Citation
An X-ray scattering study of ordering in block copolymers

Material Information

Title:
An X-ray scattering study of ordering in block copolymers
Creator:
Harkless, Curtis Ray, 1963-
Publication Date:
Language:
English
Physical Description:
xiii, 249 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Block copolymers ( jstor )
Conceptual lattices ( jstor )
Data lines ( jstor )
Free energy ( jstor )
Furnaces ( jstor )
Heating ( jstor )
Kinetics ( jstor )
Nucleation ( jstor )
Polymers ( jstor )
Temperature dependence ( jstor )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 241-248).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Curtis Ray Harkless.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
001659177 ( ALEPH )
AHX0904 ( NOTIS )
24529648 ( OCLC )

Downloads

This item has the following downloads:


Full Text











AN X-RAY SCATTERING STUDY OF ORDERING IN BLOCK COPOLYMERS


By

CURTIS RAY HARKLESS














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

UNIVERSITY OF FLORIDA


1990











ACKNOWLEDGMENTS


Firstly, I would like to express my gratitude to my advisor, Stephen E. Nagler.

His assistance in the preparation of this dissertation is greatly appreciated, and without

his supervision this work would not have been possible.

I would like to express my thanks to the beamline staff, Brian Stephenson and

Jean Jordan-Sweet, for their valuable technical assistance and to DARPA and DOE for

financial support in the early and later stages of this work, respectively.

Much of this work could not have been performed without the outstanding

technical support offered by Ward Ruby. His creativity and resourcefulness were second

only to his jokes of questionable taste.

I thank Paul Lyman and Liz Seiberling not only for their helpful discussions and

honest opinions, but also for late night support and indefatigable good humor that I could

always count on.

I would like to thank Robert F. Shannon, Jr., for all of his help. Writing this

dissertation was made a great deal easier by his company and the knowledge that

someone else was suffering as much as I.

The assistance given to me by Marsha A. Singh in all aspects of the performance

of these experiments cannot be overstated. Our trip to Brookhaven to take the data, upon

which this dissertation is based, was made less arduous by her companionship and

experimental expertise. In addition, her attention to detail was invaluable during the

writing of this dissertation.

I thank the Lord for supporting me through this challenging time.









Finally, I thank my family. The advice offered by my parents: Les and Nancy

(like "Get a job." ) and the stimulating questions posed by my sisters: Lisa, Kim, Jen,

and Stefanie (like "Aren't you done yet?" ) were greatly appreciated. But most of all, I

am grateful for their unfailing support in all of my endeavors.












TABLE OF CONTENTS




ACKNOWLEDGEMENTS.............................. ....................................................ii

LIST OF FIGU RES .............................................................................................. vii

ABSTRACT......................................... .............................................................. xii

CHAPTERS

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

2 PHASE TRANSITION KINETICS................................ ........... .... 9
Phase Transition Kinetics--General Concepts ...................................... 9
Nucleation Theory............................................................................ 17
Spinodal Decomposition ................................................................ 21
Domain Growth ........................................................................ 23
Transformation Curves........................................................... ..... 24
Late Stage Growth--Coarsening .................................... ............ 32
Universal Classification..................................................... .......... 34

3 X-RAY DIFFRACTION............................................... ............ 36
Fundamentals of X-ray Diffraction................................. .......... 36
Small Angle X-ray Scattering............................................................. 41
Scattering From Liquids............................................... ............ 47
Scattering From Ordered Structures .............................................. 50
The Effect of Finite Size...................... .......... ........... ... 52
The Crystal Structure Factor............................. ........... .. 53
The Effect of Particle Motion .............................................. 54
The Effect of Strain ............................................................ 55
The Powder Pattern ........................................... ........... ... 56

4 REVIEW OF BLOCK COPOLYMER LITERATURE...................... 59
Polymers--General Background.................................... ........... .... 59
The Glass and Melting Transitions................................................. 61
Phase Separation in Polymer Blends................................ ............ 62
Block Copolymers--Bulk Properties.................................................... 64
Block Copolymers--Kinetic Studies .................................................... 69







Block Copolymer Solutions.......................................................... 70
Equilibrium Measurements ................................................... 72
Time-Dependent Experiments............................................ 75

5 THE EXPERIMENTAL APPARATUS ............................................ 79
System Overview ........................................................................... 79
The Polymer X-ray Scattering Furnace............................................ 82
The Furnace Body .............................................................. 82
The Translational Stage ...................................................... 85
The Sample Mount ............................................................. 86
The Quench Technique............................................................ 89
The X-ray Scattering Apparatus ..................................................... 92
The National Synchrotron Light Source.................................. 92
The Collimation System ....................................................... 94
The pre-hutch collimation system............................... 96
The in-hutch collimation system................................. 97
The Detection System .......................................................... 98
The detector operating principle................................. 98
The dark count ......................................................... 99
Preliminary Experimental Procedures............................ .............. 100
Sample Preparation.............. .............................. ........ 100
X-ray Beam Alignment and Collimation.............. ........... 105
Normalization for Incident Intensity ................................. 108
Measurement of the Parasitic Scattering ............... ............... 109
Equilibrium Data Acquisition Procedure .................................... 112
Kinetic Data Acquisition Procedure ................................................... 113

6 EQUILIBRIUM MEASUREMENTS: ANALYSIS
AND RESULTS.................................................................... 117
Presentation of the Static Data................................ 117
Identification of the X-ray Scattering Features .............................. 130
Preliminary Discussion of the Static Data................................ 131
The Effects of Experimental Resolution........................ ........ .. 133
Determination of the Structural Parameters ................................. 134
The Microdomain Parameters................................. ... 134
Analysis of the Macrolattice Structure and Dimensions ....... 137
The background contribution............................ 139
The amorphous contribution........................ 139
The crystalline contribution..................... ........ 140
Discussion of the Macrolattice Characterization..................... 140
Finite Size and Strain Effects............................................... 147
The Ordered Volume Fraction.................. ............. 148
Discussion of the Fine Structure in the SAXS Profiles......... 149







Concentration and Temperature Dependence
of the Structure............................................. ............................... 152
Behavior of the Crystalline Component ................................. 154
Behavior of the Amorphous Component.............................. 161


7 KINETIC MEASUREMENTS: ANALYSIS AND RESULTS ........ 168
Preliminary Discussion...................................................................... 168
The Ordering Kinetics................................................................... 174
The Master Curve............................................................. 193
The Crossover Behavior ...................................................... 198
Shallow Quench Oscillatory Behavior ................................... 200

8 SUMMARY AND CONCLUSIONS................................................. 206

APPENDICES

A DATA ACQUISITION PROGRAMS ............................................... 209

B THE HEATING CIRCUIT .......................................................... 227

C THE SOLENOID INTERFACE .................................................. 229

D THE DEVELOPMENT OF THE SAMPLE MOUNT....................... 231

REFERENCES .................................................................................................. ... 241

BIOGRAPHICAL SKETCH ............................................................................... 249














LIST OF FIGURES


Figure

1-1 The diblock copolymer molecule and an illustration of
microphase separation........................................... .............................. 3

1-2 The three domain morphologies commonly observed in diblock
copolymer system s...................................................................................... 5

1-3 An illustration of the ordering of spherical microdomains onto
a cubic lattice......................................................................................................... 6

2-1 An example of a phase diagram for a binary alloy exhibiting a phase
transition....................................................................................... ............ 10

2-2 The shape of the free energy curve versus order parameter for various
temperatures, above and below the transition point......................................... 13

2-3 An illustration of the different temporal regimes in the transformation
process............................................................. ................................. 16

2-4 A contrast of the nucleation and spinodal decomposition transformation
mechanisms ................................................................................................. 22

2-5 A simulated transformation curve showing the induction time and the
half-completion time........................................................................... 26

2-6 An ideal Master Curve for a transformation which obeys Cahn's
relation for nucleation on two dimensional nucleation sites............................ 31

3-1 An illustration of diffraction by two scattering centers and the definition
of the scattering angle 20 ............................................................................. 38

3-2 The scattering form factors for ideal spheres having radii: 80, 100,
and 120 A .......................................................................................... ......... 42







3-3 A plot showing the effect of polydispersity on the scattering form factor
for a gaussian distribution of spherical particles having a width parameter,
= 1, 10, and 20 A ........................................................................................ 44

3-4 The modifications to the scattered profile which result from spheres
having imperfect boundaries................................................................. 46

3-5 Model scattering profiles are shown for systems of spheres interacting
via the hard core potential at varying particle concentrations, v4/v0 ..........................49

3-6 An illustration of Bragg scattering from a crystal lattice ................................. 51

5-1 A schematic of the data acquisition system............................................... 80

5-2 A top view of the polymer x-ray scattering furnace illustrating the
relative placement of its various features as described in the text ................. 83

5-3 The final sample mount design is illustrated in a fron and top view ............. 87

5-4 A typical quench profile illustrating the base and sample temperatures as a
function of time following the quench ........................................................... 90

5-5 A schematic of the SAXS collimation system............................................ 95

5-6 The chemical formulas of polystyrene and polybutadiene along with
the selective solvent used in these studies, n-tetradecane............................. 101

5-7 A graph of the beam height (out of scattering plane ) and width
(in plane) profiles ................................................................................. 107

5-8 A graph of the measured parasitic scattering................................................. 110

6-1 Raw scattering profiles for the SB15 sample upon cooling.......................... 118

6-2 Raw scattering profiles for the SB15 sample upon heating.......................... 119

6-3 Raw scattering profiles for the SB25 sample upon cooling.......................... 120

6-4 Raw scattering profiles for the SB25 sample upon heating.......................... 121

6-5 Raw scattering profiles for the SB35 sample upon cooling.......................... 122

6-6 Raw scattering profiles for the SB35 sample upon heating.......................... 123

6-7 Raw scattering profiles for the SB50 sample upon cooling........................ 124







6-8 Raw scattering profiles for the SB50 sample upon heating........................... 125

6-9 Raw scattering profiles for the SBS25 sample upon cooling ......................... 126

6-10 Raw scattering profiles for the SBS25 sample upon heating........................ 127

6-11 Raw scattering profiles for the SBS35 sample upon cooling ....................... 128

6-12 Raw scattering profiles for the SBS35 sample upon heating........................ 129

6-13 A plot of the scattered profile for the SB50 sample at 44.0 C
illustrating the estimated spherical form factor............................................ 135

6-14 A plot showing the three components of the parameterization of
the low Q range of the scattering spectra..................................................... 138

6-15 A comparison of the simple cubic and body-centered cubic structure
specific fits to the scattered spectrum for the SB35 sample at 44.0 C......... 144

6-16 An illustration of the splitting of the first-order Bragg reflection
in the SB15 solution at 62.5 C................................................................ 150

6-17 Plots of the microdomains radius and lattice constants versus
polymer concentration at approximately 44 C ......................................... 153

6-18 Graphs of the normalized peak intensities of the primary Bragg peak vs.
temperature for each of the four polymer solutions..................................... 155

6-19 Graphs of the measured Bragg widths vs. temperature for the four
polym er solutions ......................................................................................... 158

6-20 Graphs of the measured lattice constants vs. temperature for the four
polymer solutions ................................................................................... 160

6-21 Variation of the amorphous peak amplitude vs. temperature for the
SB15 solution ............................................................................................... 162

6-22 Plots of the amorphous peak position and width vs. temperature for the
SB50 solution ......................................................................................... 164

6-23 The measured phase diagram for the polystyrene-polybutadiene /C14
solutions ................................................................................................. 166

7-1 A plot of the final spectrum as a function of quench depth for the
SB15 solution ............................................................................................... 170







7-2 A plot of the final spectrum as a function of quench depth for the
SB25 solution ............................................................................................... 171

7-3 A plot of the final spectrum as a function of quench depth for the
SB35 solution ............................................................................................... 172

7-4 A plot of the final spectrum as a function of quench depth for the
SB50 solution ............................................................................................... 169

7-5 A plot of the raw SAXS spectra as a function of time following
a quench from 159.6 C to 112.2 C on the SB25 solution............................ 175

7-6 A plot of the raw SAXS spectra as a function of time following
a quench from 130.6 C to 68.5 C on the SB15 solution............................ 177

7-7 A plot of the raw SAXS spectra as a function of time following
a quench from 159.6 C to 94.0 C on the SB25 solution............................ 178

7-8 A plot of the raw SAXS spectra as a function of time following
a quench from 170.0 C to 86.5 C on the SB35 solution............................ 179

7-9 A plot of the raw SAXS spectra as a function of time following
a quench from 201.6 C to 98.5 C on the SB50 solution............................ 180

7-10 Graphs of the peak maximum as a function of time following a
shallow quench ( upper graph ) and a quench to below the
ordering temperature ( lower graph )........................................................... 182

7-11 A 3-D plot of the temporal peak development of the scattered profile........... 183

7-12 Graphs of the fitted amorphous peak position and FWHM along
with the Bragg FWHM and the measured lattice constant vs. time
following a quench from 159.6 C to 94.0 C on the SB25
solution............................................. .................................................... 185

7-13 A plot of the normalized Bragg and amorphous fitted amplitudes
as a function of time following a quench from 159.6 OC to
94.0 C on the SB25 solution.................................................................. 188

7-14 A plot of the normalized Bragg fitted amplitudes as a function of
time following quenches from 160 oC to 102.2 OC, 94.0 oC,
82.0 oC, and 58.7 oC on the SB35 solution ................................................. 192

7-15 A plot of the quantity: In ( In ( ( 1 C )-1 ) vs. the natural
log of the time following quenches from 160 C to 102.2 OC,







94.0 C, 82.0 C, and 58.7 C on the SB35 solution...................................... 194

7-16 A master curve of all of the quench data on all of the samples
as described in the text.................................................................................. 196

7-17 The time constants, t, for the ordering transition for the SB15,
SB25, SB35, and SB50 solutions are plotted as a function of
quench depth................................................................................................. 197

7-18 A schematic illustrating the proposed origin of the observed
crossover in ordering behavior.......................................................................... 199

7-19 Graphs of the fitted and normalized Bragg amplitudes as a function
of time following quenches from ~ 202 C to 152.0 oC, 143.5 OC,
and 137.5 OC on the SB50 solution ............................................................. 201

7-20 Graphs of the raw data from a quench from 201.8 C to 152.0 C
on the SB50 solution at various times following the quench.......................... 202

7-21 Plots of the fitted Bragg widths and positions along with the
normalized intensity, Z, as a function of time following the
quench from 201.8 OC to 152.0 C on the SB50 solution............................. 204

B-l A schematic of the heating circuit........................................................... 228

C-1 A schematic of the solenoid interface............................................................ 230

D-1 An illustration of the initial sample mount design....................................... 232












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

AN X-RAY SCATTERING STUDY OF ORDERING IN BLOCK COPOLYMERS

By

Curtis Ray Harkless

December 1990

Chairman: Stephen E. Nagler
Major Department: Physics


The block copolymer, a novel system for studying the kinetics of first-order phase

transitions, is investigated. Solutions of the block copolymer polystyrene-polybutadiene

exhibit two types of phase transitions presently of great interest to the science

community. Studies of the process by which these transformations occur can broaden

our understanding of kinetic phenomena and aid in the identification of universal features

such as nonequilibrium scaling. This thesis represents the first attempt to probe the

kinetics of these transitions using synchrotron x-ray diffraction.

The block copolymer molecule is composed of two different polymer chains

joined by a covalent bond. When the chains are incompatible mesophases form through

the process of microphase separation. The system also exhibits an ordering transition

which results in a characteristic superlattice of the microdomains. A brief discussion of

first-order phase transition kinetics is given followed by a detailed review of the relevant

literature on block copolymers.








High quality diblock and triblock copolymer solutions were prepared. The

structure of each system was determined from the x-ray scattering profiles as a function

of temperature after which kinetic measurements were performed. Each kinetic

measurement involved annealing the sample above the dissolution temperature and

rapidly quenching the sample temperature to a fixed point below. The subsequent

transformation process was observed through the x-ray scattering profile.

Due to the resolution obtained at the synchrotron, the scattering contributions

from the ordered and disordered states are identified and separated for the first time. As

a result several new features are observed such as the presence of fine structure in the

x-ray scattering profile. Fast kinetic measurements reveal that transformation occurs as a

two-stage process and that the ordering transition exhibits an unexpected crossover in

behavior consistent with two dimensional nucleation. In addition, the velocities of

transformation, as determined from the kinetic data, follow trends expected from

fundamental thermodynamic considerations. Finally, novel oscillatory growth of the

Bragg component is observed in the shallow quench limit.














CHAPTER 1
INTRODUCTION



One of the most fascinating and active fields in statistical physics is the study of

phase transition kinetics.1 Specifically, there is much interest in understanding the non-

linear and cooperative phenomena involved in first-order phase transitions (FOPTs).2

Although instances of FOPTs in nature are abundant, much of the essential physics of

these processes remains unclear. Examples of these phenomena include phase separation

in binary alloys,3 45, 6 and binary fluids, ordering in alloys,7. 8, 9.10.11,12.13 ferromagnetic

and antiferromagnetic systems,14,15 and melting/crystallization transitions.16 One of the

foremost efforts at this time is the identification of relevant parameters to form a

universal classification scheme for these systems.17 Consequently, there is a need to

study more diverse systems and polymer systems are and ideal and novel system for this

purpose.1

A polymer is a long chain molecule composed of a number of repeat units called

monomers. Generally, each monomer itself is an organic molecule joined to adjacent

molecules by a covalent bond. Common examples of polymers include polystyrene,

polyisoprene, and poly (vinyl chloride). The excellent mechanical properties of these

materials lend themselves to a variety of industrial applications. The interesting

statistical physics involved in understanding these materials has attracted the attention of

many notable physicists including de Gennes, Debye,18 Kramers,19 and Flory. Excellent

references on the static and dynamic properties of polymers have been written by Doi

and Edwards,20 Flory,21 and de Gennes.22









The block copolymer molecule is formed by joining two chemically distinct

polymer chains end to end with a covalent bond and is the subject of the experimental

studies to be presented here. A typical block copolymer is illustrated in Figure 1-1.

Much recent work has been done toward understanding these complicated systems.23

When the monomers, A and B, are incompatible ( A-A B-B, nearest neighbor

configurations are favored over A-B configurations ), the system attempts to minimize

internal energy by forming A- or B-rich domains thereby reducing the surface to volume

ratio between A and B regions. This type of arrangement leads to a loss of

conformational entropy, resulting in an equilibrium structure that is a balance between

these two opposing forces. At high temperatures, the entropic force dominates and the

system is a homogeneous melt as shown in Figure 1-1. As the temperature is lowered,
the balance shifts to formation of domains at the dissolution temperature, Td. This

domain formation is similar to the phase separation observed in binary alloys. However,

the junction of A and B monomers in the block copolymer limits the ultimate size of the

phase separated regions to microscopic dimensions. This process is appropriately termed

microphase separation and results in the different domain morphologies observed in these

materials.24

Phase separation in binary systems has been well studied and has led to the

formulation of very general relations describing the phase transition kinetics in conserved

order parameter systems.' Microphase separation is an intriguing analog of this process

involving both the complex process of polymer interdiffusion and an inherent constraint

to the size of the phase separated regions. For this reason, block copolymers are useful

systems for broadening our knowledge of kinetic phenomena and testing the limitations

of present models.

A variety of domain morphologies has been observed in block copolymer systems

through x-ray scattering23 and transmission electron microscopy2.26 and the type of

morphology was found to depend on the fractional composition of the individual block














THE BLOCK COPOLYMER MOLECULE

AAAA----AAAABBBB----BBBB






MICROPHASE SEPARATION


T > Td










T < Td


Figure 1-1 The diblock copolymer molecule and an illustration of microphase
separation.








copolymer chains. For example, polymers having a small A block relative to the B block

are known to form spherical A-rich microdomains in a B-rich matrix. Remarkably, these

systems are seen to exhibit long-range order in the form of a cubic lattice of the spherical

microdomains. The specific type of cubic structure is presently the subject of some

debate27 and, for this reason, structure determination is discussed in detail in this work.

In those copolymers where the A block is somewhat smaller than the B block, a

2-dimensional hexagonal closed packed ( HCP ) arrangement of cylindrical domains is

formed. A lamellar structure results if the A and B blocks are of comparable size.

These features are summarized in Figure 1-2.

Often in studying these materials, it is useful to employ a solvent which

selectively dissolves one block of the copolymer. In this way the interaction potential

between unlike monomers is partially screened resulting in a lower, experimentally

accessible dissolution temperature. An added benefit of using selective solvents is the

ability to control the effective fractional composition of the chain by selectively swelling

one block relative to the other. The study of these polymer solutions is also important

from an industrial standpoint in that additives are commonly incorporated in pure

polymer materials to improve their already advantageous mechanical properties.

Further interesting effects were observed in small angle x-ray scattering (SAXS)

studies of polystyrene(PS)-polybutadiene (PB) diblock copolymer in the selective solvent

n-tetradecane (C14).28 The system forms spherical PS domains within a PB/C14 matrix.

However, a temperature range was found below the dissolution point where the expected

higher order Bragg reflections were missing from the SAXS profile indicating the

absence of long-range order. At lower temperatures, evidence of higher order Bragg
reflections was found leading the authors to propose an ordering temperature, T,, below

which the spherical microdomains arrange themselves to form a cubic lattice. This

ordering process is illustrated in Figure 1-3.















---4c
CD


C-)


0
-<


m
U)


! MA


II


(I)

I-I

C)



CD
-H
II


The three domain morphologies commonly observed in diblock
copolymer systems.


0
C-<
F-
> :z
I
CO


U)
-D


m
U)


Figure 1-2


0

7
CD

0
--U
0
IF





0
-<




IF-
7>



0



I-
0
C)
r-H
o

rT

CO


























A DISORDERED ARRANGEMENT OF SPHERES


B / Cli

' 4
5 / cli


B / C14


B / 1
5 / Cli


AN ORDERED LATTICE OF SPHERES





Figure 1-3 An illustration of the ordering of spherical microdomains onto a cubic
lattice.








Microdomain ordering in block copolymers is an interesting analog to melting-

crystallization phenomena and as such is characterized by a nonconserved order

parameter (NCOP). Other systems such as ordering in "polyballs"29 have contributed

much to our understanding of these phenomena. The presence of this second type of

transformation in block copolymer solutions provides an additional opportunity to test

the applicability of present models of crystallization and behavior in NCOP systems.

An important feature of block copolymer materials is their great diversity in

industrial applications. These include their use as synthetic rubber, additives to plastics

and adhesives, and hydrophilic coatings.30 Much theoretical and experimental research

has been done in attempting to relate the mechanical properties of block copolymers to

their structural properties.31 32 For this reason, a better understanding of the structural

and thermal behavior of these materials is of direct industrial significance.

The SAXS technique is an excellent method for analyzing the structure of block

copolymers because the scattered intensity is directly related to the positional correlations

of the particles in the system. It was the goal of these studies to perform high resolution

x-ray scattering measurements of the equilibrium structure of block copolymer solutions

and to probe the microphase separation and ordering processes with fast kinetic

measurements. The successful completion of this experiment was dependent on three

major requirements. These were the use of high quality samples, a specially designed

polymer x-ray scattering furnace that allowed for rapid in-situ quenches, and a very

intense x-ray source enabling high resolution measurements. Ultimately, x-rays

produced by synchrotron radiation were utilized. The experimental methods employed

and the novel results obtained are discussed in the following chapters.

This dissertation is divided into seven chapters. Chapter 2 is a summary of recent

advances in our understanding of first order phase transitions and provides a background

for understanding the significance of these measurements. The third chapter is an in-

depth discussion of the general principles of x-ray diffraction and relates the relevant





8


physical parameters to experimentally measured quantities. Chapter 4 is a brief review

of the relevant literature on block copolymers. Chapter 5 is a detailed description of the

experimental apparatus and procedures performed in these studies. The

parameterization, analysis, and interpretation of the equilibrium results are presented in

Chapter 6 in preparation for discussion of the kinetic results in Chapter 7. Finally, the

important conclusions which are drawn from this work will be summarized in Chapter 8.













CHAPTER 2
PHASE TRANSITION KINETICS


This chapter is a brief review of the major developments in first-order phase

transition kinetics. The first section provides the definitions of many of the basic

concepts as well as an introduction to some of the general behavior. Broad comparisons

are made with higher order critical phenomena. The transformation process is divided

into three temporal regimes which are then discussed individually. These are the early,

growth, and coarsening regimes. Two distinct mechanisms are identified in the early

regime: nucleation and spinodal decomposition. During the growth regime,

transformation is characterized by growth laws and transformation curves. Finally, the

late stage of growth, coarsening, is described.


Phase Transition Kinetics General Concepts


Figure 2-1 is an example of the phase diagram for a binary alloy exhibiting a

phase transformation. The solid line on the plot represents the transition temperature as a

function of sample composition. The dashed curve is called the spinodal curve and will

be discussed shortly. The objective of kinetic studies is to observe the process by which

the transformation from one phase to another occurs. One method of probing these

phenomena is through the thermal quench. In a typical quench experiment, the system is
held at a temperature, Ti ( or other extensive variable ), above the transition temperature,

T, and then rapidly lowered to a fixed value below T,, Tf. If the quench occurs in a time

short relative to the time required for transformation, the system will still have the

















LU -_ T DISORDERED

COEXISTENCE
r0 [ CURVE
Ln SPINODAL
- -i- \- CURVE
LU
I __ UNSTABLE
SMETASTABLE

COMPOSITION








Figure 2-1 An example of a phase diagram for a binary alloy exhibiting a phase
transition.









structure of the initial phase. This phase is now unstable ( or metastable ) and will decay

to its equilibrium phase as time proceeds. The processes by which transformation occur

are often complex and the study of these processes is one of the most challenging topics

in non-equilibrium statistical physics.'
In order to quantify the transition process a quantity called the order parameter, 4,

is defined. This parameter takes on values between zero and one, being identically zero

in the disordered phase and one in the fully ordered phase. In some systems the order
parameter is a conserved quantity and in these cases, 4 changes only locally and remains

on average a constant for the entire system. Systems of this type are called conserved

order parameter ( COP) systems. An example of a COP system is a binary alloy which

exhibits phase separation.3 4, 5,6

A typical non-conserved order parameter ( NCOP ) system is a binary alloy

which exhibits an order-disorder transition.7, 8.9. 1.12.13 An example of such a system is

Cu3Au.7, 10 In the disordered state the crystal structure for this system is face-centered

cubic (FCC ) where the Cu and Au atoms are randomly distributed on all sites. In the

ordered state, the structure remains FCC with the Au atoms dominating the conventional

cube covers and the Cu atoms occupying the cube faces. The order parameter is related

to fraction of each species on a sublattice. Since the order parameter in this case is not

constrained by conservation laws, this is an example of an NCOP system. Other

examples of transitions characterized by NCOPs are melting/crystallization phenomena16

and antiferromagnetic ordering.14,15
The driving force for transformation is the difference in free energies between the

initial and final phases. Phase transitions may be classified as first, second, or higher

order based on a fundamental thermodynamic criterion according to the Ehrenfest

classification scheme.33 If the first derivative of the free energy of the system with

respect to the order parameter is singular, the transition is said to be first order. One

characteristic of first order phase transitions ( FOPTs ) which follows directly from the








discontinuity in the free energy is the existence of a latent heat (i. e. the heats of melting

and vaporization ). In addition transitions involving a change of symmetry from one

symmetry group to another that is not a subgroup of the first can be identified as FOPTs

because it is not possible for the system to go from one phase to the other continuously.

Discontinuity in the second and higher derivatives of the free energy with respect to the

order parameter without singularity in the lower derivatives implies second and higher

order transitions. Second and higher order ( also called critical or multicritical)

phenomena are in general much better understood than first order phenomena.1.2 This is

because critical phenomena arise from fluctuations in the order parameter which can be

characterized by a single correlation length. In fact, near the critical point the correlation

length diverges and only phenomena occurring on this length scale are relevant to the

transition behavior. It is the existence of this dominant length scale which has led to the

success of the renormalization group approach and the observation of scaling and self-

similarity in these systems.34 First order phase transitions are different because there is

in general no dominant length scale and transformation is determined by processes at the

interface between transformed and untransformed regions.

Mean field theories express the free energy, F, as a functional of the order

parameter, however, approximations are required in evaluating this expression. The

equilibrium state of the system can be determined by minimizing the free energy with

respect to <. The Ginzburg-Landau35 formalism describes an expansion of the coarse-

grained free energy in powers of the order parameter. In general, the order parameter

may be a scalar, vector or a tensor as is the orientational ordering of liquid crystals.2

The prototypical shape of the free energy curve for a scalar order parameter and
fixed composition is shown in Figure 2-2 for several temperature ranges. F{j) has the
double well structure characteristic of phase transformations. At temperatures above Te,

the lowest energy state corresponds to <=0. As the temperature is lowered, the difference

in free energies between the ordered, 4=1 and disordered, 4=0 configurations is reduced





























T > Tc



T Tc

Tc > T > Ts

Ts > T


The shape of the free energy curve versus order parameter for various
temperatures, above and below the transition point


Figure 2-2









until T=T(, at which point the free energy is equivalent for both phases. Below T, the

ordered phase is favored, but the state 0=0 is at a local minimum of free energy and is

therefore metastable. If the temperature is lowered further, the potential barrier

separating the ordered and disordered phases disappears and the <=0 state becomes

unstable. This instability limit which separates the metastable and unstable regions of the

phase diagram is commonly called the spinodal curve.'.2. 35 In fact it is no longer

believed that a sharp spinodal point exists2 as will be discussed momentarily. The

process by which the disordered state decays is qualitatively different for quenches into

the metastable and unstable regions of the phase diagram. In fact two distinct behaviors

are associated with these two regimes. When the initial phase is metastable,

transformation is said to result from the mechanism of nucleation. During nucleation,

localized droplets of the ordered phase spontaneously form within a disordered matrix.

These droplets grow independently until separate domains impinge on one another.

When the initial phase is unstable, it will decay through the mechanism of spinodal

decomposition. During spinodal decomposition long wavelength, infinitesimal

oscillations of the order parameter appear which are characterized by an amplitude and

wavelength that grow with time.

Many of the essential features observed in phase transition kinetics have been

elucidated through analysis of very simple models.' Perhaps the most useful of these is

the Ising Model because of the ease of calculation and the straightforward analogies to

physical systems.16 Within the Ising model, a lattice is composed of spins having values
of o = +- 1. By choosing the correct spin dynamics and an appropriate order parameter

this model may be applied to both NCOP and COP processes. For example, if changes in

the lattice distribution of spins occur by exchanging adjacent spins the system is said to

obey Kawasaki spin dynamics.' The total magnetization
M=Yoa (2-1)
i









remains constant throughout the transformation and is therefore analogous to the order

parameter in a COP system such as phase separation in binary alloys.36 In this case "up"

spins correspond to one type of atom while "down" spins correspond to the other type of

atom. The order-disorder transition in binary alloys can also be modeled using Kawasaki

dynamics if the order parameter is defined to be a sub=lattice magnetization:
M,= Doi (2-2)
sublattice

In this event, the total magnetization is conserved, but the order parameter is

nonconserved.37 38 Another way to model NCOP systems is with Glauber spin

dynamics. Within this model spins are able to change sign independent of adjacent spins

so that the order parameter which is taken to be the total magnetization is nonconserved.

In general spin dynamical models have shown good agreement with theory.16

Conceptually the transformation process can be divided into three temporal

regimes: early stage growth characterized by either nucleation or spinodal

decomposition, intermediate growth, and late stage growth commonly called coarsening.

These three stages are illustrated in Figure 2-3. During late stage growth, essentially all

of the system is in small fully transformed regions. During this regime the larger of

these domains will grow at the expense of the smaller domains resulting in an increase in

the average size of the domains. This process is termed coarsening and is illustrated in

frames c and d of Figure 2-3. The different time regimes and transformation

mechanisms will now be discussed separately beginning with nucleation and spinodal

decomposition, followed by a brief discussion of intermediate stage growth and finishing

with a description of coarsening behavior. The chapter will conclude with some

comments on recent efforts to develop a universal classification scheme.








Nucleation




Growth
0

Growth


0*


Early Coarsening


Late Coarsening


An illustration of the different temporal regimes in the transformation
process.


Figure 2-3








Nucleation Theory


The rate of transformation from the old to new phase is controlled by the

combination of the total nucleation rate and the growth rate. The total nucleation rate per

unit time, IT(t), is the product of the nucleation rate, I(t), and the volume available for
nucleation, V,. Conceptually, the behavior of the total nucleation rate can be divided

into three regimes. Often these three regimes are distinct but in some cases or there may

be significant overlap. In addition, not all of the regimes are necessarily observed. The

first regime is characterized by a transient nucleation rate.39 Following a quench from

above the transition point I(t) will be zero initially and then grow smoothly to its steady-

state value. Once the nucleation rate reaches its steady-state value, Is', it will remain

unchanged throughout the transformation. However, as the transformation proceeds the
nucleation volume, V,, decreases leading to a decrease in the total nucleation rate, IT(t),

in the late time regime. The process by which the nucleation volume is exhausted

depends in detail on the type of site in which nucleation occurs.40 Several of these

processes will be discussed in a later section. Each of these three regimes will be

discussed in detail in the following sections.

The driving force for transformation from the metastable state to the equilibrium

state is the free energy difference between the two phases.39 However, when a finite

sized nucleus forms, there exists an interfacial region between the nucleus and the bulk

metastable phase. The surface tension serves to raise the free energy of the droplet. The

total free energy of the embryo is typically written as the sum of bulk and surface terms:


AF, = n (f f) + So (2-3)


or AF,= [ (f f- f ) + 4xcr2 (2-4)










In the above expression, the droplet of radius, r, is composed of n particles. The product
of the surface area of the droplet, S,, and the surface tension, a, gives the surface

contribution to the free energy. The bulk free energies of the equilibrium and metastable

phases per particle are f" and f", respectively. The volume per particle in the

equilibrium phase is designated v". The volume free energy term f" is generally assumed
to be independent of droplet size. Similarly, a is assumed to be independent of size and

calculations of the surface tension exhibited by crystalline clusters suspended in liquid

support the validity of this assumption.41

The energies, fe and f", are in general a function of the reduced temperature,
t=(To-T)/To where To is the transition temperature. Above To, fe > fm and AFn is a

monotonically increasing function of n and fluctuations of any size tend to evaporate. In

this case, the equilibrium form of the distribution of sizes of nuclei as determined by

Boltzmann statistics is given by


f-AFn1
N. = N' exp (2-5)



which is only of significant magnitude at very low values of n. However, below To, f <

fn and AF, goes through a maximum at the critical radius, r,.


2ov*
rc =(ife) (2-6)



Classically, nuclei of this radius are in unstable equilibrium whereas nuclei with r < r,

will evaporate and droplets with r > r, will grow. This model, while being somewhat

simplified, exhibits the important feature that nuclei having radii near r, control the









nucleation rate. In other words, it is the rate at which fluctuations reach the critical size

that determines the rate of decay of the metastable state.

Immediately following a quench the majority of nuclei are of a very small size

and until the first droplets reach the critical size the nucleation rate is effectively zero.
This period is called the induction or incubation time, to.

There are two commonly used approximations for I(t) in the transient period.

The first assumes an incubation time followed by a slowly growing nucleation rate which

is approximated by a power law,


I(t) = ( t to ) B,t (2-7)


Here the step function, O(t), incorporates the effect of a finite incubation time. This

approximation will be most valid when the nucleation rate is a slowly changing function

of the time and the transformation nears completion before the nucleation rate

approaches its steady-state value. The second approximation incorporates an induction

time and a step function to describe I(t) and will be most accurate when, following the

induction period, I(t) grows rapidly to its steady-state value. Typically, the effect of the

changing nucleation rate is most obvious in the early stages of the transformation.

Experimental measurements are not overly sensitive to the exact form chosen to

approximate the time dependence of I(t) in this regime.38

In the second time regime the nucleation rate is constant at its steady-state value,

I". While there have been many attempts to calculate I" by far the most successful

treatment is that of Becker and Doring.42 The key to this model is its treatment of the

problem as a kinetic phenomenon. A set of difference equations is used to describe the
number of droplets as a function of size and time, Z,. The rate of change of the number

of droplets composed of n particles is expressed as the difference between the rate of

condensation and the rate of evaporation of particles from the nucleus. The condensation








rate is assumed to be proportional to the surface area of the nucleus, S., and otherwise

independent of n. As time proceeds, the number of embryos of size n approaches its
quasi-steady state value ( Z., = Z ). This implies that I, = I, a constant independent of

n so that the distribution of nuclei, Z7, is constant. There are two boundary conditions

for the kinetic process. The first is that for small n the distribution of nuclei approaches
its equilibrium form as determined by Boltzmann statistics (Z, approaches N, from

Equation 2-5 ). The other boundary condition is that embryos containing a very large
number of particles leave the system completely ( Z,= 0 n>>n,). By equating the rates

of production and loss of embryos of size n, the following result is obtained:



I-s = v (2-8)

n=l


where v is the rate of condensation per unit surface area of the nucleus. Since AF
exhibits a maximum at nc, the quantity N,-1 from Equation 2-5 will be sharply peaked

near nc. Becker and Doring reasoned that only values of n near the critical number
would contribute significantly to the above sum. Taking the continuum limit of N, and

AF, the standard Becker-Doring form for the steady-state nucleation rate is obtained:


I- = A exp (2-9)


fNvSc] AF(n.)
where A= 3J B = AF(n) (2-10)

In the above relations, both the prefactor A and the energy term, B, depend on the

reduced temperature. In those cases where these parameters are only weakly dependent

on the temperature, the steady-state value of the nucleation rate takes the Arhenneus35









form with an activation energy equal to the free energy difference between a critically

sized nucleus and the bulk metastable phase.


Spinodal Decomposition


The different modes of transformation observed for metastable and unstable

systems can be viewed as instabilities against different types of fluctuations as discussed

by many authors.1' 2.16 The method of decay of a metastable state is through the

nucleation process. This indicates an instability against localized, large amplitude

fluctuations of the order parameter, the size of which grows with time. This process is

illustrated in Figure 2-4. In contrast, the method of decay of an unstable system is

through a process termed spinodal decomposition. This process is also shown in Figure

2-4. This mode of decay indicates an instablility to nonlocalized or long wavelength

fluctuations of the order parameter of very small amplitude. The amplitude of these

fluctuations grows with time. While nucleation gives rise to growing droplets, spinodal

decomposition leads to complex interconnected patterns of the transformed phase.43 The

interfacial region between transformed and untransformed regions is initially much less

defined than that of nucleated droplets but sharpens as time proceeds. As the size of the

transformed domains increases, the patterns become less interconnected until the domains

are similar in shape to those observed for nucleated transformations.

The significance of the spinodal curve has recently been questioned by many

researchers2 17 because the concept of an instability limit is an artifact of mean field

theory used to formulate the free energy functional. However, real systems do show the

types of behavior described above. And although the transition from nucleated to

spinodal decomposition behavior occurs gradually26 instead of suddenly at the spinodal

point, the concept of a spinodal limit is not totally without merit.













NUCLEATION


ti \ t2 t t3

Distance





SPINODAL DECOMPOSITION


Distance


A contrast of the nucleation and spinodal decomposition transformation
mechanisms.


Figure 2-4








Domain Growth


For the types of systems we are considering, the growth of domains of the

equilibrium phase is said to be thermally activated. This implies that the growth velocity

is a function of temperature, being greatly reduced at temperatures low relative to the

transition temperature. One consequence of this behavior is the ability to "quench in" the

high temperature phase by rapidly lowering the system temperature to a value where the

velocity of growth is extremely small.
In addition to the dependence on temperature, the growth velocity, F, will also

have a characteristic dependence on the size of the domain ( or equivalently, on the time

elapsed since the birth of the domain). In general, thermally activated transitions can be

divided into two groups: interface controlled and diffusion controlled reactions.39 These

two groups are characterized by functionally distinct forms for the growth velocity.

Generally, transformations between two phases having approximately the same

particle concentrations are controlled by the energetic of processes occurring at the

interface. This being the case, the growth velocity, F, is independent of particle size and

time so that the average dimension of a growing droplet, L(t) satisfies the simple

equation.44,45
L(t) = ro t, F(t) = r (2-11)

Christian39 has estimated the dependence of F on the temperature, T, for the decay of

metastable phases. This calculation assumes the difference in free energy between the
old and new phases to be small relative to the thermal energy, kBT, where kB is the

Boltzmann constant. In addition, the characteristic frequency for particle motion, vo, is

assumed to be identical in both the transformed and untransformed regions. The

resulting relation is
(T) = ( AFv) e (2-12)
r(T) =(vo) kgT exp [- T(2-12)
^B1 I B1








Here 8 is the thickness of the interface and AH, is the activation energy separating the

equilibrium and metastable states which is generally a function of the reduced
temperature, T.

In the event that there is a significant concentration difference between the initial

and final phases, it is possible for the propagation of the domain boundary to be diffusion

limited. If the concentration of particles is higher in the transformed domain, there exists

a region adjacent to the interface called the depletion zone' in which the particle

concentration falls below its average value in the untransformed volume. In order for the

boundary to propagate, particles must diffuse through the depletion zone and then

"condense" on the transformed domain. If the processes occurring at the interface are

much slower than the diffusion of particles through the depleted region, the
transformation is interface controlled (i. e.. L(t) = F0 t). However, if the condensation

process is rapid, transformation will be limited to the rate of diffusion through the

depletion zone. From the diffusion equation and dimensional analysis Christian finds
D
L(t) ( Dt )z r(t) (2-13)

where D is the diffusion constant. This parabolic growth law is verified in more exact

calculations,46.47 and in computer simulations.48'49

It is noted that the above growth laws for interface and diffusion controlled

growth are only valid prior to impingement of one transformed domain on another.

Different growth laws govern behavior in the post-impingement or coarsening regime

and these will be discussed shortly.


Transformation Curves


Now that the processes of nucleation and growth have been discussed, a method

for combining the results to yield a rate of transformation will be presented. A measure









of the degree to which transformation has progressed is the fractional volume in the
transformed state, C(t). From the previous discussion of the nucleation rate we expect

this quantity to grow slowly at first and then more rapidly as a larger number of embryos

reach the critical size and begin to grow. Eventually, the surfaces of different domains
will impinge on each other, slowing the growth of ((t) at later times. The shape of a

typical transformation curve is shown in Figure 2-5. The relevant features include the
induction time, to, and the time to half-completion, ;,.

The earliest formal theories of transformation kinetics are due to Kolmogorov,50

Avrami,51-52.53 and Mehl, and Johnson.54 The argument of Avrami goes as follows. If a

nucleus forms at some time, t, then the transformed volume which is contained in the

nucleus is given by


V(t) =rlyxyY (t-t)3 (t>t) (2-14)
=0 (t
Here Yx, y,, and yz are the growth velocities in the x, y, and z directions respectively and
4L
Tr is a shape factor (for isotropic growth, rl = and y, = yy = y, = y). The above

expression is valid for a system exhibiting a linear growth law, L(t) t, such as is the

case for an interface controlled transformation as was previously discussed. In a system

where growth is diffusion limited, the growth law is given by L(t) tIA. In this case

Equations 2-14 would be replaced by


V = TlyYy z ( t -t)3' (t>r) (2-15)

=0 (t
For the remainder of this section only systems exhibiting a linear growth law will be

discussed with the understanding that the appropriate results for different growth laws

can be derived simply by making the modification shown in Equations 2-15.

































_ _
1/2 /


0-


TIME


A simulated transformation curve showing the induction time and the
half-completion time.


S-


C.


Figure 2-5


i 1 I


I








In early times there is little impingement of domains, so that each droplet grows

independently. In this limit the transformed volume is given by


V= 3-]f I(t) (t- T)3 d' (2-16)
0


where isotropic growth was assumed. For a constant nucleation rate, I(t) (I(t) reaches

the steady-state value, I", immediately ) Equation 2-16 is reduced to



= V 3 I3 t4 (2-17)



This confirms the rapid increase of C at early times.

Unimpeded growth will not continue for all times, however, due to impingement
by other droplets. To deal with this complication, a concept called the extended volume,
Ve is introduced. The extended volume is the volume of transformed material assuming

that nuclei continue to form not only in untransformed regions of the system, but in
previously transformed regions as well. Similarly, growth of domains is assumed not to

stop when impingement occurs; rather, each domain continues to grow through the
others. In this view, Ve is equivalent to VB in Equation 2-17 for all times. In any

increment of time, dt, the probability that the incremental increase in the extended
volume occurs in a previously untransformed region is ( 1 VP/V ). It follows that


dVP= (1 VB/V) dVe (2-18)

or = vP/V = 1 exp( -Ve) = 1 exp(-T_ I" t4)








The shape of this curve is shown in Figure 2-5 and in the early time this expression

reduces to Equation 2-17.

In the above argument, the nucleation rate I(t) was assumed to be a constant. In

general, the total nucleation rate is a function of the time, either due to transient

nucleation in the early regime or due to nucleation site saturation in the later regime.
The corresponding transformation laws may be derived in a straight forward way by

inserting the appropriate expression for I(t) into Equation 2-16. These calculations

generally lead to an expression similar to that in Equation 2-18 except that the time

exponent is different from 4. This led Mehl, Johnson, and Avrami ( MJA) to propose

the following general transformation law39:


= 1- exp( -ktn) (2-19)

For a constant nucleation rate it has been shown that n=4. For an increasing nucleation

rate ( early regime ) n>4, and for a decreasing nucleation rate ( site saturation) n<4.

For homogeneous nucleation, often the nucleation rate may be approximated by a
step function being zero during the induction period, to, and then taking on its steady-

state value Is. In this case, the transformation curve is given by Equation 2-19 with n=4
and the time, t, is replaced by t-to.

Often nucleation occurs not homogeneously, but at impurity or defect sites within

the sample. The transformation curves for heterogeneous nucleation can be calculated in

much the same way as for the homogeneous case if the appropriate form for the
nucleation rate can be found. For instance, when nucleation occurs spontaneously at No

impurity sites the effective nucleation rate, I,(t), may be represented by a delta function at

t=O having a strength, No; i. e..


I(t)= No 8(t)


(2-20a)








so that from Equation 2-17,


= 1 exp( N 'y t3) (2-20b)


The form of the transformation curve is unchanged, with n=3 reflecting the nature of the

nucleation process.

Interesting dimensional effects result if nucleation is spontaneous and growth is

restricted to either a lamellar or rod-like volume. For growth occurring in a sheet ( 2-D

growth ), as might be the case in a thin film, the extended volume is given by :


VP = (47cV6)) No y2 ( t x )2 (2-21)



where 8 is the thickness of the sheet. The result for a constant nucleation rate is the MJA

expression with n=2. Similarly, it is seen that for growth in a rod-like volume ( 1-D ) the

result is MJA with n=l.

More complicated transformation curves may result if nucleation occurs

heterogeneously, but not spontaneously. If defects are located randomly, the growth in

early times should be indistinguishable from the homogeneous case ( Equation 2-19 with

n=4). In time the available nucleation sites may be exhausted before transformation of

the entire volume is complete. In this event, different behavior is expected following

saturation of the available sites. Cahn derived an expression for nucleation occurring on

grain boundaries, edge dislocations and point defects.40 Analogous to the concept of an

extended volume for transformation, an extended volume for nucleation is introduced. In

this way, the space available for nucleation at time, t, is calculated as a function of the

growth velocity, steady-state nucleation rate, and the initial volume available for
nucleation, Vn. The resulting transformation curve for nucleation occurring on two

dimensional defects is :










S= 1- exp(-b f(t)) (2-22)
1
where f(t) = ( 1 exp(-' ( (1-32+2l3)]d
0


The coefficients a and b are related to the physical quantities y, I", and V, in the

following way:


a= ( Iss )1t3 (2-23)

and b = 2Vn l



The function f(t) has two limiting forms, at late times f(t) t, and at early times f(t) t
as expected. The transition from n=4 before site saturation to n=l after saturation is best
illustrated by a plot of In( ln( 1/(1-) ) ) vs In( t) as shown in Fig. 2-6. In this plot the

slope of the curve is equivalent to the exponent in the MJA form, and the bend in the line
corresponds to site saturation. From Equations 2-21 it is evident that if we offset the plot
vertically by In( b ) and horizontally by ln( a) we have a master curve for this type of
transformation. In other words, plots for a number of data sets will fall upon this master
curve if offset by the appropriate selection of a and b on this scale.
Similarly, the expressions for the transformation curves for nucleation on line and
point defects have been derived. In both cases the early time behavior agrees with the
MJA form having n=4. In late times, n crosses over to 2 for nucleation on line, and 3
for point defects. It is apparent that the spontaneous nucleation of point defects discussed
previously is just a special case of this result where only the n=3 regime of the



















The Cahn Model


8

S6


4


2

0


-2


C
-4


-6
- -6 -


-8
-1


Master Curve


0 1 2 3 4


In (


Figure 2-6


time )


An ideal Master Curve for a transformation which obeys Cahn's
relation for nucleation on two dimensional nucleation sites.


n = 1


n = 4









transformation curve is observable. In fact, the easily observable range of C in

experiment is typically from about 0.01 to 0.99. Although it seems that this range should

encompass the full range of kinetic behavior, this range corresponds to only a relatively

small portion of the master curve. If the nucleation rate is too high, saturation occurs

very early in the transformation and only the behavior of saturated regime is evident.

Conversely, if the nucleation rate is too low, saturation occurs late in the reaction and

only the exponent corresponding to the unsaturated or homogeneous regime is seen. In
fact, the crossover is only visible if saturation occurs when t 0.5. The steady-state

nucleation rate within the Cahn theory for nucleation on 2-D defects which corresponds

to this point is given by :

r i3
I-s = V3 (2-24)




Systems exhibiting the required balance between the nucleation rate, growth velocity,

and nucleation volume appear to be exceedingly rare. As will be discussed, the

observation of this phenomenon is one of the more fascinating features in the study of

ordering in block copolymers.


Late Stage Growth Coarsening


Often the equilibrium state toward which the system is evolving is degenerate.
For example, in the Cu3Au system described previously there are four identical sites in

the unit cell. The lattice can be defined by choosing any one of these four sites as the

corer site. In this way it is seen that the ground state degeneracy of the ordered phase is

p=4. For the crystallization process, p=o reflecting the fact that the crystal may form in

any orientation within the liquid.39








In the event the ground state is degenerate, in the late time regime the system will

be composed of many fully transformed domains distributed in the p degenerate states.

As time proceeds, the number of domains will decrease with a corresponding increase in

the average size of the ordered domains. This process is called coarsening.

Within the coarsening regime, the system is self-similar with respect to scaling by

length and time. This can be illustrated by zooming in on frame (c) of Figure 2-3 and

comparing the result with frame (d). The relevant features in each case will on average

be identical. A well known consequence of this type of scaling is the existence of a

characteristic length37, L(t), (the domain size) where

L(t) t (2-25)

Here t is the time following the quench. The exponent a has been seen to be different in

COP and NCOP systems. Specifically, theoretical calculations55 and experimental

observations56 of COP systems support the assertion a=l/3. In NCOP systems the

growth rate of domains in the coarsening stage is determined by energetic at the

interface and is often curvature driven. Theoretical calculations predict a corresponding

growth exponent, a=l/2.57 This result is supported by a growing number of experimental

findings7.58 and computer simulations59,60. It should be noted that the growth exponents

are different in the coarsening regime from those discussed for the growth regime. This

is because during the early stages, growth of domains is relatively uninhibited by

impingement. In the coarsening regime, the domains are strongly interacting with the

major mode of growth being the absorption of smaller domains by larger ones. Also, the

dynamic quantity during coarsening is the average size of a domain, while the volume

fraction of the system in the transformed phase remains approximately unchanged.

In the early stages of transformation it was seen that impurities and defects have a

profound effect on the nucleation process by providing heterogeneous nucleation sites.

These entities can also lead to interesting effects in the coarsening regime. Lai et al.61








show that random fields which are generated by impurities and defects slow the

coarsening process so that the growth law becomes:

L(t) (log(t) )m (2-26)

where m depends on the detailed nature of the imperfections in the system. Evidence
supporting this assertion has been provided by experiments on Cu3+xAu where excess Cu

atoms appear to behave as diffusive impurities.10


Universal Classification


The non-equilibrium scaling behavior observed at late times is reminiscent of the

scaling seen in critical phenomena. For this reason, one of the foremost efforts in the

study of phase transition kinetics has been the attempt at developing a universal

classification scheme analogous to that for critical phenomena.17 In this way it is hoped

that the relevant parameters such as ground state degeneracy, dimensionality, and

conservation property of the order parameter can be identified so that the behavior of

widely diverse systems can be better understood. The coarsening growth law is one

feature which may be useful in identifying these universality classes. For systems

exhibiting random fields, the growth law is logarithmic suggesting that systems with

different types of impurities might form separate universality classes. The growth

exponent a has been seen to depend on the type of order parameter. It is, however,

independent of ground state degeneracy, and dimensionality in many cases.

In order to better distinguish the relevant parameters for classification and

provide a deeper understanding of the fundamental processes which govern phase

transformation, a broad range of physical systems must be investigated. Polymers and

block copolymers in particular are excellent systems for this purpose.1 An excellent

method for observing the transformation process in these materials is through the small





35


angle x-ray scattering spectrum. The fundamentals of the x-ray scattering technique are

now presented.











CHAPTER 3
THE X-RAY SCATTERING TECHNIQUE




The x-ray scattering technique is an excellent probe of inter- and intraparticle

correlations. Many excellent reviews of general scattering theory can be found in the
literature.62.63.64,65.66 A brief discussion of the fundamentals of x-ray diffraction is given

here followed by some details of intraparticle effects in small angle scattering.

Interparticle interference effects resulting from a system of particles with liquid-like

correlations are then described. Finally a description of the scattering from an ordered

array is presented including the effects of grain size, strain, and temperature.


Fundamentals of X-ray Diffraction


When a beam of x-rays is incident on a sample, a characteristic scattering pattern

results. In the systems with which we are dealing, the diffraction of x-rays is due almost

entirely to elastic scattering from the electrons in the material. The scattering

distribution from a single electron is given by the Thomson formula:67
Ie(20) = I0 m2 (3-1)

The angle 20 is called the scattering angle and is illustrated in Figure 3-1. Here OP is a

polarization factor which equals 1 for an incident beam polarized perpendicular to the

plane of scattering ( as is the case for synchrotron radiation). For an unpolarized beam,
l+cos2(20)
= 2 (3-2)








In either case the polarization factor is approximately equal to unity in small angles.
Compton and inelastic scattering are negligibly small at small angles (i.e. 20 < 20 ).64

The elastically scattered waves are coherent, meaning that the total scattered

amplitude is the linear superposition of the electric field vectors for each scattered wave.

The scattering from two scattering centers is illustrated in Figure 3-1. If the amplitudes

of the two scattered waves are equal and of unit magnitude, the total scattering amplitude
will be determined by the relative phases of the two waves. The phase for each wave at
the point of detection equals 2nx/ times the optical path for each of the two waves.

Actually, there is an additional factor of n which results from the phase inversion

incurred by scattering from an electron. Since all scattered waves are offset by this

factor, the total phase shift is given by
27t
= r.- (- so )=-r.(k-ko ) (3-3)

Here, so and s are unit vectors in the incident and scattered directions, and k k are wave

vectors in the incident and scattered directions. At this point it is convenient to define
the scattering vector, Q, which is the difference in incoming and outgoing wave vectors

so that
[20 4E f29
Q = k-ko, Q= 2k sin = sinm (3-4)

The scattering vector is the variable typically used in x-ray diffraction and henceforth the

scattering will be described in terms of this quantity. This is done with the understanding
that Q can be related to the scattering angle, 20, by Equations 3-4. The transformation

from Q to scattering angle is approximately linear at small Q with 10 = 0.07121 A-1 for

an incident wavelength of 1.541 A.
With these definitions, the amplitude of scattering from two scattering centers is
A = 1 + exp(-i4) = 1 + exp( -iQ-r) (3-5)

































So















An illustration of diffraction by two scattering centers and the definition
of the scattering angle, 20.


Figure 3-1


7-(s-So)








This result can be extended to the case of scattering from a continuous system
characterized by an electron density, p(r). The resulting amplitude is given by

A(Q) = fdVp(r) exp( -iQ-r) (3-6)


Hence, the scattering amplitude is equivalent to the Fourier transformation of the electron

density. The observed scattered intensity is related to the amplitude in the following

way:


I(Q) c AA*= f dV1 dVz p(r1) p(rz) exp(-iQ-(rl-r2)) (3-7)



Redefining r = r, r2, in Equation 3-7 yields the result that the scattered intensity is

equal to the Fourier transform of the auto-correlation function of the electron density,
-2
p (r):

I(Q) f dV 2(r) exp( -i Q-r) (3-8)


where p (r) = dV, p(r) p(r2)


This is a very general result which relates the positional correlations of electron density

fluctuations in the system to the measured scattering profile in the absence of

approximations. In cases where a discrete treatment is more appropriate, a particle-
2
particle correlation function replaces the term, p (r), in Equation 3-8.68

Often the systems studied with scattering methods are composed of particles
having an approximately uniform electron density, p,. These particles are suspended in a

gel or solution characterized by another electron density, P2. Neglecting inhomogeneities

in the charge density, the scattering is equivalent to that from a system of particles with
density Ap = p, P2 superimposed on a background of density P2. There will be no








observable scattering from the background in the relevant scattering range. The problem

is therefore reduced to the much simpler case of calculating the scattering from particles
of density Ap suspended in a vacuum.64

The scattering amplitude for a system of N identical particles can be written
N
A(Q) = fp(r) exp( -iQ-( R + r )) dV (3-9)

k=1
N
= F(Q) exp(-iQ.Rk)
k=1
F(Q) = fp(r) exp(-iQ-r) dV

Here Rk is a vector extending from the origin to the center of symmetry of the kth

particle and p(r) is the electron density distribution of a single particle about its center of

symmetry. The quantity, F(Q), called the form factor for the particle, contains all

information regarding intraparticle correlations. The structure factor, S(Q), is the

product of the scattered amplitude, A, with its complex conjugate and is a weighted sum

of the phase factors corresponding to the center of mass positions of all of the particles.

S(Q) therefore contains information concerning interparticle correlations. The scattered
intensity is proportional to S(Q) which can be written :
I(Q) c S(Q) = F(Q) F*(Q) cos( Q.( R, R) ) (3-10)

k
After averaging over all particle orientations and separating those terms where

j=k, this expression ( Equation 3-10) for S(Q) becomes
S(Q) = N< IF(Q)2 > + NI< F(Q) >12 C cos( Q-(Rk-RJ)) (3-11)
kj
The pair correlation function, P(r), introduced by Zernicke and Prins69 represents the

probability that two particles are separated by a distance r. In the case of an isotropic

distribution of particle positions and orientations, this can be used to evaluate the double

summation in Equation 3-11. The resulting form for S(Q) is








00
< F(Q) >2 sin(Qr)
S(Q) = N < F2(Q) > 0
where v, is the average volume available per particle in the system. As v, decreases,

interparticle interference effects become more pronounced. With v, large S(Q) is

dominated by the first term in Equation 3-12. In this case, the observable scattering is

just that of a single particle multiplied by the number of particles in the system. This is
the limit typically explored in SAXS and the detailed properties of the scattered intensity

in this regime will now be discussed.


Small Angle X-ray Scattering


For particles with dimensions on the order of hundreds of angstroms and with a

typical incident x-ray wavelength of 1.5 A, the majority of scattering features appear in
the range from 0 to 2 degrees. This is typical of block copolymer solutions. Assuming a
dilute, isotropic system of particles, the scattered intensity can be calculated from

Equations 3-8. In all but the simplest cases this must be done numerically. For the

important case of a hard sphere, the structure factor, first derived by Rayleigh,70 is
S= Ap )2 V 3 sin(QR) QR cos( QR)2
ere (Q12 ( )2 V2 (QR) (3-13)

where V is the volume and R the radius of the sphere. A plot of this function for three

radii is shown on a linear and on a log scales in Figure 3-2. For the larger radius, the

scattering function decays more rapidly reflecting the inverse relation between length

scale and scattering vector. This function exhibits well defined maxima at the locations

QR = 5.77, 9.10, 12.32, etc. from which the sphere radius can be determined.

A more general method for determining the average size of particles exhibiting
small anisotropy was developed by Guinier.71 In the limit of very small angles, Equation











0
4-j
0
0
L-

E
0

0
L-
O



0












O
LL)
0)







0
U-






Ou
t-
LL
O
U


L
UL)


10


1 -1


10-2


10-3
10


10-5 L
10.00
0.00


1.00


0.80


0.60


0.40


0.20


0.00 -
0.00


0.02 0.04 0.06 0.08


Scattering Vector


0.13


A-1
A


Figure 3-2 The scattering form factors for and ideal spheres having radii: 80, 100,
and 120 A.


3.02 0.04 0.06 0.08

Scattering Vector -1
Scattering Vector ( A )








3-8 reduces to
-2<
S(Q) = ( Ap )2 Vexp (3-14)

fp(r) r2 dV
where Rg = the radius of gyration = M

Here the integral is taken over a single particle and M is the total mass of the

particle. Equation 3-14 shows that if the log of the scattered intensity is plotted versus

the square of the scattering vector ( called a Guinier plot), the slope of the resulting line

will be proportional to the square of the radius of gyration in the limit of very small

angles. Thus if the shape of the particle is known, the parameters describing its size may

be determined from the radius of gyration as obtained from the scattering profile. For
example, the radius of a sphere is related to its radius of gyration by Rg = [3/5 R. The

Guinier approximation breaks down for severely anisotropic particles (i. e. plates

or rods) and is invalid in the presence of interparticle interference effects.

In practice, all of the particles do not have identical radii. Rather, there is a

distribution of radii centered about a mean. In the case of block copolymers this effect

arises from the fact that the polymer chains themselves are polydisperse. For example

the scattering which results from a system of spheres characterized by a gaussian size
distribution, having a mean radius of R0 = 100 A, and a standard deviations, a = 1, 10,

and 20A are plotted in Figure 3-3. The number of particles, N, having radius, R, is

given by

N(R)= exp R-)2 (3-15)

and the total structure factor is the superposition of the structure factors from each

particle in the distribution,
S(Q) = fN(R) S pbe(Q,R) dR (3-16)











1.00

0.80

0.60

0.40

0.20


L-
0
4-a
0
0
LL
E
L-
0
LL

0
0

-c
L_
/)


0
I,
0
LL

E
0
L-


0


CL
c--


100


10


10-2


10-3
10


10-4 1
0.00


0.02 0.04 0.06 0.08 0.10

Scattering Vector ( A )


0.02 0.04 0.06 0.08 0.10


Scattering Vector ( A ')
A plot showing the effect of polydispersity on the scattering form factor
for a gaussian distribution of spherical particles having a width parameter,
a = 1, 10, and 20 A.


0.00 1
0.00


Figure 3-3








The major effect of polydispersity is a weakening of the scattering minima which is more
pronounced with larger o.

Typically, the interface between the particles and background is of finite width.
For block copolymers this region is where the two covalently joined blocks meet. If
these junction points form a perfect surface, the interface is ideal. It is often assumed
that the locations of these junction points are described by a gaussian distribution of
width, This leads to a density profile
p(r) = Ap 1 erf[ (3-17)

The structure factor calculated from Equations 3-6 and 3-17, incorporating the effect of
the interfacial thickness, 4, are shown in Figure 3-4. The spheres in this calculation have
a mean radius of 100 A and values of = 1, 10, and 20 A. There is relatively little

perturbation of the profile at very low Q, but the scattering at higher Q is greatly
suppressed for large values of 4. In addition slight shifts in the locations of the scattering
minima are observed.
At large values of the scattering vector, a different limit is found. In the high Q
region, the expression for the structure factor takes on a much simplified form
27n( Ap )2
S(Q) S (3-18)

In this equation, Sa is the surface area of the scattering particle. This power law

dependence is called Porod's Law.72 The radius of gyration and the surface area of the
particles can be determined from the limiting forms given by Eqs. 3-14 and 3-18 and are
very useful in deducing the shape and dimensions of the particle. If the surface of the
particle is of finite thickness the scattering at high Q will exhibit systematic deviations
from Porod's Law which depend on the interfacial thickness 4.
In a method developed by Vonk73 and Ruland,74 the effect of finite 4 is taken into
account by considering the electron density of the particle. p(r), as the convolution of an
ideal electron density profile, p,(r), and a smoothing function, h(r). Since the scattering











0

0


E
L-
0
O


0
.-
O





L..
-4




U
0
i-

E
L-
Q.
0
U-






U)


1.00
-

0.80

0.60

0.40

0.20

0.00
0.00



10


10 -
10.

ic2
10-2


o-3


-4
10 -

10-4
0.00


Scattering


Vector


The modifications to the scattered profile which result from spheres
having imperfect boundaries are illustrated in these plots for a system of
spheres having a radius of 100 A and values of x = 1, 10, and 20 A as
described in the text.


0.02 0.04- 0.06 0.08 0.10
Scattering Vector ( A
Scattering Vector ( A )


0.02 0.04 0.06 0.08 0.10


Figure 3-4








form factor, F(Q), is the Fourier transform of the electron density, p(r), it can be

expressed as the product of the form factor for the ideal particle and the Fourier
transform of h(r), Fh(Q). As described previously, h(r) is often approximated by a

Gaussian having a width parameter equal to In this case the total structure factor takes

the form
S(Q) C Q4 exp( -4 X2 2 Q2) (3-19)

Consequently, a plot of Q4 I(Q) vs Q2 yields the parameter, Application of this

method to the study of block copolymers is discussed in Chapter 4.
The above discussion was based on the assumption of widely spaced scattering

centers so that the total scattering is the sum of the individual contributions. Now, the

effect of a close packing of particles and interparticle interference will be discussed.


Scattering From Liquids


In the event that interparticle interference is present the scattering is given by

Equation 3-12. In general, the pair correlation function, P(r), depends on the average
volume per particle, v1, the temperature, T, and the interaction potential, Q(r). c(r) is

the two particle potential, i. e. the potential energy of a system of two particles separated

by a distance r. The determination of P(r) from the interaction potential is an extremely
difficult problem which has been extensively studied.7577 The approach developed by
Born and Green76 assumes a modified Maxwell-Boltzman dependence of P(r) on Q(r).

Their final result for the liquid structure factor is

S(Q) = N < F(Q)2 > + < F(Q) >2 (2-)3l(Q) 1- (3-20)

where the function 3(Q) is defined as



0








Here e is a constant of the system which is on the order of unity.
As an illustration of the above form, Equation 3-25 is evaluated for the hard
sphere potential :
I(r) = 0 r =0 r>Ro

The resulting profiles for four values of where vo is the particle volume are plotted in

Figure 3-5. Interference effects show up in the form of a reduction in intensity at very
low Q, and an enhancement at intermediate Q (Q = 0.03 A-, ). The result is a broad
maximum which grows with increasing concentration. Scattering at higher Q ( Q >
0.045 A-' ) remains mostly unaffected with the exception of a subtle shift in the position
of the primary spherical scattering maximum (Q = 0.06 A- ).
The Van der Waals force is the origin for the interaction commonly occurring in
many systems. The potential describing this interaction is well approximated by the
Lennard-Jones ( L-J) form :

(](r) = (o 12 (3-23)

This potential along with Equations 3-20 and 3-21 was successfully used by Fournet78'79
to predict the scattering from liquid and gaseous argon. Qualitatively, the scattering
curves are very similar to those of Figure 3-5. One distinguishing feature is a sharp
upturn in the scattering at small angles resulting from the long range of the L-J potential.
In addition, a secondary interparticle interference maximum appears on the high Q side
of the primary interference maximum.
From the above discussion, it is apparent that the scattering at high Q in a
concentrated system shows little deviation from single particle scattering. At lower Q,
strong oscillations about the single particle scattering become evident and take the shape
of broad peaks in the spectrum. In the case where the free energy favors a crystalline






49








The Hard Sphere


101


0
O
0

U-

0



CO

O"

-J













Figure 3-5


0
10
B

C
D
-ff


10
E



10-2





0.00
0.00


0.02 0.04 0.06 0.08

Sctterin Vec-1or ( A
Scattering Vector ( A )


".10


Model scattering profiles are shown for systems of spheres interacting
via the hard core potential at varying particle concentrations, vl/vo.


Model









arrangement of particles, the assumption of an isotropic distribution of particles is no

longer valid. The scattering from ordered structures will now be discussed.


Scattering From Ordered Structures


A crystal is a three dimensional periodic structure.0 The repeat unit for the

structure is called the unit cell. The lattice is constructed by translating the unit cell by
linear combinations of the basis vectors ( designated a,, a2, a3 ). A structure constructed

in this way can be considered to be composed of planes of particles as is shown in Figure

3-6. The scattering amplitude from such a structure will be determined by the phase

difference between waves diffracted from adjacent planes. If the path difference is equal

to an integer number of wavelengths, there will be constructive interference otherwise

the scattering amplitude will be zero. This is a qualitative statement of Bragg's Law62

which in its most common form is

mX = 2d sin( 20/2) (3-24)

Here d designates the distance between adjacent planes, X the wavelength of the scattered

waves, and m the order of scattering. In general, adjacent planes are separated by
al a2 a3
h k and in the three crystallographic directions. The integers (h,k,l) are called the

Miller indices and are used to specify crystallographic planes.

Calculating the distance between and the orientation of every group of
crystallographic planes is a formidable task. The concept of a reciprocal lattice is useful

in dealing with this type of calculation.80 In this theoretical construction three reciprocal

lattice vectors are defined


2( a2 x a3 ) 2x( a3 x a ) 27( aI x a2 )
a,* a2xa3 a *a2 xa3' a a, axa3



















































An illustration of Bragg scattering from a crystal lattice.


Figure 3-6








From vector algebra, one is able to show that the Bragg condition is equivalent to the
requirement that the scattering vector, Q, be a linear combination of the reciprocal lattice
vectors:
S= hb1 + kb2 + lb3 (3-26)

Within this construction, Bragg peaks are easily identified and designated by the Miller
indices h, k, and 1. For Bragg's law to be satisfied, the angle of incidence relative to the
set of scattering planes must equal the angle of reflection. This implies that Q is required
to be perpendicular to these planes and, for an orthorhombic system, has magnitude
Q = = 2r h + +2 1 )2 (3-27)


Thus, constructive interference from a perfect crystal will only occur at specific values of
the scattering angle, 28, and for specific orientations of the crystal relative to the

incident beam.


The Effect of Finite Size


A real crystal is generally composed of a large number of crystallites of finite
size. The scattered intensity from a single finite-sized crystallite having n particles in the
unit cell, and N,, N2, and N3 unit cells in the x, y, and z directions can be calculated62

and is proportional to the structure factor S(Q),


S(Q)= A A* sin2( N Q'al / 2) sin2( N2 Qa / 2) sin2( N3 Qa / 2)
sin2( QaI /2) sin2( Q-a / 2) sin2( Qa3 / 2)



where A(Q) = F(Q) eiQri









In the above expression the scattering amplitude, A(Q) is a function of the form factors,

Fi(Q), for n particles at positions ri within the unit cell. This function is peaked when the

following three conditions are satisfied:
Q-ai = 2xh, Q-a2 = 2xk, Q-a3 = 2rl (3-29)

or equivalently, when Q is a linear combination of the reciprocal lattice vectors.

In deriving Eq. 3-28 it was assumed that the grains were perfect parallelpipedons
of uniform size. In the limit of large N1, N2, and N3 and in real systems having a

distribution of grain sizes and shapes, the scattering lineshape is typically well
approximated by the gaussian, having an amplitude, A and a width, af.



The Crystal Structure Factor


From the above discussion it would appear that there exists a peak in intensity

corresponding to all integer values of h,k, and 1. This is true for a simple cubic structure.

However, for any structure having more than one particle in the unit cell, the structure
factor vanishes for certain combinations of h,k, and 1. The vector, r, from Equation 3-33

which specifies the location of each particle within the unit cell can be written as
ri= x, a + y, a2 + zi a3 (3-30)

where xi, Yi, and zi are belong to the interval: [ 0, 1]

The scattering amplitude can now be rewritten as


A(Q) = 2Fi(Q) ei( hxi +y + ) (3-31)
i=1

For a body-centered cubic structure ( BCC), there are two particles in the unit
cell having coordinates : x, = y, = z, = 0, and x2 = y2 = zz = 1/2. The scattering

amplitude for identical particles is








A(Q) = 0, h+k+1 odd (3-32)

A(Q) = 2 F(Q) h+k+1 even
So the crystal structure factor, SBCC(Q) = A A*, vanishes at those peak locations where

h+k+l is odd, and at those locations where h+k+1 is even, the peak is present having an

effective form factor equal to twice that of the constituent particles.

In a similar way, the scattering amplitude can be determined for the face-centered

cubic (FCC) structure,
A(Q) = 0 h, k, 1 mixed (3-33)

A(Q) = 4 F(Q) h, k, 1 unmixed
Only those peaks denoted by indices that are all even or all odd will be present ( i. e.

(111), (200), (311) etc.).


The Effect of Particle Motion


Until now it has been assumed that the particles are stationary on their lattice

positions. In fact, the particles exhibit displacement oscillations about their equilibrium

lattice positions. In conventional crystals, these motions are the thermal vibrations of the

atoms. For the block copolymer system, thermal oscillations for the massive

polystyrene domains suspended within the viscous polybutadiene matrix are negligible.

In these materials, the displacements result from diffusive motion of the domains in the

vicinity of their lattice positions. The change in the diffraction profile due to particle

motion can be described as a modification of the form factor in those cases where the
time characterizing particle motion is much shorter than observation times.
At any time the position of the kth particle can be written Rky = Rk + B(t). In

x-ray scattering we measure the temporal average (indicated by triangular brackets ) of

the scattered intensity, which is proportional to the structure factor:









S(Q) iQ-(Rk-RI) iQ-(S(t)-(8(t))
S(Q)= Fk2(Q) e < e > (3-34)

k
If uk(t) and u,(t) are defined to be the amplitudes of oscillation in the direction parallel to
Q, then equation 3-34 can be rewritten as
S(Q) = S(Q) + Sf(Q) (3-35)

2 iQ-(Rk-RI)
SI(Q) = > ^F,k(Q)e- J IFi(Q)e Je

k
The second contribution in Equation 3-35, Sd, contains the cross term, < ukUl > which is

vanishingly small unless the kth and Ith particles are very close. The result is a scattering
component that is slowly varying with Q and of very low amplitude. This contribution
to the scattering is called the thermal diffuse scattering and is approximately constant in
the high Q region of the spectrum.
The first term, SI, is identical to the result for the stationary finite-sized crystal

with the form factor replaced by an effective form factor
Feff(Q) = F(Q) exp( -- Q2< u2>) (3-36)

The exponential in the above equation is called the Debye-Waller factor. The peak
widths and positions are unchanged by displacement oscillations, but the peak amplitudes
are modulated by this factor.


The Effect of Strain


In a typical experiment scattering occurs not from one, but from many
crystallites. In general there exist random strain fields within the sample. The result is a
collection of grains having a narrow distribution of lattice constants centered about the








equilibrium value. If we define the strain, e = Aa/a, then the distribution of strains may

be approximated by
1 e2
P(e) ---exp( ) (3-37)
2 OY 20,

For simplicity we have assumed an isotropic gaussian distribution of strains and a cubic
lattice. The magnitude of the scattering vector at the bragg peak position is related to the
lattice constant via Eq. 3-27. The Bragg peak will be shifted by, AQ, if the lattice
constant is stretched by Aa. The peak shift and strain are related in the following way:

AQ = Q (/a) Aa = Q e (3-38)

so that the distribution of peak offsets is given by
1 (AQ)2
P(AQ)= exp(- 2 ) OQ = Q (3-39)
V2itOQ 20Q

The scattered intensity is the convolution of this function with the profile for a strain-free
system. The scattering profile for a strained system in this approximation is the
convolution of two gaussians which is itself a gaussian having a width equal to the sum
of the individual widths added in quadrature:
(Q-Qo)2 2
I(Q) = A (f/o) exp(- 2a ), O = ( (o+ (a.Q)2 )1 (3-40)


So the effect of strain is to broaden the peak by aoQ with an accompanying decrease in

the Bragg amplitude so that the integrated intensity of the peak is conserved. It is evident
that strain broadening is much more pronounced in the higher order peaks.


The Powder Pattern


If the orientation of the crystallites is random, the sample is called a powder. The
scattering will then be a function of the magnitude of Q only. Assuming a Gaussian








lineshape for the non-powder averaged Bragg reflection, the powder averaged structure
factor can be calculated


(Q) = (Q-Q0)21
Q (27 2)3/2 < exp- 22 > (3-41)

Here the brackets indicate an average of the structure factor over all orientations of the
Bragg center, Q0, relative to the scattering vector, Q. The form factor which has been

incorporated into A(Q) is assumed to be isotropic. Without loss of generality, Q is
chosen to lie in the z direction:


Q = Q Z, Qo = Qosin(0)cos(()) + Qosin(0)sin( )) + Q0sin(0)2 (3-42)


Here 0 and 4 are the conventional spherical coordinates. The average over 4 and 0
yields


Ao (Q-Qo)2 (Q+Q)2
S(Q)= Q [exp -Q +exp [- ) (3-43)



The second exponential is negligible for all positive values of Q.
There is an additional factor called the multiplicity, mnh which arises from the

fact that Bragg peaks having different indices h, k, 1 appear in the same location in the
powder spectrum. The intensity is proportional to the degeneracy of the Bragg
reflection. As examples, the multiplicities of several peaks for a simple cubic lattice are
listed below
(hkl)= (100) m= 6 (3-44)
(110) m= 12
(321) m= 48








It is possible to identify different crystal structures by comparing the relative amplitudes
of the observed Bragg peaks due to differences in the multiplicities of the different
structures. This will be discussed in greater detail in Chapter 6.
If the effects of temperature, strain, and powder-averaging are included, the
resulting structure factor for a single Bragg peak is
S(Q) = Ah (Q) exp[-( 02 (3-45)
mlw e F(Q)2 exp(-Q2)
where Ag(Q) = Ao ,) QQ
4[2(Q) QQ%
and o(Q) =[ o + ( oQ)2 '

The application of these results to the block copolymer SAXS spectra will be discussed
in Chapter 6 following a review of the block copolymer literature in Chapter 4 and a
discussion of the experimental apparatus in Chapter 5.











CHAPTER 4
BACKGROUND AND LITERATURE REVIEW OF BLOCK COPOLYMERS


The polymer molecule is a long flexible chain composed of covalently bonded

repeat units called monomers. Polymeric materials have long been of interest to chemists

and materials scientists due largely to their advantageous mechanical properties and were

only much later recognized as an excellent system for the study of physics. Major

experimental tools that have proved to be extremely useful in recent studies of polymer

structure, morphology, and dynamical behavior include small angle neutron scattering

( SANS ) and synchrotron x-ray diffraction. Outstanding theoretical advancements have

been made, especially through the application of renormalization group ideas to the study

of chain conformations, due primarily to the original work by de Gennes in this field.22


Polymers--General Background


The size of a single polymer molecule is typically described by the

polymerization index, N,
N=- (4-1)
MO
where M is the mass of the entire chain and Mo is the mass of a single monomer.

Generally, a polymer sample is composed of a distribution of chains having varying total

masses. The mass distribution can be characterized by three parameters: the number and
weight average molecular weights, M. and M,, respectively, and the polydispersity

index, n. The number average molecular weight is defined as
IN. M.
Mn =, (4-2)
Mf. XN.








Here Ni is the number of polymer chains of mass, Mi, in the distribution. Similarly, the

weight average molecular weight is given by
Ni, Mi2
M. = (4-3)

The polydispersity is a measure of the width of the mass distribution and is defined to be
the ratio of Mw and Mn. It is easily related to the standard deviation of the distribution,

o, through the equation
S= (< M2 >- < M >2)12 = M, (n-l)l2 (4-4)

Together the above parameters describe the mass distribution of a collection of
polymer chains. The size of individual chains is described by the end-to-end distance, r.
If each monomer is of length I then the rms value of r can be determined by considering a

polymer chain as a random walk containing N steps of length a. It is well known2 that

for an unconstrained or "ideal" random walk the result is

= N P (4-5)
Within this model, the entropy of a single polymer can be calculated and at fixed end to

end distance is
3r2
S(r) = S(0) 2 N (4-6)

From the free energy relation, F=U-TS, it is evident that the chain behaves as a Hooke's

Law spring, when the entropy term dominates the free energy. The effective spring

constant increases with increasing temperature. This illustrates the physical basis for the

contraction of polymer materials which occurs upon heating.

The above calculation is flawed in that chain conformations are allowed in which
the chain "loops back" on itself so that two or more monomers occupy the same physical

space. Calculations which do not allow these conformations are called self avoiding
walks ( SAWs ) and, for dilute polymer solutions, yield a result similar to that for the

ideal random walk:








1 = INv (4-7)

Here v is called the Flory exponent. Theoretical and experimental studies show v=3/s5.s

In the concentrated limit, chains can no longer be considered as independent

random walks and the result, although much more difficult to obtain22 is the same as that

for the ideal random walk, Equation 4-5. Polymer chains are therefore swollen in dilute

solutions containing a good solvent and exhibit ideal ( i. e. random walk ) conformations

in the limit of a concentrated polymer solution or a pure polymer melt.
The radius of gyration, R,, is a parameter conveniently measured in both light

scattering and small angle x-ray scattering experiments and is defined as
Mr? 1/2
Rg I (4-8)


For an ideal random walk the radius of gyration is equal to one sixth of the rms end-to-

end distance.82


The Glass and Melting Transitions


Many polymer systems exhibit a melting temperature, T., above which the

molecules are in constant relative motion. Below Tm, the polymer chains crystallize by

forming helical or layered structures. In many cases, "bulky" monomers prevent efficient

stacking of the polymer chains so that crystallization is prevented. These materials
exhibit a glass transition temperature, Tg. Below T,, relative motion between chains is

hindered by entanglement effects. Often small molecules called plasticizerss" are added
to the material to increase flow by providing lubrication and lowering T,.

Above the glass or melting temperatures and in the absence of chemical

crosslinking, polymer molecules are constantly moving relative to one another. For a

long time, the mode of interdiffusion for polymer molecules was not well understood.








Presently, this process has been clarified and is described by the reputation model

developed by de Gennes.22 Within this model, polymer chains move by "worming" their
way through an open ended tube in which they are partially encased. The mean time

required for a single chain to reptate through a length of tube equivalent to its own length
is called the terminal time, ,, and the following dependence on the polymerization index,

N, has been observed:22
S- NX x = 3 3.3 (4-9)

The translational diffusion coefficient, D,, is given by
2
D N-2 (4-10)

assuming gaussian chain statistics.


Phase Separation in Polymer Blends


Many of the interesting and useful properties of polymer materials result from

segregated structures in polymer mixtures where the component polymers are

incompatible. In this section, the phase separation process in polymer blends is discussed

as an introduction to the more complex block copolymer case.

Many of the features of polymer phase separation are elucidated in a mean field

treatment developed by Flory21 and Huggins.83 Within this model, the free energy of a

mixture of polymers A and B is expressed in terms of the Flory-Huggins interaction
parameter, x:

X=k (EAB-( EAA+EBB) (4-11)

Here Exy is the energy corresponding to an X-Y nearest-neighbor configuration and kBT

is the thermal energy. The polymer system is typically assumed to be incompressible84
so that the volume fractions of A and B monomers can be written: ^A = ( and








(g = 1-0. The free energy for the mixture, including entropy of mixing terms,22 is
I1-1
F=kBTN lnc +kBTNl n(l-0)+X (l-I-) (4-12)

Here and are the A and B chain concentrations, respectively, in dimensionless
NA B

units. It is trivial22 to construct the phase diagram for this system including both the

coexistence and spinodal curves. The result for the symmetric case is very similar in

form to the phase diagram illustrated in Figure 2-1.

From the phase diagram it is evident that at temperatures above the coexistence

curve the equilibrium state of the system is a homogeneous mixture. Below the

coexistence curve, the equilibrium state consists of phase separated A- and B-rich

regions. As described in Chapter 2, many interesting kinetic effects can be observed in a

system exhibiting this type of phase diagram. Specifically, upon lowering the

temperature through the coexistence temperature to a final value in the metastable region,

there exists an interfacial energy associated with formation of A- or B-rich droplets. The

mechanism of transformation is nucleation and growth. When the final temperature is in

the unstable region of Figure 2-1, the interfacial energy disappears and transformation

occurs through spinodal decomposition. However, as mentioned in Chapter 2, a sharp

spinodal line is an artifact of mean field theories such as the Flory-Huggins model and

therefore its true physical significance is unclear.

Nucleation and spinodal decomposition have been observed in studies of phase

separation in blends of polystyrene and poly-( vinyl methyl ether ) by Nishi et al.85 They

used light transmission, optical microscopy, and NMR techniques. They were able to

identify a range for the spinodal point above and below which morphological structures

corresponding to nucleation and spinodal decomposition, respectively, were observed. It

is expected that qualitatively different behavior should be observed for phase separation

in polymer blends in comparison to binary alloys since diffusion in polymer systems is

believed to occur through the reputation mechanism. Reptation yields a chain mobility








which is strongly dependent on the reciprocal space vector, Q, and, as a result, alters the

equations of motion for the system. Nevertheless, Nishi's group found that early time

spinodal decomposition behavior agrees well with Cahn's linearized model of spinodal

decomposition,86 a result which is expected in the presence of long-range interactions,

but never observed in metals. McMaster87 also observed a spinodal point in his light

scattering study of polymer blends. In addition, the coarsening behavior observed

following quenches into the metastable temperature regime, was consistent with the

Lifshitz-Slyozov theory55 described in Chapter 2. However, coarsening following

quenches into the unstable regime occurred much more rapidly. McMaster accounted

for this behavior, invoking a viscous flow mechanism based on models developed by

Tomotika.88

Much work has yet to be done on phase separation in polymer blends, but present

results indicate that studies of these systems, because of their slow molecular diffusion,

can yield valuable information toward understanding kinetic effects.'


Block Copolvmers Bulk Properties


As introduced in Chapter 1, the block copolymer ( BCP) is composed of two or

more chemically distinct polymer chains joined by a covalent bond. When the block

chains are incompatible, phase separation similar to that occurring in polymer blends

takes place. The fundamental properties of bulk block copolymers are summarized in a

review article by Hashimoto et. al.84 As described in Chapter 1, block copolymers
exhibit a dissolution temperature, Td, below which microphase separation transition

( MST ) results in the lamellar, cylindrical, and spherical microstuctures shown in Figure
1-2 and above which the system is homogeneous. The morphology of the microstucture

has been found both experimentally23 and theoretically89 to depend only on the fractional

composition, f, of the block copolymer and is believed to be independent of total








polymerization index, N. In addition, the microphase separated domains are known to

form macrolattices: a lamellar arrangement, a 2-D hexagonal closed packed array of

cylinders and a cubic lattice of spherical domains.

Early theoretical work toward understanding this process was performed by

Meier.90 The driving force for phase separation in a BCP is the repulsive interaction

between the two types of monomers. Since the polymer system is highly

incompressible,28 only those domain configurations which fill all space are allowed. If

the size of the domains exceed the sums of the radii of gyration of the block chains

composing the domain, then that chain must be stretched to satisfy the requirement of

uniform space filling. This results in a loss of conformational entropy and therefore

limits the size of the phase separated domains. There is an additional entropy loss due to

confinement of the junction points (the location of the covalent bond between block

chains) to the interfacial region at the surface of the domain. Meier first identified these

three factors which determine the equilibrium morphology and dimensions of the system.

In addition, Meier recognized that the domains themselves interact through a

potential of entropic origin which in turn is responsible for the macrolattices which the

microdomains are known to form. The physical basis for this interaction can be

understood in the following way. Consider an AB diblock copolymer which forms

spherical domains of the A block in a B matrix. If two A spheres are too far apart, then a

density deficiency is created between them which must be filled by stretching B chains.

In this case, the free energy is increased due to the loss of conformational entropy of the

stretched B chains. Similarly, if two A domains are too close, the volume available to

the B chains between them is too small, again resulting in an increased free energy due to

the loss of conformational entropy. As discussed earlier in this chapter, the contribution

to the free energy from stretching or compressing a polymer chain is quasi-elastic in

form.









One of the first attempts to predict the dimensions of the equilibrium structure is

that of Helfand and Wasserman.91 Within their model, the structure is composed of

chains individually constructed from self avoiding random walks confined to the

appropriate domain space. In the limit of strong segregation for the lamellar

morphology, they found the following scaling relation between the polymerization index,

N, and the microdomain periodicity, d:
d N, 0 = 0.636 (4-13)

Similar results for the value of 0 were obtained for the cylindrical and spherical

morphologies.9293

The result given in Equation 4-13 was supported by studies of the polystyrene-

polyisoprene system by Hashimoto et al.94,95 They cast polymer films from solutions

having fractional compositions that yield the lamellar and spherical structures at varying

total degrees of polymerization. The films were studied with the small angle x-ray

scattering technique. From the SAXS profiles, the periodicities of the lamellar

structures, D, were calculated as well as the lattice constants and spherical radii, R, of

the spherical structures. In addition, Hashimoto's group studied the systematic deviations

from Porod's Law in the high Q region of the scattered profiles ( discussed in Chapter 3)

to estimate the thickness of the interfacial region, 4, separating polystyrene and

polyisoprene domains. Their results indicate the following scaling relations for the

lamellar samples:
d Na (4-14)
S-N

For the spherical structure, the same relations were found to hold and, in addition, the

following empirical relationships were observed:
R N4 (4-15)

p,- N'13








In these equations, pj is the density of junction points on the surface of the spherical

microdomains calculated from the sphere radius, the bulk density of polystyrene, and the

molecular weight of a single polystyrene block. For both morphologies, the periodicity

scales with molecular weight in fair agreement with theoretical predictions.93

The exponent in this relation is greater than the exponent of 1/2 predicted for the
radius of gyration, R, of the chain in a homogeneous melt.22 This indicates that the

individual chains are stretched in the direction perpendicular to the domain surface. A

more recent study confirms this assertion. Hasegawa et. al.9 studied lamellar

polystyrene-polyisoprene using small angle neutron scattering ( SANS ). By deuterating

the polystyrene block of the polymer, they were able to determine the radius of gyration

of these chains from the SANS profile. In their solvent cast films, the lamellae

preferentially lie parallel to the film surface. By scattering from the film edge and
surface they could measure R, both parallel and perpendicular to the domain surface,

respectively. Their results indicate an elongation of the chains by 60% perpendicular to

the lamellar surface and a contraction of 30% in each of the parallel directions.

A vast amount of work on similar systems also supports these scaling relations for

lamellar,25.97,98.99. '0101 cylindrical, and sphericallo0lo02z,03,104.l'1 microdomain

morphologies in the strong segregation limit.

Leibler89 presented a Landau mean field35 analysis of the MST occurring in an

AB diblock copolymer having a fractional composition f, a Flory-Huggins interaction
parameter X, and a total polymerization index N. The order parameter Nf(r), was defined
to be the local density deviation of the A monomer, WA(r), from its mean value in the

homogeneous state; W(r) = < yA(r) f >. The static structure factor, S(Q), is defined as


S(Q) = f exp( -i Q-r) < AA(r) AA(O) > d3r


(4-16)








Leibler expands the free energy in powers of the order parameter. Using the random
phase approximation,22 S(Q) and the free energy are evaluated to fourth order in W(Q)

( the fourier transform of V(i) ). By assuming that the minimized free energy near the

transition is dominated by fluctuations with Q = Q* (Q* Rg- ), Leibler is able to

evaluate the coexistence and spinodal curves in the (XN) f plane and determine the

limits of stability of the different polymer morphologies.

The stable morphology is found to depend not only on the fractional composition,

f (as shown in figure 1-2 ), but also on the temperature. Such transitions, i. e.. from

spherical to cylindrical to lamellar, with changing temperature were later observed

experimentally.23 In addition, Leibler argues that the stable macrolattice for the

cylindrical morphology is the two dimensional hexagonal close-packed structure (HCP)

and for spherical domains is the BCC structure. There have since been reports of a

variety of different macrolattices ( BCC as well as FCC and SC) observed for the

spherical domains.'06

The above formulation yields a scaling relation between the periodicity, d = Q*-',

and the polymerization index, N,
d N1I (4-17)

for the weak segregation limit (where the density profile is a varies smoothly ), distinct

from the relation derived for the strong segregation limit, where well defined interfaces

exist ( Equations 4-13 and 4-14 ). Recent support has been given to this scaling relation

by Chakrabarti et. al.107 who performed Monte Carlo simulations using a two

dimensional lattice model composed of restricted self-avoiding random walks. Their

results and experimental measurements by Shibayama et al.108 also yield the relation

given in Equation 4-17.

Ohta and Kawasaki'09 extended Leibler's theory by including the effects of the

long-range elastic interaction. Using the random phase approximation in the strong









segregation limit they obtain results similar to those of Leibler, and calculate the correct

scaling relation for this limit (Equation 4-13).
More recently, Frederickson and Helfand10 have attempted to improve upon

Leibler's theory by including the effects of concentration fluctuations in the self-

consistent Hartree approximation."1 Their results are similar to those of Leibler, but
they find the transition temperature to be no longer linear in N, but proportional to N1+I,

where (3 is small. The differences between these predictions and those of Leibler are,

however, small and have not yet been resolved experimentally.

In addition to the three commonly observed morphologies: lamellar, cylindrical,

and spherical, a new morphology called the ordered bicontinuous double-diamond

(OBDD) structure has recently been observed23-27 in the polystyrene-polyisoprene

diblock copolymer. The structure is composed of tetrapod domains probably residing on

the double-diamond lattice. This structure was observed in SAXS and electron

microscopy studies when the volume fraction of polystyrene is in the narrow range

between 0.62 and 0.66. The OBDD morphology appears between the cylindrical and

lamellar morphologies in the accepted morphology scheme( Figure 1-2). This discovery

shows the possibility of finding more interesting and technologically useful

morphologies.


Bulk Block Copolymers Kinetic Studies


The theory of the kinetics of the MST is not well developed at the present time.

Kawasaki and Sekimotol12 present a complex model of this process within the reputation

model of polymer diffusion. These results are, however, still very preliminary.

Oono and Bahiana'13 employed a cell dynamical model to describe the MST in a

symmetric diblock copolymer in two dimensions. The partial differential equation used

to describe the time development of the order parameter was a modification of the Cahn-









Hilliard equation.' The two dimensional patterns resulting from their computations

accurately reproduced those observed in electron microscopy studies.84 Using the results
of their computations, dimensional analysis, and comparisons to equilibrium theories
89.109 they obtain a relation between the exponent, 0, describing the equilibrium

periodicity of the microphase structure (Equation 4-13) and the exponent, 4, describing

growth of the microdomains during spinodal decomposition:
L-to (4-18)

That relation is
8 = 2) (4-19)
This implies that in the strong segregation limit, where 0 is believed to be 2/3 for the

lamellar morphology, f = 1/3, in agreement with the Lifshitz-Slyozov result described in

Chapter 2.


Block Copolymer Solutions


The first observation of an ordered structure in block copolymer solutions

occurred in 1966 by Vanzo.114 In studies of the optical reflectance of diblock copolymer

solutions as a function of increasing polymer concentration, C, a critical concentration,

C*, was found at which point a sharp increase in reflectance was observed. In addition,
above C the peak in optical reflectance moved to larger wavelength with increasing C.

From these data Vanzo postulated the presence of an ordered structure at concentrations

above C with an identity period that increases with increasing polymer content.

Early work by Sadron and Gallot25 helped to clarify these effects. They studied a

polystyrene-polybutadiene diblock copolymer mixed with styrene monomer, a selective
solvent ( one in which only the styrene blocks are soluble). After the solutions were

prepared they were illuminated with UV light to polymerize the styrene monomers.

Through this "post-polymerization" technique the polymer solutions are converted into a








gel for electron microscopy studies presumably without altering the microstructure of the

solution. The resulting sample were stained with osmium tetroxide and studied with

transmission electron microscopy (TEM). From these experiments, Sadron and Gallot
identified two critical concentrations, C1" < C"'. Below C,*, the mixture was

homogeneous. In the region between C1" and C2*, cylindrical polybutadiene domains

were observed in a disordered arrangement. Above C2*, the cylindrical domains were

observed in and ordered arrangement. From these results, it was inferred that C2*

corresponds to the quantity C* observed by Vanzo.
Sadron and Gallot25 and Skoulios97 also observed morphological transitions

(i. e. from spherical to cylindrical, or cylindrical to lamellar microstructures ) with

changing polymer concentration in solutions. Using the same "post-polymerization"

technique they were also able to observe morphological transformations upon altering the

fractional composition, f, of the polymer chain.

Pico and Williamsn15 studied the ordering transition in triblock copolymer

solutions using theological measurements. A transition from non-Newtonian to

Newtonian viscous flow was observed upon increasing the polymer concentration above

the measured critical value.

The properties of BCP solutions have been systematically explored by many

researchers, 03, 116.117 using a variety of techniques, but perhaps the most complete

description of these complex systems is presented in a series of five papers by Hashimoto

et al.28, 8,119,120.121 They performed theological and small angle scattering measurements

of the equilibrium structure of diblock copolymers in both selective and nonselective

solvents as a function of temperature and polymer concentration. A brief discussion of

the equilibrium and kinetic results reported by Hashimoto et. al. is given here.








Equilibrium Measurements


The first two papers of this series28 118 report studies of the diblock copolymer

polystyrene (PS )-polybutadiene (PB ) in the selective solvent n-tetradecane (C14)
which dissolves the PB block. The BCP had a number average molecular weight, M. =

5.2 x 104 and a fractional composition, f = 0.30. The solutions studied had the following

polymer concentrations: 8, 11, 20, 35, and 60 wt.% polymer. At these concentrations

and at this fractional composition the MST results in spherical PS domains within a
PB/C14 matrix. X-ray scattering profiles were acquired at varying temperatures upon

heating and cooling. The x-ray spectra were corrected for parasitic scattering and

collimation errors resulting from instrumental resolution.

In these experiments, Bragg peaks were evident in the low temperature spectra for

all of the solutions in the relative locations Nl/, "f2, '3, and v4 after collimation

correction. The lattice constants were determined from the first Bragg peak location and

the average radius of the PS spheres was determined from the location of the first

maximum in the spherical form factor as described in Chapter 3. It was speculated that

the macrolattice of PS domains was simple cubic ( SC ) rather than body-centered cubic

( BCC) from comparisons of the calculated and stoichiometric PS volume fractions for

these structures. This type of calculation is described in detail in Chapter 6.

As the temperature, T, for each solution was increased, the Bragg peak amplitude

decreased continuously while the full width at half maximum ( FWHM ) remained

unchanged. In addition, the calculated lattice constants decreased slowly with increasing
temperature. At a polymer concentration dependent temperature To, the Bragg amplitude

decreased abruptly, accompanied by an increase in the FWHM and a drop in the lattice
constant. As the temperature was increased further above T,, the peak amplitude and

lattice constant continued to decrease, while the width increased rapidly. Finally above
T = Td, no further evidence of the peak structure remained.








Hashimoto et al. associated the temperature To with the disordering of the

macrolattice resulting in a "liquid-like" arrangement of the PS spheres. The temperature,
Td, is identified as the dissolution temperature at which point the PS domains totally

dissolve. In order to clarify these interpretation, theological measurements were made on

the polymer solutions as a function of temperature. Two transitions in the mechanical
properties were identified. Upon raising T above To, a transition from non-linear plastic

flow to linear non-Newtonian flow was observed in support of their identification of T,.

Also, at T = Td, a transformation from linear non-Newtonian flow to linear Newtonian

flow was seen in agreement with earlier studies15 identifying this behavior with the

dissolution point.

The authors identify a number of quantitative trends in the thermal behavior of

the system. The dependence of the apparent lattice spacing, a, on temperature both

above and below the ordering point were consistent with the following empirical relation
above and below the discontinuity at T,:

a T- (4-20)

The discontinuity in a at T = To is thought to be due to the change in structure occurring

at this temperature. Above T,, the peak is no longer a Bragg peak, and therefore,

determining a lattice constant from the peak position is erroneous. The decreasing trend

of a with T is the result of two factors. First, the PB chains between the domains

contract upon heating to increase their conformational entropy. Second, at higher
temperatures the effective interaction, given by the Flory-Huggins parameter, X 1/kBT,

is reduced resulting in greater intermixing of the PS and PB chains which in turn causes a

contraction of the lattice. The authors believe the second of these two is the more

important effect at all temperatures.
The Bragg widths below To appear to be independent of temperature. The

authors offer an interesting explanation for this result. The domain-domain interaction in

BCP solutions is assumed to be much the same as described by Meier for the pure








polymer system, being quasi-elastic. The change in free energy, AF, due to small

elongations, Aa, of the lattice constant from its equilibrium value can therefore be

expressed as
AF f(C, T) (Aa)2 (4-21)

Here f(C,T) the spring constant of the PB chains, is a function of the polymer

concentration, C, and temperature. The authors argue that if the distribution of strains

satisfies the Boltzmann distribution, then the distribution of lattice constants is
characterized by a gaussian form having a width parameter, o,,
kBT
a2 (CT)-f(CT) (4-22)

The "spring constant" for the PB chains is entropic in origin as described earlier in this

chapter so that
f ( C, T) kBT (4-23)

Together 4-22 and 4-23 imply a strain distribution which is independent of temperature.

Therefore the Bragg peak width is expected to be independent of temperature when strain

is the dominant contributor to the width.
The authors propose the following intuitive explanation for disordering of the

macrolattice. As the temperature is increased and there is greater intermixing of the PS

and PB chains, the PS blocks extend further into the interstitial region between the

spheres. The PB chains are thus relaxed by intermixing and the effective potential well

of equation 4-21 becomes more shallow until melting of the macrolattice results. When

the demixing becomes pronounced, the domain-domain interaction is no longer purely

entropic. The energetic contribution to the interaction is claimed to be the cause of the

increasing trend in the peak width with temperature just below the dissolution

temperature.

Hashimoto et al. also observed several trends with increasing polymer
concentration. Both To and Td increased with increasing concentration due to reduced








screening of the PS-PB interaction which results in a higher Flory-Huggins interaction

parameter, X. The measured lattice constants were consistent with the scaling relation
a C-'" (4-24)

Again, the data extend significantly less than a decade in both a and C. The PS sphere

radius, R, was seen to increase slowly with increasing C. This trend was said to be the

result of a shifting in the relative weights of the energetic and entropic terms in the free

energy. At higher concentrations, more chains are added to each domain increasing the

sphere radius. As a result the PS chains are stretched and some conformational entropy

is lost, but the interaction energy from the surface to volume ratio is reduced.


Time-Dependent Experiments


As an extension of their earlier work on equilibrium properties, Hashimoto et. al.

performed what appear to be the only kinetic studies of the block copolymer system to

date.122,123.124 They again studied PS PB diblock copolymer in the selective solvent C14.
The fractional composition was f = 0.30, and the molecular weight was Mn = 5.2 x 104.

The polymer concentrations studied were 20, 25, 30, 35, and 40 wt.% polymer in

solvent. They attempted to measure the polymer diffusion coefficient as a function of

temperature by performing temperature jump experiments using the time-dependent

small angle x-ray scattering technique. These measurements are described in some detail

to provide a comparison with the experiments reported in this thesis.

The polymer solutions were initially held in equilibrium at room temperature.

The temperature was then rapidly raised to a point above the dissolution point. With the

thermodynamic driving force for the MST removed, it was thought that the system would

relax to the homogeneous state through diffusion of the center of mass positions of the

diblock chains. By studying the SAXS profile as a function of time, it was their goal to
measure the diffusion constant, DC, for the diblock chains in solution.








In fact, even at elevated temperatures, the interaction between PS and PB

monomers will still affect the dissolution process. In addition, the diffusivity of the

whole chain is a function of the diffusivities for each of the individual blocks. As a
result, the measured quantity is an effective diffusivity, De.

The polymer solutions were placed in an aluminum sample cell, which was sealed

with mica windows, and held at room temperature. The sample cell was then rapidly

transferred manually to a large copper heating block which was controlled at the desired

final temperature. The sample temperature exhibited a sigmoidal variation with time and

a time constant of 6 seconds. Placement of the cell in the heating block triggered the

start of data acquisition. Sixty-four consecutive scattered spectra were taken in real time

following the temperature jump, each with an exposure time of 2 seconds. Due to poor

counting statistics, temperature jumps to each final temperature were repeated 10 times

and for each iteration, the spectra corresponding to the same time slice were summed.

The resulting SAXS profiles exhibit a first-order peak which decays rapidly with

time following the temperature jump. A relation between the scattered intensity, I(Q,t),
and Df. was derived involving three assumptions: an isotropic particle distribution,

Fickian diffusion of the diblock chains, and no center of mass motion for the

microdomains while they dissolve, i. e.. no macrolattice disordering. Their result is

I(Q,t) = I(Q,t=0) e-2 Q2 Dl t (4-25)
From this expression, the intensity decays exponentially with time and so D.f can be

determined from the slope of the semi-log plot.

The interpretation of these results is complicated by the speed with which

transformation takes place. Much of the transition occurs non-isothermally. In order to

account for these non-isothermal effects, a crude model for the time development of the

scattered intensity was derived and is described by the following recursion relation:

I(Q,til) = [ I(Q,) I(Q,T) ] e-2 Q2 Ddf(T) At + I(Q,Tj) (4-26)








Here the temperature jump is divided into time increments, At. The intensity at time, tj.

is determined from the intensity at the previous time increment, tj. During each

increment, the intensity moves toward the equilibrium intensity distribution, Ie

corresponding to the temperature Tj which is the temperature of the sample at time tj.

In order to further simplify the problem, D, is assumed to follow an Arrhenius

temperature dependence:
Dff = Do exp (-AH ) (4-27)

Their model for the temporal development of the intensity at any scattering vector, Q, is
a function of the two parameters, Do, and the activation energy, AH,.

The authors plot the logarithm of the scattered intensity at a specific scattering

vector, Q*, near the first-order maximum as a function of time following the quench.

The data are generally non-exponential at early times becoming exponential at later

times. This behavior is accurately reproduced in their model described above. The

authors present two methods of extracting the desired parameters from the data. First,
the diffusion constants Def are determined from the late time exponential region of the

data and from a plot of Dff versus temperature, AH, and Do can be calculated. The

second method, involves varying Do and AH. and individually matching the model

( Equation 4-26 ) to the measured data. The two methods gave similar numerical results
for AHa and Do. The data were found to be consistent with the following relation:

Do- C'.75 (4-28)

Results for the activation energy, AHa, show no consistent trend, but yield results for all

concentrations in the vicinity of 10 kcal / mole.

As can be seen from this brief discussion of the current literature, experimental

studies of BCP transition phenomena are still quite limited. These systems are rich in

their diverse microstructural morphologies and the transitions which occur between them





78


and the homogeneous melt. This complex behavior has been the stimulus for the series

of experiments reported here.













CHAPTER 5
THE POLYMER XRAY SCATTERING EXPERIMENT


In this chapter, the experimental apparatus is described beginning with a brief

overview of the data acquisition system. This is followed by a more detailed discussion

of the major components of the apparatus: the polymer x-ray scattering furnace and the

SAXS apparatus. The chapter is concluded with a discussion the experimental

procedures employed in these studies.


System Overview


The objective of these studies was to observe structural changes in block

copolymer solutions as a function of temperature and time by measuring the SAXS

profile. The data acquisition system which was devised for this purpose is shown in

Figure 5-1. The control system is centered around an IBM PC AT operating in the

ASYST environment. The programs and routines written to control the data acquisition

system are described in Appendix A. The computer controls the x-ray detector and

polymer x-ray scattering furnace through four devices controlled via the General Purpose

Interface Bus. These devices are a PAR 1461 detector controller, a Stanford Research

Systems Model DG535 digital delay generator, a Micristar temperature controller, and a

Keithley 197 digital multimeter.

The PAR detector interface controls the PAR 1412 XR position sensitive x-ray

detector. Upon receipt of commands from the computer, the detector interface relays

those commands to the detector, then stores up to 512 acquired spectra for DMA transfer






















































A schematic of the data acquisition system.


Figure 5-1









to the computer at a later time. The silicon diode array detector is evacuated using a

Leibold D8A two-stage roughing pump and requires a cooling system ( A Fisher Model

900 Circulator ). The details of the SAXS apparatus including the detection system are

discussed momentarily.

The sample temperature is monitored and controlled by a Micristar temperature

controller operating in the proportional mode of control. In this mode, the Micristar

compares the sample temperature to the programmed setpoint and outputs a DC voltage

between 0 and 5V which is proportional to the difference. This voltage is sent to a

specially designed heating circuit ( described in Appendix B ) which in turn controls the

current to heating resistors in the polymer x-ray scattering furnace. In this way, the

sample temperature is controlled with a stability of +- 0.3 oC. The details of the x-ray

scattering furnace are discussed momentarily. The sample temperature as determined

from the thermocouple voltage emitted by a thermocouple embedded in the sample is

also monitored by a Linseis Model L4100 chart recorder and the Keithley multimeter.

The time-dependent measurements were performed by annealing the BCP
solution above its dissolution point, Td, and then rapidly lowering the temperature to a

fixed point below Td and observing the structural transition through the SAXS profile.

This thermal quench is achieved by forcing a fixed amount of coolant rapidly through a

quenching channel in the sample mount. The measured coolant was injected into the

aperture of the quenching channel inlet to the furnace. Polypropylene tubing extending

from a cannister of helium was then connected to the quenching channel inlet. A

solenoid valve placed between the furnace and the helium was used to control the time

and duration of the helium burst which was initiates the quench. The solenoid valve is

controlled by the digital delay generator through a specially designed solenoid interface

circuit, a schematic of which is shown in Appendix C.








A description of the procedures performed in these experiments is given later in

this chapter. Now, a more detailed discussion of the major components to the data

acquisition system, the polymer furnace and the SAXS apparatus, is presented.


The Polymer X-ray Scattering Furnace


The polymer quenching furnace is composed of three parts : the furnace body, the

translational stage and the sample mount. The furnace body and the translational stage

were originally designed for use in other experiments and were used here with relatively

minor modifications. The sample mount, however, was specifically designed for these

experiments and will therefore be described in slightly more detail.


The Furnace Body


The body of the furnace is essentially a vacuum chamber composed of a stainless

steel base and an aluminum lid. These two parts are joined with an o-ring seal and a

number of bolts arranged in an octagonal pattern. The furnace base as seen from above

is shown in Figure 5-2.

The lid is a cylindrical aluminum shell with two aluminum windows for entrance

of the incident x-ray beam and exit of the scattered radiation. The windows are
aluminum plates with 21/4" diameter holes. The window frames are bolted to the lid for

easy replacement. The frames are sealed to the lid with o-rings. A number of window

materials were tried and the two optimal materials were beryllium and Kapton. Kapton

is the tradename for a polyimide material commonly distributed in sheets and used as

substrates or vacuum windows. Beryllium is the standard window material used in wide

angle x-ray scattering and offers the best percentage transmission ( 96.17% for a 5 mil

sheet). However, it is a horrendous parasitic scatterer in the region of interest in these



















































A top view of the polymer x-ray scattering furnace illustrating the relative
placement of its various features described in the text.


Figure 5-2









experiments. Parasitic scattering is defined as that unwanted component to the scattering

profile which is a result of scattering from slits and window materials. This quantity was

measured as the integrated number of counts per second above background. This

quantity for the 5 mil beryllium sheet was 8.0 times larger than for the 5 mil Kapton

sheet in the 0.05 to 1.0 degree range and 5.2 times larger in the 0.05 to 0.35 degree

range. Although the percentage transmission was slightly less (transmission = 91.13%)

for the Kapton, the reduction in parasitic scattering was much more important. The

Kapton windows were sealed to the window frame with Omegabond epoxy and resin.

The chamber was evacuated with an Edwards ETPG pumping station which

incorporated a roughing pump and a turbomolecular pump. The ultimate pressure

typically achieved under experimental conditions was between 1 and 10 millitorr. The

pressure was much lower ( below the level which could be read from the Varian

thermocouple gauge controller ) when the windows were replaced with solid aluminum

plates. For this reason, it was concluded that the ultimate pressure was being limited by

permeation through the Kapton windows. The vacuum pressure probably could have

been improved with thicker Kapton windows, but the pressure was satisfactory for these

measurements and further improvement was unnecessary.

The furnace body has a number of vacuum feedthroughs. These are shown in

figure 5-2 and include the following. An eight-pin electrical feed through is used to pass

current to the resistors which heat the sample. The feed through was a standard high

vacuum flange which bolted to the side of the furnace and was sealed with teflon tape.
Two 1/4" outer diameter, I/g" inner diameter copper tubes were passed through the

furnace wall and were sealed with standard Swadgelock fittings. These tubes were used

to pass coolant through the sample mount to initiate the thermal quench during the

kinetic measurements. One feedthrough passes four thermocouple connections into the

chamber: two chrome and two alumel. One chromel and one alumel connection are

used to form the thermocouple junction which compose a single thermal sensor. Both








sets of connections were used, one to monitor the sample temperature and one to monitor

the sample mount temperature. A Varian Type 0531 thermocouple gauge head was

attached in the location shown in figure 5-2 with teflon tape. The sensor was connected

to a Varian 843 gauge controller.
The chamber was evacuated through the 1/2 vacuum port shown in the Figure.

Two additional high vacuum flanges are available on the scattering furnace, but are not

being used at this time.


The Translational Stage


The base of the furnace body is cut into a dovetail shape. The base mates with a

brass translational stage, the bottom of which is itself cut into the dovetail shape in the

orthogonal direction. An additional brass plate mates with this piece, and by sliding the

dovetail within its track, two independent translations are provided for centering the

sample in the x-ray beam. The translational stages are mounted on a Huber Four Circle

Diffractometer ( the four independent orientation angles were a little overkill in a small

angle scattering experiment ) via the standard Huber Goniometer connecting ring. The

connecting ring can be translated vertically. Along with those translations allowed by the

brass translational stage, three degrees of freedom are provided for centering the sample

in the beam. The connecting ring is typically adequate for small samples or sample

chambers, but is inadequate for a furnace of this size. For this reason, appropriate

spacers were chosen and the vertical translation of the connecting ring was set so that the

furnace bottom rested on the spacers. Then four additional screws were used to further

secure the furnace to the diffractometer.

As stated previously, the furnace body and translational stage were originally
designed for use in the Cu3Au experiments being performed in the laboratory and were

used with minor modifications (i. e. installation of the quenching loop feedthoughs ).








However, the sample holder was designed specifically for these experiments. Many

iterations were required to optimize the performance of the sample mount. The

development of the sample mount is described in Appendix D. Only the final version is

described here.


The Sample Mount


The sample mount is composed of two parts, the heating unit and the sample cell

both of which are shown in Figure 5-3. the heating unit was mounted in the furnace and

remained there throughout the experiments, while the sample cell was replaced for each

polymer solution.

The sample cell is a 27 mm x 12 mm x 3 mm copper block. A slot of dimensions

2 mm x 5 mm was cut in the copper block to serve as the sample cavity. The polymer

solutions were held in this cavity by two 2 mil Kapton windows. The Kapton windows

were sealed using Omegabond high temperature epoxy. Two 1/16" stainless steel masks

were bolted to the front and back of the sample cell to provide structural support to the

seal. these masks were found to be essential in preserving the integrity of the seal at the

elevated temperatures used in these experiments.

The sample temperature was measured by inserting an Omegaclad K-type

thermocouple directly into the sample cavity through a 1/16" diameter hole. these

thermocouples are composed of one 3 mil chromel and one 3 mil alumel wire extended

parallel to each other and surrounded by a cylindrical stainless steel shell. This

cylindrical sheath is packed with ceramic powder which prevents contact between either

of the alloy wires and the outer shell. The metal coating and ceramic powder are sheared

off at both ends of a 2" long section. At one end, the leads are welded together to form

the thermocouple junction. at the other end, the wires are connected to the thermocouple


















































The final sample mount is illustrated in a front and top view. Evident in
this drawing are the quenching tubes, the heating resistors, the
thermocouple, the stainless steel masks which support the Kapton
windows, and the stainless steel brace which firmly clamps the sample
cell into the sample mount.


Figure 5-3




Full Text

PAGE 2

$1 ;5$< 6&$77(5,1* 678'< 2) 25'(5,1* ,1 %/2&. &232/<0(56 %\ &857,6 5$< +$5./(66 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 3

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

PAGE 4

)LQDOO\ WKDQN P\ IDPLO\ 7KH DGYLFH RIIHUHG E\ P\ SDUHQWV /HV DQG 1DQF\ OLNH *HW D MREf DQG WKH VWLPXODWLQJ TXHVGRQV SRVHG E\ P\ VLVWHUV /LVD .LP -HQ DQG 6WHIDQLH OLNH $UHQnW \RX GRQH \HW"f ZHUH JUHDG\ DSSUHFLDWHG %XW PRVW RI DOO DP JUDWHIXO IRU WKHLU XQIDLOLQJ VXSSRUW LQ DOO RI P\ HQGHDYRUV

PAGE 5

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f§%XON 3URSHUWLHV %ORFN &RSRO\PHUVf§.LQHWLF 6WXGLHV ,9

PAGE 6

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

PAGE 7

&RQFHQWUDWLRQ DQG 7HPSHUDWXUH 'HSHQGHQFH RI WKH 6WUXFWXUH %HKDYLRU RI WKH &U\VWDOOLQH &RPSRQHQW %HKDYLRU RI WKH $PRUSKRXV &RPSRQHQW .,1(7,& 0($685(0(176 $1$/<6,6 $1' 5(68/76 3UHOLPLQDU\ 'LVFXVVLRQ 7KH 2UGHULQJ .LQHWLFV 7KH 0DVWHU &XUYH 7KH &URVVRYHU %HKDYLRU 6KDOORZ 4XHQFK 2VFLOODWRU\ %HKDYLRU 6800$5< $1' &21&/86,216 $33(1',&(6 $ '$7$ $&48,6,7,21 352*5$06 % 7+( +($7,1* &,5&8,7 & 7+( 62/(12,' ,17(5)$&( 7+( '(9(/230(17 2) 7+( 6$03/( 02817 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ YL

PAGE 8

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nV UHODWLRQ IRU QXFOHDWLRQ RQ WZR GLPHQVLRQDO QXFOHDWLRQ VLWHV $Q LOOXVWUDWLRQ RI GLIIUDFWLRQ E\ WZR VFDWWHULQJ FHQWHUV DQG WKH GHILQLWLRQ RI WKH VFDWWHULQJ DQJOH 7KH VFDWWHULQJ IRUP IDFWRUV IRU LGHDO VSKHUHV KDYLQJ UDGLL DQG ƒ 9OO

PAGE 9

$ SORW VKRZLQJ WKH HIIHFW RI SRO\GLVSHUVLW\ RQ WKH VFDWWHULQJ IRUP IDFWRU IRU D JDXVVLDQ GLVWULEXWLRQ RI VSKHULFDO SDUWLFOHV KDYLQJ D ZLGWK SDUDPHWHU R O DQG ƒ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f DQG ZLGWK LQ SODQH f SURILOHV $ JUDSK RI WKH PHDVXUHG SDUDVLWLF VFDWWHULQJ 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ FRROLQJ 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ KHDWLQJ 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ FRROLQJ 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ KHDWLQJ 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ FRROLQJ 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ KHDWLQJ 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ FRROLQJ YP

PAGE 10

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r& 3ORWV RI WKH PLFURGRPDLQV UDGLXV DQG ODWWLFH FRQVWDQWV YHUVXV SRO\PHU FRQFHQWUDWLRQ DW DSSUR[LPDWHO\ r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

PAGE 11

$ SORW RI WKH ILQDO VSHFWUXP DV D IXQFWLRQ RI TXHQFK GHSWK IRU WKH 6% VROXWLRQ $ SORW RI WKH ILQDO VSHFWUXP DV D IXQFWLRQ RI TXHQFK GHSWK IRU WKH 6% VROXWLRQ $ SORW RI WKH ILQDO VSHFWUXP DV D IXQFWLRQ RI TXHQFK GHSWK IRU WKH 6% VROXWLRQ $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ *UDSKV RI WKH SHDN PD[LPXP DV D IXQFWLRQ RI WLPH IROORZLQJ D VKDOORZ TXHQFK XSSHU JUDSK f DQG D TXHQFK WR EHORZ WKH RUGHULQJ WHPSHUDWXUH ORZHU JUDSK f $ SORW RI WKH WHPSRUDO SHDN GHYHORSPHQW RI WKH VFDWWHUHG SURILOH *UDSKV RI WKH ILWWHG DPRUSKRXV SHDN SRVLWLRQ DQG ):+0 DORQJ ZLWK WKH %UDJJ ):+0 DQG WKH PHDVXUHG ODWWLFH FRQVWDQW YV WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ $ SORW RI WKH QRUPDOL]HG %UDJJ DQG DPRUSKRXV ILWWHG DPSOLWXGHV DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ $ SORW RI WKH QRUPDOL]HG %UDJJ ILWWHG DPSOLWXGHV DV D IXQFWLRQ RI WLPH IROORZLQJ TXHQFKHV IURP a r& WR r& r& r& DQG r& RQ WKH 6% VROXWLRQ $ SORW RI WKH TXDQWLW\ ,Q ,Q e fOff YV WKH QDWXUDO ORJ RI WKH WLPH IROORZLQJ TXHQFKHV IURP r& WR r& [

PAGE 12

r& r& DQG r& RQ WKH 6% VROXWLRQ $ PDVWHU FXUYH RI DOO RI WKH TXHQFK GDWD RQ DOO RI WKH VDPSOHV DV GHVFULEHG LQ WKH WH[W 7KH WLPH FRQVWDQWV W IRU WKH RUGHULQJ WUDQVLWLRQ IRU WKH 6% 6% 6% DQG 6% VROXWLRQV DUH SORWWHG DV D IXQFWLRQ RI TXHQFK GHSWK $ VFKHPDWLF LOOXVWUDWLQJ WKH SURSRVHG RULJLQ RI WKH REVHUYHG FURVVRYHU LQ RUGHULQJ EHKDYLRU *UDSKV RI WKH ILWWHG DQG QRUPDOL]HG %UDJJ DPSOLWXGHV DV D IXQFWLRQ RI WLPH IROORZLQJ TXHQFKHV IURP a r& WR r& r& DQG r& RQ WKH 6% VROXWLRQ *UDSKV RI WKH UDZ GDWD IURP D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ DW YDULRXV WLPHV IROORZLQJ WKH TXHQFK 3ORWV RI WKH ILWWHG %UDJJ ZLGWKV DQG SRVLWLRQV DORQJ ZLWK WKH QRUPDOL]HG LQWHQVLW\ = DV D IXQFWLRQ RI WLPH IROORZLQJ WKH TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ % $ VFKHPDWLF RI WKH KHDULQJ FLUFXLW &O $ VFKHPDWLF RI WKH VROHQRLG LQWHUIDFH $Q LOOXVWUDWLRQ RI WKH LQLWLDO VDPSOH PRXQW GHVLJQ [L

PAGE 13

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

PAGE 14

+LJK TXDOLW\ GLEORFN DQG WULEORFN FRSRO\PHU VROXWLRQV ZHUH SUHSDUHG 7KH VWUXFWXUH RI HDFK V\VWHP ZDV GHWHUPLQHG IURP WKH [UD\ VFDWWHULQJ SURILOHV DV D IXQFWLRQ RI WHPSHUDWXUH DIWHU ZKLFK NLQHWLF PHDVXUHPHQWV ZHUH SHUIRUPHG (DFK NLQHWLF PHDVXUHPHQW LQYROYHG DQQHDOLQJ WKH VDPSOH DERYH WKH GLVVROXWLRQ WHPSHUDWXUH DQG UDSLGO\ TXHQFKLQJ WKH VDPSOH WHPSHUDWXUH WR D IL[HG SRLQW EHORZ 7KH VXEVHTXHQW WUDQVIRUPDWLRQ SURFHVV ZDV REVHUYHG WKURXJK WKH [UD\ VFDWWHULQJ SURILOH 'XH WR WKH UHVROXWLRQ REWDLQHG DW WKH V\QFKURWURQ WKH VFDWWHULQJ FRQWULEXWLRQV IURP WKH RUGHUHG DQG GLVRUGHUHG VWDWHV DUH LGHQWLILHG DQG VHSDUDWHG IRU WKH ILUVW WLPH $V D UHVXOW VHYHUDO QHZ IHDWXUHV DUH REVHUYHG VXFK DV WKH SUHVHQFH RI ILQH VWUXFWXUH LQ WKH [UD\ VFDWWHULQJ SURILOH )DVW NLQHWLF PHDVXUHPHQWV UHYHDO WKDW WUDQVIRUPDWLRQ RFFXUV DV D WZRVWDJH SURFHVV DQG WKDW WKH RUGHULQJ WUDQVLWLRQ H[KLELWV DQ XQH[SHFWHG FURVVRYHU LQ EHKDYLRU FRQVLVWHQW ZLWK WZR GLPHQVLRQDO QXFOHDWLRQ ,Q DGGLWLRQ WKH YHORFLWLHV RI WUDQVIRUPDWLRQ DV GHWHUPLQHG IURP WKH NLQHWLF GDWD IROORZ WUHQGV H[SHFWHG IURP IXQGDPHQWDO WKHUPRG\QDPLF FRQVLGHUDWLRQV )LQDOO\ QRYHO RVFLOODWRU\ JURZWK RI WKH %UDJJ FRPSRQHQW LV REVHUYHG LQ WKH VKDOORZ TXHQFK OLPLW [P

PAGE 15

&+$37(5 ,1752'8&7,21 2QH RI WKH PRVW IDVFLQDWLQJ DQG DFWLYH ILHOGV LQ VWDWLVWLFDO SK\VLFV LV WKH VWXG\ RI SKDVH WUDQVLWLRQ NLQHWLFV 6SHFLILFDOO\ WKHUH LV PXFK LQWHUHVW LQ XQGHUVWDQGLQJ WKH QRQn OLQHDU DQG FRRSHUDWLYH SKHQRPHQD LQYROYHG LQ ILUVWRUGHU SKDVH WUDQVLWLRQV )237Vf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f 7KH H[FHOOHQW PHFKDQLFDO SURSHUWLHV RI WKHVH PDWHULDOV OHQG WKHPVHOYHV WR D YDULHW\ RI LQGXVWULDO DSSOLFDWLRQV 7KH LQWHUHVWLQJ VWDWLVWLFDO SK\VLFV LQYROYHG LQ XQGHUVWDQGLQJ WKHVH PDWHULDOV KDV DWWUDFWHG WKH DWWHQWLRQ RI PDQ\ QRWDEOH SK\VLFLVWV LQFOXGLQJ GH *HQQHV 'HE\H .UDPHUV DQG )ORU\ ([FHOOHQW UHIHUHQFHV RQ WKH VWDWLF DQG G\QDPLF SURSHUWLHV RI SRO\PHUV KDYH EHHQ ZULWWHQ E\ 'RL DQG (GZDUGV )ORU\ DQG GH *HQQHV

PAGE 16

7KH EORFN FRSRO\PHU PROHFXOH LV IRUPHG E\ MRLQLQJ WZR FKHPLFDOO\ GLVWLQFW SRO\PHU FKDLQV HQG WR HQG ZLWK D FRYDOHQW ERQG DQG LV WKH VXEMHFW RI WKH H[SHULPHQWDO VWXGLHV WR EH SUHVHQWHG KHUH $ W\SLFDO EORFN FRSRO\PHU LV LOOXVWUDWHG LQ )LJXUH 0XFK UHFHQW ZRUN KDV EHHQ GRQH WRZDUG XQGHUVWDQGLQJ WKHVH FRPSOLFDWHG V\VWHPV :KHQ WKH PRQRPHUV $ DQG % DUH LQFRPSDWLEOH $$ %% QHDUHVW QHLJKERU FRQILJXUDWLRQV DUH IDYRUHG RYHU $% FRQILJXUDWLRQV f WKH V\VWHP DWWHPSWV WR PLQLPL]H LQWHUQDO HQHUJ\ E\ IRUPLQJ $ RU %ULFK GRPDLQV WKHUHE\ UHGXFLQJ WKH VXUIDFH WR YROXPH UDWLR EHWZHHQ $ DQG % UHJLRQV 7KLV W\SH RI DUUDQJHPHQW OHDGV WR D ORVV RI FRQIRUPDWLRQDO HQWURS\ UHVXOWLQJ LQ DQ HTXLOLEULXP VWUXFWXUH WKDW LV D EDODQFH EHWZHHQ WKHVH WZR RSSRVLQJ IRUFHV $W KLJK WHPSHUDWXUHV WKH HQWURSLF IRUFH GRPLQDWHV DQG WKH V\VWHP LV D KRPRJHQHRXV PHOW DV VKRZQ LQ )LJXUH $V WKH WHPSHUDWXUH LV ORZHUHG WKH EDODQFH VKLIWV WR IRUPDWLRQ RI GRPDLQV DW WKH GLVVROXWLRQ WHPSHUDWXUH 7G 7KLV GRPDLQ IRUPDWLRQ LV VLPLODU WR WKH SKDVH VHSDUDWLRQ REVHUYHG LQ ELQDU\ DOOR\V +RZHYHU WKH MXQFWLRQ RI $ DQG % PRQRPHUV LQ WKH EORFN FRSRO\PHU OLPLWV WKH XOWLPDWH VL]H RI WKH SKDVH VHSDUDWHG UHJLRQV WR PLFURVFRSLF GLPHQVLRQV 7KLV SURFHVV LV DSSURSULDWHO\ WHUPHG PLFURSKDVH VHSDUDWLRQ DQG UHVXOWV LQ WKH GLIIHUHQW GRPDLQ PRUSKRORJLHV REVHUYHG LQ WKHVH PDWHULDOV 3KDVH VHSDUDWLRQ LQ ELQDU\ V\VWHPV KDV EHHQ ZHOO VWXGLHG DQG KDV OHG WR WKH IRUPXODWLRQ RI YHU\ JHQHUDO UHODWLRQV GHVFULELQJ WKH SKDVH WUDQVLWLRQ NLQHWLFV LQ FRQVHUYHG RUGHU SDUDPHWHU V\VWHPV 0LFURSKDVH VHSDUDWLRQ LV DQ LQWULJXLQJ DQDORJ RI WKLV SURFHVV LQYROYLQJ ERWK WKH FRPSOH[ SURFHVV RI SRO\PHU LQWHUGLIIXVLRQ DQG DQ LQKHUHQW FRQVWUDLQW WR WKH VL]H RI WKH SKDVH VHSDUDWHG UHJLRQV )RU WKLV UHDVRQ EORFN FRSRO\PHUV DUH XVHIXO V\VWHPV IRU EURDGHQLQJ RXU NQRZOHGJH RI NLQHWLF SKHQRPHQD DQG WHVWLQJ WKH OLPLWDGRQV RI SUHVHQW PRGHOV $ YDULHW\ RI GRPDLQ PRUSKRORJLHV KDV EHHQ REVHUYHG LQ EORFN FRSRO\PHU V\VWHPV WKURXJK [UD\ VFDWWHULQJ DQG WUDQVPLVVLRQ HOHFWURQ PLFURVFRS\ DQG WKH W\SH RI PRUSKRORJ\ ZDV IRXQG WR GHSHQG RQ WKH IUDFWLRQDO FRPSRVLWLRQ RI WKH LQGLYLGXDO EORFN

PAGE 17

7+( %/2&. &232/<0(5 02/(&8/( $$$$ $$$$%%%% %%%% 0,&523+$6( 6(3$5$7,21 )LJXUH 7KH GLEORFN FRSRO\PHU PROHFXOH DQG DQ LOOXVWUDWLRQ RI PLFURSKDVH VHSDUDWLRQ

PAGE 18

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f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f VWXGLHV RI SRO\VW\UHQH36fSRO\EXWDGLHQH 3%f GLEORFN FRSRO\PHU LQ WKH VHOHFWLYH VROYHQW QWHWUDGHFDQH &f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

PAGE 19

%/2&. &232/<0(5 '20$,1 0253+2/2*,(6 LQ $ 63+(5(6 % 0$75,; $ $% % % &
PAGE 20

$ ',625'(5(' $55$1*(0(17 2) 63+(5(6 $1 25'(5(' /$77,&( 2) 63+(5(6 )LJXUH $Q LOOXVWUDWLRQ RI WKH RUGHULQJ RI VSKHULFDO PLFURGRPDLQV RQWR D FXELF ODWWLFH

PAGE 21

0LFURGRPDLQ RUGHULQJ LQ EORFN FRSRO\PHUV LV DQ LQWHUHVWLQJ DQDORJ WR PHOWLQJ FU\VWDOOL]DWLRQ SKHQRPHQD DQG DV VXFK LV FKDUDFWHUL]HG E\ D QRQFRQVHUYHG RUGHU SDUDPHWHU 1&23f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

PAGE 22

SK\VLFDO SDUDPHWHUV WR H[SHULPHQWDOO\ PHDVXUHG TXDQWLWLHV &KDSWHU LV D EULHI UHYLHZ RI WKH UHOHYDQW OLWHUDWXUH RQ EORFN FRSRO\PHUV &KDSWHU LV D GHWDLOHG GHVFULSWLRQ RI WKH H[SHULPHQWDO DSSDUDWXV DQG SURFHGXUHV SHUIRUPHG LQ WKHVH VWXGLHV 7KH SDUDPHWHUL]DWLRQ DQDO\VLV DQG LQWHUSUHWDWLRQ RI WKH HTXLOLEULXP UHVXOWV DUH SUHVHQWHG LQ &KDSWHU LQ SUHSDUDWLRQ IRU GLVFXVVLRQ RI WKH NLQHWLF UHVXOWV LQ &KDSWHU )LQDOO\ WKH LPSRUWDQW FRQFOXVLRQV ZKLFK DUH GUDZQ IURP WKLV ZRUN ZLOO EH VXPPDUL]HG LQ &KDSWHU

PAGE 23

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c RU RWKHU H[WHQVLYH YDULDEOH f DERYH WKH WUDQVLWLRQ WHPSHUDWXUH 7F DQG WKHQ UDSLGO\ ORZHUHG WR D IL[HG YDOXH EHORZ 7F 7I ,I WKH TXHQFK RFFXUV LQ D WLPH VKRUW UHODWLYH WR WKH WLPH UHTXLUHG IRU WUDQVIRUPDWLRQ WKH V\VWHP ZLOO VWLOO KDYH WKH

PAGE 24

7(03(5$785( &20326,7,21 )LJXUH $Q H[DPSOH RI D SKDVH GLDJUDP IRU D ELQDU\ DOOR\ H[KLELWLQJ D SKDVH WUDQVLWLRQ

PAGE 25

VWUXFWXUH RI WKH LQLWLDO SKDVH 7KLV SKDVH LV QRZ XQVWDEOH RU PHWDVWDEOH f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f V\VWHPV $Q H[DPSOH RI D &23 V\VWHP LV D ELQDU\ DOOR\ ZKLFK H[KLELWV SKDVH VHSDUDWLRQ $ W\SLFDO QRQFRQVHUYHG RUGHU SDUDPHWHU 1&23 f V\VWHP LV D ELQDU\ DOOR\ ZKLFK H[KLELWV DQ RUGHUGLVRUGHU WUDQVLWLRQ Q $Q H[DPSOH RI VXFK D V\VWHP LV &X$X ,Q WKH GLVRUGHUHG VWDWH WKH FU\VWDO VWUXFWXUH IRU WKLV V\VWHP LV IDFHFHQWHUHG FXELF )&& f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f ZKLFK IROORZV GLUHFWO\ IURP WKH

PAGE 26

GLVFRQWLQXLW\ LQ WKH IUHH HQHUJ\ LV WKH H[LVWHQFH RI D ODWHQW KHDW L H WKH KHDWV RI PHOWLQJ DQG YDSRUL]DWLRQ f ,Q DGGLWLRQ WUDQVLWLRQV LQYROYLQJ D FKDQJH RI V\PPHWU\ IURP RQH V\PPHWU\ JURXS WR DQRWKHU WKDW LV QRW D VXEJURXS RI WKH ILUVW FDQ EH LGHQWLILHG DV )237V EHFDXVH LW LV QRW SRVVLEOH IRU WKH V\VWHP WR JR IURP RQH SKDVH WR WKH RWKHU FRQWLQXRXVO\ 'LVFRQWLQXLW\ LQ WKH VHFRQG DQG KLJKHU GHULYDWLYHV RI WKH IUHH HQHUJ\ ZLWK UHVSHFW WR WKH RUGHU SDUDPHWHU ZLWKRXW VLQJXODULW\ LQ WKH ORZHU GHULYDWLYHV LPSOLHV VHFRQG DQG KLJKHU RUGHU WUDQVLWLRQV 6HFRQG DQG KLJKHU RUGHU DOVR FDOOHG FULWLFDO RU PXOWLFULWLFDOf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n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n JUDLQHG IUHH HQHUJ\ LQ SRZHUV RI WKH RUGHU SDUDPHWHU ,Q JHQHUDO WKH RUGHU SDUDPHWHU PD\ EH D VFDODU YHFWRU RU D WHQVRU DV LV WKH RULHQWDWLRQDO RUGHULQJ RI OLTXLG FU\VWDOV 7KH SURWRW\SLFDO VKDSH RI WKH IUHH HQHUJ\ FXUYH IRU D VFDODU RUGHU SDUDPHWHU DQG IL[HG FRPSRVLWLRQ LV VKRZQ LQ )LJXUH IRU VHYHUDO WHPSHUDWXUH UDQJHV )^^!` KDV WKH GRXEOH ZHOO VWUXFWXUH FKDUDFWHULVWLF RI SKDVH WUDQVIRUPDWLRQV $W WHPSHUDWXUHV DERYH 7F WKH ORZHVW HQHUJ\ VWDWH FRUUHVSRQGV WR ^! $V WKH WHPSHUDWXUH LV ORZHUHG WKH GLIIHUHQFH LQ IUHH HQHUJLHV EHWZHHQ WKH RUGHUHG `f DQG GLVRUGHUHG M! FRQILJXUDWLRQV LV UHGXFHG

PAGE 27

)LJXUH 7KH VKDSH RI WKH IUHH HQHUJ\ FXUYH YHUVXV RUGHU SDUDPHWHU IRU YDULRXV WHPSHUDWXUHV DERYH DQG EHORZ WKH WUDQVLWLRQ SRLQW

PAGE 28

XQWLO 7 7& DW ZKLFK SRLQW WKH IUHH HQHUJ\ LV HTXLYDOHQW IRU ERWK SKDVHV %HORZ 7F WKH RUGHUHG SKDVH LV IDYRUHG EXW WKH VWDWH -f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‘ f

PAGE 29

UHPDLQV FRQVWDQW WKURXJKRXW WKH WUDQVIRUPDWLRQ DQG LV WKHUHIRUH DQDORJRXV WR WKH RUGHU SDUDPHWHU LQ D &23 V\VWHP VXFK DV SKDVH VHSDUDWLRQ LQ ELQDU\ DOOR\V ,Q WKLV FDVH XS VSLQV FRUUHVSRQG WR RQH W\SH RI DWRP ZKLOH GRZQ VSLQV FRUUHVSRQG WR WKH RWKHU W\SH RI DWRP 7KH RUGHUGLVRUGHU WUDQVLWLRQ LQ ELQDU\ DOOR\V FDQ DOVR EH PRGHOHG XVLQJ .DZDVDNL G\QDPLFV LI WKH RUGHU SDUDPHWHU LV GHILQHG WR EH D VXE ODWWLFH PDJQHWL]DWLRQ 0V ;D f VXEODWWLFH ,Q WKLV HYHQW WKH WRWDO PDJQHWL]DWLRQ LV FRQVHUYHG EXW WKH RUGHU SDUDPHWHU LV QRQFRQVHUYHGn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

PAGE 30

)LJXUH 1XFOHDWLRQ (DUO\ &RDUVHQLQJ Gf $Q LOOXVWUDWLRQ RI WKH GLIIHUHQW WHPSRUDO UHJLPHV LQ WKH WUDQVIRUPDWLRQ SURFHVV

PAGE 31

1XFOHDWLRQ 7KHRU\ 7KH UDWH RI WUDQVIRUPDWLRQ IURP WKH ROG WR QHZ SKDVH LV FRQWUROOHG E\ WKH FRPELQDWLRQ RI WKH WRWDO QXFOHDWLRQ UDWH DQG WKH JURZWK UDWH 7KH WRWDO QXFOHDWLRQ UDWH SHU XQLW WLPH )Wf LV WKH SURGXFW RI WKH QXFOHDWLRQ UDWH ,Wf DQG WKH YROXPH DYDLODEOH IRU QXFOHDWLRQ 9Q &RQFHSWXDOO\ WKH EHKDYLRU RI WKH WRWDO QXFOHDWLRQ UDWH FDQ EH GLYLGHG LQWR WKUHH UHJLPHV 2IWHQ WKHVH WKUHH UHJLPHV DUH GLVWLQFW EXW LQ VRPH FDVHV RU WKHUH PD\ EH VLJQLILFDQW RYHUODS ,Q DGGLWLRQ QRW DOO RI WKH UHJLPHV DUH QHFHVVDULO\ REVHUYHG 7KH ILUVW UHJLPH LV FKDUDFWHUL]HG E\ D WUDQVLHQW QXFOHDGRQ UDWH )ROORZLQJ D TXHQFK IURP DERYH WKH WUDQVLWLRQ SRLQW ,Wf ZLOO EH ]HUR LQLWLDOO\ DQG WKHQ JURZ VPRRWKO\ WR LWV VWHDG\ VWDWH YDOXH 2QFH WKH QXFOHDGRQ UDWH UHDFKHV LWV VWHDG\VWDWH YDOXH ,VV LW ZLOO UHPDLQ XQFKDQJHG WKURXJKRXW WKH WUDQVIRUPDWLRQ +RZHYHU DV WKH WUDQVIRUPDWLRQ SURFHHGV WKH QXFOHDWLRQ YROXPH 9Q GHFUHDVHV OHDGLQJ WR D GHFUHDVH LQ WKH WRWDO QXFOHDWLRQ UDWH ,7Wf LQ WKH ODWH WLPH UHJLPH 7KH SURFHVV E\ ZKLFK WKH QXFOHDWLRQ YROXPH LV H[KDXVWHG GHSHQGV LQ GHWDLO RQ WKH W\SH RI VLWH LQ ZKLFK QXFOHDWLRQ RFFXUV 6HYHUDO RI WKHVH SURFHVVHV ZLOO EH GLVFXVVHG LQ D ODWHU VHFWLRQ (DFK RI WKHVH WKUHH UHJLPHV ZLOO EH GLVFXVVHG LQ GHWDLO LQ WKH IROORZLQJ VHFWLRQV 7KH GULYLQJ IRUFH IRU WUDQVIRUPDWLRQ IURP WKH PHWDVWDEOH VWDWH WR WKH HTXLOLEULXP VWDWH LV WKH IUHH HQHUJ\ GLIIHUHQFH EHWZHHQ WKH WZR SKDVHV +RZHYHU ZKHQ D ILQLWH VL]HG QXFOHXV IRUPV WKHUH H[LVWV DQ LQWHUIDFLDO UHJLRQ EHWZHHQ WKH QXFOHXV DQG WKH EXON PHWDVWDEOH SKDVH 7KH VXUIDFH WHQVLRQ VHUYHV WR UDLVH WKH IUHH HQHUJ\ RI WKH GURSOHW 7KH WRWDO IUHH HQHUJ\ RI WKH HPEU\R LV W\SLFDOO\ ZULWWHQ DV WKH VXP RI EXON DQG VXUIDFH WHUPV $)Q QIHIWQf 6QD f RU f

PAGE 32

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f7 ZKHUH 7 LV WKH WUDQVLWLRQ WHPSHUDWXUH $ERYH 7 IH IP DQG $)Q LV D PRQRWRQLFDOO\ LQFUHDVLQJ IXQFWLRQ RI Q DQG IOXFWXDWLRQV RI DQ\ VL]H WHQG WR HYDSRUDWH ,Q WKLV FDVH WKH HTXLOLEULXP IRUP RI WKH GLVWULEXWLRQ RI VL]HV RI QXFOHL DV GHWHUPLQHG E\ %ROW]PDQQ VWDWLVWLFV LV JLYHQ E\ 1Q 1Y H[S $)f NE7 f ZKLFK LV RQO\ RI VLJQLILFDQW PDJQLWXGH DW YHU\ ORZ YDOXHV RI Q +RZHYHU EHORZ 7 IH 3 DQG $)Q JRHV WKURXJK D PD[LPXP DW WKH FULWLFDO UDGLXV UF A IPIHf f &ODVVLFDOO\ QXFOHL RI WKLV UDGLXV DUH LQ XQVWDEOH HTXLOLEULXP ZKHUHDV QXFOHL ZLWK U UF ZLOO HYDSRUDWH DQG GURSOHWV ZLWK U UF ZLOO JURZ 7KLV PRGHO ZKLOH EHLQJ VRPHZKDW VLPSOLILHG H[KLELWV WKH LPSRUWDQW IHDWXUH WKDW QXFOHL KDYLQJ UDGLL QHDU UF FRQWURO WKH

PAGE 33

QXFOHDWLRQ UDWH ,Q RWKHU ZRUGV LW LV WKH UDWH DW ZKLFK IOXFWXDWLRQV UHDFK WKH FULWLFDO VL]H WKDW GHWHUPLQHV WKH UDWH RI GHFD\ RI WKH PHWDVWDEOH VWDWH ,PPHGLDWHO\ IROORZLQJ D TXHQFK WKH PDMRULW\ RI QXFOHL DUH RI D YHU\ VPDOO VL]H DQG XQWLO WKH ILUVW GURSOHWV UHDFK WKH FULWLFDO VL]H WKH QXFOHDWLRQ UDWH LV HIIHFWLYHO\ ]HUR 7KLV SHULRG LV FDOOHG WKH LQGXFWLRQ RU LQFXEDWLRQ WLPH WA 7KHUH DUH WZR FRPPRQO\ XVHG DSSUR[LPDWLRQV IRU ,Wf LQ WKH WUDQVLHQW SHULRG 7KH ILUVW DVVXPHV DQ LQFXEDWLRQ WLPH IROORZHG E\ D VORZO\ JURZLQJ QXFOHDWLRQ UDWH ZKLFK LV DSSUR[LPDWHG E\ D SRZHU ODZ ,Wf W WRf %VWQ f +HUH WKH VWHS IXQFWLRQ #Wf LQFRUSRUDWHV WKH HIIHFW RI D ILQLWH LQFXEDWLRQ WLPH 7KLV DSSUR[LPDWLRQ ZLOO EH PRVW YDOLG ZKHQ WKH QXFOHDWLRQ UDWH LV D VORZO\ FKDQJLQJ IXQFWLRQ RI WKH WLPH DQG WKH WUDQVIRUPDWLRQ QHDUV FRPSOHWLRQ EHIRUH WKH QXFOHDWLRQ UDWH DSSURDFKHV LWV VWHDG\VWDWH YDOXH 7KH VHFRQG DSSUR[LPDWLRQ LQFRUSRUDWHV DQ LQGXFWLRQ WLPH DQG D VWHS IXQFWLRQ WR GHVFULEH ,Wf DQG ZLOO EH PRVW DFFXUDWH ZKHQ IROORZLQJ WKH LQGXFWLRQ SHULRG ,Wf JURZV UDSLGO\ WR LWV VWHDG\VWDWH YDOXH 7\SLFDOO\ WKH HIIHFW RI WKH FKDQJLQJ QXFOHDWLRQ UDWH LV PRVW REYLRXV LQ WKH HDUO\ VWDJHV RI WKH WUDQVIRUPDWLRQ ([SHULPHQWDO PHDVXUHPHQWV DUH QRW RYHUO\ VHQVLWLYH WR WKH H[DFW IRUP FKRVHQ WR DSSUR[LPDWH WKH WLPH GHSHQGHQFH RI ,Wf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

PAGE 34

UDWH LV DVVXPHG WR EH SURSRUWLRQDO WR WKH VXUIDFH DUHD RI WKH QXFOHXV 6Q DQG RWKHUZLVH LQGHSHQGHQW RI Q $V WLPH SURFHHGV WKH QXPEHU RI HPEU\RV RI VL]H Q DSSURDFKHV LWV TXDVLVWHDG\ VWDWH YDOXH =AW Af 7KLV LPSOLHV WKDW ,AW D FRQVWDQW LQGHSHQGHQW RI Q VR WKDW WKH GLVWULEXWLRQ RI QXFOHL =LV FRQVWDQW 7KHUH DUH WZR ERXQGDU\ FRQGLWLRQV IRU WKH NLQHWLF SURFHVV 7KH ILUVW LV WKDW IRU VPDOO Q WKH GLVWULEXWLRQ RI QXFOHL DSSURDFKHV LWV HTXLOLEULXP IRUP DV GHWHUPLQHG E\ %ROW]PDQQ VWDWLVWLFV =Q DSSURDFKHV 1Q IURP (TXDWLRQ f 7KH RWKHU ERXQGDU\ FRQGLWLRQ LV WKDW HPEU\RV FRQWDLQLQJ D YHU\ ODUJH QXPEHU RI SDUWLFOHV OHDYH WKH V\VWHP FRPSOHWHO\ =f Q}QFf %\ HTXDWLQJ WKH UDWHV RI SURGXFWLRQ DQG ORVV RI HPEU\RV RI VL]H Q WKH IROORZLQJ UHVXOW LV REWDLQHG f ZKHUH Y LV WKH UDWH RI FRQGHQVDWLRQ SHU XQLW VXUIDFH DUHD RI WKH QXFOHXV 6LQFH $) H[KLELWV D PD[LPXP DW QF WKH TXDQWLW\ 1QB, IURP (TXDWLRQ ZLOO EH VKDUSO\ SHDNHG QHDU QF %HFNHU DQG 'RULQJ UHDVRQHG WKDW RQO\ YDOXHV RI Q QHDU WKH FULWLFDO QXPEHU ZRXOG FRQWULEXWH VLJQLILFDQWO\ WR WKH DERYH VXP 7DNLQJ WKH FRQWLQXXP OLPLW RI 1Q DQG $) WKH VWDQGDUG %HFNHU'RULQJ IRUP IRU WKH VWHDG\VWDWH QXFOHDWLRQ UDWH LV REWDLQHG f ZKHUH 1Y6F@ _n$)QFf % $)QFf f ,Q WKH DERYH UHODWLRQV ERWK WKH SUHIDFWRU $ DQG WKH HQHUJ\ WHUP % GHSHQG RQ WKH UHGXFHG WHPSHUDWXUH ,Q WKRVH FDVHV ZKHUH WKHVH SDUDPHWHUV DUH RQO\ ZHDNO\ GHSHQGHQW RQ WKH WHPSHUDWXUH WKH VWHDG\VWDWH YDOXH RI WKH QXFOHDWLRQ UDWH WDNHV WKH $UKHQQHXV

PAGE 35

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

PAGE 36

)LJXUH 18&/($7,21 2 & &' f§ ? ? A A 9 9 'LVWDQFH 63,12'$/ '(&20326,7,21 $ FRQWUDVW RI WKH QXFOHDWLRQ DQG VSLQRGDO GHFRPSRVLWLRQ WUDQVIRUPDWLRQ PHFKDQLVPV

PAGE 37

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f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f VDWLVILHV WKH VLPSOH HTXDWLRQ /Wf U UWf U &KULVWLDQ KDV HVWLPDWHG WKH GHSHQGHQFH RI 7 RQ WKH WHPSHUDWXUH 7 IRU WKH GHFD\ RI PHWDVWDEOH SKDVHV 7KLV FDOFXODWLRQ DVVXPHV WKH GLIIHUHQFH LQ IUHH HQHUJ\ EHWZHHQ WKH ROG DQG QHZ SKDVHV WR EH VPDOO UHODWLYH WR WKH WKHUPDO HQHUJ\ N%7 ZKHUH N% LV WKH %ROW]PDQQ FRQVWDQW ,Q DGGLWLRQ WKH FKDUDFWHULVWLF IUHTXHQF\ IRU SDUWLFOH PRWLRQ Y LV DVVXPHG WR EH LGHQWLFDO LQ ERWK WKH WUDQVIRUPHG DQG XQWUDQVIRUPHG UHJLRQV 7KH UHVXOWLQJ UHODWLRQ LV H $) $+Dn 77f Y f S H[S S f

PAGE 38

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f )f +RZHYHU LI WKH FRQGHQVDWLRQ SURFHVV LV UDSLG WUDQVIRUPDWLRQ ZLOO EH OLPLWHG WR WKH UDWH RI GLIIXVLRQ WKURXJK WKH GHSOHWLRQ ]RQH )URP WKH GLIIXVLRQ HTXDWLRQ DQG GLPHQVLRQDO DQDO\VLV &KULVWLDQ ILQGV /Wf 'W f UWfaS f ZKHUH LV WKH GLIIXVLRQ FRQVWDQW 7KLV SDUDEROLF JURZWK ODZ LV YHULILHG LQ PRUH H[DFW FDOFXODWLRQV DQG LQ FRPSXWHU VLPXODWLRQVf ,W LV QRWHG WKDW WKH DERYH JURZWK ODZV IRU LQWHUIDFH DQG GLIIXVLRQ FRQWUROOHG JURZWK DUH RQO\ YDOLG SULRU WR LPSLQJHPHQW RI RQH WUDQVIRUPHG GRPDLQ RQ DQRWKHU 'LIIHUHQW JURZWK ODZV JRYHUQ EHKDYLRU LQ WKH SRVWLPSLQJHPHQW RU FRDUVHQLQJ UHJLPH DQG WKHVH ZLOO EH GLVFXVVHG VKRUWO\ 7UDQVIRUPDWLRQ &XUYHV 1RZ WKDW WKH SURFHVVHV RI QXFOHDWLRQ DQG JURZWK KDYH EHHQ GLVFXVVHG D PHWKRG IRU FRPELQLQJ WKH UHVXOWV WR \LHOG D UDWH RI WUDQVIRUPDWLRQ ZLOO EH SUHVHQWHG $ PHDVXUH

PAGE 39

RI WKH GHJUHH WR ZKLFK WUDQVIRUPDWLRQ KDV SURJUHVVHG LV WKH IUDFWLRQDO YROXPH LQ WKH WUDQVIRUPHG VWDWH eWf )URP WKH SUHYLRXV GLVFXVVLRQ RI WKH QXFOHDWLRQ UDWH ZH H[SHFW WKLV TXDQWLW\ WR JURZ VORZO\ DW ILUVW DQG WKHQ PRUH UDSLGO\ DV D ODUJHU QXPEHU RI HPEU\RV UHDFK WKH FULWLFDO VL]H DQG EHJLQ WR JURZ (YHQWXDOO\ WKH VXUIDFHV RI GLIIHUHQW GRPDLQV ZLOO LPSLQJH RQ HDFK RWKHU VORZLQJ WKH JURZWK RI eWf DW ODWHU WLPHV 7KH VKDSH RI D W\SLFDO WUDQVIRUPDWLRQ FXUYH LV VKRZQ LQ )LJXUH 7KH UHOHYDQW IHDWXUHV LQFOXGH WKH LQGXFWLRQ WLPH WR DQG WKH WLPH WR KDOIFRPSOHWLRQ [ 7KH HDUOLHVW IRUPDO WKHRULHV RI WUDQVIRUPDWLRQ NLQHWLFV DUH GXH WR .ROPRJRURY $YUDPLf DQG 0HKO DQG -RKQVRQ 7KH DUJXPHQW RI $YUDPL JRHV DV IROORZV ,I D QXFOHXV IRUPV DW VRPH WLPH [ WKHQ WKH WUDQVIRUPHG YROXPH ZKLFK LV FRQWDLQHG LQ WKH QXFOHXV LV JLYHQ E\ 9Wf 7_ <[<\ \] W [ f W [f nf W [ f +HUH \[ \\ DQG \] DUH WKH JURZWK YHORFLWLHV LQ WKH [ \ DQG ] GLUHFWLRQV UHVSHFWLYHO\ DQG W 7_ LV D VKDSH IDFWRU IRU LVRWURSLF JURZWK 7_ f§ DQG <[ <\ <] < ff 7KH DERYH H[SUHVVLRQ LV YDOLG IRU D V\VWHP H[KLELWLQJ D OLQHDU JURZWK ODZ /Wf a W VXFK DV LV WKH FDVH IRU DQ LQWHUIDFH FRQWUROOHG WUDQVIRUPDWLRQ DV ZDV SUHYLRXVO\ GLVFXVVHG ,Q D V\VWHP ZKHUH JURZWK LV GLIIXVLRQ OLPLWHG WKH JURZWK ODZ LV JLYHQ E\ /Wf W ,Q WKLV FDVH (TXDWLRQV ZRXOG EH UHSODFHG E\ Y 7 <[<\<]W[f W ![f f W [f )RU WKH UHPDLQGHU RI WKLV VHFWLRQ RQO\ V\VWHPV H[KLELWLQJ D OLQHDU JURZWK ODZ ZLOO EH GLVFXVVHG ZLWK WKH XQGHUVWDQGLQJ WKDW WKH DSSURSULDWH UHVXOWV IRU GLIIHUHQW JURZWK ODZV FDQ EH GHULYHG VLPSO\ E\ PDNLQJ WKH PRGLILFDWLRQ VKRZQ LQ (TXDWLRQV

PAGE 40

2UGHU 3DUDPHWHU )LJXUH 7,0( $ VLPXODWHG WUDQVIRUPDWLRQ FXUYH VKRZLQJ WKH LQGXFWLRQ WLPH DQG WKH KDOIFRPSOHWLRQ WLPH

PAGE 41

,Q HDUO\ WLPHV WKHUH LV OLWWOH LPSLQJHPHQW RI GRPDLQV VR WKDW HDFK GURSOHW JURZV LQGHSHQGHQWO\ ,Q WKLV OLPLW WKH WUDQVIRUPHG YROXPH LV JLYHQ E\ f R ZKHUH LVRWURSLF JURZWK ZDV DVVXPHG )RU D FRQVWDQW QXFOHDWLRQ UDWH ,Wf ,Wf UHDFKHV WKH VWHDG\VWDWH YDOXH ,VV LPPHGLDWHO\ f (TXDWLRQ LV UHGXFHG WR f 7KLV FRQILUPV WKH UDSLG LQFUHDVH RI e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f ,W IROORZV WKDW RU GY3 939 f G9H & Y39 H[S 9H f H[SI \ ,VV Wf f

PAGE 42

7KH VKDSH RI WKLV FXUYH LV VKRZQ LQ )LJXUH DQG LQ WKH HDUO\ WLPH WKLV H[SUHVVLRQ UHGXFHV WR (TXDWLRQ ,Q WKH DERYH DUJXPHQW WKH QXFOHDWLRQ UDWH ,Wf ZDV DVVXPHG WR EH D FRQVWDQW ,Q JHQHUDO WKH WRWDO QXFOHDWLRQ UDWH LV D IXQFWLRQ RI WKH WLPH HLWKHU GXH WR WUDQVLHQW QXFOHDWLRQ LQ WKH HDUO\ UHJLPH RU GXH WR QXFOHDWLRQ VLWH VDWXUDWLRQ LQ WKH ODWHU UHJLPH 7KH FRUUHVSRQGLQJ WUDQVIRUPDWLRQ ODZV PD\ EH GHULYHG LQ D VWUDLJKW IRUZDUG ZD\ E\ LQVHUWLQJ WKH DSSURSULDWH H[SUHVVLRQ IRU ,Wf LQWR (TXDWLRQ 7KHVH FDOFXODWLRQV JHQHUDOO\ OHDG WR DQ H[SUHVVLRQ VLPLODU WR WKDW LQ (TXDWLRQ H[FHSW WKDW WKH WLPH H[SRQHQW LV GLIIHUHQW IURP 7KLV OHG 0HKO -RKQVRQ DQG $YUDPL 0-$ f WR SURSRVH WKH IROORZLQJ JHQHUDO WUDQVIRUPDWLRQ ODZ & H[SNWQf f )RU D FRQVWDQW QXFOHDWLRQ UDWH LW KDV EHHQ VKRZQ WKDW Q )RU DQ LQFUHDVLQJ QXFOHDWLRQ UDWH HDUO\ UHJLPH f Q! DQG IRU D GHFUHDVLQJ QXFOHDWLRQ UDWH VLWH VDWXUDWLRQ f Q )RU KRPRJHQHRXV QXFOHDWLRQ RIWHQ WKH QXFOHDWLRQ UDWH PD\ EH DSSUR[LPDWHG E\ D VWHS IXQFWLRQ EHLQJ ]HUR GXULQJ WKH LQGXFWLRQ SHULRG WR DQG WKHQ WDNLQJ RQ LWV VWHDG\ VWDWH YDOXH ,VV ,Q WKLV FDVH WKH WUDQVIRUPDWLRQ FXUYH LV JLYHQ E\ (TXDWLRQ ZLWK Q DQG WKH WLPH W LV UHSODFHG E\ WWRr 2IWHQ QXFOHDWLRQ RFFXUV QRW KRPRJHQHRXVO\ EXW DW LPSXULW\ RU GHIHFW VLWHV ZLWKLQ WKH VDPSOH 7KH WUDQVIRUPDWLRQ FXUYHV IRU KHWHURJHQHRXV QXFOHDWLRQ FDQ EH FDOFXODWHG LQ PXFK WKH VDPH ZD\ DV IRU WKH KRPRJHQHRXV FDVH LI WKH DSSURSULDWH IRUP IRU WKH QXFOHDWLRQ UDWH FDQ EH IRXQG )RU LQVWDQFH ZKHQ QXFOHDWLRQ RFFXUV VSRQWDQHRXVO\ DW 1 LPSXULW\ VLWHV WKH HIIHFWLYH QXFOHDWLRQ UDWH ,cWf PD\ EH UHSUHVHQWHG E\ D GHOWD IXQFWLRQ DW W KDYLQJ D VWUHQJWK 1 L H ,LWf 1Wf Df

PAGE 43

VR WKDW IURP (TXDWLRQ e H[S 1 \f Ef 7KH IRUP RI WKH WUDQVIRUPDWLRQ FXUYH LV XQFKDQJHG ZLWK Q UHIOHFWLQJ WKH QDWXUH RI WKH QXFOHDWLRQ SURFHVV ,QWHUHVWLQJ GLPHQVLRQDO HIIHFWV UHVXOW LI QXFOHDWLRQ LV VSRQWDQHRXV DQG JURZWK LV UHVWULFWHG WR HLWKHU D ODPHOODU RU URGOLNH YROXPH )RU JURZWK RFFXUULQJ LQ D VKHHW JURZWK f DV PLJKW EH WKH FDVH LQ D WKLQ ILOP WKH H[WHQGHG YROXPH LV JLYHQ E\ 9S W9ff1
PAGE 44

& H[S E IWff f L LI W@ W ZKHUH e ,, H[S >,W-T RAf-R 7KH FRHIILFLHQWV D DQG E DUH UHODWHG WR WKH SK\VLFDO TXDQWLWLHV \ ,Vr DQG 9Q LQ WKH IROORZLQJ ZD\ DQG D ,VV \ f: E 9Q f 7KH IXQFWLRQ IWf KDV WZR OLPLWLQJ IRUPV DW ODWH WLPHV IWf W DQG DW HDUO\ WLPHV IWf W DV H[SHFWHG 7KH WUDQVLWLRQ IURP Q EHIRUH VLWH VDWXUDWLRQ WR Q O DIWHU VDWXUDWLRQ LV EHVW LOOXVWUDWHG E\ D SORW RI OQ OQ 4 ff YV OQ Wf DV VKRZQ LQ )LJ ,Q WKLV SORW WKH VORSH RI WKH FXUYH LV HTXLYDOHQW WR WKH H[SRQHQW LQ WKH 0-$ IRUP DQG WKH EHQG LQ WKH OLQH FRUUHVSRQGV WR VLWH VDWXUDWLRQ )URP (TXDWLRQV LW LV HYLGHQW WKDW LI ZH RIIVHW WKH SORW YHUWLFDOO\ E\ OQ E f DQG KRUL]RQWDOO\ E\ OQ D f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

PAGE 45

7KH &RKQ 0RGHO 0DVWHU &XUYH F F ,Q WLPH f )LJXUH $Q LGHDO 0DVWHU &XUYH IRU D WUDQVIRUPDWLRQ ZKLFK REH\V &DKQfV UHODWLRQ IRU QXFOHDWLRQ RQ WZR GLPHQVLRQDO QXFOHDWLRQ VLWHV

PAGE 46

WUDQVIRUPDWLRQ FXUYH LV REVHUYDEOH ,Q IDFW WKH HDVLO\ REVHUYDEOH UDQJH RI e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e 7KH VWHDG\VWDWH QXFOHDWLRQ UDWH ZLWKLQ WKH &DKQ WKHRU\ IRU QXFOHDWLRQ RQ GHIHFWV ZKLFK FRUUHVSRQGV WR WKLV SRLQW LV JLYHQ E\ -VV f§ BBB OQ f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rr UHIOHFWLQJ WKH IDFW WKDW WKH FU\VWDO PD\ IRUP LQ DQ\ RULHQWDWLRQ ZLWKLQ WKH OLTXLG

PAGE 47

,Q WKH HYHQW WKH JURXQG VWDWH LV GHJHQHUDWH LQ WKH ODWH WLPH UHJLPH WKH V\VWHP ZLOO EH FRPSRVHG RI PDQ\ IXOO\ WUDQVIRUPHG GRPDLQV GLVWULEXWHG LQ WKH S GHJHQHUDWH VWDWHV $V WLPH SURFHHGV WKH QXPEHU RI GRPDLQV ZLOO GHFUHDVH ZLWK D FRUUHVSRQGLQJ LQFUHDVH LQ WKH DYHUDJH VL]H RI WKH RUGHUHG GRPDLQV 7KLV SURFHVV LV FDOOHG FRDUVHQLQJ :LWKLQ WKH FRDUVHQLQJ UHJLPH WKH V\VWHP LV VHOIVLPLODU ZLWK UHVSHFW WR VFDOLQJ E\ OHQJWK DQG WLPH 7KLV FDQ EH LOOXVWUDWHG E\ ]RRPLQJ LQ RQ IUDPH Ff RI )LJXUH DQG FRPSDULQJ WKH UHVXOW ZLWK IUDPH Gf 7KH UHOHYDQW IHDWXUHV LQ HDFK FDVH ZLOO RQ DYHUDJH EH LGHQWLFDO $ ZHOO NQRZQ FRQVHTXHQFH RI WKLV W\SH RI VFDOLQJ LV WKH H[LVWHQFH RI D FKDUDFWHULVWLF OHQJWK /Wf WKH GRPDLQ VL]H f ZKHUH /Wf WD f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f ,W VKRXOG EH QRWHG WKDW WKH JURZWK H[SRQHQWV DUH GLIIHUHQW LQ WKH FRDUVHQLQJ UHJLPH IURP WKRVH GLVFXVVHG IRU WKH JURZWK UHJLPH 7KLV LV EHFDXVH GXULQJ WKH HDUO\ VWDJHV JURZWK RI GRPDLQV LV UHODWLYHO\ XQLQKLELWHG E\ LPSLQJHPHQW ,Q WKH FRDUVHQLQJ UHJLPH WKH GRPDLQV DUH VWURQJO\ LQWHUDFWLQJ ZLWK WKH PDMRU PRGH RI JURZWK EHLQJ WKH DEVRUSWLRQ RI VPDOOHU GRPDLQV E\ ODUJHU RQHV $OVR WKH G\QDPLF TXDQWLW\ GXULQJ FRDUVHQLQJ LV WKH DYHUDJH VL]H RI D GRPDLQ ZKLOH WKH YROXPH IUDFWLRQ RI WKH V\VWHP LQ WKH WUDQVIRUPHG SKDVH UHPDLQV DSSUR[LPDWHO\ XQFKDQJHG ,Q WKH HDUO\ VWDJHV RI WUDQVIRUPDWLRQ LW ZDV VHHQ WKDW LPSXULWLHV DQG GHIHFWV KDYH D SURIRXQG HIIHFW RQ WKH QXFOHDWLRQ SURFHVV E\ SURYLGLQJ KHWHURJHQHRXV QXFOHDWLRQ VLWHV 7KHVH HQWLWLHV FDQ DOVR OHDG WR LQWHUHVWLQJ HIIHFWV LQ WKH FRDUVHQLQJ UHJLPH /DL HW DO

PAGE 48

VKRZ WKDW UDQGRP ILHOGV ZKLFK DUH JHQHUDWHG E\ LPSXULWLHV DQG GHIHFWV VORZ WKH FRDUVHQLQJ SURFHVV VR WKDW WKH JURZWK ODZ EHFRPHV /Wf ORJWf fP f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

PAGE 49

DQJOH [UD\ VFDWWHULQJ VSHFWUXP 7KH IXQGDPHQWDOV RI WKH [UD\ VFDWWHULQJ WHFKQLTXH DUH QRZ SUHVHQWHG

PAGE 50

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f ,PFA m 3 f 7KH DQJOH LV FDOOHG WKH VFDWWHULQJ DQJOH DQG LV LOOXVWUDWHG LQ )LJXUH +HUH ,3 LV D SRODUL]DWLRQ IDFWRU ZKLFK HTXDOV IRU DQ LQFLGHQW EHDP SRODUL]HG SHUSHQGLFXODU WR WKH SODQH RI VFDWWHULQJ DV LV WKH FDVH IRU V\QFKURWURQ UDGLDWLRQ f )RU DQ XQSRODUL]HG EHDP OFRVf ,3 f

PAGE 51

,Q HLWKHU FDVH WKH SRODUL]DWLRQ IDFWRU LV DSSUR[LPDWHO\ HTXDO WR XQLW\ LQ VPDOO DQJOHV &RPSWRQ DQG LQHODVWLF VFDWWHULQJ DUH QHJOLJLEO\ VPDOO DW VPDOO DQJOHV LH r f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f UNNf f +HUH V DQG V DUH XQLW YHFWRUV LQ WKH LQFLGHQW DQG VFDWWHUHG GLUHFWLRQV DQG N N DUH ZDYH YHFWRUV LQ WKH LQFLGHQW DQG VFDWWHUHG GLUHFWLRQV $W WKLV SRLQW LW LV FRQYHQLHQW WR GHILQH WKH VFDWWHULQJ YHFWRU 4 ZKLFK LV WKH GLIIHUHQFH LQ LQFRPLQJ DQG RXWJRLQJ ZDYH YHFWRUV VR WKDW 4 NN 4 N VLQ >@ WW > ; VLQ O f 7KH VFDWWHULQJ YHFWRU LV WKH YDULDEOH W\SLFDOO\ XVHG LQ [UD\ GLIIUDFWLRQ DQG KHQFHIRUWK WKH VFDWWHULQJ ZLOO EH GHVFULEHG LQ WHUPV RI WKLV TXDQWLW\ 7KLV LV GRQH ZLWK WKH XQGHUVWDQGLQJ WKDW 4 FDQ EH UHODWHG WR WKH VFDWWHULQJ DQJOH E\ (TXDWLRQV 7KH WUDQVIRUPDWLRQ IURP 4 WR VFDWWHULQJ DQJOH LV DSSUR[LPDWHO\ OLQHDU DW VPDOO 4 ZLWK ,R a ƒn IRU DQ LQFLGHQW ZDYHOHQJWK RI ƒ :LWK WKHVH GHILQLWLRQV WKH DPSOLWXGH RI VFDWWHULQJ IURP WZR VFDWWHULQJ FHQWHUV LV $ H[S 2f H[S Ln4Uf f

PAGE 52

)LJXUH $Q LOOXVWUDWLRQ RI GLIIUDFWLRQ E\ WZR VFDWWHULQJ FHQWHUV DQG WKH GHILQLWLRQ RI WKH VFDWWHULQJ DQJOH

PAGE 53

7KLV UHVXOW FDQ EH H[WHQGHG WR WKH FDVH RI VFDWWHULQJ IURP D FRQWLQXRXV V\VWHP FKDUDFWHUL]HG E\ DQ HOHFWURQ GHQVLW\ SUf 7KH UHVXOWLQJ DPSOLWXGH LV JLYHQ E\ $4f -G9SUf H[S L4 Uf f +HQFH WKH VFDWWHULQJ DPSOLWXGH LV HTXLYDOHQW WR WKH )RXULHU WUDQVIRUPDWLRQ RI WKH HOHFWURQ GHQVLW\ 7KH REVHUYHG VFDWWHUHG LQWHQVLW\ LV UHODWHG WR WKH DPSOLWXGH LQ WKH IROORZLQJ ZD\ ,4f rF $$ I G9 G9 SU[f SUf H[Sn4UUUff f 5HGHILQLQJ U U U LQ (TXDWLRQ \LHOGV WKH UHVXOW WKDW WKH VFDWWHUHG LQWHQVLW\ LV HTXDO WR WKH )RXULHU WUDQVIRUP RI WKH DXWRFRUUHODWLRQ IXQFWLRQ RI WKH HOHFWURQ GHQVLW\ S9f ,4f f G9 SUf H[S L 4 Uf f B U ZKHUH S Uf G9M SUf SUf 7KLV LV D YHU\ JHQHUDO UHVXOW ZKLFK UHODWHV WKH SRVLWLRQDO FRUUHODWLRQV RI HOHFWURQ GHQVLW\ IOXFWXDWLRQV LQ WKH V\VWHP WR WKH PHDVXUHG VFDWWHULQJ SURILOH LQ WKH DEVHQFH RI DSSUR[LPDWLRQV ,Q FDVHV ZKHUH D GLVFUHWH WUHDWPHQW LV PRUH DSSURSULDWH D SDUWLFOH SDUWLFOH FRUUHODWLRQ IXQFWLRQ UHSODFHV WKH WHUP S Uf LQ (TXDWLRQ 2IWHQ WKH V\VWHPV VWXGLHG ZLWK VFDWWHULQJ PHWKRGV DUH FRPSRVHG RI SDUWLFOHV KDYLQJ DQ DSSUR[LPDWHO\ XQLIRUP HOHFWURQ GHQVLW\ S 7KHVH SDUWLFOHV DUH VXVSHQGHG LQ D JHO RU VROXWLRQ FKDUDFWHUL]HG E\ DQRWKHU HOHFWURQ GHQVLW\ S 1HJOHFWLQJ LQKRPRJHQHLWLHV LQ WKH FKDUJH GHQVLW\ WKH VFDWWHULQJ LV HTXLYDOHQW WR WKDW IURP D V\VWHP RI SDUWLFOHV ZLWK GHQVLW\ $S S S VXSHULPSRVHG RQ D EDFNJURXQG RI GHQVLW\ S 7KHUH ZLOO EH QR

PAGE 54

REVHUYDEOH VFDWWHULQJ IURP WKH EDFNJURXQG LQ WKH UHOHYDQW VFDWWHULQJ UDQJH 7KH SUREOHP LV WKHUHIRUH UHGXFHG WR WKH PXFK VLPSOHU FDVH RI FDOFXODWLQJ WKH VFDWWHULQJ IURP SDUWLFOHV RI GHQVLW\ $S VXVSHQGHG LQ D YDFXXP 7KH VFDWWHULQJ DPSOLWXGH IRU D V\VWHP RI 1 LGHQWLFDO SDUWLFOHV FDQ EH ZULWWHQ f $4f -SUf H[S 4 5N Uff G9 N O 1 A)4fH[SL45Nf N O )4f -SUf H[S L4 Uf G9 +HUH 5N LV D YHFWRU H[WHQGLQJ IURP WKH RULJLQ WR WKH FHQWHU RI V\PPHWU\ RI WKH NWK SDUWLFOH DQG SUf LV WKH HOHFWURQ GHQVLW\ GLVWULEXWLRQ RI D VLQJOH SDUWLFOH DERXW LWV FHQWHU RI V\PPHWU\ 7KH TXDQWLW\ )4f FDOOHG WKH IRUP IDFWRU IRU WKH SDUWLFOH FRQWDLQV DOO LQIRUPDWLRQ UHJDUGLQJ LQWUDSDUWLFOH FRUUHODWLRQV 7KH VWUXFWXUH IDFWRU 64f LV WKH SURGXFW RI WKH VFDWWHUHG DPSOLWXGH $ ZLWK LWV FRPSOH[ FRQMXJDWH DQG LV D ZHLJKWHG VXP RI WKH SKDVH IDFWRUV FRUUHVSRQGLQJ WR WKH FHQWHU RI PDVV SRVLWLRQV RI DOO RI WKH SDUWLFOHV 64f WKHUHIRUH FRQWDLQV LQIRUPDWLRQ FRQFHUQLQJ LQWHUSDUWLFOH FRUUHODWLRQV 7KH VFDWWHUHG LQWHQVLW\ LV SURSRUWLRQDO WR 64f ZKLFK FDQ EH ZULWWHQ ,4f 64f )4f )r4f A;FRV 4 5N 5Mff f $IWHU DYHUDJLQJ RYHU DOO SDUWLFOH RULHQWDWLRQV DQG VHSDUDWLQJ WKRVH WHUPV ZKHUH f M N WKLV H[SUHVVLRQ (TXDWLRQ f IRU 64f EHFRPHV 64f 1 ,)4f, 1N )4f !O ] V FRV 45N5Mff N M 7KH SDLU FRUUHODWLRQ IXQFWLRQ 3Uf LQWURGXFHG E\ =HPLFNH DQG 3ULQV UHSUHVHQWV WKH SUREDELOLW\ WKDW WZR SDUWLFOHV DUH VHSDUDWHG E\ D GLVWDQFH U ,Q WKH FDVH RI DQ LVRWURSLF GLVWULEXWLRQ RI SDUWLFOH SRVLWLRQV DQG RULHQWDWLRQV WKLV FDQ EH XVHG WR HYDOXDWH WKH GRXEOH VXPPDWLRQ LQ (TXDWLRQ 7KH UHVXOWLQJ IRUP IRU 64f LV

PAGE 55

64f 1^ )4f!)AfA_A(OO3UffFUGU ` f R ZKHUH Y LV WKH DYHUDJH YROXPH DYDLODEOH SHU SDUWLFOH LQ WKH V\VWHP $V Y GHFUHDVHV LQWHUSDUWLFOH LQWHUIHUHQFH HIIHFWV EHFRPH PRUH SURQRXQFHG :LWK 9M ODUJH 64f LV GRPLQDWHG E\ WKH ILUVW WHUP LQ (TXDWLRQ ,Q WKLV FDVH WKH REVHUYDEOH VFDWWHULQJ LV MXVW WKDW RI D VLQJOH SDUWLFOH PXOWLSOLHG E\ WKH QXPEHU RI SDUWLFOHV LQ WKH V\VWHP 7KLV LV WKH OLPLW W\SLFDOO\ H[SORUHG LQ 6$;6 DQG WKH GHWDLOHG SURSHUWLHV RI WKH VFDWWHUHG LQWHQVLW\ LQ WKLV UHJLPH ZLOO QRZ EH GLVFXVVHG 6PDOO $QJOH ;UDY 6FDWWHULQJ )RU SDUWLFOHV ZLWK GLPHQVLRQV RQ WKH RUGHU RI KXQGUHGV RI DQJVWURPV DQG ZLWK D W\SLFDO LQFLGHQW [UD\ ZDYHOHQJWK RI ƒ WKH PDMRULW\ RI VFDWWHULQJ IHDWXUHV DSSHDU LQ WKH UDQJH IURP WR GHJUHHV 7KLV LV W\SLFDO RI EORFN FRSRO\PHU VROXWLRQV $VVXPLQJ D GLOXWH LVRWURSLF V\VWHP RI SDUWLFOHV WKH VFDWWHUHG LQWHQVLW\ FDQ EH FDOFXODWHG IURP (TXDWLRQV ,Q DOO EXW WKH VLPSOHVW FDVHV WKLV PXVW EH GRQH QXPHULFDOO\ )RU WKH LPSRUWDQW FDVH RI D KDUG VSKHUH WKH VWUXFWXUH IDFWRU ILUVW GHULYHG E\ 5D\OHLJK LV 6VSKHUH&4f n )VSKHUH&4fn 3 f 9 n VLQ45f 45 FRV 45 45f f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

PAGE 56

6SKHULFDO )RUP )DFWRU 6SKHULFDO )RUP )DFWRU 6FDWWHULQJ 9HFWRU ƒ f 6FDWWHULQJ 9HFWRU ƒ f )LJXUH 7KH VFDWWHULQJ IRUP IDFWRUV IRU DQG LGHDO VSKHUHV KDYLQJ UDGLL DQG ƒ

PAGE 57

UHGXFHV WR 64f $S f 9H[S 4A nf f ZKHUH -SUf U G9 5J WKH UDGLXV RI J\UDWLRQ A +HUH WKH LQWHJUDO LV WDNHQ RYHU D VLQJOH SDUWLFOH DQG 0 LV WKH WRWDO PDVV RI WKH SDUWLFOH (TXDWLRQ VKRZV WKDW LI WKH ORJ RI WKH VFDWWHUHG LQWHQVLW\ LV SORWWHG YHUVXV WKH VTXDUH RI WKH VFDWWHULQJ YHFWRU FDOOHG D *XLQLHU SORWf WKH VORSH RI WKH UHVXOWLQJ OLQH ZLOO EH SURSRUWLRQDO WR WKH VTXDUH RI WKH UDGLXV RI J\UDWLRQ LQ WKH OLPLW RI YHU\ VPDOO DQJOHV 7KXV LI WKH VKDSH RI WKH SDUWLFOH LV NQRZQ WKH SDUDPHWHUV GHVFULELQJ LWV VL]H PD\ EH GHWHUPLQHG IURP WKH UDGLXV RI J\UDWLRQ DV REWDLQHG IURP WKH VFDWWHULQJ SURILOH )RU H[DPSOH WKH UDGLXV RI D VSKHUH LV UHODWHG WR LWV UDGLXV RI J\UDWLRQ E\ 5J ?I 5 7KH *XLQLHU DSSUR[LPDWLRQ EUHDNV GRZQ IRU VHYHUHO\ DQLVRWURSLF SDUWLFOHV L H SODWHV RU URGV f DQG LV LQYDOLG LQ WKH SUHVHQFH RI LQWHUSDUWLFOH LQWHUIHUHQFH HIIHFWV ,Q SUDFWLFH DOO RI WKH SDUWLFOHV GR QRW KDYH LGHQWLFDO UDGLL 5DWKHU WKHUH LV D GLVWULEXWLRQ RI UDGLL FHQWHUHG DERXW D PHDQ ,Q WKH FDVH RI EORFN FRSRO\PHUV WKLV HIIHFW DULVHV IURP WKH IDFW WKDW WKH SRO\PHU FKDLQV WKHPVHOYHV DUH SRO\GLVSHUVH )RU H[DPSOH WKH VFDWWHULQJ ZKLFK UHVXOWV IURP D V\VWHP RI VSKHUHV FKDUDFWHUL]HG E\ D JDXVVLDQ VL]H GLVWULEXWLRQ KDYLQJ D PHDQ UDGLXV RI 5T ƒ DQG D VWDQGDUG GHYLDWLRQV D DQG ƒ DUH SORWWHG LQ )LJXUH 7KH QXPEHU RI SDUWLFOHV 1 KDYLQJ UDGLXV 5 LV JLYHQ E\ 15f 55Rf H[S D f DQG WKH WRWDO VWUXFWXUH IDFWRU LV WKH VXSHUSRVLWLRQ RI WKH VWUXFWXUH IDFWRUV IURP HDFK SDUWLFOH LQ WKH GLVWULEXWLRQ 64f -15f 6VSKHUH45fG5 f

PAGE 58

6SKHULFDO )RUP )DFWRU 6SKHULFDO )RUP )DFWRU 6FDWWHULQJ 9HFWRU ƒ f 6FDWWHULQJ 9HFWRU ƒ f )LJXUH $ SORW VKRZLQJ WKH HIIHFW RI SRO\GLVSHUVLW\ RQ WKH VFDWWHULQJ IRUP IDFWRU IRU D JDXVVLDQ GLVWULEXWLRQ RI VSKHULFDO SDUWLFOHV KDYLQJ D ZLGWK SDUDPHWHU &7 DQG ƒ

PAGE 59

7KH PDMRU HIIHFW RI SRO\GLVSHUVLW\ LV D ZHDNHQLQJ RI WKH VFDWWHULQJ PLQLPD ZKLFK LV PRUH SURQRXQFHG ZLWK ODUJHU R 7\SLFDOO\ WKH LQWHUIDFH EHWZHHQ WKH SDUWLFOHV DQG EDFNJURXQG LV RI ILQLWH ZLGWK )RU EORFN FRSRO\PHUV WKLV UHJLRQ LV ZKHUH WKH WZR FRYDOHQWO\ MRLQHG EORFNV PHHW ,I WKHVH MXQFWLRQ SRLQWV IRUP D SHUIHFW VXUIDFH WKH LQWHUIDFH LV LGHDO ,W LV RIWHQ DVVXPHG WKDW WKH ORFDWLRQV RI WKHVH MXQFWLRQ SRLQWV DUH GHVFULEHG E\ D JDXVVLDQ GLVWULEXWLRQ RI f ZLGWK 7KLV OHDGV WR D GHQVLW\ SURILOH SUf $S HUI Un5rOO ‹ -7KH VWUXFWXUH IDFWRU FDOFXODWHG IURP (TXDWLRQV DQG LQFRUSRUDWLQJ WKH HIIHFW RI WKH LQWHUIDFLDO WKLFNQHVV DUH VKRZQ LQ )LJXUH 7KH VSKHUHV LQ WKLV FDOFXODWLRQ KDYH D PHDQ UDGLXV RI ƒ DQG YDOXHV RI e DQG ƒ 7KHUH LV UHODWLYHO\ OLWWOH SHUWXUEDWLRQ RI WKH SURILOH DW YHU\ ORZ 4 EXW WKH VFDWWHULQJ DW KLJKHU 4 LV JUHDWO\ VXSSUHVVHG IRU ODUJH YDOXHV RI ,Q DGGLWLRQ VOLJKW VKLIWV LQ WKH ORFDWLRQV RI WKH VFDWWHULQJ PLQLPD DUH REVHUYHG $W ODUJH YDOXHV RI WKH VFDWWHULQJ YHFWRU D GLIIHUHQW OLPLW LV IRXQG ,Q WKH KLJK 4 UHJLRQ WKH H[SUHVVLRQ IRU WKH VWUXFWXUH IDFWRU WDNHV RQ D PXFK VLPSOLILHG IRUP Q $S f 64f 4 6D f ,Q WKLV HTXDWLRQ 6D LV WKH VXUIDFH DUHD RI WKH VFDWWHULQJ SDUWLFOH 7KLV SRZHU ODZ GHSHQGHQFH LV FDOOHG 3RURGnV /DZ 7KH UDGLXV RI J\UDWLRQ DQG WKH VXUIDFH DUHD RI WKH SDUWLFOHV FDQ EH GHWHUPLQHG IURP WKH OLPLWLQJ IRUPV JLYHQ E\ (TV DQG DQG DUH YHU\ XVHIXO LQ GHGXFLQJ WKH VKDSH DQG GLPHQVLRQV RI WKH SDUWLFOH ,I WKH VXUIDFH RI WKH SDUWLFOH LV RI ILQLWH WKLFNQHVV WKH VFDWWHULQJ DW KLJK 4 ZLOO H[KLELW V\VWHPDWLF GHYLDWLRQV IURP 3RURGnV /DZ ZKLFK GHSHQG RQ WKH LQWHUIDFLDO WKLFNQHVV n(f ,Q D PHWKRG GHYHORSHG E\ 9RQN DQG 5XODQG WKH HIIHFW RI ILQLWH A LV WDNHQ LQWR DFFRXQW E\ FRQVLGHULQJ WKH HOHFWURQ GHQVLW\ RI WKH SDUWLFOH SUf DV WKH FRQYROXWLRQ RI DQ LGHDO HOHFWURQ GHQVLW\ SURILOH ScUf DQG D VPRRWKLQJ IXQFWLRQ KUf 6LQFH WKH VFDWWHULQJ

PAGE 60

6SKHULFDO )RUP )DFWRU 6SKHULFDO )RUP )DFWRU 6FDWWHULQJ 9HFWRU ƒ f 7KH PRGLILFDWLRQV WR WKH VFDWWHUHG SURILOH ZKLFK UHVXOW IURP VSKHUHV KDYLQJ LPSHUIHFW ERXQGDULHV DUH LOOXVWUDWHG LQ WKHVH SORWV IRU D V\VWHP RI VSKHUHV KDYLQJ D UDGLXV RI ƒ DQG YDOXHV RI [ DQG ƒ DV GHVFULEHG LQ WKH WH[W )LJXUH

PAGE 61

IRUP IDFWRU )4f LV WKH )RXULHU WUDQVIRUP RI WKH HOHFWURQ GHQVLW\ SUf LW FDQ EH H[SUHVVHG DV WKH SURGXFW RI WKH IRUP IDFWRU IRU WKH LGHDO SDUWLFOH DQG WKH )RXULHU WUDQVIRUP RI KUf )K4f $V GHVFULEHG SUHYLRXVO\ KUf LV RIWHQ DSSUR[LPDWHG E\ D *DXVVLDQ KDYLQJ D ZLGWK SDUDPHWHU HTXDO WR T ,Q WKLV FDVH WKH WRWDO VWUXFWXUH IDFWRU WDNHV WKH IRUP 64f a & 4 H[S N 4f f &RQVHTXHQWO\ D SORW RI 4 ,4f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f GHSHQGV RQ WKH DYHUDJH YROXPH SHU SDUWLFOH 9M WKH WHPSHUDWXUH 7 DQG WKH LQWHUDFWLRQ SRWHQWLDO -!Uf G!Uf LV WKH WZR SDUWLFOH SRWHQWLDO L H WKH SRWHQWLDO HQHUJ\ RI D V\VWHP RI WZR SDUWLFOHV VHSDUDWHG E\ D GLVWDQFH U 7KH GHWHUPLQDWLRQ RI 3Uf IURP WKH LQWHUDFWLRQ SRWHQWLDO LV DQ H[WUHPHO\ GLIILFXOW SUREOHP ZKLFK KDV EHHQ H[WHQVLYHO\ VWXGLHGA 7KH DSSURDFK GHYHORSHG E\ %RP DQG *UHHQ DVVXPHV D PRGLILHG 0D[ZHOO%ROW]PDQ GHSHQGHQFH RI 3Uf RQ 2Uf 7KHLU ILQDO UHVXOW IRU WKH OLTXLG VWUXFWXUH IDFWRU LV 64f 1 )4f )4f 8MWfbS4f f ZKHUH WKH IXQFWLRQ 4f LV GHILQHG DV H[S 3Uf .E7M U VLQ 4Uf GU f

PAGE 62

+HUH H LV D FRQVWDQW RI WKH V\VWHP ZKLFK LV RQ WKH RUGHU RI XQLW\ $V DQ LOOXVWUDWLRQ RI WKH DERYH IRUP (TXDWLRQ LV HYDOXDWHG IRU WKH KDUG VSKHUH SRWHQWLDO 2Uf } U 5Q U 5Q f Y 7KH UHVXOWLQJ SURILOHV IRU IRXU YDOXHV RI f§ ZKHUH Y LV WKH SDUWLFOH YROXPH DUH SORWWHG LQ )LJXUH ,QWHUIHUHQFH HIIHFWV VKRZ XS LQ WKH IRUP RI D UHGXFWLRQ LQ LQWHQVLW\ DW YHU\ ORZ 4 DQG DQ HQKDQFHPHQW DW LQWHUPHGLDWH 4 4 ƒnf 7KH UHVXOW LV D EURDG PD[LPXP ZKLFK JURZV ZLWK LQFUHDVLQJ FRQFHQWUDWLRQ 6FDWWHULQJ DW KLJKHU 4 4 ƒnf UHPDLQV PRVWO\ XQDIIHFWHG ZLWK WKH H[FHSWLRQ RI D VXEWOH VKLIW LQ WKH SRVLWLRQ RI WKH SULPDU\ VSKHULFDO VFDWWHULQJ PD[LPXP 4 ƒn f 7KH 9DQ GHU :DDOV IRUFH LV WKH RULJLQ IRU WKH LQWHUDFWLRQ FRPPRQO\ RFFXUULQJ LQ PDQ\ V\VWHPV 7KH SRWHQWLDO GHVFULELQJ WKLV LQWHUDFWLRQ LV ZHOO DSSUR[LPDWHG E\ WKH /HQQDUG-RQHV /f IRUP ? ? n UU F f f R f n 2 2 f§ f§ N,? 7KLV SRWHQWLDO DORQJ ZLWK (TXDWLRQV DQG ZDV VXFFHVVIXOO\ XVHG E\ )RXPHWf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

PAGE 63

/LTXLG 6WUXFWXUH )DFWRU )LJXUH 7KH +DUG 6SKHUH 0RGHO 6FDWWHULQJ 9HFWRU ƒ f 0RGHO VFDWWHULQJ SURILOHV DUH VKRZQ IRU V\VWHPV RI VSKHUHV LQWHUDFWLQJ YLD WKH KDUG FRUH SRWHQWLDO DW YDU\LQJ SDUWLFOH FRQFHQWUDWLRQV YA9T

PAGE 64

DUUDQJHPHQW RI SDUWLFOHV WKH DVVXPSWLRQ RI DQ LVRWURSLF GLVWULEXWLRQ RI SDUWLFOHV LV QR ORQJHU YDOLG 7KH VFDWWHULQJ IURP RUGHUHG VWUXFWXUHV ZLOO QRZ EH GLVFXVVHG 6FDWWHULQJ )URP 2UGHUHG 6WUXFWXUHV $ FU\VWDO LV D WKUHH GLPHQVLRQDO SHULRGLF VWUXFWXUH 7KH UHSHDW XQLW IRU WKH VWUXFWXUH LV FDOOHG WKH XQLW FHOO 7KH ODWWLFH LV FRQVWUXFWHG E\ WUDQVODWLQJ WKH XQLW FHOO E\ OLQHDU FRPELQDWLRQV RI WKH EDVLV YHFWRUV GHVLJQDWHG D D D f $ VWUXFWXUH FRQVWUXFWHG LQ WKLV ZD\ FDQ EH FRQVLGHUHG WR EH FRPSRVHG RI SODQHV RI SDUWLFOHV DV LV VKRZQ LQ )LJXUH 7KH VFDWWHULQJ DPSOLWXGH IURP VXFK D VWUXFWXUH ZLOO EH GHWHUPLQHG E\ WKH SKDVH GLIIHUHQFH EHWZHHQ ZDYHV GLIIUDFWHG IURP DGMDFHQW SODQHV ,I WKH SDWK GLIIHUHQFH LV HTXDO WR DQ LQWHJHU QXPEHU RI ZDYHOHQJWKV WKHUH ZLOO EH FRQVWUXFWLYH LQWHUIHUHQFH RWKHUZLVH WKH VFDWWHULQJ DPSOLWXGH ZLOO EH ]HUR 7KLV LV D TXDOLWDWLYH VWDWHPHQW RI %UDJJnV /DZ ZKLFK LQ LWV PRVW FRPPRQ IRUP LV P; G VLQ f f +HUH G GHVLJQDWHV WKH GLVWDQFH EHWZHHQ DGMDFHQW SODQHV ; WKH ZDYHOHQJWK RI WKH VFDWWHUHG ZDYHV DQG P WKH RUGHU RI VFDWWHULQJ ,Q JHQHUDO DGMDFHQW SODQHV DUH VHSDUDWHG E\ ‘e DQG M LQ WKH WKUHH FU\VWDOORJUDSKLF GLUHFWLRQV 7KH LQWHJHUV KNOf DUH FDOOHG WKH 0LOOHU LQGLFHV DQG DUH XVHG WR VSHFLI\ FU\VWDOORJUDSKLF SODQHV &DOFXODWLQJ WKH GLVWDQFH EHWZHHQ DQG WKH RULHQWDWLRQ RI HYHU\ JURXS RI FU\VWDOORJUDSKLF SODQHV LV D IRUPLGDEOH WDVN 7KH FRQFHSW RI D UHFLSURFDO ODWWLFH LV XVHIXO LQ GHDOLQJ ZLWK WKLV W\SH RI FDOFXODWLRQ ,Q WKLV WKHRUHWLFDO FRQVWUXFWLRQ WKUHH UHFLSURFDO ODWWLFH YHFWRUV DUH GHILQHG N D [ D f EM E W D[Df DL f D [ D f W D [ Df E D f D [ D f DOn D [ D f

PAGE 65

4 )LJXUH $Q LOOXVWUDWLRQ RI %UDJJ VFDWWHULQJ IURP D FU\VWDO ODWWLFH

PAGE 66

)URP YHFWRU DOJHEUD RQH LV DEOH WR VKRZ WKDW WKH %UDJJ FRQGLWLRQ LV HTXLYDOHQW WR WKH UHTXLUHPHQW WKDW WKH VFDWWHULQJ YHFWRU 4 EH D OLQHDU FRPELQDWLRQ RI WKH UHFLSURFDO ODWWLFH YHFWRUV 4 KEL NE OE f :LWKLQ WKLV FRQVWUXFWLRQ %UDJJ SHDNV DUH HDVLO\ LGHQWLILHG DQG GHVLJQDWHG E\ WKH 0LOOHU LQGLFHV K N DQG )RU %UDJJnV ODZ WR EH VDWLVILHG WKH DQJOH RI LQFLGHQFH UHODWLYH WR WKH VHW RI VFDWWHULQJ SODQHV PXVW HTXDO WKH DQJOH RI UHIOHFWLRQ 7KLV LPSOLHV WKDW 4 LV UHTXLUHG WR EH SHUSHQGLFXODU WR WKHVH SODQHV DQG IRU DQ RUWKRUKRPELF V\VWHP KDV PDJQLWXGH A N 4 G OW 7K BN 7 8L r D f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f 64f $ $ VLQ 1 4D f VLQ 1 4D f VLQ 1 4D f VLQ 4DM f VLQ 4D f VLQ 4D f f ZKHUH $4f 9 4U =M+4f H A c L

PAGE 67

,Q WKH DERYH H[SUHVVLRQ WKH VFDWWHULQJ DPSOLWXGH $4f LV D IXQFWLRQ RI WKH IRUP IDFWRUV )4f IRU Q SDUWLFOHV DW SRVLWLRQV U ZLWKLQ WKH XQLW FHOO 7KLV IXQFWLRQ LV SHDNHG ZKHQ WKH IROORZLQJ WKUHH FRQGLWLRQV DUH VDWLVILHG 4DM QK 4 D UFN 4D Q? f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cD \cD ]LD f ZKHUH [c \c DQG ]c DUH EHORQJ WR WKH LQWHUYDO >@ 7KH VFDWWHULQJ DPSOLWXGH FDQ QRZ EH UHZULWWHQ DV $4f LL ]r 4fH LQ K[c N\c O]cf f c L )RU D ERG\FHQWHUHG FXELF VWUXFWXUH %&& f WKHUH DUH WZR SDUWLFOHV LQ WKH XQLW FHOO KDYLQJ FRRUGLQDWHV [ \ ] DQG [ \ = 7KH VFDWWHULQJ DPSOLWXGH IRU LGHQWLFDO SDUWLFOHV LV

PAGE 68

$4f K N RGG f $4f )4f K N HYHQ 6R WKH FU\VWDO VWUXFWXUH IDFWRU 6%&&4f $ $ YDQLVKHV DW WKRVH SHDN ORFDWLRQV ZKHUH KN LV RGG DQG DW WKRVH ORFDWLRQV ZKHUH KN LV HYHQ WKH SHDN LV SUHVHQW KDYLQJ DQ HIIHFWLYH IRUP IDFWRU HTXDO WR WZLFH WKDW RI WKH FRQVWLWXHQW SDUWLFOHV ,Q D VLPLODU ZD\ WKH VFDWWHULQJ DPSOLWXGH FDQ EH GHWHUPLQHG IRU WKH IDFHFHQWHUHG FXELF )&& f VWUXFWXUH $4f K N PL[HG f $4f )4f K N XQPL[HG 2QO\ WKRVH SHDNV GHQRWHG E\ LQGLFHV WKDW DUH DOO HYHQ RU DOO RGG ZLOO EH SUHVHQW L H ,OOf f f HWFf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f ,Q [UD\ VFDWWHULQJ ZH PHDVXUH WKH WHPSRUDO DYHUDJH LQGLFDWHG E\ WULDQJXODU EUDFNHWV f RI WKH VFDWWHUHG LQWHQVLW\ ZKLFK LV SURSRUWLRQDO WR WKH VWUXFWXUH IDFWRU

PAGE 69

45N5f L&.A2AWff 64f ? =Z)LF4f)4f H H f ,I XNWf DQG 8MWf DUH GHILQHG WR EH WKH DPSOLWXGHV RI RVFLOODWLRQ LQ WKH GLUHFWLRQ SDUDOOHO WR 4 WKHQ HTXDWLRQ FDQ EH UHZULWWHQ DV 64f 64f 6GLII4f f 64f O)N4fHn c4 XN r4 UUULf )4fH H N 7KH VHFRQG FRQWULEXWLRQ LQ (TXDWLRQ 6GLII FRQWDLQV WKH FURVV WHUP XNX ZKLFK LV YDQLVKLQJO\ VPDOO XQOHVV WKH NWK DQG WK SDUWLFOHV DUH YHU\ FORVH 7KH UHVXOW LV D VFDWWHULQJ FRPSRQHQW WKDW LV VORZO\ YDU\LQJ ZLWK 4 DQG RI YHU\ ORZ DPSOLWXGH 7KLV FRQWULEXWLRQ WR WKH VFDWWHULQJ LV FDOOHG WKH WKHUPDO GLIIXVH VFDWWHULQJ DQG LV DSSUR[LPDWHO\ FRQVWDQW LQ WKH KLJK 4 UHJLRQ RI WKH VSHFWUXP 7KH ILUVW WHUP 6M LV LGHQWLFDO WR WKH UHVXOW IRU WKH VWDWLRQDU\ ILQLWHVL]HG FU\VWDO ZLWK WKH IRUP IDFWRU UHSODFHG E\ DQ HIIHFWLYH IRUP IDFWRU )HII4f )4f H[S 4 X !f f 7KH H[SRQHQWLDO LQ WKH DERYH HTXDWLRQ LV FDOOHG WKH 'HE\H:DOOHU IDFWRU 7KH SHDN ZLGWKV DQG SRVLWLRQV DUH XQFKDQJHG E\ GLVSODFHPHQW RVFLOODWLRQV EXW WKH SHDN DPSOLWXGHV DUH PRGXODWHG E\ WKLV IDFWRU 7KH (IIHFW RI 6WUDLQ ,Q D W\SLFDO H[SHULPHQW VFDWWHULQJ RFFXUV QRW IURP RQH EXW IURP PDQ\ FU\VWDOOLWHV ,Q JHQHUDO WKHUH H[LVW UDQGRP VWUDLQ ILHOGV ZLWKLQ WKH VDPSOH 7KH UHVXOW LV D FROOHFWLRQ RI JUDLQV KDYLQJ D QDUURZ GLVWULEXWLRQ RI ODWWLFH FRQVWDQWV FHQWHUHG DERXW WKH

PAGE 70

HTXLOLEULXP YDOXH ,I ZH GHILQH WKH VWUDLQ H $DD WKHQ WKH GLVWULEXWLRQ RI VWUDLQV PD\ EH DSSUR[LPDWHG E\ 3Hf H[S f ?WRH D f )RU VLPSOLFLW\ ZH KDYH DVVXPHG DQ LVRWURSLF JDXVVLDQ GLVWULEXWLRQ RI VWUDLQV DQG D FXELF ODWWLFH 7KH PDJQLWXGH RI WKH VFDWWHULQJ YHFWRU DW WKH EUDJJ SHDN SRVLWLRQ LV UHODWHG WR WKH ODWWLFH FRQVWDQW YLD (T 7KH %UDJJ SHDN ZLOO EH VKLIWHG E\ $4 LI WKH ODWWLFH FRQVWDQW LV VWUHWFKHG E\ $D 7KH SHDN VKLIW DQG VWUDLQ DUH UHODWHG LQ WKH IROORZLQJ ZD\ $4 4 Df $D 4 H f VR WKDW WKH GLVWULEXWLRQ RI SHDN RIIVHWV LV JLYHQ E\ $4f 3$4f f§ H[S a f rT 4DH f ?7*T RT 7KH VFDWWHUHG LQWHQVLW\ LV WKH FRQYROXWLRQ RI WKLV IXQFWLRQ ZLWK WKH SURILOH IRU D VWUDLQIUHH V\VWHP 7KH VFDWWHULQJ SURILOH IRU D VWUDLQHG V\VWHP LQ WKLV DSSUR[LPDWLRQ LV WKH FRQYROXWLRQ RI WZR JDXVVLDQV ZKLFK LV LWVHOI D JDXVVLDQ KDYLQJ D ZLGWK HTXDO WR WKH VXP RI WKH LQGLYLGXDO ZLGWKV DGGHG LQ TXDGUDWXUH 44Rf ,4f $ 2IDf H[Sf§f D *I D4f f f 6R WKH HIIHFW RI VWUDLQ LV WR EURDGHQ WKH SHDN E\ DV4 ZLWK DQ DFFRPSDQ\LQJ GHFUHDVH LQ WKH %UDJJ DPSOLWXGH VR WKDW WKH LQWHJUDWHG LQWHQVLW\ RI WKH SHDN LV FRQVHUYHG ,W LV HYLGHQW WKDW VWUDLQ EURDGHQLQJ LV PXFK PRUH SURQRXQFHG LQ WKH KLJKHU RUGHU SHDNV 7KH 3RZGHU 3DWWHUQ ,I WKH RULHQWDWLRQ RI WKH FU\VWDOOLWHV LV UDQGRP WKH VDPSOH LV FDOOHG D SRZGHU 7KH VFDWWHULQJ ZLOO WKHQ EH D IXQFWLRQ RI WKH PDJQLWXGH RI 4 RQO\ $VVXPLQJ D *DXVVLDQ

PAGE 71

OLQHVKDSH IRU WKH QRQSRZGHU DYHUDJHG %UDJJ UHIOHFWLRQ WKH SRZGHU DYHUDJHG VWUXFWXUH IDFWRU FDQ EH FDOFXODWHG 64f WL f H[3 44Rfn R f +HUH WKH EUDFNHWV LQGLFDWH DQ DYHUDJH RI WKH VWUXFWXUH IDFWRU RYHU DOO RULHQWDWLRQV RI WKH %UDJJ FHQWHU 4 UHODWLYH WR WKH VFDWWHULQJ YHFWRU 4 7KH IRUP IDFWRU ZKLFK KDV EHHQ LQFRUSRUDWHG LQWR $4f LV DVVXPHG WR EH LVRWURSLF :LWKRXW ORVV RI JHQHUDOLW\ 4 LV FKRVHQ WR OLH LQ WKH ] GLUHFWLRQ 4 4] 4 4RVLQfFRV^!f [ 4RVLQfVLQW!f\ 4RVLQf] f +HUH DQG `f DUH WKH FRQYHQWLRQDO VSKHULFDO FRRUGLQDWHV 7KH DYHUDJH RYHU ?! DQG \LHOGV 64f DWFJ44 I 44Rf@ 44Rf@ H[S R H[S D f 7KH VHFRQG H[SRQHQWLDO LV QHJOLJLEOH IRU DOO SRVLWLYH YDOXHV RI 4 7KHUH LV DQ DGGLWLRQDO IDFWRU FDOOHG WKH PXOWLSOLFLW\ PKNf ZKLFK DULVHV IURP WKH IDFW WKDW %UDJJ SHDNV KDYLQJ GLIIHUHQW LQGLFHV K N DSSHDU LQ WKH VDPH ORFDWLRQ LQ WKH SRZGHU VSHFWUXP 7KH LQWHQVLW\ LV SURSRUWLRQDO WR WKH GHJHQHUDF\ RI WKH %UDJJ UHIOHFWLRQ $V H[DPSOHV WKH PXOWLSOLFLWLHV RI VHYHUDO SHDNV IRU D VLPSOH FXELF ODWWLFH DUH OLVWHG EHORZ P P KNOf f +2f f P f

PAGE 72

,W LV SRVVLEOH WR LGHQWLI\ GLIIHUHQW FU\VWDO VWUXFWXUHV E\ FRPSDULQJ WKH UHODWLYH DPSOLWXGHV RI WKH REVHUYHG %UDJJ SHDNV GXH WR GLIIHUHQFHV LQ WKH PXOWLSOLFLWLHV RI WKH GLIIHUHQW VWUXFWXUHV 7KLV ZLOO EH GLVFXVVHG LQ JUHDWHU GHWDLO LQ &KDSWHU ,I WKH HIIHFWV RI WHPSHUDWXUH VWUDLQ DQG SRZGHUDYHUDJLQJ DUH LQFOXGHG WKH UHVXOWLQJ VWUXFWXUH IDFWRU IRU D VLQJOH %UDJJ SHDN LV 44RfO 64f $KNO4f H[S ZKHUH DQG R PKN> )4f H[SX!4f $KX4f$R \-].*4f 44 R4f >F" RV4fn_ f 7KH DSSOLFDWLRQ RI WKHVH UHVXOWV WR WKH EORFN FRSRO\PHU 6$;6 VSHFWUD ZLOO EH GLVFXVVHG LQ &KDSWHU IROORZLQJ D UHYLHZ RI WKH EORFN FRSRO\PHU OLWHUDWXUH LQ &KDSWHU DQG D GLVFXVVLRQ RI WKH H[SHULPHQWDO DSSDUDWXV LQ &KDSWHU

PAGE 73

&+$37(5 %$&.*5281' $1' /,7(5$785( 5(9,(: 2) %/2&. &232/<0(56 7KH SRO\PHU PROHFXOH LV D ORQJ IOH[LEOH FKDLQ FRPSRVHG RI FRYDOHQWO\ ERQGHG UHSHDW XQLWV FDOOHG PRQRPHUV 3RO\PHULF PDWHULDOV KDYH ORQJ EHHQ RI LQWHUHVW WR FKHPLVWV DQG PDWHULDOV VFLHQWLVWV GXH ODUJHO\ WR WKHLU DGYDQWDJHRXV PHFKDQLFDO SURSHUWLHV DQG ZHUH RQO\ PXFK ODWHU UHFRJQL]HG DV DQ H[FHOOHQW V\VWHP IRU WKH VWXG\ RI SK\VLFV 0DMRU H[SHULPHQWDO WRROV WKDW KDYH SURYHG WR EH H[WUHPHO\ XVHIXO LQ UHFHQW VWXGLHV RI SRO\PHU VWUXFWXUH PRUSKRORJ\ DQG G\QDPLFDO EHKDYLRU LQFOXGH VPDOO DQJOH QHXWURQ VFDWWHULQJ 6$16 f DQG V\QFKURWURQ [UD\ GLIIUDFWLRQ 2XWVWDQGLQJ WKHRUHWLFDO DGYDQFHPHQWV KDYH EHHQ PDGH HVSHFLDOO\ WKURXJK WKH DSSOLFDWLRQ RI UHQRUPDOL]DWLRQ JURXS LGHDV WR WKH VWXG\ RI FKDLQ FRQIRUPDWLRQV GXH SULPDULO\ WR WKH RULJLQDO ZRUN E\ GH *HQQHV LQ WKLV ILHOG 3RO\PHUVf§*HQHUDO %DFNJURXQG 7KH VL]H RI D VLQJOH SRO\PHU PROHFXOH LV W\SLFDOO\ GHVFULEHG E\ WKH SRO\PHUL]DWLRQ LQGH[ 1 f ZKHUH 0 LV WKH PDVV RI WKH HQWLUH FKDLQ DQG 0T LV WKH PDVV RI D VLQJOH PRQRPHU *HQHUDOO\ D SRO\PHU VDPSOH LV FRPSRVHG RI D GLVWULEXWLRQ RI FKDLQV KDYLQJ YDU\LQJ WRWDO PDVVHV 7KH PDVV GLVWULEXWLRQ FDQ EH FKDUDFWHUL]HG E\ WKUHH SDUDPHWHUV WKH QXPEHU DQG ZHLJKW DYHUDJH PROHFXODU ZHLJKWV 0f DQG 0Z UHVSHFWLYHO\ DQG WKH SRO\GLVSHUVLW\ LQGH[ Q 7KH QXPEHU DYHUDJH PROHFXODU ZHLJKW LV GHILQHG DV ;QP (1c f

PAGE 74

+HUH 1c LV WKH QXPEHU RI SRO\PHU FKDLQV RI PDVV 0c LQ WKH GLVWULEXWLRQ 6LPLODUO\ WKH ZHLJKW DYHUDJH PROHFXODU ZHLJKW LV JLYHQ E\ ;1L 0M 0 ;1c0c f 7KH SRO\GLVSHUVLW\ LV D PHDVXUH RI WKH ZLGWK RI WKH PDVV GLVWULEXWLRQ DQG LV GHILQHG WR EH WKH UDWLR RI 0Z DQG 0 ,W LV HDVLO\ UHODWHG WR WKH VWDQGDUG GHYLDWLRQ RI WKH GLVWULEXWLRQ D WKURXJK WKH HTXDWLRQ R 0 0 f Q f f 7RJHWKHU WKH DERYH SDUDPHWHUV GHVFULEH WKH PDVV GLVWULEXWLRQ RI D FROOHFWLRQ RI SRO\PHU FKDLQV 7KH VL]H RI LQGLYLGXDO FKDLQV LV GHVFULEHG E\ WKH HQGWRHQG GLVWDQFH U ,I HDFK PRQRPHU LV RI OHQJWK WKHQ WKH UPV YDOXH RI U FDQ EH GHWHUPLQHG E\ FRQVLGHULQJ D SRO\PHU FKDLQ DV D UDQGRP ZDON FRQWDLQLQJ 1 VWHSV RI OHQJWK D ,W LV ZHOO NQRZQ WKDW IRU DQ XQFRQVWUDLQHG RU LGHDO UDQGRP ZDON WKH UHVXOW LV U 1 3 f :LWKLQ WKLV PRGHO WKH HQWURS\ RI D VLQJOH SRO\PHU FDQ EH FDOFXODWHG DQG DW IL[HG HQG WR HQG GLVWDQFH LV 6Uf 6fA f )URP WKH IUHH HQHUJ\ UHODWLRQ ) 876 LW LV HYLGHQW WKDW WKH FKDLQ EHKDYHV DV D +RRNHfV /DZ VSULQJ ZKHQ WKH HQWURS\ WHUP GRPLQDWHV WKH IUHH HQHUJ\ 7KH HIIHFWLYH VSULQJ FRQVWDQW LQFUHDVHV ZLWK LQFUHDVLQJ WHPSHUDWXUH 7KLV LOOXVWUDWHV WKH SK\VLFDO EDVLV IRU WKH FRQWUDFWLRQ RI SRO\PHU PDWHULDOV ZKLFK RFFXUV XSRQ KHDWLQJ 7KH DERYH FDOFXODWLRQ LV IODZHG LQ WKDW FKDLQ FRQIRUPDWLRQV DUH DOORZHG LQ ZKLFK WKH FKDLQ ORRSV EDFN RQ LWVHOI VR WKDW WZR RU PRUH PRQRPHUV RFFXS\ WKH VDPH SK\VLFDO VSDFH &DOFXODWLRQV ZKLFK GR QRW DOORZ WKHVH FRQIRUPDWLRQV DUH FDOOHG VHOI DYRLGLQJ ZDONV 6$:V f DQG IRU GLOXWH SRO\PHU VROXWLRQV \LHOG D UHVXOW VLPLODU WR WKDW IRU WKH LGHDO UDQGRP ZDON

PAGE 75

U !!m 1Y f +HUH Y LV FDOOHG WKH )ORU\ H[SRQHQW 7KHRUHWLFDO DQG H[SHULPHQWDO VWXGLHV VKRZ Y ,Q WKH FRQFHQWUDWHG OLPLW FKDLQV FDQ QR ORQJHU EH FRQVLGHUHG DV LQGHSHQGHQW UDQGRP ZDONV DQG WKH UHVXOW DOWKRXJK PXFK PRUH GLIILFXOW WR REWDLQ LV WKH VDPH DV WKDW IRU WKH LGHDO UDQGRP ZDON (TXDWLRQ 3RO\PHU FKDLQV DUH WKHUHIRUH VZROOHQ LQ GLOXWH VROXWLRQV FRQWDLQLQJ D JRRG VROYHQW DQG H[KLELW LGHDO L H UDQGRP ZDON f FRQIRUPDWLRQV LQ WKH OLPLW RI D FRQFHQWUDWHG SRO\PHU VROXWLRQ RU D SXUH SRO\PHU PHOW 7KH UDGLXV RI J\UDWLRQ 5J LV D SDUDPHWHU FRQYHQLHQWO\ PHDVXUHG LQ ERWK OLJKW VFDWWHULQJ DQG VPDOO DQJOH [UD\ VFDWWHULQJ H[SHULPHQWV DQG LV GHILQHG DV f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

PAGE 76

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f f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f +HUH (;< LV WKH HQHUJ\ FRUUHVSRQGLQJ WR DQ ;< QHDUHVWQHLJKERU FRQILJXUDWLRQ DQG N%7 LV WKH WKHUPDO HQHUJ\ 7KH SRO\PHU V\VWHP LV W\SLFDOO\ DVVXPHG WR EH LQFRPSUHVVLEOH VR WKDW WKH YROXPH IUDFWLRQV RI $ DQG % PRQRPHUV FDQ EH ZULWWHQ 2D -! DQG

PAGE 77

2E 7KH IUHH HQHUJ\ IRU WKH PL[WXUH LQFOXGLQJ HQWURS\ RI PL[LQJ WHUPV LV ) N%7 OQ r N%7 ,Q f [ 2 f f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f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

PAGE 78

ZKLFK LV VWURQJO\ GHSHQGHQW RQ WKH UHFLSURFDO VSDFH YHFWRU 4 DQG DV D UHVXOW DOWHUV WKH HTXDWLRQV RI PRWLRQ IRU WKH V\VWHP 1HYHUWKHOHVV 1LVKLnV JURXS IRXQG WKDW HDUO\ WLPH VSLQRGDO GHFRPSRVLWLRQ EHKDYLRU DJUHHV ZHOO ZLWK &DKQnV OLQHDUL]HG PRGHO RI VSLQRGDO GHFRPSRVLWLRQ D UHVXOW ZKLFK LV H[SHFWHG LQ WKH SUHVHQFH RI ORQJUDQJH LQWHUDFWLRQV EXW QHYHU REVHUYHG LQ PHWDOV 0F0DVWHU DOVR REVHUYHG D VSLQRGDO SRLQW LQ KLV OLJKW VFDWWHULQJ VWXG\ RI SRO\PHU EOHQGV ,Q DGGLWLRQ WKH FRDUVHQLQJ EHKDYLRU REVHUYHG IROORZLQJ TXHQFKHV LQWR WKH PHWDVWDEOH WHPSHUDWXUH UHJLPH ZDV FRQVLVWHQW ZLWK WKH /LIVKLW]6O\R]RY WKHRU\ GHVFULEHG LQ &KDSWHU +RZHYHU FRDUVHQLQJ IROORZLQJ TXHQFKHV LQWR WKH XQVWDEOH UHJLPH RFFXUUHG PXFK PRUH UDSLGO\ 0F0DVWHU DFFRXQWHG IRU WKLV EHKDYLRU LQYRNLQJ D YLVFRXV IORZ PHFKDQLVP EDVHG RQ PRGHOV GHYHORSHG E\ 7RPRWLND 0XFK ZRUN KDV \HW WR EH GRQH RQ SKDVH VHSDUDWLRQ LQ SRO\PHU EOHQGV EXW SUHVHQW UHVXOWV LQGLFDWH WKDW VWXGLHV RI WKHVH V\VWHPV EHFDXVH RI WKHLU VORZ PROHFXODU GLIIXVLRQ FDQ \LHOG YDOXDEOH LQIRUPDWLRQ WRZDUG XQGHUVWDQGLQJ NLQHWLF HIIHFWV %ORFN &RSRO\PHUV %XON 3URSHUWLHV $V LQWURGXFHG LQ &KDSWHU WKH EORFN FRSRO\PHU %&3f LV FRPSRVHG RI WZR RU PRUH FKHPLFDOO\ GLVWLQFW SRO\PHU FKDLQV MRLQHG E\ D FRYDOHQW ERQG :KHQ WKH EORFN FKDLQV DUH LQFRPSDWLEOH SKDVH VHSDUDWLRQ VLPLODU WR WKDW RFFXUULQJ LQ SRO\PHU EOHQGV WDNHV SODFH 7KH IXQGDPHQWDO SURSHUWLHV RI EXON EORFN FRSRO\PHUV DUH VXPPDUL]HG LQ D UHYLHZ DUWLFOH E\ +DVKLPRWR HW DO $V GHVFULEHG LQ &KDSWHU EORFN FRSRO\PHUV H[KLELW D GLVVROXWLRQ WHPSHUDWXUH 7G EHORZ ZKLFK PLFURSKDVH VHSDUDWLRQ WUDQVLWLRQ 067 f UHVXOWV LQ WKH ODPHOODU F\OLQGULFDO DQG VSKHULFDO PLFURVWXFWXUHV VKRZQ LQ )LJXUH DQG DERYH ZKLFK WKH V\VWHP LV KRPRJHQHRXV 7KH PRUSKRORJ\ RI WKH PLFURVWXFWXUH KDV EHHQ IRXQG ERWK H[SHULPHQWDOO\ DQG WKHRUHWLFDOO\ WR GHSHQG RQO\ RQ WKH IUDFWLRQDO FRPSRVLWLRQ I RI WKH EORFN FRSRO\PHU DQG LV EHOLHYHG WR EH LQGHSHQGHQW RI WRWDO

PAGE 79

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f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

PAGE 80

2QH RI WKH ILUVW DWWHPSWV WR SUHGLFW WKH GLPHQVLRQV RI WKH HTXLOLEULXP VWUXFWXUH LV WKDW RI +HOIDQG DQG :DVVHUPDQ :LWKLQ WKHLU PRGHO WKH VWUXFWXUH LV FRPSRVHG RI FKDLQV LQGLYLGXDOO\ FRQVWUXFWHG IURP VHOI DYRLGLQJ UDQGRP ZDONV FRQILQHG WR WKH DSSURSULDWH GRPDLQ VSDFH ,Q WKH OLPLW RI VWURQJ VHJUHJDWLRQ IRU WKH ODPHOODU PRUSKRORJ\ WKH\ IRXQG WKH IROORZLQJ VFDOLQJ UHODWLRQ EHWZHHQ WKH SRO\PHUL]DWLRQ LQGH[ 1 DQG WKH PLFURGRPDLQ SHULRGLFLW\ G G1 f 6LPLODU UHVXOWV IRU WKH YDOXH RI ZHUH REWDLQHG IRU WKH F\OLQGULFDO DQG VSKHULFDO PRUSKRORJLHV 7KH UHVXOW JLYHQ LQ (TXDWLRQ ZDV VXSSRUWHG E\ VWXGLHV RI WKH SRO\VW\UHQH SRO\LVRSUHQH V\VWHP E\ +DVKLPRWR HW DO 7KH\ FDVW SRO\PHU ILOPV IURP VROXWLRQV KDYLQJ IUDFWLRQDO FRPSRVLWLRQV WKDW \LHOG WKH ODPHOODU DQG VSKHULFDO VWUXFWXUHV DW YDU\LQJ WRWDO GHJUHHV RI SRO\PHUL]DWLRQ 7KH ILOPV ZHUH VWXGLHG ZLWK WKH VPDOO DQJOH [UD\ VFDWWHULQJ WHFKQLTXH )URP WKH 6$;6 SURILOHV WKH SHULRGLFLWLHV RI WKH ODPHOODU VWUXFWXUHV ZHUH FDOFXODWHG DV ZHOO DV WKH ODWWLFH FRQVWDQWV DQG VSKHULFDO UDGLL 5 RI WKH VSKHULFDO VWUXFWXUHV ,Q DGGLWLRQ +DVKLPRWRfV JURXS VWXGLHG WKH V\VWHPDWLF GHYLDWLRQV IURP 3RURGnV /DZ LQ WKH KLJK 4 UHJLRQ RI WKH VFDWWHUHG SURILOHV GLVFXVVHG LQ &KDSWHU f WR HVWLPDWH WKH WKLFNQHVV RI WKH LQWHUIDFLDO UHJLRQ VHSDUDWLQJ SRO\VW\UHQH DQG SRO\LVRSUHQH GRPDLQV 7KHLU UHVXOWV LQGLFDWH WKH IROORZLQJ VFDOLQJ UHODWLRQV IRU WKH ODPHOODU VDPSOHV Ga10 f A 1r )RU WKH VSKHULFDO VWUXFWXUH WKH VDPH UHODWLRQV ZHUH IRXQG WR KROG DQG LQ DGGLWLRQ WKH IROORZLQJ HPSLULFDO UHODWLRQVKLSV ZHUH REVHUYHG 5a1 3M a 1 f

PAGE 81

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f %\ GHXWHUDWLQJ WKH SRO\VW\UHQH EORFN RI WKH SRO\PHU WKH\ ZHUH DEOH WR GHWHUPLQH WKH UDGLXV RI J\UDWLRQ RI WKHVH FKDLQV IURP WKH 6$16 SURILOH ,Q WKHLU VROYHQW FDVW ILOPV WKH ODPHOODH SUHIHUHQWLDOO\ OLH SDUDOOHO WR WKH ILOP VXUIDFH %\ VFDWWHULQJ IURP WKH ILOP HGJH DQG VXUIDFH WKH\ FRXOG PHDVXUH 5J ERWK SDUDOOHO DQG SHUSHQGLFXODU WR WKH GRPDLQ VXUIDFH UHVSHFWLYHO\ 7KHLU UHVXOWV LQGLFDWH DQ HORQJDWLRQ RI WKH FKDLQV E\ b SHUSHQGLFXODU WR WKH ODPHOODU VXUIDFH DQG D FRQWUDFWLRQ RI b LQ HDFK RI WKH SDUDOOHO GLUHFWLRQV $ YDVW DPRXQW RI ZRUN RQ VLPLODU V\VWHPV DOVR VXSSRUWV WKHVH VFDOLQJ UHODWLRQV IRU ODPHOODUn F\OLQGULFDO DQG VSKHULFDO PLFURGRPDLQ PRUSKRORJLHV LQ WKH VWURQJ VHJUHJDWLRQ OLPLW /HLEOHU SUHVHQWHG D /DQGDX PHDQ ILHOG DQDO\VLV RI WKH 067 RFFXUULQJ LQ DQ $% GLEORFN FRSRO\PHU KDYLQJ D IUDFWLRQDO FRPSRVLWLRQ I D )ORU\+XJJLQV LQWHUDFWLRQ SDUDPHWHU b DQG D WRWDO SRO\PHUL]DWLRQ LQGH[ 1 7KH RUGHU SDUDPHWHU \Uf ZDV GHILQHG WR EH WKH ORFDO GHQVLW\ GHYLDWLRQ RI WKH $ PRQRPHU ?\$Uf IURP LWV PHDQ YDOXH LQ WKH KRPRJHQHRXV VWDWH ?_Uf ?\$Uf I 7KH VWDWLF VWUXFWXUH IDFWRU 64f LV GHILQHG DV 64f H[S L 4Uf $\$Uf $\$f GU f

PAGE 82

/HLEOHU H[SDQGV WKH IUHH HQHUJ\ LQ SRZHUV RI WKH RUGHU SDUDPHWHU 8VLQJ WKH UDQGRP SKDVH DSSUR[LPDWLRQ 64f DQG WKH IUHH HQHUJ\ DUH HYDOXDWHG WR IRXUWK RUGHU LQ \4f WKH IRXULHU WUDQVIRUP RI ?_Lff %\ DVVXPLQJ WKDW WKH PLQLPL]HG IUHH HQHUJ\ QHDU WKH WUDQVLWLRQ LV GRPLQDWHG E\ IOXFWXDWLRQV ZLWK 4 4r 4r 5Jf /HLEOHU LV DEOH WR HYDOXDWH WKH FRH[LVWHQFH DQG VSLQRGDO FXUYHV LQ WKH b1f I SODQH DQG GHWHUPLQH WKH OLPLWV RI VWDELOLW\ RI WKH GLIIHUHQW SRO\PHU PRUSKRORJLHV 7KH VWDEOH PRUSKRORJ\ LV IRXQG WR GHSHQG QRW RQO\ RQ WKH IUDFWLRQDO FRPSRVLWLRQ I DV VKRZQ LQ ILJXUH f EXW DOVR RQ WKH WHPSHUDWXUH 6XFK WUDQVLWLRQV L H IURP VSKHULFDO WR F\OLQGULFDO WR ODPHOODU ZLWK FKDQJLQJ WHPSHUDWXUH ZHUH ODWHU REVHUYHG H[SHULPHQWDOO\ ,Q DGGLWLRQ /HLEOHU DUJXHV WKDW WKH VWDEOH PDFURODWWLFH IRU WKH F\OLQGULFDO PRUSKRORJ\ LV WKH WZR GLPHQVLRQDO KH[DJRQDO FORVHSDFNHG VWUXFWXUH +&3f DQG IRU VSKHULFDO GRPDLQV LV WKH %&& VWUXFWXUH 7KHUH KDYH VLQFH EHHQ UHSRUWV RI D YDULHW\ RI GLIIHUHQW PDFURODWWLFHV %&& DV ZHOO DV )&& DQG 6& f REVHUYHG IRU WKH VSKHULFDO GRPDLQV 7KH DERYH IRUPXODWLRQ \LHOGV D VFDOLQJ UHODWLRQ EHWZHHQ WKH SHULRGLFLW\ G 4r DQG WKH SRO\PHUL]DWLRQ LQGH[ 1 G 1 f IRU WKH ZHDN VHJUHJDWLRQ OLPLW ZKHUH WKH GHQVLW\ SURILOH LV D YDULHV VPRRWKO\ f GLVWLQFW IURP WKH UHODWLRQ GHULYHG IRU WKH VWURQJ VHJUHJDWLRQ OLPLW ZKHUH ZHOO GHILQHG LQWHUIDFHV H[LVW (TXDWLRQV DQG f 5HFHQW VXSSRUW KDV EHHQ JLYHQ WR WKLV VFDOLQJ UHODWLRQ E\ &KDNUDEDUWL HW DO ZKR SHUIRUPHG 0RQWH &DUOR VLPXODWLRQV XVLQJ D WZR GLPHQVLRQDO ODWWLFH PRGHO FRPSRVHG RI UHVWULFWHG VHOIDYRLGLQJ UDQGRP ZDONV 7KHLU UHVXOWV DQG H[SHULPHQWDO PHDVXUHPHQWV E\ 6KLED\DPD HW DO DOVR \LHOG WKH UHODWLRQ JLYHQ LQ (TXDWLRQ 2KWD DQG .DZDVDNL H[WHQGHG /HLEOHUnV WKHRU\ E\ LQFOXGLQJ WKH HIIHFWV RI WKH ORQJUDQJH HODVWLF LQWHUDFWLRQ 8VLQJ WKH UDQGRP SKDVH DSSUR[LPDWLRQ LQ WKH VWURQJ

PAGE 83

VHJUHJDWLRQ OLPLW WKH\ REWDLQ UHVXOWV VLPLODU WR WKRVH RI /HLEOHU DQG FDOFXODWH WKH FRUUHFW VFDOLQJ UHODWLRQ IRU WKLV OLPLW (TXDWLRQ f 0RUH UHFHQWO\ )UHGHULFNVRQ DQG +HOIDQG KDYH DWWHPSWHG WR LPSURYH XSRQ /HLEOHUn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f VWUXFWXUH KDV UHFHQWO\ EHHQ REVHUYHG LQ WKH SRO\VW\UHQHSRO\LVRSUHQH GLEORFN FRSRO\PHU 7KH VWUXFWXUH LV FRPSRVHG RI WHWUDSRG GRPDLQV SUREDEO\ UHVLGLQJ RQ WKH GRXEOHGLDPRQG ODWWLFH 7KLV VWUXFWXUH ZDV REVHUYHG LQ 6$;6 DQG HOHFWURQ PLFURVFRS\ VWXGLHV ZKHQ WKH YROXPH IUDFWLRQ RI SRO\VW\UHQH LV LQ WKH QDUURZ UDQJH EHWZHHQ DQG 7KH 2%'' PRUSKRORJ\ DSSHDUV EHWZHHQ WKH F\OLQGULFDO DQG ODPHOODU PRUSKRORJLHV LQ WKH DFFHSWHG PRUSKRORJ\ VFKHPH )LJXUH f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

PAGE 84

+LOOLDUG HTXDWLRQ 7KH WZR GLPHQVLRQDO SDWWHUQV UHVXOWLQJ IURP WKHLU FRPSXWDWLRQV DFFXUDWHO\ UHSURGXFHG WKRVH REVHUYHG LQ HOHFWURQ PLFURVFRS\ VWXGLHV 8VLQJ WKH UHVXOWV RI WKHLU FRPSXWDWLRQV GLPHQVLRQDO DQDO\VLV DQG FRPSDULVRQV WR HTXLOLEULXP WKHRULHV WKH\ REWDLQ D UHODWLRQ EHWZHHQ WKH H[SRQHQW GHVFULELQJ WKH HTXLOLEULXP SHULRGLFLW\ RI WKH PLFURSKDVH VWUXFWXUH (TXDWLRQ f DQG WKH H[SRQHQW GHVFULELQJ JURZWK RI WKH PLFURGRPDLQV GXULQJ VSLQRGDO GHFRPSRVLWLRQ /aWr f 7KDW UHODWLRQ LV ^f f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f $IWHU WKH VROXWLRQV ZHUH SUHSDUHG WKH\ ZHUH LOOXPLQDWHG ZLWK 89 OLJKW WR SRO\PHUL]H WKH VW\UHQH PRQRPHUV 7KURXJK WKLV SRVWSRO\PHUL]DWLRQ WHFKQLTXH WKH SRO\PHU VROXWLRQV DUH FRQYHUWHG LQWR D

PAGE 85

JHO IRU HOHFWURQ PLFURVFRS\ VWXGLHV SUHVXPDEO\ ZLWKRXW DOWHULQJ WKH PLFURVWUXFWXUH RI WKH VROXWLRQ 7KH UHVXOWLQJ VDPSOH ZHUH VWDLQHG ZLWK RVPLXP WHWUR[LGH DQG VWXGLHG ZLWK WUDQVPLVVLRQ HOHFWURQ PLFURVFRS\ 7(0 f )URP WKHVH H[SHULPHQWV 6DGURQ DQG *DOORW LGHQWLILHG WZR FULWLFDO FRQFHQWUDWLRQV &r & r‘ %HORZ &r WKH PL[WXUH ZDV KRPRJHQHRXV ,Q WKH UHJLRQ EHWZHHQ &M DQG & F\OLQGULFDO SRO\EXWDGLHQH GRPDLQV ZHUH REVHUYHG LQ D GLVRUGHUHG DUUDQJHPHQW $ERYH & WKH F\OLQGULFDO GRPDLQV ZHUH REVHUYHG LQ DQG RUGHUHG DUUDQJHPHQW )URP WKHVH UHVXOWV LW ZDV LQIHUUHG WKDW &r FRUUHVSRQGV WR WKH TXDQWLW\ &r REVHUYHG E\ 9DQ]R 6DGURQ DQG *DOORW DQG 6NRXOLRV DOVR REVHUYHG PRUSKRORJLFDO WUDQVLWLRQV L H IURP VSKHULFDO WR F\OLQGULFDO RU F\OLQGULFDO WR ODPHOODU PLFURVWUXFWXUHV f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r ,LLLLLL 7KH\ SHUIRUPHG UKHRORJLFDO DQG VPDOO DQJOH VFDWWHULQJ PHDVXUHPHQWV RI WKH HTXLOLEULXP VWUXFWXUH RI GLEORFN FRSRO\PHUV LQ ERWK VHOHFWLYH DQG QRQVHOHFWLYH VROYHQWV DV D IXQFWLRQ RI WHPSHUDWXUH DQG SRO\PHU FRQFHQWUDWLRQ $ EULHI GLVFXVVLRQ RI WKH HTXLOLEULXP DQG NLQHWLF UHVXOWV UHSRUWHG E\ +DVKLPRWR HW DO LV JLYHQ KHUH

PAGE 86

(TXLOLEULXP 0HDVXUHPHQWV 7KH ILUVW WZR SDSHUV RI WKLV VHULHV UHSRUW VWXGLHV RI WKH GLEORFN FRSRO\PHU SRO\VW\UHQH 36 fSRO\EXWDGLHQH 3% f LQ WKH VHOHFWLYH VROYHQW QWHWUDGHFDQH & f ZKLFK GLVVROYHV WKH 3% EORFN 7KH %&3 KDG D QXPEHU DYHUDJH PROHFXODU ZHLJKW 0f [ DQG D IUDFWLRQDO FRPSRVLWLRQ I 7KH VROXWLRQV VWXGLHG KDG WKH IROORZLQJ SRO\PHU FRQFHQWUDWLRQV DQG ZWb SRO\PHU $W WKHVH FRQFHQWUDWLRQV DQG DW WKLV IUDFWLRQDO FRPSRVLWLRQ WKH 067 UHVXOWV LQ VSKHULFDO 36 GRPDLQV ZLWKLQ D 3%& PDWUL[ ;UD\ VFDWWHULQJ SURILOHV ZHUH DFTXLUHG DW YDU\LQJ WHPSHUDWXUHV XSRQ KHDWLQJ DQG FRROLQJ 7KH [UD\ VSHFWUD ZHUH FRUUHFWHG IRU SDUDVLWLF VFDWWHULQJ DQG FROOLPDWLRQ HUURUV UHVXOWLQJ IURP LQVWUXPHQWDO UHVROXWLRQ ,Q WKHVH H[SHULPHQWV %UDJJ SHDNV ZHUH HYLGHQW LQ WKH ORZ WHPSHUDWXUH VSHFWUD IRU DOO RI WKH VROXWLRQV LQ WKH UHODWLYH ORFDWLRQV ?>? \c ?c DQG DIWHU FROOLPDWLRQ FRUUHFWLRQ 7KH ODWWLFH FRQVWDQWV ZHUH GHWHUPLQHG IURP WKH ILUVW %UDJJ SHDN ORFDWLRQ DQG WKH DYHUDJH UDGLXV RI WKH 36 VSKHUHV ZDV GHWHUPLQHG IURP WKH ORFDWLRQ RI WKH ILUVW PD[LPXP LQ WKH VSKHULFDO IRUP IDFWRU DV GHVFULEHG LQ &KDSWHU ,W ZDV VSHFXODWHG WKDW WKH PDFURODWWLFH RI 36 GRPDLQV ZDV VLPSOH FXELF 6& f UDWKHU WKDQ ERG\FHQWHUHG FXELF %&& f IURP FRPSDULVRQV RI WKH FDOFXODWHG DQG VWRLFKLRPHWULF 36 YROXPH IUDFWLRQV IRU WKHVH VWUXFWXUHV 7KLV W\SH RI FDOFXODWLRQ LV GHVFULEHG LQ GHWDLO LQ &KDSWHU $V WKH WHPSHUDWXUH 7 IRU HDFK VROXWLRQ ZDV LQFUHDVHG WKH %UDJJ SHDN DPSOLWXGH GHFUHDVHG FRQWLQXRXVO\ ZKLOH WKH IXOO ZLGWK DW KDOI PD[LPXP ):+0 f UHPDLQHG XQFKDQJHG ,Q DGGLWLRQ WKH FDOFXODWHG ODWWLFH FRQVWDQWV GHFUHDVHG VORZO\ ZLWK LQFUHDVLQJ WHPSHUDWXUH $W D SRO\PHU FRQFHQWUDWLRQ GHSHQGHQW WHPSHUDWXUH 7 WKH %UDJJ DPSOLWXGH GHFUHDVHG DEUXSWO\ DFFRPSDQLHG E\ DQ LQFUHDVH LQ WKH ):+0 DQG D GURS LQ WKH ODWWLFH FRQVWDQW $V WKH WHPSHUDWXUH ZDV LQFUHDVHG IXUWKHU DERYH 7 WKH SHDN DPSOLWXGH DQG ODWWLFH FRQVWDQW FRQWLQXHG WR GHFUHDVH ZKLOH WKH ZLGWK LQFUHDVHG UDSLGO\ )LQDOO\ DERYH 7 7G QR IXUWKHU HYLGHQFH RI WKH SHDN VWUXFWXUH UHPDLQHG

PAGE 87

+DVKLPRWR HW DO DVVRFLDWHG WKH WHPSHUDWXUH 7 ZLWK WKH GLVRUGHULQJ RI WKH PDFURODWWLFH UHVXOWLQJ LQ D OLTXLGOLNH DUUDQJHPHQW RI WKH 36 VSKHUHV 7KH WHPSHUDWXUH 7G LV LGHQWLILHG DV WKH GLVVROXWLRQ WHPSHUDWXUH DW ZKLFK SRLQW WKH 36 GRPDLQV WRWDOO\ GLVVROYH ,Q RUGHU WR FODULI\ WKHVH LQWHUSUHWDWLRQ UKHRORJLFDO PHDVXUHPHQWV ZHUH PDGH RQ WKH SRO\PHU VROXWLRQV DV D IXQFWLRQ RI WHPSHUDWXUH 7ZR WUDQVLWLRQV LQ WKH PHFKDQLFDO SURSHUWLHV ZHUH LGHQWLILHG 8SRQ UDLVLQJ 7 DERYH 7 D WUDQVLWLRQ IURP QRQOLQHDU SODVWLF IORZ WR OLQHDU QRQ1HZWRQLDQ IORZ ZDV REVHUYHG LQ VXSSRUW RI WKHLU LGHQWLILFDWLRQ RI 7 $OVR DW 7 7G D WUDQVIRUPDWLRQ IURP OLQHDU QRQ1HZWRQLDQ IORZ WR OLQHDU 1HZWRQLDQ IORZ ZDV VHHQ LQ DJUHHPHQW ZLWK HDUOLHU VWXGLHV LGHQWLI\LQJ WKLV EHKDYLRU ZLWK WKH GLVVROXWLRQ SRLQW 7KH DXWKRUV LGHQWLI\ D QXPEHU RI TXDQWLWDWLYH WUHQGV LQ WKH WKHUPDO EHKDYLRU RI WKH V\VWHP 7KH GHSHQGHQFH RI WKH DSSDUHQW ODWWLFH VSDFLQJ D RQ WHPSHUDWXUH ERWK DERYH DQG EHORZ WKH RUGHULQJ SRLQW ZHUH FRQVLVWHQW ZLWK WKH IROORZLQJ HPSLULFDO UHODWLRQ DERYH DQG EHORZ WKH GLVFRQWLQXLW\ DW 7 D a 7f f 7KH GLVFRQWLQXLW\ LQ D DW 7 7 LV WKRXJKW WR EH GXH WR WKH FKDQJH LQ VWUXFWXUH RFFXUULQJ DW WKLV WHPSHUDWXUH $ERYH 7 WKH SHDN LV QR ORQJHU D %UDJJ SHDN DQG WKHUHIRUH GHWHUPLQLQJ D ODWWLFH FRQVWDQW IURP WKH SHDN SRVLWLRQ LV HUURQHRXV 7KH GHFUHDVLQJ WUHQG RI D ZLWK 7 LV WKH UHVXOW RI WZR IDFWRUV )LUVW WKH 3% FKDLQV EHWZHHQ WKH GRPDLQV FRQWUDFW XSRQ KHDWLQJ WR LQFUHDVH WKHLU FRQIRUPDWLRQDO HQWURS\ 6HFRQG DW KLJKHU WHPSHUDWXUHV WKH HIIHFWLYH LQWHUDFWLRQ JLYHQ E\ WKH )ORU\+XJJLQV SDUDPHWHU [ a ON%7 LV UHGXFHG UHVXOWLQJ LQ JUHDWHU LQWHUPL[LQJ RI WKH 36 DQG 3% FKDLQV ZKLFK LQ WXUQ FDXVHV D FRQWUDFWLRQ RI WKH ODWWLFH 7KH DXWKRUV EHOLHYH WKH VHFRQG RI WKHVH WZR LV WKH PRUH LPSRUWDQW HIIHFW DW DOO WHPSHUDWXUHV 7KH %UDJJ ZLGWKV EHORZ 7 DSSHDU WR EH LQGHSHQGHQW RI WHPSHUDWXUH 7KH DXWKRUV RIIHU DQ LQWHUHVWLQJ H[SODQDWLRQ IRU WKLV UHVXOW 7KH GRPDLQGRPDLQ LQWHUDFWLRQ LQ %&3 VROXWLRQV LV DVVXPHG WR EH PXFK WKH VDPH DV GHVFULEHG E\ 0HLHU IRU WKH SXUH

PAGE 88

SRO\PHU V\VWHP EHLQJ TXDVLHODVWLF 7KH FKDQJH LQ IUHH HQHUJ\ $) GXH WR VPDOO HORQJDWLRQV $D RI WKH ODWWLFH FRQVWDQW IURP LWV HTXLOLEULXP YDOXH FDQ WKHUHIRUH EH H[SUHVVHG DV $)aI&7f $Df f +HUH I&7f WKH VSULQJ FRQVWDQW RI WKH 3% FKDLQV LV D IXQFWLRQ RI WKH SRO\PHU FRQFHQWUDWLRQ & DQG WHPSHUDWXUH 7KH DXWKRUV DUJXH WKDW LI WKH GLVWULEXWLRQ RI VWUDLQV VDWLVILHV WKH %ROW]PDQQ GLVWULEXWLRQ WKHQ WKH GLVWULEXWLRQ RI ODWWLFH FRQVWDQWV LV FKDUDFWHUL]HG E\ D JDXVVLDQ IRUP KDYLQJ D ZLGWK SDUDPHWHU *D N57 *D&7fI&7f f 7KH VSULQJ FRQVWDQW IRU WKH 3% FKDLQV LV HQWURSLF LQ RULJLQ DV GHVFULEHG HDUOLHU LQ WKLV FKDSWHU VR WKDW I & 7 f N%7 f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

PAGE 89

VFUHHQLQJ RI WKH 363% LQWHUDFWLRQ ZKLFK UHVXOWV LQ D KLJKHU )ORU\+XJJLQV LQWHUDFWLRQ SDUDPHWHU b‘ 7KH PHDVXUHG ODWWLFH FRQVWDQWV ZHUH FRQVLVWHQW ZLWK WKH VFDOLQJ UHODWLRQ D &nP f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f [ 7KH SRO\PHU FRQFHQWUDWLRQV VWXGLHG ZHUH DQG ZWb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

PAGE 90

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f DQG 'HII ZDV GHULYHG LQYROYLQJ WKUHH DVVXPSWLRQV DQ LVRWURSLF SDUWLFOH GLVWULEXWLRQ )LFNLDQ GLIIXVLRQ RI WKH GLEORFN FKDLQV DQG QR FHQWHU RI PDVV PRWLRQ IRU WKH PLFURGRPDLQV ZKLOH WKH\ GLVVROYH L H QR PDFURODWWLFH GLVRUGHULQJ 7KHLU UHVXOW LV ,4Wf ,4W f Hn 2n 'rII f )URP WKLV H[SUHVVLRQ WKH LQWHQVLW\ GHFD\V H[SRQHQWLDOO\ ZLWK WLPH DQG VR 'HII FDQ EH GHWHUPLQHG IURP WKH VORSH RI WKH VHPLORJ SORW 7KH LQWHUSUHWDWLRQ RI WKHVH UHVXOWV LV FRPSOLFDWHG E\ WKH VSHHG ZLWK ZKLFK WUDQVIRUPDWLRQ WDNHV SODFH 0XFK RI WKH WUDQVLWLRQ RFFXUV QRQLVRWKHUPDOO\ ,Q RUGHU WR DFFRXQW IRU WKHVH QRQLVRWKHUPDO HIIHFWV D FUXGH PRGHO IRU WKH WLPH GHYHORSPHQW RI WKH VFDWWHUHG LQWHQVLW\ ZDV GHULYHG DQG LV GHVFULEHG E\ WKH IROORZLQJ UHFXUVLRQ UHODWLRQ .49Lf > ,4WMf ,H47Mf @ Hn 2n$W OF4\ rf

PAGE 91

+HUH WKH WHPSHUDWXUH MXPS LV GLYLGHG LQWR WLPH LQFUHPHQWV $W 7KH LQWHQVLW\ DW WLPH WM LV GHWHUPLQHG IURP WKH LQWHQVLW\ DW WKH SUHYLRXV WLPH LQFUHPHQW WM 'XULQJ HDFK LQFUHPHQW WKH LQWHQVLW\ PRYHV WRZDUG WKH HTXLOLEULXP LQWHQVLW\ GLVWULEXWLRQ ,H FRUUHVSRQGLQJ WR WKH WHPSHUDWXUH 7M ZKLFK LV WKH WHPSHUDWXUH RI WKH VDPSOH DW WLPH WM ,Q RUGHU WR IXUWKHU VLPSOLI\ WKH SUREOHP 'HII LV DVVXPHG WR IROORZ DQ $UUKHQLXV WHPSHUDWXUH GHSHQGHQFH 'HII rR H[3 $'}f f 7KHLU PRGHO IRU WKH WHPSRUDO GHYHORSPHQW RI WKH LQWHQVLW\ DW DQ\ VFDWWHULQJ YHFWRU 4 LV D IXQFWLRQ RI WKH WZR SDUDPHWHUV DQG WKH DFWLYDWLRQ HQHUJ\ $+D 7KH DXWKRUV SORW WKH ORJDULWKP RI WKH VFDWWHUHG LQWHQVLW\ DW D VSHFLILF VFDWWHULQJ YHFWRU 4r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f WR WKH PHDVXUHG GDWD 7KH WZR PHWKRGV JDYH VLPLODU QXPHULFDO UHVXOWV IRU $+D DQG 7KH GDWD ZHUH IRXQG WR EH FRQVLVWHQW ZLWK WKH IROORZLQJ UHODWLRQ & f 5HVXOWV IRU WKH DFWLYDWLRQ HQHUJ\ $+D VKRZ QR FRQVLVWHQW WUHQG EXW \LHOG UHVXOWV IRU DOO FRQFHQWUDWLRQV LQ WKH YLFLQLW\ RI NFDO PROH $V FDQ EH VHHQ IURP WKLV EULHI GLVFXVVLRQ RI WKH FXUUHQW OLWHUDWXUH H[SHULPHQWDO VWXGLHV RI %&3 WUDQVLWLRQ SKHQRPHQD DUH VWLOO TXLWH OLPLWHG 7KHVH V\VWHPV DUH ULFK LQ WKHLU GLYHUVH PLFURVWUXFWXUDO PRUSKRORJLHV DQG WKH WUDQVLWLRQV ZKLFK RFFXU EHWZHHQ WKHP

PAGE 92

DQG WKH KRPRJHQHRXV PHOW 7KLV FRPSOH[ EHKDYLRU KDV EHHQ WKH VWLPXOXV IRU WKH VHULHV RI H[SHULPHQWV UHSRUWHG KHUH

PAGE 93

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

PAGE 94

'(7, &227 (&725 2//(5 Â’ &&087%, D PWXHL Â’ U &2&,7$/ 2(/$7 *(!L$75 7HI%:25( &21752//(5 '(7(&725 7!HZ$/ UXQQp /07 %$7+ 62/(12,' '77 %,)$&( )($7'& &,5&8,7 2 >? \\,D',*,7$/ m-/7,3(7%, &+$57 5(&252%7 326,7,21 6%,7,9( ;5$7 '(7(&725 /URWQH" [UDW 6&$77%7,1& )851$&( A;m$76 7/502/(&8/$5 SXP[F /07 )LJXUH $ VFKHPDWLF RI WKH GDWD DFTXLVLWLRQ V\VWHP

PAGE 95

WR WKH FRPSXWHU DW D ODWHU WLPH 7KH VLOLFRQ GLRGH DUUD\ GHWHFWRU LV HYDFXDWHG XVLQJ D /HLEROG '$ WZRVWDJH URXJKLQJ SXPS DQG UHTXLUHV D FRROLQJ V\VWHP $ )LVKHU 0RGHO &LUFXODWRUf 7KH GHWDLOV RI WKH 6$;6 DSSDUDWXV LQFOXGLQJ WKH GHWHFWLRQ V\VWHP DUH GLVFXVVHG PRPHQWDULO\ 7KH VDPSOH WHPSHUDWXUH LV PRQLWRUHG DQG FRQWUROOHG E\ D 0LFULVWDU WHPSHUDWXUH FRQWUROOHU RSHUDWLQJ LQ WKH SURSRUWLRQDO PRGH RI FRQWURO ,Q WKLV PRGH WKH 0LFULVWDU FRPSDUHV WKH VDPSOH WHPSHUDWXUH WR WKH SURJUDPPHG VHWSRLQW DQG RXWSXWV D '& YROWDJH EHWZHHQ DQG 9 ZKLFK LV SURSRUWLRQDO WR WKH GLIIHUHQFH 7KLV YROWDJH LV VHQW WR D VSHFLDOO\ GHVLJQHG KHDWLQJ FLUFXLW GHVFULEHG LQ $SSHQGL[ % f ZKLFK LQ WXUQ FRQWUROV WKH FXUUHQW WR KHDWLQJ UHVLVWRUV LQ WKH SRO\PHU [UD\ VFDWWHULQJ IXUQDFH ,Q WKLV ZD\ WKH VDPSOH WHPSHUDWXUH LV FRQWUROOHG ZLWK D VWDELOLW\ RI r& 7KH GHWDLOV RI WKH [UD\ VFDWWHULQJ IXUQDFH DUH GLVFXVVHG PRPHQWDULO\ 7KH VDPSOH WHPSHUDWXUH DV GHWHUPLQHG IURP WKH WKHUPRFRXSOH YROWDJH HPLWWHG E\ D WKHUPRFRXSOH HPEHGGHG LQ WKH VDPSOH LV DOVR PRQLWRUHG E\ D /LQVHLV 0RGHO / FKDUW UHFRUGHU DQG WKH .HLWKOH\ PXOWLPHWHU 7KH WLPHGHSHQGHQW PHDVXUHPHQWV ZHUH SHUIRUPHG E\ DQQHDOLQJ WKH %&3 VROXWLRQ DERYH LWV GLVVROXWLRQ SRLQW 7G DQG WKHQ UDSLGO\ ORZHULQJ WKH WHPSHUDWXUH WR D IL[HG SRLQW EHORZ 7G DQG REVHUYLQJ WKH VWUXFWXUDO WUDQVLWLRQ WKURXJK WKH 6$;6 SURILOH 7KLV WKHUPDO TXHQFK LV DFKLHYHG E\ IRUFLQJ D IL[HG DPRXQW RI FRRODQW UDSLGO\ WKURXJK D TXHQFKLQJ FKDQQHO LQ WKH VDPSOH PRXQW 7KH PHDVXUHG FRRODQW ZDV LQMHFWHG LQWR WKH DSHUWXUH RI WKH TXHQFKLQJ FKDQQHO LQOHW WR WKH IXUQDFH 3RO\SURS\OHQH WXELQJ H[WHQGLQJ IURP D FDQQLVWHU RI KHOLXP ZDV WKHQ FRQQHFWHG WR WKH TXHQFKLQJ FKDQQHO LQOHW $ VROHQRLG YDOYH SODFHG EHWZHHQ WKH IXUQDFH DQG WKH KHOLXP ZDV XVHG WR FRQWURO WKH WLPH DQG GXUDWLRQ RI WKH KHOLXP EXUVW ZKLFK ZDV LQLWLDWHV WKH TXHQFK 7KH VROHQRLG YDOYH LV FRQWUROOHG E\ WKH GLJLWDO GHOD\ JHQHUDWRU WKURXJK D VSHFLDOO\ GHVLJQHG VROHQRLG LQWHUIDFH FLUFXLW D VFKHPDWLF RI ZKLFK LV VKRZQ LQ $SSHQGL[ &

PAGE 96

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b IRU D PLO VKHHWf +RZHYHU LW LV D KRUUHQGRXV SDUDVLWLF VFDWWHUHU LQ WKH UHJLRQ RI LQWHUHVW LQ WKHVH

PAGE 97

)LJXUH $ WRS YLHZ RI WKH SRO\PHU [UD\ VFDWWHULQJ IXUQDFH LOOXVWUDWLQJ WKH UHODWLYH SODFHPHQW RI LWV YDULRXV IHDWXUHV GHVFLEHG LQ WKH WH[W

PAGE 98

H[SHULPHQWV 3DUDVLWLF VFDWWHULQJ LV GHILQHG DV WKDW XQZDQWHG FRPSRQHQW WR WKH VFDWWHULQJ SURILOH ZKLFK LV D UHVXOW RI VFDWWHULQJ IURP VOLWV DQG ZLQGRZ PDWHULDOV 7KLV TXDQWLW\ ZDV PHDVXUHG DV WKH LQWHJUDWHG QXPEHU RI FRXQWV SHU VHFRQG DERYH EDFNJURXQG 7KLV TXDQWLW\ IRU WKH PLO EHU\OOLXP VKHHW ZDV WLPHV ODUJHU WKDQ IRU WKH PLO .DSWRQ VKHHW LQ WKH WR GHJUHH UDQJH DQG WLPHV ODUJHU LQ WKH WR GHJUHH UDQJH $OWKRXJK WKH SHUFHQWDJH WUDQVPLVVLRQ ZDV VOLJKWO\ OHVV WUDQVPLVVLRQ b f IRU WKH .DSWRQ WKH UHGXFWLRQ LQ SDUDVLWLF VFDWWHULQJ ZDV PXFK PRUH LPSRUWDQW 7KH .DSWRQ ZLQGRZV ZHUH VHDOHG WR WKH ZLQGRZ IUDPH ZLWK 2PHJDERQG HSR[\ DQG UHVLQ 7KH FKDPEHU ZDV HYDFXDWHG ZLWK DQ (GZDUGV (73* SXPSLQJ VWDWLRQ ZKLFK LQFRUSRUDWHG D URXJKLQJ SXPS DQG D WXUERPROHFXODU SXPS 7KH XOWLPDWH SUHVVXUH W\SLFDOO\ DFKLHYHG XQGHU H[SHULPHQWDO FRQGLWLRQV ZDV EHWZHHQ DQG PLOOLWRUU 7KH SUHVVXUH ZDV PXFK ORZHU EHORZ WKH OHYHO ZKLFK FRXOG EH UHDG IURP WKH 9DUDQ WKHUPRFRXSOH JDXJH FRQWUROOHU f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

PAGE 99

VHWV RI FRQQHFWLRQV ZHUH XVHG RQH WR PRQLWRU WKH VDPSOH WHPSHUDWXUH DQG RQH WR PRQLWRU WKH VDPSOH PRXQW WHPSHUDWXUH $ 9DUDQ 7\SH WKHUPRFRXSOH JDXJH KHDG ZDV DWWDFKHG LQ WKH ORFDWLRQ VKRZQ LQ ILJXUH ZLWK WHIORQ WDSH 7KH VHQVRU ZDV FRQQHFWHG WR D 9DUDQ JDXJH FRQWUROOHU ,W 7KH FKDPEHU ZDV HYDFXDWHG WKURXJK WKH YDFXXP SRUW VKRZQ LQ WKH )LJXUH 7ZR DGGLWLRQDO KLJK YDFXXP IODQJHV DUH DYDLODEOH RQ WKH VFDWWHULQJ IXUQDFH EXW DUH QRW EHLQJ XVHG DW WKLV WLPH 7KH 7UDQVODWLRQDO 6WDJH 7KH EDVH RI WKH IXUQDFH ERG\ LV FXW LQWR D GRYHWDLO VKDSH 7KH EDVH PDWHV ZLWK D EUDVV WUDQVODWLRQDO VWDJH WKH ERWWRP RI ZKLFK LV LWVHOI FXW LQWR WKH GRYHWDLO VKDSH LQ WKH RUWKRJRQDO GLUHFGRQ $Q DGGLWLRQDO EUDVV SODWH PDWHV ZLWK WKLV SLHFH DQG E\ VOLGLQJ WKH GRYHWDLO ZLWKLQ LWV WUDFN WZR LQGHSHQGHQW WUDQVODWLRQV DUH SURYLGHG IRU FHQWHULQJ WKH VDPSOH LQ WKH [UD\ EHDP 7KH WUDQVODWLRQDO VWDJHV DUH PRXQWHG RQ D +XEHU )RXU &LUFOH 'LIIUDFWRPHWHU WKH IRXU LQGHSHQGHQW RULHQWDWLRQ DQJOHV ZHUH D OLWWOH RYHUNLOO LQ D VPDOO DQJOH VFDWWHULQJ H[SHULPHQWf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f

PAGE 100

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

PAGE 101

)LJXUH 7KH ILQDO VDPSOH PRXQW LV LOOXVWUDWHG LQ D IURQW DQG WRS YLHZ (YLGHQW LQ WKLV GUDZLQJ DUH WKH TXHQFKLQJ WXEHV WKH KHDWLQJ UHVLVWRUV WKH WKHUPRFRXSOH WKH VWDLQOHVV VWHHO PDVNV ZKLFK VXSSRUW WKH .DSWRQ ZLQGRZV DQG WKH VWDLQOHVV VWHHO EUDFH ZKLFK ILUPO\ FODPSV WKH VDPSOH FHOO LQWR WKH VDPSOH PRXQW

PAGE 102

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f 7ZR GLDPHWHU KROHV DUH PDGH LQ WKH FRSSHU EORFNV DQG DUH LQWHUFRQQHFWHG E\ RXWHU GLDPHWHU LQQHU GLDPHWHU FRSSHU WXELQJ DV VKRZQ LQ WKH ILJXUH 7KH FRSSHU WXELQJ LV VHDOHG WR WKH KHDWLQJ XQLW XVLQJ VLOYHU VROGHU 6WDLQOHVV VWHHO WXELQJ RI WKH VDPH VL]H LV XVHG WR FRQQHFW WKH TXHQFKLQJ FKDQQHO WR WKH YDFXXP IHHGWKURXJKV 7KLV VWDLQOHVV WXELQJ ZDV WKH VROH VXSSRUW IRU WKH VDPSOH PRXQW LQ WKH IXUQDFH DQG ZDV XVHG EHFDXVH LWV SRRU WKHUPDO FRQGXFWLYLW\ UHODWLYH WR FRSSHUf KHOSV WR LVRODWH WKH VDPSOH PRXQW IURP WKH H[WHUQDO HQYLURQPHQW $Q DGGLWLRQDO WKHUPRFRXSOH FDOOHG WKH EDVH WKHUPRFRXSOH LV FODPSHG WR WKH FRSSHU KHDWLQJ XQLW 7KH WHPSHUDWXUH RI WKH EDVH ZDV IRXQG WR UHDFW PXFK PRUH TXLFNO\ WKDQ WKDW RI WKH VDPSOH )RU WKLV UHDVRQ WKH EDVH LV XVHG DV WKH FRQWURO WHPSHUDWXUH E\ WKH 0LFULVWDU WHPSHUDWXUH FRQWUROOHU :LWK WKLV GHVLJQ D WHPSHUDWXUH VWDELOLW\ RI r& LV DFKLHYHG

PAGE 103

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c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f $W $7V 7E7Vf f

PAGE 104

7HPSHUDWXUH 7LPH VHFRQGV f )LJXUH $ W\SLFDO TXHQFK SURILOH LOOXVWUDWLQJ WKH EDVH DQG VDPSOH WHPSHUDWXUHV DV D IXQFWLRQ RI WLPH IROORZLQJ WKH TXHQFK 7KH GDVKHG OLQH LV D PRGHO GHVFULEHG LQ WKH WH[Wf RI WKH TXHQFKLQJ EHKDYLRU H[SHFWHG ZLWK LGHQWLFDO EDVH DQG VDPSOH FKDUDFWHULVWLFV EXW ZLWKRXW WKH XQGHUVKRRW LQ EDVH WHPSHUDWXUH

PAGE 105

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f 7 7If H[S W[ f 7I f 7KH GDVKHG OLQH LQ )LJXUH LV D SORW RI WKLV HTXDWLRQ DVVXPLQJ WKH VDPH WLPH FRQVWDQW HVWLPDWHG IURP PHDVXUHG VDPSOH DQG EDVH WHPSHUDWXUH SURILOHV [ VHF ,W LV HYLGHQW IURP WKH JUDSK WKDW WKH HTXLOLEUDWLRQ WLPH GHILQHG DV WKH WLPH UHTXLUHG WR UHDFK ZLWKLQ r& RI WKH ILQDO WHPSHUDWXUH XVLQJ WKH XQGHUVKRRW LQ EDVH WHPSHUDWXUH VHFRQGV f LV VLJQLILFDQWO\ EHWWHU WKDQ WKDW REWDLQHG ZLWKRXW WKH XQGHUVKRRW VHFRQGV f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r& EXW GXH WR WKH ODJ LQ WKH VDPSOH WHPSHUDWXUH WKH SHDN WR SHDN ULSSOH LQ 7V LV W\SLFDOO\ OHVV WKDQ r& ,Q WKLV ZD\ TXHQFKHV WR ZLWKLQ r& RI WKH ILQDO WHPSHUDWXUH DUH REWDLQHG ZLWKRXW XQGHUVKRRW LQ DSSUR[LPDWHO\ VHFRQGV ZLWK D VXEVHTXHQW WKHUPDO YDULDWLRQ RI r&

PAGE 106

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

PAGE 107

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nV RUELW 7KLV LQWULQVLF FROOLPDWLRQ OHDGV WR [UD\ EHDPV KDYLQJ UHODWLYHO\ VPDOO GLPHQVLRQV a PPf RI H[WUHPHO\ KLJK EULOOLDQFH 7KLV SURSHUW\ RIWHQ WUDQVODWHV LQWR H[WUDRUGLQDU\ LPSURYHPHQWV LQ H[SHULPHQWDO UHVROXWLRQ 7KH 1DWLRQDO 6\QFKURWURQ /LJKW 6RXUFH )DFLOLW\ DFFHOHUDWHV HOHFWURQV WKURXJK D WRURLGDO YDFXXP FKDPEHU PHWHUV LQ UDGLXV $ QXPEHU RI EHDPOLQHV DUH VLWXDWHG XQLIRUPO\ DURXQG WKH ULQJ %HQGLQJ PDJQHWV H[KLELWLQJ D ILHOG RI WHVOD DUH VLWXDWHG DW HDFK EHDPOLQH 7KH UHVXOWLQJ WUDMHFWRU\ RI WKH HOHFWURQ EHDP KDV D FXUYDWXUH RI

PAGE 108

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ƒ f 7KH &ROOLPDWLRQ 6\VWHP 7KH SXUSRVH RI WKH FROOLPDWLRQ V\VWHP LV WR SURGXFH D PRQRFKURPDWLF [2UD\ EHDP KDYLQJ PLQLPDO DQJXODU GLYHUJHQFH DQG SDUDVLWLF VFDWWHULQJ 7KH V\VWHP HPSOR\HG LQ WKHVH PHDVXUHPHQWV FDQ EH GLYLGHG LQWR SUHKXWFK SUHSDUHG E\ EHDPOLQH SHUVRQQHOf DQG LQKXWFK FRPSRQHQWV SUHSDUHG E\ WKH XVHUf 7KHVH FRPSRQHQWV DUH LOOXVWUDWHG LQ ILJXUH 7KH SUHKXWFK V\VWHP LQFOXGHV D IRFXVVLQJ PLUURU DQG D KLJK LQWHQVLW\ PRQRFKURPDWRU 7KH PDMRU FRPSRQHQWV RI WKH LQKXWFK FROOLPDWLRQ V\VWHP DUH WZR LQn YDFXXP VOLWV GHVLJQDWHG 6, DQG 6 7KH DQJXODU GLYHUJHQFH RI WKH EHDP LQFLGHQW RQ WKH VDPSOH LV GHWHUPLQHG E\ WKH ZLGWK RI 6 DQG WKHLU GLVWDQFH IURP WKH V\QFKURWURQ VRXUFH ,Q WKLV H[SHULPHQW LW LV WKH DQJXODU GLYHUJHQFH ZKLFK XOWLPDWHO\ GHWHUPLQHV WKH V\VWHP UHVROXWLRQ 7KH VHFRQG VHW RI VOLWV 6 DUH XVHG WR HOLPLQDWH SDUDVLWLF VFDWWHULQJ 7KH SUHKXWFK DQG SRVWKXWFK FRPSRQHQWV DUH QRZ GHVFULEHG LQ JUHDWHU GHWDLO

PAGE 109

B%HQW &\OLQGHU )RFXVLQJ 0LUURU 6, 6 %HDP (QWUDQFH WR +XWFK 3RO\PHU )XUQDFH 3RVLWLRQ 6HQVLWLYH 'HWHFWRU )LJXUH $ VFKHPDWLF RI WKH 6$;6 FROOLPDWLRQ V\VWHP $ERYH LV WKH SUHKXWFK FROOLPDWLRQ V\VWHP LQFOXGLQJ WKH IRFXVVLQJ PLUURU DQG WKH PXOWLOD\HU PRQRFKURPDWRU %HORZ LV WKH LQKXWFK FROOLPDWLRQ V\VWHP LQFOXGLQJ VOLWV 6 DQG 6

PAGE 110

7KH SUHKXWFK FROOLPDWLRQ V\VWHP 8SRQ OHDYLQJ WKH V\QFKURWURQ ULQJ WKH [UD\ EHDP SDVVHV WKURXJK D EHU\OOLXP ZLQGRZ DQG LV LQFLGHQW XSRQ D KRUL]RQWDOO\ LQ WKH VFDWWHULQJ SODQH f DQG YHUWLFDOO\ IRFXVVLQJ PLUURU KDYLQJ D KRUL]RQWDO DQJXODU DFFHSWDQFH RI PUDG 7KH PLUURU \LHOGV D IRFDO VSRW RI DSSUR[LPDWHO\ PP [ PP ):+0 7KH IRFXVVHG EHDP WKHQ UHIOHFWV IURP D VSHFLDOO\ GHVLJQHG KLJK LQWHQVLW\ PXOWLOD\HU PRQRFKURPDWRU EHIRUH HQWHULQJ WKH [UD\ VFDWWHULQJ KXWFK D VKLHOGHG HQFORVXUH f 6RPH RI WKH GHWDLOV ZKLFK PDNH WKH PDNH WKLV PRQRFKURPDWRU HVSHFLDOO\ XVHIXO DUH QRZ GHVFULEHG &RQYHQWLRQDO FU\VWDO PRQRFKURPDWRUV KDYH DQ HQHUJ\ EDQGSDVV $( ( O2 ,Q DSSOLFDWLRQV ZKLFK GRQn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ƒ DQG D :OD\HU WKLFNQHVV RI ,G ,Q WKHVH PHDVXUHPHQWV WKH UHIOHFWHG HQHUJ\ ZDV WXQHG WR NH9 ZKLFK FRUUHVSRQGV WR D GLIIUDFWLRQ DQJOH RI r DQG DQ LQFLGHQW ZDYHOHQJWK RI ƒ 7KH PHDVXUHG HQHUJ\ EDQGSDVV RI WKLV PRQRFKURPDWRU ZDV PHDVXUHG WR E\ $)) [ ):+0 DQG DQ DFFRPSDQ\LQJ LQFUHDVH LQ EULOOLDQFH SKRWRQV PP f RI DSSUR[LPDWHO\ WZR RUGHUV RI PDJQLWXGH RYHU FRQYHQWLRQDO PRQRFKURPDWRUV

PAGE 111

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f 7KH IRFDO SRLQW RI WKH V\VWHP ZDV IRXQG WR OLH EHWZHHQ 6, DQG 6 )URP WKH DERYH GLVFXVVLRQ LW LV HYLGHQW WKDW DQ XSSHU OLPLW RQ WKH DQJXODU GLYHUJHQFH GM RI WKH EHDP LQFLGHQW RQ WKH VDPSOH LV JLYHQ E\ D^ WDQ : G6f f ZKHUH : WKH ):+0 RI WKH EHDP DV PHDVXUHG E\ WKH GHWHFWRU DQG G6 WKH GLVWDQFH VHSDUDWLQJ 6 DQG WKH GHWHFWRU 7KH VOLWWLQJ SURFHGXUH DQG UHVXOWLQJ DQJXODU GLYHUJHQFH DUH GHVFULEHG PRPHQWDULO\ 7KH VHFRQG VHW RI VOLWV DUH DGMXVWHG VR WKDW LWV EODGHV MXVW PDVN WKH RXWHU HGJH RI WKH EHDP ,Q WKLV ZD\ SDUDVLWLF VFDWWHULQJ IURP VOLWV 6 LV PDVNHG ZLWKRXW SURGXFLQJ D ODUJH DPRXQW RI SDUDVLWLF VFDWWHULQJ IURP 6

PAGE 112

7KH 'HWHFWLRQ 6\VWHP 7KH [UD\ GHWHFWRU XVHG LQ WKHVH H[SHULPHQWV ZDV DQ (*t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f LV XVHG WR VWRUH WKH GDWD LQ PHPRU\ 7KH JDLQ RI WKH $'& LV DGMXVWHG VR WKDW RQH SKRWRQ FRUUHVSRQGV WR RQH FRXQW 7KLV \LHOGV D G\QDPLF UDQJH RI FRXQWVFKDQQHOVFDQ 7KH PD[LPXP VFDQ UDWH IRU WKH GHWHFWRU LV PVHF SL[HOV 7KH VFDQ WLPH FDQ EH VHW IURP PVHF DQG XS 7KH GDWD IURP VXFFHVVLYH VFDQV FDQ EH DFFXPXODWHG LQ D VLQJOH VSHFWUXP

PAGE 113

,Q WKLV ZD\ WKH PD[LPXP FRXQW UDWH LV [ VHF PVHF f [ FRXQWVVHFSL[HO 7KH HIILFLHQF\ RI WKH GHWHFWRU LV GHWHUPLQHG E\ WKH WKLFNQHVV RI WKH SURWHFWLYH VLOLFRQ R[LGH FRDWLQJ RQ WKH VXUIDFH RI WKH GLRGHV DQG LV TXRWHG DV b IRU WKH NH9 [UD\V XVHG LQ WKLV H[SHULPHQW 7KH GDUN FRXQW 'HIHFWV ZLWKLQ WKH VLOLFRQ GLRGHV UHVXOW LQ D OHDNDJH FXUUHQW ZKLFK VKRZV XS DV D GDUN FRXQW LQ WKH PHDVXUHG VSHFWUXP HYHQ LQ WKH DEVHQFH RI LQFLGHQW UDGLDWLRQ 7KH PDJQLWXGH RI WKH GDUN FXUUHQW LV YHU\ WHPSHUDWXUH GHSHQGHQW )RU WKLV UHDVRQ WKH GHWHFWRU LV W\SLFDOO\ FRROHG WR WHPSHUDWXUHV EHORZ r& 7KLV FRROLQJ ZDV SHUIRUPHG E\ FLUFXODWLQJ r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r& LQFUHDVLQJ WR FRXQWV FKDQQHO VHF DW r& 7KHVH QXPEHUV DUH LQ URXJK DJUHHPHQW ZLWK WKH TXRWHG WHPSHUDWXUH GHSHQGHQFH RI WKH GDUN FXUUHQW DSSUR[LPDWHO\ D IDFWRU RI SHU r& f 7KH PDJQLWXGH RI WKH GDUN FXUUHQW KDV EHHQ IRXQG WR GHSHQG RQ WKH DFFXPXODWHG UDGLDWLRQ GRVH IRU HDFK SL[HO 7KH OHYHO RI WKH GDUN FXUUHQW FDQ LQFUHDVH WR DSSUR[LPDWHO\ WHQ WLPHV LWV LQLWLDO OHYHO GXH WR UDGLDWLRQ GDPDJH $V ZLOO EH GLVFXVVHG WKH QRLVH LQWURGXFHG E\ WKH SUHVHQFH RI WKLV GDUN FXUUHQW LV PLQLPDO EHFDXVH WKH OHYHO LV YHU\ UHSHDWDEOH DQG WKH XQFHUWDLQW\ LQ WKH GDUN FRXQW LV PXFK OHVV WKDQ SUHGLFWHG E\ 3RLVVRQ VWDWLVWLFV

PAGE 114

7R DFFRXQW IRU WKH GDUN FXUUHQW WKH W\SLFDO PRGH RI RSHUDWLRQ LQYROYHV DFTXLULQJ RQH GDUN VSHFWUXP QR LQFLGHQW EHDP f IRU HDFK GDWD VSHFWUXP WDNHQ XQGHU DV QHDU LGHQWLFDO FRQGLWLRQV DV SRVVLEOH 6XEWUDFWLRQ RI WKH GDUN VSHFWUXP IURP WKH GDWD LQWURGXFHV D V\VWHP QRLVH FRQWULEXWLRQ LQWR WKH FRUUHFWHG GDWD 7KH PDJQLWXGH RI WKLV HUURU KDV EHHQ HVWLPDWHG E\ DFTXLULQJ GDUN VSHFWUD XQGHU WKH LGHQWLFDO FRQGLWLRQV XVHG LQ DFTXLULQJ HDFK RI WKH HTXLOLEULXP GDWD IRU WKHVH H[SHULPHQWV VFDQV # VHFVFDQ VHF H[SRVXUH f 7KH VWDQGDUG GHYLDWLRQ LQ HDFK FKDQQHO ZDV FDOFXODWHG DQG IRXQG WR EH LQGHSHQGHQW RI FKDQQHO QXPEHU DQG KDG D PHDQ YDOXH RI FRXQWVFKDQQHO 7KH W\SLFDO PLQLPXP FRXQW YDOXH LQ WKH FRUUHFWHG HTXLOLEULXP SURILOHV LV DSSUR[LPDWHO\ [ FRXQWV 7KH FRQWULEXWLRQ WR WKH XQFHUWDLQW\ LQ WKLV SRLQW GXH WR FRXQWLQJ VWDWLVWLFV DVVXPLQJ D 3RLVVRQ GLVWULEXWLRQ f LV DSSUR[LPDWHO\ FRXQWV ,Q WKLV ZD\ LW LV VHHQ WKDW WKH QRLVH LQWURGXFHG GXH WR WKH GDUN FXUUHQW LV D VPDOO FRQWULEXWLRQ WR WKH XQFHUWDLQW\ LQ WKHVH GDWD 3UHOLPLQDU\ ([SHULPHQWDO 3URFHGXUHV 6DPSOH 3UHSDUDWLRQ 7KH SRO\VW\UHQHSRO\EXWDGLHQH GLEORFN FRSRO\PHU 6% f XVHG LQ WKLV H[SHULPHQW ZDV V\QWKHVL]HG E\ 3RO\PHU /DERUDWRULHV /7' 7KH SRO\PHU ZDV SUHSDUHG WKURXJK WKH DQLRQLF SRO\PHUL]DWLRQ SURFHGXUH XVLQJ FRQYHQWLRQDO YDFXXP OLQH WHFKQLTXHV LQ DQ DOO JODVV UHDFWLRQ DSSDUDWXV 0RQRPHUV ZHUH DGGHG VHTXHQWLDOO\ WR DQ RUJDQROLWKLXP LQLWLDWRU 7KH SRO\VW\UHQH DQG SRO\EXWDGLHQH PROHFXOHV DUH LOOXVWUDWHG LQ )LJXUH DORQJ ZLWK WKH VROYHQW PROHFXOHV XVHG LQ WKHVH H[SHULPHQWV 7KH JHO SHUPHDWLRQ FKURPRWRJUDSK\ *3&f WHFKQLTXH ZDV XVHG ERWK WR SUHSDUH DQG FKDUDFWHUL]H WKH KLJKO\ PRQRGLVSHUVH SRO\PHU VDPSOH 7KH *3& WHFKQLTXH LQYROYHV SDVVLQJ WKH SRO\PHU LQ VROXWLRQ WKURXJK D QDUURZ WXEH ILOOHG ZLWK SRURXV PHGLD 7KH

PAGE 115

7KH 3RO\VW\UHQH 0ROHFXOH ? &+f§&+ ? Q 0T $08 7KH 3RO\EXWDGLHQH 0ROHFXOH ? ? &+ &+ &+ &+ 9Q 0 $08 7KH Q7HWUDGHFDQH 0ROHFXOH A % A 0 $08 )LJXUH 7KH FKHPLFDO IRUPXODV RI SRO\VW\UHQH DQG SRO\EXWDGLHQH DORQJ ZLWK WKH VHOHFWLYH VROYHQW XVHG LQ WKHVH VWXGLHV QWHWUDGHFDQH $OVR OLVWHG DUH WKH PROHFXODU ZHLJKWV RI WKH LQGLYLGXDO PRQRPHUV 0

PAGE 116

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f EXWDGLHQHf [ Q EXWDGLHQH f 0Q6%f [ Q 6%f 7KH SRO\PHU KDV D IUDFWLRQDO FRPSRVLWLRQ I RI ZW b SRO\VW\UHQH 7KHVH YDOXHV RI I DQG ZHUH FKRVHQ VSHFLILFDOO\ EHFDXVH WKLV LV D SRO\PHU V\VWHP ZKRVH HTXLOLEULXP SURSHUWLHV KDYH SUHYLRXVO\ EHHQ VWXGLHG 7KH SRO\GLVSHUVLW\ RI WKLV PDWHULDO LQGLFDWHV D VL]H GLVWULEXWLRQ VLJQLILFDQWO\ QDUURZHU WKDQ WKDW RI WKH SRO\PHUV SUHYLRXVO\ XVHG IRU 6$;6 LQYHVWLJDWLRQV f

PAGE 117

7KH SRO\VW\UHQHSRO\EXWDGLHQH WULEORFN FRSRO\PHU 6%6 f DOVR XVHG LQ WKHVH LQYHVWLJDWLRQV ZDV GRQDWHG E\ (QLFKHP ,QF DQG KDV WKH WUDGHQDPH 62/ 7 7KH SRO\PHU ZDV TXRWHG DV KDYLQJ WKH IROORZLQJ FKDUDFWHULVWLFV I ZWb SRO\VW\UHQH 0f [ Q +HQFH WKH WULEORFN VDPSOHV DUH PXFK PRUH SRO\GLVSHUVH WKDQ WKH GLEORFN VDPSOHV ,Q DGGLWLRQ VPDOO DPRXQWV RI SKHQROLF DQG XOWUDYLROHW VWDELOL]HUV DUH SUHVHQW LQ WKH WULEORFN VSHFLPHQV 7KH WULEORFN VL]H DQG FRPSRVLWLRQ ZHUH IRUWXLWRXV LQ WKDW HDFK FKDLQ LV URXJKO\ HTXLYDOHQW WR WZR RI WKH GLEORFN FKDLQV KDYLQJ WKHLU EXWDGLHQH FKDLQV MRLQHG HQG WR HQG 6ROXWLRQV RI WKH SXUH SRO\PHU ZHUH SUHSDUHG LQ WKH VHOHFWLYHVROYHQW Q WHWUDGHFDQH & f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f WROXHQH &+&+f GLPHWK\O IRUPDPLGH +&21 &+ ff PHWK\OHQH FKORULGH &+&f DQG WHWUDK\GUDIXUDQ &+ &+f f 7KH SRO\PHU& PL[WXUH GLG QRW GLVVROYH FRPSOHWHO\ LQ WKH GLPHWK\O IRUPDPLGH RU

PAGE 118

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b GLEORFN RU WULEORFN LQ & UHVSHFWLYHO\ )RU H[DPSOH WKH ZWb GLEORFN FRSRO\PHU VDPSOH LV GHVLJQDWHG 6% 7DEOH 7KH 3RO\PHU 6ROXWLRQ &RQFHQWUDWLRQV 6ROXWLRQ 0DVV RI 3RO\PHU Jf 0DVV RI &O Jf b 3RO\PHU 6% 6% 6% 6% 6%6 6%6

PAGE 119

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f UHJLVWHUHG RQ WKH FHQWHU FKDQQHO f RI WKH GLRGH DUUD\ 7KLV SRVLWLRQ LQ VFDWWHULQJ DQJOH ZDV GHILQHG WR EH r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n VFDWWHULQJ SODQHf GLUHFWLRQ 6LQFH WKH IRFDO SRLQW RI WKH SUHKXWFK FROOLPDWLRQ V\VWHP OLHV EHWZHHQ 6, DQG 6 DQ XSSHU OLPLW RQ WKH DQJXODU GLYHUJHQFH RI WKH EHDP LQ WKH FULWLFDO LQSODQH GLUHFWLRQ LV JLYHQ E\ 3 WDQ ):+0 GV f f ZKHUH GA LV WKH GLVWDQFH IURP VOLWV 6 DQG WKH GHWHFWRU :LWK GV PP WKLV \LHOGV D YDOXH RI 3 r PUDG 7KH WRS DQG ERWWRP EODGHV RI VOLWV 6, ZHUH WKHQ DGMXVWHG XQWLO WKH LQWHJUDWHG LQWHQVLW\ RI WKH EHDP DV UHJLVWHUHG E\ WKH GHWHFWRU GHFUHDVHG E\ b

PAGE 120

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b ZDV REVHUYHG ,Q WKLV SRVLWLRQ VOLWV 6 PDVN WKH SDUDVLWLF VFDWWHULQJ RI 6 ZLWKRXW JHQHUDWLQJ D ODUJH DPRXQW RI SDUDVLWLF VFDWWHULQJ LWVHOI 1H[W WKH UHODWLRQV EHWZHHQ VFDWWHULQJ DQJOH VFDWWHULQJ YHFWRU DQG FKDQQHO QXPEHU ZHUH GHWHUPLQHG 6WDUWLQJ ZLWK WKH VFDWWHULQJ DUP LQ WKH r SRVLWLRQ DQG LQFUHPHQWLQJ E\ r WKH SRVLWLRQ RI WKH FHQWHU RI WKH PDLQ EHDP ZDV UHFRUGHG DV D IXQFWLRQ RI GHWHFWRU SRVLWLRQ ,Q VPDOO DQJOHV WKH UHODWLRQ EHWZHHQ FKDQQHO QXPEHU DQG VFDWWHULQJ DQJOH LV H[SHFWHG WR EH OLQHDU 7KHVH GDWD JDYH WKH IROORZLQJ OLQHDU ILW &KDQQHO rf f )URP WKHVH GDWD LW LV HYLGHQW WKDW FKDQQHOV VSDQ DSSUR[LPDWHO\ ,R LQ VFDWWHULQJ DQJOH ,Q WKH DERYH PHDVXUHPHQW WKH DPSOLWXGH RI WKH ]HUR EHDP SURILOH DW HDFK SRVLWLRQ ZDV XVHG DV D FKHFN RI WKH GHWHFWRU XQLIRUPLW\ 7KH GHWHFWRU ZDV IRXQG WR EH XQLIRUP WR EHWWHU WKDQ b /HDG WDSH ZDV DIIL[HG WR WKH IDFH RI WKH GHWHFWRU VR WKDW FKDQQHOV WR ZHUH PDVNHG RII ,W ZDV GHWHUPLQHG IURP (TXDWLRQ WKDW SODFLQJ WKH VFDWWHULQJ DUP DW r ZRXOG \LHOG D ]HUR EHDP SRVLWLRQ WKDW FRUUHVSRQGV WR FKDQQHO ,Q WKLV SRVLWLRQ WKH OHDG WDSH EORFNV WKH PDLQ EHDP ZKLOH DOORZLQJ PHDVXUHPHQW RI WKH

PAGE 121

1RUPDOL]HG &RXQWV 7KH %HDP 3URILOH 'LVWDQFH PP f )LJXUH $ JUDSK RI WKH EHDP KHLJKW RXW RI WKH VFDWWHULQJ SODQH f DQG ZLGWK LQ SODQH f SURILOHV 7KH ):+0 RI WKH ZLGWK SURILOH LV DSSUR[LPDWHO\ PP 7KH ):+0 RI WKH RXW RI SODQH SURILOH LV VHHQ WR EH PRVWO\ GXH WR WKH KHLJKW RI WKH LQGLYLGXDO SL[HOV LQ WKH GLRGH DUUD\

PAGE 122

VFDWWHUHG SURILOH RYHU WKH UDQJH IURP FKDQQHO WR FKDQQHOV WR GR QRW SHUIRUP UHOLDEO\ f 7KLV FRUUHVSRQGV WR D UDQJH IURP r WR r LQ VFDWWHULQJ DQJOH 8VLQJ (TXDWLRQV DQG WKH IROORZLQJ UHODWLRQ EHWZHHQ FKDQQHO DQG VFDWWHULQJ YHFWRU LV GHULYHG FKDQQHO 4 f [ + FKDQQHOf $r 7KH ZDYHOHQJWK RI WKH LQFLGHQW EHDP XVHG ZDV A DV GLVFXVVHG SUHYLRXVO\ ,W LV FRQYHQLHQW WR GHILQH WZR GHWHFWRU FRQILJXUDWLRQV 7KH ILUVW LV WKH DUP2 FRQILJXUDWLRQ LQ ZKLFK WKH VFDWWHULQJ DUP LV SODFHG DW r DQG WKUHH PLO &X DEVRUEHUV DUH SODFHG LQ WKH EHDP DW WKH SRLQW ZKHUH WKH EHDP HQWHUV WKH KXWFK 7KLV FRQILJXUDWLRQ LV XVHIXO IRU PHDVXULQJ WKH QRUPDOL]DWLRQ SDUDPHWHU WR EH GLVFXVVHG PRPHQWDULO\ 7KH VHFRQG FRQILJXUDWLRQ LV FDOOHG WKH DFTXLVLWLRQ FRQILJXUDWLRQ LQ ZKLFK QR DEVRUEHUV DUH XVHG DQG WKH VFDWWHULQJ DUP LV SODFHG DW r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f LQ WKH EHDP 7KLV PHWKRG KDV WKH GLVDGYDQWDJH WKDW LV RQO\ SURSRUWLRQDO WR LI WKH

PAGE 123

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

PAGE 124

FQ + e c 2 X 3DUDVLWLF 6FDWWHULQJ [O 6FDWWHULQJ 9HFWRU 4 $n f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f ZKLFK KDV EHHQ VFDOHG WR FRLQFLGH ZLWK WKH RWKHUV 7KH WUHDWPHQW RI WKLV JOLWFK LV GHVFULEHG LQ WKH WH[W

PAGE 125

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f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r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

PAGE 126

,W ,P DS D. f :KHQ WKH SDUDVLWLF VFDWWHULQJ PHDVXUHPHQWV ZHUH WDNHQ WKLV TXDQWLW\ ZDV PHDVXUHG DQG LV SURSRUWLRQDO WR WKH LQFLGHQW LQWHQVLW\ DW WKDW WLPH ,%* DQG WKH WUDQVPLVVLRQ FRHIILFLHQW RI WKH .DSWRQ DORQH ,WEJ rr ,EJ D. f )URP WKH DERYH GLVFXVVLRQ LW LV DSSDUHQW WKDW WKH VFDOLQJ IDFWRU $%* LV JLYHQ E\ WKH UHODWLRQ ,[ @ ,WEJ O%* ZKHUH DOO RI WKH YDOXHV RQ WKH ULJKW DUH PHDVXUHG RU UHFRUGHG TXDQWLWLHV f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r& LQFUHPHQWV XQWLO DOO SHDN VWUXFWXUH GLVDSSHDUV &DUHIXO HTXLOLEULXP PHDVXUHPHQWV ZHUH WKHQ LQLWLDWHG 7KH VDPSOH WHPSHUDWXUH ZDV ORZHUHG LQ r& LQFUHPHQWV $W HDFK WHPSHUDWXUH WKH LQWHJUDWHG LQWHQVLW\ RI WKH EHDP WUDQVPLWWHG WKURXJK WKH VDPSOH ZDV PHDVXUHG ZLWK WKH GHWHFWRU LQ WKH DUP2 FRQILJXUDWLRQ 7KH VFDWWHUHG SURILOH ZDV DFTXLUHG ZLWK WKH GHWHFWRU LQ WKH DFTXLVLWLRQ SRVLWLRQ 7\SLFDO H[SRVXUHV ZHUH VFDQV

PAGE 127

# VHF VFDQ VHF H[SRVXUH RU VFDQV # VHF VFDQ VHF H[SRVXUH ,W ZDV QHFHVVDU\ WR DFFXPXODWH WKH VSHFWUD LQ WKLV ZD\ EHFDXVH WKH DFFXPXODWHG FRXQWV H[FHHG WKH G\QDPLF UDQJH RI WKH GHWHFWRU FRXQWV FKDQQHOf $W HDFK WHPSHUDWXUH WKH HOHFWURQ EHDP FXUUHQW ZDV UHFRUGHG 8SRQ UHDFKLQJ URRP WHPSHUDWXUH WKH VROXWLRQ ZDV KHDWHG LQ LQFUHPHQWV RI r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f LV LQMHFWHG LQWR WKH TXHQFKLQJ FKDQQHO LQOHW DV SUHYLRXVO\ GLVFXVVHG 7KH GHSWK RI TXHQFK LV GHWHUPLQHG E\ WKH TXDQWLW\ RI FRRODQW XVHG DQG WKH GXUDWLRQ RI WKH SURSHOODQW EXUVW 7DEOH OLVWV WKHVH YDOXHV DORQJ ZLWK WKH LQLWLDO DQG ILQDO WHPSHUDWXUHV IRU DOO RI WKH TXHQFKHV SHUIRUPHG LQ WKHVH VWXGLHV $IWHU ORDGLQJ WKH FRRODQW WKH WXELQJ FRQQHFWLQJ WKH IXUQDFH DQG WKH +H FDQQLVWHU LV SXW LQ SODFH

PAGE 128

7DEOH 7KH ([SHULPHQWDO 4XHQFK 3DUDPHWHUV 6ROXWLRQ 7L r&f 7I r&f &RRODQW FFf +H %XUVW VHFf ([SRVXUH VHFf 6% 6% 6% 6%

PAGE 129

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c 7KH VSHFWUD DUH WUDQVIHUUHG WR WKH FRPSXWHU IURP WKH GHWHFWRU LQWHUIDFH LQ SDLUV L H VSHFWUD DQG DQG HWFf 7KH GLIIHUHQFH EHWZHHQ WKHVH GDWD DQG GDUN VSHFWUD LV FDOFXODWHG DQG WKH UHVXOW LV VDYHG WR IORSS\ GLVN

PAGE 130

7KLV SURFHGXUH ZDV UHSHDWHG IRU DOO RI WKH TXHQFKHV OLVWHG LQ 7DEOH 7KH DQDO\VLV RI WKH GDWD DFTXLUHG LQ WKH HTXLOLEULXP VWXGLHV LV GHVFULEHG LQ WKH IROORZLQJ FKDSWHU ZLWK WKH IDVFLQDWLQJ UHVXOWV RI WKH NLQHWLF PHDVXUHPHQWV GHIHUUHG WR &KDSWHU

PAGE 131

&+$37(5 $1$/<6,6 $1' 67$7,& 5(68/76 ,Q WKLV FKDSWHU WKH UHVXOWV RI WKH VWDWLF PHDVXUHPHQWV DUH GHVFULEHG 7KH WHPSHUDWXUH GHSHQGHQW 6$;6 SURILOHV DUH SUHVHQWHG DQG LW LV VKRZQ WKDW WKH UHVROXWLRQ DFKLHYHG HQDEOHV VHSDUDWLRQ RI WKH VSHFWUD LQWR RUGHUHG DQG GLVRUGHUHG FRPSRQHQWV FRUUHVSRQGLQJ WR WKH RUGHUHG DQG GLVRUGHUHG DUUDQJHPHQWV RI PLFURGRPDLQV 7KLV LV WKH ILUVW WLPH WKDW VXFK D VHSDUDWLRQ LV SRVVLEOH 'HWHUPLQDWLRQ RI WKH VWUXFWXUDO SDUDPHWHUV IURP PRGHOV RI WKH [UD\ VFDWWHULQJ DUH GLVFXVVHG IROORZHG E\ DQ LQGHSWK FKDUDFWHUL]DWLRQ RI WKH PDFURODWWLFHV REVHUYHG LQ WKHVH VROXWLRQV 6RPH FRPPHQWV RQ WKH HIIHFWV RI UHVROXWLRQ DUH GLVFXVVHG 9DULDWLRQ RI WKH VWUXFWXUDO SDUDPHWHUV ZLWK SRO\PHU FRQFHQWUDWLRQ DUH FRPSDUHG ZLWK WKH UHVXOWV RI SUHYLRXV VWXGLHV )LQDOO\ WKH WHPSHUDWXUH GHSHQGHQFH RI WKH PLFURGRPDLQ VWUXFWXUH LV H[DPLQHG DQG WKH SKDVH GLDJUDP RI WKH V\VWHP LV GLVFXVVHG LQ SUHSDUDWLRQ IRU SUHVHQWDWLRQ RI WKH NLQHWLF UHVXOWV 3UHVHQWDWLRQ RI WKH 6WDWLF 'DWD 7KH VFDWWHUHG LQWHQVLWLHV ,4f DUH SORWWHG YHUVXV VFDWWHULQJ YHFWRU IRU WKH 6% VROXWLRQ DV D IXQFWLRQ RI WHPSHUDWXUH XSRQ FRROLQJ )LJXUH f DQG KHDWLQJ )LJXUH f 7KHVH SURILOHV DUH SORWWHG RQ D ORJ VFDOH DQG DUH RIIVHW IURP HDFK RWKHU E\ RQH GHFDGH IRU FODULW\ VWDUWLQJ ZLWK WKH ORZ WHPSHUDWXUH SURILOHV 6LPLODU UHVXOWV IRU WKH 6% 6% 6% 6%6 DQG 6%6 VROXWLRQV DUH SUHVHQWHG LQ )LJXUHV WKURXJK 7KHVH ILJXUHV DUH SUHVHQWHG WRJHWKHU KHUH IRU FRQYHQLHQFH DQG DUH UHIHUUHG WR WKURXJKRXW WKH FKDSWHU

PAGE 132

&28176 R 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ FRROLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP

PAGE 133

&28176 R 6FDWWHULQJ 9HFWRU 4 $ f 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ KHDWLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP )LJXUH

PAGE 134

&28176 R 6FDWWHULQJ 9HFWRU 4 $ f 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ FRROLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP )LJXUH

PAGE 135

&28176 r f§ O ? 6FDWWHULQJ 9HFWRU 4 $ f 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ KHDWLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP )LJXUH

PAGE 136

&28176 D A 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ FRROLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP

PAGE 137

&28176 r 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ KHDWLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP

PAGE 138

&28176 R 6FDWWHULQJ 9HFWRU 4 $ f 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ FRROLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP )LJXUH

PAGE 139

&28176 R A 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH 5DZ VFDWWHULQJ SURILOHV IRU WKH 6% VDPSOH XSRQ KHDWLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP

PAGE 140

&28176 R A 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH 5DZ VFDWWHULQJ SURILOHV IRU WKH 6%6 VDPSOH XSRQ FRROLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP

PAGE 141

&28176 r f§ ? 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH 5DZ VFDWWHULQJ SURILOHV IRU WKH 6%6 VDPSOH XSRQ KHDWLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP

PAGE 142

&28176 r f§ ? 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH 5DZ VFDWWHULQJ SURILOHV IRU WKH 6%6 VDPSOH XSRQ FRROLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP

PAGE 143

&28176 R 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH 5DZ VFDWWHULQJ SURILOHV IRU WKH 6%6 VDPSOH XSRQ KHDWLQJ 7KH GDWD DUH UHSUHVHQWHG E\ D VROLG OLQH GXH WR WKH GHQVLW\ RI SRLQWV RQ WKHVH FXUYHV DQG DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ RI SUHVHQWDWLRQ 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP

PAGE 144

,GHQWLILFDWLRQ RI WKH ;UDY 6FDWWHULQJ )HDWXUHV 0DQ\ RI WKH IHDWXUHV RI [UD\ VSHFWUD GHVFULEHG LQ FKDSWHU DUH HYLGHQW LQ WKHVH ILJXUHV $W KLJK WHPSHUDWXUHV WKH VFDWWHULQJ FXUYHV DUH PRQRWRQLFDOO\ GHFUHDVLQJ IXQFWLRQ RI WKH VFDWWHULQJ YHFWRU 4f 7KH VWHS ZKLFK FDQ EH VHHQ DW DURXQG [ ƒ LQ )LJXUH LV D IHDWXUH FRQWDLQHG LQ WKH SDUDVLWLF VFDWWHULQJ DQG LWV WUHDWPHQW ZDV GLVFXVVHG LQ &KDSWHU $W 7 r& WKH VDPSOH LV LQ WKH KRPRJHQHRXV VWDWH DQG H[KLELWV RQO\ FRQWLQXRXV VFDWWHULQJ 7KH ORZ 4 UHJLRQ RI WKLV VORSLQJ FXUYH LV ZHOO DSSUR[LPDWHG E\ WKH VXP RI WKH PHDVXUHG SDUDVLWLF VFDWWHULQJ IURP WKH NDSWRQ ZLQGRZV DQG D JDXVVLDQ RI YDULDEOH DPSOLWXGH DQG ZLGWK SDUDPHWHUV $V WKH WHPSHUDWXUH LV ORZHUHG EURDG SHDNV EHFRPH HYLGHQW DV WKH UHVXOW RI WKH PLFURSKDVH VHSDUDWLRQ WUDQVLWLRQ 067f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r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

PAGE 145

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f ,Q WKHVH VROXWLRQV QR %UDJJ SHDNV EHFRPH HYLGHQW HYHQ DW WKH ORZHVW WHPSHUDWXUHV VWXGLHG LQ WKHVH H[SHULPHQWV 7 &f 7KH DEVHQFH RI RUGHULQJ LQ WKHVH VROXWLRQV LV EHOLHYHG WR EH WKH UHVXOW RI WKH ODUJH SRO\GLVSHUVLW\ RI WKH LQGLYLGXDO FKDLQV ZKLFK JLYHV ULVH WR D ODUJH SRO\GLVSHUVLW\ LQ WKH UDGLL RI WKH VSKHULFDO VW\UHQH GRPDLQV 7KH VL]H GLVWULEXWLRQ KDV EHHQ HVWLPDWHG IURP PRGHOOLQJ RI WKH VSKHULFDO IRUP IDFWRU LQ WKH KLJK 4 UHJLRQ RI WKH 6$;6 VSHFWUD VHH 7DEOH f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

PAGE 146

PDFURODWWLFH VWUXFWXUH LV IRXQG WR EH ERG\FHQWHUHGFXELF %&& f DV ZLOO EH GLVFXVVHG ZLWK WZR H[FHSWLRQV ,Q WKH 6% VROXWLRQ WKH ILUVWRUGHU EUDJJ UHIOHFWLRQ H[KLELWV ILQH VWUXFWXUH DW DOO WHPSHUDWXUHV EHORZ 7 DQG LQ WKH 6% VDPSOH D FRPSRQHQW ZLWK VLPSOH FXELF 6& f VWUXFWXUH LV REVHUYHG LQ WKH r& VSHFWUXP VHH )LJXUH f ,QWHUSUHWDWLRQ RI WKHVH UHVXOWV LV GLVFXVVHG ODWHU LQ WKLV FKDSWHU )URP WKH VFDWWHULQJ GDWD DORQH LW LV QRW FOHDU WKDW LQWHUSUHWDWLRQ RI WKH GLVRUGHUHG VWUXFWXUH DV D OLTXLG LV GHILQLWLYH ,Q IDFW WKH VWUXFWXUH IDFWRUV IRU RWKHU W\SHV RI V\VWHPV OLNH DPRUSKRXV VROLGV DQG LPSHUIHFW FU\VWDOV FDQ DOVR \LHOG VLPLODU VFDWWHULQJ SDWWHUQV 7KH REVHUYHG VFDWWHULQJ PD\ EH FRQVLVWHQW ZLWK WKDW H[SHFWHG IRU DQ LPSHUIHFW FU\VWDO DOWKRXJK WKH FRQFHQWUDWLRQ RI GHIHFWV LQ VXFK D VWUXFWXUH ZRXOG KDYH WR EH YHU\ KLJK 7KH GLVWLQFWLRQ EHWZHHQ D FU\VWDO ZLWK D ODUJH FRQFHQWUDWLRQ RI GHIHFWV DQG DQ DPRUSKRXV VROLG EHFRPHV VRPHZKDW DUELWUDU\ LQ WKLV OLPLW )URP D SK\VLFDO VWDQGSRLQW LW VHHPV XQOLNHO\ WKDW WKH VWUXFWXUH FKDUDFWHULVWLF RI WKH LQWHUPHGLDWH WHPSHUDWXUH UDQJH 7G 7 7 f LV WKDW RI DQ DPRUSKRXV VROLG EHFDXVH FRROLQJ WKH VDPSOH IURP WKLV WHPSHUDWXUH UDQJH HYHQWXDOO\ UHVXOWV LQ D FU\VWDOOLQH VWUXFWXUH *HQHUDOO\ WKH DPRUSKRXV SKDVH LV HLWKHU WKH ORZ WHPSHUDWXUH SKDVH RU D PHWDVWDEOH VWDWH WKDW UHVXOWV IURP VXSHUFRROLQJ RI WKH VSHFLPHQ ,Q WKH ODWWHU FDVH WKH FU\VWDOOLQH SKDVH LV RQO\ DWWDLQHG E\ D VXEVHTXHQW KHDWLQJ RI WKH VDPSOH WR DOORZ UHGLVWULEXWLRQ RI WKH FRPSRVLWH SDUWLFOHV RQWR D ODWWLFH 7KH DERYH LV WUXH LQ PHWDOV ZKHUH WKH VROLG LV FRPSRVHG RI DWRPV PRYLQJ DERXW LQ YDFXXP ,W LV QRW FOHDU WKDW VLPLODU EHKDYLRU VKRXOG EH H[SHFWHG LQ WKH %&3 V\VWHP ZKHUH WKH SDUWLFOHV DUH ODUJH VW\UHQH GRPDLQV ZKLFK PRYH DERXW LQ DQ H[WUHPHO\ YLVFRXV PHGLXP 7KH GLVFHUQLQJ TXDQWLW\ EHWZHHQ WKH DPRUSKRXV OLTXLG DQG VROLG SKDVHV LV RI FRXUVH WKH GHJUHH RI PRELOLW\ RI WKH SDUWLFOHV $QG WR WKH H[WHQW WKDW WKLV PRELOLW\ LV QRW PHDVXUHG LQ WKHVH H[SHULPHQWV FODVVLILFDWLRQ RI WKH GLVRUGHUHG VWDWH LV QRW SRVVLEOH IURP WKHVH GDWD ,Q DQ\ HYHQW LW LV EHOLHYHG WKDW WKH GLVRUGHUHG SKDVH FDQ EH LGHQWLILHG DV DQ

PAGE 147

DPRUSKRXV VWUXFWXUH HLWKHU VROLG RU OLTXLG f FRPSRVHG RI SRO\VW\UHQH PLFURGRPDLQV DQG WKDW WKH V\VWHP XQGHUJRHV ZKDW LV DQDORJRXV WR D FU\VWDOOL]DWLRQ WUDQVLWLRQ DW 7 7KH (IIHFWV RI ([SHULPHQWDO 5HVROXWLRQ ,Q 6$;6 WKH HIIHFWV RI H[SHULPHQWDO UHVROXWLRQ DUH W\SLFDOO\ GHVFULEHG WKURXJK WKH VPHDULQJ LQWHJUDO 84f :[f +\f ,4 \ fP [ f G[ G\ f ZKHUH A4f LV WKH PHDVXUHG VFDWWHUHG LQWHQVLW\ DW 4 DQG LV WKH WUXH VFDWWHUHG LQWHQVLW\ ZLWKRXW WKH HIIHFWV RI UHVROXWLRQ 7KH IXQFWLRQ :[f LV FDOOHG WKH VOLW ZLGWK ZHLJKWLQJ IXQFWLRQ :[f LV XVXDOO\ WDNHQ WR EH WKH PHDVXUHG EHDP SURILOH :%4f LQ WKH SODQH RI VFDWWHULQJ DW WKH GHWHFWRU SRVLWLRQ 6LPLODUO\ +\f LV FDOOHG WKH VOLW KHLJKW ZHLJKWLQJ IXQFWLRQ ZKLFK LV WDNHQ WR EH WKH PHDVXUHG EHDP SURILOH +%4f LQ WKH SODQH SHUSHQGLFXODU WR WKH VFDWWHULQJ SODQH ([WUDFWLQJ ,4f IURP WKH PHDVXUHG VFDWWHULQJ ,P4f DQG (TXDWLRQ f LV WHUPHG GHVPHDULQJ 7KHVH ZHLJKWLQJ IXQFWLRQV FDQ EH FRUUHFW LI WKH LQFLGHQW EHDP LV FKDUDFWHUL]HG E\ DQ DQJXODU GLYHUJHQFH D SHUIHFW FROOLPDWLRQ f :KHQ WKLV LV QRW WKH FDVH WKH GHULYDWLRQ RI WKH VPHDULQJ LQWHJUDO EHFRPHV PXFK PRUH FRPSOH[ DQG D GHWDLOHG NQRZOHGJH RI WKH DQJXODU GLVWULEXWLRQ RI WKH LQFLGHQW EHDP LV UHTXLUHG (YLGHQFH WKDW (TXDWLRQ f ZLWK :%4f DQG +%4f GRHV QRW FRUUHFWO\ GHVFULEH WKH UHVROXWLRQ HIIHFWV LQ WKHVH GDWD LV WKDW WKH ):+0 RI :%4f LV [ ƒ ZKLOH %UDJJ SHDNV PHDVXUHG LQ WKHVH H[SHULPHQWV VKRZ YDOXHV RI WKH ):7,0 RI OHVV WKDQ KDOI RI WKLV YDOXH [ O2 ƒnf 7KHVH GDWD DUH WKHUHIRUH QRW FRQVLVWHQW ZLWK (TXDWLRQ DQG WKLV FKRLFH RI ZHLJKWLQJ IXQFWLRQV :LWKRXW D PRUH GHWDLOHG DQDO\VLV RI WKH HIIHFW RI DQJXODU GLYHUJHQFH RQ WKH VPHDULQJ LQWHJUDO TXDQWLWDWLYH H[WUDFWLRQ RI WKH UHVROXWLRQ LV QRW SRVVLEOH +RZHYHU WKH

PAGE 148

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e WKH IUDFWLRQ RI PDWHULDO LQ WKH RUGHUHG VWDWH 7KH 0LFURGRPDLQ 3DUDPHWHUV )LJXUH VKRZV WKH 6$;6 SURILOH IRU WKH 6% VDPSOH DW r& RQ D ORJ VFDOH 7KH GDWD DUH UHSUHVHQWHG E\ SL[HOV 7KH SRO\GLVSHUVH PRGHO RI VSKHULFDO VFDWWHULQJ

PAGE 149

&28176 6SKHULFDO )RUP )DFWRU 2 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH $ SORW RI WKH VFDWWHUHG SURILOH IRU WKH 6% VDPSOH DW r& 7KH GDWD DUF UHSUHVHQWHG E\ SL[HOV DQG WKH VTXDUH RI WKH VSKHULFDO IRUP IDFWRU HVWLPDWHG DV GHVFULEHG LQ WKH WH[W LV GLVSOD\HG DV D VROLG OLQH

PAGE 150

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ƒ E\ YDU\LQJ 5 XQWLO DQ REYLRXV PLVPDWFK ZDV VHHQ 6LPLODUO\ WKH HVWLPDWHG XQFHUWDLQW\ LQ R5 ZDV IRXQG WR EH ƒ 7KH VWDQGDUG GHYLDWLRQ LQ FKDLQ OHQJWK LV UHODWHG WR WKH SRO\GLVSHUVLW\ WKURXJK (TXDWLRQ DQG LV FDOFXODWHG WR b RI WKH WRWDO FKDLQ OHQJWK IRU D SRO\GLVSHUVLW\ RI 7KLV YDOXH LV LQ DJUHHPHQW ZLWK WKH YDOXHV RI D5 VKRZQ LQ 7DEOH LQGLFDWLQJ WKDW WKH GLVWULEXWLRQ LQ VSKHUH VL]HV URXJKO\ VFDOHV ZLWK WKH SRO\GLVSHUVLW\ LQ FKDLQ VL]HV 7KH PD[LPD RI WKH VSKHULFDO IRUP IDFWRU DUH RQO\ FOHDUO\ LGHQWLILDEOH DW WKH ORZHU WHPSHUDWXUHV $W KLJKHU WHPSHUDWXUHV WKHUPDO GLIIXVH VFDWWHULQJ ZDVKHV RXW WKH PLQLPD SUHYHQWLQJ UHOLDEOH HVWLPDWLRQ RI WKH VSKHULFDO SDUDPHWHUVf $ GHWHUPLQDWLRQ RI WKH LQWHUIDFLDO WKLFNQHVV LV RIWHQ SRVVLEOH E\ DQDO\]LQJ WKH V\VWHPDWLF GHYLDWLRQ RI WKH VFDWWHULQJ IURP 3RURGf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

PAGE 151

7DEOH 7KH (VWLPDWHG 0LFURGRPDLQ DQG /DWWLFH 3DUDPHWHUV 6ROXWLRQ 7 r& f 5 ƒ f &5 ƒ f D ƒ f 6% 6% 6% 6% 6%6 6%6 $QDO\VLV RI WKH 0DFURODWWLFH 6WUXFWXUH DQG 'LPHQVLRQV 4XDQWLWDWLYH DQDO\VLV RI WKH LQWHUSDUWLFOH VFDWWHULQJ ZDV SHUIRUPHG E\ SDUDPHWHUL]LQJ D VXEUDQJH RI WKH GDWD IURP [ OIU ƒ :LWKLQ WKLV UDQJH WKH REVHUYHG VFDWWHUHG LQWHQVLW\ ,4f LV GRPLQDWHG E\ WKUHH VFDWWHULQJ FRPSRQHQWV ,4f 6D4f 6F4f %4f f ZKHUH %4f LV WKH EDFNJURXQG ZKLFK DULVHV SULPDULO\ IURP SDUDVLWLF VFDWWHULQJ E\ WKH FROOLPDWLQJ VOLWV DQG WKH .DSWRQ ZLQGRZV 6D4f DQG 6&4f DUH WKH DPRUSKRXV DQG FU\VWDOOLQH VWUXFWXUH IDFWRUV UHVSHFWLYHO\ ZKLFK DUH ERWK SURSRUWLRQDO WR WKH VTXDUH RI WKH IRUP IDFWRU ,)4f, IRU WKH PLFURGRPDLQV 7KHVH FRPSRQHQWV DUH LOOXVWUDWHG LQ )LJXUH ZKHUH VSHFWUD FRUUHVSRQGLQJ WR WKUHH GLIIHUHQW WHPSHUDWXUHV DUH VKRZQ 7KH VSHFWUD DUH RIIVHW E\ WZR GHFDGHV IRU FODULW\ DQG WKH GDWD SRLQWV DUH LQGLFDWHG DV SL[HOV 7KH VROLG OLQHV DUH ILWV RI WKH VXP RI WKH WKUHH FRPSRQHQWV WR WKH GDWD 7KH VKRUWGDVKHG DQG ORQJ GDVKHG OLQHV LOOXVWUDWH WKH DPRUSKRXV DQG EDFNJURXQG VFDWWHULQJ FRPSRQHQWV WR WKH r& VSHFWUXP UHVSHFWLYHO\

PAGE 152

&28176 r f§ 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH $ SORW VKRZLQJ WKH WKUHH FRPSRQHQWV RI WKH SDUDPHWHUL]DWLRQ RI WKH ORZ 4 UDQJH RI WKH VFDWWHUHG VSHFWUD 7KH GDWD DW WKH IROORZLQJ WHPSHUDWXUHV r& r& DQG r& DUH UHSUHVHQWHG E\ SL[HOV DQG RIIVHW IURP HDFK RWKHU E\ WZR GHFDGHV IRU FODULW\ 7KH VROLG OLQHV DUH WKH IXOO ILWV WR WKH GDWD LQFOXGLQJ DOO WKUHH FRPSRQHQWV DV GHVFULEHG LQ WKH WH[W 7KH VKRUW DQG ORQJ GDVKHG OLQHV LQ WKH ORZ WHPSHUDWXUH FXUYH UHSUHVHQW WKH DPRUSKRXV DQG EDFNJURXQG FRQWULEXWLRQV WR WKLV VSHFWUXP UHVSHFWLYHO\

PAGE 153

7KH EDFNJURXQG FRQWULEXWLRQ 7KH EDFNJURXQG VFDWWHULQJ FRQWULEXWLRQ ZDV GHWHUPLQHG DV GHVFULEHG LQ &KDSWHU DQG VFDOHG E\ D PHDVXUHG IDFWRU VR WKDW WKHUH DUH QR YDULDEOH SDUDPHWHUV IRU WKLV FRQWULEXWLRQ 7KH DPRUSKRXV FRQWULEXWLRQ 7KH VHFRQG WHUP LQ (TXDWLRQ f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}4f $T H[3 IAO >2@ @ L f n *R $ U ‘ $ Nf 7KLV LV LQ DJUHHPHQW ZLWK WKH 2PVWHLQ=HPLFNH UHVXOW ZKLFK SUHGLFWV /RUHQW]LDQ SHDNV LQ 64f IRU V\VWHPV FRQWDLQLQJ VKRUWUDQJH RUGHU ,Q DGGLWLRQ WKH PRGHOV IRU OLTXLG VFDWWHULQJ SUHVHQWHG LQ &KDSWHU ZHUH HPSLULFDOO\ IRXQG WR EH ZHOO DSSUR[LPDWHG E\ WKH VDPH IRUP (TXDWLRQ f 7KH DPRUSKRXV FRPSRQHQW LV VKRZQ LQ )LJXUH DV D VKRUWGDVKHG OLQH DQG LV HDVLO\ LGHQWLILHG HYHQ LQ WKH SUHVHQFH RI %UDJJ SHDNV GXH WR WKH EURDG WDLOV FKDUDFWHULVWLF RI WKH /RUHQW]LDQ OLQHVKDSH

PAGE 154

7KH &U\VWDOOLQH &RQWULEXWLRQ 7KH WKLUG WHUP LQ (TXDWLRQ UHSUHVHQWV WKH RUGHUHG FRQWULEXWLRQ ZKLFK DSSHDUV DV %UDJJ SHDNV LQ WKH VFDWWHUHG VSHFWUXP 7KH %UDJJ SHDNV ZHUH IRXQG WR EH ZHOO FKDUDFWHUL]HG E\ D VXP RI *DXVVLDQV 6&4f f L O +HUH WKH $c DQG c DUH YDULDEOH DPSOLWXGH DQG ZLGWK SDUDPHWHUV UHVSHFWLYHO\ 7KH SRVLWLRQV RI WKH SHDNV 4c ZHUH DOORZHG WR YDU\ DQG IRXQG WR EH UHODWHG LQ WKH IROORZLQJ ZD\ 4 \c 4X 4 ?I"! 4DQG 4 4L 'HYLDWLRQV IURP WKHVH UHODWLYH SRVLWLRQV DUH W\SLFDOO\ OHVV WKDQ b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f

PAGE 155

IDFHFHQWHUHG FXELF )&& f DQG VLPSOH FXELF 6& f 0RGHOV RI %&3V LQ EXON JHQHUDOO\ SUHGLFW WKH %&& VWUXFWXUH WR EH VOLJKWO\ IDYRUHG RYHU WKH )&& VWUXFWXUH ZLWK WKH 6& VWUXFWXUH KDYLQJ D FRQVLGHUDEO\ KLJKHU HQHUJ\ ,Q VROXWLRQV RI 363% LQ &O DW WKH FRQFHQWUDWLRQV VWXGLHG KHUH WKH VWUXFWXUH ZDV SUHYLRXVO\ UHSRUWHG DV 6& 7KH UHODWLYH ORZHU RUGHU SHDN ORFDWLRQV REVHUYHG \>? \c \c DQG ? DUH FRQVLVWHQW ZLWK ERWK WKH 6& DQG %&& VWUXFWXUHV EXW QRW WKH )&& VWUXFWXUH 2I WKH UHPDLQLQJ PRVW OLNHO\ VWUXFWXUHV %&& DQG 6& WKHVH GDWD ZHUH IRXQG WR VWURQJO\ IDYRU WKH %&& VWUXFWXUH LQ WKUHH ZD\V YROXPH IUDFWLRQ FRPSDULVRQV VWUXFWXUHVSHFLILF PRGHO ILWV DQG WKH DSSHDUDQFH RI WKH \IM SHDN ZKLFK LV GLVDOORZHG IRU WKH 6& VWUXFWXUH )URP WKH REVHUYHG SHDN ORFDWLRQV WKH ODWWLFH FRQVWDQW D FDQ EH GHWHUPLQHG LI D VSHFLILF ODWWLFH W\SH LV DVVXPHG )RU WKH %&& ODWWLFH f ZKHUH S LV WKH ORFDWLRQ RI WKH ORZHVW RUGHU SHDN REVHUYHG )RU WKH %&& ODWWLFH WKLV FRUUHVSRQGV WR WKH f JURXS RI UHIOHFWLRQV )RU WKH 6& ODWWLFH f VLQFH WKH ORZHVW RUGHU SHDN IRU WKH 6& VWUXFWXUH FRUUHVSRQGV WR WKH f JURXS RI UHIOHFWLRQV )URP WKH HVWLPDWHG VSKHULFDO UDGLL 5 LQ 7DEOH WKH 36 YROXPH IUDFWLRQ _fV FDQ EH FDOFXODWHG LI FHUWDLQ DVVXPSWLRQV DUH PDGH )LUVW WKH VSKHULFDO GRPDLQV DUH DVVXPHG WR KDYH ]HUR LQWHUIDFLDO WKLFNQHVV DQG EH FRPSRVHG SXUHO\ RI 36 6LPLODUO\ WKH LQWHUVWLWLDO YROXPH LV DVVXPHG WR EH RFFXSLHG H[FOXVLYHO\ E\ 3% DQG &O 7KHVH DVVXPSWLRQV LPSO\ D SHUIHFW VHOHFWLYLW\ RI WKH & VROYHQW 8VLQJ WKHVH DSSUR[LPDWLRQV WKH ffV IRU WKH %&& DQG 6& VWUXFWXUHV DUH N5 f S 5f Df A D%&& f f UHVSHFWLYHO\

PAGE 156

7KHVH YDOXHV FDQ EH FRPSDUHG WR WKH LGHDO YDOXH _fVMGHDO! ZKLFK LV FDOFXODWHG IURP WKH VWRLFKLRPHWU\ RI WKH VROXWLRQV 7KHUH LV DVVXPHG WR EH QR YROXPH FKDQJH XSRQ PL[LQJ RI WKH 3% DQG & DQG WKH GHQVLWLHV RI WKH FRPSRQHQWV DUH WDNHQ WR EH WKHLU EXON YDOXHV DQG JFP IRU 36 3% DQG & UHVSHFWLYHO\ 7KH VWRLFKLRPHWULF YROXPH IUDFWLRQ LV WKHQ I& 3V W!6,'($/ I& OIf& &f n ` 3V 3E 3FL ZKHUH I LV WKH VW\UHQH IUDFWLRQ RI WKH %&3 E\ ZHLJKW DQG & LV WKH SRO\PHU FRQFHQWUDWLRQ LQ VROXWLRQ 7KH LGHDO YDOXHV DUH VKRZQ LQ 7DEOH DORQJ ZLWK WKRVH YDOXHV FDOFXODWHG IRU WKH %&& DQG 6& VWUXFWXUHV 1RWH WKH YDOXH RI D LQ 7DEOH LV WKH FRQYHQWLRQDO ODWWLFH FRQVWDQW DVVXPLQJ D %&& VWUXFWXUH (TXDWLRQ f $ GLIIHUHQW YDOXH (TXDWLRQ f ZDV XVHG LQ WKH SUHFHGLQJ FDOFXODWLRQ RI MfVVF 7KHVH YDOXHV FOHDUO\ IDYRU WKH %&& VWUXFWXUH 7KH GLVFUHSDQF\ EHWZHHQ `fV ,'($/ DQG c!6!%&& EHOLHYHG WR EH FDXVHG PDLQO\ E\ LPSHUIHFW VHOHFWLYLW\ RI WKH & VRPH RI WKH VSKHULFDO YROXPH LV RFFXSLHG E\ & PDNLQJ LW ODUJHU WKDQ H[SHFWHG IURP VWRLFKLRPHWU\ f DQG D QHJDWLYH YROXPH FKDQJH XSRQ PL[LQJ 3% DQG & &RUUHFWLQJ IRU ERWK RI WKHVH HIIHFWV ZRXOG WHQG WR LPSURYH WKH DJUHHPHQW EHWZHHQ WKH LGHDO DQG %&& FDOFXODWHG YDOXHV 7DEOH &DOFXODWHG 6W\UHQH 9ROXPH )UDFWLRQV 6ROXWLRQ ,'($/ A6%&& AVVF 5 ƒ f D ƒ f 6% 6% 6% 6%

PAGE 157

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f $,)4fOH[S%4f f ZKHUH % X f 4L 9L 4R D R@ R 4R Lf +HUH % LV WKH PHDQ VTXDUH GLVSODFHPHQW RI WKH 'HE\H:DOOHU IDFWRU 4 LV WKH ORFDWLRQ RI WKH ORZHVW RUGHU PD[LPXP DQG $ LV DQ RYHUDOO DPSOLWXGH IDFWRU 7KH ZLGWK SDUDPHWHU D LQFRUSRUDWHV WKH UHVROXWLRQ DQG ILQLWH VL]H HIIHFWV ZKLOH WKH VHFRQG ZLGWK FRQWULEXWLRQ 2M LV D PHDVXUH RI WKH VWUDLQ LQ WKH ODWWLFH DV GHVFULEHG LQ &KDSWHU %\ IL[LQJ WKH Pc DW WKRVH YDOXH IRU WKH %&& RU 6& ODWWLFH W\SHV JLYHQ LQ 7DEOH VWUXFWXUH VSHFLILF PRGHOV DUH FRQVWUXFWHG KDYLQJ YDULDEOH SDUDPHWHUV $ % 4T D DQG D 7KH DERYH IRUP FRPELQHG ZLWK WKH DPRUSKRXV DQG EDFNJURXQG VFDWWHULQJ FRPSRQHQWV SUHYLRXVO\ GLVFXVVHG ZDV ILW WR WKH GDWD IRU WKH 6% VROXWLRQ 7KH UHVXOWV IRU WKH r& VSHFWUXP IRU HDFK ODWWLFH W\SH DUH VKRZQ LQ )LJXUH &OHDUO\ WKH %&&

PAGE 158

6WUXFWXUH 'HWHUPLQDWLRQ 6FDWWHULQJ 9HFWRU $f )LJXUH $ FRPSDULVRQ RI WKH VLPSOH FXELF DQG ERG\FHQWHUHG FXELF VWUXFWXUH VSHFLILF ILWV GHVFULEHG LQ WKH WH[Wf WR WKH VFDWWHUHG VSHFWUXP IRU WKH 6% VROXWLRQ DW r&

PAGE 159

FXUYH LV D PXFK EHWWHU ILW WR WKH GDWD 7KH UHGXFHG FKLVTXDUHG SDUDPHWHU LV XVHG WR WHVW WKH JRRGQHVV RI ILW RI WKHVH WZR PRGHOV \I8n\Lf Q f f \c  7DEOH 0XOWLSOLFLWLHV DQG 5HODWLYH 3RVLWLRQV RI %UDJJ 3HDNV IRU WKH 6& DQG %&& 6WUXFWXUHV KNO 6& P 3 KNO %&& P 3 r r 7KHUH LV QR SHDN IRU WKH 6& VWUXFWXUH LQ WKH S UHODWLYH SRVLWLRQ %HFDXVH P LV OHVV WKDQ P IRU WKH 6& VWUXFWXUH WKH 6& PRGHO LV XQDEOH WR VLPXOWDQHRXVO\ PDWFK WKH ILUVW VHFRQG DQG WKLUG RUGHU SHDNV 7KLV LV QRW WKH FDVH IRU WKH %&& PRGHO ZKHUH P LV VLJQLILFDQWO\ ODUJHU WKDQ P )RU WKLV UHDVRQ D PXFK EHWWHU ILW LV DFKLHYHG ZLWK WKH %&& PRGHO 7KH VWUXFWXUH VSHFLILF ILWV IRU WKH 6% DQG 6% VROXWLRQV DW YDU\LQJ WHPSHUDWXUHV \LHOG UHVXOWV FRQVLVWHQW ZLWK WKHVH

PAGE 160

7KH VWUXFWXUH VSHFLILF PRGHO RI WKH FU\VWDOOLQH [UD\ VFDWWHULQJ FRPSRQHQW LQ JHQHUDO \LHOG VOLJKWO\ ZRUVH ILWV WR WKH GDWD WKDQ WKH IRUP JLYHQ LQ (TXDWLRQ IRU WKH 6% 6% DQG 6% VROXWLRQV 2QH SRVVLEOH VRXUFH RI GHYLDWLRQ IURP WKH %&& PRGHO LV WKH DVVXPSWLRQ RI DQ LVRWURSLF VWUDLQ )URP WKLV DVVXPSWLRQ WKH H[SUHVVLRQ IRU WKH %UDJJ ZLGWK JLYHQ LQ (TXDWLRQ UHVXOWV ZKLFK LV PRQRWRQLFDOO\ LQFUHDVLQJ IXQFWLRQ RI WKH RUGHU RI WKH UHIOHFWLRQ L :KHQ WKH VWUDLQ LV DQLVRWURSLF WKH SHDN ZLGWKV FDQ EH ODUJHU LQ FHUWDLQ KNO GLUHFWLRQV WKDQ LQ RWKHUV $V D UHVXOW WKH ZLGWK RI WKH SHDN PD\ EH ODUJHU WKDQ WKDW IRU WKH SHDN HYHQ WKRXJK WKH VHFRQG SHDN DSSHDUV DW KLJKHU VFDWWHULQJ YHFWRU ,Q RUGHU WR DFFRXQW IRU WKLV DGGLWLRQDO ZLGWK SDUDPHWHUV ZRXOG EH UHTXLUHG LQ WKH ILW L H RARRf &7M L L! HWF $ PRUH LPSRUWDQW GLIILFXOW\ ZLWK WKH %&& ILWV LV WKH LQFRUSRUDWLRQ RI WKH IRUP IDFWRU ,)4f, 7KH YDOXHV RI WKH IRUP IDFWRU LQ WKH UHOHYDQW VFDWWHULQJ UDQJH PXVW EH GHWHUPLQHG IURP DQ H[WUDSRODWLRQ IURP WKH LQWHUPHGLDWH DQG KLJK 4 UHJLRQV DV VKRZQ LQ )LJXUH 7KHUH LV H[SHFWHG WR EH VRPH HUURU LQ WKH YDOXHV RI ,)4f, GHWHUPLQHG LQ WKLV ZD\ WKH PDJQLWXGH RI ZKLFK FDQ EH IDLUO\ ODUJH DQG LV GLIILFXOW WR HVWLPDWH GXH WR WKH ODUJH QHJDWLYH VORSH RI ,)4f, LQ WKH UHJLRQ RI WKH KLJKHU RUGHU %UDJJ SHDNV $ SRVVLEOH LQIOXHQFH RI WKLV XQFHUWDLQW\ LQ WKH IRUP IDFWRU LV VHHQ LQ WKH 'HE\H:DOOHU ILW SDUDPHWHU % LQ WKH VWUXFWXUH VSHFLILF ILWV 7KLV IDFWRU LV H[SHFWHG WR LQFUHDVH ZLWK LQFUHDVLQJ WHPSHUDWXUH EHORZ WKH RUGHULQJ SRLQW XQLW WKLV IDFWRU H[FHHGV WKH /LQGHPDQ PHOWLQJ FULWHULRQ DW ZKLFK SRLQW WKH V\VWHP GLVRUGHUV 7KH ILWWHG YDOXHV VKRZ QR FRQVLVWHQW WUHQG DQG DUH LQ VRPH LQVWDQFHV VHHQ WR GHFUHDVH MXVW EHORZ WKH RUGHULQJ WHPSHUDWXUH ,W WKHUHIRUH VHHPV OLNHO\ WKDW WKH ILWWHG 'HE\H:DOOHU IDFWRU PD\ EH LQDFFXUDWH GXH WR WKH XQFHUWDLQW\ LQ WKH IRUP IDFWRU )RU WKHVH UHDVRQV WKH PRUH EDVLF PRGHO RI (TXDWLRQ ZDV XVHG WR SDUDPHWHUL]H ERWK WKH VWDWLF DQG NLQHWLF VSHFWUD IRU DOO RI WKH GLEORFN VROXWLRQV )LQDO HYLGHQFH LQ VXSSRUW RI WKH %&& VWUXFWXUH LV WKH PRVW IXQGDPHQWDO DQG WKHUHIRUH WKH PRVW FRQYLQFLQJ 8SRQ H[DPLQDWLRQ RI 7DEOH LW FDQ EH VHHQ WKDW WKHUH

PAGE 161

VKRXOG QRW EH D SHDN LQ WKH \MO UHODWLYH SRVLWLRQ IRU WKH 6& VFDWWHULQJ VSHFWUXP ,Q IDFW WKLV SHDN LV REVHUYHG LQ WKH 6% VSHFWUD RI )LJXUH DQG DW [ O2 ƒ DV ZHOO DV LQ WKH 6% VSHFWUD RI )LJXUHV DQG DW [ ƒ )RU WKHVH WKUHH UHDVRQV LW LV FRQFOXGHG WKDW WKH PDFURODWWLFH LV RI WKH %&& W\SH ZLWK D ODWWLFH FRQVWDQW JLYHQ E\ (TXDWLRQ DQG OLVWHG LQ 7DEOH IRU WKH GLEORFN VROXWLRQV QHDU r& )LQLWH 6L]H DQG 6WUDLQ (IIHFWV $V GHVFULEHG LQ &KDSWHU WKH ZLGWK RI D %UDJJ SHDN LV GHWHUPLQHG SULPDULO\ E\ WKUHH TXDQWLWLHV WKH V\VWHP UHVROXWLRQ WKH PHDQ VL]H RI DQ RUGHUHG UHJLRQ WKH ILQLWH VL]H HIIHFWf DQG WKH VWUDLQ LQ WKH ODWWLFH 6LQFH WKH REVHUYHG %UDJJ SHDNV DUH ZHOO DSSUR[LPDWHG LQ PRVW FDVHV E\ WKH *DXVVLDQ OLQHVKDSH LW LV FRQYHQLHQW WR DVVXPH D *DXVVLDQ OLQHVKDSH WR HDFK RI WKH LQGLYLGXDO FRPSRQHQWV RI WKH ZLGWK IRU VLPSOLFLW\ ,Q WKLV FDVH WKH ZLGWK RI WKH SHDN D FDQ EH H[SUHVVHG DV R RWJ IRM f ZKHUH DU DI DQG DV DUH WKH UHVROXWLRQ ILQLWH VL]H DQG VWUDLQ SDUDPHWHUV GHVFULEHG LQ &KDSWHU UHVSHFWLYHO\ $Q XSSHU OLPLW RQ WKH V\VWHP UHVROXWLRQ LV ):+0 [ ƒn 6LQFH WKH PHDVXUHG SHDN ZLGWKV DUH LQ PRVW FDVHV ODUJHU WKDQ WKLV VRPH FRQWULEXWLRQ WR WKH ZLGWK IURP ILQLWH VL]H DQG VWUDLQ HIIHFWV DUH REVHUYHG )RU D V\VWHP H[KLELWLQJ DQLVRWURSLF VWUDLQ WKH VWUDLQ FRQWULEXWLRQ WR WKH ZLGWK FDQ EH ZULWWHQ 2VAKNO4 f ,Q SULQFLSOH WKH ILQLWH VL]H DQG VWUDLQ FRQWULEXWLRQV FDQ EH VHSDUDWHG E\ XWLOL]LQJ WKH GHSHQGHQFH RI D RQ 4 +RZHYHU WKH GHWHUPLQDWLRQ RI WKH VWUDLQ FRHIILFLHQW D+./

PAGE 162

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r,SHDN4f G4 f R DQG LV UHODWHG WR WKH YROXPH RI PDWHULDO LQ WKH RUGHUHG VWDWH WKDW JLYHV ULVH WR WKH %UDJJ SHDN 8VLQJ WKH UHVXOWV RI &KDSWHU ,7 FDQ EH ZULWWHQ f LQ WHUPV RI WKH IRUP IDFWRU )4f WKH LQFLGHQW LQWHQVLW\ SDUDPHWHU ,n WKH %UDJJ SHDN SRVLWLRQ 4 DQG WKH QXPEHU RI SDUWLFOHV WKDW RFFXS\ WKH ODWWLFH ZKLFK JLYHV ULVH WR WKH %UDJJ SHDN 17Wf 7KH DSSUR[LPDWLRQ LV PDGH WKDW WKH 'HE\H:DOOHU IDFWRU DW WKH SHDN SRVLWLRQ 4 UHPDLQV XQFKDQJHG ZLWKLQ WKH UHOHYDQW WLPH RU WHPSHUDWXUH UDQJH DQG WKDW )4f YDULHV VORZO\ ZLWK 4 LQ WKH YLFLQLW\ RI WKH %UDJJ SRVLWLRQ 4T )4f LV DOVR DVVXPHG WR EH LQGHSHQGHQW RI WHPSHUDWXUH $OWKRXJK WKH IRUP IDFWRU LV H[SHFWHG WR FKDQJH GXH WR WKLFNHQLQJ RI WKH VW\UHQHEXWDGLHQH LQWHUIDFLDO UHJLRQ WKHVH FKDQJHV PDLQO\ DIIHFW WKH KLJK 4 UHJLRQ RI )4f DV GLVFXVVHG LQ &KDSWHU 7KH ORZ 4 UHJLRQ RI )4f LV H[SHFWHG WR UHPDLQ PXFK XQFKDQJHG XQWLO YHU\ FORVH WR WKH

PAGE 163

GLVVROXWLRQ WHPSHUDWXUH 7KH PDJQLWXGH RI )4Rf LV HVWLPDWHG E\ H[WUDSRODWLQJ IURP WKH LQWHUPHGLDWH DQG KLJK 4 UHJLRQV RI WKH VFDWWHULQJ SURILOH DV VKRZQ LQ )LJXUH ,W LV H[SHFWHG WKDW )4f GHWHUPLQHG LQ WKLV ZD\ LV D UHDVRQDEOH DSSUR[LPDWLRQ LQ WKH QDUURZ UHJLRQ RI 4 QHDU 4 7KH TXDQWLW\ ,W LV WKH QRUPDOL]DWLRQ IDFWRU ZKLFK LV SURSRUWLRQDO WR WKH LQFLGHQW EHDP LQWHQVLW\ DV GHVFULEHG LQ &KDSWHU :LWK WKH DERYH DSSUR[LPDWLRQV WKH QRUPDOL]HG LQWHQVLW\ = LV GHILQHG LQ WKH IROORZLQJ ZD\ 4R = :f _)4 f_ 17ff f 7KH TXDQWLW\ = DV GHILQHG LQ (TXDWLRQ f LV SURSRUWLRQDO WR 17Wf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

PAGE 164

&281,6 &28176 & U W & K & I U U U I & I U / / n L ‘M M , W R 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH $Q LOOXVWUDWLRQ RI WKH VSOLWWLQJ RI WKH ILUVWRUGHU %UDJJ UHIOHFWLRQ LQ WKH 6% VROXWLRQ DW r& 7KH XSSHU FXUYH LV WKH UDZ GDWD LQ WKH YLFLQLW\ RI WKH PDLQ SHDN ZKLOH WKH ORZHU FXUYH LV WKH UDZ GDWD LQ WKH YLFLQLW\ RI WKH KLJKHU RUGHU SHDNV 7KH VROLG OLQH LV D JXLGH WR WKH H\H

PAGE 165

WKH XQLW FHOO D E s F < A D 3 f $OWKRXJK WKH FRUUHFW %UDYDLV ODWWLFH IRU WKLV VWUXFWXUH LV WKH IDFHFHQWHUHG RUWKRUKRPELF WKH ERG\FHQWHUHG PRQRFOLQLF JLYHV D FOHDUHU SLFWXUH RI WKH GLVWRUWLRQ LQYROYHG 8VLQJ WKLV PRGHO IRU WKH ODWWLFH LW LV SRVVLEOH WR FRUUHFWO\ SUHGLFW WKH VSOLWWLQJ REVHUYHG LQ WKH UHIOHFWLRQ ZLWK \ r 7KLV GLVWRUWLRQ UHVXOWV LQ D VWUHWFKLQJ RI WKH LQWHUSODQDU GLVWDQFH G DQG D FRQWUDFWLRQ LQ WKH LQWHUSODQDU GLVWDQFH GB RI PDJQLWXGH f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f ,Q WKLV HYHQW D ODWWLFH GLVWRUWLRQ VLPLODU WR WKH RQH GHVFULEHG DERYH FRXOG UHVXOW ,Q DGGLWLRQ WKH DQLVRWURS\ LQ WKH IRUP IDFWRU WKDW UHVXOWV IURP WKH HOOLSVRLGDO VKDSH FRXOG VXSSUHVV FHUWDLQ %UDJJ UHIOHFWLRQV VR WKDW VSOLWWLQJ LQ WKH KLJKHU RUGHU SHDNV LV QRW REVHUYHG 2QH DGGLWLRQDO ODWWLFH W\SH ZDV REVHUYHG XSRQ FRROLQJ WKH 6% VROXWLRQ WR MXVW EHORZ WKH RUGHULQJ WHPSHUDWXUH 7KH ILUVW RUGHU SHDN LQ WKH r& VSHFWUXP RI )LJXUH

PAGE 166

LV DW D PXFK ORZHU VFDWWHULQJ YHFWRU WKDQ WKDW REVHUYHG DW RU r& 7KH VW\UHQH YROXPH IUDFWLRQ ZDV FDOFXODWHG DV GHVFULEHG SUHYLRXVO\ IRU WKH 6& DQG %&& VWUXFWXUHV 7KH UHVXOWV `!,'($/ ^!VF DQG `!%&& f IDYRU WKH 6& VWUXFWXUH ,Q DGGLWLRQ WKH UHODWLYH SHDN LQWHQVLWLHV IRU WKH ILUVW DQG VHFRQG RUGHU SHDNV DUH YHU\ FORVH WR WKRVH H[SHFWHG IURP WKH PXOWLSOLFLWLHV RI WKH 6& VWUXFWXUH 7KHVH UHVXOWV VHHP WR LQGLFDWH WKDW QHDU WKH RUGHULQJ WHPSHUDWXUH WKH 6& DQG %&& VWUXFWXUHV DUH FORVH LQ HQHUJ\ VR WKDW SUHSDUDWLRQ RI WKH PLFURGRPDLQV LQ D 6& DUUDQJHPHQW LV SRVVLEOH +RZHYHU WKH 6& VWUXFWXUH ZDV QRW REVHUYHG XSRQ KHDWLQJ DW WKH WHPSHUDWXUHV DQG r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r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

PAGE 167

3RO\PHU &RPSRVLWLRQ b f 3RO\PHU &RPSRVLWLRQ b f )LJXUH 3ORWV RI WKH PLFURGRPDLQ UDGLXV DQG ODWWLFH FRQVWDQWV YHUVXV SRO\PHU FRQFHQWUDWLRQ DW 7 r&

PAGE 168

WKH FRQFHQWUDWLRQ RI SRO\PHU LQ WKH VROXWLRQ LQFUHDVHV b LQFUHDVHV EHFDXVH WKH UHSXOVLYH LQWHUDFWLRQ LV QRW DV HIIHFWLYHO\ VFUHHQHG $V D UHVXOW WKH HQHUJHWLF FRPSRQHQW LV LQFUHDVHG UHODWLYH WR WKH HQWURSLF FRPSRQHQW OHDGLQJ WR ODUJHU YDOXHV RI 5 $OVR VKRZ LQ )LJXUH LV D JUDSK RI WKH ODWWLFH FRQVWDQW D DW 7 r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

PAGE 169

1RUPDOL]HG %UDJJ Q WHQ VL WLHV U? L 9 Y M L 9 r r ? U 9 9 r f§ R &0 £ 9 a 6% LL K L U? ?6% r‘ M _ T L c L  62 n 62 7HPSHUDWXUH r& f )LJXUH *UDSKV RI WKH QRUPDOL]HG SHDN LQWHQVLWLHV RI WKH SULPDU\ %UDJJ SHDN YV WHPSHUDWXUH IRU HDFK RI WKH IRXU SRO\PHU FRQFHQWUDWLRQV 7KH VROLG OLQH LV D ILW WR DQ HPSLULFDO PRGHO GHVFULEHG LQ WKH WH[W

PAGE 170

DQG VRPH UHJLRQV RI RUGHU EHJLQ WR DSSHDU VR WKDW WKHUH LV D FRH[LVWHQFH RI RUGHUHG DQG DPRUSKRXV UHJLRQV $V WKH WHPSHUDWXUH LV ORZHUHG IXUWKHU EHORZ WKH RUGHULQJ WHPSHUDWXUH WKH FRH[LVWHQFH VKLIWV LQ IDYRU RI PRUH RUGHUHG SKDVH +RZHYHU HYHQ DW WKH ORZHVW WHPSHUDWXUHV PHDVXUHG WKH VFDWWHULQJ VSHFWUD VKRZ ERWK WKH RUGHUHG DQG DPRUSKRXV VFDWWHULQJ FRPSRQHQWV 7KH VROLG OLQHV LQ WKHVH ILJXUHV DUH DQ HPSLULFDO PRGHO WR WKH GDWD 7 7 =P7f $H[S%nWf f +HUH $ DQG % DUH YDULDEOH SDUDPHWHUV 7KH YDOXH RI 7 ZDV WDNHQ DV WKDW YDOXH PLG ZD\ EHWZHHQ WKH KLJKHVW WHPSHUDWXUH ZKHUH %UDJJ VWUXFWXUH LV HYLGHQW DQG WKH QH[W KLJKHU WHPSHUDWXUH ZKHUH WKLV VWUXFWXUH ZDV DEVHQW 7KH UHVXOWLQJ ILW SDUDPHWHUV DUH VKRZQ LQ 7DEOH $OWKRXJK WKH PRGHO LV SXUHO\ HPSLULFDO LW KHOSV WR LGHQWLI\ D WUHQG ZLWK LQFUHDVLQJ SRO\PHU FRQFHQWUDWLRQ 7KH SDUDPHWHU % IURP 7DEOH LQFUHDVHV GUDPDWLFDOO\ ZLWK LQFUHDVLQJ FRQFHQWUDWLRQ LQGLFDWLQJ WKDW WKH WUDQVLWLRQ LV VKDUSHU DW WKH ORZHU FRQFHQWUDWLRQV 7R WKH H[WHQW WKDW WKH RUGHUHG YROXPH IUDFWLRQ LV D PHDVXUH RI WKH HQHUJ\ GLIIHUHQFH $) EHWZHHQ WKH DPRUSKRXV DQG RUGHUHG SKDVHV WKLV LQGLFDWHV WKDW WKH UDWH RI FKDQJH RI $) ZLWK 7 LV PXFK ODUJHU DW ORZ SRO\PHU FRQFHQWUDWLRQV 7KH SK\VLFDO EDVLV IRU WKLV UHVXOW UHPDLQV XQFOHDU 7DEOH 7KH )LW 3DUDPHWHUV $ DQG % 'HVFULELQJ WKH &KDQJH RI WKH 2UGHUHG 9ROXPH )UDFWLRQ ZLWK 7HPSHUDWXUH 6ROXWLRQ $ % 6% 6% 6% 6%

PAGE 171

7KH ):+0 RI WKH ILUVWRUGHU %UDJJ UHIOHFWLRQV DUH VKRZQ DV D IXQFWLRQ RI WHPSHUDWXUH IRU HDFK RI WKH IRXU SRO\PHU FRQFHQWUDWLRQV LQ )LJXUH 7KHVH GDWD IRU WKH 6% VDPSOH DUH HIIHFWLYH ZLGWKV FDOFXODWHG IURP WKH VHFRQG PRPHQW RI WKH ILQH VWUXFWXUH SHDNV 7KH XSSHU OLPLW WR WKH V\VWHP UHVROXWLRQ LV [ 2 ƒ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

PAGE 172

% 7ULJJ :LFL8O6 R ; W & £ ] £ 9$ £ $ 6% W! Z 0 / L L O 6% O L L L L L  ; L L Y 9 9 £ L 6% c L 9  62 n n ; L 9 9 £ 9 £ f ? ] f§; 9 Y f 6% /L n n 7HPSHUDWXUH r & f )LJXUH *UDSKV RI WKH PHDVXUH %UDJJ ZLGWKV YV WHPSHUDWXUH IRU HDFK RI WKH IRXU SRO\PHU VROXWLRQV 7KH RSHQ WULDQJOHV UHSUHVHQW WKH GDWD WDNHQ XSRQ FRROLQJ ZKLOH WKH VROLG WULDQJOHV UHSUHVHQW WKH GDWD WDNHQ XSRQ KHDWLQJ $Q XSSHU ERXQG RQ WKH LQVWUXPHQWDO UHVROXWLRQ LV [ 2 ƒ LQGLFDWLQJ WKDW WKHVH YDOXHV DUH QRW UHVROXWLRQ OLPLWHG

PAGE 173

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r& ZKHUH WKH ODWWLFH FRQVWDQW GURSV UDSLGO\ ZLWK LQFUHDVLQJ WHPSHUDWXUH HYHQWXDOO\ FRLQFLGLQJ ZLWK WKRVH YDOXHV GHWHUPLQHG XSRQ FRROLQJ QHDU WKH RUGHULQJ WHPSHUDWXUH 7KH K\VWHUHVLV LV PRVW SURQRXQFHG DW ORZHU WHPSHUDWXUHV 7 r& f ZKHUH PROHFXODU PRWLRQ LV VORZHG ,W WKHUHIRUH VHHPV OLNHO\ WKDW WKH REVHUYHG K\VWHUHVLV LV D QRQHTXLOLEULXP HIIHFW WKDW UHVXOWV EHFDXVH LQVXIILFLHQW WLPH ZDV DOORZHG IRU HTXLOLEUDWLRQ 7KH SUHFLVH RULJLQ RI WKH K\VWHUHVLV EHKDYLRU UHPDLQV XQFOHDU

PAGE 174

/DWWLFH &RQVWDQWV $ f f§ £ Y $ DD f £ Y 9 $ 6% f§ n        mR  L U  0      9 9 9 %  n 7HPSHUDWXUH r& )LJXUH *UDSKV RI WKH PHDVXUHG ODWWLFH FRQVWDQWV YV WHPSHUDWXUH IRU HDFK RI WKH IRXU SRO\PHU VROXWLRQV 7KH RSHQ WULDQJOHV UHSUHVHQW WKH GDWD WDNHQ XSRQ FRROLQJ ZKLOH WKH VROLG WULDQJOHV UHSUHVHQW WKH GDWD WDNHQ XSRQ KHDWLQJ

PAGE 175

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f ,4faWLR7A nf +DVKLPRWR H[WHQGV /HLEOHUfV SUHGLFWLRQ DQG ILQGV ,4Rf$%7L f ZKHUH 4 LV WKH ORFDWLRQ RI WKH VFDWWHULQJ PD[LPXP 7KH TXDQWLWLHV $ DQG % GHSHQG RQ WKH SRO\PHUL]DWLRQ LQGH[ 1 DQG WKH )ORU\+XJJLQV LQWHUDFWLRQ SDUDPHWHU ; 8VLQJ WKLV UHODWLRQ +DVKLPRWR LGHQWLILHV WKH GLVVROXWLRQ WHPSHUDWXUH E\ SORWWLQJ ,r YV 7n $V VKRZQ LQ WKH ORZHU FXUYH RI )LJXUH D OLQHDU UHJLRQ LV URXJKO\ HYLGHQW DW ORZ 7r

PAGE 176

$PRUSKRXV $PSOLWXGH f $PRUSKRXV $PSOLWXGH )LJXUH 7HPSHUDWXUH r & f 7 f 9DULDWLRQ RI WKH DPRUSKRXV SHDN DPSOLWXGH YV WHPSHUDWXUH IRU WKH 6% VROXWLRQ 7KH XSSHU JUDSK VKRZV WKH ILWWHG DQG QRUPDOL]HG SHDN DPSOLWXGH YV WHPSHUDWXUH RQ D OLQHDU VFDOH 7KH ORZHU JUDSK LV D SORW RI WKH UHFLSURFDO RI WKLV YDOXH YV WKH UHFLSURFDO RI WKH WHPSHUDWXUH DQG LV XVHG WR GHWHUPLQH WKH GLVVROXWLRQ WHPSHUDWXUH 7KH VROLG OLQH LV D OLQHDU ILW WR WKH ILUVW WKUHH SRLQWV DV GLVFXVVHG LQ WKH WH[W

PAGE 177

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nV ZRUN LV WKDW WKH SHDN SRVLWLRQ LV LQGHSHQGHQW RI WHPSHUDWXUH DERYH WKH GLVVROXWLRQ SRLQW 7KLV DSSHDUV WR EH WUXH IRU WKHVH GDWD DV LOOXVWUDWHG LQ )LJXUH ZKHUH WKH DPRUSKRXV SHDN ZLGWK DQG SRVLWLRQ DUH SORWWHG YV WHPSHUDWXUH IRU WKH 6% VROXWLRQ ,Q WKH ORZHU JUDSK WKH SHDN SRVLWLRQ LV LQGHSHQGHQW RI WHPSHUDWXUH DERYH 7G r& $V WKH WHPSHUDWXUH LV ORZHUHG WKH SHDN SRVLWLRQ PRYHV WR ORZHU 4 XQWLO DSSUR[LPDWHO\ r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f r& f 7KLV PLJKW DFFRXQW IRU WKH REVHUYHG WUHQG LQ WKH DPRUSKRXV SHDN SRVLWLRQ DW ORZ WHPSHUDWXUHV 7KH ):+0 RI WKH DPRUSKRXV SHDN LV SORWWHG LQ WKH XSSHU JUDSK RI )LJXUH IRU WKH 6% VROXWLRQ DV D IXQFWLRQ RI WHPSHUDWXUH ,W LV REVHUYHG WR EH UHODWLYHO\ IODW DW

PAGE 178

$PRUSKRXV ):+0 $ HPSHUFLXUH r& f )LJXUH 3ORWV RI WKH DPRUSKRXV SHDN SRVLWLRQ DQG ZLGWK YV WHPSHUDWXUH IRU WKH 6% VROXWLRQ 7KH RSHQ WULDQJOHV UHSUHVHQW WKRVH GDWD WDNHQ XSRQ FRROLQJ ZKLOH WKH VROLG WULDQJOHV UHSUHVHQW WKH GDWD WDNHQ XSRQ KHDWLQJ

PAGE 179

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f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b f WKDQ HDUO\ PHDVXUHPHQWV RQ WKHVH VROXWLRQV EXW DUH LQ EHWWHU DJUHHPHQW ZLWK PRUH UHFHQW PHDVXUHPHQWV b f PDGH E\ WKH VDPH JURXS 7KH RUGHULQJ WHPSHUDWXUHV GHWHUPLQHG SUHYLRXVO\ ZHUH EDVHG RQ D TXDOLWDWLYH LGHQWLILFDWLRQ RI KLJKHU RUGHU %UDJJ SHDNV IURP GHVPHDUHG VSHFWUD ,W LV OLNHO\ WKDW VXFK D GHWHUPLQDWLRQ FRXOG \LHOG VLJQLILFDQWO\ ORZHU UHVXOWV GXH WR WKH QDWXUH RI WKH FROOLPDWLRQ FRUUHFWLRQ DQG LV IDU OHVV UHOLDEOH WKDQ WKH GHWHUPLQDWLRQ PDGH KHUH EDVHG RQ UDZ GDWD

PAGE 180

nH P SF UDLQ UH 3KDVH 'LDJUDP U L & n  3RO\PHU &RQFHQWUDWLRQ & b f )LJXUH 7KH PHDVXUHG SKDVH GLDJUDP IRU WKH SRO\VW\UHQHSRO\EXWDGLHQH & VROXWLRQV 7KH RSHQ VTXDUHV DQG FLUFOHV UHSUHVHQW WKH GLVVROXWLRQ DQG RUGHULQJ WHPSHUDWXUHV UHVSHFWLYHO\ 7KHLU GHWHUPLQDWLRQ LV GHVFULEHG LQ WKH WH[W

PAGE 181

7KH REMHFWLYH RI WKH NLQHWLF PHDVXUHPHQWV ZDV WR REVHUYH WKH WHPSRUDO GHYHORSPHQW RI RUGHU LQ WKH V\VWHP 7R VWXG\ WKLV SURFHVV HDFK VROXWLRQ ZDV DQQHDOHG DERYH WKH GLVVROXWLRQ WHPSHUDWXUH VKRZQ LQ )LJXUH 7KH WHPSHUDWXUH ZDV WKHQ UDSLGO\ ORZHUHG WR YDU\LQJ ILQDO WHPSHUDWXUHV ERWK DERYH DQG EHORZ WKH RUGHULQJ SRLQW 7KH DSSURDFK WR HTXLOLEULXP ZDV PRQLWRUHG ZLWK WKH WKURXJK WKH 6$;6 SURILOH 7KH UHVXOWV RI WKHVH ULFK PHDVXUHPHQWV DUH QRZ SUHVHQWHG

PAGE 182

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f DQG ERWK SRVLWLYH DQG QHJDWLYH YDOXHV RI $7 ZHUH VWXGLHG )LJXUHV WKURXJK SUHVHQW WKH UDZ VSHFWUD IRU WKH ODWHVW WLPHV PHDVXUHG LQ WKH YDULRXV TXHQFKHV 7KH TXHQFK GHSWK $7 LV LQGLFDWHG RQ WKH OHIW DQG WKH HODSVHG

PAGE 183

HQ (f§ = 2 2 6FDWWHULQJ 9HFWRU $ f )LJXUH $ SORW RI WKH ILQDO VSHFWUD DV D IXQFWLRQ RI TXHQFK GHSWK IRU WKH 6% VROXWLRQ 7KH TXHQFK GHSWK $7 LV LQGLFDWHG RQ WKH OHIW DQG WKH HODSVHG WLPH IURP WKH TXHQFK LV LQGLFDWHG RQ WKH ULJKW 7KH GDWD DUH FRUUHFWHG IRU GLIIHULQJ H[SRVXUH WLPHV DQG RIIVHW IURP HDFK RWKHU E\ RQH GHFDGH IRU FODULW\ ZLWK WKH GHHSHVW TXHQFK GDWD XQDOWHUHG

PAGE 184

6FDWWHULQJ 9HFWRU $ f )LJXUH $ SORW RI WKH ILQDO VSHFWUD DV D IXQFWLRQ RI TXHQFK GHSWK IRU WKH 6% VROXWLRQ 7KH TXHQFK GHSWK $7 LV LQGLFDWHG RQ WKH OHIW DQG WKH HODSVHG WLPH IURP WKH TXHQFK LV LQGLFDWHG RQ WKH ULJKW 7KH GDWD DUH FRUUHFWHG IRU GLIIHULQJ H[SRVXUH WLPHV DQG RIIVHW IURP HDFK RWKHU E\ RQH GHFDGH IRU FODULW\ ZLWK WKH GHHSHVW TXHQFK GDWD XQDOWHUHG

PAGE 185

&28176 )LJXUH $ SORW RI WKH ILQDO VSHFWUD DV D IXQFWLRQ RI TXHQFK GHSWK IRU WKH 6% VROXWLRQ 7KH TXHQFK GHSWK $7 LV LQGLFDWHG RQ WKH OHIW DQG WKH HODSVHG WLPH IURP WKH TXHQFK LV LQGLFDWHG RQ WKH ULJKW 7KH GDWD DUF FRUUHFWHG IRU GLIIHULQJ H[SRVXUH WLPHV DQG RIIVHW IURP HDFK RWKHU E\ RQH GHFDGH IRU FODULW\ ZLWK WKH GHHSHVW TXHQFK GDWD XQDOWHUHG

PAGE 186

6FDWWHULQJ 9HFWRU $ f )LJXUH $ SORW RI WKH ILQDO VSHFWUD DV D IXQFWLRQ RI TXHQFK GHSWK IRU WKH 6% VROXWLRQ 7KH TXHQFK GHSWK $7 LV LQGLFDWHG RQ WKH OHIW DQG WKH HODSVHG WLPH IURP WKH TXHQFK LV LQGLFDWHG RQ WKH ULJKW 7KH GDWD DUH FRUUHFWHG IRU GLIIHULQJ H[SRVXUH WLPHV DQG RIIVHW IURP HDFK RWKHU E\ RQH GHFDGH IRU FODULW\ ZLWK WKH GHHSHVW TXHQFK GDWD XQDOWHUHG

PAGE 187

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f§ 91 ZKHUH 1 LV WKH QXPEHU RI FRXQWV f $ QXPEHU RI WUHQGV DUH LPPHGLDWHO\ HYLGHQW LQ WKH GDWD DV D IXQFWLRQ RI TXHQFK GHSWK DQG DUH QRZ GHVFULEHG $V H[SHFWHG WKH ODWHWLPH VSHFWUD IROORZLQJ TXHQFKHV WR ILQDO WHPSHUDWXUHV DERYH 7 QHJDWLYH $7f H[KLELW RQO\ DPRUSKRXV VFDWWHULQJ 7KH DEVHQFH RI %UDJJ SHDN GHYHORSPHQW LV LQGLFDWLYH RI WKH LQVWDELOLW\ RI WKH RUGHUHG SKDVH LQ WKLV WHPSHUDWXUH UHJLPH 6LPLODU EHKDYLRU LV DOVR REVHUYHG LQ WZR TXHQFKHV WR ILQDO WHPSHUDWXUHV MXVW EHORZ WKH RUGHULQJ SRLQW $7 r& IRU 6% DQG $7 r& IRU 6% 2UGHULQJ RI WKH PLFURGRPDLQV LV H[SHFWHG EDVHG RQ WKH UHVXOWV RI WKH HTXLOLEULXP PHDVXUHPHQWV ,W LV EHOLHYHG WKDW WKH DEVHQFH RI RUGHULQJ LQ WKH VKDOORZHVW TXHQFKHV LV GXH WR D UHGXFHG GULYLQJ IRUFH IRU WUDQVIRUPDWLRQ RU HTXLYDOHQWO\ D FULWLFDO VORZLQJ GRZQ RI WKH RUGHULQJ SURFHVV f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

PAGE 188

RI WKHVH PHDVXUHPHQWV 7KH SRVVLEOH RULJLQV RI WKH HIIHFW DUH LQFRPSOHWH PLFURSKDVH VHSDUDWLRQ RU UHGXFHG WKHUPDO DFWLYDWLRQ UHVXOWLQJ LQ VORZHU JURZWK RI WKH RUGHUHG SKDVH 7KH ILQDO WHPSHUDWXUHV LQ WKH GHHS TXHQFK OLPLW IDOO EHORZ WKH JODVV WUDQVLWLRQ WHPSHUDWXUH RI WKH SRO\VW\UHQH EORFN r& f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f 7KH UHVXOWV RI D TXHQFK IURP r& WR r& IRU WKH 6% VROXWLRQ DUH LOOXVWUDWHG LQ )LJXUH 7KH IXOO ILW WR WKH GDWD ZKLFK IRU WKHVH VSHFWUD FRQWDLQ RQO\ WKH

PAGE 189

&28176 X 6FDWWHULQJ 9HFWRU $ f )LJXUH $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& 7KH GDWD DUH UHSUHVHQWHG E\ GRWV DQG WKH IXOO ILW WR WKH GDWD LQFOXGLQJ DPRUSKRXV DQG EDFNJURXQG FRQWULEXWLRQV LV LQGLFDWHG E\ D VROLG OLQH 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP DQG WKH GDWD IRU HDFK WLPH VOLFH DUH RIIVHW E\ RQH GHFDGH VWDUWLQJ ZLWK WKH HDUOLHVW WLPH

PAGE 190

DPRUSKRXV DQG EDFNJURXQG FRPSRQHQWV LV LQGLFDWHG E\ D VROLG OLQH IRU HDFK VSHFWUXP 7KH GDWD IRU HDFK WLPH VOLFH DUH RIIVHW E\ RQH GHFDGH IRU FODULW\ 7KLV ILQDO WHPSHUDWXUH LV DERYH WKH RUGHULQJ WHPSHUDWXUH 76%f r& f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r& WR r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

PAGE 191

&28176 )LJXUH $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& IRU WKH 6% VROXWLRQ 7KH GDWD DUH UHSUHVHQWHG E\ GRWV DQG WKH IXOO ILW WR WKH GDWD LQFOXGLQJ RUGHUHG DPRUSKRXV DQG EDFNJURXQG FRQWULEXWLRQV LV LQGLFDWHG E\ D VROLG OLQH 7KH DPRUSKRXV FRQWULEXWLRQ WR WKH ILW LV VKRZQ DV D VKRUWGDVKHG OLQH 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP DQG WKH GDWD IRU HDFK WLPH VOLFH DUH RIIVHW E\ RQH GHFDGH VWDUWLQJ ZLWK WKH HDUOLHVW WLPH

PAGE 192

&28176 X M 6FDWWHULQJ 9HFWRU $ f )LJXUH $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& IRU WKH 6% VROXWLRQ 7KH GDWD DUH UHSUHVHQWHG E\ GRWV DQG WKH IXOO ILW WR WKH GDWD LQFOXGLQJ RUGHUHG DPRUSKRXV DQG EDFNJURXQG FRQWULEXWLRQV LV LQGLFDWHG E\ D VROLG OLQH 7KH DPRUSKRXV FRQWULEXWLRQ WR WKH ILW LV VKRZQ DV D VKRUWGDVKHG OLQH 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP DQG WKH GDWD IRU HDFK WLPH VOLFH DUH RIIVHW E\ RQH GHFDGH VWDUWLQJ ZLWK WKH HDUOLHVW WLPH

PAGE 193

&28176 X M 6FDWWHULQJ 9HFWRU $ f )LJXUH $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& IRU WKH 6% VROXWLRQ 7KH GDWD DUH UHSUHVHQWHG E\ GRWV DQG WKH IXOO ILWWR WKH GDWD LQFOXGLQJ RUGHUHG DPRUSKRXV DQG EDFNJURXQG FRQWULEXWLRQV LV LQGLFDWHG E\ D VROLG OLQH 7KH DPRUSKRXV FRQWULEXWLRQ WR WKH ILW LV VKRZQ DV D VKRUWGDVKHG OLQH 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP DQG WKH GDWD IRU HDFK WLPH VOLFH DUF RIIVHW E\ RQH GHFDGH VWDUWLQJ ZLWK WKH HDUOLHVW WLPH

PAGE 194

X M 6FDWWHULQJ 9HFWRU $ f )LJXUH $ SORW RI WKH UDZ 6$;6 VSHFWUD DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& IRU WKH 6% VROXWLRQ 7KH GDWD DUH UHSUHVHQWHG E\ GRWV DQG WKH IXOO ILWWR WKH GDWD LQFOXGLQJ RUGHUHG DPRUSKRXV DQG EDFNJURXQG FRQWULEXWLRQV LV LQGLFDWHG E\ D VROLG OLQH 7KH DPRUSKRXV FRQWULEXWLRQ WR WKH ILW LV VKRZQ DV D VKRUWGDVKHG OLQH 7KH H[SRVXUH WLPH ZDV VHF VSHFWUXP DQG WKH GDWD IRU HDFK WLPH VOLFH DUH RIIVHW E\ RQH GHFDGH VWDUWLQJ ZLWK WKH HDUOLHVW WLPH

PAGE 195

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f %XW DV WLPH SURFHHGV WKH SHDN PD[LPXP DJDLQ EHJLQV WR JURZ LQGLFDWLQJ WKH VWDUW RI WKH RUGHULQJ SURFHVV DQG WKH IRUPDWLRQ RI WKH %UDJJ SHDN 6LPLODU EHKDYLRU LV REVHUYHG LQ DOO RI WKRVH TXHQFKHV ZKHUH %UDJJ SHDN IRUPDWLRQ LV HYLGHQW LQGLFDWLQJ WKH GLIIHUHQW WLPH VFDOHV RQ ZKLFK PLFURSKDVH VHSDUDWLRQ DQG RUGHULQJ RFFXU 7KH WHPSRUDO GHYHORSPHQW RI WKH VFDWWHUHG SURILOH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ LV VXPPDUL]HG LQ WKH SORW RI )LJXUH 7KH GDWD DUH VKRZQ OLQHDU VFDOH ,Q WKH HDUO\ WLPHV WKH DPRUSKRXV SHDN VKLIWV WR ORZHU ZDYH YHFWRU DQG QDUURZV ZKLOH JURZLQJ LQ DPSOLWXGH $V VWDWHG SUHYLRXVO\ WKLV EHKDYLRU LV VLPLODU WR WKDW REVHUYHG LQ WKH HDUO\ VWDJHV RI SKDVH VHSDUDWLRQ LQ ELQDU\ DOOR\V *URZWK RI WKH DPRUSKRXV SHDN LV DOPRVW FRPSOHWH DIWHU VHFRQGV 6XEVHTXHQWO\ WKH %UDJJ FRPSRQHQW DSSHDUV DQG JURZV ZLWK WLPH LQGLFDWLQJ WKH IRUPDWLRQ RI WKH %&& PDFURODWWLFH

PAGE 196

7LPH VHF f 7LPH VHF f )LJXUH *UDSKV RI WKH SHDN PD[LPXP DV D IXQFWLRQ RI WLPH IROORZLQJ D VKDOORZ TXHQFK XSSHU JUDSK f DQG D TXHQFK WR EHORZ WKH RUGHULQJ WHPSHUDWXUH ORZHU JUDSK f 7KH SHDN YDOXHV DUH WDNHQ GLUHFWO\ IURP WKH UDZ VSHFWUD DQG DUH LQGLFDWHG E\ FLUFOHV 7KH VDPSOH WHPSHUDWXUH SURILOH LV LQGLFDWHG E\ VROLG VTXDUHV

PAGE 197

&28176 )LJXUH $ SORW RI WKH WHPSRUDO GHYHORSPHQW RI WKH VFDWWHUHG SURILOH 7KH GDWD DUH VKRZQ RQ D OLQHDU VFDOH IRU D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ

PAGE 198

7KH QRUPDOL]HG LQWHQVLW\ RI D %UDJJ SHDN = ZDV VKRZQ WR EH SURSRUWLRQDO WR WKH IUDFWLRQ RI PDWHULDO LQ WKH RUGHUHG SKDVH e VHH (TXDWLRQ f 7KH QRUPDOL]HG LQWHQVLW\ LV SURSRUWLRQDO WR WKH IROORZLQJ ILYH IDFWRUV WKH %UDJJ DPSOLWXGH $Wf DQG ZLGWK RWf WKH VTXDUH RI WKH IRUP IDFWRU ,)4f, WKH SRZGHU DYHUDJLQJ IDFWRU P 4f DQG WKH 'HE\H:DOOHU IDFWRU :KHQ WKH ODWWHU IRXU RI WKHVH IDFWRUV DUH FRQVWDQW e LV SURSRUWLRQDO WR WKH %UDJJ DPSOLWXGH $Wf ,W ZLOO QRZ EH DUJXHG WKDW LQ WKH PDMRULW\ TXHQFK GDWD WR EH SUHVHQWHG KHUH WKHVH IRXU IDFWRUV FDQ EH DVVXPHG WR EH FRQVWDQW WKURXJKRXW WKH RUGHULQJ SURFHVV DQG $Wf LV D XVHIXO PHDVXUH RI e 6KRZQ LQ )LJXUH DUH WKH ILWWHG %UDJJ ZLGWKV DQG %&& ODWWLFH FRQVWDQWV FDOFXODWHG IURP WKH %UDJJ SRVLWLRQV DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r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

PAGE 199

/DWWLFH &RQVWDQW $ f $PRUSKRXV 3HDN 3RVLWLRQ F 7LPH VHFRQGV f 7LPH VHFRQGV f )LJXUH *UDSKV RI WKH ILWWHG DPRUSKRXV SHDN SRVLWLRQ DQG ):+0 DORQJ ZLWK WKH %UDJJ ):+0 DQG WKH PHDVXUHG ODWWLFH FRQVWDQW DV GHWHUPLQHG IURP WKH ILWWHG %UDJJ SRVLWLRQ DUH SORWWHG YHUVXV WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ 'XH WR WKH QDWXUH RI WKH SDUDPHWHUL]DWLRQ WKH EHVW HVWLPDWH RI WKH XQFHUWDLQW\ LQ WKHVH YDOXHV LV WKH IOXFWXDWLRQ LQ WKH SRLQWV

PAGE 200

7KH DPRUSKRXV SHDN ZLGWK DOVR GHFUHDVHV UDSLGO\ ZLWK GHFUHDVLQJ WHPSHUDWXUH DV D UHVXOW RI LQFUHDVLQJ SRVLWLRQDO FRUUHODWLRQV EHWZHHQ WKH PLFURGRPDLQV $Q LQWHUHVWLQJ UHVXOW HYLGHQW LQ WKH PDMRULW\ RI WKH TXHQFK GDWD LV D PLQLPXP LQ WKH DPRUSKRXV ZGGWK SULRU WR WKH RQVHW RI RUGHULQJ WKLV HIIHFW LV QRW XQGHUVWRRG DW SUHVHQW EXW VHHPV WR LQGLFDWH D UHGXFWLRQ LQ WKH SRVLWLRQDO FRUUHODWLRQV RI WKH GRPDLQV LQ WKH DPRUSKRXV SKDVH DV D UHVXOW RI LQFUHDVLQJ RUGHUHG YROXPH IUDFWLRQ 7KH ODWWLFH VSDFLQJV VKRZQ LQ WKH ILJXUH VKRZ D GHFUHDVLQJ WUHQG ZLWK WLPH 7KLV WUHQG LV RSSRVLWH WR WKH WUHQG LQIHUUHG IURP WKH DPRUSKRXV SHDN SRVLWLRQ ZKLFK LPSOLHV DQ LQFUHDVLQJ GRPDLQ VHSDUDWLRQ LQ WKH DPRUSKRXV SKDVH 7KH SK\VLFDO RULJLQ RI WKLV HIIHFW LV QRW NQRZQ DW WKH SUHVHQW WLPH $OO RI WKH TXHQFK GDWD DSSHDU WR VKRZ D VLPLODU GHFUHDVLQJ WUHQG RI WKH ODWWLFH FRQVWDQW ZLWK WLPH WR YDU\LQJ GHJUHHV 7KH RYHUDOO PDJQLWXGH RI WKH VKLIW LQ ODWWLFH FRQVWDQW IURP EHJLQQLQJ WR HQG LV DOZD\V VPDOO bf 7KH %UDJJ ZLGWKV LQ WKLV ILJXUH DSSHDU WR VKRZ D VPDOO GHFUHDVLQJ WUHQG ZLWK WLPH EXW WKLV WUHQG LV QRW FRQVLVWHQW IRU DOO RI WKH TXHQFK GDWD *HQHUDOO\ LW FDQ EH VDLG WKDW WKH RYHUDOO YDULDWLRQ RI WKH ZLGWK ZLWK WLPH IURP EHJLQQLQJ WR HQG LV VPDOO bf UHODWLYH WR WKH IOXFWXDWLRQV LQ WKH ILWWHG YDOXHV 7KH ODUJHVW FKDQJHV LQ WKH %UDJJ ZLGWK RFFXU DW HDUO\ WLPHV ZKHUH WKH SHDN LV VPDOO DQG WKH ILWWHG DPSOLWXGH DQG ZLGWK DUH WKH OHDVW DFFXUDWH 1R FRQVLVWHQW WUHQG ZLWK WLPH LV REVHUYHG EXW WKH XSSHU HVWLPDWH RI WKH LQVWUXPHQWDO UHVROXWLRQ [ ƒrf LV VLJQLILFDQWO\ ORZHU WKDQ WKH YDOXHV W\SLFDOO\ REVHUYHG LQGLFDWLQJ WKDW WKH GDWD DUH QRW UHVROXWLRQ OLPLWHG )URP WKH SUHFHGLQJ GLVFXVVLRQ RI WKH DPRUSKRXV SHDN SDUDPHWHUV WKH WUHQGV DUH FRQVLVWHQW ZLWK D UDSLG PLFURSKDVH VHSDUDWLRQ WUDQVLWLRQ ZKLFK LV HVVHQWLDOO\ FRPSOHWH EHIRUH WKH RQVHW RI RUGHULQJ $V D UHVXOW WKH IRUP IDFWRU )4f RI WKH PLFURGRPDLQV FDQ EH DVVXPHG WR EH XQFKDQJLQJ GXULQJ WKH RUGHULQJ VWDJH 7KH SRVLWLRQ RI WKH %UDJJ SHDN 4R JHQHUDOO\ PRYHV E\ OHVV WKDQ b WKURXJKRXW WKH WUDQVLWLRQ VR WKDW WKH SRZGHU DYHUDJLQJ IDFWRU QA 4R DOVR UHPDLQV XQFKDQJHG WKURXJKRXW WKH RUGHULQJ SURFHVV ,Q DGGLWLRQ WKH WHPSHUDWXUH DQG GRPDLQGRPDLQ LQWHUDFWLRQ DV LQIHUUHG IURP WKH

PAGE 201

DPRUSKRXV SHDN ZLGWK UHPDLQ PRVWO\ XQFKDQJHG WKURXJKRXW WKH RUGHULQJ SURFHVV ,W LV WKHUHIRUH DVVXPHG WKDW WKH 'HE\H:DOOHU IDFWRU ZKLFK DULVHV IURP RVFLOODWRU\ PRWLRQ RI WKH SDUWLFOHV DERXW WKHLU ODWWLFH VLWHV DOVR LV LQGHSHQGHQW RI WLPH GXULQJ RUGHULQJ ILQDOO\ VLQFH WKH %UDJJ ZLGWK LV PXFK XQFKDQJHG GXULQJ WUDQVIRUPDWLRQ WKH %UDJJ DPSOLWXGH VKRXOG EH D XVHIXO PHDVXUH RI WKH RUGHUHG YROXPH IUDFWLRQ e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f VWHDG\VWDWH QXFOHDWLRQ WKHRULHV GHVFULEHG LQ &KDSWHU SUHGLFW WKDW WKH IUDFWLRQ RI WKH VSHFLPHQ LQ WKH RUGHUHG SKDVH e VKRXOG JURZ ZLWK WLPH DV & H[S E WQ ff f ZKHUH IRU D FRQVWDQW KRPRJHQHRXV QXFOHDWLRQ UDWH Q 7KH ORQJ GDVKHG OLQH LV D ILW RI WKLV IXQFWLRQ WR WKH HDUO\ WLPH SRUWLRQ RI WKH GDWD IURP WR VHFf $V VHHQ LQ )LJXUH WKH HDUO\ WLPH GDWD DUH IRXQG WR EH ZHOO DSSUR[LPDWHG E\ WKLV IRUP 7KH GDWD KRZHYHU GHYLDWH IURP WKLV IRUP DW LQWHUPHGLDWH WLPHV FURVVLQJ RYHU WR VORZHU JURZWK LQ WKH ODWH WLPH OLPLW 7KH ODWH WLPH EHKDYLRU LV IRXQG WR EH ZHOO DSSUR[LPDWHG E\ D SXUH H[SRQHQWLDO Q O LQ (TXDWLRQ DV VKRZQ E\ WKH VKRUW GDVKHG OLQH ZKLFK LV D ILW RI WKH ODWH WLPH GDWD IURP WR VHFRQGV f WR WKLV IXQFWLRQ $V GHWDLOHG LQ &KDSWHU WKH 0-$ H[SRQHQW Q GHSHQGV RQ WKH W\SH RI QXFOHDWLRQ KRPRJHQHRXV RU KHWHURJHQHRXV f DQG WKH GLPHQVLRQDOLW\ RI JURZWK $ EHVW ILW WR WKH ZKROH UDQJH RI GDWD LQ )LJXUH \LHOGV DQ H[SRQHQW Q 7KH GDWD FOHDUO\ GHYLDWH

PAGE 202

1RUPDOL]HG $PSLL/XGH 7LPH VHFRQGV f )LJXUH $ SORW RW WKH QRUPDOL]HG %UDJJ DQG DPRUSKRXV ILWWHG DPSOLWXGHV DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ 7KH ORQJGDVKHG OLQH LV D ILW RI WKH 0-$ IRUP ZLWK Q WR WKH HDUO\ WLPH %UDJJ DPSOLWXGHV DV GHVFULEHG LQ WKH WH[W 6LPLODUO\ WKH VKRUWGDVKHG OLQH LV D ILW RI WKH ODWH WLPH GDWD ZLWK Q O 7KH PHGLXP GDVKHG OLQH LV D ILW WR WKH IXOO UDQJH RI WKH GDWD DOORZLQJ WKH $YUDPL H[SRQHQW Q WR YDU\ )LQDOO\ WKH &DKQ PRGHO LV UHSUHVHQWHG E\ WKH VROLG OLQH DQG DJUHHV ZHOO ZLWK WKH GDWD IRU DOO WLPHV 'XH WR WKH QDWXUH RI WKH SDUDPHWHUL]DWLRQ WKH EHVW HVWLPDWH RI WKH XQFHUWDLQW\ LQ WKHVH GDWD LV WKH IOXFWXDWLRQ LQ WKH SRLQWV

PAGE 203

IURP WKLV ILW HVSHFLDOO\ LQ WKH ODWH WLPHV 7KH DSSDUHQW VORZLQJ RI JURZWK DW ODWH WLPHV PD\ DULVH IURP WKH HIIHFWV RI VLWH VDWXUDWLRQ DQG LPSLQJHPHQW ZKLFK DUH DFFRXQWHG IRU LQ D PRGHO GHYHORSHG E\ &DKQ DV GHVFULEHG LQ &KDSWHU &DKQn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nV PRGHO IRU QXFOHDWLRQ RQ WZR GLPHQVLRQDO VLWHV DVVXPLQJ FRQVWDQW LVRWURSLF JURZWK \LHOGV WKH H[SUHVVLRQ &Wf m $Wf $f H[S E IWWRfff f ZKHUH I W U & IW D H[S n OD OA Af f R 7KH IXQFWLRQ IWf H[KLELWV WKH IROORZLQJ DV\PSWRWLF EHKDYLRU IWf DW HDUO\ WLPHV f IWf W DW ODWH WLPHV 7KH QXFOHDWLRQ UDWH LQ WKLV PRGHO LV DSSUR[LPDWHG E\ D VWHS IXQFWLRQ ,Wf k W WRf ,f f ZKLFK LV ]HUR WKURXJKRXW WKH LQFXEDWLRQ WLPH WR ULVLQJ LQVWDQWDQHRXVO\ WR LWV VWHDG\VWDWH YDOXH ,-6 DW WKDW WLPH ,Q IDFW WKH EHKDYLRU RI WKH QXFOHDWLRQ UDWH LV H[SHFWHG WR EH PRUH FRPSOLFDWHG VHH &KDSWHU f +RZHYHU WKH HIIHFWV RI WKH WLPH GHSHQGHQFH RI WKH

PAGE 204

WUDQVLHQW QXFOHDWLRQ UDWH DSSHDU LQ WKH YHU\ HDUO\ WLPH UHJLRQ RI WKH WUDQVIRUPDWLRQ FXUYH ZKHUH WKH GDWD DUH OHDVW DFFXUDWH )RU WKLV UHDVRQ WKH PRVW EDVLF IRUP (TXDWLRQ f KDV EHHQ DVVXPHG IRU WKH QXFOHDWLRQ UDWH 7KH )LW SDUDPHWHUV D DQG E GHSHQG RQ WKH JURZWK YHORFLW\ 7 WKH QXFOHDWLRQ YROXPH DQG WKH VWHDG\VWDWH QXFOHDWLRQ UDWH ,VV 7KH VROLG OLQH LQ )LJXUH LV D ILW RI WKH &DKQ PRGHO WR WKH GDWD ZLWK YDULDEOH SDUDPHWHUV D E A DQG $f ,Q )LJXUH WKH QRUPDOL]HG %UDJJ DPSOLWXGHV DUH SORWWHG DV D IXQFWLRQ RI WLPH IROORZLQJ TXHQFKHV IURP r& WR r& r& r& DQG r& RQ WKH 6% VDPSOH 7KH Q DQG Q O WHPSRUDO UHJLPHV DUH UHDGLO\ LGHQWLILHG E\ WKH ORQJ DQG VKRUWGDVKHG OLQHV UHVSHFWLYHO\ 7KH &DKQ PRGHO VROLG OLQH f DJUHHV ZHOO ZLWK WKH GDWD IRU HDFK RI WKHVH TXHQFKHV DW DOO WLPHV 7KH PRGHO ILWV WKH GDWD YHU\ ZHOO IRU DOO WLPHV DQG VLPLODU DJUHHPHQW LV REWDLQHG ZLWK DOO RI WKH TXHQFK GDWD ZKLFK H[KLELW %UDJJ DPSOLWXGHV WKDW LQFUHDVH VPRRWKO\ ZLWK WLPH 7KLV FDYHDW LV QHFHVVDU\ GXH WR WKH REVHUYDWLRQ RI RVFLOODWRU\ EHKDYLRU LQ WKH YDULDWLRQ RI WKH %UDJJ DPSOLWXGH ZLWK WLPH IROORZLQJ VKDOORZ TXHQFKHV LQ WKH 6% DQG 6% VROXWLRQV 6RPH HYLGHQFH RI RVFLOODWRU\ EHKDYLRU LV DOVR REVHUYHG LQ WKH r& TXHQFK RI )LJXUH 7KLV VXUSULVLQJ EHKDYLRU LV GLVFXVVHG LQ PRUH GHWDLO DW WKH HQG RI WKLV FKDSWHU 7KH ILW SDUDPHWHUV D E WR DQG $f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

PAGE 205

7DEOH 4XHQFK 5HVXOWV 6ROXWLRQ 6% 6% 6% 6% 7L 7I $W D E WR 7 DE %HKDYLRU 1R 2UGHULQJ 1R 2UGHULQJ 1R 2UGHULQJ 1R 2UGHULQJ 0XOWLSOH 3NV 2VFLOODWRU\ 6PRRWK 6PRRWK 6PRRWK 1R 2UGHULQJ 1R 2UGHULQJ 1R 2UGHULQJ 6PRRWK 6PRRWK 6PRRWK 6PRRWK 6PRRWK 1R 2UGHULQJ 1R 2UGHULQJ 1R 2UGHULQJ 2VFLOODWRU\ 6PRRWK 6PRRWK 6PRRWK 6PRRWK /LPLWHG 2UG 1R 2UGHULQJ 1R 2UGHULQJ 2VFLOODWRU\ 1R 2UGHULQJ 2VFLOODWRU\ 6PRRWK 2VFLOODWRU\ 6PRRWK 6PRRWK /LPLWHG 2UG

PAGE 206

1RUPXOL]RG $ L Q S XGH % & 7LPH VHFRQGV f 7LPH VHFRQGV f )LJXUH $ SORW RI WKH QRUPDOL]HG %UDJJ ILWWHG DPSOLWXGHV DV D IXQFWLRQ RI WLPH IROORZLQJ TXHQFKHV IURP r& WR r& r& r& DQG r& RQ WKH 6% VROXWLRQ 7KH &DKQ PRGHO LV UHSUHVHQWHG E\ WKH VROLG OLQH DQG DJUHHV ZHOO ZLWK WKH GDWD IRU DOO WLPHV 7KH ORQJGDVKHG OLQH LV D ILW RI WKH HDUO\ WLPH UHJLRQ VKRZLQJ WKH Q EHKDYLRU ZKLOH WKH VKRUW GDVKHG OLQH LV D ILW RI WKH ODWH WLPH UHJLRQ VKRZLQJ WKH Q O UHJLPH 'XH WR WKH QDWXUH RI WKH SDUDPHWHUL]DWLRQ WKH EHVW HVWLPDWH RI WKH XQFHUWDLQW\ LQ WKHVH YDOXHV LV WKH IOXFWXDWLRQ LQ WKH SRLQWV

PAGE 207

)URP 7DEOH LW LV HYLGHQW WKDW QR RUGHULQJ LV REVHUYHG LQ WKRVH TXHQFKHV WR ILQDO WHPSHUDWXUHV DERYH 7 ,Q DGGLWLRQ TXHQFKHV DUH OLVWHG ZKLFK VKRZHG OLPLWHG RU QR RUGHULQJ RQ WKH WLPH VFDOH RI WKH H[SHULPHQW EHORZ 7T LQ WKH YHU\ VKDOORZ DQG YHU\ GHHS TXHQFK OLPLWV 2VFLOODWRU\ EHKDYLRU DSSHDUV LQ VKDOORZ TXHQFKHV RI WKH 6% DQG 6% VROXWLRQV $QG VPRRWK JURZWK RI WKH %UDJJ SHDN LV REVHUYHG LQ WKH LQWHUPHGLDWH UHJLPH 7KH 0DVWHU &XUYH ,Q D SORW RI WKH TXDQWLW\ ,Q ,Q e fBff YHUVXV OQ Wf WKH VORSH RI WKH UHVXOWLQJ FXUYH FRUUHVSRQGV WR WKH H[SRQHQW Q LQ WKH 0-$ IRUP (TXDWLRQ f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f RU WKH QXFOHDWLRQ UDWH LV WRR ORZ VR WKDW VLWH VDWXUDWLRQ RFFXUV ODWH LQ WKH WUDQVIRUPDWLRQ DQG RQO\ 0-$ EHKDYLRU ZLWK Q LV VHHQ ,Q IDFW WKH GDWD LQ )LJXUHV DQG VKRZ WKDW WKH SRLQW ZKHUH WKH VORSH FKDQJHV FDOOHG WKH VDWXUDWLRQ SRLQW YDULHV ZLWK TXHQFK GHSWK UHIOHFWLQJ WKH YDULDWLRQ RI QXFOHDWLRQ UDWH DQG JURZWK YHORFLW\ ZLWK TXHQFK GHSWK 7KH EHKDYLRU RI WKH JURZWK YHORFLW\ 7 ZLWK TXHQFK GHSWK LV GLVFXVVHG PRPHQWDULO\

PAGE 208

f f } ,Q W D f ,Q W D f ,Q W D f )LJXUH $ SORW RI WKH TXDQWLW\ ,Q ,Q e frrff YV WKH QDWXUDO ORJ RI WKH WLPH IROORZLQJ TXHQFKHV IURP r& WR r& r& r& DQG r& RQ WKH 6% VROXWLRQ 7KH VORSH RI WKH FXUYH RQ WKLV VFDOH FRUUHVSRQGV WR WKH H[SRQHQW Q LQ WKH 0-$ IRUP 7KH Q DQG Q O UHJLPHV DQG WKH FURVVRYHU DUH HYLGHQW RQ WKHVH SORWV

PAGE 209

2QH SUHGLFWLRQ ZKLFK IROORZV IURP WKH &DKQ PRGHO LV WKH H[LVWHQFH RI D PDVWHU FXUYH GLVFXVVHG LQ &KDSWHU f ,I WKH WUDQVIRUPDWLRQ FXUYHV DUH SORWWHG LQ WKH ZD\ GHVFULEHG DERYH DQG WUDQVODWHG YHUWLFDOO\ E\ OQ E f DQG KRUL]RQWDOO\ E\ OQ D f WKH UHVXOWLQJ FXUYHV VKRXOG FRLQFLGH IRU DOO RI WKH TXHQFK GDWD 7R WHVW WKLV DVVHUWLRQ WKH TXDQWLW\ < ,Q n OQ & 9 f a E f LV SORWWHG YV OQ W D f LQ )LJXUH &OHDUO\ WKH GDWD H[KLELW WKH SUHGLFWHG VFDOLQJ EHKDYLRU VHH QH[W SDJH f )URP WKH ODWHWLPH OLPLW RI (TXDWLRQV DQG LW LV DSSDUHQW WKDW WKH TXDQWLW\ [ DE SOD\V WKH UROH RI DQ HIIHFWLYH WLPH FRQVWDQW IRU WKH WUDQVIRUPDWLRQ SURFHVV LQ WKLV WLPH UHJLPH :LWKLQ WKH &DKQ PRGHO WKLV TXDQWLW\ LV LQYHUVHO\ SURSRUWLRQDO WR WKH JURZWK YHORFLW\ 7 WKH YHORFLW\ RI SURSDJDWLRQ RI WKH GRPDLQ ZDOO VHSDUDWLQJ WKH RUGHUHG DQG GLVRUGHUHG SKDVHV DV GHVFULEHG LQ FKDSWHU f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

PAGE 210

7KH 0DVWHU &XUYH ,Q WD f )LJXUH $ PDVWHU FXUYH RI WKH TXHQFK GDWD IURP DOO RI WKH VDPSOHV DV GHVFULEHG LQ WKH WH[W 7KH GDWD IURP GLIIHUHQW TXHQFKHV IRU VROXWLRQV 6% 6% 6% DQG 6% DUH UHSUHVHQWHG E\ RSHQ GLDPRQGV FLUFOHV VTXDUHV DQG WULDQJOHV UHVSHFWLYHO\

PAGE 211

LfLUQH &RQVODQO W VHF 4XHQFK 'HSWK $7 r& f )LJXUH 7KH WLPH FRQVWDQWV 7 IRU WKH RUGHULQJ WUDQVLWLRQ IRU WKH 6% 6% 6% DQG 6% VROXWLRQV DUH SORWWHG DV D IXQFWLRQ RI TXHQFK GHSWK 7KH VROLG OLQHV DUH JXLGHV WR WKH H\H

PAGE 212

7KH &URVVRYHU %HKDYLRU 7KH SK\VLFDO RULJLQ RI WKH FURVVRYHU LQ 0-$ EHKDYLRU ZKLFK LV REVHUYHG LQ WKLV V\VWHP LV LOOXVWUDWHG LQ )LJXUH WKH GRXEOH VROLG OLQH UHSUHVHQWV WKH QXFOHDWLRQ VXUIDFH DQG WKH VROLG FLUFOHV UHSUHVHQW QXFOHDWLRQ FHQWHUV 7KH VKRUW GDVKHG DUFV UHSUHVHQW WKH DGYDQFLQJ GRPDLQ ZDOOV VHSDUDWLQJ WKH RUGHUHG GURSOHW IURP WKH GLVRUGHUHG SKDVH $W YHU\ HDUO\ WLPHV XSSHU SLFWXUH f QXFOHL IRUP UDQGRPO\ RQ WKH QXFOHDWLRQ VXUIDFH DQG DUH VHSDUDWHG E\ ODUJH GLVWDQFHV RQ DYHUDJH VR WKDW QXFOHDWLRQ DQG JURZWK RFFXU IUHHO\ RQ WKH VXUIDFH 7KLV VLWXDWLRQ FRUUHVSRQGV WR WKUHH GLPHQVLRQDO KRPRJHQHRXV QXFOHDWLRQ DQG JURZWK DV GHVFULEHG LQ &KDSWHU 7KH WUDQVIRUPDWLRQ EHKDYLRU LV WKHUHIRUH GHVFULEHG E\ WKH 0-$ HTXDWLRQ (TXDWLRQ f ZLWK DQ H[SRQHQW Q $W D ODWHU WLPH WKH ORZHU SLFWXUH f WKH JURZLQJ GRPDLQV LPSLQJH RQ RQH DQRWKHU LQ WKH GLUHFWLRQV SDUDOOHO WR WKH QXFOHDWLRQ VXUIDFH ,Q DGGLWLRQ WKH VXUIDFH DYDLODEOH IRU QXFOHDWLRQ LV IXOO\ WUDQVIRUPHG VDWXUDWHG f ZKLOH WKH UHJLRQ IDU IURP WKH VXUIDFH LV DV \HW XQWUDQVIRUPHG $W WKLV WLPH QXFOHL KDYH FHDVHG WR IRUP DQG WKH JURZWK KDV EHFRPH RQH GLPHQVLRQDO LQ WKH GLUHFWLRQ QRUPDO WR WKH QXFOHDWLRQ VXUIDFH f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

PAGE 213

(DUO\ 7LPH 1XFOHDWORQ &HQWHUV 6XUIDFH /DWH 7LPH )LJXUH $ VFKHPDWLF LOOXVWUDWLQJ WKH SURSRVHG RULJLQ RI WKH REVHUYHG FURVVRYHU LQ RUGHULQJ EHKDYLRU 7KH VROLG FLUFOHV UHSUHVHQW QXFOHDWLRQ VLWHV DQG WKH VKRUWGDVKHG OLQHV UHSUHVHQW WKH DGYDQFLQJ GRPDLQ ZDOO VHSDUDWLQJ WKH RUGHUHG GURSOHW IURP WKH GLVRUGHUHG SKDVH

PAGE 214

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r& TXHQFK RQ WKH 6% VROXWLRQ DQG WKH $7 r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r& TXHQFK IRU WKH IROORZLQJ WLPHV DQG VHFRQGV DUH SORWWHG LQ )LJXUH 7KHVH WLPHV HQFRPSDVV RQH RI WKH JURZWK DQG GHFD\ F\FOHV VKRZQ LQ )LJXUH 2Q WKH XSSHU JUDSK WKH GDWD DUH SORWWHG RQ WKH UDQJH IURP [ O2 ƒ ,Q WKLV SORW WKH %UDJJ SHDN LV VHHQ WR JURZ DQG WKHQ GHFD\ ZLWK LQFUHDVLQJ WLPH +RZHYHU WKH OLTXLG SHDN VWUXFWXUH UHPDLQV XQFKDQJHG WKURXJKRXW WKLV F\FOH 2Q WKH ORZHU JUDSK WKH UDQJH LV UHGXFHG WR [ ƒ 2Q WKLV VFDOH JURZWK DQG VXEVHTXHQW GHFD\ RI WKH %UDJJ FRPSRQHQW DUH HYHQ

PAGE 215

)LJXUH 2 7LPH VHH f *UDSKV RI WKH ILWWHG DQG QRUPDOL]HG %UDJJ DPSOLWXGHV DV D IXQFWLRQ RI WLPH IROORZLQJ TXHQFKHV IURP r& WR r& r& DQG r& RQ WKH 6% VROXWLRQ 7KH GDWD DUF UHSUHVHQWHG E\ RSHQ FLUFOHV DQG WKH VROLG OLQHV DUH JXLGHV WR WKH H\H ,Q WKH ORZHU FXUYHV WKH GDVKHG OLQH LV D ILW RI WKH &DKQ PRGHO WR WKH GDWD DQG KHOSV WR LOOXVWUDWH WKH RVFLOODWRU\ EHKDYLRU 'XH WR WKH QDWXUH RI WKH SDUDPHWHUL]DWLRQ WKH EHVW HVWLPDWH RI WKH XQFHUWDLQW\ LQ WKHVH YDOXHV LV WKH IOXFWXDWLRQ LQ WKH SRLQWV LQ WKH HDUO\ WLPH IODWf UHJLRQ RI WKH FXUYH

PAGE 216

R VHF ’ VHF R VHF $ VHF ‘ VHF A VHF r+ 6FDWWHULQJ 9HFWRU 4 $ f )LJXUH *UDSKV RI WKH UDZ GDWD IURP D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ IRU YDULRXV WLPHV IROORZLQJ WKH TXHQFK 7KH XSSHU JUDSK DQG ORZHU JUDSKV VKRZ ODUJHU DQG VPDOOHU UDQJHV RI WKH GDWD UHVSHFWLYHO\ 7KH OLQHV RQ WKHVH JUDSKV DUH JXLGHV WR WKH H\H 7KH XQFHUWDLQWLHV LQ WKH SRLQWV DUH GRPLQDWHG E\ FRXQWLQJ VWDWLVWLFDO HUURU

PAGE 217

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f LV VPDOO HQRXJK WKDW WKLV JUDGLHQW VKRXOG EH QHJOLJLEOH +RZHYHU LW ZDV VSHFXODWHG WKDW WKH IORZ RI FRRODQW WKURXJK WKH TXHQFKLQJ FKDQQHO PLJKW LQGXFH

PAGE 218

IH 7I RQ 6L2 F2  P f K R ; & R FL R &8 t &2 8 '4 f F?M ; 7LPH VHF f )LJXUH 3ORWV RI WKH ILWWHG %UDJJ ZLGWKV DQG SRVLWLRQV DORQJ ZLWK WKH QRUPDOL]HG LQWHQVLW\ = DV D IXQFWLRQ RI WLPH IROORZLQJ D TXHQFK IURP r& WR r& RQ WKH 6% VROXWLRQ 7KH GDWD DUH UHSUHVHQWHG E\ RSHQ FLUFOHV DQG WKH VROLG OLQHV DUH JXLGHV WR WKH H\H

PAGE 219

YLEUDWLRQV LQ WKH VDPSOH KROGHU 7KHVH YLEUDWLRQV FRXOG FDXVH WKH EHDP WR LOOXPLQDWH SRUWLRQV RI WKH VDPSOH DW VOLJKWO\ GLIIHUHQW WHPSHUDWXUHV ZKLFK PLJKW H[KLELW VLJQLILFDQWO\ GLIIHUHQW YDOXHV RI e ZKHQ WKH TXHQFK GHSWK LV VPDOO 7KLV SRVVLELOLW\ ZDV H[SORUHG E\ PDQXDOO\ DWWHPSWLQJ WR LQGXFH WKH UHOHYDQW PRWLRQ PHFKDQLFDOO\ 7KHVH DWWHPSWV ZHUH XQVXFFHVVIXO EHFDXVH WKH LQGXFHG YLEUDWLRQ GDPSHG RXW TXLFNO\ W VHFRQGV ZKLOH RVFLOODWLRQV LQ WKH %UDJJ LQWHQVLW\ SHUVLVW IRU W VHFRQGV f DQG WKH IUHTXHQF\ RI WKH YLEUDWLRQV LV RI FRXUVH PXFK ODUJHU WKDQ UHTXLUHG WR SURGXFH WKH REVHUYH EHKDYLRU RI WKH %UDJJ DPSOLWXGH ,Q DGGLWLRQ LW VHHPV XQOLNHO\ WKDW DQ\ FDXVH IRU YLEUDWLRQ RI WKH VDPSOH FHOO ZRXOG H[KLELW D IUHTXHQF\ WKDW LV FRUUHODWHG WR WKH TXHQFK GHSWK )URP WKH DERYH GLVFXVVLRQ LW DSSHDUV WKDW WKH PHFKDQLVP IRU WKH REVHUYHG RVFLOODWLRQV LV XQOLNHO\ WR EH H[WULQVLF WR WKH V\VWHP 2QH SRVVLEOH LQWULQVLF PHFKDQLVP LV FRQVLGHUHG KHUH EXW WKH IROORZLQJ GLVFXVVLRQ LV KLJKO\ VSHFXODWLYH DQG TXDOLWDWLYH 7KLV SRVWXODWLRQ LV EDVHG RQ WKH ORZ WKHUPDO FRQGXFWLYLW\ RI WKH SRO\PHU VROXWLRQV $V GHVFULEHG LQ &KDSWHU WKH JURZWK RI DQ RUGHUHG GURSOHW RFFXUV ZKHQ WKH IUHTXHQF\ RI FRQGHQVDWLRQ RI SDUWLFOHV RQWR WKH GURSOHW LV JUHDWHU WKDQ WKH IUHTXHQF\ RI HYDSRUDWLRQ RI SDUWLFOHV IURP WKH GURSOHW 7\SLFDOO\ LW LV DVVXPHG WKDW DV SDUWLFOHV FRQGHQVH RQ WKH VXUIDFH RI WKH GURSOHW WKH KHDW RI FU\VWDOOL]DWLRQ LV UHPRYHG UDSLGO\ IURP WKH YLFLQLW\ RI WKH GURSOHW VR WKDW WKH ORFDO WHPSHUDWXUH UHPDLQV XQFKDQJHG ,Q WKHVH VROXWLRQV WKLV PLJKW QRW EH WKH FDVH $V WKH VXUIDFH RI WKH RUGHUHG GURSOHW SURSDJDWHV RXWZDUG WKH KHDW RI FU\VWDOOL]DWLRQ PLJKW UDLVH WKH WHPSHUDWXUH RI WKH VROXWLRQ DERYH WKH RUGHULQJ SRLQW LQ WKH YLFLQLW\ RI WKH GURSOHW $V D UHVXOW LQ WKDW VPDOO YROXPH RI WKH VROXWLRQ QHDU WKH GURSOHW WKH GLVRUGHUHG DUUDQJHPHQW LV IDYRUHG DQG WKH HYDSRUDWLRQ UDWH RI SDUWLFOHV IURP WKH GURSOHW WHPSRUDULO\ H[FHHGV WKH FRQGHQVDWLRQ UDWH &RQVHTXHQWO\ WKH GURSOHW EHJLQV WR GLVVROYH +RZHYHU DV WKH ORFDO WHPSHUDWXUH UHWXUQV WR WKDW RI WKH EXON WKH RUGHUHG SKDVH LV DJDLQ WKH VWDEOH SKDVH DQG GURSOHW EHJLQV WR JURZ ,Q WKLV ZD\ WKH RVFLOODWLRQ LQ WKH %UDJJ SHDN LQWHQVLW\ LV WKH UHVXOW RI WUXH RVFLOODWLRQV LQ WKH IUDFWLRQ RI RUGHUHG SKDVH LQ WKH VROXWLRQ

PAGE 220

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

PAGE 221

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b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

PAGE 222

FRXOG EH WDNHQ WR UHGXFH WKH FRQFHQWUDWLRQ RI WKH LPSXULWLHV LQ WKH VROXWLRQ )RU LQVWDQFH WKH VROXWLRQV FRXOG EH SDVVHG WKURXJK D PLOOLSRUH ILOWHU 6HFRQG WKH HIIHFW RI LPSXULW\ VL]H DQG FRQFHQWUDWLRQ FRXOG EH WHVWHG E\ LQWURGXFLQJ D NQRZQ TXDQWLW\ RI LPSXULW\ SDUWLFOHV L H VLOYHU SRZGHUf LQWR WKH VROXWLRQV ,Q WKLV ZD\ LW LV EHOLHYHG WKDW WKH HQWLUH UDQJH RI WUDQVIRUPDWLRQ EHKDYLRU FRXOG EH VWXGLHG $QRWKHU LQWHUHVWLQJ GLUHFWLRQ IRU IXWXUH UHVHDUFK LV WKH VWXG\ RI WKH RUGHULQJ NLQHWLFV LQ D GLEORFN FRSRO\PHU ZKLFK PLFURSKDVH VHSDUDWHV WR IRUP F\OLQGULFDO GRPDLQV 7KH F\OLQGULFDO GRPDLQV DUH NQRZQ WR IRUP D KH[DJRQDO FORVHG SDFNHG ODWWLFH ,Q WKH HYHQW WKDW PLFURSKDVH VHSDUDWLRQ LQ WKHVH V\VWHPV RFFXUV UDSLGO\ WKH VXEVHTXHQW RUGHULQJ RI WKH GRPDLQV ZRXOG EH RI LQWHUHVW GXH WR WKH QDWXUH RI WKH PDFURODWWLFH )LQDOO\ IXUWKHU H[SHULPHQWDWLRQ RQ WKH VROXWLRQV VWXGLHG KHUH DUH UHTXLUHG WR XQGHUVWDQG WKH QRYHO RVFLOODWRL\ EHKDYLRU REVHUYHG LQ WKH VKDOORZ TXHQFK OLPLW

PAGE 223

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f WR WKH TXHQFKLQJ IXUQDFH ,Q WKDW HYHQW WKH

PAGE 224

VROHQRLG YDOYH PXVW EH FRQQHFWHG WR WKH ''* $% SXOVH FRQQHFWLRQ YLD WKH VROHQRLG LQWHUIDFH DUUD\ %,1 7KLV IXQFWLRQ WDNHV DV DQ DUJXPHQW RQH DUUD\ RI GLPHQVLRQ DQG ELQV LW LQ JURXSV RI DQG OHDYHV WKH UHVXOWLQJ DUUD\ RI GLPHQVLRQ RU RQ WKH QXPEHU VWDFN UHDOVFDODU &'38/6( 7KLV IXQFWLRQ FDXVHV WKH 656 GLJLWDO GHOD\ JHQHUDWRU WR HPLW D SXOVH RI GXUDWLRQ UHDOVFDODU VHFRQGV IURP WKH &' SXOVH RXWSXW 127( WKLV FRPPDQG LV W\SLFDOO\ XVHG WR LQLWLDWH D EXUVW RI FRRODQW JDV OLTXLG f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f VR LW LV DGYLVHG WKDW WKHVH YDOXHV EH FKHFNHG EHIRUH LQLWLDWLRQ RI WKH VFDQ &2817 UHWXUQV WKH FRPSXWHU SURPSW DV VRRQ DV WKH LQVWUXFWLRQV DUH UHFHLYHG E\ WKH FRQWUROOHU DQG WKH YDOXH RI WKH '$7$ DUUD\ GRHV QRW FRQWDLQ WKH UHVXOW RI WKLV DFTXLVLWLRQ XQWLO WKH *(7 FRPPDQG KDV EHHQ XVHG '(7581 XVDJH &2817 *(7

PAGE 225

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f LV QRW ILQLVKHG WKH FRPSXWHU ZLOO SROO WKH 3$5 FRQWUROOHU XQWLO FRXQWLQJ LV FRPSOHWH DQG ZLOO WKHUHIRUH EH XQDYDLODEOH IRU IXUWKHU FRPPDQGV XQWLO FRXQWLQJ LV GRQH *(7&0' 7KLV FRPPDQG LV XVHG E\ *2 DQG LV QRW XVHG DV D FRPPDQG DORQH *(7,-. 7KLV FRPPDQG JHWV WKH YDOXHV RI DQG IURP WKH 3$5 GHWHFWRU FRQWUROOHU DQG SODFHV WKHP RQ WKH VWDFN LQ RUGHU 7KH\ DUH UHWULHYHG IURP WKH VWDFN LQ UHYHUVH RUGHU L H WKHQ DQG ILQDOO\ 127( ,Q WKH QRUPDO PRGH RI GDWD DFTXLVLWLRQ DQG DUH WKH IROORZLQJ WKH RI VFDQV WR EH SODFHG LQ RQH VSHFWUXP WKH RI VSHFWUD WR EH DFFXPXODWHG WKH QXPEHU RI GHOD\ VFDQV EHWZHHQ HDFK VSHFWUXP *(7 0(0 7KLV FRPPDQG JHWV WKH YDOXH RI 0(0 IURP WKH 3$5 GHWHFWRU FRQWUROOHU DQG SODFHV LW RQ WKH QXPEHU VWDFN 0(0 LV WKH LQWHJHU FRUUHVSRQGLQJ WR WKH ILUVW VSHFWUXP WR EH DFTXLUHG E\ WKH GHWHFWRU FRQWUROOHU QRUPDOO\ IRU DFTXLVLWLRQ RI VLQJOH VSHFWUD 0(0 LV VHW WR VHH 6(70(0 f LQWHJHUVFDODU *(7 021 7KLV IXQFWLRQ UHWULHYHV WKH PRQLWRU FRXQW IURP WKH GHWHFWRU FRQWUROOHU IRU VSHFWUXP LQWHJHUVFDODU 127( 7KH GHWHFWRU PXVW EH LQ HLWKHU WKH VRXUFH FRPSHQVDWLRQ RU H[WHUQDO DQDORJ PRGH 6HH ;$02'( DQG 6&02'(

PAGE 226

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r DW WKH SURPSW XVDJH *2 f(17(5 &200$1' r 72 (;,7 f'7 nn n(17(5 &200$1' r 72 (;,7 f r f/($9,1* '(7(&725 &20081,&$7,216f +&23< 7KLV FRPPDQG LV XVHG WR JHW KDUG FRSLHV RI JUDSKV IURP WKH $6<67 JUDSKLFV GLVSOD\ 6HH <$ DQG <' f )LUVW SORW WKH GDWD RI ZKLFK \RX ZLVK WR JHW D KDUG FRS\ 7KHQ H[HFXWH +&23< 7KH WH[W ZLQGRZ ZLOO EH FOHDUHG DQG WKH QH[W IGH QDPH ZLOO EH GLVSOD\HG 6HH 0.),/ f ,1,73*0 7KLV LV D URXWLQH ZKLFK LV FDOOHG E\ 67$5783 DQG LV W\SLFDOO\ QRW XVHG DORQH ,17&0' 7KLV LV D URXWLQH ZKLFK LV FDOOHG E\ *2 DQG LV W\SLFDOO\ QRW XVHG DORQH

PAGE 227

VWULQJ /2$'&21 7KH W\SLFDO IRUPDW LQ ZKLFK GDWD LV DQDO\]HG LQ WKH ODE KDV EHHQ WHUPHG FRQILOH IRUPDW $ ILOH LQ WKLV IRUPDW FRQWDLQV IRXU OLQHV RI FRPPHQWV IROORZHG E\ D VLQJOH OLQH FRQWDLQLQJ DQ LQWHJHU GHVLJQDWLQJ WKH QXPEHU RI SRLQWV LQ WKH ILOH DQG ILQDOO\ WKH GDWD YDOXHV [ \ DQG G\ ZKHUH WKHUH LV RQH OLQH SHU GDWD SRLQW 7KLV FRPPDQG ORDGV WKH GDWD YDOXHV IURP WKH ILOH VWULQJ ZKLFK LV LQ FRQILOH IRUPDWf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f 1RWH WKLV FRPPDQG DSSOLHV RQO\ WR FKDQQHO RI WKH WHPSHUDWXUH FRQWUROOHU 0,&0$1 7KH 0LFULVWDU WHPSHUDWXUH FRQWUROOHU FDQ SHUIRUP LQ HLWKHU IXOO\ DXWRPDWLF RU PDQXDO PRGH 7KLV FRPPDQG VHOHFWV WKH PDQXDO PRGH ZKHUH WKH SRZHU RXWSXW WR WKH IXUQDFH LV VHW XVLQJ 0$1287 6HH 0,&$872 DQG 0,&287f 1RWH 7KLV FRPPDQG DSSOLHV RQO\ WR FKDQQHO RI WKH WHPSHUDWXUH FRQWUROOHU UHDOVFDODU 0,&287 ,Q WKH PDQXDO PRGH WKLV FRPPDQG VHWV WKH SHUFHQWDJH RI IXOO SRZHU RXWSXW E\ WKH 0LFULVWDU WHPSHUDWXUH FRQWUROOHU WR WKH IXUQDFH 6HH 0,&$872 DQG 0,&0$1f 1RWH 7KLV FRPPDQG DSSOLHV RQO\ WR FKDQQHO RI WKH WHPSHUDWXUH FRQWUROOHU XVDJH 0,&0$1 0,&287 LQWHJHUVFDODU 0.)/

PAGE 228

7KLV FRPPDQG FRPSRVHV D ILOH QDPH IURP LQWHJHUVFDODU DQG WKH WKUHH OHWWHU ILOH SUHIL[ VWRUHG LQ WKH VWULQJ YDULDEOH ),/(1$0( )RU H[DPSOH WKH IROORZLQJ VHTXHQFH &$%& ),/(1$0( 0.),/ ZRXOG SODFH &$%&'$7 RQ WKH V\PERO VWDFN 7KLV VWULQJ LV WKHQ XVHG E\ RWKHU FRPPDQGV VXFK DV 6$9 +&23< DQG %,16$9 029($502 7KLV FRPPDQG LQVWUXFWV WKH .OLQJHU VWHSSLQJ PRWRU FRQWUROOHU WR SRVLWLRQ WKH DUP RI WKH +XEHU GLIIUDFWRPHWHU DW ]HUR GHJUHHV 7KLV FRPPDQG LV XVHIXO IRU PRYLQJ EDFN DQG IRUWK EHWZHHQ WKH GDWD DFTXLVLWLRQ SRVLWLRQ 6HH 029(%6 f DQG WKH ]HUR DQJOH SRVLWLRQ ZKHUH UHVROXWLRQ DQG SRO\PHU WUDQVPLVVLRQ PHDVXUHPHQWV DUH PDGH 029(%6 7KLV FRPPDQG LQVWUXFWV WKH .OLQJHU VWHSSLQJ PRWRU FRQWUROOHU WR SRVLWLRQ WKH k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f IURP WKH 2UWHF FRXQWHU WLPHU DQG SULQWV WKH YDOXH WR WKH VFUHHQ 7KLV FRPPDQG LV XVHG LQ FRQMXQFWLRQ ZLWK 67$57257(& ZKLFK LQLWLDWHV WKH 2UWHF FRXQWLQJ VHTXHQFH XVDJH 67$57257(& 35,17257(& 35,17563 f(;32685( n n021,725 n

PAGE 229

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f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f XVDJH ),/(180 &$%& ),/(1$0( 6$9 7KLV VHTXHQFH VDYHV WKH VSHFWUXP LQ WKH ILOH &$%&'$7 ),/(180 LV LQFUHPHQWHG IROORZLQJ H[HFXWLRQ RI 6$9 UHDODUUD\ VWULQJ 6$9('26 7KLV FRPPDQG VDYHV WKH FRQWHQWV RI UHDODUUD\ WR WKH GDWD ILOH GHVLJQDWHG E\ VWULQJ LQ VWDQGDUG $6&,, IRUPDW 6$9('26 LV FDOOHG E\ 6$9 EXW 6$9('26 GRHV QRW LQFUHPHQW WKH ILOH LQGH[ ),/(180 XVDJH '$7$ %* &$%& '$7 6$9('26 UHDOVFDODU 6&$1

PAGE 230

7KLV SURJUDP VHWV WKH H[SRVXUH WLPH WR EH UHDOVFDODU WKHQ LQLWLDWHV WKH FRXQWLQJ VHTXHQFH RQ WKH 3$5 GHWHFWRU FRUUHVSRQGLQJ WR WKH SUHVHW YDOXHV RI WKH SDUDPHWHUV DQG 0(0 6HH 6(7,-. *(7,-. 6(70(0 DQG *(70(0f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nn VWULQJ 6(1'&0' 7KLV FRPPDQG SROOV WKH VHOHFWHG GHYLFH XQWLO LW LV SUHSDUHG WR UHFHLYH D *3(% PHVVDJH DQG WKHQ VHQGV VWULQJ WR WKH GHYLFH 7KLV FRPPDQG LV XVHG LQ RWKHU SURJUDPV VXFK DV *2 DQG &2817 6(532/ 7KLV FRPPDQG SHUIRUPV D VHULDO SROO RI WKH *3,% EXV DQG GLYLGHV WKH UHVXOW LQWR LQGLYLGXDO ELWV ZKLFK LQGLFDWH WKH VWDWXV RI WKH GHYLFH 7KLV FRPPDQG LV XVHG E\ 6(1'&0' DQG 5($'&0' LQ RUGHU WR GHWHUPLQH LI WKH GHYLFH LV UHDG\ WR VHQG RU UHFHLYH LQIRUPDWLRQ UHDOVFDODU 6(7'(/$< 7KLV IXQFWLRQ VHWV XS WKH 656 GLJLWDO GHOD\ JHQHUDWRU WR RXWSXW D SXOVH RI GXUDWLRQ UHDOVFDODU VHFRQGV IURP WKH &' RXWSXW 7KH GLJLWDO GHOD\ JHQHUDWRU LV SODFHG LQ WKH VLQJOHVKRW PRGH DQG LV WULJJHUHG E\ SUHVVLQJ WKH (;& EXWWRQ RQ WKH IURQW SDQHO RU

PAGE 231

VXEVHTXHQW H[HFXWLRQ RI WKH 75,*''* FRPPDQG 7KLV FRPPDQG LV XVHIXO IRU VHQGLQJ FRQWUROOHG EXUVWV RI FRRODQW WKURXJK WKH IXUQDFH DQG LV PRVW FRPPRQO\ XVHG GXULQJ TXHQFKLQJ PHDVXUHPHQWV XVDJH 6(7'(/$< 7KLV LV IROORZHG E\ GHSUHVVLRQ RI (;& RU H[HFXWLRQ RI WKH 75,*''* FRPPDQGf UHDOVFDODU 6(7(7 7KLV URXWLQH VHWV WKH H[SRVXUH WLPH IRU WKH 3$5 GHWHFWRU FRQWUROOHU WR EH UHDOVFDODU 1RWH WKLV LV QRW QHFHVVDULO\ HTXLYDOHQW WR WKH WRWDO FRXQWLQJ WLPH IRU D VSHFWUXP ZKLFK DOVR GHSHQGV RQ WKH YDOXHV RI DQG 6HH 6(78. *(78. 6(70(0 DQG *(70(0 f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f LQWHJHUVFDODU LQWHJHUVFDODU 67$57257(& 7KLV FRPPDQG VHWV WKH FRXQWLQJ WLPH IRU WKH 257(& FRXQWHUWLPHU DQG LQLWLDWHV DFTXLVLWLRQ RI WKH PRQLWRU FRXQW 7KH FRXQWLQJ WLPH LV [ LQWHJHUVFDODUO [O2OULWHJHUVFDODU ZKHQ FRXQWLQJ LV FRPSOHWHG WKH PRQLWRU FRXQW LV WUDQVIHUUHG WR WKH FRPSXWHU E\ XVLQJ WKH 35,17257(& FRPPDQG XVDJH 67$57257(& &RXQW WLPH VHFRQGV f 67$5783

PAGE 232

7KLV SURJUDP PXVW EH UXQ XSRQ HQWHULQJ WKH $6<67 HQYLURQPHQW IRU WKH SXUSRVH RI XVLQJ WKH GHWHFWRU SURJUDPV 67$5783 SHUIRUPV PDQ\ WDVNV )LUVWO\ WKH YDOXHV RI ),/(1$0( DQG ),/(180 DUH UHTXHVWHG DQG WKHQ WKH GDWD DFTXLVLWLRQ SRVLWLRQ IRU WKH GHWHFWRU %6326 f LV UHTXHVWHG 1H[W WKH SURJUDP DVNV IRU WKH QDPH DQG DGGUHVV IRU DOO RI WKH *3,% GHYLFHV XS WR )LYH f )LQDOO\ WKH SURJUDPV FRQWUROOLQJ WKH 257(& FRXQWHU DQG .OLQJHU PRWRU FRQWUROOHU DUH DXWRPDWLFDOO\ ORDGHG XVDJH 67$5783 )ROORZ WKH LQVWUXFWLRQVf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f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

PAGE 233

SUHVHQWHG LQ $6<67nV JUDSKLFVUHDGRXW IRUPDW :LWKLQ WKLV GLVSOD\ WKH IROORZLQJ NH\V KDYH WKHVH PHDQLQJV /HIW RU 5LJKWDUURZ! 0RYHV WKH FXUVRU OHIW RU ULJKW +RPH! 'LVSOD\V WKH SUHVHQW FXUVRU SRVLWLRQ 'HO! ([LWV WKH JUDSKLFVUHDGRXWf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r *3,%'(9,&( &75 *3,%'(9,&( 027 *3% '(9,&( '(7 *3,% '(9 ,&( 0,& *3,% '(9 ,&( ''* ,17(*(5 6&$/$5 632// ,17(*(5 6&$/$5 ,1',17(*(5 6&$/$5 ,1', ,17(*(5 ',0O @ $55$< 632//%,7 675,1* &0'287 675,1* 5(63216( ',0O f '0$$55$< '0$'$7$ 5($/ ',0> @ $55$< ;; 5($/ ',0O $55$< << 5($/ ',0O f $55$< (55 5($/ ',0O f $55$< '$7$ 5($/ ',0O @ $55$< %* 5($/ ',0O $55$< $ 5($/ ',0O @ $55$< $ 5($/ ',0O @ $55$< $ 5($/ ',0O $55$< $ 5($/ ',0O $55$< $ 5($/ ',0O $55$< $ ,17(*(5 ',0O $55$< &+$1

PAGE 234

,17(*(5 6&$/$5 563&75 ,17(*(5 6&$/$5 2/'&75 675,1* &03 &85 f ff &03&+5 f %* $ &+$1 -),// ? ? ? ,1773*0 ),; 3250$7 &5 &5 (17(5 7+( *3,% $''5(66 2) '(9,&( 81'(5 7(67 r f,1387 &5 f180%(5 f *3,% '(9 ,&( &5 r (17(5 7+( 1$0( 2) 7+( '(9,&( r f,1387 n&$7 f(;(& 7,0(287 6(1',17(5)$&(&/($5 5(027(1$%/%21 (26 &+$5$&7(5 ?86(&5 $6 (26 (26 21 n '' r &0'287 f ? 86( &5 $6 '(/,0,7(5 %(7:((1 ? 5(63216(6 62 08/7,3/( ? 5(63216(6 :,// %( 6$9(' ,1 ? ',))(5(17 9$5,$%/(6 ),; )250$7 *(7&0' &5 ‘ (17(5 &200$1' f 72 (;,7 \ r f,1387 &0'287 ,17&0' &03&+5 &0'287 f/()7 ,) &0'287 /(1 f5,*+7 &0'287 f 758( (/6( )$/6( 7+(1 6(532/ 6(5,$/ 32// '83632//} !0$6. '83 $1' 0$6.! 632//%,7 > '83 $1' 0$6.! 632//%87 > f '83 $1' 0$6.! 632//%,7 > '83 $1' 0$6.! 6 32// %,7 > '83 $1' 0$6.! 6 32// %,7 > @ '83 $1' 0$6.! 6 32// %,7 > @ U '83 $1' 0$6.! 6 32// %,7 $1' 0$6.! 6 32// %,7 > 6(1'&0' %(*,1 6(532/ 632//%,7 > I!0$6. ? :$77 )25 7+( &0' '21( 817,/ *3,% :5,7( 5($'563 %(*,1 6(532/ 632//%,7 > !0$6. ? 287387 5($'<" ,) 5(63216( *3,% 5($' 7+(1 6(532/

PAGE 235

632//%,7 f !0$6. ?&0' '21( 817,/ *(7 563 0( /,67(1(5 '(7 7$/.(5 '0$'$7$ '0$*3,% %8))(5 563&75 2/'&75 563&75 %(*,1 6(532/ 632//%,7 @ !0$6. ?287387 5($'< ,) 563&75 r 563&75 0( /,67(1(5 '(7 7$/.(5 '0$ /,67(1 7+(1 6(532/ 6 32// %,7 > @ !0$6. ?&0' '21( 817,/ 35,17563 632//%,7 > !0$6. ,) %(// &5 &200$1' (5525f &5 2/'&75 563&75 (/6( &5 5(63216( 7<3( &5 7+(1 *2 '(7 1250$/',63/$< ),; )250$7 *(7&0' %(*,1 &0'287 6(1'&0' 5($'563 35,17 563 *(7&0' f ff &0'287 f/()7 r 817,/ &5 &5 f /($9,1* '(7(&725 &20081,&$7,216 f &5 ),; )250$7 *( 7 ),;)250$7 '(7 f %'817 f 9 f&$7 f f f&$7 6(1'&0' *(7563 '0$'$7$ '83 68%> @ f 6:$3 68% A 02'8/2 A '$7$ ),;)250$7 &2817 ),; )250$7 '(7 f (7 f&$7 6(1'&0' r 581 f 6(1'&0' <' ;<'$7$3/27 $55$<5($'287 1250$/ &225'6 5($'287!326,7, 21 :25/' &225'6 <$ ;< $872 3/27 $55$<5($'287 1250$/ &225'6 5($'287!326P21

PAGE 236

:25/'&225'6 6&$1 &2817 *(7 2 ,$1 '$7$ %* <$ )'& )250$7 6$9('26 )'& )250$7 f'83 ‘ 287!),/( n f6:$3 f&$7 f(;(& &2162/( 2)) &5 287!)7/( &/26( ‘ ',5 f 6:$3 &$7 (;(& )'& )250$7 %,1 '83 68%> 6:$3 '83 68% f 6:$3 '83 68% @ 6:$3 '83 68% f 6:$3 '83 68% @ 6:$3 '83 68%^ @ 6:$3 '83 68% 6:$3 68% @ %,1 '83 68% f 6:$3 '83 68% f f 6:$3 '83 68% f 6:$3 68% A A A %,1 '8368%> f 6:$3 68% f A 4827( $6&,, 6:$3 &$7 $6&2 &$7 23(16+877(5 0,& /(7 (9f 4827( &$7 *3,%:5,7( 5(63216( *3,% 5($' D 06 (&'(/$< &/ 26(6+877(5 0,& /(7 (9f ‘ 4827( &$7 *3,% :5,7( 5(63216( *3,% 5($' 06 (&'(/$< ;6&$1 %* D $U )'& )250$7 A '2 23(1 6+877(5 '(7 '83 &2817 *(7 '$7$ $ A $ &/26(' +877(5 '(7 '83 &2817 *(7 '$7$ %* A %2 /223 &+$1 $ %* <$ $ '$7$ '523

PAGE 237

),; )250$7 /2$' 36' 2,1'n'83 '()(5! %$6,&23(1 &5 r /2$',1* r 7<3( &5 %$6,&5($' 67$&.&/($5 %(*,1 ,1'A ,1', %$6 &5($' ,1'A ,1',1', ,1', '2 '$7$ #> (17(5 W @ '523 A/223 ,1'817,/ %$6,&&/26( /2$'&21 ‘'83 ,1''()(5! %$6,&23(1 &5f/2$',1* f 7<3( &5 %$6 &5($' %$6,&5($' %$6,&5($' %$6,&5($' 67$&.&/($5 %$6,&5($' '523 %(*,1 ,1'A ,1'%$6,&5($' 127 ,) &5 5HDG (LWRW %$6,& &/26( 48,7 7+(1 ,1'(55 # f (17(5 '523 ,1'<<#@ (17(5 f '523 ,1', ;; # f (17(5 '523 '83 ,1'817,/ %$6,&&/26% &5 r 7+( 180%(5 23 (/(0(176 &5 0.)7/ 9 f5,*+7 ;" /()7 '83 f f ,) f'523 f 2n 7+(1 f6:$3 f5,*+7 f'83 f/()7 f'83 f f ,) f'523 f 2f 7+(1 f6:$3 f5,*+7 f&$7 f&$7 ),/(1$0( f6:$3 f&$7 f '$7f f&$7 +&23< 6&5((1 &/($5 ),/(180 0.),/ 7<3( &5 6&5((135,17 6$9 1250$/',63/$< '$7$ %* ),/(180 0.),/ 6$9('26 ),/(180 A ),/(180 6(1' 7% :5,7( 5(63216( *3,%5($' 0295$502 027 f 5= f *3,%:5,7( f 3= f *3,% :5,7( f 7= f *3IIL:5,7%

PAGE 238

r 3=2 f *),% :5,7( 029(56 027 f 5= r *3,%:5,7( r <7 %6326 A 9 n&$7 *),% :5,7( r 7= r *3,% :5,7( f )7 %6326 9 f&$7 *3,%:5,7( 67$57021 &75 f (1B5(0 f 6(1' f ,1,7 r *3,%:5,7( 06(&'(/$< 5(63216( *3,% 5($' f &/$// r 6(1' f 6(7B02'B6(& 6(1' f 6(7B228B)5 r 6:$3 9 r&$7 &$7 9 f&$7 6(1' f 67$ r 6(1' 35,17021 &75 r (15(0 r 6(1' r 6+B&28 r 6(1' f7,0( f5(63216( f/()7 7<3( r 021,725 5(63216( f5,*+7f/()77<3(&5 5(63216( *3,% 5($' r (1B/2& f 6(1' 67$57 83 &5 (17(5 ),/( 1$0( 35(),; f,1387 ),/(1$0( f &5 &5 f (17(5 67$57,1* ),/( f,1387 f180%(5 ),/(180 ,)f'523 7+(1 &5 &5 r (17(5 7+( %($0 6723 326,7,21 f f,1387 f180%(5 %6 326 ,)f'523 7+(1 ,1,7 3*0 ,1,7 3*0 ,1,73*0 ,1773*0 ,1,7 3*0 &75 (26 &+$5$&7(5 /2$' &2817(5 3*0 /2$' 027256 3*0 (;,7 ) )81&7,21 -&(<'2(6 +&23< ) )81&71.(<'2(6 6$9 287!),/( &/26( '(/(7( %2277;7 6&5((1 &/($5 6(78. '(7 ),;)250$7 f ff f&$7 *3,%:5,7( f f&$7 *3,%:5,7( r fr f&$7 *3,%:5,7( ),; )250$7 *(7--. '(7 ),;3250$7 f f *3,%:5,7( 5(63216( *3 %5($' 5(63216( f180%(5 ,3 7+(1 f'523 f f *3,% :5,7( 5(63216( *3,% 5($' 5(63216( 180%(5 ,) 7+(1 f'523 f f *3,%:5,7( 5(63216( *3,%5($' 5(63216( 180%(5 ,3 7+(1 n'523 ),;3250$7 6(7(7 '(7 ),;)250$7 r (7 f&$7 *3,% :5,7(

PAGE 239

),; )250$7 6(70(0 '(7 ),;)250$7 f 0(0 9 &$7 *3,% :5,7( ),; )250$7 *(70(0 '(7 ),;)250$7 r 0(0 r *3,%:5,7( 5(63216( *3,%5($' 5(63216( 180%(5 ,) 7+(1 '523 ),; )250$7 '(7581 '(7 581 f *3,% :5,7( 6&02'( '(7 6& r *3,% :5,7( ;$02'( '(7 ;$ *3,%:5,7( 0,&0$1 0,& /(7 /6f f 0n 4827( &$7 *3,% :5,7( 5(63216( *3,% 5($' 0,&$872 0,& /(7 /6f $ 4827( &$7 *3,% :5,7( 5(63216( *3,% 5($' 0.=287 0,& ),;)250$7 /(7 52Of 4827( &$7 *3,%:5-7( 5(63216( *3,% 5($' ),; )250$7 *(739 0,& 35 39fr *3,%:5,7( 5(63216( *3,% 5($' 5(63216( 180%(5 ,) 7+(1 '523 *(739 0,& 35 39f *3,%:5,7( 5(63216( *3,% 5($' 5(63216( 180%(5 ,) 7+(1 '523 &+$1*(63 0,& ),; )250$7 /(7 63f 4827( &$7 *3,%:5,7( 5(63216( *3,% 5($' ),; )250$7 6(7'(/$< ''* ),;)250$7 '7 r *3,%:5,7( '7 *3,%:5-7( '7 *3,%:5-7( '7 A 9 &$7 *3,%:5-7(

PAGE 240

r 70 f *),% :5,7( ),; )250$7 75,* ''* ''* f 66 n *3,%:5,7( &(7021 '(7 ),; )250$7 f '& f 9 n&$7 f f&$7 *3 ,% :5,7( 5(63216( *3 ,% 5($' 5(63216( 180%(5 ,) 7+(1 n'523 ),; )250$7 &' 38/6( ''* ),;)250$7 f '7 f *3,% :5,7( f '7 f *3,%:5,7( f '7 f *3,%:5,7( f '7 9 f&$7 *3,% :5,7( r 70 f *3,%:5,7( ),; )250$7 75,* ''* $%5-/6( ''* ),; )250$7 r '7 f *3,% :5,7( r '7 9 &$7 *3,%:5,7( f '7 r *3,% :5,7( f '7 r *3,% :5,7( r 70 *3,% :5,7( ),; )250$7 75,* ''*

PAGE 241

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

PAGE 242

)LJXUH %O 2 7 P Q &7! ,' f§ a 2 f' 2 &af $ VFKHPDWLF RI WKH KHDWLQJ FLUFXLW XVHG WR SRZHU WKH SRO\PHU [UD\ VFDWWHULQJ IXUQDFH 7KH HOHPHQWV RI WKH FLUFXLW DUH GHVFULEHG LQ WKH WH[W

PAGE 243

$33(1',; & 7+( 62/(12,' ,17(5)$&( $ 6WDQIRUG 5HVHDFK 6\VWHPV GLJLWDO GHOD\ JHQHUDWRU ''* f ZDV XVHG WR FRQWURO WKH VROHQRLG YDOYH ZKLFK LQ WXUQ FRQWUROV WKH IORZ RI +H WKURXJK WKH TXHQFKLQJ FKDQQHO RI WKH IXUQDFH $ FLUFXLW ZDV UHTXLUHG ZKLFK ZRXOG LQSXW D 77/ SXOVH IURP WKH ''* DQG HPLW DQ 9 SXOVH WR WKH VROHQRLG YDOYHV 7KH UHTXLUHG FXUUHQW WR WKH YDOYHV LV DSSUR[LPDWHO\ $ 7KH FLUFXLW ZKLFK ZDV GHYHORSHG LV VKRZQ LQ )LJXUH &O WKH IROORZLQJ LV D OLVW RI WKH FRPSRQHQWV LQ WKLV FLUFXLW 7 9 $ 7UDQVIRUPHU ), $ )XVH 6: 0LQLDWXUH 7RJJOH 6ZLWFKHV &O P) 9 &DSDFLWRUV ' 'LRGHV 1 55 .: : b &DUERQ 5HVLVWRUV &5 029 9/$% ', 'LRGHV 1 44 'DUOLQJWRQ 7UDQVLVWRUV 17,3 /3O 9$& 1HRQ /DPS

PAGE 244

92& &O &&7+ &RUDPRQ If§ ’ 7 f OSL L 6:O SL ‘ 9$& % 9$& f 5DWLPL 7KH VROHQRLG LQWHUIDFH FLUFXLW 7KH FRPSRQHQWV DQG XVH LV GHVFULEHG LQ WKH WH[W )LJXUH &O

PAGE 245

$33(1',; 7+( '(9(/230(17 2) 7+( 6$03/( 02817 7KH 7KHUPDO &KDUDFWHULVWLFV RI WKH 6DPSOH 0RXQW 7KH LQLWLDO VDPSOH PRXQW GHVLJQ LV VKRZQ LQ )LJXUH 'O 7KH KHDWLQJFRROLQJ XQLW DQG WKH VDPSOH FHOO DUH VHSDUDWH 7KH KHDWLQJ XQLW LV D VROLG FRSSHU EORFN RI GLPHQVLRQV Of [ n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f &RRODQW LV WUDQVSRUWHG WR DQG IURP WKH VDPSOH PRXQW WKURXJK 9 RXWHU GLDPHWHU n LQQHU GLDPHWHU VWDLQOHVV VWHHO WXELQJ 6WDLQOHVV VWHHO LV XVHG EHFDXVH LWV SRRU UHODWLYH WR FRSSHUf WKHUPDO FRQGXFWLYLW\ KHOSV WR WKHUPDOO\ LVRODWH WKH VDPSOH PRXQW IURP WKH H[WHUQDO HQYLURQPHQW 6ZDJHORFN FRQQHFWLRQV DUH XVHG WR MRLQ WKH VWDLQOHVV WXELQJ WR WKH FRSSHU WXELQJ ZKLFK LV IHG WKURXJK WKH IXUQDFH ZDOO VKRZQ LQ ILJXUH f 7KH TXHQFKLQJ FKDQQHO LV FRPSRVHG RI WZR r GLDPHWHU KROHV H[WHQGLQJ IURP WKH WRS WR ERWWRP DQG FRQQHFWHG YLD DQ n GLDPHWHU KROH GULOOHG IURP

PAGE 246

L )LJXUH 'O $Q LOOXVWUDWLRQ RI WKH LQLWLDO VDPSOH PRXQW GHVLJQ 6KRZQ DUH WKH KHDWLQJ XQLW LQFOXGLQJ WKH KHDWLQJ UHVLVWRUV DQG WKH TXHQFKLQJ FKDQQHO WKH .DSWRQ ZLQGRZV DQG WKH VDPSOH FHOO DV GHWDLOHG LQ WKH WH[W

PAGE 247

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

PAGE 248

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

PAGE 249

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f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

PAGE 250

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f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

PAGE 251

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r& DQG PDLQWDLQ LWV VWUXFWXUDO LQWHJULW\ ZLWKRXW RXWJDVVLQJ XQGHU YDFXXP 6HYHUDO W\SHV RI HSR[\ DQG KHDW WUHDWPHQWV ZHUH WULHG LQ DQ DWWHPSW WR RYHUFRPH WKLV SUREOHP 2XU ILUVW DWWHPSW ZDV WR SODFH WKH SRO\PHU VROXWLRQ LQ WKH KROGHU DQG DIIL[ WKH .DSWRQ ZLQGRZV ZLWK RPHJDERQG HSR[\ 7KH VDPSOH KROGHU ZDV WKHQ FXUHG DW r& XQGHU QRUPDO DWPRVSKHUH IRU KRXUV $ VHFRQG DWWHPSW ZDV WKH VDPH FRQILJXUDWLRQ XVLQJ 2PHJD && +LJK 7HPSHUDWXUH FHUDPLF FHPHQW LQVWHDG RI WKH HSR[\ ,Q ERWK RI WKHVH FDVHV WKH VDPSOHV ZHUH VHDOHG DW URRP WHPSHUDWXUH EXW OHDNHG DV WKH VDPSOHV ZHUH KHDWHG WR YDU\LQJ WHPSHUDWXUHV EHWZHHQ DQG r& 7KH OHDN ZDV GHWHFWHG DV D ULVH LQ FKDPEHU SUHVVXUH IURP LWV W\SLFDO YDOXH EHORZ PLOOLWRUU WR DERYH PLOOLWRUU 7KH PRGH RI IDLOXUH IRU ERWK RI WKHVH DWWHPSWV ZDV WKH IRUPDWLRQ RI D FUDFN LQ WKH HSR[\ H[WHQGLQJ SDUDOOHO WR WKH .DSWRQ VXUIDFH 8VH RI WKLFNHU RU WKLQQHU OD\HUV RI HSR[\ FHPHQWf JDYH WKH VDPH UHVXOW 7KH FUDFNLQJ RI WKH HSR[\ ZDV HYLGHQWO\ FDXVHG E\ D FRPELQDWLRQ RI WKH LQWHUQDO SUHVVXUH DSSOLHG E\ WKH SRO\PHU VROXWLRQ DQG WKH EULWWOHQHVV RI WKH HSR[\ RU FHPHQWf

PAGE 252

)RU WKLV UHDVRQ 3HUPDWH[ +LJK 7HPSHUDWXUH 579 6LOLFRQH FHPHQW ZDV WULHG ,W ZDV EHOLHYHG WKDW WKH HODVWLF QDWXUH RI WKH FHPHQW PLJKW DOORZ UHOLHI RI WKH VWUHVV ZLWKRXW OHDNDJH 7KLV ZDV QRW WKH FDVH KRZHYHU LQ WKDW VPDOO SLQKROH OHDNV ZHUH IRUPHG LQ WKH 579 FHPHQW DW WHPSHUDWXUHV EHWZHHQ DQG r& 1H[W ZH GHFLGHG WR WU\ WKH RPHJDERQG HSR[\ ZLWK D KLJKHU FXULQJ WHPSHUDWXUH r& f ,W ZDV KRSHG WKDW WKLV PLJKW DGG WR LWV VWUXFWXUDO VWDELOLW\ DW KLJK WHPSHUDWXUHV +RZHYHU DQQHDOLQJ DW WKLV WHPSHUDWXUH ZRXOG KDYH UHVXOWHG LQ VLJQLILFDQW ORVV RI & GXULQJ WKH FXULQJ SURFHVV )RU WKLV UHDVRQ D PHWKRG RI ORDGLQJ WKH VDPSOH DIWHU WKH HSR[\ ZDV FXUHG KDG WR EH GHYHORSHG $ 9 GLDPHWHU KROH ZDV GULOOHG DQG WDSSHG LQ WKH WRS RI WKH VDPSOH FHOO DV VKRZQ LQ ILJXUH ; 7KH HSR[\ ZDV WKHQ FXUHG DW r& IRU KRXUV 2QFH FRRO WKH SRO\PHU VROXWLRQ ZDV LQVHUWHG WKURXJK WKH ORDGLQJ KROH DQG WKH KROH ZDV VHDOHG ZLWK D VFUHZ ZUDSSHG LQ 7HIORQ WDSH 7KLV FXULQJ WUHDWPHQW GLG HQKDQFH WKH VWUHQJWK RI WKH HSR[\ LQ WKDW LW ZDV REVHUYHG WR IDLO DW KLJKHU WHPSHUDWXUHV &f $W WKLV SRLQW DQRWKHU WDFN ZDV WULHG ,W ZDV UHDVRQHG WKDW VLQFH WKH SUHVVXUH GLIIHUHQFH EHWZHHQ WKH KRW SRO\PHU VROXWLRQ LQVLGH WKH FHOO DQG WKH YDFXXP RXWVLGH WKH FHOO ZDV UHVSRQVLEOH IRU EUHDNLQJ WKH VHDO WKHQ EDFNILOOLQJ WKH IXUQDFH ZLWK RQH DWPRVSKHUH RI DUJRQ VKRXOG SURYLGH VXIILFLHQW VXSSRUW WR SUHYHQW OHDNDJH WKLV DUUDQJHPHQW GLG LQ IDFW SUHYHQW OHDNDJH RI WKH VDPSOH WR WHPSHUDWXUHV LQ H[FHVV RI r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

PAGE 253

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r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f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

PAGE 254

7KH 2SWLPDO 6DPSOH 7KLFNQHVV 7KH LQWHQVLW\ RI UDGLDWLRQ VFDWWHUHG DW VPDOO DQJOHV ,V LV OLQHDUO\ SURSRUWLRQDO WR WKH SURGXFW RI WKH VDPSOH WKLFNQHVV W DQG WKH DEVRUSWLRQ IDFWRU L ,V RF W Hn0r 'nf +HUH WKH DEVRUSWLRQ OHQJWK L LV WKH WKLFNQHVV RI PDWHULDO ZKLFK UHGXFHV WKH LQWHQVLW\ RI WKH LQFLGHQW LQWHQVLW\ E\ D IDFWRU H 7KLV H[SUHVVLRQ LV D PD[LPXP ZKHQ W WPD[ S ,Q RUGHU WR PD[LPL]H WKH VFDWWHUHG LQWHQVLW\ IURP RXW SRO\PHU VROXWLRQV DEVRUSWLRQ PHDVXUHPHQWV ZHUH SHUIRUPHG RQ IRXU ZWb 6%6 WULEORFN LQ & VDPSOHV KDYLQJ WKLFNQHVVHV W DQG )URP WKHVH PHDVXUHPHQWV _O ZDV GHWHUPLQHG WR EH LQFKHV DQG FRQVHTXHQWO\ WPD[ f $V H[SHFWHG IURP WKLV UHVXOW SUHOLPLQDU\ PHDVXUHPHQWV RI WKH VFDWWHUHG SURILOH VKRZHG D PD[LPXP LQWHUIHUHQFH SHDN LQWHQVLW\ IRU WKH WKLFN VDPSOH 6LQFH WKH IXQFWLRQ RI (TXDWLRQ 'O LV UHODWLYHO\ IODW LQ WKH UHJLRQ RI WAA DQG WKH GLIIHUHQW SRO\PHU VROXWLRQV H[KLELW VOLJKWO\ GLIIHUHQW WUDQVPLVVLRQ FRHIILFLHQWV D VDPSOH WKLFNQHVV RI PP ZDV FKRVHQ IRU FRQYHQLHQFH

PAGE 255

5()(5(1&(6 *XQWRQ 0 6 0LJXHO DQG 3 6 6DKQL LQ 3KDVH 7UDQVLWLRQV DQG &ULWLFDO 3KHQRPHQD HGLWHG E\ & 'RPE DQG / /HERZLW] $FDGHPLF 3UHVV /RQGRQ e f %LQGHU 5HS 3URJ 3KYV f 0 +HQQLRQ 5RQ]DXG DQG 3 *X\RW $FWD 0HW 4 f 6 .DWDQR 0 ,L]XPL 5 0 1LFNORZ DQG + 5 &KLOG 3K\V 5HY % f % *DXOLQ 6 6SRRQHU DQG < 0RULL 3K\V 5HY /HWW  f 0 6XWWRQ DQG % 6WHSKHQVRQ 3K\VLFV LQ &DQDGD f 6 ( 1DJOHU 5 ) 6KDQQRQ -U & 5 +DUNOHVV 0 $ 6LQJK DQG 5 0 1LFNORZ 3K\V 5HY /HWW f < 1RGD 6 1LVKLKDUD DQG <
PAGE 256

6 : .RFK '\QDPLFV RI )LUVW2UGHU 3KDVH 7UDQVLWLRQV LQ (TXLOLEULXP DQG 1RQ 3LTQLOLKULQP 6\VWHPV 6SULQJHU9HUODJ %HUOLQ f ) 0D]HQNR 2 7 9DLOV DQG ) & =KDQJ 3K\V 5HY % f 3 'HE\H 3K\V &ROORLG &KHP  f .UDPHUV 3K\V &ROORLG &KHP  f 0 'RL DQG 6 ) (GZDUGV 7KH 7KHRU\ RI 3ROYPHU '\QDPLFV 2[IRUG 6FLHQFH 3XEOLFDWLRQV 2[IRUG f 3 )ORU\ 3ULQFLSOHV RI 3ROYPHU &KHPLVWU\ &RUQHOO 8QLYHUVLW\ 3UHVV ,WKDFD 1HZ
PAGE 257

3 (KUHQIHVW DQG 7 (KUHQIHVW 7KH &RQFHSWXDO )RXQGDWLRQV RI WKH 6WDWLVWLFDO $SSURDFK WR 0HFKDQLFV &RUQHOO 8QLYHUVLW\ 3UHVV ,WKDFD f 3 3IHXW\ DQG 7RXORXVH ,QWURGXFWLRQ WR WKH 5HQRUPDOL]DWLRQ *URXS DQG WR &ULWLFDO 3KHQRPHQD -RKQ :LOH\ t 6RQV 1HZ
PAGE 258

0 $YUDPL &KHP 3K\V t f 0 $YUDPL &KHP 3K\V f : $ -RKQVRQ DQG 5 ) 0HKO 7UDQV $P ,QVW 0LQ 0HWDOO f (QJUV f 0 /LIVKLW] DQG 9 9 6O\R]RY 3K\V &KHP 6ROLGV 1 & :RQJ DQG & 0 .QREOHU &KHP 3K\V e f 6 0 $OOHQ DQG : &DKQ $FWD 0HW f 6 0 $OOHQ DQG : &DKQ $FWD 0HW f 0 3KDQL / /HERZLW] 0 + .DORV DQG 2 3HQURVH 3K\V 5HY /HWW e f 3 6 6DKQL 'HH *XQWRQ 0 3KDQL / /HERZLW] DQG 0 + .DORV 3K\V 5HY % f = : /DL ) 0D]HQNR DQG 2 7 9DLOV 3K\V 5HY % f % ( :DUUHQ ;UDY 'LIIUDFWLRQ $GGLVRQ:HVOH\ 3XEOLVKLQJ 5HDGLQJ f $ *XLQLHU ;UDY 'LIIUDFWLRQ : + )UHHPDQ DQG &RPSDQ\ 6DQ )UDQVLVFR f 2 *ODWWHU DQG 2 .UDWN\ 6PDOO $QJOH ;UD\ 6FDWWHULQJ $FDGHPLF 3UHVV 1HZ
PAGE 259

$ *XLQLHU $QQ 3K\V f 3RURG .ROORLG= f & 9RQN $SSO &U\VW e f : 5XODQG $SSO &U\VW f + 6 *UHHQ 3URF 5R\ 6RF /RQGRQ f $ f 0 %RP + 6 *UHHQ 3URF 5R\ 6RF /RQGRQ f $ f 0 %RP + 6 *UHHQ 3URF 5R\ 6RF /RQGRQ f $ f )RXPHW &RPSW 5HQG f )RXPHW $FWD &U\VW f & .LWWHO ,QWURGXFWLRQ WR 6ROLG 6WDWH 3K\VLFV )LWIWK (G -RKQ :LOH\ DQG 6RQV 1HZ
PAGE 260

( +HOIDQG DQG = 5 :DVVHUPDQ 0DFURPROHFXOHV -B/ f ( +HOIDQG DQG = 5 :DVVHUPDQ 0DFURPROHFXOHV f 7 +DVKLPRWR 0 6KLED\DPD DQG + .DZDL 0DFURPROHFXOHV f 7 +DVKLPRWR 0 6KLED\DPD DQG + .DZDL 0DFURPROHFXOHV f + +DVHJDZD 7 +DVKLPRWR + .DZDL 7 3 /RGJH ( $YLV & *OLQND DQG & & +DQ 0DFURPROHFXOHV f $ ( 6NRXOLRV LQ %ORFN DQG *UDIW &RSRO\PHUV HGLWHG E\ %XUNH DQG 9 :HLVV 6\UDFXVH 8QLYHUVLW\ 3UHVV f 0 )RONHV DQG $ .HOOHU LQ 7KH 3K\VLFV RI *ODVV\ 3RO\PHUV HGLWHG E\ 5 1 +DZDUG $SSOLHG 6FLHQFH 3XEOLVKHUV /RQGRQ f 5 0D\HU 3RO\PHU f $ 'RX\ 5 0D\HU 5RVVL DQG % *DOORW 0RO &U\VW /LT &U\VW f ( 3HGHPRQWH DQG & $OIRQVR 0DFURPROHFXOHV e f 0 +RIIPDQ .DPSI + .URPHU DQG 3DPSXV $GY &KHP 6HU f 0HLHU 3RO\P 6FL 3DUW & f 3 5 /HZLV DQG & 3ULFH 3RO\PHU -B f ( &DPSRV/RSH] 0F,QW\UH DQG / )HWWHUV 0DFURPROHFXOHV f + 0DWVXRND + 7DQDND 7 +DVKLPRWR DQG 1 ,VH 3K\V 5HY % f $ &KDNUDEDUWL 5 7RUDO DQG *XQWRQ 3K\V 5HY /HWW f 0 6KLED\DPD 7 +DVKLPRWR DQG + .DZDL 0DFURPROHFXOHV f 7 2KWD DQG .DZDVDNL 0DFURPROHFXOHV f + )UHGHULFNVRQ DQG ( +HOIDQG &KHP 3K\V e f 6 $ %UD]RYVNLL 6RY 3K\V -(73 f

PAGE 261

. .DZDVDNL DQG 6HNLPRWR 3K\VLFD $ f < 2RQR DQG 0 %DKLDQD 3K\V 5HY /HWW e f ( 9DQ]R 3RO\P 6FL 3DUW $O f ( 5 3LFR DQG 0 & :LOOLDPV 3RO\P (QJ 6FL f 7 ,QRXH 7 6RHQ 7 +DVKLPRWR DQG + .DZDL 0DFURPROHFXOHV f ( +HOIDQG 0DFURPROHFXOHV t f 7 +DVKLPRWR 0 6KLED\DPD + .DZDL + :DWDQDEH DQG 7 .RWDND 0DFURPROHFXOHV f 0 6KLED\DPD 7 +DVKLPRWR + +DVHJDZD DQG + .DZDL 0DFURPROHFXOHV f 7 +DVKLPRWR 0 6KLED\DPD DQG + .DZDL 0DFURPROHFXOHV f 0 6KLED\DPD 7 +DVKLPRW DQG + .DZDL 0DFURPROHFXOHV f 7 +DVKLPRWR < 7VXNDKDUD DQG + .DZDL 0DFURPROHFXOHV -$ f 7 +DVKLPRWR .RZVDND 0 6KLED\DPD DQG 6 6XHKLUR 0DFURPROHFXOHV f 7 +DVKLPRWR .RZVDND 0 6KLED\DPD DQG + .DZDL 0DFURPROHFXOHV f % 6WHSKHQVRQ -RUGDQ /XGZLJ 0 6XWWRQ DQG <
PAGE 262

%URRNKDYHQ 1DWLRQDO /DERUDWRU\ 8SWRQ 1 < f % 6WHSKHQVRQ 1XFO ,QVWUXP 0HWKRGV 3K\V 5HV 6HFW $ f $ *ODWWHU $SSO &U\VW-= f ( / 7KRPDV .LQQLQJ % $OZDUG DQG & 6 +HQNHH 0DFURPROHFXOHV 4 f < ,VKLEDVKL DQG < 7DNDJL 3K\V 6RF -SQ f 7KH DXWKRU ZRXOG OLNH WR DFNQRZOHGJH WKLV VXJJHVWLRQ IURP DQG XVHIXO GLVFXVVLRQV ZLWK 0 %XPV

PAGE 263

%,2*5$3+,&$/ 6.(7&+ &XUWLV +DUNOHVV ZDV ERP LQ (ULH 3HQQV\OYDQLD QHDU WKH HQG RI +H KDG DQ HQMR\DEOH FKLOGKRRG LQ QRUWKZHVWHUQ 3HQQV\OYDQLD EHIRUH HQWHULQJ FROOHJH LQ +H UHFHLYHG KLV %DFKHORU RI 6FLHQFH LQ $SSOLHG 3K\VLFV IURP &DVH :HVWHUQ 5HVHUYH 8QLYHUVLW\ LQ 0D\ RI

PAGE 264

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ e 6WHSKHQ ( 1DJOHU &KDLUPDQ $VVRFLDWH 3URIHVVRU RI 3K\VLFV FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ A Ff§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n9" $ 3 &KULVWRSKHU %DWLFK 3URIHVVRU RI 0DWHULDOV 6FLHQFH DQG (QJLQHHULQJ 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH 'HSDUWPHQW RI 3K\VLFV LQ WKH &ROOHJH RI /LEHUDO $UWV DQG 6FLHQFHV DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ $XJXVW 'HDQ *UDGXDWH 6FKRRO

PAGE 265

81,9(56,7< 2) )/25,'$


xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID ELZ4CLYQC_PR0JUK INGEST_TIME 2017-07-13T21:35:40Z PACKAGE AA00003746_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES