k}_{N}$, and so fits to
(
6.18) will also be insensitive to
${k}_{N}$
when that limit is approached.
This is reasonable, as all the nuclei have already
formed before the time scale of observation.
10 Analysis of
$q$
$t$ Data
To begin the analysis of the data in figure
6.8,
we fit each time slice to a Lorentzian line shape.
A Lorentzian is the lowest order approximation to the structure factor
for any system with only short range positional order.
Some representative time slices and fits are shown by the thin lines in
figure
6.9.
From these fits, we extract the
halfwidth at halfmaximum (
$\Delta $)
and integrated intensity vs. time.
These are shown as circles in figure
6.10b.
As expected for growing islands,
$\Delta $ decreases with time.
At the same time,
the integrated intensity (proportional to the coverage)
grows monotonically.
Figure 9:
Fits of data in figure
6.8 at representative times
$t=$ 12, 14.24, 16.48, and 18.72 seconds.
The thin lines
are from fits to the individual slices,
while the thick lines
are from the fit to (
6.23).
Figure 10:
(a) Contours of constant intensity of the data set (thin lines) in
figure
6.8 and from the best fit to (
6.23)
(thick lines). (b) Best fit results for the halfwidth at halfmaximum
(
$\Delta $; descending) and the integrated intensity (
${I}_{0}$; ascending)
vs. time. Circles are for separate Lorentzian fits to each
time slice and the
solid lines are from a fit to (
6.23).
We can continue our analysis by incorporating the simple nucleation
and growth model considered previously.
Instead of fitting each timeslice independently, we now
want to fit the entire data set from figure
6.8
with a simple function of
few parameters.
This intensity function
$I(q,t)$ we choose should satisfy the
following conditions.
First,
$I(q,t)$ has a Lorentzian line shape
at any fixed time
$t$.
This Lorentzian function is written
$L({I}_{0}(t),\Delta (t),{q}_{0};q)=\frac{1}{\pi \Delta (t)}\frac{{I}_{0}(t)}{1+\frac{(q{q}_{0}{)}^{2}}{\Delta (t{)}^{2}}}$ 
$(19)$

where
${I}_{0}$ is the integrated intensity,
$\Delta $ is the halfwidth at half
maximum, and
${q}_{0}$ is the peak position in
$q$.
Second,
the integrated intensities
${I}_{0}(t)$ are
proportional to the coverage
$\theta (t)$.
This coverage follows the Avrami form (
6.12)
$\theta =1\mathrm{exp}({\theta}_{\mathrm{ext}})$ 
$(20)$

where
${\theta}_{\mathrm{ext}}(t)=\pi {N}_{\infty}{\left(\frac{{k}_{g}}{{k}_{N}\rho}\right)}^{2}[{k}_{N}^{2}{t}^{2}2{k}_{N}t+22{e}^{{k}_{N}t}]$ 
$(21)$

was given by (
6.18).
Third,
the halfwidth at halfmaxima
$\Delta (t)$ depend on the mean island
size.
The individual length scale of each
island
$i$
should be a function only of the island growth rate
${k}_{g}$ and elapsed
time for growth
$tt{\text{'}}_{i}$. However, the average length scale will in general be a complicated
function of several
parameters. Defining the average island size as
$\u27e8A\u27e9=\theta /N$, we expect that the typical correlation length
${\xi}_{\mathrm{typ}}$ will be proportional to
$\u27e8A{\u27e9}^{1/2}$ and
inversely proportional to
$\Delta $. So we constrain
$\Delta (t)$ to be
$\Delta (t)=\sqrt{{C}_{N}N(t)/\theta (t)}$ 
$(22)$

with the proportionality constant
${C}_{N}$.
Combining (
6.19), (
6.20),
(
6.21),
and (
6.22), the twodimensional model function is
$I(q,t)={C}_{I}L(\theta (t),\Delta (t),{q}_{0};q,t)+{b}_{0}+{b}_{1}(q{q}_{0})$ 
$(23)$

where
${C}_{I}$ is just a proportionality constant relating coverage and xray
intensity, and
${b}_{0}$ and
${b}_{1}$ parameterize the linear background.
We can now refit the entire data set (
$2\times {10}^{4}$ points) shown in
figure
6.8 to the single function (
6.23).
The best fit to this simple model produces
${\chi}^{2}=1.04$.
The intensities from the model and the data are compared in
figure
6.11, and appear to agree.
However, it is easier to compare the contours
of constant intensity
that are shown in figure
6.10a.
The contours of constant intensity for the model and the data agree very
well.
The generated
integrated intensity
${I}_{0}(t)$
and
$\Delta (t)$ functions are plotted as solid lines in
figure
6.10b.
They agree with our previous results
(plotted as circles) where
each time slice was fit independently.
Returning to figure
6.9,
we can also compare the intensities
from the twodimensional model (thick lines) with our previous results (thin
lines) and the data itself (circles).
Both of the lines fit the data quite well; the minor discrepancies between
them are due to a difference in the form of the background function.
In sum, these
kinetic data are welldescribed as the nucleation and growth of a
twodimensional film.
Figure 11: Comparison of (top) intensity generated from a fit
to (
6.23)
with (bottom) measured xray intensity; time bins have been merged for
clarity.
All of the fit parameters are shown in
table
6.1; the physically interesting
ones are summarized in the top portion.
The initial time
${t}_{0}$, against which
times
$t$ are measured, was allowed to
float above 10 seconds (when the voltage was stepped), but fit to
10 seconds. There seems to be no
time delay before the nucleation process begins.
The growth parameter
${N}_{\infty}({k}_{g}/\rho {)}^{2}$ is a product of various
parameters from (
6.21)
that cannot be separated. The saturation island number
${N}_{\infty}$ is
likewise coupled with a proportionality factor that cannot be isolated.
From the uncertainties
shown in the table,
all of the fit parameters are welldetermined except for
${k}_{N}$, to which the
fit is relatively
insensitive.
This is an indication that either the data at early times (when there is very
low signal) is insufficient to fix this parameter, or that the observations
are in
the instantaneous limit (
6.15),
where
$\theta $ does not depend on
${k}_{N}$.
Table 1: Parameters obtained from fits to figure
6.8.
The physically interesting parameters are shown in the top portion, and the
remainder are shown in the bottom portion.
parameter  variable  fit value  units 
initial time 
${t}_{0}$ 
$10.0\pm 0.32$  s

growth parameter 
${N}_{\infty}({k}_{g}/\rho {)}^{2}$ 
$0.0404\pm 0.0069$
 Å
${}^{2}$ s
${}^{2}$

saturation island number

${C}_{N}{N}_{\infty}$ 
$1.644\times {10}^{5}\pm 3.2\times {10}^{7}$
 Å
${}^{2}$ 
nucleation rate constant 
${k}_{N}$ 
$1.33\pm 1.33$  s
${}^{1}$

peak intensity coefficient 
${C}_{I}$ 
$4.367\pm 0.035$  arbitrary 
peak position 
${q}_{\perp}$ 
$0.00117\pm 3\times {10}^{5}$
 Å
${}^{1}$ 
background constant 
${b}_{0}$ 
$962.1\pm 0.4$  counts s
${}^{1}$ 
background slope 
${b}_{1}$ 
$443\pm 14$  Å counts s
${}^{1}$ 

We have attempted to test the assumption of firstorder nucleation
kinetics.
Rearranging (
6.9), we have
${k}_{N}t=\mathrm{ln}\left(\frac{{N}_{\infty}}{{N}_{\infty}N(t)}\right)$ 
$(24)$

and noting from (
6.22)
that
$N\propto {\Delta}^{2}\theta $, we obtain
${k}_{N}t=\mathrm{ln}\left(\frac{{\Delta}^{2}(\infty )\theta (\infty )}{{\Delta}^{2}(\infty )\theta (\infty ){\Delta}^{2}(t)\theta (t)}\right)$ 
$(25)$

where
$\Delta (t=\infty )$ and
$\theta (t=\infty )$
are the saturation values.
If the firstorder assumption given by (
6.9) is correct, then
a plot of the righthand side of (
6.25) vs.
$t$ should be linear
and provide a measure of
${k}_{N}$. This plot is shown in
figure
6.12.
Unfortunately, the noise in our data makes such a determination inconclusive.
Better data may be able to prove or disprove this
hypothesis.
Figure 12: Plot of the expression on the righthand side of (
6.25)
vs.
$t$. The noise in the
$y$values are primarily due to the
uncertainty in
$\Delta (t)$, and make a test of (
6.25)
impossible.
11 DensityDriven Nucleation and Growth Kinetics
Previously, we posited a voltagedependent Gibbs free energy (
6.6).
For the case of twodimensional circular clusters,
$\Delta {G}_{\mathrm{crit}}=\frac{(\gamma a{)}^{2}}{4ze\eta}=\frac{{\gamma}^{2}\pi}{\rho ze\eta}$ 
$(26)$

where we have solved (
6.4) for the geometrical
constant
$a=2\sqrt{\pi /\rho}$.
One way to obtain this form is to assume (as Schmickler
[
117] does) the deposition of metal ions from
solution
${M\_\text{sol}}^{+}$ directly into a crystalline phase on the surface
$M\_\text{cry}$, as shown by the right side of
figure
6.13.
In this section, an alternative route to obtaining (
6.26)
is demonstrated. (This treatment was initially developed by
Joel Brock [
48].)
Consider a model in which adsorption (described by a Langmuir isotherm)
is driven by the change in potential. As the potential is varied the coverage
varies, and the increasing coverage then drives a conventional phase
transition. This is shown by the left side of
figure
6.13.
This densitydriven phase transition may occur well before the adsorption
transition is completed, in which case the system may still be in the linear
region of the Langmuir isotherm (
2.59).
Figure 13:
Schematic of two possible deposition processes. The solid lines represent
mass flow (particle transfer) and dotted lines represent
current flow (charge transfer).
Upon the formation of a circular cluster with radius
$r$,
the Gibbs free energy of the system changes by
$\Delta G$.
$\Delta G$ has contributions from the difference of
chemical potentials of the lattice phase
${\mu}_{\mathrm{lat}}$
and disordered adsorbed phase
${\mu}_{\mathrm{ad}}$ and from the surface energy.
$\Delta G=\pi {r}^{2}\rho ({\mu}_{\mathrm{lat}}{\mu}_{\mathrm{ad}})+2\pi \gamma r$ 
$(27)$

where
$\rho $ is the atom/area density in the lattice phase (section
6.7), and
$\gamma $ is the energy per unit length of the interface
(section
6.3).
Setting
$\partial \Delta G/\partial r=0$ we obtain the
critical radius
${r}_{c}$,
${r}_{c}=\frac{\gamma}{\rho ({\mu}_{\mathrm{ad}}{\mu}_{\mathrm{lat}})}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(28)$

For simplicity,
treat the adsorbed phase as an ideal gas.
Define
${P}_{r}$ to be the pressure at which a circular nucleus
of radius
$r$ is in equilibrium with the gas phase.
At
${P}_{\infty}$,
${\mu}_{\mathrm{ad}}={\mu}_{\mathrm{lat}}$.
At
${P}_{r}$,
${\mu}_{\mathrm{ad}}{\mu}_{\mathrm{lat}}=\frac{\gamma}{\rho r}$.
From thermodynamics we have
$\mathrm{dg}=s\hspace{0.5em}\mathrm{dT}+\frac{1}{\rho}\hspace{0.5em}\mathrm{dP}$ 
$(29)$

where
$g$ and
$s$ are the Gibbs energy and entropy per particle.
In equilibrium,
$(\mathrm{dg}{)}_{\mathrm{ad}}=(\mathrm{dg}{)}_{\mathrm{lat}}$ so
at constant temperature we obtain,
$(1/{\rho}_{\mathrm{ad}}1/\rho )\mathrm{dP}=\frac{\gamma}{\rho {r}^{2}}\hspace{0.5em}\mathrm{dr}$ 
$(30)$

Now assume that
${\rho}_{\mathrm{ad}}<<\rho $ and use the ideal gas law
$P={\rho}_{\mathrm{ad}}kT$.
$kT\frac{\mathrm{dP}}{P}=\frac{\gamma}{\rho {r}^{2}}\hspace{0.5em}\mathrm{dr}$ 
$(31)$

Integrating from
$\infty $ to
${r}_{c}$ (
${P}_{\infty}$ to
${P}_{{r}_{c}}$),
$kT\mathrm{ln}\left(\frac{{P}_{{r}_{c}}}{{P}_{\infty}}\right)=\frac{\gamma}{\rho r}$ 
$(32)$

and
using this value of
${r}_{c}$ in (
6.27),
$\Delta {G}_{c}=\frac{\pi {\gamma}^{2}}{kT\rho \mathrm{ln}\hspace{0.5em}({P}_{{r}_{c}}/{P}_{\infty})}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(33)$

Using the ideal gas relation again,
$P=\Gamma kT$, where
$\Gamma $ is the surface excess concentration
(section
2.8)
of the adsorbed phase. Then,
$\frac{{P}_{r}}{{P}_{\infty}}=\frac{{\Gamma}_{r}}{{\Gamma}_{\infty}}=\mathrm{exp}(ze\eta /kT)$ 
$(34)$

where the latter equality assumes a linearized
isotherm (
2.59).
Substituting into (
6.33),
$\Delta {G}_{c}=\frac{\pi {\gamma}^{2}}{\rho ze\eta}$ 
$(35)$

which duplicates (
6.26) exactly.
Therefore,
in this limit of low coverages, the energy barrier for a densitydriven phase
transition and the potentialdriven phase transitions are identical.
The primary difference is that the current transfer precedes
the nucleation and growth in a densitydriven transition, while the two
processes occur in tandem in a potentialdriven phase transition.
The wide separation in time scales between the current transient and the onset
of ordering (section
6.2 and
figure
6.1) indicate a densitydriven transition.
12 Step Chronoamperometry of an Ideal System
In section
2.5.2, we considered the electrode to be a
perfect sink for ions: any ions that arrive at
$z=0$ are deposited onto the
electrode irreversibly. This is the meaning of the boundary
condition (
2.45)
$c(z={0}^{+},t>0)=0$ 
$(36)$

which is reasonable for bulk diffusion where the deposition kinetics are very
fast, and so the ratelimiting step is the diffusion of ions to the electrode
surface.
However, this is not a reasonable approximation for UPD. Firstly, the
coverage
$\theta $ will not exceed unity. Secondly, the kinetics of deposition
can not be neglected. As in section
5.2.1, a
simple model is one with a Langmuir isotherm. We will make a further
assumption, that we can linearize the Langmuir isotherm, as
in (
2.59),
$\theta \approx c\hspace{0.5em}\mathrm{exp}(\frac{\Delta {G}_{0}}{\mathrm{kT}})$ 
$(37)$

where we are now keeping only the linear term. This will be adequate for
sufficiently low coverages, and has the advantage of simplicity.
In the particular Cu/Cl/Pt(111) UPD system studied in this dissertation, a
phase transition occurs once the coverage reaches a certain point. So the
behavior will deviate from the Langmuir result in that case, anyway.
The treatment follows the derivation of the Cottrell equation
(section
2.5.2). We arrive at the same result
as (
2.48),
$c(z,s)=A(s)\mathrm{exp}[(s/D{)}^{1/2}z]+B(s)\mathrm{exp}[(s/D{)}^{1/2}z]+{c}_{\infty}/s$ 
$(38)$

where
$A(s)=0$ because
$c(z\to \infty ,t)={c}_{\infty}$.
The boundary condition at
$z=0$ is not
$c(z={0}^{+},t>0)=0$ as for the Cottrell equation.
To complete the solution, we need to find the appropriate
boundary condition. (The following mathematics are borrowed from
[
48].)
The surface excess is
$\Gamma (t)={\int}_{0}^{t}\mathrm{\hspace{0.5em}\hspace{0.5em}}{\mathrm{dt}}^{\text{'}}D{\frac{\partial c(z,{t}^{\text{'}})}{\partial z}}_{z=0}$ 
$(39)$

and its Laplace transform is
$\Gamma (s)=\left(\frac{D}{s}\right){\frac{\partial c(z,s)}{\partial z}}_{z=0}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(40)$

Using (
6.37), we can relate
$\Gamma $ to the
concentration,
$\Gamma (t)=\Gamma \_\text{sat}\theta =\Gamma \_\text{sat}\hspace{0.5em}c(z=0,t)\hspace{0.5em}\mathrm{exp}(\frac{\Delta {G}_{0}}{\mathrm{kT}})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(41)$

Defining
$b(t)=\Gamma (t)/c(t)$
we can substitute (
6.38) into (
6.41)
and
Laplace transform to obtain
$\Gamma (s)=b[B(s)+\frac{{c}_{\infty}}{s}]\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(42)$

Similarly,
substituting (
6.38) into (
6.40)
we obtain
$\Gamma (s)={\left(\frac{D}{s}\right)}^{1/2}B(s)\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(43)$

Now we can equate (
6.42) and (
6.43)
and solve for
$B(s)$:
$B(s)=\frac{b\hspace{0.5em}{c}_{\infty}}{b\hspace{0.5em}s+{\left(Ds\right)}^{1/2}}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(44)$

We now substitute (
6.44) into our previous
solution (
6.38), and obtain the Laplace transform
of
$c(z,t)$.
$c(z,s)=\frac{b\hspace{0.5em}{c}_{\infty}}{b\hspace{0.5em}s+{\left(Ds\right)}^{1/2}}\mathrm{exp}[{(s/D)}^{1/2}z]+\frac{{c}_{\infty}}{s}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(45)$

We can now obtain the current density by substituting (
6.45)
into Fick's Law (
2.35):
$j(s)=neD{\frac{\partial c}{\partial z}}_{z=0}=D[\frac{b\hspace{0.5em}{c}_{\infty}}{b\hspace{0.5em}s+{\left(Ds\right)}^{1/2}}][{\left(\frac{s}{D}\right)}^{1/2}]=\frac{b\hspace{0.5em}{c}_{\infty}}{b\hspace{0.5em}(s/D{)}^{1/2}+1}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(46)$

The last step is to perform the inverse Laplace transform on (
6.46).
The following transform [
55]
$L[\frac{1}{\sqrt{\pi t}}a{e}^{{a}^{2}t}\mathrm{erfc}\left(a\sqrt{t}\right)]=\frac{1}{\sqrt{s}+a}$ 
$(47)$

completes the solution:
$j(t)=ne{c}_{\infty}\sqrt{D}[\frac{1}{\sqrt{\pi t}}\frac{\sqrt{D}}{b}\mathrm{exp}\left(\frac{\mathrm{Dt}}{{b}^{2}}\right)\mathrm{erfc}\left(\frac{\sqrt{\mathrm{Dt}}}{b}\right)]\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(48)$

In the limit that adsorption kinetics play no role (
$\Gamma \_\text{sat}\to \infty $, or
$b\to \infty $), then the second term
in (
6.48) vanishes and we recover the Cottrell current (
2.53).
Equation (
6.48) is plotted in
figure
6.14 for various values of
$b$.
Figure 14: Current density at the electrode surface
from (
6.48) with
$b=0.0001$ (dotted),
$b=0.001$
(dotdashed), and
$b=0.01$ (dashed). Note that
$b=0.01$ is nearly
indistinguishable from the Cottrell result (solid). I have chosen
$D=9\times {10}^{6}{\mathrm{cm}}^{2}/s$, which is typical of aqueous
solutions.
13 Discussion
The basic scenario is that the applied potential step drives copper off of the
surface and the CuCl grains then nucleate and grow.
Upon reversal of the potential step, the CuCl bilayer is destroyed
as the commensurate copper layer is formed.
From the exponential dependence of the time scale on voltage
(figure
6.7), it is now clear why Cu/Cl/Pt(111) cyclic
voltammetry (figure
5.2) always shows hysteresis,
even for extremely slow sweep rates.
Incorporating the nucleation and growth model into the cyclic voltammetry
formula should lead to quantitative predictions for the degree of hysteresis
as a function of sweep rate.
However, it is unlikely that cyclic voltammetry alone would provide sufficient
evidence for nucleation and growth; there are many processes that can affect
electrode kinetics and cause similar responses.
Instead, electrochemists typically study nucleation with chronoamperometry
[
79,
80].
We have carefully considered possible effects of the thin solution layer
on the ordering kinetics after a potential step.
If the thinlayer were an inhibiting factor, we would expect to
find an upper limit to the rate at which the nucleation process could take
place.
This is not apparent in our data.
As shown in figure
6.7,
the fastest observable time scale is
limited by the constraint that the voltage quench not reach into phase A,
beyond the incommensurate phase B.
Furthermore,
there is no evidence of any rollover in the voltage dependence
of the ordering time constant.
Finally, the ordering time constant is always longer than the
electronic time constant.
In standard depositionnucleation problems [
117],
the voltage dependence in the Gibbs free energy (
6.3)
is a result of the differing electric potentials in the bulk solution and at
the
surface of the electrode. In our problem, the incommensurate and commensurate
phases are both near the electrode surface. The charge transfer from the
platinum surface precedes the ordering and may be voltage dependent. In
particular, how much of the charge is shared between the Cu and Cl atoms is
unknown. It has been shown that the ionization state does not jump directly
from Cu
${}^{0}$ to Cu
${}^{2+}$
[
78,
4].
Another possibility is that the Cu or Cl ions change their positions
(especially along the
$z$ direction) as a precursor to desorption out of the
ordered state. Expanded or compressed layers, which are voltagecontrolled,
have been observed in several UPD systems.
To resolve these underlying issues, more data on this transition must be
acquired.
Timeresolved
in situ reflectivity, both specular and nonspecular,
would help to
clarify the positions and occupancies of the Cu and Cl layers throughout the
desorption process.
Xray standing waves, which have been applied to UPD
[
5,
35,
3],
are particularly sensitive to the positions of ordered layers above the
electrode surface.
It may also be helpful to take data on a related UPD process:
Cu on Au(111) in H
${}_{2}$SO
${}_{4}$ [
133]. This one of the most studied
systems and has been extremely wellcharacterized. The gold surface is much
easier to work with and does not oxidize as readily as platinum, simplifying
the sample transfer (section
4.3.7) and subsequent data
acquisition.
Scattering from the commensurate overlayer occurs at the same
$H$,
$K$ positions as the Pt(111) crystal truncation rods and will be
difficult to observe.
Thus, experimental information on the reverse reaction,
formation of the commensurate layer, will be difficult to obtain.
Finally, a comparison of the electrochemical current transients in the
hangingmeniscus cell and the xray cell is necessary. Hangingmeniscus data
show distinct nucleation bumps that are absent in the xray cell. While
capacitive charging effects (which are enhanced in the xray cell) may mask
some features, a systematic study of each would resolve the discrepancy and
clarify the voltage dependence in (
6.3).
Chapter 7
Conclusions
In conclusion, we have studied a system wherein desorption (rather than
deposition) is followed by ordering. The chargetransfer process is much
faster than the development of longrange order. The xray data are well
described by a nucleation and growth model with only a few parameters. The
potentialstep experiments demonstrate that the rate of ordering agrees well
with nucleation models over two decades in time, and is not limited by the
thinlayer geometry.
Electrochemistry is a good model system for studying growth phenomena in
general. In comparison with
in vacuo systems, it has the advantage
that
heteroepitaxial material can be removed to recover the initial substrate. So
the same deposition (or desorption) processes can be studied repeatedly, under
identical conditions, and without having to change samples. Also,
electrochemical systems can be simpler, and components are less expensive,
than for ultrahigh vacuum systems. This is more practical for traveling to
distant locations (such as synchrotrons).
Despite the "conventional wisdom", it is possible to perform good kinetic
xray measurements in a thinlayer electrochemical cell. The limiting rates
are not specifically constrained by the cell itself. Voltammograms of ideal
quality are a necessary condition to detecting the xray signal. Crystal
quality is a determining factor in both voltammetry and the bilayer xray
scattering. To this end, we developed a polishing / annealing procedure and
apparatus that have now propagated to other research groups in Clark Hall.
We have simultaneously measured
in situ xray scattering
from the adsorbed incommensurate bilayer and current transients. This allows
us to directly address the kinetics of the nonequilibrium desorption/ordering
process. Upon a positive voltage step, there is Cu desorption, and the
commensurate structure transforms into an incommensurate structure, with a
larger inplane lattice constant. During this process, we see a current
transient and the emergence of an xray scattering peak. The current
transients have two components: the capacitive charging of the doublelayer,
and the Faradaic charge transfer due to desorption/adsorption of ions. We
have not yet been able to separate the two. The xray scattering intensity
indicates the ordering of the incommensurate bilayer. The rise in integrated
intensity is proportional to the increasing number density in the ordered
phase. The narrowing of the peak corresponds to a increasing correlation
length, which implies growing phase domains.
Using a nucleation and growth model, we can fit the entire
$q$
$t$
data set
(
$2\times {10}^{4}$ points) to a function of a few variables.
Extending these arguments, we demonstrate that the ordering time scales with
voltage over the entire range, in quantitative agreement with the nucleation
and growth of twodimensional islands.
All we know about the desorption process is contained in the
current transient data.
One could better determine when the copper ions leave the surface by
performing a similar kinetic xray experiment but monitoring the specular
reflectivity instead of the intensity of the CuCl order parameter.
The information gained would be very interesting; however,
the experiment would be a major undertaking, requiring a different
cell design and many months of experimenting.
Furthermore, the results would not affect the conclusions of the
current experiment.
Rather, they might shed some light on what happens before the nucleation
and growth process begins.
Further studies could explore the relationship between
geometrydependent diffusion processes, charge transfer at the interface, and
nucleation mechanisms.
Some future experimental directions have already been detailed in
section
4.6. Additional chronoamperometric
measurements in a cell where the thickness of the solution layer could be
systematically varied would allow one to investigate whether the desorption
process is diffusionlimited in one limit and reactionratelimited in the
other. These measurements are also well beyond the scope of the current
investigation, requiring new experimental cells and, again, the results would
not affect our current conclusions.
Appendix 1
Gaussian Distributions
This appendix summarizes some relevant results on Gaussian distributions for
the reader who may not be familiar with them.
Consider the cumulant expansion [
70] for
$\u27e8{e}^{x}\u27e9$, where
$x$ is considered small.
Here, we will only expand to second order, though the extension to higher
orders is straightforward.
To derive the cumulant expansion,
begin with
$\mathrm{ln}\hspace{0.5em}(1+x)=x{x}^{2}/2+O({x}^{3})$ 
$(1)$

for small
$x$
and expand the exponential
$\u27e8{e}^{x}\u27e9=1+\u27e8x\u27e9+\u27e8{x}^{2}\u27e9/2+O({x}^{3})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(2)$

Inserting (
A.2) into (
A.1)
we obtain
$\mathrm{ln}\hspace{0.5em}\u27e8{e}^{x}\u27e9=\u27e8x\u27e9+\u27e8{x}^{2}\u27e9/2\u27e8x{\u27e9}^{2}/2+O({x}^{3})\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(3)$

Thus
$\u27e8{e}^{x}\u27e9=\mathrm{exp}\{\u27e8x\u27e9+\frac{1}{2}[\u27e8{x}^{2}\u27e9\u27e8x{\u27e9}^{2}]+O({x}^{3})\}\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(4)$

A "Gaussian random variable" is a random variable with a Gaussian
distribution. For a Gaussian random variable
$x$, all of the terms of order
${x}^{3}$ and higher are identically zero.
This follows from
considering a Gaussian distribution of the form
$p(x)=\frac{\alpha}{\sqrt{\pi}}\mathrm{exp}[{\alpha}^{2}(x{x}_{0}{)}^{2}]$ 
$(5)$

which is normalized such that
${\int}_{\infty}^{+\infty}\mathrm{dx}\hspace{0.5em}p(x)=1$.
By integrating, it is easy to show that
$\begin{array}{cccc}\multicolumn{1}{c}{\u27e8x\u27e9}& =\hfill & {\int}_{\infty}^{+\infty}\mathrm{dx}\hspace{0.5em}p(x)\hspace{0.5em}x={x}_{0}\mathrm{\hspace{0.5em}\hspace{0.5em}},\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(6)\\ \multicolumn{1}{c}{\u27e8{x}^{2}\u27e9}& =\hfill & {\int}_{\infty}^{+\infty}\mathrm{dx}\hspace{0.5em}p(x)\hspace{0.5em}{x}^{2}={x}_{0}^{2}+\frac{1}{2{\alpha}^{2}}\mathrm{\hspace{0.5em}\hspace{0.5em}}.\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(7)\end{array}$

Now, the expectation value of
${e}^{x}$ is
$\u27e8{e}^{x}\u27e9={\int}_{\infty}^{+\infty}\mathrm{dx}\hspace{0.5em}p(x)\hspace{0.5em}{e}^{x}={\int}_{\infty}^{+\infty}\mathrm{dx}\hspace{1em}\frac{\alpha}{\sqrt{\pi}}\mathrm{exp}[{\alpha}^{2}(x{x}_{0}{)}^{2}+x]\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(8)$

The easiest way to solve this is by factoring judiciously to "complete
the square" in the exponential:
$\begin{array}{cccc}\multicolumn{1}{c}{\u27e8{e}^{x}\u27e9}& =\hfill & {\int}_{\infty}^{+\infty}\mathrm{dx}\hspace{1em}\frac{\alpha}{\sqrt{\pi}}\mathrm{exp}[{\alpha}^{2}(x{x}_{0}{)}^{2}+x]\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(9)\\ \multicolumn{1}{c}{}& =\hfill & {e}^{{x}_{0}}{e}^{1/(4{\alpha}^{2})}{\int}_{\infty}^{+\infty}\mathrm{dx}\hspace{0.5em}\frac{\alpha}{\sqrt{\pi}}\mathrm{exp}[{\alpha}^{2}(x{x}_{0}{)}^{2}+(x{x}_{0})\frac{1}{4{\alpha}^{2}}]\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(10)\\ \multicolumn{1}{c}{}& =\hfill & {e}^{{x}_{0}}{e}^{1/(4{\alpha}^{2})}{\int}_{\infty}^{+\infty}\mathrm{dx}\hspace{0.5em}\frac{\alpha}{\sqrt{\pi}}\mathrm{exp}{[\alpha (x{x}_{0})\frac{1}{2\alpha}]}^{2}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(11)\\ \multicolumn{1}{c}{}& =\hfill & {e}^{\u27e8x\u27e9}{e}^{(1/2)(\u27e8{x}^{2}\u27e9\u27e8x{\u27e9}^{2})}\mathrm{\hspace{0.5em}\hspace{0.5em}}.\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(12)\end{array}$

For a Gaussian distribution, all of the terms of the cumulant
expansion beyond second order are identically zero. For this reason, the
assumption of a Gaussian distribution is equivalent to the assumption that
$x$
is sufficiently small (so that terms beyond second order can be neglected).
I mention one more useful relation, used in
section
3.10.
Assuming that the Gaussian approximation is valid and
$\u27e8A+B\u27e9=0$,
$\u27e8{e}^{A}{e}^{B}\u27e9=\u27e8{e}^{A+B}\u27e9={e}^{\frac{1}{2}\u27e8A+B{\u27e9}^{2}}={e}^{\frac{1}{2}\u27e8{A}^{2}+2AB+{B}^{2}\u27e9}$ 
$(13)$

follows directly from (
A.4).
Appendix 2
Diffusion Equation
This treatment expands upon the threedimensional treatment by McQuarrie
[
99] and extends it to arbitrary dimensionality.
McQuarrie actually suggests two ways to give the
result (
B.14); I am using the first.
The diffusion equation for
$G$ is
$\frac{\partial G(r,t)}{\partial t}=D{\nabla}^{2}G(r,t)$ 
$(1)$

and we take the initial condition
$G(r,0)=\delta (r)\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(2)$

This can be solved most easily by taking a Fourier transform
$\frac{\partial G(q,t)}{\partial t}=D{q}^{2}G(q,t)$ 
$(3)$

and the initial condition becomes
$G(q,0)=1\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(4)$

The solution of this is
$G(q,t)=\mathrm{exp}({q}^{2}Dt)\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(5)$

The Fourier transform approach is useful in another way.
We can differentiate the Fourier transform
expression (
3.21) twice to find
$\begin{array}{cccc}\multicolumn{1}{c}{\frac{{\partial}^{2}}{\partial {q}^{2}}G(q,t)}& =\hfill & \frac{{\partial}^{2}}{\partial {q}^{2}}\int dr\hspace{1em}{e}^{iq\xb7r}G(r,t)\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(6)\\ \multicolumn{1}{c}{}& =\hfill & \frac{{\partial}^{2}}{\partial {q}^{2}}\int dr\hspace{1em}{e}^{iqr\mathrm{cos}\theta}G(r,t)\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(7)\\ \multicolumn{1}{c}{}& =\hfill & \int dr\hspace{1em}{r}^{2}{\mathrm{cos}}^{2}\theta \hspace{1em}{e}^{iqr\mathrm{cos}\theta}G(r,t)\mathrm{\hspace{0.5em}\hspace{0.5em}}.\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(8)\end{array}$

Take the
$q=0$ case,
${\frac{{\partial}^{2}G(q,t)}{\partial {q}^{2}}}_{q=0}=\int dr\hspace{1em}{r}^{2}{\mathrm{cos}}^{2}\theta \hspace{1em}G(r,t)\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(9)$

The righthand side of this equation is just
$\u27e8{r}^{2}{\mathrm{cos}}^{2}\theta \u27e9$, which can be separated into radial and
angular parts
$\u27e8{r}^{2}\u27e9\u27e8{\mathrm{cos}}^{2}\theta \u27e9$.
The latter factor
$\u27e8{\mathrm{cos}}^{2}\theta \u27e9$ is just the average value of
$\u27e8{x}^{2}/{r}^{2}\u27e9$ over a
$d$dimensional spherical shell.
Since the equation of that shell is
$x}^{2}+{y}^{2}+{z}^{2}+\dots ={r}^{2$ 
$(10)$

and taking the average value,
$\u27e8\frac{{x}^{2}}{{r}^{2}}\u27e9+\u27e8\frac{{y}^{2}}{{r}^{2}}\u27e9+\u27e8\frac{{z}^{2}}{{r}^{2}}\u27e9+\dots =1$ 
$(11)$

then the average value of each component (they are all equivalent) is just
$1/d$. Then we have
$\u27e8{\mathrm{cos}}^{2}\theta \u27e9$ =
$1/d$ and
${\frac{{\partial}^{2}G(q,t)}{\partial {q}^{2}}}_{q=0}=\frac{1}{d}\int dr\hspace{1em}{r}^{2}\hspace{1em}G(r,t)=\frac{1}{d}\u27e8{r}^{2}\u27e9\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(12)$

Now, from (
B.5) we know that
${\frac{{\partial}^{2}G(q,t)}{\partial {q}^{2}}}_{q=0}=2Dt$ 
$(13)$

so
we can equate these to find the important result
$\u27e8{r}^{2}(t)\u27e9=2dDt\mathrm{\hspace{0.5em}\hspace{0.5em}}.$ 
$(14)$

where
$d$ is the spatial
dimensionality. This proves (
2.37).
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Footnotes:
${}^{1}$I will not discuss quasicrystals, fascinating systems
wherein the diffraction pattern exhibits (at least) fivefold symmetry.
This implies that the realspace structure cannot even be even remotely
described by a (threedimensional) Bravais lattice.
${}^{2}$The factor of two for the instantaneous case is
required by the definition of the
$\delta $function, which runs from
$\infty $ to
$\infty $.
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