UNIVERSITY OF CALIFORNIASanta Barbara
Fluctuations and Spatio-temporal Chaos in
Electroconvection of Nematic Liquid Crystals
A Dissertation submitted in partial satisfaction
of the requirements for the degree of
Doctor of Philosophy
in
Physics
by
Xiaochao Xu
Committee in Charge:
Professor Guenter Ahlers, Chair
Professor David Cannell
Professor James Langer
June 2005
The Dissertation ofXiaochao Xu is approved:
Professor David Cannell
Professor James Langer
Professor Guenter Ahlers, Committee Chairperson
June 2005
Fluctuations and Spatio-temporal Chaos in Electroconvection of Nematic
Liquid Crystals
Copyright c© 2005
by
Xiaochao Xu
iii
To my wife Chen, for her constant support.
iv
Acknowledgements
First, I would like to thank my adviser, Prof. Guenter Ahlers, for his
mentorship and constant support, especially for the freedom to choose topics
and to pursue some of my own ideas, although they may seem to be naive. And
for his encouragement when I was afraid to do something because they were
too complicated or too far from my field. I had wonderful working experience
at UCSB, being to able to access many great facilities and got to know many
people. I could not have had these without his support.
A special thank goes to Prof. David Cannell, for his readiness to discuss.
Actually some of the ideas in this thesis were initially proposed by him. I also
thank him and Prof. James Langer for serving on my Ph.D. committee.
I thank Dr. Kapil Bajaj and Michael Scherer for guidance in early stage of
my PhD. Thank Dr. Dan Murphy for discussion of many electronics and soft-
ware problems, especially for introducing me some low-temperature electronics
which found their applications in my research. I thank my colleagues at Ahlers
group at UCSB, Dr. Kerry Kuehn, Urs Bisang, Edgar Genio, Jaechul Oh,
Nathalie Mukolobwiez, Sarabjit Mehta, Alessio Guarino, Denis Funfschilling,
Tahar Aouaroun, Xinliang Qiu, Shenqi Zhou, Mr. Nathan Becker, Eric Brown,
Alexey Nicolaenco, Worawat Meevasana, John Royer, Patrick O’Neill, Jeff
Danciger and Ms. Kim Thompson, for their help and discussion. Specially
thank Mr. Nathan Becker for allowing me to use some of his data in this
thesis. I also thank some of my fellow graduate students at UCSB, Mr. Alex
Small and Craig Maloney for their help and discussion.
I benefit enormously from discussion with Prof. Michael Dennin, Werner
Pesch, Penger Tong, Walter Goldburg, James Gleeson, Hermann Riecke and
Dr. Jurgen Vollmer.
And thank physics machine shop staff members for building and teaching
me to build my apparatus. And UCSB cleanroom staff members for training
me to use the equipment and their support, specially Brian Thibeault for his
processing support.
v
Finally, I would like thank for my family for constant support in all these
years. I hope the achievement of this thesis is a partial satisfaction for their
expectations. And very special thanks for my wife, Chen Guo, for her love,
care and sacrifice of her time and career, and for some real work in this thesis,
pretty graphs were from her, and ugly ones were made by myself.
vi
Curriculum VitæXiaochao Xu
Education
1995 B.S. Jilin University, China.
1998 M.S. Beijing Normal University, China.
2005 Ph.D. (expected) University of California, Santa Barbara.
Publications
G. Ahlers and X. Xu, “Prandtl-number Dependence of Heat
Transport in Turbulent Rayleigh-Benard Convection.” Phys.
Rev. Lett. 86 , 3320 (2001).
X.Xu, K.M.S. Bajaj, and G. Ahlers, “Heat Transport in Tur-
bulent Rayleigh-Benard Convection.” Phys. Rev. Lett. 84 ,
4357 (2000).
vii
Abstract
Fluctuations and Spatio-temporal Chaos in
Electroconvection of Nematic Liquid Crystals
Xiaochao Xu
We have studied two fundamental issues in driven nonequilibrium systems
using electroconvection in nematic liquid crystal I52 and N4.
We first report experimental results for electroconvection of the nematic
Liquid Crystal I52 with planar alignment and a conductivity of 1.0×10−8 (Ω m)−1.
The cell spacing was 19.4 µm and the driving frequency was 25.0 Hz. Spatio-
temporal chaos consisting of a superposition of zig and zag oblique rolls evolved
by means of a supercritical Hopf bifurcation from the uniform conduction
state[14]. For small ε ≡ V 2/V 2c −1 (V is the applied voltage amplitude and Vc
the value of V at the onset of convection), we measured the correlation lengths
of the envelopes of both zig and zag patterns. These lengths could be fit to a
power law in ε with an exponent smaller than that predicted from amplitude
equations. The disagreement with theory is similar to that found previously
for domain chaos in rotating Rayleigh-Benard convection[82].
In the following part, we developed a way to measure local current fluctu-
ations in eletroconvection. Several special cells were made and each cell had a
small local detecting electrode. The detecting electrodes were squares of width
8, 16, 32, 48 and128 µm at the center of one of the two large electrodes. The
spacing of these cells was close to 20 µm. We used the NLC Merck phase IV
(N4) with planar alignment. When the driving ε was from 0.8 to 6, we found
that the distribution of the current fluctuations was strongly skewed towards
larger values for detecting electrodes smaller than the cell spacing and slightly
skewed towards smaller values for the rest of the cells. This is compared with
global current-fluctuation measurements in similar cells which showed a gaus-
sian distribution. For the small electrodes, large fluctuations that extended
viii
below the current expected for the conduction state were found and a possible
connection with the Gallavotti-Cohen Fluctuation Theorem is discussed.
This dissertation and all the supporting materials are also available on my
dissertation webpage[151]. For some large files which are not printed in this
dissertation, the reader should refer to this webpage.
ix
Contents
Acknowledgements v
Curriculum Vitæ vii
Abstract viii
List of Figures xiii
List of Tables xvi
I Preliminary 1
1 Introduction: Electroconvection in nematic liquid crystals as a driven non-equilibrium1.1 Nematic liquid crystal and Nematics . . . . . . . . . . . . . . 41.2 Electroconvection . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 General Experimental Setup for All Experiments 132.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Shadowgraph . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 Temperature Control . . . . . . . . . . . . . . . . . . . 202.1.3 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.1 N4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.2 I52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Cell Construction . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.1 Glass Preparation . . . . . . . . . . . . . . . . . . . . . 362.3.2 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.3 Sealing and filling . . . . . . . . . . . . . . . . . . . . . 43
x
II Spatiotemporal Chaos(STC) Right Above Onset in I5247
3 Introduction: Spatio-temporal chaotic state right above a pattern forming transition
4 Length Scales in STC Phase 534.1 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . 534.2 EC onset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Determination of correlation length . . . . . . . . . . . . . . . 594.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 65
III Local current fluctuations at large driving in Electroconvection in70
5 Introduction 715.1 Previous work on global current fluctuation measurements in EC 715.2 Local current fluctuations in EC . . . . . . . . . . . . . . . . . 75
6 Cell Preparation and Circuits 776.1 Effective Circuit of the Cell . . . . . . . . . . . . . . . . . . . 77
6.1.1 Effective Circuit . . . . . . . . . . . . . . . . . . . . . . 776.1.2 Analysis of the Effective Circuit . . . . . . . . . . . . . 83
6.2 Processing Procedures . . . . . . . . . . . . . . . . . . . . . . 886.2.1 Processing slide with Local and Main Electrode . . . . 896.2.2 Processing Slide with Collector Electrode . . . . . . . . 1036.2.3 Finish up a cell. . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Measuring Circuit . . . . . . . . . . . . . . . . . . . . . . . . . 106
7 Current Fluctuations in Cells with Local Electrodes 1117.1 Electroconvection Onset and Conduction State . . . . . . . . . 1137.2 Current fluctuations . . . . . . . . . . . . . . . . . . . . . . . 119
7.2.1 The general features at different driving . . . . . . . . 1207.2.2 Current fluctuations at high driving . . . . . . . . . . . 121
7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.3.1 Stationary state temperature kBTss . . . . . . . . . . . 1337.3.2 The cause of global current fluctuations at intermediate driving (ε from to 0.19 to
A Numerical values for plots 137
xi
B References 187References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
xii
List of Figures
1.1 Schematic diagram of how to measure σ‖ (a) and σ⊥ (b). Not to scale. This only works1.2 Typical patterns in EC. (a) normal rolls, (b) oblique rolls, (c) spatio-temporal chaos. For1.3 Carr-Helfrich mechanism for EC. Adapted from Ref. [5]. The NLC molecules (rods) are
2.1 Schematic of the apparatus. Adapted from [7]. . . . . . . . . . 142.2 Schematic of the imaging system (shadowgraph), light source, and temperature control stage.2.3 Schematic of light source . . . . . . . . . . . . . . . . . . . . . 162.4 γ2 in [21] Eq. 13 for Setup 2, with l = 1.0mm, f = 16mm, pattern wavelength is 20µm .2.5 Average intensity Ioff for different software offset O at gain=0.524. The incoming light in2.6 Schematic of bridge circuit to drive the cell and monitor cell conducitivty. 222.7 Response curves for driving source of Setup 1 at ratio 1:13.71. Input voltage is 0.1 Vrms.2.8 A simple approximation of the bridge (Figure 2.6) similar to Figure3-12 in [34]. Note that2.9 Conductivity and capacitance of Cell X801 1 filled with I52 at 50C. Applied voltage is2.10 Schematic of UV exposure system . . . . . . . . . . . . . . . 392.11 Spectrum of an MB-100 UV lamp without black glass filter. The arrows show the peaks2.12 Spectrum of an MJB 3 UV400 mask aligner in the UCSB cleanroom, with the permission2.13 Schematic of first few steps of the assembling process. . . . . . 45
4.1 Average of 2048 images at ε ≈ −0.1 (V = 9.740 V) for run 102804 of cell X801 1 as bac4.2 One raw image at V = 10.349 V of run 102804 of cell X801 1. 554.3 The raw image Figure 4.2 divided by the background image Figure 4.1. 564.4 Fourier amplitude image of Figure 4.3. . . . . . . . . . . . . . 564.5 Filtered Fourier amplitude image from Figure 4.4, only fundamental peaks left. 574.6 Total fundamental power versus V 2 for run 102804 of cell X801 1. Up pointing triangles4.7 Filtered Fourier amplitude image from Figure 4.4 ( V = 10.349 V , ε = 0.013), only a zig4.8 (a): zig amplitude background image for ε = 0.013. (b) is a contour plot showing the values4.9 (a): one zig amplitude image for ε = 0.013. (b) is a contour plot showing the values of the4.10 Autocorrelation of zig pattern amplitude of the image in Figure 4.3. The center is normalized4.11 Average autocorrelation of zig pattern amplitude of 2048 images at ε = 0.013. 64
xiii
4.12 (a): contour plot of the average correlation function at ε = 0.013 (central part of Figure4.13 Circles: ξx along long axis for cell X801 1 for different ε; squares: correlation length (extracted4.14 ξy along short axis for cell X801 1 for different ε. . . . . . . . . 684.15 Long axis angle relative to the x-axis for different ε. . . . . . . 68
6.1 Schematic of a cell with a local electrode measuring current passing through the electrode.6.2 Realistic schematic diagram of a cell with local electrodes, leads and local electrode arrangemen6.3 Effective circuit of Figure 6.2. Only components affecting measurements are shown. 796.4 Circuit to measure the effective impedance of the preamplifier at 10−9A/V setting. Lock-in6.5 Simplified effective circuit diagram of Figure 6.2. Some of the capacitors in Figure 6.3 ha6.6 Complex block diagram of Figure 6.5 . . . . . . . . . . . . . . 846.7 From (a) to (e): Steps of processing the slide with local and main electrodes. 916.8 A possible explanation of why lift-off procedure does not work, while etching does, for making6.9 Microscope images of a 8 µm window for local electrode. Left: reflective image; right: transmission6.10 Relative position of the local electrode and the main electrode. The main electrode covers6.11 A broken local electrode lead caused by ESD. The broken point is almost always at the6.12 From (a) to (e): Steps of processing the slide collector electrodes. 1026.13 Relative position of the collector electrode and collector shielding electrode. 1046.14 Schematic diagram of an assembled cell. . . . . . . . . . . . . 1056.15 Schematic diagram of circuit used to measure local current, similar to Fig. 2.6 in Sect. 2.1.36.16 The location of the cell in temperature controlled apparatus. Ground wiring scheme is sho6.17 Circuit use to measure the phase shift by preamplifer. The preamplifier output is measured
7.1 Images at different voltages for cell X817 2 (8 µm). For each voltage (a)-(f), the left image7.2 (a): average Fourier power of 64 images at V = 6.302 V , cell X817 2 (8 µm). (b): circular7.3 Fourier power of fundamental peaks for cell X817 2 (8 µm). Up pointing triangles: applied7.4 Conductance through the 8 µm window of X817 2. Up pointing triangles (black in color7.5 Conductance through the 48 µm window of X817 5. Up pointing triangles (black in color7.6 Section of time series for different driving voltages of cell X817 2 (8 µm). From bottom to7.7 Autocorrelation function of in-phase current Ix for cell X817 2 (8 µm) at ε = 0.93. 1227.8 Circles: X817 2 (8 µm); squares: X817 3 (16 µm); diamonds: X817 4 (32 µm); pluses: X817
7.9 Probability distribution function of In-phase current for X817 2 (8 µm). Obtained by binning
7.10 Probability distribution function of In-phase current for X817 5 (48 µm). Obtained by binning
7.11 In-phase current skewness for X817 2 (8 µm) (circles) and X817 5 (48 µm)(squares). Dash7.12 In-phase current kurtosis for X817 2 (8 µm) (circles) and X817 5 (48 µm) (squares). Dash7.13 Skewness vs. the size of the cell. Circles: ε ≈ 1.4; squares: ε ≈ 3.0. Points at left side of7.14 Up pointing triangles and down pointing triangles are the same as in Figure 7.4. Up poin7.15 Up pointing triangles and down pointing triangles are the same as in Figure 7.5. Up poin
xiv
7.16 ln(
π(P0−∆P )π(P+∆P )
)
for a small range of ∆P for cell X817 2 (8 µm). The solid (red) line is a linear
xv
List of Tables
2.1 Properties of N4 of different dopant concentrations at 30C. Cell SB01 was filled and measured
6.1 Measured and estimated values of effective circuit components of all cells measured and
7.1 Onset Vc and conduction state conductance c0 of different cells. 1187.2 Statistics of in-phase current of X817 2 (8 µm). . . . . . . . . 1237.3 Statistics of in-phase current of X817 5 (48 µm). . . . . . . . 124
xvi
Part I
Preliminary
1
Chapter 1
Introduction: Electroconvectionin nematic liquid crystals as adriven non-equilibrium patternforming system
Statistical physics has been very successful in dealing with equilibrium
problems. There is a general guideline for dealing with such problems. First
find out the energy levels of constituent elements (like molecules), in the pres-
ence of interactions (this step may be very difficult, but it is not really the
problem of statistical physics). Then we can write down the partition function.
This partition function may be solvable or insolvable. If it is solvable, then
it is the end of the problem. If insolvable, most of the physics is to compare
different scales to find which ones are important in the problem and made
proper approximations to bring down the problem to a solvable one.
When we deal with nonequilibrium problems, we are still lacking general
rules in attacking them. In systems near equilibrium, we have Onsager re-
lationships and the fluctuation-dissipation theorem (FDT). But they are not
generally applicable in systems far from equilibrium. For such systems, we
usually deal with them case by case, try to learn special things about them
and apply our ideas and methods learned in other system to them to solve
2
them individually. After partial success at solving the problem, we want to
ask whether we have learned from such specific system can help us understand
more general nonequilibrium problems.
One class of extremely complicated nonequilibrium problems appear in spa-
tially extended driven systems. And in such a system, often time, when the
external driving force reaches certain critical value, instability (instabilities)
occurs and some spatial and temporal structures spontaneously appear. This
process is referred to as pattern formation in driven non-equilibrium systems
and occurs on many length scales. Examples of patterns formed in such sys-
tems include the stripes and spots on animal coats, clouds streets, patterns
formed in etching and deposition processes and dendritic formation. For a
very thorough review on pattern formation, see [1].
Among the pattern forming systems, Rayleigh-Benard convection (RBC)[2,
1] is a particularly interesting system for its easy access for lab studies, well-
known governing hydrodynamic equations, high stability and reproducibility.
In a RBC system, a simple Newtonian fluid (water, methanol, ethanol, CO2
and SF6 gases are commonly used in this group) is confined between two paral-
lel conducting plates positioned horizontally and heated from below and cooled
from above to keep the two plates at constant temperatures. The underlying
mechanism of the pattern forming instability is the balance between fluid buoy-
ancy and viscous dissipation and thermal diffusion. The control parameter,
which characterizes the strength of external driving, is the Rayleigh number
R ≡ αg∆Td3/(νκ). Here g is the gravity acceleration, ∆T is the temperature
difference between the plates, d is the distance between the plates, α is the
thermal expansion coefficient, ν is the kinematic viscosity and κ the thermal
diffusivity. Below the critical Rayeligh number Rc, which is equal to 1708 for a
layer with no lateral boundaries [3], heat is transported by conduction and the
system is spatially uniform. Above Rc, the fluid begins to flow, and convection
rolls form. Depending on the details of the system, there is a great variety
of possible patterns including straight rolls, hexagons, squares and spirals. If
3
in addition there is rotation about a vertical axis, an extra instability which
is called the Kuppurts-Lortz (KL) instability[4] comes in when the rotation
rate is larger than a critical value[2]. This instability can lead to a state of
spatio-temporal chaos right above the onset of convection. We will discuss this
state further in Part II.
RBC serves as a paradigm of pattern formation in isotropic systems. A
complementary system from the viewpoint of symmetry, and a very important
one, is electroconvection (EC) in nematic liquid crystals (NLC). EC in NLC is
a paradigm for pattern formation in anisotropic systems. A NLC cell consists
of a thin NLC layer confined between two parallel capacitance plates (see the
schematic diagram Figure 1.3). The surfaces of the plates are treated to align
the NLC director parallel to the the plates. Usually an AC voltage is used to
drive the convection.
EC in NLC will be the focus of this dissertation. In the following, I will give
a brief introduction to EC. The detailed introductions of the main problems
will be put in the two main parts (Part II and III) of this dissertation.
1.1 Nematic liquid crystal and Nematics
Liquid cystals (LC) are a thermodynamical phase of matter which has
both fluid like properpties (they flow) and crystal like properties (long range
positional or orientational order) [5, 6]. All the work in this dissertation is
about nematic liquid cystals. Usually NLC molecules are rode-like molecules
(for example, the two NLCs we used, N4 and I52, see Sect. 2.2). In the nematic
phase, these molecules have an inherent long-range orientational order, but no
long-range positional order. The direction of average molecular alignment
is given by the director, which can be a function of space and time. Upon
heating a NLC, at a critical temperature called the clearing point, the NLC
loses its positional order and becomes an isotropic liquid (ordinary liquid).
Upon cooling, a NLC will usually have a few transitions to other LC phases
4
which possess various degrees of positional order (smectic A, smectic C, etc.),
and eventually, the NLC will crystalize.
Various aspects of NLC are relevant to EC: the anisotropy of material
parameters, the elastic properties of NLC, and the viscosity tensor. For a
complete discussion of all these the reader can refer to [5, 6], and for a brief
introduction to M. Dennin and M. Treiber’s dissertations[7, 8]. Here I will
only give a brief introduction to the most important property of NLC: the
anisotropy of material parameters.
The director is represented as a unit vector n in the direction of the local
average molecular alignment. But it is not a real vector as n and −n are
equivalent. Mathematically, any physical variable (such as material proper-
ties which will be mentioned below, elastic energy, and viscosity properties)
describing the NLC which contains n can only have even power of n. The
local average of the molecule alignment or any other macroscopic property is
obtained by averaging over scales that are large compared with the scales of
the molecules while small comparing to the macroscopic scales in the system
we study, e.g., the spacing of our cells d. In all our work, the molecules have
scales of order nm and d has scales of tens of µm and therefore they are well
separated and this local average is always doable. When boundaries and/or
external fields are applied, n will usually be a function of space and time and
will be a dynamical variable in the full description of a NLC.
The material properties of a NLC, like the electrical and thermal conduc-
tivities σ and λ, the dielectric constant ǫ, the magnetic susceptibility χ, the
index of refraction n, are generally anisotropic and are represented by ten-
sors. Since all the NLCs we considered have a cylindrical symmetry, i. e. the
axes perpendicular to the director are equivalent, the tensors are uniaxial and
there are only two independent components, the component measured parallel
to the director is b‖ and the component measure perpendicular to the director
is b⊥, where b represents one of the above mentioned properties. We can write
5
V
I
E
(a)
V
I
E
(b)
Figure 1.1: Schematic diagram of how to measure σ‖ (a) and σ⊥ (b). Not toscale. This only works for director fields not distorted by the applied field, i.e. without Frederiks transitions or EC.
component ij of tensor b in the general form:
bij = b⊥δij + baninj , (1.1)
where ni is the a component of the director, and ba = b‖−b⊥ is the anisotropy
of b.
Two properties mostly related to the EC instability are the electrical con-
ductivity σ and the dielectric constant ǫ. Figure 1.1 shows schematically how
to measure σ‖ (a) and σ⊥ (b) using a set of parallel capacitor plates. From the
measured values of resistance and geometry of the plates, one can calculate
the two values. In a similar fashion, by measuring the capacitance at these
two configurations, one can get ǫ‖ and ǫ⊥.
1.2 Electroconvection
Some of the typical patterns of EC are shown in Figure 1.2. In (a), the
roll axes are perpendicular to the director and they are called normal rolls. In
(b), the roll axes form oblique angles to the director, they are called oblique
rolls. There are two oblique roll domains and they form a domain wall at
their bounbary. These pattern are called zig and zag patterns[15]. In (c), the
6
Director
(a)
(b)
(c)
Figure 1.2: Typical patterns in EC. (a) normal rolls, (b) obliquerolls, (c) spatio-temporal chaos. For all images, the director isaligned along the horizontal direction. (a) and (b) are fromhttp://www.nls.physics.ucsb.edu/image_pages/picturepage6.html.
7
φ
E
δEj
jx
B'
+ ++ +A
+
E
j
jx
B
x
z
δE+
Figure 1.3: Carr-Helfrich mechanism for EC. Adapted from Ref. [5]. TheNLC molecules (rods) are initially aligned along the x axis before applyingthe voltage (planar alignment).
amplitudes of the zig and zag patterns vary as a function of time and space
chaotically, this is a spatio-temporal chaotic state[14].
There are two configurations for EC, for a complete discussion of both, see
Ref. [5] Chap. 5.3.3. In this dissertation, we only deal with the case where
the director is planarly aligned and σa > 0 and ǫa < 0 (It is possible to use
NLC whose ǫa is slightly positive. But if ǫa is large and positive, the Frederiks
transition dominates[5]. To avoid any complications, we only consider ǫa < 0
and this is the case we always use in our experiments.).
We first assume that the electric conduction of the NLC is Ohmic, i. e.
j(x) = σE(x) + ρe(x)v (1.2)
where σ is the a constant uniaxial tensor.
The mechanism which produces EC is known as the Carr-Helfrich mechanism[10].
We follow the treatment in Ref. [5] to treat the unidemensional model intro-
duced by Helfrich[9]. In Figure 1.3, before applied a voltage, the director is
8
uniformly aligned along the x direction, n = [1, 0, 0]. After applying a voltage,
consider a small periodic distortion in director field n = [1, 0, φ(x)]. Due to the
distortion of the director field, a current density along the x axis is generated,
jx = σaEzφ(x). jx in turn induces a charge density ρe(x). In Figure 1.3, more
positive charges pile up at region A. The electric field then applies an electric
force upon the fluid around A and thus can induce the flow pattern shown in
Figure 1.3. This flow then drag the director at B to increase the distortion.
Another effect at point B is that, due to the pile-up of position charge at A,
there is an extra component of the electric field along x, δE that adds to the
original electric field. Since ǫa < 0, the director at B tends to be orthogonal to
the local electric field, this gives a torque also tends to increase the initial dis-
tortion. Of course, this distortion cannot just distort the initial director field
at an arbitrarily small electric field since a small distortion is balanced by the
elastic torque of the NLC (for the elastic forces, see Chap. 3 in Ref. [5]). Only
when the distortion torque is large enough to overcome the restoring torque,
the convection happens.
From the above unidimentional Helfrich mechanism, we see that the onset
will be determined by the strength of three torques, and therefore it is obvious
that the elastic constant K, σ’s and ǫ’s will affect the onset. An interesting
result from the balancing of the three torques is that the onset voltage is
independent of the thickness of the cell[5]. And the control parameter for EC
instability is V 2.
The above mechanism is described assuming DC driving. Usually DC
driving is avoided as the NLC is doped to have ionic conduction and therefore
a DC current can destroy the cell by electrolysis. We usually use an AC voltage
and put an insulating boundary condition on the electrodes, i.e. put a thin
insulating layer (in most of our cells, a thin SiO2 layer, see 2.3.1). With a low
frequency, the charge density is able to follow the applied field. And since the
three torques in Figure 1.3 are unchanged under change of the polarization of
the applied voltage, the mechanism is unchanged. As the driving frequency
9
increases, at constant driving voltage, there are fewer charges that can pile
up at point A, and the onset voltage will increase (for a discuss on frequency
dependence of the Helfrich model, see Ref. [5]). When the frequency increases
to the point where the charge density ρe(x) is not able to follow the applied
field (due to finite charge relaxation time τq = ǫ0ǫ⊥/σ⊥), the Carr-Helfrich
mechanism onset diverges. This frequency is the cut-off frequency. Below the
cut-off frequency, it is called conduction region and the pattern wavelength is
comparable to the thickness of the cell d. Above the cut-off frequency, in the
so-called dielectric region, another kind of pattern called ”chevron pattern”
comes in[5] and the wavelength is much smaller than d. For all the work
discussed in this dissertation, we will only work in the conduction region.
The unidimentional Helfrich model can predict the onset voltage qualita-
tively. For a more accurate prediction of onset voltage and to explain the
appearance of oblique patterns, one has to do a full 3-D calculation and it was
done in [11].
The above model can be called the standard model(SM) in EC. But it
always predicts a stationary pattern right above onset. The Weak electrolyte
model (WEM) was proposed by M. Treiber and L. Kramer[12, 8] to explain
the origin of the Hopf bifurcation in EC observed in I52[13]. It has been very
successful in predicting the Hopf frequencies for that system. In the WEM,
instead of pure Ohmic conduction (constant conducting ion concentration),
there are two species of ionic charge carriers with charges ±e and they couple
to each other with a simple dissociation-recombination reaction. The densities
of positive and negative ions are n+(x, t) and n−(x, t). The charge density
field is ρe(x, t) = e[n+(x, t) − n−(x, t)]. The ion mobilities are µ±⊥ and µ±
‖ and
these give the conductivity field σ⊥(x, t) = e(µ+⊥n+ +µ−
⊥n−) (and similarly for
σ‖(x, t)) . Therefore, in the WEM, besides the basic equations for the velocity
field v(x, t), director field n(x, t), and charge density field ρe(x, t), we have
an extra dynamical variable, the perpendicular conductivity field σ⊥(x, t) (
σ‖(x, t) is not independent of σ⊥(x, t))). All these variables are coupled and
10
it is more complicated than the already pretty complicated SM equations.
Nevertheless, it gives rise the new phenomenon, the Hopf bifurcation and
this is very important for our discussion of the spatio-temporal chaos state
in Part II.
There are a few general advantages of EC for as a paradigm for studying
pattern formation or general driven systems. First of all, the size of the cell.
The cell thickness ranges from a few µm to about 100 µm and the size of the
electrode can be as big as centimeters. This enables us to study systems of
very large aspect ratio (order of 1000). The large aspect ratios are closer to
the theoretical idealization of a system of infinite lateral extent and in many
cases, we can simply ignore the effects of boundaries. In addition, the scale
in EC is very suitable for modern photolithography technology to control the
size of the electrodes to achieve special designs. In Part III, we will discuss
the fabrication of small local electrodes to measure local current fluctuations
in an EC system.
The typical time scales in EC are of order 10−3 to 10−1 s. This is sig-
nificantly faster than the typical RBC experiments which have time scales of
the order of minutes. This can shorten the time required to obtain reasonable
statistics.
Another advantage is that driving with electrical current enables us to
measure electrical signals in the system which can be much more convenient
than measuring other thermal or optical signals. In Part III, we will utilize
this advantage.
The disadvantages of EC systems are mostly the stability and uniformity of
the cells. LCs are more complicated than ordinary liquids. In addition, in EC,
we need to dope the LCs to get enough electrical conductivity which is required
in EC. Usually the cells’ properties (most prominently, the conductance of the
cell) drift. Spatially, due to the small thickness of the cell, a micron scale
nonuniformity is big relatively. Most importantly, due to the tricky nature of
alignment (see Sect. 2.3.2), it is very hard to achieve uniformity across a big
11
area which is required in our work in Part II. In Chapter 2, we will discuss
some of our work in dealing with these disadvantages.
In the following chapters, the general setup for all the experiments will be
discussed first. Then two slightly different subjects will be addressed. One
is about the spatio-temporal chaotic (STC) state in NLC I52 convection first
discovered by M. Dennin et al[14]. The other one is about the local current
fluctuation measurements in NLC N4 convection. In studying the two special
cases, we hope we can shed some light on two important subjects in general
driven non-equilibrium systems: the STC in spatially extended driven systems
and fluctuations in general non-equilibrium systems.
12
Chapter 2
General Experimental Setup forAll Experiments
The setup for NLC electroconvection experiments in our group was first
introduced by M. Dennin. In my experiments, the general line of setup follows
the original design. In the following, I will focus on changes I made, along with
a brief description of the unchanged features for completeness. For a complete
reference of original setups, see Dennin’s dissertation[7] and [16].
Two sets of apparatus are used in my experiments, Setup 1 is modified
from the one in [16]. The difference between this one and the one described in
[7] is that this one does not have circulating water cooling, and instead uses
room air cooling. Setup 2 is a modified commercial microscope from Meiji
Techno[20]. The working principles are the same, but this is just a lot smaller
than Setup 1. The emphasis will be on the first one. In the following, without
mentioning, the apparatus means Setup 1.
2.1 Apparatus
Following [7], the whole apparatus is split into three parts, the imaging
system, temperature control stage and electronic controls, which are shown in
Figure 2.1 similar to Figure 3.1 in [7]. Among them, the imaging system is
13
Computer 1( bellman)
Frame Grabber
Temperaturecontrol
MultimeterPowersupply
Bridge circuitFig. 2.6
Computer 2 (ehc)
Figure 2.1: Schematic of the apparatus. Adapted from [7].
modified very little, temperature control stage is modified, and the electronic
controls are completely different from the one in [7].
2.1.1 Shadowgraph
The shadowgraph technique is a well developed method of visualizing vari-
ation in the index of refraction of a fluid. In RBC, the variation is due to the
density variation caused by the temperature field. In EC, the variation is due
to director tilt. For the shadowgraph technique in EC, the reader can refer to
[17, 18, 19].
14
Nikon Camera Lens
CCD Camera
lower lens
Temperature- Control Stage
lightsource
cell
Translation Stage
electronics in/out
shadow graph system
Figure 2.2: Schematic of the imaging system (shadowgraph), light source,and temperature control stage. Adapted from [7].
The shadowgraph apparatus (as shown in Figure 2.2) is a modified version
of RBC a shadowgraph tower[21, 7]. Along the optical path, we have the light
source, a rotatable polarizer, the cell, a lower lens, a Nikon camera lens and a
15
Figure 2.3: Schematic of light source
CCD camera. In Setup 2, an objective lens replaces the lower lens and there
is no camera lens used, and the size is much smaller than this setup. Other
than that, the scheme is almost identical to the one in Figure 2.2.
The light source consists of a power supply, an ultra bright red LED, an
optical fiber patch cord, and a collimating lens(Figure 2.3).
The power supply is a current source which can put out 0 ∼ 40 mA cur-
rent and it is made of a simple op amp circuit. The LED is an InGaAlP (peak
wavelength 630nm) one, model LZE12W from SunLED Corporation[22]. Com-
paring with ordinary LEDs used in previous setups, this one has a luminosity
of 7000 mcd minium and 18000 mcd maximum at 20mA driving current. Be-
cause of its brightness, no special coupling was made between the LED and
the optical fiber. There are cases when brighter light is desirable. The top of
the LED can then be cut off and polished and carefully aligned with the fiber.
This is done in other setups in our group[23]. A commercial LED-to-fiber cou-
pler (model HULED-25-630-M-BE-CSP) from OZ Optics[24] was also tested
in another setup[25] and it significantly increased the coupling efficiency.
16
The optical fiber patch cord is a step index multimode fiber for visible and
infrared wavelengths from 350 ∼ 2100 nm. Its core and cladding diameters are
50µm and 125µm respectively. Two SMA 905 fiber optic connectors are used
to connect the fiber patch cord to the light source and the collimator.
In Setup 2, a red LED (GaAlAs, peak wavelength 650nm) from Industrial
Fiber Optics, Inc. [26] is used. The advantage of this LED is that it has a
connector built in it and one can plug in a special fiber directly. The fiber, also
from Industrial Fiber Optics, Inc. [26],has a plastic core. The core diameter
is 1mm, which is much thicker than any other fibers we used in this group.
The reason for choosing this is that the microscope objects have much smaller
apertures than any of the shadowgraph towers and a 50µm fiber just cannot
provide enough transmitting light for the microscope. A thick fiber causes the
optical sensitivity of the system to have very few oscillations before it drops
to zero[21]. Figure 2.4 shows γ2 = 2J1(qz1Θa)/(qz1Θa) in [21] Eq. 13 versus
optical distance z1 for this microscope optical setup. (Eq. 13 A = 2dγ1γ2q2z1
is about the sensitivity A of the optical system and it does not apply exactly to
the electroconvection case, but the factor γ2 due to a distributed light source
is still valid.) To use this microscope setup, one has to observe the pattern by
focusing the imaging plane very close to the convecting LC layer. In most cases,
this has not been a problem. Because one can show [19] that the term linear
to director tilt responding to shadowgraph imaging forming is the strongest
near the imaging plane and the linear term is what really needed. In practice,
this actually makes it easier to adjust the optic setup. One usually can see
where the image signal is the strongest by eyes and this almost coincides with
the focusing on the LC layer. So one can either let the cell convects and adjust
the microscope and visually find the best focusing position, or one can focus
a dust particle in the LC layer before letting it convect to determine that.
In each apparatus, a dichroic sheet polarizer of diameter 20.6mm [28] is
used to generate polarized light. The direction of the polarization is adjustable,
and we adjust it to be parallel to LC director in the cell.
17
z1(mm)
γ2
0.0 1.0 2.0 3.0 4.0
0.0
0.4
0.8
Figure 2.4: γ2 in [21] Eq. 13 for Setup 2, with l = 1.0mm, f = 16mm,pattern wavelength is 20µm .
The light enters the cell perpendicularly and the cell is in a temperature-
control stage. We will discuss the cell construction in the following sections.
The lower lens in Figure 2.2 is a 60mm-focal-length achromatic lens1[29].
In the microscope, the original objectives are used. Most of them have rather
small focal length and they have to be rather close to the cell, while with a
60mm-focal-length lens, the lens is about 65 ∼ 75mm from the cell. This gives
us more space to work with.
In Figure 2.2, the Nikon lens and CCD camera are mounted on separate
movable carriages. The design of the carriages allows the relative position
of the Nikon lens and CCD camera and the position of the Nikon lens and
camera as a unit to be adjusted independently (see [7]). The lens is a 50 mm
f/1.4 Nikon camera lens. And the camera is a monochromatic Cohu[30] model
1The light source is almost monochromatic. One chooses a doublet achromat because italso has less other aberrations than a singlet.
18
4915-3000. The image is digitized with an 8-bit framegrabber. The maximum
size of the image is 640 × 480 pixels.
For Setup 2, the microscope has its own camera mount and a digital 12-bit
CCD camera from QImaging[31] is mounted directly on it. The digital images
can be downloaded from the camera via a firewire (IEEE 1394) cable.
In Setup 1, digital gain and offset can be controlled by the framegrabber.
For Setup 2, the camera’s gain, offset, exposure time and triggering time can
be controlled. Before one starts a run taking a series of images, one has to
adjust light intensity, and camera’s gain and offset (for the QImaging camera,
exposure can be adjusted as well, but it is all fixed to be 20ms for all the
following experiments) to make sure the histogram of the pixels fills almost
entire range (0 ∼ 255 for COHU and 0 ∼ 4095 for QImaging), but no or very
few pixels overflow the range for any of the images. The framegrabber’s driver
does not provide a mechanism to take a series fast images with accurate timing.
For each image, it opens the framegrabber, takes a image and closes it, and
then uses sleep function to wait for integer number of seconds before starting to
take next image. This process is very slow and accurate timing is impossible as
all the timing of the steps in the process depends on other processes running in
the computer. The fastest rate for this setup is about 3 ∼ 4 frames per second
(use sleep 0 second) for image size 256 × 256. QImaging camera provides
a mode of external triggering which enables one to use an external square
signal to synchronize the image-taking process. Therefore if accurate timing
is required, one should use this mode. The fastest rate for taking 256 × 256
images is about 20 frames per second.
The first step for quantitative image analysis is image division and it follows
reference [21]. In a run, we collect a series of images at different voltages, at
least one voltage is far enough from the onset of electroconvection at which
one does not see any pattern. This provides a background image I0(x). Then
the rest of the images I(x) are divided pixel by pixel by the background image.
This is to remove inhomogeneities in the optical path. The shadowgraph signal
19
is then defined to be
I(x) =I(x)−Ioff
I0(x)−Ioff(2.1)
where Ioff is the offset of the camera and it was measured at incoming light
intensity equals zero(covering the camera). It was found that, for QImaging
camera, at different gains, the offset values are slightly different. For QImaging
driver Version 1.68.6, the gain has a range from 0.524 to 18.3 and the software
offset O is from -1460 to 2635. Note that this offset value is provided by the
software and for each individual camera, one has to calibrate its own offset
Ioff in Eq. 2.1. Since the light source intensity is strong enough for all the
experiment conditions, we always set the gain to be the smallest which is 0.524
and when needed, vary light intensity to make the histogram cover a proper
range. Figure 2.5 shows calibration data for different software offset O at
gain=0.524. In most of runs, software offset is set to 473 and the corresponding
Ioff is 429.9.
2.1.2 Temperature Control
In Setup 1, the temperature control stage was slightly modified from that
in [16]. We use a bigger aluminum cylinder (12.95 cm high with a diameter of
13.11 cm) to keep the temperature at the center of cylinder where the cell is
placed more stable. The whole aluminum is wrapped with Insulating foam.
In [7], the stage temperature is controlled by a home-made analog con-
troller. Our group has switched all our near-room temperature controllers
to software controllers. The temperature is measured with a thermistor and
read out by a Keithley 196 multimeter to a computer (bellman in Figure 2.1).
The control algorithm is a standard PID control [32, 147] and we usually set
the parameter D=0 (PI control). The computer outputs a 0 ∼ 5V voltage
through a D/A converter to drive a power supply to adjust heat output to five
heaters[33] attached on to the aluminum cylinder to keep the readings of the
thermistor constant.
20
0.0 1000.0 2000.0 3000.00
500
1000
1500
2000
Software Offset O
Ave
rage
Inte
nsity
I~
off
Figure 2.5: Average intensity Ioff for different software offset O atgain=0.524. The incoming light intensity is zero. The solid line is a linearfit to the measured data, Ioff = 152.3 + 0.587O
In Setup 2, a Lauda water circulator is used to keep a copper block at
constant temperature and the cell is placed inside of the block.
2.1.3 Electronics
The AC driving source and measuring circuit are completely replace with
an AC bridge based system. The electronics in [7] has the advantage of low-cost
and of being computer-controllable. But it cannot monitor the cell conduc-
tivity during a run. So usually, one first measures the cell conductivity and
capacitance before a run, then after the run finishes, one measures it again.
But one does not have any idea of the history of the conductivity. Both the-
ory and our experience suggest that the cell conductivity is the most crucial
parameter that controls the behavior of the pattern formation. Firstly, the
conductivity anisotropy σa gives the driving force for the electroconvection
21
Figure 2.6: Schematic of bridge circuit to drive the cell and monitor cellconducitivty.
instability. Secondly, the charge relaxation time τq = ǫ0ǫ⊥/σ⊥ determines the
cutoff frequency of the convection. And it also comes into other places to
determine the details of the patterns. All these considerations motivate us
to pursue a better way to monitor the cell conductivity changes, and even to
control them when necessary.
We used an AC bridge similar to those used in our group’s low-temperature
labs[34, 35] to achieve this goal(also, the precision is improved greatly). Fig-
ure 2.6 shows the schematic of the circuit. The driving source comes from
the internal oscillator of a lock-in amplifier. It goes into a unity gain buffer
to lower its output impedance that enables it to drive a step-up transformer.
The driving voltage comes from the secondary and drives the AC bridge. The
bridge consists of a ratio transformer, at one arm a reference resistor decade
box Rr plus a reference fixed capacitor Cr, another arm is the working EC
cell, and a lock-in amplifier measuring the bridge signal. The following is the
details of the circuit and each of its components:
22
Circuit Components
The lock-in oscillator outputs a sinusoidal wave of 1 mHz to 250 kHz, with
amplitude 1µV to 5V, and the output impedance is 50 Ω which is too high
to drive the step-up transformer directly. A unity gain buffer is in between
these two components. It is just a typical op amp follower. The op amp is
a PA07 from Apex Microtechnology Corp. [36]. It is a 5A power op amp
having a relatively low offset voltage (0.5 mV typical, can be trimmed using
the 10 kΩ pot if needed.) and low offset drift (10 µV/C typical) which is
good for driving voltage stability and reducing DC component applied onto
the step-up transformer(a detailed comparison of Apex power op amps can be
found at their webpages[37]). A 1MΩ resistor is put in parallel with the input
to reduce the input impedance. The input impedance of the PA07 is of order
of 1011 Ω. If the input is open without this resistor, an induced voltage can
bring the output to quite high level (possibly close to ±15 V) and saturate
the step-up transformer. A big DC current directly applied to the transformer
should always be avoided since that saturates the core material. Once this
happens, a degaussing procedure should be performed[38]. This is true for the
ratio transformer as well[41].
The step-up transformer serves to raise the driving voltage, to isolate the
driving source from the bridge, so one can ground the central tap of the bridge
at the ratio transformer which is desirable. It is a model PAT4002 toroidal
transformer originally designed for tube amplifier as a push-pull ultra wide
bandwidth output transformer by Plitron Manufacturing Inc. [39]. It is used
reversely as a step-up transformer, i.e. the original primary with more winding
is used here as our secondary, and the original secondary with less winding as
our primary. There are a few reasons to choose this transformer: It works
at audio frequencies(±0.1dB range 3.9 Hz to 24 kHz[40]), which cover all
possible frequencies we work on. Unlike some other transformers used in our
low-temperature labs[42], it can provide big enough output voltages without
23
saturating. It can provide up to 65 V(rms) with ratio 1:13.71. Higher voltage
should be possible with the biggest ratio 1:34.28, but since 65 V is big enough
for all practical purpose, that is not tested. There are 5 tabs[40] available for
different output ratios, so there are 10 possible combinations to choose from.
Some of them give the same ratios. Excluding those, we can have 6 different
ratios and they can be chosen by a rotary switch. But it was found that only
two ratios which are symmetrical to the central taps work properly for the
bridge. They are 1:13.71 using the 40% ultra linear taps and 1:34.28, the
highest ratio, using the full secondary coil. This is possibly related to that the
transformer was manufactured with two windings method. In practice, only
1:13.71 is used.
The ratio transformer is a model 1000 Gertsch AC ratio standard[41]. It
has two units, 0.35f (meaning applied voltage should not exceed 0.35 times
the applied frequency) unit for high frequency; 2.5f unit for low frequecy. We
have two common frequencies in use, 25 Hz and 591 Hz. For 591 Hz, we use
only 0.35f unit; for 25 Hz, we primary use 2.5f unit. One advantage of using a
ratio transformer is that it gives very little phase error comparing with other
voltage dividers. For all applicable ratio range, the phase error caused by ratio
transformer[41] is much smaller than that caused by step-up transformer plus
unity gain buffer(see below). Therefore we will ignore this phase error in all
occasions. Another advantage is that it simplifies the circuit design and enable
us to keep the cell driving voltage constant(see below for circuit analysis).
The lock-in amplifier is from Ametek Signal Recovery (formerly Princeton
Applied Research, EG&G, Perkin Elmer), model 7265[43]. It is a DSP dual
phase lock-in amplifer which enables us to measure both in-phase and out-
of-phase components simutaneously. In choosing this one, we have compared
others in the market. Two models suit our purpose similarly, besides 7265,
Stanford Research Systems 830 DSP dual phase lock-in amplifier[45] is roughly
comparable in performance and price. We chose 7265 mostly because its in-
ternal oscillator has a better resolution (125 µV for 4 mV to 500 mV and 0.5
24
mV for 501 mV to 2 V, versus SR830’s 2 mV resolution, see [43] and [45] ) and
a better resolution is important as after passing through the step-up trans-
former, this resolution will become about 13 times bigger. The 7265 lock-in
can be controlled via GPIB bus and a complete command list can be found in
its manual[44]. The C codes “PE7265.c”[151] contains all the commands used
in experiments described in this thesis.
Mostly, we use the current mode instead of the voltage mode to measure
the imbalance signal of the bridge. We will discuss the reason in the next sub-
section. The 7265 lock-in has its own internal current-to-voltage preamplifier
and it has two modes for that: wide bandwidth and low noise[44]. For most
bridge balancing, we use the wide bandwidth mode. In this mode, its -3 dB
point is 50 kHz and the phase shift caused by the preamplifier is negligible for
frequencies below 100 Hz. The -3 dB point of the low noise mode is only 500
Hz and we seldom use it. In case a low-noise measurement is needed, we use
an external current-to-voltage preamplifier from DL Instruments[46]. Details
about this preamplifier will be discussed in Chap. 6.3.
The two arms of the bridge is the measuring cell and reference resistor and
capacitor. Details about the cell will be discussed in 2.3. The reference resistor
is a General Radio GR1433H decade box which covers the 1 Ω to 11.11111 MΩ
range. Sometimes, it is desirable to expand the range to 111 MΩ, a one-decade
box was made using one 10 MΩ, two 20 MΩ and one 100 MΩ resistors. They
are model TF626R and TF656R Low temperature coefficient precision resistors
from Caddock Electronics, Inc.[47]. The reference capacitor is a General Radio
1403 series standard air capacitor [48]. Their values are fixed, they are 1000pF,
100pF, 10pF, 1pF, 0.1pF and 0.01pF with a nominal error of 0.1%. When we
measure whole cell properties, we usually use 1000pF.
Analysis and Characterization of the Circuit
We first characterize the driving source of the bridge focusing on the phase
shift caused by the source circuit. Firstly, the op amp PA07 has a very small
25
output impedance(should be smaller than 1 Ω). The nominal impedance for
the primary of the step-up transformer is 5 Ω, the output impedance at the
1:13.71 ratio secondary should be less than 1 kΩ and this is very small com-
paring with either the ratio transformer’s input impedance (see [41]), or the
cell and references’ input impedance at our experiment frequencies. Therefore,
the bridge should not affect the source output much.
For phase shift relative to the oscillator phase, ideally, we like the output
of the step-up transformer to be in-phase with the lock-in internal oscillator
(i.e. tap 1 of the ratio transformer in phase, tap 3 out phase by 180o, as
the central tap 2 is grounded, see Figure 2.6). But generally this is not the
case as the source circuit will cause a phase shift relative to internal oscillator.
Since we usually run below 1 kHz, the unity gain buffer won’t cause too much
phase shift; the shift will mostly be due to the step-up transformer. In the
transformer’s data sheet[40], they provided the phase response curve covering
all working frequencies. But we use it reversely, we have to measure it directly.
This is done by treating the whole source as a black box and measuring at
the output. We measured at step-up ratio 13.71, set oscillator to be 0.1 V(the
maximum range of the lock-in is 1 V which has a 3 V limit, so we cannot use any
voltage too much higher.) and measure the output of the step-up transformer
directly with the lock-in amplifier(with different output filter time constants
for different frequencies). Figure 2.7 shows the output voltage and phase shift
against frequencies for Setup 1. For Setup 2, since we only run at 25 Hz, only
this one point is measured, which is 2.51o.
Next, we turn our attention to the bridge itself. We will follow the analysis
in [34]. But in most cases, our bridge is much simplier than that in [34] as our
cell and reference have much larger impedance, therefore, it drains very little
current from the source and the ratio transformer. Also, the wire resistance is
negligible. Although usually stray capacitance is not too much smaller than
the cell capacitance, we do not really need to know the absolute values of the
cell capacitance very precisely. Therefore, it does not do too much harm. As
26
Frequency(Hz)10 100 1000 10000
-4.0
0.0
4.0
8.0
phas
e sh
ift(o
)
freq_phase_all.data
1.36
1.40
1.44S
tep-
up o
utpu
t V
olta
ge(V
)
400 500 600 700 800-0.1
0.0
0.1
Frequency(Hz)
phas
e sh
ift(o
)
(a)
(b)
Figure 2.7: Response curves for driving source of Setup 1 at ratio 1:13.71.Input voltage is 0.1 Vrms. up pointing triangles: frequency ramps up; downpointing triangles: frequency ramps down. (a) amplitude response. (b) phaseshift relative to lock-in internal oscillator. Inset is a blow up around 600 Hzwhere the shift is close to zero.
27
Figure 2.8: A simple approximation of the bridge (Figure 2.6) similar toFigure3-12 in [34]. Note that the actual ground is not as shown here.
stated before, the driving source’s output impedance is much smaller than the
bridge’s input impedance, therefore, we lump the driving source as an ideal
voltage source with output impedance Zc = 0. For the above reasons, we can
simplify the bridge into the circuit shown in Figure 2.8, similar to Figure 3-12
in [34].
Here we use the lock-in to measure current. The input impedance of the
lock-in current preamplifier is very small (< 250 Ω at 1 kHz for wide band-
width mode, see [44], much smaller than the cell and reference impedance.),
therefore, it is a good approximation to treat the potential at the two ends of
the lock-in to be the same. This greatly simplify the analysis of the circuit
28
and most importantly, keep the driving voltage applied onto the cell constant
disregarding the cell impedance changes2. We first balance the bridge by ad-
justing both the ratio R of the ratio transformer and the reference resistance
Rr to let ILI to be zero, assume the current through the cell is I0, we have
I0 = (1 −R)VIN/Z0 = Ir = RVIN/Zr (2.2)
where the balancing cell impedance Z0 is an effective resistance R0 in parallel
with an effective capacitance C0
Z0 = 11
R0
+iωC0
= R0
1+iωR0C0
= 1G0+iωC0
(2.3)
where G0 = 1/R0 is the electrical conductance, and reference impedance Zr =Rr
1+iωRrCr= 1
Gr+iωCrFrom Eq. 2.2, we have the balance conditions:
R0 = 1−RR Rr (G0 = R
1−RGr); C0 = R1−RCr (2.4)
When the cell’s resistance (conductance) changes to R = R0 + ∆R(G = G0 +
∆G) and capacitance changes to C = C0 + ∆C, the current flowing through
the lock-in ILI becomes
ILI = I − I0 = (1 −R)VIN (∆G + iω∆C) (2.5)
2Initially, we used voltage mode of the lock-in just like most AC bridges do. Since we usethe imbalancing signal to measure the conductivity and capacitance of the cell, the inputimpedance of the lock-in came in play as it is of the same order (10 MΩ for FET input[44])as that of the cell, and this was solved by carefully characterizing the impedance. Then thedriving voltage on the cell will be changing as the impedance of the cell changes, this wassolved by first calculating the voltage drop over the cell, then using a feedback loop to adjustthe oscillator output voltage to set the cell driving voltage constant. Another problem camein that the oscillator voltage can only be set in discrete steps which is 0.5 mV and 2 mVin our working voltages, and they become 6.8 mV and 27 mV after passing the step-uptransformer at ratio 1:13.71. This is too much for us, and it sometimes caused control loopunstable. This was also solved by adding a smoothing circuit similar to Figure 3.9 in [7]to divide the 0.5 mV (or 2 mV) step into 4096 smaller steps. It took a lot of effort andeventually it was working. But using current mode circumvents all these problems andmakes it much easier to operate and analyze.
29
We therefore can monitor the change of the cell’s conductance and capacitance
from the real and imaginary part of ILI
∆G = ℜ(ILI )(1−R)VIN
; ∆C = ℑ(ILI)(1−R)ωVIN
(2.6)
Then the cell’s conductivity σ and dielectric constant ǫ can be calculated
σ = GdA
; ǫ = Cdǫ0A
(2.7)
where A and d are the cell’s electrode area and thickness respectively.
An example of cell conductivity and dielectric constant time series plot can
be found in Sect. 2.2.
Note that unlike using the lock-in in a voltage mode where the cell resis-
tance (as well as capacitance) comes into both in-phase(real part) and out-
phase(imaginary part) of the lock-in readings, here we have a complete sep-
aration of resistive and capacitive components. This is another advantage of
using the current mode. But for most frequencies, the VIN in Figure 2.8 is
not exactly in-phase with lock-in internal oscillator due to the driving source
phase shift mentioned above. This can be fixed by setting the reference angle
of the lock-in at the angle measured above, or following the following easy
procedure: first balance the bridge, then vary the reference resistor by a small
amount to give an imbalancing signal, then use the lock-in’s built-in “Auto
Phase” function to bring the out-phase component to zero, then rebalance the
bridge.
After being able to measure the cell conductivity in real time, we then are
able to control its conductivity by changing the cell temperature. This is done
in similar way as temperature control in 2.1.2: instead of using a themistor
reading as control variable, we now use the lock-in in-phase reading. We need
to know roughly the coefficient of conductivity change versus temperature.
This is done by manually testing a few coefficients.
30
2.2 Liquid Crystals
Two NLC are used in all experiments, I52 and Merck Phase IV(N4)[49]. I52
has been described in [7] in detail. We are mostly just follow the steps there.
N4 is a eutectic mixture of two forms of p-Methoxy-p-n-ButylAzoxyBenzene,
namely CH3O-C6H4-NO=N-C6H4-C4H9 and CH3OC6H4-N=NO-C6H4-C4H9.
It has a clearing point of 76oC. At 20C its dielectric anisotropy and optical
anisotropy are ǫa = −0.2 and na = n‖ − n⊥ = 0.28 respectively[49]. In this
dissertation, some of the properties of N4 are taken from [50, 51] and references
therein.
The main issue is to dope the NLCs to let them have proper conductivi-
ties. In the following, we will discuss some techniques we use for I52 and N4
sepeartely.
2.2.1 N4
For N4, it is relatively easy. Following [50], we dope it with tetra butylam-
monium bromide [TBAB, (C4H9)4NBr]. The doping process works like the
following:
1. Choose a few small glass bottles with tight caps and a small Teflon coated
magnetic stirring bar. Ultrasound them in acetone for 3 minutes, then
ultrasound in Isopropanol for 3 minutes, then completely rinse inside of
the bottles and the bar with DI water. Blow dry. Put the bar in one
bottle. And let them stay in 110C oven for one hour.
2. Put the bottle with the stirring bar on a balance of precision of 0.1mg
and zero the balance, put TBAB small crystals one by one until it weighs
0.0021g.
3. Fill in N4 NLC, until it measures 2.1604g, when it is close to the weight
one needs, fill N4 drop by drop.
31
Table 2.1: Properties of N4 of different dopant concentrations at 30C. CellSB01 was filled and measured by S. Bowers.
Bott. Conc. Date Date Cell Date 10−6σ⊥ ǫ⊥ note
No. (ppm) doped filled name measured (Ω · m)−1
0 970 4/11/02 7/9/02 SB01 7/17/02 1.02 5.08 1000Hz
1 293 4/18/02 10/10/02 XX85 6/3/03 0.434 5.07 500Hz
2 81 4/18/02 10/10/02 XX88 5/30/03 0.240 5.05 500Hz
3 28 4/22/02 10/10/02 XX89 N/A N/A N/A not
measured
4 14 4/22/02 10/10/02 XX92 11/14/02 0.0966 5.03 500Hz
5 3.8 4/29/02 10/10/02 XX93 11/7/02 0.019 5.02 150Hz
6 1.2 4/29/02 10/10/02 XX95 11/04/02 0.015 4.99 200Hz
-1 0 N/A 3/19/02 XX49 4/11/02 < 0.001 ∼ 5.0 rough
estimate
at 22C
4. put the solution on a stirring plate, stir for a few days, but do not
heat[50]. We then have a 970 ppm (by weight) solution.
5. put another clean bottle on the balance, zero it. Fill in desired amount
of pure N4, then the 970ppm solution. Let it stay for days before use.
6. repeat the last step to get even more diluted solutions. The original
solution can be the 970ppm solution or the solution made in the last
step depending on which one is easier.
Solutions up to 0.3%(by weight) concentration were tested, and TBAB was
able to completely dissolve in N4, but its conducitivity is too large(more than
2 × 10−5(Ω · m)−1), and the resistive heating created many small bubbles in
the cell. Therefore, we did not try anymore on concentrations higher than
970ppm.
Table 2.1 lists a few concentrations tested and measured conductivities and
dielectric constants at 30C with applied cell voltage 2.0V.
32
The conductivity of N4 is relatively stable.
2.2.2 I52
As noted in [7], many factors affect properties of doped I52 and it is an
art to dope it to get a conductivity of desired ranges. In our many trials,
we find that the following procedure usually gets to a conductivity in a more
controlled way:
1. Make a highly doped solution with I2. For example, our I52-XX1 bottle,
0.100g I2 was added into 0.9958 I52(I2/I52=0.10). The cleaning and
mixing procedures are the same as doping N4. Then the bottle and
the stirring plate are all put in an oven of 50 ∼ 55C (we do not use
stirring hot plate as the heat is not uniform and the solution temperature
is hard to tell). Stir the solution for a few days. The solution will
become very dark. We initially intended to use this solution to get
a big enough conductivity to last a long time. We indeed got a big
conductivity, around 3 × 10−8(Ω · m)−1. However, we could not observe
any convection. We also notice a possible change of its melting point.
We kept a few doped I52 solution bottles in a fridge and noticed that
those lightly doped froze and those highly doped stayed in a dark-colored
fluid like state. The exact mechanism of this change is unknown. One
guess is that some chemical reactions happened during the heating and
stirring process3, and the resulting product(s) may not be a NLC at our
measuring temperature (around 50C), although it is highly conductive
compared to the original NLC.
2. Use the dark-colored solution produced in the last step to increase con-
ductivity of a solution without enough conductivity. In our case, we used
this solution to charge our I52-XX10 solution(I2:I52=0.017, total mass is
3It seems stirring is affecting the color of the solution. Without stirring, even a highlydoped solution only get a pink color, the color then fades away in a year or so.
33
0.604g)4. We added 0.2801g I52-XX1 (I52-XX1:I52-XX10 ∼ 0.46 5). A
trick in estimating the mass the solution added is to use a 28 gauge nee-
dle to add drop by drop and one drop is about 7mg and its fair constant
during a couple of trials.
3. The solution is then stirred in the same condition as step 1 for one day
and put in a fridge to keep I2 from evaporating.
The solution prepared by the above procedure initially had a conductivity
of 1.6 × 10−8(Ω · m)−1. It then drifted nearly exponentially, see Fig. 2.9. In
about 9 days, it only had less than half of the initial value. It was expected
to settle down at 0.7 × 10−8(Ω · m)−1
2.3 Cell Construction
All our EC cells consist of two glass plates separated by a spacer and
sealed on all four sides after filling in the LCs. The steps are (as in [7])
glass preparation(including electrode process and patterning if applicable),
alignment, sealing and filling. For local detecting electrode cells, most of the
effort was spent in process and patterning the electrodes and we will discuss
this separately in Sect. 6.2. For all the cells, alignment is critical and we
explored a couple of methods and I will focus on photo-alignment using Rolic’s
[63] linearly photopolymerizable polymer (LPP) technology[61, 62].
In our experiments, we also used commercial LC cells from E.H.C. Japan
[52]. The structures of these cells are similar to those made in our lab.
4We chose to recharge old solutions which did not work just for convenience. Presumably,using pure I52 should work as well.
5It should be slightly bigger than this, as we took out a few drops to test during theprocess.
34
Figure 2.9: Conductivity and capacitance of Cell X801 1 filled with I52 at50C. Applied voltage is 11.9V, 25Hz, around onset of EHC. (a): perpendic-ular conductivity σ⊥ vs. time, the line is a fit to σ⊥ = (6.94 × 10−9 + 9.58 ×10−9 exp(−t/2.52 × 105))(Ω · m)−1. (b): perpendicular capacitance C⊥, theline is a fit to C⊥ = (720.8 + 4.64 exp(−t/1.97 × 105))pF
35
2.3.1 Glass Preparation
All the glass slides we used are coated with indium-tin oxide (ITO) at one
side. Initially, we used those described in [7] and they are 2.5 mm thick soda-
lime glass free samples from Libbey-Owen. But we later switched to 1.1 mm
thick glass from Delta Tech. Ltd. [53] exclusively. They have several choices
in glass types, sizes and ITO resistivity. We have tried both polished float
(soda-lime) and Corning 1737 aluminosilicate glass with nominal ITO coating
thickness 120∼160 nm( sheet resistance Rs = 8 ∼ 12 Ω for float glass CG-51IN,
and 5 ∼ 15 Ω for 1737 glass CB-50IN.). We prefer 1737 as it has a smaller
coefficient of thermal expansion(CTE) (37.6× 10−7/C vs. 77 ∼ 85× 10−7/C
for float glass[54]), therefore can stand thermal treatment better. They have
different sizes to choose from. Initially, we chose 25 mm × 25 mm size as all
the cells are made of slides of this size. Later on, we chose 14′′ × 14′′ size and
cut it into 53mm × 53mm pieces. The reasons for this are that this makes it
much cheaper than buying smaller size, and we can process four 25 mm×25
mm areas each time and then cut it into 4 pieces. To cut the glass, we can
use a glass cutter with a cutting wheel, but this only gives us a very crude
size, and makes it is harder for assembling two slides into one cell. For precise
cutting, we use a Disco dicing saw in the UCSB cleanroom[55].
One must wear a pair of gloves for the following operations. The gloves
should be dust free.
We first clean the 53 mm × 53 mm glass slides. We basically follow the
procedure in [7] with slight modifications.
1. Put the slides in a wafer basket[57] made of PFA from Entegris Inc.[56].
The basket can hold up to 12 slides, but we usually do not put so many
due to the limitation of beaker diameter, and beaker diameter is in turn
limited by the size of the ultrasonic bath we have.
36
2. Dip the basket together with the slide into home liquid detergent solution
for a few hours to overnight, rinse with tap water, then deionized(DI)
water. This step is optional.
3. Ultrasound the slides in Liquinox and ammonium hydroxide solution(350
ml DI water, 50 ml ammonium hydroxide and 40 ml Liquinox) for 15 ∼30 min.
4. Rinse with DI water.
5. The above steps are in our own labs. The following steps sometimes
are in our own labs, sometimes in cleanroom depending on what steps
follow this clean procedure. If next operate in cleanroom, blow dry with
nitrogen gas and store each slide in a 4′′ (H22-40) or 3′′ (H22-35) wafer
shipper[58].
6. Ultrasound for 20 min in acetone(certified ACS grade). Do 3 min if in
cleanroom.
7. Ultrasound for 20 min in Isopropanol(certified ACS grade). Do 3 min if
in cleanroom.
8. Rinse with DI water.
9. Blow dry with nitrogen gas
10. Dry the slides on 110C hotplate for 2 min.
Each time we begin a new process on the slides, we need to clean the slides
with acetone and isopropanol following the above step 6 to step 10.
Next, we need to evaporate insulating layer(s) onto the ITO side. [7]
described three types of evaporated layers: uniform insulating layer to form a
non-injecting boundary condition for EC, ”beaches” used to confine convecting
region and alignment layer. We do not use evaporation to align LC molecules,
37
so we never used the alignment layer. And we seldom use the beaches. If they
are needed, one can refer to [7] for the details. Unlike the ITO coated glass
used in [7], the glass we used does not have an insulating layer initially, we
therefore need to apply one by ourselves. One can use the method in [7] to
evaporate a layer of SiO. An easier way is to used plasma enhanced chemical
vapor deposition(PECVD)[141] to grow a layer of SiO2. A PECVD apparatus
is available at the UCSB cleanroom[55] and it takes much less time to deposit
the film. The deposition is done at 250C and the rate of growth is about 40
nm/min. For this insulating layer, we usually grow 400 nm and the total time
for this process is about 20 min and it is done automatically. The PECVD
chamber can hold up to four 53 mm × 53 mm slides. After the deposition,
the thickness of the resulting SiO2 layer can be checked by an F20 Filmetrics
Thin-Film Measurement System[59] also available at the UCSB cleanroom[55].
And the measurements show that the thickness is fairly close to 400 nm.
If needed, we can also used the PECVD to evaporate “beaches”. For this,
we have to use a different scheme from that in [7], i.e. first grow a thicker
layer of SiO2, then apply the beach pattern[7] either by photolithography,
or painting photoresist directly(the pattern is big, paint directly should be
doable), then controlled etch away the a predetermined thickness in the region
not covered. The etch rate for PECVD grown SiO2 is 520 nm/min in buffered
HF solution[60].
2.3.2 Alignment
We only used planar alignment in our experiments. In [7], it had been
shown that rubbing polyimide film worked for aligning I52. We tested and
found that it worked for N4 as well. Oblique-angle evaporation of SiO was
briefly tested for N4 and it was not successful, and it was not carried on any
further. For a long time, we have stuck to rubbing polyimide which is also
standard method of alignment for LCD industry.
38
Shu tter UG5 filter WG280 +
HNP ’B +
WG280
SampleUV lamp
Figure 2.10: Schematic of UV exposure system
But this alignment is dirty and unpredictable sometimes, makes the result-
ing EC cell nonuniform, this is especially a serious problem for our experiment
with I52 at relatively low-conductivity where it was found that the pattern
tends to start locally. For our purpose of study STC in I52 convection, it is
desirable to have a big area of uniform convection and we found it is extremely
difficult to achieve this using the previous methods. We then turned our ef-
fort to a relatively new method: photo-alignment using Rolic’s [63] linearly
photopolymerizable polymer(LPP) technology[61, 62].
Photo-alignment works by using LPP materials in which an anisotropy can
be induced when illuminated with polarized UV light. The anisotropy in turn
aligns the liquid crystal in contact. The LPP material we used is polymer
solution ROP 203/2 CP[64] from Rolic and a detailed technical data sheet on
the material and how to use it can be found in [65]. Below I will describe the
detailed setup and operations here at UCSB.
The setup consists of a UV mercury lamp, a few filters, a UV polarizer, a
shutter apparatus to control exposure time and a sample holder(Fig. 2.10.
The LPP is sensitive to 280 ∼ 330nm UV light[65]. Therefore, we need
to use a UV light source which provides enough light in that range. We
were recommended to use a doped medium pressure mercury lamp[66]. But
39
250 300 350 400 450 5000
500
1000
1500
2000
2500
3000
3500
4000
Wavelength(nm)
Inte
nsity
(a. u
.)
Figure 2.11: Spectrum of an MB-100 UV lamp without black glass filter.The arrows show the peaks useful to us.
that is rather expensive. We have tested an MB-100 UV lamp(without its
original black glass filter) from Spectronics Corporation[67], and an MJB 3
UV400 mask aligner in the UCSB cleanroom[55]. With proper filters, they
work pretty well. Fig. 2.11 and Fig. 2.12 are the spectra of the MB-100 lamp
and MJB 3 UV400 aligner. We see that the strongest peaks are i-line (365nm)
and other visible peaks which we do not want. But we do have some power
at 313nm and 335nm. We use a UG5[68] black glass to filter out most visible
light.
The UV polarizer is HNP’B type from 3M[70]. It is a plastic polarizer,
rather delicate and expensive, so we sandwiched it between two WG280[69]
filters to protect it from scratch and overexposure.
Two setups were tested using the MB-100 lamp and the MJB3 UV400
aligner respectively. The one using the MB-100 lamp was made in our lab.
A 8” long metal tube houses the components in Fig. 2.10. It keeps UV light
40
Figure 2.12: Spectrum of an MJB 3 UV400 mask aligner in the UCSB clean-room, with the permission of Brian Thibeault.
from coming into the lab. The tube should be long enough to let the UV
polarizer far from the lamp bulb as it can get very hot in less than 3 minutes
of exposure to melt the plastic polarizer. The shutter is a simple movable
aluminum plate and the exposure is controlled manually. Using the aligner
makes the operation much easier. It has a build-in exposure control system
to control UV light intensity and exposure time. In this case, the WG280
filters and the polarizer are fixed in an aluminum plate of the size of a photo
lithography mask and clamped onto the aligner like a mask and the UG5 filter
is put into the light path after finishing alignment(because it is dark, one
cannot make alignment with it in the optical path).
With enough exposure, the LPP gives a planar alignment with zero tilt
angle[65] which is exactly we need. We are looking for an exposure energy of
50 ∼ 200mJ/cm2[65]. Too much more than this amount may possibly damage
the LPP layer. We do not have the ability to directly measure the absolute
41
light intensity in the 280 ∼ 330nm range. But since the exposure energy range
is rather wide, we can roughly estimate that from their spectra in Fig. 2.11
and Fig. 2.12. For the aligner, we know that the 405 nm line is controlled
to be 7.5mW/cm2[71]. By measuring the areas (subtract the constant part)
covered by 298, 302,313, 335 and 405nm peaks, we estimate that the usable
light power (below 340nm) is roughly 5.5mW/cm2. Then after passing each
filter and polarizer, there is some energy loss, at the useful range, UG5 and
WG280 can pass about 90% of the incoming light, the UV polarizer passes
20%. So the final polarized light intensity is about 0.8mW/cm2. We therefore
need to expose 60 ∼ 250 seconds. We fix this time to be 135 seconds in
all our operations. For the home-made system, the absolute light intensity
is unknown, but it is quite strong, testing showed that using 2 minutes was
enough.
The exact procedure for alignment is as following:
1. Clean glass slide: ultrasound in acetone and isopropanol for 3 minutes
each, rinse with DI water and blow dry.
2. Dehydrate the slides on a 130C hotplate for 1.5 minutes, cool for a few
minutes before the next step.
3. Spin-coat LPP solution at 3000rpm for 60 seconds.
4. Dry the slides on a 130C hotplate for 10 minutes to remove residual
solvent.
5. Put one slide in the exposure apparatus, align the desired director direc-
tion to be parallel to the polarization direction. Slide in the UG5 black
glass if using an aligner.
6. Expose for designated time (135 seconds for aligner, about 2 minutes for
using MB-100 lamp).
42
After the above steps, we can go on to assemble the finished slides into
cells. The LPP solution is stored in a refridgerator between 2C and 18C in
the original bottle. No light should be shined on the solution.
We have tested this method for I52 and N4 and they all work very well.
The patterns near onset in these cells are much more uniform than any other
cells made by other methods tested by us.
2.3.3 Sealing and filling
The sealing and filling procedures mostly are similar to those in [7] and
most cases, simpler than those.
We used two methods to make spacers. First one is that describe in
[7], cutting spacers out of some thin plastic films. Two plastic films from
Goodfellow[72] are used, mylar and polyvinylfluoride (PVF). The film was cut
into many small pieces of about 7mm × 50mm6, washed with DI water and
stored in DI water for future use.
We can use the above method to make cells of most thicknesses we needed,
from a few microns to 100 microns. But when the film is very thin(< 10µm),
it is hard to handle. Also, it was believed that absorption of dopant by spacers
is one of the most important reasons for cell conductance drift[7], and in [7],
Dennin found that glass did not absorb I2 significantly. An alternative way
of making spacers which we just briefly tested is by depositing SiO2. This is
done before we make the alignment layer. We first cover the region of slides
where NLC will be filled in with high temperature vacuum grease and leave the
spacer region uncovered. Then we deposit SiO2 to designated thickness using
PECVD or Unaxis VLR Etch and Deposition tool in the UCSB cleanroom[55]
at 100C7. Then the deposited slides are soaked in toluene for a few hours
6narrower ones were tried, but it was found that cells with wider ones had better thicknessuniformity.
7It is desirable to use a vacuum grease which works well at 250C since deposition at250C is better and one does not have to change the default temperature which takes verylong time, but we could not find one.
43
(ultrasound them if needed) to take away the vacuum grease. Then the slides
are throughly cleaned to go onto alignment layer step. I tested this to make
4 µm spacers and it works fairly well. But it probably will not work for thick
spacers as the stress build-up in depositing thick film will cause the film to
break and fall off. Also, due to the film growth rated limitation(about 40
nm/min for PECVD and less for the Unaxis tool), it will cost too much time.
Fig. 2.13 shows the assembling process. Two short lines denote two equiv-
alent positions on the two slides (inscribed on the nonconducting sides for as-
sembly identification). In photo-alignment process, these two slides are jointed
as in Fig. 2.13 and then broken along the middle line, or in rubbing process,
these two slides are holding on the same direction relative to these two short
lines. The directions of the two short lines are the polarization direction or
rubbing direction. If using plastic film spacers, blow dry two spacers, put them
along the two edges parallel to the short lines. Flip the other slide to let the
conducting side face down, align it with the first slide, leave about 5 mm not
overlapping for attaching wires. Use two paper clips to clamp them together.
Leave them in 110C oven overnight. This is to get rid of the water absorbed
by the spacers, also it was found that heated cells are slightly thinner than
not heated ones. For instance, using 23 µm mylar spacers, heated cells usually
measure about 20 µm thickness, while non-heated ones are about 26 ∼ 27µm.
Presumably, a thinner cell shows that spacers are well in contact with slides
and the final cell will be sturdier.
Then with the clips still on, the two edges parallel to director are sealed
with Torr Seal[73]. Heat at 60C for two hours. Take off the two clips and heat
another couple of hours. Visually check the interference fringes under room
light to have a general idea of the thickness variation across the whole cell.
Usually at the center of the cell, there should be very few fringes. Measure
the empty cell thicknesses at a few positions of the cell with an F20 Filmetrics
Thin-Film Measurement System[59].
44
Torr seal l
spacer
Figure 2.13: Schematic of first few steps of the assembling process.
Filling is done by putting a few drops of NLC at one of two edges not
sealed. The NLC will be sucked in by capillary force rather quickly. Since the
opening is big, one usually does not have any bubbles at initial filling. After
sealing and in use, sometimes some bubbles appear. As long as they are not
in regions of interest, the cell is still usable. If there are no bubbles or the
bubbles do not affect the observed areas, there is no need to put the cell in
vacuum.
After the cell is filled, use a clean room wipe to wipe clean the remaining
NLC. Seal it with Torr seal. Leave in 50C oven for 2 hours to let it cure. Clean
the whole cell with a cleanroom wipe and acetone to remove any remaining
NLC residue.
The remaining task is to attach wires. Since we deposited a layer of in-
sulating SiO2, we need first break a small area of the film using a scriber to
expose the conducting ITO layer. Put a little indium on the exposed area,
press with a hard object and indium will stick on the surface. Put a stripped
30 gauge single-strand wire on the flattened indium, put a little more indium
on the wire, press and the wire will stick onto the glass. If needed, one or
more extra joints can be make by repeating the above procedure. Finally,
apply Torr seal to cover all the indium joints to protect them and let it cure.
This finishes making a cell.
45
We can inspected the alignment by putting the cell in between a pair of
cross polarizers during and after the filling process. The cell acts like a (not
very good) polarizer and it allows light to pass through when the director is
45o to the other polarization directions. Alignment defects can be views at
some positions. This also gives us a general idea how uniform the cell is.
Newly made cells usually experience a fast conductivity drop during the
first ten days or so(see Fig. 2.9), possibly due to the absorption of ions by
all the surfaces. And conductivity changes are much more in I52 cells than
N4 cells. After initial fast drop, the conductivity mostly will still drop slowly,
and depending on its applied voltages history, it may increase as well. If time
allows, before we make any measurements, we can store the cell for some days,
apply a voltage or not.
46
Part II
Spatiotemporal Chaos(STC)
Right Above Onset in I52
47
Chapter 3
Introduction: Spatio-temporalchaotic state right above apattern forming transition
Chaos usually denotes persistent temporal irregular behavior of a low-
dimensional deterministic system, and it has been extensively studied in the
past few decades and methods have been developed for analyzing chaotic be-
haviors, such as measuring Lyapunov exponents and fractal dimensions of
strange attractors[74].
When we talk about spatially extended deterministic systems, we can
extend the concept of chaos to those states which show irregular behaviors
in both spatial and temporal domains and call them spatial-temporal chaos
(STC)[75, 1]. One of best known examples of STC in controlled experimental
conditions is the spiral defect chaos[76] state in small Prandtl number RBC.
The fascinating thing about STC is that it appears so often in nature, and
yet it is very hard to explain quantitatively. To study it, as a first step, one
natural way is to extend the successful methods of temporal chaos. First, in
a small system (for classifying small system and large system, see[1], roughly
speaking, the correlation length of the system of order of the system size),
it is expected that some of the methods in temporal chaos may apply. And
there has been a lot of work on this aspect[1]. For example, we can calculated
48
the Lyapunov exponents of the system, generalized dimensions of strange at-
tractors in the system. For large systems, there have been attempts to apply
similar ideas in temporal chaos studies, a recent example is the application of
Lyapunov spectral method in spiral defect chaos[78]. Another direction is to
consider the case where the system size is much larger than the correlation
length of the system, and it is expected, some statistical description should
apply, an example of this kind work is shown in [77].
One of the difficulties in studying STC is that most of it occur as a result of
a transition (or bifurcation) from a base state that already has an intricate but
nonchaotic spatial and/or temporal behavior. Therefore, our most systematic
theoretical tool, expanding around a simple base state by one (or more) small
parameter(s) will not work.
Fortunately, there have been a few cases where STC evolves directly from
the uniform state as the system is driven from the first bifurcation point. One
of the most studied cases is the domain chaos[79, 80, 81, 82, 83] state for ro-
tating RBC after undergoing Kuppurts-Lortz (KL) instability. KL [4] showed
that when the rotation rate is greater than some critical value, a straight set of
roll pattern will lose its stability to another set of rolls at some angle θc relative
to the original set. The new set will in turn lose its stability to another set
at a further angle of θc, and so so. So that in this system, there is no stable
steady pattern. A few other systems have similar mechanism to destabilize
the original roll pattern. In RBC with free-free boundaries, for sufficiently
low Prandtl numbers σ < 0.543, straight rolls are unstable with respect to
the skewed-varicose instability immediately above onset [84, 85, 86, 87]. In
homeotropically aligned NLC EC, the presence of an additional Goldstone
mode destabilizes the original rolls[89] . In homeotropically aligned NLC with
σa < 0 and ǫa > 0, it was predicted[92] that there was a direct transition
to a stationary pattern that is destabilized by zig-zag instability leading to
disorder or to an ordered quasiperiodic pattern. And finally, in our STC state
in our plannar EC in NLC I52[14], isolated zig and zag rolls are unstable with
49
respect to a superposition to traveling rectangles[93], and this leads to a direct
transition to STC state.
In the above systems, RBC with free-free boundaries with small Prandtl
numbers is hard to realize in a lab experiment, there is only a simulation
result[88]. In the case for homeotropically aligned NLC with σa < 0 and
ǫa > 0, no definite result on STC has been reported[92]. There has been
experimental report on homeotropically aligned NLC EC with MBBA[90, 91].
In this dissertation, we will discuss and compare the two cases which we have
studied in our group, the KL domain chaos and STC in plannar EC in I52.
In the KL domain chaos state, the system has a rotational symmetry and
therefore, there are infinite modes in this continuous system. But based on that
θc is close to 2π/3, Busse and Heikes[79] first considered three couple dynamical
equations for three amplitudes of three modes with relative orientation 2π/3
to each other. Tu and Cross[80] introduced spatial dependence in the three
amplitudes and wrote down three coupled amplitude equations.
1τ0
∂tA1 = ξ20∂
2x1
A1 + A1(ε − gA21 − g−A2
2 − g+A23), (3.1)
1τ0
∂tA2 = ξ20∂
2x2
A2 + A2(ε − gA22 − g−A2
3 − g+A21), (3.2)
1τ0
∂tA3 = ξ20∂
2x3
A3 + A3(ε − gA23 − g−A2
1 − g+A22). (3.3)
where A1, A2 and A3 are the amplitudes of the three modes and assumed to
be real, ε = RRc
− 1 is the reduced control parameter. Due to the rotation and
therefore lack of reflection symmetry, the coupling constants g− and g+ are
generally not equal. Making scaling transformation t → τ0t/ε, xi → ξ0xi/√
ε,
Ai →√
εAi, we have the scaled equations
∂tA1 = ∂2x1
A1 + A1(1 − gA21 − g−A2
2 − g+A23), (3.4)
and equations for A2 and A3.
With this simplified model, Tu and Cross[80] could simulate the domain
chaotic behavior.
50
With the three scaled coupled Ginzburg-Landau (GL) type equations, the
simplest predictions is that the correlation length ξ ∝ ε−1/2 and correlation
time τ ∝ ε−1. Their further simulation using generalized Swift-Hohenberg
(SH) equations for this rotating convection problem and the predictions from
SH are consistent with the GL equation approach, although the simulation
data seem to be insufficient to infer precise exponents.
Hu et al[82] experimentally measured the correlation length ξ and switching
frequency ωa of the domain dynamics. Surprisingly, the experimental data
could be fit by power laws in ε only when exponent values much smaller than
those predicted from the amplitude equations were used. In [82], they got
approximately 0.17 for the exponent of ξ and 0.6 for the exponent of τ .
Here we have a case of clear contradiction between theoretical models and
experimental data, for the simplest possible things in a STC state, the cor-
relation length and correlation time. This is quite puzzling. Further efforts
were taken in both experimental and simulation aspects. One consideration in
experimental system is the effect of system sidewall, especially the injection of
defects from the sidewall was considered. Cells with ramped sidewall[94, 95]
were used to eliminate this effect and this does not seem make the experimental
results different too much[96]. In simulation aspect, Cross et al [83] measured
correlation length ξ for cell of different aspect ratio Γ and they suggested that
the discrepancy between experiment and theory might be due to the finite size
of the experimental system.
Vigorous experimental and numerical investigation on the correlation length
and time scaling is still going on in this group[97]. While we keep our effort on
this system, it is very helpful to look at a different system of with completely
different symmetry while still having a direct transition to STC. And STC in
plannar EC in I52 is good choice for this comparison.
In [14], Dennin et al described this state and they pointed out that this
STC state happens right above onset and that it is a supercritical bifurcation.
They also identified the region where this state appears[15, 7] (EC1 region
51
in Figure 5 in [15], for their I52 cell of thickness of 28 µm, it appeared at
conductivity around 1 × 10−8 (Ω m)−1). At this conductivity, right above
onset, it enters EC1 region. At about ε = 0.055, it enters SO1 region at which
stationary pattern appears.
The weakly nonlinear analysis has been carried out for EC based on WEM
(see Chap. 1) by Treiber and Kramer[93, 8]. The details of the coupled complex
GL equation are rather complicate. We can write down the GL equation
symbolically without knowing all the nonlinear coupling constants. There are
four modes in our system, right-traveling zig, left-traveling zig, right-traveling
zag and left-traveling zag rolls. Their amplitudes are A1, A2, A3 and A4
respectively.
τ0∂tA1 = εA1 + ξ20∇2A1 − gA1(|A1|2 + b|A2|2 + c|A3|2 + d|A4|2) (3.5)
+ three other equations for A2, A3, A4.
where ε = V 2/V 2c −1 is the reduced control parameter. We can make the same
transformation as for Eq. 3.1 ∼ 3.3 and reach a form without ε, therefore, in
these approach, we can predict that we have similar scaling relationships as
for Tu-Cross model: the correlation length ξ ∝ ε−1/2 and correlation time
τ ∝ ε−1.
In the next chapter, we will discuss our experimental setup and results of
our measurement of the correlation length ξ to compare with this prediction
and the results from KL domain chaos measurements.
52
Chapter 4
Length Scales in STC Phase
4.1 Experimental Details
All the experiments on STC are done using cells filled with NLC I52. We
have discussed the NLC (Sect. 2.2) and the making of the cells (Sect. 2.3).
A few batches of cells were tested. Since the most critical step in mak-
ing cells is the alignment, we tested both rubbing polyimide (PI) and photo-
alignment using linearly photopolymerizable polymer technology (see Sect. 2.3.2).
It was found that using photo-alignment gives much better uniformity than
using rubbing PI. Since all the considerations about GL equations and corre-
lation length in Chap. 3 are based on the assumption that the pattern forming
system has a translational symmetry, a uniform cell is strongly desirable. At a
small ε, we can take a visually uniform image over an area of 0.8 mm × 0.8 mm
for our best rubbing-PI cell and 3.4 mm × 3.4 mm for our best photo-alignment
cell X801 1 (the cells are similar in thickness 19 ∼ 26 µm). In the following,
we will only discuss the results obtained from cell X801 1 run 102804 (means
that it started on Oct. 28, 2004). The thickness of the cell is d = 19.4 µm. All
the images were taken at temperature of 55C, and at this temperature, the
conductivity is about 1.0 × 10−8 (Ω m)−1.
We use the microscope setup (setup 2, see Chap. 2), in order to observe
a big area, a 1X lens is used. To avoid any interference from ambient light,
53
a big black plastic trash can is used to cover the whole microscope when we
have a run for taking data.
We first visually estimate the onset of the cell. We start from a voltage
below onset, take 2048 images, wait for 1800 seconds, increase the voltage to
another point and take another 2048 images. We repeat this step until we
reach our largest set voltage. Then we decrease the voltage and do the same
thing until we reach one voltage below our onset point. In our cell X801 1,
the EC1 to SO1 transition happens at about ε = 0.075, we therefore choose
our largest voltage to let ε ≈ 0.05. Our image size is 512 pixel × 512 pixel
(we bin four neighboring pixels into one, therefore on the CCD array, the
image covers an area of 1024 pixel × 1024 pixel which is the largest square our
camera can take). In our image, one pixel distance corresponds to 6.65µm
physical distance in the cell which is 0.342d (CCD pixel physical distance is
6.7µm).
4.2 EC onset
First the onset of the convection is determined. We choose a low voltage
among our 50 voltage points, which is 9.740V, then average all the 2048 im-
ages and obtain an average image as our background image for division (see
Sect. 2.1.1) which is shown in Figure 4.11. We then process the images at differ-
ent voltage separately. First, divide all the images (an example in Figure 4.2)
by the background image and obtain the divided images (see Figure 4.3). A
FFT is then performed on the divided images (Figure. 4.4). Note that this
FFT image has all the harmonics and low wave-number noise near k = 0, all
these are filtered out in Figure 4.5 and we only leave four fundamental peaks.
In Figure 4.4, the left-traveling and right-traveling components are not sepa-
1All the images shown here have 256 gray levels, but in actual image processing, theimages have 4096 gray levels.
54
x
y
Figure 4.1: Average of 2048 images at ε ≈ −0.1 (V = 9.740 V) for run102804 of cell X801 1 as background image.
x
y
Figure 4.2: One raw image at V = 10.349 V of run 102804 of cell X801 1.
55
x
y
Figure 4.3: The raw image Figure 4.2 divided by the background imageFigure 4.1.
kx
ky
Figure 4.4: Fourier amplitude image of Figure 4.3.
56
kx
ky
Figure 4.5: Filtered Fourier amplitude image from Figure 4.4, only funda-mental peaks left.
57
95 100 105 1100
4.10-4
8.10-4
1.10-3
V2 (V2)
pow
er (
a. u
.)
Figure 4.6: Total fundamental power versus V 2 for run 102804 of cell X801 1.Up pointing triangles (color version: black): voltage going up; down pointingtriangles (red): voltage going down. Solid line along up pointing triangles(black), a fit to Eq. 4.1 for voltage-up data, with P0 = 1.41 × 10−6 (fixed),a = 0.040 and Vc = 10.266 V ; Solid line along down pointing triangles (red), afit to Eq. 4.1 for voltage-down data, with P0 = 1.41 × 10−6 (fixed), a = 0.041and Vc = 10.284 V .
rated, but the sum of the power of these peaks is proportional to the sum of
the power (square amplitude) of the four modes in Eq. 3.5.
The power of the fundamentals are fit to:
P =
4∑
i=1
|Ai|2 =
P0 if V < Vc
P0 + a(
V 2
V 2c− 1
)
if V ≥ Vc
(4.1)
where P0 is a small power due to noise below onset and its exact value does
not affect our determination of the onset too much.
Figure 4.6 shows the fit to points below and slight above onset. We can see
that above onset, the lines fit rather well. This confirms that the bifurcation
58
is supercritical. The up-points and down-points do not collapse because the
material properties drift. This is not hysteresis because that would mean that
the down points have a lower onset, but our measurements shows an increase
of the onset.
Since there is a drift in onset voltage and we need to know ε = V 2/V 2c − 1
accurately to infer an accurate scaling exponent, we should make a correction
to the drift. The simplest correction is to assume the onset voltage square
drifts linearly with time. We have the time when each point was taken, which
is listed along with numerical values of Figure 4.6 in Appendix A. The time
dependent onset voltage Vc(t) is given by
V 2c (t) = 105.39 + 1.49871 × 10−6 (t − 26838099) (t in seconds). (4.2)
4.3 Determination of correlation length
We need to determine the correlation lengths of the zig and zag modes.
Since there are both right-traveling and left traveling modes, strictly, we need
to separate them, but most times, only one zig and one zag mode dominate[98],
we only need to separate them into zig and zag modes and determine their
correlation lengths separately. Since Fourier peaks above onset have rather
narrow width which only cover a few pixels, it is not easy to determine the
correlation length from Fourier space. We will separate them in Fourier space
and then measure the correlation length in real space.
We use the same images for determining onset for this purpose. Image
division and Fourier transformation are the same as above. We use a different
Fourier filter to leave only one zig fundamental peak in Figure 4.7.2 The filter
2The way to take away the main oscillation in the pattern and get the amplitude: Assumethe zig pattern is described by
A(r) = A0(r) cos (k0 · r + φ) = 12A0(r)
(
eiφeik0·r + e−iφe−ik0·r)
, (4.3)
where A0(r) > 0 contains the small amplitude modulation. Suppose we can write A0 inFourier integral (we will use an integral form for connivence. For image processing, replace
59
is centered around the zig peak position and it tapers off at 0.1k0 and becomes
zero at 0.2k0. The zag peak is processed separately following the exact same
procedures and we will only discuss the processing of zig peak in the following.
The single zig peak is then back Fourier transformed and the amplitude of
the complex modulus is taken. As mentioned before, we have tried very hard to
make the cell uniform, but unfortunately, there is always some nonuniformity
present in our cell (over 3.4 mm × 3.4 mm which is a huge area). To minimize
the effect of this nonuniformity, we first average all the 2048 amplitude images
and get an average image, Figure 4.8, which should contain the nonuniformity
information of the zig pattern. This background image is then subtracted from
each individual amplitude image pixel by pixel and the zig amplitude image
for Figure 4.3 is shown in Figure 4.9.
all the integral by summations)
A0(r) =
∫ ∞
−∞
A0(k)eik·rd2k (4.4)
Since we have put the fast oscillation in the cosine function, A0(k) contains only slowvariations and is nonzero only for |k| ≪ |k0|. The Fourier transform of A(r) is
A(q) = 18π2
∫ ∞
−∞
d2r e−iq·r
∫ ∞
−∞
A0(k)(
eiφei(k+k0)·r + e−iφei(k−k0)·r)
d2k
= 12
(
eiφA0(q − k0) + e−iφA0(q + k0))
(4.5)
Since the Fourier amplitudes of our images are centered around q ≈ k0 and q ≈ −k0 , ifwe keep only positive q (qx > 0 and qy > 0, first quardent) and cut off the negative peak(leaving only negative peak will get the same result). We have
A′(q) = 12eiφA0(q − k0) (4.6)
Take the back Fourier transform,
A′(r) =
∫ ∞
−∞
A′(q)eiq·rd2q = 12eiφeik0·rA0(r) (4.7)
And|A′(r)| = 1
4 |A0(r)| (4.8)
Thus we get the amplitude of the zig pattern.
60
kx
ky
Figure 4.7: Filtered Fourier amplitude image from Figure 4.4 ( V =10.349 V , ε = 0.013), only a zig fundamental peak left.
Then autocorrelation of the amplitude field is taken by FFT algorithm and
normalized.
C(δx, δy) =ab
RR
(A(x,y)−A)(A(x+δx,y+δy)−A)dxdy
(a−|δx|)(b−|δy|)RR
(A(x,y)−A)2
dxdy(4.9)
where the integration in the numerator is over where A(x, y) and A(x+δx, y+
δy) are both defined, and the integration in the denominator is over the whole
region. And a and b are length and width of the integration area, in our case,
a = b = 512 pixel. A is the average value of A over the whole region. The factorab
(a−|δx|)(b−|δy|) is due to that the integration is confined in a finite region and
for different (δx, δy), the integration area is different. Figure 4.10 shows the
correlation image of zig pattern amplitude of the sample image in Figure 4.3,
and Figure 4.11 is the average of 2048 correlation images at ε = 0.013.
61
x
y
(a)
500
400
300
200
100
0
5004003002001000
0.02 0.018
0.017
0.017
0.016 0.016
0.015
0.015 0.015
0.015
0.015
0.014
0.014
0.014
0.014
0.014
0.014
0.013
0.013
0.013
0.013
0.013
0.012
0.012
0.012
0.012
0.011
0.011
0.011
0.011 0.011
0.01
0.01
0.01
0.01
0.01
0.01
0.009
0.008
(b)
Figure 4.8: (a): zig amplitude background image for ε = 0.013. (b) is acontour plot showing the values of the nonuniformity. Its average is 0.0119,standard deviation is 0.0019.
62
x
y
(a)
500
400
300
200
100
0
5004003002001000
0.01
0.01 0.01 0.01
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005 0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005 0.005
0.005
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0
0
0 0
-0.005
-0.005
-0.005
-0.005
-0.005 -0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005 -0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01 -0.01
-0.01
-0.01 -0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01 -0.01
-0.01
(b)
Figure 4.9: (a): one zig amplitude image for ε = 0.013. (b) is a contourplot showing the values of the amplitude. Its average is -0.000192, standarddeviation is 0.0052.
63
δx
δy
Figure 4.10: Autocorrelation of zig pattern amplitude of the image in Fig-ure 4.3. The center is normalized to be one.
δx
δy
Figure 4.11: Average autocorrelation of zig pattern amplitude of 2048 imagesat ε = 0.013.
64
We need to extract some length scale(s) from the average correlation image.
Usually, we assume correlation function has the form[1]
C(δr)∼
δr→∞ exp(−δr/ξ) (4.10)
For our experimental system, we are not able to measure correlation at very
large δr because we are limited by the system nonuniformity. Although we
have tried to correct it, our measurements at large δr are still not very reliable,
one thing is that they do not approach zero as expected (see, Figure 4.12 (b)).
Therefore, we are forced to use values at small δr to deduce our length scale(s).
For that, we do not have theoretical prediction in either real space or Fourier
space as Eq. 3.5 is not solvable for ε > 0. We found that at small δr data fit
to Gaussian form rather well3
C(x, y) = exp(
−x′2
ξ2x− y′2
ξ2y
)
assume ξx ≥ ξy (4.11)0
B
B
B
B
@
x′
y′
1
C
C
C
C
A
=
0
B
B
B
B
@
cos(θ) − sin(θ)
sin(θ) cos(θ)
1
C
C
C
C
A
0
B
B
B
B
@
x
y
1
C
C
C
C
A
(4.12)
In next section, we will discuss our results on ξx and ξy.
4.4 Results and discussion
Figure 4.13 shows the long axis length fit from 2D Gaussian function (the
long axis angle is shown in Figure 4.15). For comparison, one set of KL do-
main chaos measurement by N. Becker is also shown here. And ε−1/2 and ε−1/4
lines are also shown. KL data are more consistent with ε−1/4. Our EC data
is also closer to ε−1/4. A direct fit yields an exponent of about 0.33. So our
3C(x, y) = exp(
−(
x′
ξx
)α
−(
y′
ξy
)α)
was tried, and it was found that α is close to 2.
Also, C(x, y) = exp
(
−√
x′2
ξ2x
+ y′2
ξ2y
)
was tried and it does not fit very well, especially, it
has a cusp at δr = 0 which does not shown in our data.
65
-60
-50
-40
-30
-20
-10
0
δy (
pixe
l)
-60 -40 -20 0 20 40 60
δx (pixel)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.6
0.4
0.1
0
Long axis
θ
(a)
-100 0 100
0.0
0.4
0.8
δr (pixel)
Cor
rela
tion
(b)
Figure 4.12: (a): contour plot of the average correlation function at ε = 0.013(central part of Figure 4.11), the dash lines are a 2D Guassian fit to Eq. 4.11which yields ξx = 24.76 pixel, ξy = 11.01 pixel and θ = 0.12 rad. (b): circles(black) are correlation function along the long axis in (a) and the line (green)along these circles is a Guassian fit with ξx = 24.88; pluses (red) are correlationfunction along the short axis (perpendicular to the long axis) and the line(blue) along these pluses is a Gaussian fit with ξy = 11.68.
66
10-3 10-2 10-14
6
8
10
20
ε
ξ /d
Figure 4.13: Circles: ξx along long axis for cell X801 1 for different ε; squares:correlation length (extracted by fitting to SH square function[99]) for KL do-main chaos for Ω = 17.7 (with permission of N. Becker); solid line is a powerlaw of ε−1/4; dash line is a power law of ε−1/2.
measurements on this system confirm the finding in KL domain chaos system.
However, the data points are rather scattered, and the scheme for determi-
nation of length scales needs further consideration. Further investigation is
needed for drawing a firm conclusion.
For the short axis, the length is rather short and does not change much for
ε region where we obtain our scaling for long-axis length. The data at region
very close to the onset are less certain.
67
-0.02 0.00 0.02 0.04 0.063
4
5
6
ε
ξ /d
Figure 4.14: ξy along short axis for cell X801 1 for different ε.
-0.02 0.00 0.02 0.04 0.06
-0.4
0.0
0.4
0.8
ε
zig
angl
e (r
ad)
Figure 4.15: Long axis angle relative to the x-axis for different ε.
68
chaos system[82]. But both of them are inconsistent with GL equation predic-
tions.
Further work is needed to improve the uniformity of the convection pattern.
A possible solution is to use more controlled and bigger stiffer plates, for
instance, polished sapphire plates of 2 in or more and evaporate uniform ITO
layer on them for electrodes. For the most critical step, the alignment, photo-
alignment is still preferred, but a bigger and more uniform linear polarized UV
light source is desirable. And finally, the thickness of the cell at different edge
should be controlled by fine-tuning screws to adjust the thickness of the cell.
This is a construction similar to the very thin RBC cells used in [100], only
that the paper side wall should be replaced with plastic spacers. Presumably,
a very uniform (in both alignment and thickness) will produce much uniform
pattern and with a very big cell, we always have the freedom to choose a
uniform region out of the whole cell. With a cell uniform over 200d or more,
we should be able to deduce the correlation function at large distance and this
could be used to fit to the structure-factor-independent correlation function
Eq. 4.10. In Eq. 4.10, the meaning of ξ is more clear and we can make a direct
comparison with ξ in GE equations 3.5.
69
Part III
Local current fluctuations at
large driving in
Electroconvection in N4
70
Chapter 5
Introduction
5.1 Previous work on global current fluctua-
tion measurements in EC
EC has been mostly studied with shadowgraph technique (see Sect. 2.1.1)
and this works very well in the vicinity of the threshold and we can quantita-
tively measure the effective index of refraction field and therefore the director
tilt field. But for driving slightly away from the threshold, the shadowgraph
image becomes hard to interpret quantitatively. This is due to the inherent
nonlinear nature of liquid crystal optics. We need to measure the tilt δn of
the director field n. Due to the invariance under the transformation of n to
−n, the optical response of the shadowgraph always has a term proportional
to δn2. There is also a linear contribution (proportional to δn) for cells with
finite thickness (see [19] for detail). But for most cases (thin cells like we used
in our experiments), even when it is only slightly away from the threshold,
the quadratic term wins and in most of images we took, second harmonics are
very strong. For small ε’s, one can still use Fourier filtering to get rid of the
higher harmonics. When ε becomes of order 1 (see. Figure 7.1 (ε = 0.93)), it
is hard to infer much information from the shadowgraph images.
71
An alternative method for studying EC is to measure the electrical current
passing through the cell. To our knowledge, this kind of work was first carried
out by Kai et al.[101]. Later Gleeson and coworkers [102, 103, 104, 105],
and Goldburg and coworkers [106, 104] did a series of work on this kind of
measurements.
In [102, 103], Gleeson and coworkers proposed to measure the global trans-
port near onset and in a turbulent regime of convection in cells filled with
MBBA. An electric Nusselt number is defined to characterize the current trans-
port:
Nr(ε) = Ir
I0r− 1 (5.1)
where I0r = σ⊥V S
d(V is the rms amplitude of the driving voltage) is the con-
duction state in-phase current and Ir is the in-phase current. It was found
that Nr is very close to zero for ε < 0 and beyond ε = 0, there is an abruptly
growth from zero. And for ε slightly larger than zero, Nr grows linearly against
ε (Figure 2 in [102]). This shows that the onset of the EC is a supercritical
bifurcation (in [102], they also showed that at this regime, Nr ∝ A2, where
A is the amplitude of the pattern.). In the turbulent convection region, they
made an analogy of the transport problem with that in turbulent RBC. In
certain regions of driving, they found a scaling relationship with respect to ε
which is similar to that found in burbulent RBC[108, 109, 110]. At very big
ε, it was observed that Nr ceases to grow which is very different from that in
turbulent RBC.
In [104, 106], Gleeson, Goldburg and coworkers looked at the fluctuations in
total power injected into their EC systems filled with MBBA and M5[111] (at
each ε, the driving rms voltage is fixed and current fluctuations are measured).
For the distribution of the power fluctuations, they did not found any sign
of deviations from Gaussian distributions for all the ε’s in all the cells they
measured. They found that in all their cells measured, as ε is increased above
≈ 0.2, the fluctuation amplitude (normalized by the average power injection)
σP / < P > increases dramatically by as much as an order of magnitude (see
72
Figure 3 in [104]). The fluctuation amplitude continues to increase and reaches
its maximum at ε ∼ 1 and starts to decrease until ε reaches it maximum
measured value.
By comparing with their light transmission measurements (for light trans-
mission measurements, also see [107]), they concluded that the strong fluctu-
ations above ε ≈ 0.2 are due to spontaneous generation and annihilation of
dislocations in the roll pattern (Figure 1 in [104]). The defects start to be
generated at ε ≈ 0.2. As ε increases, more and more defects are generated
and the fluctuation amplitude increases. When the defects almost fill the cell,
further increase of driving can only squeeze the defect and the fluctuations
average out and the amplitude starts to decrease.
They also found that the relative fluctuation amplitudes scaled by the
aspect ratio of the cells collapse on to a single curve for ε > 0.2. Along with
that the distribution of the fluctuations is Gaussian, they concluded that the
defects are spatially uncorrelated. On the other hand, they observed that
the fluctuations are temporally correlated over quite long time for ε between
0.2 and 5 revealed by oscillations in the auto-correlation function ga(t) of the
fluctuation signal (Figure 4 in [104]). And the authors were puzzled that “a
large number of spatially uncorrelated defects gives rise to oscillations in ga(t)
correlated over such long times”.
Besides studying global transport properties and mechanism of fluctuation
generation in this specific system, another motivation for study the fluctuations
is related to a series of theorems with the general name of fluctuation theorem
(FT) [112, 113, 114, 115, 116, 117, 118, 119, 120] (for a general introduction
see [121]). One of the them is the Gallavotti-Cohen FT [114, 115] and it can
be seen as a generalization of the fluctuation-dissipation theorem (FDT) for
steady state systems that are driven far from equilibrium, like our EC system
at a constant driving. The theorem is very mathematical and not easy to grasp.
We will roughly follow the formulation in [106] which may not be rigorous.
73
Gallavotti and Cohen (GC) showed that, under appropriate conditions, the
system, assumed to have microscopic reversibility, and be driven into a chaotic
state, experiences such large fluctuations in the the dissipated power P that
sometimes P can become negative; i.e., the driven, dissipative system momen-
tarily can send energy back into the power supply that generates the chaotic
fluctuations. The theory makes a prediction about the ratio π(P )/π(−P )
where π(P ) is the probability of seeing P and π(−P ) is the probability of
seeing −P .
The GC theory is formulated in terms of the entropy production rate sτ =
Sτ , which is related to the dissipated energy1 by the system during a time
period τ by sτ = Pτ/(kBTss) where the steady-state temperature kBTss is
equal to the mean kinetic energy per particle. GC show that if the chaotic
motion in the system satisfies microscopic reversibility, then
π(sτ )/π(−sτ ) = exp(τsτ ) (5.2)
The above GC FT has been generalized by Kurchan[116] to systems un-
dergoing Langevin dynamics and by Lebowitz and Spohn to general Markov
processes[117]. The FT reduces to the FDT in the limit of vanishing driving
force[122].
In [106], the authors were able to neither measure any negative power
fluctuations nor verify Equ. 5.2 directly. Instead, by assuming that the fluc-
tuations around the mean is Gaussian for the complete range of power (they
have verified that the fluctuations near the mean is Gaussian. It is still possible
1Dissipated power P and injected power by the power source have the same mean, butthey are not necessarily the same at any moment as the system can store energy (in ECsystem, the cell has capacitance). But we will loosely use the injected power as the dissipatedpower. And van Zon and Cohen[126, 127] have generalized FT to include both dissipatedpower and injected power and the formalism of the probability ratio (Equ. 5.2) for both issimilar. But generally (e.g. two examples [123, 129] for which the probability distributionscan be rigorously calculated), these two qualities have different distributions[123, 124, 125,128, 129, 130]. The discussion of the example in [129, 130] is especially relevant to our casein EC.
74
that very rare large fluctuations described by the GC FT do not obey Gaus-
sian distribution.), an effective temperature can be derived from the standard
deviation σP of the fluctuations, which were also measured in [104] (Figure 3).
πτ (P ) ∝ exp[
−P−<P>2σ2
P
]
(5.3)
then we haveπτ (P )
πτ (−P )= exp
(
2<P>Pσ2
P
)
(5.4)
They assumed that GC FT applied to EC, they got
σP
<P>=
√
2kBTss
<P>τ(5.5)
Therefore, from their measurements of σP , they could derive a stationary sta-
tion temperature and they found that this temperature is significantly higher
than the ambient temperature of the cell. They attributed this temperature
to the mean kinetic energy of the quasiparticles in this system.
In the experimental test of FT in [123], they used an optical trap to drag
a colloidal particle in water and studied the ratio of entropy production and
consumption. And their findings are in agreement with FT prediction and
the temperature they found out was just the ambient water temperature. The
measurement in EC thus has some significance in this aspect. In [123], the
relevant system constituents are the water molecules, therefore the tempera-
ture measured represent the random motion of water molecules which are at
room temperature. In [106], the relevant constituents are the excitations from
the roll state, according to [104], they are defects in roll pattern and they are
macroscopic objects, therefore, they have a much bigger mean kinetic energy.
5.2 Local current fluctuations in EC
Our work started from this point. In [106], the authors had not been able
to see any negative fluctuations. Noticing the aspect ratio dependence of the
75
fluctuation amplitude in Figure 3 of [104], if one wants to see these negative
fluctuations, a natural way for measuring very big relative fluctuations is to
reduce the aspect ratio of the cell. But as the aspect ratio is reduced to
the order of 1, the boundary effects will dominate, and we will probably be
studying a very different system from the original spatially extended system.
And as we discussed above, it was the macroscopic excitations (defects) above
the roll state that give rise to the huge effective temperature which give us a
better chance of seeing large fluctuations from the mean. While the defects are
generated from the spatially extended rolls, getting rid of the spatial freedom
of the system will certainly eliminate them.
An alternative is to keep the spatially extended system intact and measure
the fluctuations locally. As pointed in [104], the defects are spatially uncorre-
lated, the relative fluctuation σP / < P > will be proportional to√
A, where
A is the measuring area.
All the above mentioned FTs are about the fluctuations of a global variable.
Motivated by the same reason to pursue large fluctuations in a portion of
a system, there has been theoretical [131, 132] and experimental[133, 134]
considerations on local variables for similar relationship as Equ. 5.2.
Another possible important thing we want to learn from local current mea-
surement is the properties of defects. In the above, we have discussed the
importance of the role of the defects in strongly driven EC. There are still
quite a lot unknown about the defects, such as the individual properties of
defects, the cause of the long correlation time in long correlation time in the
oscillations in auto-correlation function. We hope to shed some light on these
aspect as well.
In the following two chapters, we will first discuss the fabrication and
characterization of the cells for local current measurements in Chap. 6; then
we continue in Chap. 7 to discuss our results.
76
Chapter 6
Cell Preparation and Circuits
We need to make a cell with a local detecting electrode to measure current
going through a small area. Although this seems to be rather straightforward,
there are a few tricky points needed to be solved before we could actually make
a working cell. This is due to a few reasons, mainly that we are measuring
a very small current of pico-ampere order and the tricky nature of micropro-
cessing. In this chapter, we will first analyze the effective circuit of such a cell,
then we will incorporate the analysis results into our designs. This was not
the order the cell was actually designed, instead, we had to experiment on a
simple design first and then encountered problems and modified the design to
solving the problems. And we had to repeat this cycle for a few times.
6.1 Effective Circuit of the Cell
6.1.1 Effective Circuit
Schematically in Figure 6.1 we show a simple idea how to measure a cur-
rent flowing through a small area of one of the cell electrodes. The small area,
namely the detecting electrode, is connected to an AC-current-measuring de-
vice, in our case, a current-to-voltage preamplifier and a lock-in amplifier.
The current in turn flows back to ground. If we neglect the impedance of
77
R0 R1 C1C0
Figure 6.1: Schematic of a cell with a local electrode measuring currentpassing through the electrode.
A
~
Figure 6.2: Realistic schematic diagram of a cell with local electrodes, leadsand local electrode arrangement are shown.
the preamplifier, the detecting electrode is at the same potential of the main
electrode all the time. We will call the opposite electrode “collector” which
is directly connected to the driving source. The field between the main elec-
trode plus detecting electrode and the collector will be the same as that in an
ordinary cell without a detecting electrode, therefore, our detecting electrode
will be able to measure the local current without disturbing the convection.
78
BA
V
C0 C1
Cs2 Cs3
C1p
CLp
R0 R1p
R0p
Cc
R1
Cs1
ZL
Figure 6.3: Effective circuit of Figure 6.2. Only components affecting mea-surements are shown.
It is very different to make a micron-size hole out of a 1.1 mm thick glass
and make an electrode through the hole. Also, considering lead resistances and
stray capacitances, a more realistic cell construction is shown in Figure 6.2,
and its effective circuit diagram with all relevant components are shown in
Figure 6.3. The schematic drawing of each of these components for pho-
tolithography are in Sect. 6.2. The following is a description of all the effective
components:
Cs1, Cs2 and Cs3 are capacitance due to the insulating boundary(see 2.3.1).
With 400 nm SiO2, the capacitance is about 89pF/mm2.
R0 and C0 are the effective resistance and capacitance of the main con-
vection cell. Initially, the collector and the main electrode were designed to
79
be 8 mm × 8 mm and 10 mm× 10 mm respectively, later they were reduced to
2 mm × 2 mm and 5 mm × 5 mm due to the reason which will be mentioned
below. For a typical doping level of 293 ppm in weight, and a 20 µm thick cell,
it has a resistance of 12 MΩ and and a capacitance of 10 pF for 2 mm× 2 mm.
R1 and C1 are the effective resistance and capacitance of the local elec-
trodes to the collector. For an 8 µm one of above conditions, the resistance is
about 740 GΩ and the capacitance is about 0.23 fF. This capacitance is much
smaller than stray capacitances if they are not taken care of. Therefore, one of
the most important design considerations is to take care of all possible stray
capacitances. One of them is explicitly put on the effective diagram Figure 6.3
which is C1p. This is the capacitance between the lead of the local electrode
and the lead of the collector. Although they are far away compared to the
distance between local electrode and the collector, they are much larger in
area than the local electrode. So they are usually much larger than C1. R1p is
the resistance between these two leads through a long and narrow channel of
the bulk part of LC. For the same reason as for C1p, this is an important one
to be considered. Calculation shows that for most cases, this resistance is of
the same order as R1. But we can put an extra shielding on the leads of the
collector and basically eliminate R1p and greatly reduce C1p (see Sect. 6.2.2).
With this shielding and other measures, we can reduce C1p to about 7 fF which
is still larger than C1. Since we are mostly concerned with the energy dissi-
pated in the local region and a capacitance does not dissipate energy, we can
tolerate this capacitance.
Cc is the capacitance between the local electrode and the main electrode.
This is mostly due the overlapping of the lead of the local electrode and the
main electrode. For most cells, the distance between them is 200 nm. For a
10 µm wide lead, the capacitance is about 3.9 pF.
R0p is the contact resistance of the main electrode with its lead wire plus
the wire resistance. We single out this while ignoring other contact and wire
resistances because current going through this wire is much bigger than that
80
of the local electrode and any excess resistance will cause the potential of the
main electrode (point A in Figure 6.3) and that of local electrode (point B) to
be different.
We model the preamplifier as a complex impedance ZL. It turns out that
ZL is strongly inductive at the frequency we use. This is consistent with the
frequency characteristic provide by the manufacturer[46]. This is probably due
to the gyrator circuit in the preamplifier[137].
We measure ZL using a setup shown in Figure 6.4. Lock-in 2 provides
a driving source to drive the preamplifier in series with a 10 GΩ resistor.
Lock-in 2 also measures both the in-phase and out-of-phase voltage drops
over the preamplifier. Lock-in 1 connected directly to the preamplifier output
to measure the in-phase and out-of-phase current flowing throw the preampli-
fier. For 591 Hz and 10−9A/V setting, the measured effective impedance is
ZL = 4.86 × 103 + i6.40 × 104 Ω. Not shown in the effective ciucuit, there is
also a relatively small resistance due to the narrow lead of the local electrode.
In calculations, we can add this to the effective impedance of the preamplifier.
For all our cells with gold local electrodes, the lead is slightly less than 10 µm,
about 30 nm thick and 1.9 mm long. With gold the conductivity is about
4.09×107(Ω·m)−1 at 30C, and the calculated resistance is about 155 Ω. Effec-
tively, the total impedance of the preamplifier is ZL = 5.02×103+i6.40×104 Ω.
The capacitance of the cable connecting the local electrode and the pream-
plifier must be considered as its impedance is not very much bigger than that
of the preamplifier, especially when we use its 10−9A/V setting. This capaci-
tance is denoted as CLp and its value is proportional to the length of the cable,
for 50 Ω RG58 cable, it has a capacitance about 30pF/ft. In order to be able
to attach onto the cell, we have to use a thinner coaxial cable. We chose CC-
C-50 miniature coaxial cable from Lake Shore[135]. Its capacitance is about
24pF/ft. The miniature cable then connects to the normal RG58 BNC cable
81
~
External Reference
OuputInputInput
Input
Lock-in 2
Lock-in 1
Preamplifier
Oscillator10 GΩ
Figure 6.4: Circuit to measure the effective impedance of the preamplifier at10−9A/V setting. Lock-in 2 measures the voltage drops over the preamplifier.Input uses float mode[43]. Lock-in 1 measures the output of the preamplifierwhich then converts to current flowing through the preamplifier, and its ref-erence is provided by lock-in 2. Reference angle is set to 176.45o which is tocompensate phase shift caused by the preamplifier, see Sect. 6.3.
via a shielded box. A typical value for the total cable capacitance is about
140pF.
Since values of Cs1 and Cs2 are much larger than that of C0, and Cs3 is
much larger than C1, and we mostly only concern the change of C0 and C1, we
can lump Cs2 and part of Cs1 with C0 and R0, lump Cs3 and the rest of Cs1
with C and R, shown in Figure 6.5. From now on, C0, R0, C1 and R1 denote
the lumped values.
82
BA
V
C0 C1 C1p
CLp
R0 R1p
R0p
Cc
R1
IL
ILpIcI1
ZL
Figure 6.5: Simplified effective circuit diagram of Figure 6.2. Some of thecapacitors in Figure 6.3 have been lumped into C0 and C1
6.1.2 Analysis of the Effective Circuit
Using the effective circuit, we can analyze and find out which factors will
affect our measurement most. Our goal is to measure I1 (after minimize the ef-
fects of R1p and C1p), but what we can measure is IL. We define the systematic
error r = I1/IL−1. The design goal is to minimize |r|. For simplicity purpose,
we use complex impedances Z0 and Z1, to represent the effective impedance
of the main electrode region and local electrode region of the cell(Figure 6.6).
And the complex impedances of Cc and CLp are Zc and ZLp, We have
1/Z0 = 1/R0 + iωC0, (6.1)
1/Z1 = 1/R1 + iω(C1 + C1p), (6.2)
Zc = 1iωCc
, (6.3)
ZLp = 1iωCLp
. (6.4)
83
BA
V
CLpR0p
Cc
IL
ILpIcI1
Z1
ZL
Z0
I0
I0p
Figure 6.6: Complex block diagram of Figure 6.5
Applying Kirchhoff’s Laws, we can write the following equations about this
effective ciucuit
I0 = I0p + Ic, (6.5)
I1 + Ic = ILp + IL, (6.6)
I0Z0 + I0pR0p = V, (6.7)
I1Z1 + ILZL = V, (6.8)
I0Z0 + ZcIc = I1Z1, (6.9)
ILZL = ILpZLp. (6.10)
84
We only need to know I1, IL and Vc = IcZc for the potential difference between
point A and point B.
I1 = VZ1
+
(ZL (−(V Z0 Z1 ZLp) + V ((R0p + Z0) Z1 ZLp+
(R0p Z0 + (R0p + Z0) Zc) ZLp)))/
(Z1 (−(Z1 (R0p Z0 + (R0p + Z0) Zc) (ZL + ZLp))
− ZL ((R0p + Z0) Z1 ZLp + (R0p Z0 + (R0p + Z0) Zc) ZLp))),
(6.11)
IL =(V (Zc (R0p + Z0) + R0p (Z0 + Z1)) ZLp)/
(Z0 Z1 ZL ZLp + Zc (R0p + Z0) (ZL ZLp + Z1 (ZL + ZLp))
+ R0p (Z1 ZL ZLp + Z0 (ZL ZLp + Z1 (ZL + ZLp)))),
(6.12)
Vc =(V Zc (−Z0 ZL ZLp + R0p Z1 (ZL + ZLp)))/
(Z0 Z1 ZL ZLp + Zc (R0p + Z0) (ZL ZLp + Z1 (ZL + ZLp))
+ R0p (Z1 ZL ZLp + Z0 (ZL ZLp + Z1 (ZL + ZLp)))) .
(6.13)
From I1 and IL, we have
r = I1IL
− 1 =Z0 ZL (Zc+ZLp)+R0p (Z0 ZL+Zc ZL−Z1 ZLp)
(Z0 Zc+R0p (Z0+Z1+Zc)) ZLp. (6.14)
It is obvious that the difference of I1 and IL is caused by the presence of R0p,
Cc and CLp. To see how each of them affects r, we set each of them to 0, we
have
Cc = 0, ⇒ r = ZL
ZLpeffect of CLp (6.15)
R0p = 0, ⇒ r =ZL (Zc+ZLp)
ZcZLpeffect of CLp along with Cc (6.16)
CLp = 0, ⇒ r =Z0 ZL−R0p Z1
Z0 Zc+R0p (Z0+Z1+Zc)=
ZL−r0p Z1
Zc+r0p (Z0+Z1+Zc)
effect of Cc and r0p (6.17)
where r0p = R0p/Z0.
85
The combined effects of R0p, Cc and CLp are not simply the summation of
any of the three equations. But we can clearly see that the way to reduce r is
to try to reduce CLp and keep the potential of point A and point B as close as
possible (from 6.17).
To reduce CLp, the simplest way is to reduce the length of coaxial cables
connecting the local electrode and the preamplifier. Trying to balance the
potential of point A and point B is not that easy, as we do not know the
exact values R0p, and Z1, and we may have to add components to adjust both
in-phase and out-of-phase components and this complicates the circuits. The
simplest way to minimize the potential difference is to let both of them as
close as possible to ground, i. e., to decrease |r0p| (= R0p/|Z0|) and |ZL|. Also,
reduce the coupling between the main electrode and the local electrode, i. e.,
reduce Cc.
To increase |Z0|, we can reduce the area of the collector and/or main elec-
trode (the main electrode is kept larger than the collector to shield the parasitic
lead capacitance and diagonal current) while still keep the system to be a spa-
tially extensive system. In the initial design, we chose the collector to be a
8 mm × 8 mm square, later we changed that to be 2 mm × 2 mm. For a cell of
20 µmm spacing, we still have an aspect ratio of 100.
Reducing the area of the main electrode also helps in reducing Cc since
with a smaller main electrode, the overlapping length of local electrode lead
and main electrode reduces. In initial design the overlapping length was 4.0
mm (local electrode at the center of main electrode) and later it was reduced
to 1.9 mm. The other way to reduce Cc is to reduce the width of the local
electrode lead. But by doing this, the lead is more likely to be broken during
the processing and that increases the resistance of the leads which is part of
ZL. The final design reduced the initial width of 20µmm to 10µmm.
We cannot always decrease |ZL|. For most of the fluctuation measure-
ments, we have to stick to the highest sensitivity 10−9 A/V to minimize the
preamplifier current noise. Occasionally, we use 10−8 A/V. In this case, its
86
impedance(about 6 kΩ) is negligible and the effect of CLp is also negligible for
a typical value of 140 pF.
In the first design, we made both the local electrode and the main electrode
out of ITO for its transparency. And there was always an insulating SiO2 layer
for making non-injecting boundary condition. To make a contact, we scratched
the SiO2 layer with a glass scriber at the electrode edge and attached a cable
with the help of a small amount of indium. The contacts made this way had
a rather large contact resistance, a typical value is about 240 Ω. In the final
design, we use a thin layer of gold (about 30 nm thick to let it still be semi-
transparent). We also etch out of a window (with buffered hydrofluoric acid)
on the SiO2 layer to expose the gold layer in order to make a better contact.
The contacts made this way have a typical resistance of 6 Ω.
Table 6.1 lists the measured and estimated values of all cells measured in
this study. One cell (XX300) was made out of ITO electrodes from the first
design which was not optimized. We will not discuss data measured from this
cell. The rest were based on the second design.The cell X817 1 has a design
window size of 4 µm × 4 µm and we find that its measured effective resistance
R1 is far less than expected. It is likely that this cell did not reach our design
goal and therefore we will not discuss the results on this cell. The error r due
to the cable capacitance CLp (eq. 6.15) is all about 3.3%(about 0.3%, if use
10−8A/V) for all the cells. So we see that for the second design, the error
caused by Cc and R0p is negligible. In principle, we can correct the measured
data using the estimate value of r which is all about −0.034+ i 0.0026. Notice
that the real part correction is about 10 times bigger than the imaginary part,
therefore a rough correction is to increase the measured values by about 3%.
Since our conclusions do not depend on the absolute values of measurement
very strongly and it is roughly cancelled by the AC gain error at frequency of
591 Hz (see Sect. 6.3), no corrections will be made in the following discussions.
87
Table 6.1: Measured and estimated values of effective circuit components ofall cells measured and systematic error r caused by these components. Allelectrodes of Cell XX300 were made out of ITO and it was based on the firstdesign and the rest were based on the second design in which the local andmain electrodes are made out of gold. For all these measurements, imaginarypart of preamplifier impedance ℑ(ZL) = 64000 Ω.
Cell X817 1 X817 2 X817 3 X817 4 X817 5 XX300 X727 11Spacing (µm) 22.2 20.3 21.2 26.0 18.9 27 23.9
NLC σ⊥ 0.43 0.43 0.43 0.43 0.43 0.24 0.43(106 Ω m)−1
local electrode 4 8 16 32 48 60 128size (µm)R0 (MΩ) 15.6 14.3 14.9 18.3 13.3 1.8 14.6C0 (pF) 9.3 10.2 9.8 7.9 10.9 123 8.7R1 (GΩ) 875 1080 192 58 23 32 3.6C1 (fF) 8 7 9 10 15 75 62R0p (Ω) 6 6 6 6 6 240 6Cc (pF) 3.9 3.9 3.9 3.9 3.9 17 3.9
ℜ(ZL) (Ω) 5020 5020 5020 5020 5020 10000 5020CLp (pF) 140 140 140 140 140 140 140|r|(%) 3.4 3.4 3.4 3.4 3.4 6.6 3.4
6.2 Processing Procedures
In this section, I will discuss the processing procedures following the anal-
ysis in 6.1. We will extensively utilize the UCSB cleanroom facilities[55]. I
will use schematic diagrams to illustrate the procedures. We used L-Edit[139]
to make our mask drawings and the masks were ordered from UC Berkeley
Microfabrication Lab[138]. L-Edit is available to UCSB users through a license
server. To access L-Edit, contact UCSB cleanroom[55]. All the masks files can
be found at the thesis website[151].
We usually process 4 ∼ 8 glass slides each time, and in most cases, one
slide will be cut into four final slides for assembling cells. We have to make
88
more than we need in the end since the success rate after all the procedures is
rather low (for local electrode slide, less than a quarter, and better for collector
electrode slide). But in the following, I will only mention the procedures for
one slide and most procedures are in parallel, i.e. one can process all of them
in one step.
6.2.1 Processing slide with Local and Main Electrode
The following are the procedures for the fabrication of the slide with local
electrode and main electrode:
1. We begin with a cleaned ITO slide of 53 mm× 53 mm (see Sect. 2.3.1
for glass preparations). In the end, we usually add a step to clean the
slide with oxygen plama, using the Technics plasma etching system[140]
in UCSB cleanroom[55]. The conditions for this are oxygen pressure 300
mTorr, RF power 100 W and cleaning time 1 min.
We will use gold to make our local and main electrodes, we do not
necessarily need to use ITO glass. But the ITO film serves as a shielding
layer for the local electrode which can reduce the parasitic capacitance
C1p.
2. Evaporate a gold film for making local electrode onto the non-ITO side
of the glass slide (Figure 6.7(a)). We use e-beam evaporator #3 in
UCSB cleanroom[55] for evaporating 1 or 2 slides at once and use e-
beam evaporator #4 for evaporating 3 to 10 slides at once. E-beam #3
has a load-lock system that allows very quick cycle time for evaporations
(as low as 20 minutes total time), but it can only hold maximum of 2
pieces of our slides. E-beam #4 is a multi-wafer evaporator from CHA
Industries which can hold up to 10 pieces of our slides, but it takes
much longer time than e-beam #3 to pump down to desired vacuum
(about 2× 10−6 Torr). The good thing is that almost everything can be
89
Glass substrate
Gold
30nm
(a)
Glass substrateGold 30nm
(b)
Glass substrateSiO2 Gold200nm 30nm
(c)
90
Glass substrate
Gold
200nm 30nm
30nm
4~128 µm
GoldSiO 2
(d)
Glass substrateSiO2
800 nm
30nm30nmGold
Gold
(e)
Figure 6.7: From (a) to (e): Steps of processing the slide with local and mainelectrodes.
91
programed to be done automatically after sample loading, therefore one
does not have to be there to look after the evaporating process.
Gold does not stick to oxide(in our case, our glass slide) well. We need
an intermediate layer of reactive metal for the gold to stick on and we
use a very thin (5 nm) layer of titanium. We then evaporate the 30 nm
gold layer. Since we will be evaporating a layer of SiO2 upon the gold
electrode, we need another 5 nm titanium for the SiO2 to stick onto the
electrode.
Gold and titanium are two of the most common metals used in vacuum
deposition and we follow the standard procedures[55]. Since all the three
layers are very thin, we use a rather slow deposition rate. For titanium,
we use 0.05 nm/s; for gold, 0.1 nm/s.
3. Etch the titanium and gold layers into designed shape of the local elec-
trode (Figure 6.7(b)). The mask is “detect 9” which contains 9 slightly
different parts and each is a square of 25 mm × 25 mm. The parts are
for (left to right, up to down) local square electrodes whose side length
are 128 µm, 32 µm, 48 µm, 2 µm, 4 µm, 16 µm, 64 µm, 8 µm and 24 µm
(labeled by the small bars near the leads). This is a chrome dark field
mask, i. e., small features (square electrodes, leads etc) are clear and
most area is covered with a chrome film which block the photolithogra-
phy light. One slide can hold 4 out of the 9 parts on the mask.
We use negative photoresist(PR) AZ5214 for photolithography. The
recipe is modified from the established processes of AZ5214 at UCSB
cleanroom[142]. The established recipe was tested using silicon sub-
strates, while we are using a glass substrate which has much lower ther-
mal conductivity and reflects less light. The procedures are following:
(a) After cleaning, dry the slide on a 110C hotplate for 1.5 to 2 min-
utes. Cool it down for a few minutes.
92
(b) Put the slide on the spinner[143] (use a proper size chuck). Apply
HDMS, let it stay for 20 seconds. Spin at 4000 rpm for 30 seconds.
(c) Apply AZ5214. Spin at 4000 rpm for 30 seconds.
(d) Bake the slide on a 95C hotplate for 1.5 minutes.
(e) Align the slide with 4 of the 9 parts. Leave each side about 1.5 mm
and the pattern will be on the center 50 mm × 50 mm region. We
ordered 5” masks from Berkeley[138], they are slightly bigger than
the biggest size designed for the two contact aligners[144]. But it
can fit. Use a black chuck for holding transparent substrate. Ideally,
we should use the vacuum contact option to enhance resolution, but
they do not seem to have proper chucks for holding big samples. If
smaller samples are used, this option should be used. For most of
our exposures, we used the hard contact option. Do NOT use the
UV lens provided as that only works for small samples to enhance
uniformity. For a sample as big as 53 mm square, it will only
expose a circle on it. The strength of the UV light is monitored by
a meter beneath the aligner, it should be 7.5 mW/cm2. Expose for
8 seconds.
(f) Flood exposure (no mask, directly expose to UV light) the slide for
1.3 minutes.
(g) Develop in MF701 for 45 seconds. As the unexposed PR dissolve
in the developer, one can notice some red plumes come off from the
slides. And as developing process going, one should be able to see
the pattern from side of the beaker. After about 45 seconds, put
into a beaker of DI water. Below dry, visually (both naked eyes and
microscope) check under the yellow light in the photolithography
room. If needed, a few more seconds can be used for developing.
(h) After the developing process, areas not covered by PR should have
an almost uniform color viewed obliquely.
93
Then we visually check the pattern under stronger white light. If it is
okay, we go onto etching step. Since there are three layers of metal,
we need three steps. For titanium etching, we follow the method used
by Nir Tessler[145] with the help of Yair Ganot[146]. The etchant is a
mixture of ammonium hydroxide, hydrogen peroxide and DI water by
1:1:4. The ammonium hydroxide is commercial concentration, hydrogen
peroxide is 30% concentration. There are a few ways to etch titanium,
the nice thing with this method is that it is rather slow (good for our
very thin 5 nm titanium) and it does not etch ITO on the opposite side
of the glass. For our 5 nm thick titanium, we etch 60 seconds at room
temperature, then rinse with DI water.
For gold etching, we used the commercial iodine based gold etchant avail-
able at UCSB cleanroom[55]. The rate of etching is about 2.8 nm/s at
room temperature and we etch for 11 seconds and then rinse with DI
water. And it does not seem to attack ITO on the opposite side the
glass, at least within the time of etching.
After etching, we clean away the remaining PR with acetone.
To make the local electrode, a lift-off procedure (see the processing of
main electrode next) had been tested as well since it had less steps than
etching 3 layers of metal. But it never worked, i.e. the local electrode
and the main electrode were always shorted. A possible explanation[71] is
shown in Figure 6.8. For a tutorial on lift-off, check http://www.nanotech.ucsb.edu/Processin
4. Evaporate a thin layer of SiO2 (Figure 6.7(c)). This is a very critical
insulating layer. We will then evaporate the main electrode on top of
this layer, and it has to keep the two electrodes to be insulated from each
other. A thicker layer will have a better chance fulfilling this. But we do
not want it too thick to change the electrical field distribution inside of
the cell. A few thicknesses were tested, it was very hard using 100 nm;
94
(a)
(b)
Figure 6.8: A possible explanation of why lift-off procedure does not work,while etching does, for making local electrode. In (a), etched edges taper off.In (b), the lift-off edges forms sharp wings sticking upward, and they maypierce through the thin insulation layer on top of the electrode.
150 nm has a rather low successful rate; and in the end, we chose to use
200 nm.
We first clean the slide from last step using the same way for cleaning
a new slide (including a plasma clean step). The metal film should not
peel off in ultrasonic bath. We used Unaxis VLR Etch & Deposition
tool for this deposition. It deposits better quality[60] SiO2 films than
PECVD system[141] does.
95
5. Using lift-off to make the main electrode (Figure 6.7(d)). The mask is
“main 9”. It is a clear field chrome mask (features are covered with
chrome, most areas are clear). There are 9 parts as well for 9 sizes
(see step 3) (also labeled by the small bars, but the positioning of these
bars is not very good, they should be corrected in a future version).
The alignment marks (four “+’s” on each part) are to be aligned with
previous alignment marks made during the previous photolithography
step 3.
The material of the main electrode is the same as the local electrode
Ti/Au/Ti = 5/30/5 nm. We use a lift-off process for this step as it can
take away the 3 metal layers at once.
We first make the pattern of PR. We still use negative PR AZ5214, there-
fore, the exposure and developing processes are the same as in step 3.
In addition, after developing, blow dry and inspection, we need an extra
step of descumming. This is a plasma clean step like in step 1, we only
need 30 seconds.
We then deposit the Ti/Au/Ti layers and this is the same as in step 2.
After that, we put the evaporated slide in a beaker filled with acetone
and leave it overnight. The metal at the area covered with PR before
metal evaporation will peel off, this includes the small window areas on
the mask, while metal at the clear areas will stick to the SiO2. But many
times, soaking overnight is not enough. The metal at the window areas
just does not come off. Then we have to use stronger PR stripper and/or
ultrasound to clean up the window areas. We first try soak the slide in
heated PRX1165 stripper (up to 80C), if this still does not work, try to
put the heated stripper into a heated ultrasonic bath for a few minutes.
This usually solves the problem.
96
Figure 6.9: Microscope images of a 8 µm window for local electrode. Left:reflective image; right: transmission image.
Glass substrateSiO2
Gold
Gold
800 nm
30nm30nm
Figure 6.10: Relative position of the local electrode and the main electrode.The main electrode covers the local electrode. On the main electrode, there isa small opening of 4 ∼ 128 µm square, this exposes the local electrode underthe main electrode. This square area is used to collect the local current.
After soaking in stripper, we need to clean it as in step 1. Figure 6.9
shows microscope images of an 8 µm window. Figure 6.10 shows the
relative position of the local electrode and the main electrode.
97
6. We then evaporate a relatively thick (800 nm) SiO2 layer using PECVD[141]
(Figure 6.7(e)). This is to form the non-injecting boundary condition for
the main electrode.
7. The SiO2 layers in last step and step 4 also cover the areas on the elec-
trodes for attaching wires. In order to make good contacts with the
electrodes to reduce R0p and ℜ(ZL) in Sect. 6.1.2, we need to open two
windows on the SiO2 layers to expose the gold electrodes.
The mask for this step is “SiO2 mask 9”. This is a dark field mask with
9 parts. We need a positive photoresist and we choose AZ4110. We use
a slightly modified recipe from the established one[142]. The procedures
are following:
(a) After cleaning, dry the slide on a 110C hotplate for 1.5 to 2 min-
utes. Cool it down for a few minutes.
(b) Apply HDMS, let it stay for 20 seconds. Spin at 4000 rpm for 30
seconds.
(c) Apply AZ4110. Spin at 4000 rpm for 30 seconds.
(d) Bake the slide on a 95C hotplate for 1.5 minutes.
(e) Align with the mask and expose for 8 seconds under 7.5 mW/cm2
UV light.
(f) Develop in AZ400K:DI=1:4 solution for 60 seconds.
(g) Blow dry, visually check the PR coverage. An optional bake step
can be used. Use the 95C hotplate for 1 minute.
Then check under strong white light. If everything is okay, go onto
etching step. We first cover the ITO side by applying a layer of PR
AZ4110 with a brush. This is to protect the ITO layer from being etched
next. We use buffered HF (HF is an extremely dangerous chemical,
98
special handling gears are needed) to etch the SiO2 layers. We have
800 nm SiO2 by PECVD and 200 nm SiO2 by Unaxis VLR Etch &
Deposition tool. At room temperature, the etching rate for PECVD
SiO2 is 520 nm/min and for the other is 180 nm/min[60]. We therefore
need about 160 seconds for etching both.
After etching, rinse with DI water and check with a multimeter to see
whether the gold films have been exposed. Press the electrodes very
gently on the surface, the resistance should be just a few ohms. Then
wash away the PR layers with acetone.
8. We can then move the slide out of the cleanroom and measure in a
regular lab. Pay special attention not to touch any part of the electrodes
without properly discharging yourself. This is to prevent electrostatic
discharge(ESD). Press two small pieces of indium on each of the two
windows of exposed gold films gently. They should easily stick on the
films. Measure the resistance between local and main electrodes with a
multimeter which can measure high resistance, for example, a Keithley
2001 multimeter. For a good one, the insulation resistance will be at
least a few hundred megohm.
After finding which ones are good, label them. At this point, we should
shorten the local and main electrodes with a wire and keep them short-
ened until they are actually put into the circuit which will keep the
potential between the two electrodes almost the same. This is very im-
portant to prevent ESD from breaking the insulating SiO2 layer between
the two electrodes and the thin local electrode lead. The capacitance
between these two electrodes is less than 4 pF, and small amount of
charges can charge the capacitance to the SiO2 layer breaking voltage
(about 20V, breakdown electric field for chemical vapor grown SiO2 can
be as low as 108 V/m). Before we took this measure, almost all the
local electrode leads were broken by ESD during the process of attach-
99
Figure 6.11: A broken local electrode lead caused by ESD. The broken pointis almost always at the intersection of the main electrode and the local elec-trode lead.
ing wires and cables to the cell. Figure 6.11 shows a broken one and
it almost always happens at the intersection of the lead and the main
electrode.
6.2.2 Processing Slide with Collector Electrode
The fabrication of the collector electrode slide is similar but much easier.
The following are the steps:
1. We start with a clean ITO glass and we will be using the ITO side
(Figure 6.12 (a)).
100
100nm
Glass substrate
ITO
(a)
100nm
Glass substrate
ITO
(b)
100nm
Glass substrate
SiO2700nm ITO
(c)
101
100nm
100nm
Glass substrate
SiO2700nm
Gold
ITO
(d)
1000nm
Glass substrate
SiO2700nm
Gold
ITO
(e)
Figure 6.12: From (a) to (e): Steps of processing the slide collector electrodes.
102
2. Etching out the collector electrode shape (Figure 6.12 (b)). We use
mask collector 4 which has four repeating parts. This is a 2-inch dark
field mask. We use negative PR AZ5214 and the procedures are the same
as above.
For etching ITO, we used a mixture of 250 ml DI water, 450 ml hy-
drochloride acid and 50 ml nitric acid. All commercial concentrations.
Mix water and hydrochloride acid first, then add nitric acid. It will be
warm. Let it cool down for about 15 minutes or more. The color will
become yellowish. One should test to see how long it takes to etch the
ITO film. For our CB-50IN glass from Delta Tech[53], we need about 8
minutes.
3. Using Unaxis VLR Etch & Deposition tool deposit a SiO2 layer of 700
nm (Figure 6.12 (c)).
4. Using AZ5214 lift-off make the collector shielding electrode (Figure 6.12
(d)). The mask is collector shield 4 and the electrode materials are
Ti/Au/Ti=10/100/10 nm.
5. Cover part of gold film for attaching wire with a small piece of Al foil. Use
PECVD to deposit a 300 nm SiO2 layer on top of the collector shielding
(Figure 6.12 (e)). Take away the Al foil to expose the uncovered gold
film.
The relative position of the ITO collector electrode and the collector
shielding electrode can be seen in Figure 6.13. Most of the lead of
the ITO electrode is shielded. The leads would not collect any diag-
onal parasitic current. This therefore completely eliminates the effective
component R1p in Sect. 6.1.1 and reduces C1p.
6. We do not need to etch the SiO2 layer to attach the cable to the ITO
electrode as this contact resistance does not matter that much. We just
103
Glass substrate
700nm
Gold
ITO
Figure 6.13: Relative position of the collector electrode and collector shield-ing electrode.
use a glass scriber to break the SiO2 layer and attach a cable using small
amount of indium.
6.2.3 Finish up a cell.
After making enough of the two kinds of working slides, we then go on to
the assemblying process. First we need to cut each of 53 mm × 53 mm slides
into four 25 mm × 25 mm pieces. We use a Disco dicing saw to do that. The
glass is 1.1 mm thick and the saw can cut narrow grooves of 0.7 mm deep.
We just need to have 6 cuts along the boundary lines formed after the above
processing. DO NOT break the glass yet at this stage.
Clean the slide. Make alignment polymer layer (See Sect. 2.3.2) on 4
smaller pieces which are still connected. We do not cut after alignment because
cutting is a very dirty step and we cannot clean the surface after alignment.
Now we can break the slide into 4 pieces. A small trick in breaking the
1.5 mm wide edge: put a blade into the cut groove, hold the slide onto a
cleanroom paper, apply a force through the blade and it can usually break the
edge nicely.
104
ITO
Gold
Glass substrate
SiO2
Gold
Gold
30nm200 nm
20µ m
4 ~128 µm800 nm
Alignment polymer
Figure 6.14: Schematic diagram of an assembled cell.
Following the general description in Sect. 2.3.3, assemble two slides into
a cell, fill with NLC N4 and seal it with Torr Seal. A schematic diagram of
an assembled cell is shown in Figure 6.14. Three scales are important in our
design: distance of local electrode and main electrode d1 = 200 nm, window
size a = 4 ∼ 128 µm and cell spacing d = 20 µm. In order for the local
electrode to collect almost all the current flowing through the window (not be
shielded by the main electrode), we need d1 << a. In order not to disturb the
electric field inside the cell, we need d1 << d. In addition, it is possible that
the thickness of the insulating layer for the main electrode d2 = 800 nm is also
important. We know that there is a very thin charge boundary layer near the
interface of the NLC and the contacting insulating layer[8]. If d2 is too much
thinner than d1, then possibly the presence of the local electrode will disturb
the charge boundary layer, and in turn disturb the electric field inside the cell.
We therefore made d2 > d1 to minimize this possible disturbance.
105
The last step, we attach wires and coaxial cables (see Figure 6.16). There
are five electrodes in total. The local electrode and the collector need coax-
ial cables. We use Lakeshore CC-C-50 miniature coaxial cable[135] for local
electrode and regular RG174/U cable for collector. Shielding braids of both
cables are kept float. For normal measurement, main electrode is grounded,
it does not need coaxial cable. But for measurement of cell resistance R0 and
capacitance C0, we need the wire to be shielded. Therefore, we usually connect
main electrode with a CC-C-50 miniature cable as well. The collector shielding
electrode and the ITO side of the local electrode slide are always grounded,
two regular 30-gauge single-strand wires are used.
6.3 Measuring Circuit
We use an AC bridge similar to the one described in Sect. 2.1.3. The differ-
ence is that in Sect. 2.1.3, the current is µA or bigger, here we are measuring
current of pA order, therefore shielding and grounding[147, 148] have to be
done very carefully. Fig. 6.15 is a schematic diagram of the circuit used to
measure the local current. The driving source unit-gain follower and step-up
transformer (see Sect. 2.1.3) are housed in one shielded case. All the other
components have their own shielding cases. We should try to float these com-
ponents from ground and each other, then each with one single grounding
point connecting to one common ground[147, 148]. The red thick lines denote
coaxial cables (cable shields not shown), their cable shields should be properly
grounded following [147, 148].
The shielding of the cell is shown in Fig. 6.16. The slides with main and
local electrodes also has an ITO shielding in the other side, it should be put
on top and the ITO shielding is then connect to the temperature controlled
housing of the cell which also provides an enclosed shield for the cell.
106
Step-up Trans.
2
Ratio Trans.
Rr
Preamplifier
Cr
Follower
7265 Lock-in
OscillatorShield
this point
Cell
Collector
Shield
Figure 6.15: Schematic diagram of circuit used to measure local current,similar to Fig. 2.6 in Sect. 2.1.3. The components are shielded from eachothers (the dash line boxes). The connecting point P(shown with an arrow)has to be shielded from others as well. The thick red lines denote coaxialcables (cable shields not shown).
An important point is to shield the connecting point P (Fig. 6.15) from
any other points in the circuit. As we are dealing with capacitance of order of
fF, any stray capacitance can easily be picked up.
The reference capacitor Cr is usually set to 0.1 pF or 1 pF. The reference
resistor Rr brings some unwanted stray capacitance added to Cr, therefore,
the measured values of the local cell capacitance C1 and resistance R1 will be
unreliable. We only use reference Rr for local electrodes bigger than 48 µm
where the in-phase current is relatively large. Sometimes, Rr is not big enough
(maximum value 111 MΩ), we can put Cr and Rr in series, but this also changes
the stray capacitance added to the Cr. For smaller electrodes, the current is
rather small, we usually do not use Rr.
The ratio transformer, step-up transformer and the follower are the same
as in Sect. 2.1.3.
107
Glass substrate
SiO2
Gold
Gold
20µ m
ITO
ITO
Gold
Aluminum BlockOptical Path
Figure 6.16: The location of the cell in temperature controlled apparatus.Ground wiring scheme is shown.
The preamplifier is a DL1212 current-to-voltage preamplifier[46] and we
have discussed its input impedance in Sect. 6.1.1. Other things we should
pay attention to are its input noise and -3 dB frequency f−3dB[46]. The input
noise is 5.0× 10−15A/√
Hz for 10−9A/V sensitivity and 1.3× 10−14A/√
Hz for
10−8A/V sensitivity. f−3dB is 4 kHz for 10−9A/V sensitivity and 12 kHz for
10−8A/V sensitivity. The ratio of gain at frequency f to that at DC is hard
to measure as we do not have an accurate AC current source of nA order. A
rough estimate of the ratio is given by [136] G = 11+(f/f−3dB )2
(The frequency
response of the preamplifier is roughly equivalent to a -12 dB/oct filter (see
[46])). We almost always use it at frequency of 591 Hz. At this frequency,
G = 0.979 for 10−9A/V sensitivity, and G = 0.998 for 10−8A/V sensitivity.
108
7265 Lock-in
Oscillator
C
Preamplifier
Figure 6.17: Circuit use to measure the phase shift by preamplifer. Thepreamplifier output is measured by a 7265 lock-in amplifier and its oscillatoris used to drive the circuit.
For 10−8A/V sensitivity, it is negligible and for 10−9A/V sensitivity, it roughly
cancels the system error due to the capacitance of the cable connecting local
electrode and the preamplifier (see Sect. 6.1.2). No gain correction is made
in the following discussions. The phase shift at 591 Hz can be measured
(Fig. 6.17). C is a small capacitor with very little loss. We use GR1403 series
General Radio air capacitors[48]. The exact value does not matter too much
as long as it does not saturate the preamplifer. For 10−9A/V sensitivity, we
use 0.01, 0.1 and 1 pF. For 10−8A/V sensitivity, we use 10 pF. We use large
enough voltages for each capacitor, but not saturating the preamplifier and
lock-in amplifier. We measure the phase of preamplifier output relative to the
driving voltage. If there is no phase shift due to the preamplifier, the phase
will be 90o (the input impedance of the preamplifier is negligible comparing
to that of C), we measured that the phase is −93.56o for 10−9A/V sensitivity
(all three capacitors values) and −93.17o for 10−8A/V sensitivity. The minus
sign is due to that the output polarity of this preamplifier is inverted. So
when we measure a current signal at 591 Hz, we should put the phase at
−93.17 − (−90) + 180 = 176.45o for 10−9A/V sensitivity and at 176.83o for
10−8A/V sensitivity.
The lock-in amplifier connecting to the preamplifier is a model 7265[43]
from Ametek Signal Recovery and it has also been discussed in Sect. 2.1.3.
109
Important things to notice are its filter time constant and filter slope. We
always use 12 dB/oct slope. For different time constants, we will get different
bandwidths. A chart can be found in Sect. 3.3.16 of its manual[43]. For a
most used time constant, 50 ms, its bandwidth is 3.335 Hz. This affects how
much noise we get from the preamplifier.
110
Chapter 7
Current Fluctuations in Cellswith Local Electrodes
In this Chapter, we present measurements for the cells described in the
last chapter. Both optical and electrical measurements are described. All the
cells we measured are listed in Table 6.1. All the measurements are done in
the apparatus setup 1 with the home-made shadowgraph tower (see Sect. 2.1),
and we always measure at 30C. Before we put one cell into the apparatus,
we always keep the local electrode cable and main electrode wire shorted (see
Sect. 6.2.1). After putting the cell in, we connect all the electronics and
disconnect the main and local electrodes shorting wire. Turn on the camera.
Apply a small voltage and roughly adjust the bridge (Figure 6.15) to close
to balance (if Rr not used, just balance out-of-phase component). Gradually
increase the voltage till one sees some pattern appearing. Balance the bridge
(or the out-of-phase component). If time allows, wait for a few days before
start a run to let the cell stablize. Readjust the bridge. Record the ratio of the
ratio transformer R, Cr and Rr and the applied voltage and any unbalanced
readings of in-phase current Ix and out-of-phase current Iy.
The applied voltage then ramps up and down to measure effective conduc-
tance and current fluctuations at different applied voltages. (See [151] for the
program running this). The conductance measurement is done together with
111
current fluctuation measurement (see Sect. 7.2). For conductance measure-
ment, after setting the driving voltage, we wait for 3 minutes, then take 8192
points at a rate of 20 points/sec. For fluctuation measurement, all are the
same except that we take much more data points (600,000 points)).
The AC driving voltage is measured by a Keithley 196 multimeter before
the current data are taken. The current data are then taken by program
“take trig00”[151]. The interval between two data points is 50 ms. The GPIB
bus is fast enough to let us transfer every data point right after we take it.
Using an external trigger source provided with a Labmaster card[149], we can
get very precise timing and lose very few triggers for 50 ms interval. If faster
data taking is needed, we need to use the memory of the lock-in amplifier
which limits the number of data points to be 16384 (both Ix and Iy. See
the description of output data curve buffer in [44]) before each transfer. The
transfer will take up to one minute to finish. During this minute, the lock-in
is frozen and cannot take any data. For taking statistics of current data, this
is okay. The program to do this is“take data xy0”[151]. Both take trig00 and
take data xy0 require one computer (ehc.physics.ucsb.edu) to dedicate to the
current data taking. Therefore, temperature control and image taking have to
be controlled by another computer (bellman.physics.ucsb.edu).
At the same time of taking current measurements, we take optical images.
This is done by the program “run fluct image”[151] which runs on bellman and
communicate with ehc via “rcp” command transferring a file containing some
simple commands. The image taking starts right after current data starts and
we usually take 64 consecutive images and the interval between these images is
equal or slightly longer than 5 seconds (set to be 5 seconds, the delay caused by
file transferring and other computer related issues averages about 0.3 seconds).
Each image-taking will finish before each current data-taking.
112
7.1 Electroconvection Onset and Conduction
State
The images are used to identify the state of convection and determine the
onset of the convection. Figure 7.1 shows images of cell X817 2 (8 µm) at
different applied voltages. To determine the onset of the convection, we first
divide the images at different voltages by a common background image (see
Sect. 2.1.1). In this case, the background image is obtained by averaging 64
images taken at V = 1.41V . Then Fourier transformation is performed on
each divided image. The average Fourier power of 64 images at V = 6.302 V
is shown in Figure 7.2 (a) and Figure 7.2 (b) shows the circular average of
the structure factor versus radial wavenumber |k|. We then filter the Fourier
images and only leave fundamental peaks at |k| around 4. The total zig and
zag power is plotted in Figure 7.3. We then take the same procedure as in
Chap. 4 to determine the onset. Notice that the transition is rather smooth.
Nevertheless, we could fit the data point below and near onset by Eq. 4.1
and obtain the onset is 6.06 V at driving frequency 591 Hz for the cell X817 2
(8 µm).
The images of other cells are essentially the same as the small local elec-
trodes do not affect the convection.
The average current divided by the applied voltage on the local electrode
defines the local conductance. Figure 7.4 and Figure 7.5 show the local con-
ductance for cell X817 2 (8 µm) and X817 5 (48 µm) respectively.
Notice that in Figure 7.4, there is a dip near the onset of convection. In
Figure 7.5, there is also one, but much narrower relatively. We found that as
the size of the local detecting electrode increases, this dip will eventually disap-
pear and we can recover curve similiar to those measured for whole cell [102].
The origin of the dip is not completely understood. There are a few points
which may be related to this. First, the onset is rather smooth (Figure 7.3).
There are visible patterns well below onset (see first and second images in
113
(a) V = 5.421V, ε = −0.2.
(b) V = 5.979V, ε = −0.03.
(c) V = 6.302V, ε = 0.08.
114
(d) V = 6.614V, ε = 0.19.
(e) V = 7.190V, ε = 0.41.
(f) V = 8.413V, ε = 0.93.
Figure 7.1: Images at different voltages for cell X817 2 (8 µm). For eachvoltage (a)-(f), the left image is a raw image, the right image is central partof the raw image divided by the background image.
115
(a)
0 2 4 6 8 100.000
0.002
0.004
0.006
0.008
0.010
k
2πkS
(k)
(b)
Figure 7.2: (a): average Fourier power of 64 images at V = 6.302 V , cellX817 2 (8 µm). (b): circular average of the structure factor versus radialwavenumber |k|
0 10 20 30
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
V2 (V2)
Fou
rier
pow
er (
a. u
.)
Figure 7.3: Fourier power of fundamental peaks for cell X817 2 (8 µm). Uppointing triangles: applied voltage ramps up; down pointing triangles: voltageramps down; solid line: a fit to Eqn. 4.1 for points near and below onset. Thefit yields Vc = 6.06V , a = 0.062, P0 is fixed at 0.0001.
116
0 2 4 6
9.5.10-13
1.0.10-12
1.0.10-12
ε
Con
duct
ance
(Ω
−1)
Figure 7.4: Conductance through the 8 µm window of X817 2. Up pointingtriangles (black in color version): applied voltage ramps up; down pointingtriangles (red in color version): voltage ramps down. Run started at thesmallest ε point along the up-voltage point at time 0 s and ended at thesmallest ε point along the down-voltage point at time 686186 s.
0 2 4 63.2.10-11
3.4.10-11
3.6.10-11
3.8.10-11
4.0.10-11
ε
Con
duct
ance
(Ω
−1)
Figure 7.5: Conductance through the 48 µm window of X817 5. Up pointingtriangles (black in color version): applied voltage ramps up; down pointingtriangles: voltage ramps down (red in color version). Run started at thesmallest ε point along the up-voltage point at time 0 s and ended at thesmallest ε point along the down-voltage point at time 716802 s.
117
Table 7.1: Onset Vc and conduction state conductance c0 of different cells.
Cell X817 2 X817 3 X817 4 X817 5 X727 11Assemble date 08/17/04 08/17/04 08/17/04 08/17/04 07/27/04
NLC σ⊥ 0.43 0.43 0.43 0.43 0.43(106 Ω m)−1
Spacing d (µm) 20.3 21.2 26.0 18.9 23.9Local electrode 8 16 32 48 128
size a (µm)Aspect ratio (s = a/d) 0.394 0.755 1.23 2.54 5.36
Vc (V) 6.06 6.38 6.43 6.13 6.28c0 (10−12Ω−1) 0.929 4.30 14.1 32.7 225Measure date 09/13/04 04/24/05 05/07/05 03/28/05 04/10/05
Figure 7.1). And from our observation[150], these patterns can reappear after
we cycle the driving voltage for a few times during a few days. Presumably,
some deterministic factors can pin the pattern near onset. One roll of such
pinned pattern will stay right at the location of the local electrode and causes
a decrease in conductance. One thing that is not clear is why it always causes
a decrease in conductance, but not an increase for all the measurements in all
the four cells shows this dip (X817 2 (8 µm), X817 3 (16 µm), X817 4 (32 µm)
and X817 5 (48 µm)). Since we are mostly concerned with current fluctuations
at high drivings, no further effort was spent on this problem.
In both of the graphs, we see that there is a drop in conductance over
time. This is due to the change of the conductivity of the doped LC over
time and it is present in all the cells. The mean value of the up and down
conductance below onset is taken as the conductance of the base state and is
listed in Table 7.1.
118
12000 14000 16000
6
8
10
12
14
time (s)
In-p
hase
Cur
rent
(pA
)
Figure 7.6: Section of time series for different driving voltages of cell X817 2(8 µm). From bottom to top curves(color name for colored version). 1 (black):ε = −0.03; 2 (blue): ε = 0.08; 3 (brown): ε = 0.19; 4 (cyan): ε = 0.41;5 (green): ε = 0.93; 6 (magenta): ε = 1.49; 7 (orange): ε = 2.35; 8 (red):ε = 3.33.
7.2 Current fluctuations
In this section, we will focus on the current measurements on cell X817 2
(8 µm, smaller than cell spacing 20.3 µm) and X817 5 (48 µm, bigger than cell
spacing 18.9 µm) as examples of the two cases (smaller and bigger than cell
spacing).
Figure 7.6 shows sections of time series for a few driving voltages (each of
them takes 8.33 hours). Some of the images at these driving voltages have
been shown in Figure 7.1.
Usually, there are drifts of the mean current during the course of the mea-
surements of 600,000 points (8.33 hours). These are usually rather small as
119
we cannot tell from Figure 7.6. The correction is done as follow: First we
use a box filter (may repeat for some times) to filter out any fast changing
components and we get a smooth curve which only changes slightly over the
whole 8.33 hours. We then subtract the filter curve from the original curve
and the remaining curve is what we need. All the statistics are done with this
curve (except the mean of the curve). The program to the de-drifting and
statistics is de drift stat 3.2.c[151].
7.2.1 The general features at different driving
1. Below and near onset, the fluctuations are small and Gaussian (see Ta-
ble 7.2). We will see that these fluctuations are mostly due to the noise
of the preamplifier (mostly Johnson current noise of the 1 GΩ feedback
resistor[46]).
According to the preamplifier manual ([46]), at 591 Hz, the 10−9A/V
sensitivity has a noise spectral density of about 10 fA/√
Hz. And the
bandwidth is 3.335 Hz (for 12 dB/oct and 50 ms filter [43]). Therefore,
the Johnson noise current is about 18 fA. The Johnson current noise of
the local region is very small (about 0.2 fA) due to its extremely large
effective resistance (R1 = 1080GΩ, Table 6.1).
The shot noise is not easy to estimate. Presumably, the ionic conduction
is due to multivalent ions which each carries more than one electron
charge, but we do not know the exact numbers. As a minimal estimate,
we use the spectral density formula of an electron charge, S = 2eI. We
have a current of about 6 pA, therefore, we have a shot noise about 2.5
fA.
Our estimate of the total current noise is about 18 fA (√
182 + 2.52 + 0.22 ≈18). This is rather close to the measured values below and near onset
(about 30 fA). For other cells, we have similar situations, see Table 7.3.
120
2. For intermediate driving (ε=0.19 and 0.41), the current oscillates with
a period of 130 seconds and it has a rather small ε dependence. By
comparing the series of images taken at the same voltage (Figure 7.1)
which give roughly the same period, we conclude that this is due to
the traveling of the rolls. Every time one roll passes though the area
of the local electrode, it gives a rise or fall of the measured current.
This oscillation serves to verify that the local electrode is measuring the
current going through the LC layer, not some artifact. As to the cause
of the small traveling of the roll, one possible scenario is: defects are
generated in this range of ε and they are well separated[104]. But the
wave number field is modified by the presence of the defects and wave
number gradient forms and it caused the roll to travel. Another possible
cause is the nonuniformity of the cell. In Figure 7.6, the two arrows
point to two spikes in the oscillating current, this is possibly associated
with the defects mentioned in [104] passing through our local electrode.
For big cells like cell X817 5 (48 µm), there are oscillations as well, only
the amplitudes are smaller.
We will discuss the possible relationship between these local oscilla-
tions and the cause of global current fluctuations[104] in our discussion
Sect. 7.3.2
3. For the biggest driving, ε equal or bigger than 0.93. The current fluc-
tuates much stronger and there is no obvious oscillatory behavior in the
time serious. We will focus our attention on this region in the rest of the
section.
7.2.2 Current fluctuations at high driving
Table 7.2 and Table 7.3 list the the statistics of the in-phase current data
for different ε’s (we still list data for small ε’s, but they are just for reference.
We should pay attention to ε ≥ 0.91). For each ε, 600,000 points of data
121
0 10 20 300.0
0.2
0.4
0.6
0.8
1.0
time (s)
Au
toco
rre
latio
n
Figure 7.7: Autocorrelation function of in-phase current Ix for cell X817 2(8 µm) at ε = 0.93.
were taken, the interval between adjacent points is 50 ms. Figure 7.7 shows
a typical autocorrelation function of Ix for X817 2 (8 µm) at ε = 0.93. It is
not a simple exponential decay, but we can see that the correlation time is
shorter than 10 s. Therefore, for each ε, we usually have data for more than
3000 correlation times.
We first look at the fluctuation amplitude. Figure 7.8 (a) shows the nor-
malized standard deviation σx/ < Ix > versus ε all five cells. We see that
they have similar behavior as measured in whole cells[104]. Only that the rel-
ative fluctuation amplitude is much larger than that in whole cells (by about
two orders of magnitude in X817 2 (8 µm)). We also scale σx by < Ix > /s
in Figure 7.8 (b) similar to the inset of Figure 3 in [104]. Not surprisingly,
they do not collapse onto a single curve as the fluctuating current is spatially
correlated over the small local electrode area. In Figure 7.8 (c), we attempt
122
Table 7.2: Statistics of in-phase current of X817 2 (8 µm).
ε mean (A) std. dev. σ (A) skewness kurtosis-0.03 5.55× 10−12 2.86× 10−14 -0.001 3.0020.08 5.81× 10−12 2.9× 10−14 0.002 2.9790.19 6.15× 10−12 9.6× 10−14 0.268 2.0080.41 6.79× 10−12 1.31× 10−13 0.038 1.7790.93 8.12× 10−12 2.22× 10−13 0.717 3.1321.49 9.46× 10−12 2.74× 10−13 0.511 2.8022.35 1.12× 10−11 2.93× 10−13 0.387 2.7233.33 1.3× 10−11 2.78× 10−13 0.327 2.7574.41 1.47× 10−11 2.51× 10−13 0.346 2.9026.58 1.78× 10−11 2.19× 10−13 0.280 2.9555.50 1.63× 10−11 2.31× 10−13 0.307 2.9463.87 1.39× 10−11 2.65× 10−13 0.332 2.8152.79 1.2× 10−11 2.9× 10−13 0.346 2.7241.92 1.03× 10−11 2.9× 10−13 0.430 2.7381.16 8.65× 10−12 2.52× 10−13 0.594 2.8780.62 7.23× 10−12 1.26× 10−13 0.330 2.4750.30 6.43× 10−12 1.29× 10−13 0.512 2.0190.14 5.94× 10−12 7.1× 10−14 0.248 2.2920.03 5.64× 10−12 3.25× 10−14 0.004 2.8680.06 5.72× 10−12 3.26× 10−14 -0.004 2.895
to scale σx by < Ix > /s1/2, they come much closer, except that of X727 11
(128 µm).
To have a concrete view of what the distributions of the local current fluc-
tuations look like, we choose an ε near the point of largest relative fluctuations
for each cell and plot their probability distribution functions(PDFs) in Fig-
ure 7.9 and Figure 7.10. These two cells represents the cases studied, and
we see that they are quite different except that they all clearly deviate from
a Gaussian distribution. For window size a less than the spacing d (X817 2
(8 µm)), there is a quite large positive skewness (skews towards larger values).
123
Table 7.3: Statistics of in-phase current of X817 5 (48 µm).
ε mean (A) std. dev. σ (A) skewness kurtosis-0.04 1.99× 10−10 3.83× 10−14 0.010 2.9870.07 2.09× 10−10 3.97× 10−14 0.025 3.0190.18 2.21× 10−10 5.48× 10−13 0.189 2.1440.39 2.45× 10−10 1.04× 10−12 0.663 2.7190.91 2.98× 10−10 2.81× 10−12 -0.088 3.2571.46 3.48× 10−10 3.8× 10−12 -0.194 3.2672.32 4.15× 10−10 4.16× 10−12 -0.176 3.2423.29 4.8× 10−10 3.81× 10−12 -0.126 3.2384.36 5.44× 10−10 3.28× 10−12 -0.058 3.2116.50 6.57× 10−10 2.8× 10−12 -0.000 3.0705.43 6.01× 10−10 2.99× 10−12 -0.011 3.1353.82 5.1× 10−10 3.55× 10−12 -0.097 3.2132.75 4.41× 10−10 4.06× 10−12 -0.163 3.2711.89 3.79× 10−10 4.13× 10−12 -0.192 3.2371.14 3.16× 10−10 3.36× 10−12 -0.168 3.2200.61 2.63× 10−10 1.25× 10−12 1.042 5.2030.29 2.31× 10−10 1.1× 10−12 0.338 1.7320.12 2.11× 10−10 1.39× 10−13 0.073 1.9740.02 2.00× 10−10 4.16× 10−14 -0.046 2.7690.05 2.04× 10−10 4.11× 10−14 0.024 2.854-0.95 4.58× 10−11 2.95× 10−14 -0.002 2.996
And for a > d (X817 5 (48 µm)), the current skews slightly towards smaller
values.
In these two plots, we label the conduction states determined by conduc-
tance values below onset by red (color version) arrows. One interesting ob-
servation is that for X817 2 (8 µm), we could observe some of the fluctuations
(about 0.5%) go below the conduction state. In X817 5 (48 µm), we do not
see any fluctuations go below the conduction state.
The shape of the PDF cell X817 2 (8 µm) is rather paticular. This is
possibly related to the two mechanisms for current fluctuations above onset:
124
0 2 4 60.00
0.01
0.02
0.03
ε
σ x /
<I x
>
(a)
0 2 4 60.00
0.01
0.02
0.03
ε
σ x s
/ <
I x>
(b)
0 2 4 60.00
0.00
0.01
0.01
0.02
ε
σ x s1
/2 /
<I x
>
(c)
Figure 7.8: Circles: X817 2 (8 µm); squares: X817 3 (16 µm); diamonds:X817 4 (32 µm); pluses: X817 5 (48 µm); crosses: X727 11 (128 µm). (a): in-phase current fluctuation amplitude normalized by the average of the current <Ix > for all five cells; (b): in-phase current fluctuation amplutude normalizedby < Ix > /s; (c): in-phase current fluctuation amplutude normalized by< Ix > /s1/2.
the director tilt and the fluid motion which carries charges. Relative to the
conduction state below onset, the director tilt only increases the conductance
125
(as for planar alignment at conduction state, the electric field only sees σ⊥
which is smaller than σ‖. Any tilt will give us a component of σ‖ which
increases the conductance); the fluid motion can both increases or decrease
the conductance. The most interesting part is the fluid motion. If we treat
the instantaneous director field as the base state for the convection, we could
see even more fluctuations go beyond the base state. But our measurements
cannot distinguish the two components. For reference, in Figure 7.9, we have
shown the highest possible conductance states due to pure director tilt1: arrow
(c)(blue), the director field is fully randomized in 3D, σeff = (σ‖ + 2σ⊥)/3;
arrow(b)(green), the director field is randomized but confined in x − z plane,
σeff = (σ‖ +σ⊥)/2. But for ε = 1.49, we should expect the director tilt should
be far from fully randomized.
In light of the two mechanisms for current fluctuations, we try to fit the
PDF of X817 2 with a double Gaussian function
π(Ix) = C0 exp(
(Ix−Ix1)2
σ2
1
)
+ 1−√πC0σ1√πσ2
exp(
(Ix−Ix2)2
σ2
2
)
. (7.1)
And the fit curve is plotted in Figure 7.9 along with the measured PDF and
it gives a much better fit than a single Gaussian. But we have to use five
fitting parameters instead of one for single Gaussian. We see that the double
Gaussian captures the behavior of large-probability events quite nicely. But
for rare events, there is a clear deviation. Therefore, the exact mechanism
for this particular shape is still unclear. And to further separate these two
Gaussians, we need to measure at a smaller local electrode, and possibly a
small cell spacing will help as well.
The most obvious difference in the two PDFs are their skewness, we there-
fore compare their values for various ε’s in Figure 7.11. Again, it should be
obvious that for near and below onset, the values should be close to that of a
Gaussian distribution. For intermediate ε’s, due to the oscillations, statistics
1σ‖ was measured for NLC N4 with dopant level 970 ppm and it was found that for thatbatch, σ‖/σ⊥ ≈ 1.69. We assume that the NLC N4 for other batches with different dopantlevels have the same ratio.
126
8.0 9.0 10.0 11.0 12.010-4
10-3
10-2
10-1
100
Ix (pA)
π(I x
) (p
A-1
)
Conduction State (c) (b)
Ix1
Ix2
Figure 7.9: Probability distribution function of In-phase current for X817 2(8 µm). Obtained by binning 600,000 points of data at ε = 1.49 measured in8.33 hours into 1001 bins, and bin size is 3 fA. Dash-dot (magenta) line: a
Gaussian fit to the data π(Ix) = 1.41 exp[
(
Ix−Ix0
0.40
)2]
(pA−1), Ix0 = 9.46 pA, is
the mean of the current and is held at constant for fitting. Dash (red) line: afit to double Gaussian function Eq. 7.1 with C0 = 1.348 pA−1, Ix1 = 9.295 pA,σ1 = 0.2298 pA, Ix2 = 9.660 pA and σ2 = 0.3595 pA.The left arrow (red) pointsto the conduction state of the cell which is 8.9 pA. The middle arrow (c)(blue)points to the pure conduction state for fully randomized director field in alldirection. The right arrow (b)(green) points to the pure conduction state forrandomized director field, but confined in x − z plane.
should not mean too much. For large ε’s, for X817 5 (48 µm), skewness is
slightly negative and as ε increases, converges to zero. Combing the informa-
tion on kurtosis in Figure 7.12, it is evident that the distribution is approaching
a Gaussian distribution as ε increases. For X817 2 (8 µm), it is positive for a
127
310 320 330 340 350 360 37010-6
10-5
10-4
10-3
10-2
10-1
Ix (pA)
π(I x
) (p
A-1
)
Conduction State
Figure 7.10: Probability distribution function of In-phase current for X817 5(48 µm). Obtained by binning 600,000 points of data at ε = 1.46 measured in8.33 hours into 1001 bins, and bin size is 45 fA. Dash(red) line: a Gaussian fit
to the data π(Ix) = 0.108 exp[
(
Ix−Ix0
5.18
)2]
(pA−1), Ix0 = 348 pA, is the mean of
the current and is held at constant for fitting. The arrow (red) points to theconduction state of the cell which is 315 pA. The state for fully randomizeddirector field in 3D (corresponds to arrow (c) in Figure 7.9) is 387 pA. Thestate correponds to Figure 7.9 arrow (b) is 424 pA.
large range of ε. Another run covers larger ε’s and indicts that it decreases
slowly towards zero.
In Figure 7.13, we show skewness measurements for all the cells measured
around two ε’s. We have five local electrode cells listed in Table 7.1. The
measurements on cell XX300 (60 µm) are also listed for reference. There are
two whole cells listed. One is a commercial cell from E.H.C. Japan[52] whose
electrodes are 10 mm × 10 mm. It is expected that the distribution for such a
cell with large aspect ratio should be Gaussian and the measurements verifies
128
0 2 4 6
0.0
0.4
0.8
1.2
ε
Ske
wn
ess
6 8 10 12 14 16 18 20
0.2
0.2
0.3
ε
skew
ness
Figure 7.11: In-phase current skewness for X817 2 (8 µm) (circles) andX817 5 (48 µm)(squares). Dash line (red): Gaussian distribution value 0. In-set: X817 2 (8 µm) at high ε’s (a different run).
0 2 4 61.5
2.0
2.5
3.0
3.5
ε
Kur
tosi
s
Figure 7.12: In-phase current kurtosis for X817 2 (8 µm) (circles) and X817 5(48 µm) (squares). Dash line (red): Gaussian distribution value 3.
129
100 101 102 103 104-0.2
0.0
0.2
0.4
0.6
size (µm)
ske
wn
ess
Figure 7.13: Skewness vs. the size of the cell. Circles: ε ≈ 1.4; squares:ε ≈ 3.0. Points at left side of the dash (red) line are for cells with square localelectrodes. Points at the right side of the dash line are for two whole cells,one with square electrodes about 0.5 mm × 0.5 mm, the other one with squareelectrodes about 10 mm × 10 mm. All the cells have spacing around 20 µm.
this, and it is consistent with measurements in [104]. The other whole cell is
a home-made cell with electrodes about 0.5 mm × 0.5 mm. The measurement
shows slight negative skewness. The measurements in local cells and whole
cells are consistent, and we can conclude that skewness of the distribution
depends on the size of the detecting electrodes in a non-monotonic manner.
In Figure 7.9, we have seen that some of the big fluctuations can go below
the conduction state at one ε. On Figure 7.14 and Figure 7.15, we superimpose
the ranges of the fluctuations onto the conductance plots shown in Figure 7.4
and Figure 7.5. The ranges are denoted by the verticle bars. But they are
not standard deviations of the data, but the maximum and minimum values
observed in a run of 600,000 points. To minimize any possible errors caused by
130
0 2 4 6
9.0.10-13
1.0.10-12
1.1.10-12
ε
Con
duct
ance
(Ω
−1)
Figure 7.14: Up pointing triangles and down pointing triangles are the sameas in Figure 7.4. Up pointing solid triangles: long point (average over 600,000points) for up-ramping voltage; down pointing solid triangles: long point (av-erage over 600,000 points) for down-ramping voltage. The bars for the solidpoints denote the range of fluctuations, see text.
any possible glitches in measuring instruments, 10 biggest (or smallest) values
are taken and then average to represent the maximum (or minimum) values.
For detail, see the processing program de drift stat 3.2.c[151].
131
0 2 4 63.2.10-11
3.4.10-11
3.6.10-11
3.8.10-11
4.0.10-11
ε
Con
duct
ance
(Ω
−1)
Figure 7.15: Up pointing triangles and down pointing triangles are the sameas in Figure 7.5. Up pointing solid triangles: long point (average over 600,000points) for up-ramping voltage; down pointing solid triangles: long point (av-erage over 600,000 points) for down-ramping voltage. The bars for the solidpoints denote the range of fluctuations, see text.
132
7.3 Discussion
7.3.1 Stationary state temperature kBTss
We have seen that in Figure 7.9, some of the fluctuations go below the
conduction state (but we have not found any go below the absolute zero). It
is attempting to try to make some connections between the fluctuations to the
big negative fluctuations predicted by the FT. If we make the assumption that
the excitations above the onset, which give the large fluctuations we have seen,
effectively have no interactions with the conduction state, then we should be
able to apply the FT. This assumption may not seem to be very realistic, but
it can serve as a first approximation. This is similar to the picture of phonon.
In that picture, phonons are excitations of the lattices and in many cases, we
can ignore the interactions between phonons and lattices.
In treating the conduction state as our reference state, we can calculate
the quality ln (π(P0 − ∆P ))/π(P0 + ∆P )) for certain fixed ε’s, where P =
P0 + ∆P is the power injection into the system at ε and P0 is the power
injection calculated with the conduction state conductance c0 and the applied
voltage at ε, V , P0 = V 2c0. Figure 7.16 shows the plot for cell X817 2 (8 µm).
It only covers a small range of ∆P where π(P ) has measured values (see
Figure 7.9. Not surprisingly, we could fit this range with a straight line
ln(
π(P0−∆P )π(P+∆P )
)
= −a ∆P (7.2)
and we have a = 3.62 × 1012 W−1. If we apply Eq. 5.2, we have
a = τkBTss
(7.3)
And τ = 50 ms, therefore kBTss = 1.38 × 10−14J.
For comparison, if we apply Eq. 5.5 blindly for this ε, using Ix = 9.46 ×10−13 A, σx = 2.74 × 10−13 A (Table 7.2) and V = 9.565V, we have kBTss =
1.90 × 10−15J and this is an order smaller than the above result. This sug-
gests that Eq. 5.5 may not apply to the 8 µm local cell X817 2 as the distribu-
133
-2.10-12 0 2.10-12
-10
0
10
∆P (W)
ln(π
(P0-
∆P
))/π
(P+∆
P))
Figure 7.16: ln(
π(P0−∆P )π(P+∆P )
)
for a small range of ∆P for cell X817 2 (8 µm).
The solid (red) line is a linear fit to this range of data.
tion function is not Gaussian. We can also compare with results obtained
in whole cell measurements. In a commercial cell XX54 of electrode size
10 mm × 10 mm and spacing about 25 µm filled with N4 doped with 293 ppm
TBAB. At about ε ≈ 1.25, we have Ix = 2.14 × 10−5 A and σx = 1.11 × 10−9 A
and V = 10.815V , then kBTss = 1.56 × 10−14J. This comparison suggests that
we can obtain similar Tss for both local and global measurements, and Tss is
an extensive quality.
Unfortunately, we could only observe significant portion of fluctuations
going below conduction state in cell X817 2 (8 µm) for a small range of ε’s.
To have more large fluctuations, a cell with smaller spacing and window size
is needed.
134
7.3.2 The cause of global current fluctuations at inter-
mediate driving (ε from to 0.19 to 0.9)
In [104], the authors attribute the strong global current fluctuations to
spontaneous generation and annihilation of dislocations in the roll pattern, i.
e. each dislocation changes the local director and velocity fields, and induces
some current changes as it generates and annihilates. Our local measurements
at intermediate driving (ε = 0.19 and 0.41 in Sect. 7.2.1) reveals some regular
oscillations in the local current. And we attribute some rare spikes in the
regular oscillations to the appearance of defects near the local electrode.
In Figure 7.8, we see that the local current fluctuations have similar be-
havior as the global fluctuations measured in whole cells[104]. While the local
fluctuations at intermediate driving are mostly due to the regular oscillations
in the current, if the argument in [104] is correct, then the global current fluctu-
ations and local current fluctuations originate from two different mechanisms.
This is not very natural. We speculate that both local and global fluctuations
originate from the same mechanism: instead of the dislocations directly induce
current fluctuations, they just induce some local wavenumber gradient, and in
turn causes the rolls start to travel (note in Figure 1 of Ref. [104], the rolls are
traveling very slowly as well), and this causes the oscillations in our local cur-
rent measurements. While the traveling-roll induced current changes cancel
themselves in bulk region of the cell, they cannot cancel completely near the
boundaries. As the traveling rolls are coherent over a quite long distance, may
be over the whole cell for not very big cells. The current fluctuation amplitude
should be proportional to the length of the boundary, and therefore this can
also explain collapse of the relative fluctuation amplitude scaled by the aspect
ratio of the cells. Furthermore, this can explain the puzzle (see Sect. 5.1) that
Ref. [104] could not explain, as the traveling rolls are coherent over a long
distance, the long correlation time in ga(t) is explained.
135
We propose two experiments to test this idea. One is to measure the
local and global fluctuations simultaneously to see how they are correlated.
According to Ref. [104], they should not be correlated. If they are, this will
support our idea. Another one is to keep the area of electrodes the same and to
change the length of the boundary to see how the global fluctuation amplitude
would change. If it changes proportionally to the length of the boundary, that
can confirm our speculation.
136
Appendix A
Numerical values for plots
Fig. 2.5 (-175, 46.556), (-67, 110.865), (41, 173.917), (149, 239.243), (257, 302.821),
(365, 366.126), (473, 429.861), (581, 494.335), (689, 557.667), (797, 620.793), (905, 683.579),
(1013, 747.425), (1122, 812.131), (1230, 875.880), (1338, 939.438), (1446, 1003.634), (1554,
1067.140), (1662, 1129.794), (1770, 1192.584), (1878, 1256.388), (1986, 1320.284), (2094,
1382.139), (2203, 1445.458), (2311, 1508.867), (2419, 1570.062), (2527, 1631.330), (2635,
1693.134).
Fig. 2.7 (a) up pointing triangles: (3, 1.17020), (4, 1.17060), (5, 1.22713), (6,
1.26439), (7, 1.28168), (8, 1.30326), (9, 1.30987), (10, 1.32461), (11, 1.33358), (12, 1.33623),
(13, 1.34119), (14, 1.34298), (15, 1.34802), (16, 1.34991), (17, 1.35066), (18, 1.35230),
(19, 1.35106), (20, 1.35359), (21, 1.35751), (22, 1.35658), (23, 1.35538), (24, 1.35963),
(25, 1.36007), (26, 1.36031), (27, 1.36045), (28, 1.36068), (29, 1.36099), (30, 1.36122),
(31, 1.36146), (32, 1.36203), (33, 1.36196), (34, 1.36271), (35, 1.36231), (36, 1.36257),
(37, 1.36380), (38, 1.36358), (39, 1.36339), (40, 1.36439), (42, 1.36472), (44, 1.36450),
(46, 1.36435), (48, 1.36543), (50, 1.36533), (52, 1.36561), (54, 1.36554), (56, 1.36560),
(58, 1.36575), (60, 1.36647), (62, 1.36647), (64, 1.36625), (66, 1.36638), (68, 1.36639),
(70, 1.36677), (72, 1.36676), (74, 1.36679), (76, 1.36683), (78, 1.36698), (80, 1.36703),
(82, 1.36706), (84, 1.36723), (86, 1.36729), (88, 1.36732), (90, 1.36740), (92, 1.36743), (94,
1.36749), (96, 1.36752), (98, 1.36760), (100, 1.36759), (102, 1.36758), (104, 1.36770), (106,
1.36778), (108, 1.36764), (110, 1.36791), (112, 1.36797), (114, 1.36814), (116, 1.36838), (118,
1.36800), (120, 1.36866), (122, 1.36834), (124, 1.36816), (126, 1.36768), (128, 1.36787), (130,
1.36818), (132, 1.36802), (134, 1.36819), (136, 1.36811), (138, 1.36819), (140, 1.36823), (142,
1.36820), (144, 1.36826), (146, 1.36827), (148, 1.36828), (150, 1.36831), (152, 1.36832), (154,
1.36831), (156, 1.36834), (158, 1.36836), (160, 1.36838), (162, 1.36841), (164, 1.36840), (166,
1.36837), (168, 1.36841), (170, 1.36847), (172, 1.36849), (174, 1.36856), (176, 1.36855), (178,
1.36828), (180, 1.36884), (182, 1.36870), (184, 1.36871), (186, 1.36853), (188, 1.36847), (190,
1.36850), (192, 1.36851), (194, 1.36861), (196, 1.36862), (198, 1.36857), (200, 1.36860), (203,
137
1.36861), (206, 1.36863), (209, 1.36863), (212, 1.36867), (215, 1.36867), (218, 1.36871), (221,
1.36867), (224, 1.36867), (227, 1.36877), (230, 1.36880), (233, 1.36864), (236, 1.36910), (239,
1.36835), (242, 1.36898), (245, 1.36885), (248, 1.36872), (251, 1.36873), (254, 1.36878), (257,
1.36880), (260, 1.36882), (263, 1.36886), (266, 1.36885), (269, 1.36885), (272, 1.36886), (275,
1.36887), (278, 1.36887), (281, 1.36888), (284, 1.36888), (287, 1.36892), (290, 1.36889), (293,
1.36893), (296, 1.36897), (299, 1.36889), (302, 1.36910), (305, 1.36883), (308, 1.36897), (311,
1.36894), (314, 1.36897), (317, 1.36896), (320, 1.36896), (323, 1.36898), (326, 1.36898), (329,
1.36898), (332, 1.36899), (335, 1.36901), (338, 1.36901), (341, 1.36902), (344, 1.36903), (347,
1.36907), (350, 1.36909), (353, 1.36895), (356, 1.36901), (359, 1.36875), (362, 1.36878), (365,
1.36898), (368, 1.36902), (371, 1.36902), (374, 1.36907), (377, 1.36907), (380, 1.36908), (383,
1.36908), (386, 1.36909), (389, 1.36909), (392, 1.36911), (395, 1.36910), (398, 1.36913), (401,
1.36912), (404, 1.36913), (407, 1.36913), (410, 1.36913), (413, 1.36916), (416, 1.36913), (419,
1.36920), (422, 1.36913), (425, 1.36916), (428, 1.36917), (431, 1.36915), (434, 1.36916), (437,
1.36916), (440, 1.36918), (443, 1.36918), (446, 1.36919), (449, 1.36919), (452, 1.36919), (455,
1.36920), (458, 1.36920), (461, 1.36920), (464, 1.36921), (467, 1.36922), (470, 1.36920), (473,
1.36920), (476, 1.36921), (479, 1.36924), (482, 1.36925), (485, 1.36923), (488, 1.36924), (491,
1.36923), (494, 1.36925), (497, 1.36925), (500, 1.36925), (503, 1.36927), (506, 1.36926), (509,
1.36926), (512, 1.36927), (515, 1.36928), (518, 1.36928), (521, 1.36929), (524, 1.36929), (527,
1.36929), (530, 1.36929), (533, 1.36932), (536, 1.36933), (539, 1.36931), (542, 1.36929), (545,
1.36929), (548, 1.36933), (551, 1.36931), (554, 1.36933), (557, 1.36932), (560, 1.36933), (563,
1.36932), (566, 1.36934), (569, 1.36935), (572, 1.36935), (575, 1.36935), (578, 1.36935), (581,
1.36936), (584, 1.36936), (587, 1.36935), (590, 1.36936), (593, 1.36936), (596, 1.36935), (599,
1.36938), (602, 1.36933), (605, 1.36936), (608, 1.36941), (611, 1.36940), (614, 1.36940), (617,
1.36941), (620, 1.36940), (623, 1.36940), (626, 1.36941), (629, 1.36942), (632, 1.36941), (635,
1.36942), (638, 1.36942), (641, 1.36943), (644, 1.36943), (647, 1.36943), (650, 1.36942), (653,
1.36945), (656, 1.36948), (659, 1.36941), (662, 1.36942), (665, 1.36948), (668, 1.36945), (671,
1.36945), (674, 1.36947), (677, 1.36947), (680, 1.36947), (683, 1.36947), (686, 1.36947), (689,
1.36948), (692, 1.36947), (695, 1.36948), (698, 1.36948), (701, 1.36949), (704, 1.36949), (707,
1.36950), (710, 1.36950), (713, 1.36952), (716, 1.36955), (719, 1.36948), (722, 1.36952), (725,
1.36954), (728, 1.36954), (731, 1.36951), (734, 1.36952), (737, 1.36953), (740, 1.36952), (743,
1.36953), (746, 1.36955), (749, 1.36953), (752, 1.36954), (755, 1.36954), (758, 1.36955), (761,
1.36955), (764, 1.36955), (767, 1.36955), (770, 1.36956), (773, 1.36955), (776, 1.36954), (779,
1.36955), (782, 1.36960), (785, 1.36957), (788, 1.36958), (791, 1.36958), (794, 1.36959), (797,
1.36958), (800, 1.36959), (803, 1.36959), (806, 1.36960), (809, 1.36960), (812, 1.36960), (815,
1.36961), (818, 1.36961), (821, 1.36961), (824, 1.36962), (827, 1.36961), (830, 1.36963), (833,
1.36962), (836, 1.36964), (839, 1.36968), (842, 1.36964), (845, 1.36963), (848, 1.36965), (851,
1.36965), (854, 1.36965), (857, 1.36965), (860, 1.36965), (863, 1.36966), (866, 1.36967), (869,
1.36966), (872, 1.36967), (875, 1.36967), (878, 1.36968), (881, 1.36968), (884, 1.36968), (887,
1.36969), (890, 1.36969), (893, 1.36969), (896, 1.36970), (899, 1.36968), (902, 1.36970), (905,
138
1.36972), (908, 1.36971), (911, 1.36970), (914, 1.36971), (917, 1.36972), (920, 1.36972), (923,
1.36972), (926, 1.36972), (929, 1.36973), (932, 1.36973), (935, 1.36973), (938, 1.36974), (941,
1.36974), (944, 1.36974), (947, 1.36974), (950, 1.36975), (953, 1.36976), (956, 1.36975), (959,
1.36975), (962, 1.36977), (965, 1.36976), (968, 1.36977), (971, 1.36977), (974, 1.36977), (977,
1.36978), (980, 1.36978), (983, 1.36978), (986, 1.36978), (989, 1.36979), (992, 1.36979), (995,
1.36980), (998, 1.36980), (1001, 1.36980), (1004, 1.36980), (1007, 1.36980), (1010, 1.36981),
(1013, 1.36981), (1016, 1.36981), (1019, 1.36983), (1022, 1.36982), (1025, 1.36983), (1028,
1.36983), (1031, 1.36983), (1034, 1.36983), (1037, 1.36983), (1040, 1.36983), (1043, 1.36984),
(1046, 1.36984), (1049, 1.36986), (1052, 1.36985), (1055, 1.36986), (1058, 1.36986), (1061,
1.36986), (1064, 1.36986), (1067, 1.36987), (1070, 1.36987), (1073, 1.36986), (1076, 1.36987),
(1079, 1.36989), (1082, 1.36988), (1085, 1.36989), (1088, 1.36989), (1091, 1.36988), (1094,
1.36990), (1097, 1.36989), (1100, 1.36990), (1110, 1.36991), (1120, 1.36992), (1130, 1.36993),
(1140, 1.36995), (1150, 1.36997), (1160, 1.36996), (1170, 1.36997), (1180, 1.36999), (1190,
1.37000), (1200, 1.37002), (1210, 1.37002), (1220, 1.37003), (1230, 1.37004), (1240, 1.37006),
(1250, 1.37006), (1260, 1.37008), (1270, 1.37008), (1280, 1.37009), (1290, 1.37010), (1300,
1.37011), (1310, 1.37012), (1320, 1.37013), (1330, 1.37015), (1340, 1.37015), (1350, 1.37016),
(1360, 1.37017), (1370, 1.37018), (1380, 1.37018), (1390, 1.37020), (1400, 1.37023), (1410,
1.37022), (1420, 1.37024), (1430, 1.37024), (1440, 1.37026), (1450, 1.37026), (1460, 1.37027),
(1470, 1.37029), (1480, 1.37030), (1490, 1.37030), (1500, 1.37028), (1510, 1.37029), (1520,
1.37033), (1530, 1.37030), (1540, 1.37034), (1550, 1.37033), (1560, 1.37033), (1570, 1.37035),
(1580, 1.37039), (1590, 1.37037), (1600, 1.37037), (1610, 1.37043), (1620, 1.37040), (1630,
1.37041), (1640, 1.37041), (1650, 1.37043), (1660, 1.37044), (1670, 1.37045), (1680, 1.37047),
(1690, 1.37046), (1700, 1.37048), (1710, 1.37049), (1720, 1.37049), (1730, 1.37050), (1740,
1.37053), (1750, 1.37052), (1760, 1.37053), (1770, 1.37054), (1780, 1.37055), (1790, 1.37056),
(1800, 1.37058), (1810, 1.37058), (1820, 1.37058), (1830, 1.37060), (1840, 1.37061), (1850,
1.37062), (1860, 1.37063), (1870, 1.37064), (1880, 1.37065), (1890, 1.37065), (1900, 1.37066),
(1910, 1.37068), (1920, 1.37068), (1930, 1.37070), (1940, 1.37071), (1950, 1.37072), (1960,
1.37072), (1970, 1.37073), (1980, 1.37074), (1990, 1.37075), (2000, 1.37076), (2030, 1.37078),
(2060, 1.37082), (2090, 1.37084), (2120, 1.37088), (2150, 1.37089), (2180, 1.37092), (2210,
1.37095), (2240, 1.37098), (2270, 1.37100), (2300, 1.37102), (2330, 1.37105), (2360, 1.37108),
(2390, 1.37110), (2420, 1.37113), (2450, 1.37116), (2480, 1.37119), (2510, 1.37122), (2540,
1.37124), (2570, 1.37126), (2600, 1.37129), (2630, 1.37132), (2660, 1.37134), (2690, 1.37137),
(2720, 1.37139), (2750, 1.37142), (2780, 1.37145), (2810, 1.37147), (2840, 1.37150), (2870,
1.37153), (2900, 1.37156), (2930, 1.37159), (2960, 1.37163), (2990, 1.37164), (3020, 1.37166),
(3050, 1.37172), (3080, 1.37172), (3110, 1.37175), (3140, 1.37178), (3170, 1.37183), (3200,
1.37183), (3230, 1.37186), (3260, 1.37192), (3290, 1.37192), (3320, 1.37195), (3350, 1.37198),
(3380, 1.37203), (3410, 1.37204), (3440, 1.37206), (3470, 1.37210), (3500, 1.37214), (3530,
1.37216), (3560, 1.37218), (3590, 1.37222), (3620, 1.37227), (3650, 1.37228), (3680, 1.37232),
(3710, 1.37235), (3740, 1.37238), (3770, 1.37241), (3800, 1.37245), (3830, 1.37249), (3860,
139
1.37251), (3890, 1.37254), (3920, 1.37260), (3950, 1.37260), (3980, 1.37267), (4010, 1.37268),
(4040, 1.37271), (4070, 1.37277), (4100, 1.37278), (4130, 1.37282), (4160, 1.37285), (4190,
1.37288), (4220, 1.37292), (4250, 1.37295), (4280, 1.37298), (4310, 1.37305), (4340, 1.37306),
(4370, 1.37310), (4400, 1.37315), (4430, 1.37316), (4460, 1.37320), (4490, 1.37325), (4520,
1.37328), (4550, 1.37332), (4580, 1.37336), (4610, 1.37339), (4640, 1.37342), (4670, 1.37347),
(4700, 1.37351), (4730, 1.37355), (4760, 1.37362), (4790, 1.37363), (4820, 1.37366), (4850,
1.37373), (4880, 1.37375), (4910, 1.37378), (4940, 1.37382), (4970, 1.37390), (5000, 1.37393),
(5150, 1.37411), (5300, 1.37434), (5450, 1.37456), (5600, 1.37479), (5750, 1.37504), (5900,
1.37529), (6050, 1.37554), (6200, 1.37581), (6350, 1.37608), (6500, 1.37637), (6650, 1.37665),
(6800, 1.37695), (6950, 1.37727), (7100, 1.37760), (7250, 1.37793), (7400, 1.37824), (7550,
1.37861), (7700, 1.37898), (7850, 1.37936), (8000, 1.37983), (8150, 1.38019), (8300, 1.38062),
(8450, 1.38105), (8600, 1.38148), (8750, 1.38195), (8900, 1.38243), (9050, 1.38288), (9200,
1.38337), (9350, 1.38389), (9500, 1.38443), (9650, 1.38494), (9800, 1.38557), (9950, 1.38601),
(10100, 1.38656), (10250, 1.38707), (10400, 1.38761), (10550, 1.38820), (10700, 1.38877),
(10850, 1.38933), (11000, 1.38992), (11150, 1.39050), (11300, 1.39111), (11450, 1.39165),
(11600, 1.39234), (11750, 1.39283), (11900, 1.39342), (12050, 1.39403), (12200, 1.39472),
(12350, 1.39525), (12500, 1.39586), (12650, 1.39645), (12800, 1.39708), (12950, 1.39769),
(13100, 1.39832), (13250, 1.39895), (13400, 1.39958), (13550, 1.40026), (13700, 1.40082),
(13850, 1.40146), (14000, 1.40212), (14150, 1.40277), (14300, 1.40342), (14450, 1.40406),
(14600, 1.40474), (14750, 1.40541), (14900, 1.40608), (15050, 1.40677), (15200, 1.40745),
(15350, 1.40810), (15500, 1.40880), (15650, 1.40950), (15800, 1.41020), (15950, 1.41090),
(16100, 1.41159), (16250, 1.41231), (16400, 1.41302), (16550, 1.41376), (16700, 1.41451),
(16850, 1.41522), (17000, 1.41596), (17150, 1.41671), (17300, 1.41748), (17450, 1.41825),
(17600, 1.41901), (17750, 1.41976), (17900, 1.42053), (18050, 1.42131), (18200, 1.42209),
(18350, 1.42284), (18500, 1.42363), (18650, 1.42448), (18800, 1.42522), (18950, 1.42601),
(19100, 1.42680), (19250, 1.42761), (19400, 1.42842), (19550, 1.42924), (19700, 1.43005),
(19850, 1.43089), (20000, 1.43172).
Fig. 2.7 (a) down pointing triangles: (19925, 1.43126), (19775, 1.43043), (19625,
1.42963), (19475, 1.42881), (19325, 1.42800), (19175, 1.42719), (19025, 1.42638), (18875,
1.42561), (18725, 1.42481), (18575, 1.42402), (18425, 1.42324), (18275, 1.42249), (18125,
1.42171), (17975, 1.42092), (17825, 1.42015), (17675, 1.41938), (17525, 1.41864), (17375,
1.41788), (17225, 1.41713), (17075, 1.41637), (16925, 1.41562), (16775, 1.41490), (16625,
1.41415), (16475, 1.41348), (16325, 1.41268), (16175, 1.41197), (16025, 1.41128), (15875,
1.41063), (15725, 1.40992), (15575, 1.40916), (15425, 1.40845), (15275, 1.40776), (15125,
1.40710), (14975, 1.40641), (14825, 1.40573), (14675, 1.40505), (14525, 1.40438), (14375,
1.40372), (14225, 1.40304), (14075, 1.40237), (13925, 1.40171), (13775, 1.40106), (13625,
1.40041), (13475, 1.39977), (13325, 1.39917), (13175, 1.39854), (13025, 1.39790), (12875,
1.39727), (12725, 1.39663), (12575, 1.39602), (12425, 1.39544), (12275, 1.39481), (12125,
1.39421), (11975, 1.39360), (11825, 1.39301), (11675, 1.39251), (11525, 1.39183), (11375,
140
1.39127), (11225, 1.39068), (11075, 1.39010), (10925, 1.38953), (10775, 1.38897), (10625,
1.38840), (10475, 1.38780), (10325, 1.38736), (10175, 1.38670), (10025, 1.38618), (9875,
1.38563), (9725, 1.38508), (9575, 1.38468), (9425, 1.38404), (9275, 1.38352), (9125, 1.38301),
(8975, 1.38252), (8825, 1.38208), (8675, 1.38161), (8525, 1.38115), (8375, 1.38072), (8225,
1.38030), (8075, 1.37987), (7925, 1.37947), (7775, 1.37909), (7625, 1.37872), (7475, 1.37836),
(7325, 1.37800), (7175, 1.37774), (7025, 1.37737), (6875, 1.37705), (6725, 1.37674), (6575,
1.37645), (6425, 1.37617), (6275, 1.37589), (6125, 1.37562), (5975, 1.37535), (5825, 1.37511),
(5675, 1.37485), (5525, 1.37462), (5375, 1.37439), (5225, 1.37417), (5075, 1.37397), (4925,
1.37374), (4895, 1.37370), (4865, 1.37365), (4835, 1.37362), (4805, 1.37358), (4775, 1.37354),
(4745, 1.37350), (4715, 1.37346), (4685, 1.37342), (4655, 1.37338), (4625, 1.37334), (4595,
1.37330), (4565, 1.37327), (4535, 1.37324), (4505, 1.37320), (4475, 1.37315), (4445, 1.37316),
(4415, 1.37308), (4385, 1.37305), (4355, 1.37302), (4325, 1.37298), (4295, 1.37294), (4265,
1.37290), (4235, 1.37287), (4205, 1.37284), (4175, 1.37280), (4145, 1.37276), (4115, 1.37274),
(4085, 1.37270), (4055, 1.37267), (4025, 1.37263), (3995, 1.37260), (3965, 1.37257), (3935,
1.37253), (3905, 1.37250), (3875, 1.37246), (3845, 1.37243), (3815, 1.37240), (3785, 1.37236),
(3755, 1.37233), (3725, 1.37234), (3695, 1.37226), (3665, 1.37223), (3635, 1.37220), (3605,
1.37217), (3575, 1.37213), (3545, 1.37211), (3515, 1.37208), (3485, 1.37204), (3455, 1.37201),
(3425, 1.37199), (3395, 1.37195), (3365, 1.37197), (3335, 1.37189), (3305, 1.37186), (3275,
1.37183), (3245, 1.37180), (3215, 1.37178), (3185, 1.37174), (3155, 1.37171), (3125, 1.37169),
(3095, 1.37166), (3065, 1.37169), (3035, 1.37160), (3005, 1.37157), (2975, 1.37154), (2945,
1.37152), (2915, 1.37148), (2885, 1.37145), (2855, 1.37149), (2825, 1.37139), (2795, 1.37136),
(2765, 1.37134), (2735, 1.37131), (2705, 1.37128), (2675, 1.37124), (2645, 1.37122), (2615,
1.37127), (2585, 1.37116), (2555, 1.37113), (2525, 1.37111), (2495, 1.37109), (2465, 1.37105),
(2435, 1.37101), (2405, 1.37099), (2375, 1.37096), (2345, 1.37093), (2315, 1.37091), (2285,
1.37088), (2255, 1.37085), (2225, 1.37082), (2195, 1.37079), (2165, 1.37077), (2135, 1.37074),
(2105, 1.37071), (2075, 1.37069), (2045, 1.37066), (2015, 1.37062), (1985, 1.37060), (1975,
1.37058), (1965, 1.37058), (1955, 1.37056), (1945, 1.37055), (1935, 1.37054), (1925, 1.37053),
(1915, 1.37052), (1905, 1.37052), (1895, 1.37051), (1885, 1.37050), (1875, 1.37050), (1865,
1.37049), (1855, 1.37048), (1845, 1.37047), (1835, 1.37053), (1825, 1.37044), (1815, 1.37044),
(1805, 1.37043), (1795, 1.37042), (1785, 1.37041), (1775, 1.37040), (1765, 1.37039), (1755,
1.37037), (1745, 1.37038), (1735, 1.37037), (1725, 1.37035), (1715, 1.37035), (1705, 1.37034),
(1695, 1.37033), (1685, 1.37033), (1675, 1.37031), (1665, 1.37030), (1655, 1.37029), (1645,
1.37028), (1635, 1.37026), (1625, 1.37026), (1615, 1.37024), (1605, 1.37024), (1595, 1.37023),
(1585, 1.37022), (1575, 1.37028), (1565, 1.37027), (1555, 1.37020), (1545, 1.37018), (1535,
1.37016), (1525, 1.37015), (1515, 1.37014), (1505, 1.37014), (1495, 1.37013), (1485, 1.37014),
(1475, 1.37021), (1465, 1.37013), (1455, 1.37011), (1445, 1.37017), (1435, 1.37009), (1425,
1.37008), (1415, 1.37007), (1405, 1.37006), (1395, 1.37005), (1385, 1.37004), (1375, 1.37004),
(1365, 1.37003), (1355, 1.37001), (1345, 1.37001), (1335, 1.37000), (1325, 1.37004), (1315,
1.36998), (1305, 1.36998), (1295, 1.36996), (1285, 1.36995), (1275, 1.36994), (1265, 1.36992),
141
(1255, 1.36993), (1245, 1.36992), (1235, 1.36991), (1225, 1.36989), (1215, 1.36987), (1205,
1.36987), (1195, 1.36986), (1185, 1.36985), (1175, 1.36983), (1165, 1.36983), (1155, 1.36982),
(1145, 1.36980), (1135, 1.36979), (1125, 1.36978), (1115, 1.36981), (1105, 1.36976), (1095,
1.36975), (1092, 1.36975), (1089, 1.36975), (1086, 1.36974), (1083, 1.36975), (1080, 1.36975),
(1077, 1.36973), (1074, 1.36974), (1071, 1.36973), (1068, 1.36972), (1065, 1.36973), (1062,
1.36973), (1059, 1.36972), (1056, 1.36972), (1053, 1.36971), (1050, 1.36972), (1047, 1.36971),
(1044, 1.36971), (1041, 1.36970), (1038, 1.36970), (1035, 1.36970), (1032, 1.36969), (1029,
1.36969), (1026, 1.36969), (1023, 1.36967), (1020, 1.36969), (1017, 1.36968), (1014, 1.36969),
(1011, 1.36968), (1008, 1.36967), (1005, 1.36967), (1002, 1.36966), (999, 1.36966), (996,
1.36966), (993, 1.36965), (990, 1.36964), (987, 1.36965), (984, 1.36964), (981, 1.36964),
(978, 1.36963), (975, 1.36964), (972, 1.36963), (969, 1.36962), (966, 1.36962), (963, 1.36963),
(960, 1.36960), (957, 1.36963), (954, 1.36961), (951, 1.36960), (948, 1.36960), (945, 1.36960),
(942, 1.36959), (939, 1.36959), (936, 1.36959), (933, 1.36958), (930, 1.36958), (927, 1.36959),
(924, 1.36957), (921, 1.36957), (918, 1.36957), (915, 1.36956), (912, 1.36955), (909, 1.36957),
(906, 1.36956), (903, 1.36955), (900, 1.36957), (897, 1.36954), (894, 1.36954), (891, 1.36954),
(888, 1.36954), (885, 1.36953), (882, 1.36953), (879, 1.36953), (876, 1.36952), (873, 1.36952),
(870, 1.36952), (867, 1.36952), (864, 1.36950), (861, 1.36950), (858, 1.36951), (855, 1.36950),
(852, 1.36950), (849, 1.36949), (846, 1.36948), (843, 1.36947), (840, 1.36949), (837, 1.36947),
(834, 1.36947), (831, 1.36948), (828, 1.36947), (825, 1.36947), (822, 1.36946), (819, 1.36946),
(816, 1.36946), (813, 1.36945), (810, 1.36945), (807, 1.36945), (804, 1.36944), (801, 1.36945),
(798, 1.36944), (795, 1.36944), (792, 1.36943), (789, 1.36942), (786, 1.36944), (783, 1.36942),
(780, 1.36942), (777, 1.36941), (774, 1.36941), (771, 1.36940), (768, 1.36941), (765, 1.36940),
(762, 1.36940), (759, 1.36939), (756, 1.36939), (753, 1.36939), (750, 1.36939), (747, 1.36939),
(744, 1.36938), (741, 1.36938), (738, 1.36937), (735, 1.36937), (732, 1.36937), (729, 1.36939),
(726, 1.36935), (723, 1.36941), (720, 1.36932), (717, 1.36935), (714, 1.36933), (711, 1.36934),
(708, 1.36935), (705, 1.36934), (702, 1.36933), (699, 1.36933), (696, 1.36932), (693, 1.36932),
(690, 1.36931), (687, 1.36931), (684, 1.36930), (681, 1.36931), (678, 1.36930), (675, 1.36930),
(672, 1.36930), (669, 1.36928), (666, 1.36928), (663, 1.36930), (660, 1.36925), (657, 1.36925),
(654, 1.36927), (651, 1.36925), (648, 1.36927), (645, 1.36926), (642, 1.36925), (639, 1.36926),
(636, 1.36925), (633, 1.36925), (630, 1.36923), (627, 1.36924), (624, 1.36923), (621, 1.36923),
(618, 1.36922), (615, 1.36922), (612, 1.36922), (609, 1.36922), (606, 1.36919), (603, 1.36923),
(600, 1.36915), (597, 1.36916), (594, 1.36920), (591, 1.36921), (588, 1.36919), (585, 1.36918),
(582, 1.36917), (579, 1.36917), (576, 1.36917), (573, 1.36916), (570, 1.36917), (567, 1.36916),
(564, 1.36915), (561, 1.36914), (558, 1.36915), (555, 1.36914), (552, 1.36913), (549, 1.36914),
(546, 1.36912), (543, 1.36918), (540, 1.36911), (537, 1.36911), (534, 1.36912), (531, 1.36912),
(528, 1.36911), (525, 1.36911), (522, 1.36909), (519, 1.36909), (516, 1.36909), (513, 1.36910),
(510, 1.36908), (507, 1.36906), (504, 1.36907), (501, 1.36906), (498, 1.36905), (495, 1.36906),
(492, 1.36903), (489, 1.36906), (486, 1.36908), (483, 1.36910), (480, 1.36909), (477, 1.36899),
(474, 1.36902), (471, 1.36900), (468, 1.36901), (465, 1.36899), (462, 1.36899), (459, 1.36899),
142
(456, 1.36899), (453, 1.36899), (450, 1.36897), (447, 1.36898), (444, 1.36897), (441, 1.36897),
(438, 1.36896), (435, 1.36896), (432, 1.36895), (429, 1.36895), (426, 1.36894), (423, 1.36891),
(420, 1.36895), (417, 1.36894), (414, 1.36895), (411, 1.36891), (408, 1.36891), (405, 1.36890),
(402, 1.36890), (399, 1.36890), (396, 1.36889), (393, 1.36888), (390, 1.36889), (387, 1.36887),
(384, 1.36888), (381, 1.36888), (378, 1.36887), (375, 1.36888), (372, 1.36881), (369, 1.36879),
(366, 1.36876), (363, 1.36873), (360, 1.36907), (357, 1.36885), (354, 1.36875), (351, 1.36880),
(348, 1.36883), (345, 1.36881), (342, 1.36879), (339, 1.36881), (336, 1.36879), (333, 1.36879),
(330, 1.36878), (327, 1.36878), (324, 1.36878), (321, 1.36876), (318, 1.36878), (315, 1.36877),
(312, 1.36874), (309, 1.36877), (306, 1.36867), (303, 1.36884), (300, 1.36885), (297, 1.36882),
(294, 1.36881), (291, 1.36876), (288, 1.36871), (285, 1.36870), (282, 1.36867), (279, 1.36867),
(276, 1.36867), (273, 1.36865), (270, 1.36865), (267, 1.36863), (264, 1.36862), (261, 1.36865),
(258, 1.36858), (255, 1.36861), (252, 1.36852), (249, 1.36849), (246, 1.36846), (243, 1.36846),
(240, 1.36888), (237, 1.36857), (234, 1.36874), (231, 1.36864), (228, 1.36859), (225, 1.36854),
(222, 1.36852), (219, 1.36848), (216, 1.36849), (213, 1.36847), (210, 1.36844), (207, 1.36843),
(204, 1.36842), (201, 1.36842), (198, 1.36837), (196, 1.36836), (194, 1.36840), (192, 1.36838),
(190, 1.36837), (188, 1.36840), (186, 1.36835), (184, 1.36815), (182, 1.36823), (180, 1.36824),
(178, 1.36821), (176, 1.36845), (174, 1.36835), (172, 1.36823), (170, 1.36831), (168, 1.36824),
(166, 1.36825), (164, 1.36822), (162, 1.36817), (160, 1.36817), (158, 1.36816), (156, 1.36816),
(154, 1.36814), (152, 1.36813), (150, 1.36810), (148, 1.36809), (146, 1.36808), (144, 1.36803),
(142, 1.36806), (140, 1.36798), (138, 1.36805), (136, 1.36806), (134, 1.36802), (132, 1.36799),
(130, 1.36783), (128, 1.36811), (126, 1.36783), (124, 1.36830), (122, 1.36842), (120, 1.36726),
(118, 1.36789), (116, 1.36817), (114, 1.36767), (112, 1.36788), (110, 1.36776), (108, 1.36752),
(106, 1.36763), (104, 1.36753), (102, 1.36750), (100, 1.36752), (98, 1.36746), (96, 1.36742),
(94, 1.36731), (92, 1.36731), (90, 1.36725), (88, 1.36721), (86, 1.36713), (84, 1.36706),
(82, 1.36703), (80, 1.36695), (78, 1.36679), (76, 1.36669), (74, 1.36667), (72, 1.36668),
(70, 1.36660), (68, 1.36647), (66, 1.36626), (64, 1.36606), (62, 1.36608), (60, 1.36562),
(58, 1.36577), (56, 1.36562), (54, 1.36571), (52, 1.36509), (50, 1.36485), (48, 1.36465),
(46, 1.36454), (44, 1.36464), (42, 1.36394), (40, 1.36328), (38, 1.36351), (37, 1.36334),
(36, 1.36297), (35, 1.36230), (34, 1.36285), (33, 1.36233), (32, 1.36164), (31, 1.36146),
(30, 1.36119), (29, 1.36092), (28, 1.36029), (27, 1.36051), (26, 1.35846), (25, 1.35984),
(24, 1.35679), (23, 1.35561), (22, 1.35875), (21, 1.35908), (20, 1.35183), (19, 1.35726),
(18, 1.35213), (17, 1.34835), (16, 1.34850), (15, 1.34671), (14, 1.34354), (13, 1.34029), (12,
1.33544), (11, 1.32674), (10, 1.32478), (9, 1.32566), (8, 1.30517), (7, 1.28317), (6, 1.26380),
(5, 1.23785), (4, 1.16808).
Fig. 2.7 (b) up pointing triangles: (3, 7.79), (4, 6.03), (5, 4.97), (6, 4.24), (7, 3.70),
(8, 3.29), (9, 3.02), (10, 2.77), (11, 2.54), (12, 2.36), (13, 2.19), (14, 2.05), (15, 1.92), (16,
1.80), (17, 1.75), (18, 1.65), (19, 1.57), (20, 1.52), (21, 1.45), (22, 1.38), (23, 1.32), (24,
1.26), (25, 1.26), (26, 1.21), (27, 1.16), (28, 1.11), (29, 1.07), (30, 1.06), (31, 1.02), (32,
0.98), (33, 0.99), (34, 0.96), (35, 0.92), (36, 0.90), (37, 0.86), (38, 0.84), (39, 0.80), (40,
143
0.80), (42, 0.80), (44, 0.76), (46, 0.71), (48, 0.66), (50, 0.71), (52, 0.66), (54, 0.62), (56,
0.58), (58, 0.60), (60, 0.59), (62, 0.55), (64, 0.52), (66, 0.53), (68, 0.50), (70, 0.50), (72,
0.47), (74, 0.48), (76, 0.46), (78, 0.43), (80, 0.43), (82, 0.45), (84, 0.42), (86, 0.40), (88,
0.37), (90, 0.37), (92, 0.40), (94, 0.37), (96, 0.34), (98, 0.31), (100, 0.38), (102, 0.35), (104,
0.33), (106, 0.30), (108, 0.33), (110, 0.33), (112, 0.31), (114, 0.28), (116, 0.31), (118, 0.29),
(120, 0.32), (122, 0.27), (124, 0.29), (126, 0.27), (128, 0.25), (130, 0.26), (132, 0.28), (134,
0.26), (136, 0.24), (138, 0.22), (140, 0.27), (142, 0.25), (144, 0.23), (146, 0.21), (148, 0.23),
(150, 0.25), (152, 0.23), (154, 0.21), (156, 0.24), (158, 0.22), (160, 0.22), (162, 0.20), (164,
0.23), (166, 0.21), (168, 0.19), (170, 0.20), (172, 0.18), (174, 0.20), (176, 0.18), (178, 0.16),
(180, 0.17), (182, 0.20), (184, 0.18), (186, 0.16), (188, 0.14), (190, 0.20), (192, 0.18), (194,
0.15), (196, 0.14), (198, 0.16), (200, 0.18), (203, 0.15), (206, 0.17), (209, 0.14), (212, 0.14),
(215, 0.16), (218, 0.12), (221, 0.12), (224, 0.14), (227, 0.11), (230, 0.16), (233, 0.14), (236,
0.10), (239, 0.12), (242, 0.12), (245, 0.09), (248, 0.11), (251, 0.12), (254, 0.14), (257, 0.11),
(260, 0.11), (263, 0.13), (266, 0.10), (269, 0.07), (272, 0.12), (275, 0.09), (278, 0.07), (281,
0.11), (284, 0.09), (287, 0.11), (290, 0.11), (293, 0.07), (296, 0.10), (299, 0.07), (302, 0.08),
(305, 0.10), (308, 0.07), (311, 0.07), (314, 0.09), (317, 0.06), (320, 0.11), (323, 0.08), (326,
0.05), (329, 0.07), (332, 0.07), (335, 0.05), (338, 0.06), (341, 0.07), (344, 0.05), (347, 0.07),
(350, 0.07), (353, 0.09), (356, 0.06), (359, 0.03), (362, 0.08), (365, 0.05), (368, 0.02), (371,
0.07), (374, 0.05), (377, 0.07), (380, 0.07), (383, 0.03), (386, 0.06), (389, 0.03), (392, 0.04),
(395, 0.06), (398, 0.03), (401, 0.03), (404, 0.05), (407, 0.02), (410, 0.07), (413, 0.05), (416,
0.01), (419, 0.03), (422, 0.04), (425, 0.01), (428, 0.03), (431, 0.03), (434, 0.00), (437, 0.02),
(440, 0.03), (443, 0.06), (446, 0.03), (449, 0.00), (452, 0.05), (455, 0.02), (458, 0.00), (461,
0.05), (464, 0.01), (467, 0.03), (470, 0.03), (473, 0.01), (476, 0.03), (479, 0.00), (482, 0.00),
(485, 0.02), (488, 0.00), (491, 0.01), (494, 0.03), (497, 0.00), (500, 0.05), (503, 0.02), (506,
0.00), (509, 0.01), (512, 0.01), (515, -0.01), (518, 0.01), (521, 0.01), (524, -0.01), (527, 0.00),
(530, 0.00), (533, 0.02), (536, 0.00), (539, -0.02), (542, 0.02), (545, 0.00), (548, -0.02), (551,
0.02), (554, 0.00), (557, -0.03), (560, 0.01), (563, -0.01), (566, 0.01), (569, -0.01), (572, -0.01),
(575, 0.00), (578, -0.02), (581, -0.02), (584, 0.00), (587, -0.02), (590, 0.03), (593, 0.00), (596,
-0.02), (599, 0.00), (602, 0.00), (605, -0.03), (608, -0.01), (611, 0.00), (614, -0.03), (617,
-0.01), (620, -0.01), (623, 0.00), (626, -0.02), (629, -0.05), (632, 0.00), (635, -0.02), (638,
-0.05), (641, 0.00), (644, -0.02), (647, -0.05), (650, 0.00), (653, -0.02), (656, 0.00), (659,
-0.03), (662, -0.03), (665, -0.01), (668, -0.03), (671, -0.03), (674, -0.01), (677, -0.05), (680,
0.01), (683, -0.01), (686, -0.03), (689, -0.02), (692, -0.01), (695, -0.05), (698, -0.02), (701,
-0.02), (704, -0.05), (707, -0.03), (710, -0.03), (713, -0.01), (716, -0.03), (719, -0.06), (722,
-0.01), (725, -0.04), (728, -0.07), (731, -0.01), (734, -0.03), (737, -0.06), (740, -0.01), (743,
-0.04), (746, -0.02), (749, -0.05), (752, -0.05), (755, -0.02), (758, -0.05), (761, -0.05), (764,
-0.03), (767, -0.06), (770, -0.06), (773, -0.03), (776, -0.07), (779, -0.05), (782, -0.03), (785,
-0.06), (788, -0.04), (791, -0.03), (794, -0.07), (797, -0.05), (800, -0.05), (803, -0.02), (806,
-0.05), (809, -0.08), (812, -0.03), (815, -0.06), (818, -0.08), (821, -0.03), (824, -0.06), (827,
144
-0.09), (830, -0.03), (833, -0.06), (836, -0.03), (839, -0.06), (842, -0.06), (845, -0.04), (848,
-0.07), (851, -0.07), (854, -0.05), (857, -0.07), (860, -0.07), (863, -0.05), (866, -0.08), (869,
-0.06), (872, -0.06), (875, -0.08), (878, -0.07), (881, -0.05), (884, -0.08), (887, -0.06), (890,
-0.06), (893, -0.03), (896, -0.07), (899, -0.09), (902, -0.04), (905, -0.07), (908, -0.09), (911,
-0.05), (914, -0.07), (917, -0.10), (920, -0.05), (923, -0.08), (926, -0.06), (929, -0.09), (932,
-0.07), (935, -0.06), (938, -0.08), (941, -0.08), (944, -0.06), (947, -0.09), (950, -0.09), (953,
-0.07), (956, -0.09), (959, -0.07), (962, -0.07), (965, -0.10), (968, -0.08), (971, -0.07), (974,
-0.10), (977, -0.08), (980, -0.07), (983, -0.10), (986, -0.08), (989, -0.11), (992, -0.06), (995,
-0.08), (998, -0.11), (1001, -0.06), (1004, -0.09), (1007, -0.12), (1010, -0.07), (1013, -0.09),
(1016, -0.07), (1019, -0.10), (1022, -0.09), (1025, -0.07), (1028, -0.09), (1031, -0.09), (1034,
-0.07), (1037, -0.10), (1040, -0.10), (1043, -0.08), (1046, -0.11), (1049, -0.08), (1052, -0.08),
(1055, -0.11), (1058, -0.09), (1061, -0.09), (1064, -0.12), (1067, -0.09), (1070, -0.09), (1073,
-0.12), (1076, -0.10), (1079, -0.13), (1082, -0.07), (1085, -0.09), (1088, -0.12), (1091, -0.07),
(1094, -0.10), (1097, -0.13), (1100, -0.08), (1110, -0.09), (1120, -0.11), (1130, -0.12), (1140,
-0.08), (1150, -0.09), (1160, -0.11), (1170, -0.12), (1180, -0.09), (1190, -0.10), (1200, -0.11),
(1210, -0.12), (1220, -0.13), (1230, -0.10), (1240, -0.12), (1250, -0.13), (1260, -0.13), (1270,
-0.10), (1280, -0.12), (1290, -0.13), (1300, -0.14), (1310, -0.16), (1320, -0.12), (1330, -0.13),
(1340, -0.14), (1350, -0.16), (1360, -0.12), (1370, -0.14), (1380, -0.14), (1390, -0.16), (1400,
-0.17), (1410, -0.14), (1420, -0.15), (1430, -0.16), (1440, -0.17), (1450, -0.14), (1460, -0.15),
(1470, -0.16), (1480, -0.18), (1490, -0.20), (1500, -0.15), (1510, -0.16), (1520, -0.18), (1530,
-0.20), (1540, -0.21), (1550, -0.16), (1560, -0.18), (1570, -0.20), (1580, -0.21), (1590, -0.17),
(1600, -0.19), (1610, -0.19), (1620, -0.21), (1630, -0.22), (1640, -0.19), (1650, -0.20), (1660,
-0.22), (1670, -0.22), (1680, -0.19), (1690, -0.20), (1700, -0.21), (1710, -0.23), (1720, -0.24),
(1730, -0.20), (1740, -0.21), (1750, -0.23), (1760, -0.24), (1770, -0.21), (1780, -0.22), (1790,
-0.22), (1800, -0.24), (1810, -0.25), (1820, -0.22), (1830, -0.23), (1840, -0.25), (1850, -0.25),
(1860, -0.22), (1870, -0.23), (1880, -0.25), (1890, -0.26), (1900, -0.28), (1910, -0.23), (1920,
-0.24), (1930, -0.26), (1940, -0.28), (1950, -0.24), (1960, -0.25), (1970, -0.26), (1980, -0.27),
(1990, -0.29), (2000, -0.25), (2030, -0.29), (2060, -0.28), (2090, -0.27), (2120, -0.31), (2150,
-0.29), (2180, -0.29), (2210, -0.32), (2240, -0.31), (2270, -0.30), (2300, -0.34), (2330, -0.33),
(2360, -0.32), (2390, -0.35), (2420, -0.34), (2450, -0.33), (2480, -0.37), (2510, -0.36), (2540,
-0.35), (2570, -0.38), (2600, -0.38), (2630, -0.36), (2660, -0.41), (2690, -0.39), (2720, -0.38),
(2750, -0.42), (2780, -0.41), (2810, -0.40), (2840, -0.44), (2870, -0.42), (2900, -0.42), (2930,
-0.45), (2960, -0.44), (2990, -0.47), (3020, -0.46), (3050, -0.45), (3080, -0.49), (3110, -0.48),
(3140, -0.46), (3170, -0.51), (3200, -0.49), (3230, -0.49), (3260, -0.51), (3290, -0.51), (3320,
-0.49), (3350, -0.54), (3380, -0.52), (3410, -0.51), (3440, -0.55), (3470, -0.54), (3500, -0.52),
(3530, -0.57), (3560, -0.55), (3590, -0.55), (3620, -0.58), (3650, -0.57), (3680, -0.55), (3710,
-0.60), (3740, -0.58), (3770, -0.57), (3800, -0.60), (3830, -0.60), (3860, -0.58), (3890, -0.63),
(3920, -0.60), (3950, -0.60), (3980, -0.63), (4010, -0.63), (4040, -0.61), (4070, -0.66), (4100,
-0.64), (4130, -0.63), (4160, -0.67), (4190, -0.67), (4220, -0.66), (4250, -0.69), (4280, -0.69),
145
(4310, -0.68), (4340, -0.71), (4370, -0.70), (4400, -0.70), (4430, -0.74), (4460, -0.72), (4490,
-0.71), (4520, -0.76), (4550, -0.74), (4580, -0.73), (4610, -0.78), (4640, -0.77), (4670, -0.75),
(4700, -0.79), (4730, -0.78), (4760, -0.77), (4790, -0.81), (4820, -0.80), (4850, -0.80), (4880,
-0.83), (4910, -0.82), (4940, -0.81), (4970, -0.84), (5000, -0.84), (5150, -0.89), (5300, -0.93),
(5450, -0.93), (5600, -0.97), (5750, -1.02), (5900, -1.02), (6050, -1.06), (6200, -1.10), (6350,
-1.09), (6500, -1.14), (6650, -1.19), (6800, -1.18), (6950, -1.22), (7100, -1.26), (7250, -1.25),
(7400, -1.28), (7550, -1.33), (7700, -1.37), (7850, -1.35), (8000, -1.40), (8150, -1.44), (8300,
-1.44), (8450, -1.49), (8600, -1.53), (8750, -1.53), (8900, -1.58), (9050, -1.62), (9200, -1.62),
(9350, -1.67), (9500, -1.71), (9650, -1.76), (9800, -1.74), (9950, -1.79), (10100, -1.84), (10250,
-1.82), (10400, -1.86), (10550, -1.91), (10700, -1.89), (10850, -1.94), (11000, -1.98), (11150,
-1.96), (11300, -2.01), (11450, -2.04), (11600, -2.03), (11750, -2.07), (11900, -2.11), (12050,
-2.15), (12200, -2.14), (12350, -2.19), (12500, -2.24), (12650, -2.23), (12800, -2.28), (12950,
-2.31), (13100, -2.31), (13250, -2.36), (13400, -2.39), (13550, -2.39), (13700, -2.44), (13850,
-2.48), (14000, -2.53), (14150, -2.52), (14300, -2.55), (14450, -2.60), (14600, -2.59), (14750,
-2.64), (14900, -2.68), (15050, -2.67), (15200, -2.72), (15350, -2.76), (15500, -2.75), (15650,
-2.79), (15800, -2.83), (15950, -2.82), (16100, -2.87), (16250, -2.90), (16400, -2.95), (16550,
-2.94), (16700, -2.98), (16850, -3.03), (17000, -3.02), (17150, -3.05), (17300, -3.10), (17450,
-3.10), (17600, -3.13), (17750, -3.18), (17900, -3.17), (18050, -3.22), (18200, -3.27), (18350,
-3.31), (18500, -3.31), (18650, -3.36), (18800, -3.40), (18950, -3.40), (19100, -3.45), (19250,
-3.49), (19400, -3.50), (19550, -3.55), (19700, -3.59), (19850, -3.60), (20000, -3.64).
Fig. 2.7 (b) down pointing triangles: (19925, -3.64), (19775, -3.63), (19625, -3.59),
(19475, -3.54), (19325, -3.50), (19175, -3.49), (19025, -3.44), (18875, -3.40), (18725, -3.40),
(18575, -3.35), (18425, -3.31), (18275, -3.31), (18125, -3.26), (17975, -3.22), (17825, -3.22),
(17675, -3.18), (17525, -3.14), (17375, -3.14), (17225, -3.10), (17075, -3.06), (16925, -3.01),
(16775, -3.02), (16625, -2.97), (16475, -2.93), (16325, -2.94), (16175, -2.89), (16025, -2.85),
(15875, -2.86), (15725, -2.81), (15575, -2.78), (15425, -2.78), (15275, -2.74), (15125, -2.70),
(14975, -2.66), (14825, -2.66), (14675, -2.62), (14525, -2.58), (14375, -2.58), (14225, -2.54),
(14075, -2.49), (13925, -2.50), (13775, -2.45), (13625, -2.41), (13475, -2.42), (13325, -2.37),
(13175, -2.32), (13025, -2.33), (12875, -2.29), (12725, -2.24), (12575, -2.20), (12425, -2.20),
(12275, -2.16), (12125, -2.11), (11975, -2.12), (11825, -2.08), (11675, -2.04), (11525, -2.06),
(11375, -2.02), (11225, -1.98), (11075, -1.99), (10925, -1.95), (10775, -1.91), (10625, -1.92),
(10475, -1.87), (10325, -1.84), (10175, -1.80), (10025, -1.80), (9875, -1.76), (9725, -1.72),
(9575, -1.72), (9425, -1.68), (9275, -1.64), (9125, -1.64), (8975, -1.59), (8825, -1.55), (8675,
-1.55), (8525, -1.50), (8375, -1.46), (8225, -1.41), (8075, -1.41), (7925, -1.36), (7775, -1.33),
(7625, -1.34), (7475, -1.30), (7325, -1.27), (7175, -1.28), (7025, -1.23), (6875, -1.19), (6725,
-1.20), (6575, -1.15), (6425, -1.12), (6275, -1.12), (6125, -1.08), (5975, -1.03), (5825, -0.99),
(5675, -1.00), (5525, -0.95), (5375, -0.91), (5225, -0.91), (5075, -0.86), (4925, -0.81), (4895,
-0.82), (4865, -0.84), (4835, -0.80), (4805, -0.81), (4775, -0.81), (4745, -0.78), (4715, -0.79),
(4685, -0.79), (4655, -0.76), (4625, -0.77), (4595, -0.78), (4565, -0.73), (4535, -0.75), (4505,
146
-0.76), (4475, -0.71), (4445, -0.73), (4415, -0.74), (4385, -0.70), (4355, -0.71), (4325, -0.73),
(4295, -0.68), (4265, -0.69), (4235, -0.71), (4205, -0.66), (4175, -0.67), (4145, -0.68), (4115,
-0.64), (4085, -0.65), (4055, -0.66), (4025, -0.62), (3995, -0.63), (3965, -0.60), (3935, -0.61),
(3905, -0.63), (3875, -0.58), (3845, -0.60), (3815, -0.61), (3785, -0.57), (3755, -0.58), (3725,
-0.60), (3695, -0.56), (3665, -0.57), (3635, -0.58), (3605, -0.55), (3575, -0.55), (3545, -0.57),
(3515, -0.53), (3485, -0.55), (3455, -0.55), (3425, -0.51), (3395, -0.52), (3365, -0.54), (3335,
-0.50), (3305, -0.51), (3275, -0.52), (3245, -0.49), (3215, -0.49), (3185, -0.51), (3155, -0.46),
(3125, -0.49), (3095, -0.49), (3065, -0.46), (3035, -0.46), (3005, -0.48), (2975, -0.44), (2945,
-0.45), (2915, -0.46), (2885, -0.43), (2855, -0.43), (2825, -0.45), (2795, -0.41), (2765, -0.42),
(2735, -0.43), (2705, -0.40), (2675, -0.40), (2645, -0.42), (2615, -0.37), (2585, -0.40), (2555,
-0.40), (2525, -0.36), (2495, -0.38), (2465, -0.38), (2435, -0.35), (2405, -0.36), (2375, -0.37),
(2345, -0.33), (2315, -0.35), (2285, -0.35), (2255, -0.32), (2225, -0.33), (2195, -0.34), (2165,
-0.30), (2135, -0.31), (2105, -0.32), (2075, -0.29), (2045, -0.29), (2015, -0.31), (1985, -0.27),
(1975, -0.25), (1965, -0.29), (1955, -0.28), (1945, -0.27), (1935, -0.25), (1925, -0.29), (1915,
-0.27), (1905, -0.27), (1895, -0.25), (1885, -0.24), (1875, -0.27), (1865, -0.26), (1855, -0.24),
(1845, -0.24), (1835, -0.27), (1825, -0.26), (1815, -0.24), (1805, -0.23), (1795, -0.22), (1785,
-0.26), (1775, -0.24), (1765, -0.23), (1755, -0.22), (1745, -0.20), (1735, -0.24), (1725, -0.23),
(1715, -0.22), (1705, -0.20), (1695, -0.24), (1685, -0.22), (1675, -0.21), (1665, -0.20), (1655,
-0.19), (1645, -0.22), (1635, -0.21), (1625, -0.20), (1615, -0.18), (1605, -0.22), (1595, -0.21),
(1585, -0.20), (1575, -0.18), (1565, -0.16), (1555, -0.20), (1545, -0.20), (1535, -0.19), (1525,
-0.17), (1515, -0.20), (1505, -0.19), (1495, -0.19), (1485, -0.17), (1475, -0.15), (1465, -0.19),
(1455, -0.17), (1445, -0.16), (1435, -0.16), (1425, -0.19), (1415, -0.18), (1405, -0.16), (1395,
-0.14), (1385, -0.13), (1375, -0.18), (1365, -0.16), (1355, -0.14), (1345, -0.13), (1335, -0.17),
(1325, -0.15), (1315, -0.15), (1305, -0.13), (1295, -0.12), (1285, -0.15), (1275, -0.14), (1265,
-0.12), (1255, -0.12), (1245, -0.15), (1235, -0.14), (1225, -0.12), (1215, -0.11), (1205, -0.09),
(1195, -0.14), (1185, -0.12), (1175, -0.11), (1165, -0.09), (1155, -0.13), (1145, -0.12), (1135,
-0.11), (1125, -0.10), (1115, -0.08), (1105, -0.12), (1095, -0.10), (1092, -0.07), (1089, -0.12),
(1086, -0.09), (1083, -0.07), (1080, -0.09), (1077, -0.10), (1074, -0.07), (1071, -0.09), (1068,
-0.09), (1065, -0.07), (1062, -0.09), (1059, -0.09), (1056, -0.12), (1053, -0.09), (1050, -0.06),
(1047, -0.11), (1044, -0.08), (1041, -0.05), (1038, -0.10), (1035, -0.07), (1032, -0.09), (1029,
-0.09), (1026, -0.07), (1023, -0.09), (1020, -0.06), (1017, -0.07), (1014, -0.09), (1011, -0.07),
(1008, -0.07), (1005, -0.09), (1002, -0.06), (999, -0.11), (996, -0.08), (993, -0.06), (990, -0.08),
(987, -0.08), (984, -0.05), (981, -0.07), (978, -0.08), (975, -0.06), (972, -0.08), (969, -0.08),
(966, -0.10), (963, -0.07), (960, -0.05), (957, -0.09), (954, -0.07), (951, -0.03), (948, -0.09),
(945, -0.06), (942, -0.08), (939, -0.08), (936, -0.06), (933, -0.08), (930, -0.05), (927, -0.06),
(924, -0.08), (921, -0.05), (918, -0.05), (915, -0.07), (912, -0.05), (909, -0.09), (906, -0.07),
(903, -0.04), (900, -0.06), (897, -0.07), (894, -0.03), (891, -0.06), (888, -0.06), (885, -0.03),
(882, -0.05), (879, -0.06), (876, -0.08), (873, -0.06), (870, -0.03), (867, -0.08), (864, -0.05),
(861, -0.02), (858, -0.07), (855, -0.05), (852, -0.01), (849, -0.07), (846, -0.04), (843, -0.06),
147
(840, -0.03), (837, -0.03), (834, -0.06), (831, -0.03), (828, -0.03), (825, -0.06), (822, -0.03),
(819, -0.08), (816, -0.05), (813, -0.02), (810, -0.05), (807, -0.05), (804, -0.02), (801, -0.04),
(798, -0.04), (795, -0.01), (792, -0.03), (789, -0.03), (786, -0.06), (783, -0.03), (780, 0.00),
(777, -0.06), (774, -0.03), (771, -0.01), (768, -0.06), (765, -0.03), (762, 0.00), (759, -0.05),
(756, -0.02), (753, -0.05), (750, -0.01), (747, -0.02), (744, -0.04), (741, -0.01), (738, -0.01),
(735, -0.03), (732, -0.01), (729, -0.06), (726, -0.03), (723, -0.01), (720, -0.03), (717, -0.03),
(714, -0.00), (711, -0.02), (708, -0.03), (705, 0.00), (702, -0.02), (699, -0.02), (696, -0.04),
(693, -0.01), (690, 0.01), (687, -0.03), (684, -0.01), (681, 0.01), (678, -0.04), (675, -0.01),
(672, 0.01), (669, -0.03), (666, -0.01), (663, -0.03), (660, 0.00), (657, 0.00), (654, -0.02),
(651, 0.00), (648, 0.00), (645, -0.01), (642, 0.00), (639, 0.00), (636, -0.02), (633, 0.00), (630,
-0.01), (627, -0.01), (624, 0.01), (621, -0.01), (618, -0.01), (615, 0.01), (612, 0.00), (609,
0.00), (606, -0.02), (603, 0.00), (600, 0.03), (597, -0.02), (594, 0.00), (591, 0.03), (588, -
0.02), (585, 0.00), (582, 0.02), (579, -0.01), (576, 0.00), (573, -0.01), (570, 0.01), (567, 0.01),
(564, 0.00), (561, 0.01), (558, 0.01), (555, 0.00), (552, 0.02), (549, 0.02), (546, 0.00), (543,
0.03), (540, 0.01), (537, 0.00), (534, 0.02), (531, 0.00), (528, 0.00), (525, 0.03), (522, 0.01),
(519, 0.01), (516, 0.00), (513, 0.01), (510, 0.05), (507, 0.00), (504, 0.02), (501, 0.05), (498,
0.00), (495, 0.03), (492, 0.06), (489, 0.00), (486, 0.02), (483, 0.01), (480, 0.03), (477, 0.03),
(474, 0.01), (471, 0.04), (468, 0.04), (465, 0.02), (462, 0.05), (459, 0.05), (456, 0.02), (453,
0.05), (450, 0.03), (447, 0.03), (444, 0.06), (441, 0.04), (438, 0.03), (435, 0.06), (432, 0.03),
(429, 0.03), (426, 0.06), (423, 0.05), (420, 0.07), (417, 0.02), (414, 0.05), (411, 0.08), (408,
0.03), (405, 0.06), (402, 0.09), (399, 0.03), (396, 0.07), (393, 0.05), (390, 0.07), (387, 0.06),
(384, 0.04), (381, 0.07), (378, 0.07), (375, 0.05), (372, 0.08), (369, 0.07), (366, 0.06), (363,
0.09), (360, 0.06), (357, 0.07), (354, 0.09), (351, 0.07), (348, 0.07), (345, 0.10), (342, 0.08),
(339, 0.07), (336, 0.10), (333, 0.08), (330, 0.11), (327, 0.06), (324, 0.09), (321, 0.12), (318,
0.07), (315, 0.10), (312, 0.12), (309, 0.07), (306, 0.10), (303, 0.09), (300, 0.11), (297, 0.10),
(294, 0.08), (291, 0.11), (288, 0.11), (285, 0.09), (282, 0.12), (279, 0.12), (276, 0.10), (273,
0.13), (270, 0.11), (267, 0.11), (264, 0.14), (261, 0.12), (258, 0.12), (255, 0.15), (252, 0.13),
(249, 0.12), (246, 0.15), (243, 0.13), (240, 0.14), (237, 0.11), (234, 0.14), (231, 0.17), (228,
0.12), (225, 0.15), (222, 0.18), (219, 0.13), (216, 0.16), (213, 0.19), (210, 0.17), (207, 0.18),
(204, 0.16), (201, 0.19), (198, 0.18), (196, 0.15), (194, 0.17), (192, 0.19), (190, 0.21), (188,
0.15), (186, 0.17), (184, 0.20), (182, 0.22), (180, 0.17), (178, 0.18), (176, 0.20), (174, 0.22),
(172, 0.19), (170, 0.21), (168, 0.20), (166, 0.22), (164, 0.24), (162, 0.22), (160, 0.24), (158,
0.23), (156, 0.25), (154, 0.22), (152, 0.24), (150, 0.27), (148, 0.25), (146, 0.22), (144, 0.24),
(142, 0.27), (140, 0.29), (138, 0.23), (136, 0.25), (134, 0.28), (132, 0.30), (130, 0.27), (128,
0.27), (126, 0.29), (124, 0.31), (122, 0.29), (120, 0.33), (118, 0.30), (116, 0.33), (114, 0.30),
(112, 0.33), (110, 0.35), (108, 0.35), (106, 0.31), (104, 0.35), (102, 0.37), (100, 0.40), (98,
0.33), (96, 0.36), (94, 0.38), (92, 0.41), (90, 0.38), (88, 0.38), (86, 0.41), (84, 0.43), (82, 0.46),
(80, 0.44), (78, 0.44), (76, 0.47), (74, 0.50), (72, 0.48), (70, 0.51), (68, 0.51), (66, 0.55), (64,
0.53), (62, 0.57), (60, 0.60), (58, 0.61), (56, 0.60), (54, 0.64), (52, 0.67), (50, 0.72), (48, 0.67),
148
(46, 0.72), (44, 0.77), (42, 0.82), (40, 0.82), (38, 0.85), (37, 0.88), (36, 0.91), (35, 0.94), (34,
0.98), (33, 1.01), (32, 1.00), (31, 1.03), (30, 1.07), (29, 1.08), (28, 1.13), (27, 1.17), (26, 1.22),
(25, 1.27), (24, 1.27), (23, 1.33), (22, 1.39), (21, 1.46), (20, 1.53), (19, 1.58), (18, 1.67), (17,
1.76), (16, 1.81), (15, 1.93), (14, 2.06), (13, 2.20), (12, 2.37), (11, 2.56), (10, 2.78), (9, 3.03),
(8, 3.30), (7, 3.71), (6, 4.25), (5, 4.98), (4, 6.04).
Fig. 2.11 (189.750, 0.000), (190.120, 55.000), (190.490, 53.200), (190.860, 54.000),
(191.230, 52.200), (191.600, 58.000), (191.970, 55.200), (192.340, 52.200), (192.710, 54.800),
(193.080, 57.400), (193.450, 58.800), (193.820, 58.200), (194.190, 59.200), (194.560, 60.000),
(194.930, 58.800), (195.300, 61.000), (195.670, 62.400), (196.040, 63.800), (196.410, 62.600),
(196.780, 63.600), (197.150, 64.000), (197.520, 63.200), (197.880, 63.455), (198.250, 63.491),
(198.620, 63.509), (198.990, 63.418), (199.360, 63.473), (199.730, 63.782), (200.100, 63.800),
(200.470, 63.764), (200.840, 64.036), (201.210, 63.800), (201.580, 63.927), (201.950, 63.836),
(202.320, 63.691), (202.680, 63.600), (203.050, 63.473), (203.420, 63.491), (203.790, 63.073),
(204.160, 63.200), (204.530, 63.255), (204.900, 63.109), (205.270, 63.182), (205.630, 63.018),
(206.000, 62.982), (206.370, 62.982), (206.740, 63.273), (207.110, 63.618), (207.480, 63.345),
(207.840, 63.345), (208.210, 63.582), (208.580, 63.745), (208.950, 63.709), (209.320, 63.655),
(209.690, 63.927), (210.050, 63.818), (210.420, 64.091), (210.790, 63.964), (211.160, 63.982),
(211.530, 64.109), (211.890, 64.545), (212.260, 64.345), (212.630, 64.345), (213.000, 64.418),
(213.370, 64.545), (213.730, 64.527), (214.100, 64.800), (214.470, 64.564), (214.840, 64.473),
(215.200, 64.382), (215.570, 64.473), (215.940, 64.236), (216.310, 64.291), (216.670, 64.400),
(217.040, 64.436), (217.410, 64.509), (217.780, 64.491), (218.140, 64.473), (218.510, 64.655),
(218.880, 64.964), (219.240, 64.982), (219.610, 65.055), (219.980, 65.164), (220.350, 65.127),
(220.710, 64.982), (221.080, 65.109), (221.450, 64.836), (221.810, 64.764), (222.180, 64.745),
(222.550, 64.673), (222.910, 64.491), (223.280, 64.491), (223.650, 64.291), (224.010, 64.127),
(224.380, 64.109), (224.750, 64.127), (225.110, 63.909), (225.480, 64.236), (225.850, 64.436),
(226.210, 64.273), (226.580, 64.345), (226.950, 64.364), (227.310, 64.455), (227.680, 64.509),
(228.040, 64.382), (228.410, 64.145), (228.780, 63.964), (229.140, 63.909), (229.510, 63.800),
(229.870, 63.655), (230.240, 63.655), (230.610, 63.436), (230.970, 63.618), (231.340, 63.382),
(231.700, 63.491), (232.070, 63.691), (232.440, 64.091), (232.800, 64.073), (233.170, 64.145),
(233.530, 64.182), (233.900, 64.055), (234.260, 64.091), (234.630, 64.055), (234.990, 63.836),
(235.360, 63.964), (235.730, 63.891), (236.090, 63.836), (236.460, 63.764), (236.820, 64.018),
(237.190, 64.200), (237.550, 64.127), (237.920, 64.255), (238.280, 64.291), (238.650, 64.418),
(239.010, 64.600), (239.380, 64.564), (239.740, 64.655), (240.110, 64.545), (240.470, 64.527),
(240.840, 64.527), (241.200, 64.182), (241.570, 64.073), (241.930, 63.873), (242.300, 63.982),
(242.660, 64.345), (243.020, 64.164), (243.390, 64.345), (243.750, 64.255), (244.120, 64.509),
(244.480, 64.382), (244.850, 64.200), (245.210, 64.455), (245.580, 64.473), (245.940, 64.636),
(246.300, 64.564), (246.670, 64.109), (247.030, 64.182), (247.400, 64.182), (247.760, 64.364),
(248.130, 64.127), (248.490, 64.145), (248.850, 64.164), (249.220, 64.236), (249.580, 64.400),
(249.940, 64.436), (250.310, 64.418), (250.670, 64.473), (251.040, 64.473), (251.400, 64.382),
149
(251.760, 64.164), (252.130, 64.255), (252.490, 64.400), (252.850, 64.455), (253.220, 64.145),
(253.580, 64.164), (253.940, 64.400), (254.310, 64.582), (254.670, 64.636), (255.030, 64.691),
(255.400, 64.636), (255.760, 64.891), (256.120, 64.945), (256.490, 64.891), (256.850, 64.909),
(257.210, 65.018), (257.580, 65.091), (257.940, 65.109), (258.300, 65.036), (258.660, 65.109),
(259.030, 65.018), (259.390, 64.873), (259.750, 64.691), (260.120, 64.582), (260.480, 64.527),
(260.840, 64.509), (261.200, 64.436), (261.570, 64.291), (261.930, 64.109), (262.290, 64.055),
(262.650, 64.127), (263.020, 64.164), (263.380, 64.400), (263.740, 64.436), (264.100, 64.491),
(264.470, 64.418), (264.830, 64.491), (265.190, 64.945), (265.550, 64.982), (265.910, 65.055),
(266.280, 65.145), (266.640, 65.345), (267.000, 65.345), (267.360, 65.291), (267.720, 65.200),
(268.090, 65.182), (268.450, 65.273), (268.810, 65.345), (269.170, 65.109), (269.530, 65.018),
(269.900, 64.909), (270.260, 64.855), (270.620, 64.218), (270.980, 64.000), (271.340, 63.945),
(271.700, 64.055), (272.060, 64.309), (272.430, 64.327), (272.790, 64.091), (273.150, 64.000),
(273.510, 64.036), (273.870, 64.200), (274.230, 64.109), (274.590, 64.473), (274.950, 64.818),
(275.320, 64.782), (275.680, 64.855), (276.040, 64.636), (276.400, 64.655), (276.760, 64.836),
(277.120, 64.891), (277.480, 64.818), (277.840, 64.655), (278.200, 64.636), (278.560, 64.745),
(278.920, 64.855), (279.280, 65.055), (279.650, 65.018), (280.010, 65.145), (280.370, 65.109),
(280.730, 65.255), (281.090, 65.309), (281.450, 65.418), (281.810, 65.600), (282.170, 65.436),
(282.530, 65.527), (282.890, 65.436), (283.250, 65.509), (283.610, 65.509), (283.970, 65.509),
(284.330, 65.527), (284.690, 65.255), (285.050, 65.145), (285.410, 65.291), (285.770, 65.109),
(286.130, 65.164), (286.490, 65.273), (286.850, 65.127), (287.210, 65.073), (287.570, 65.218),
(287.930, 64.982), (288.290, 64.800), (288.650, 64.618), (289.010, 64.855), (289.370, 64.891),
(289.730, 64.873), (290.090, 65.000), (290.450, 64.818), (290.800, 65.018), (291.160, 64.964),
(291.520, 64.891), (291.880, 65.109), (292.240, 65.436), (292.600, 65.745), (292.960, 65.618),
(293.320, 65.818), (293.680, 65.818), (294.040, 66.055), (294.400, 66.418), (294.760, 67.327),
(295.110, 69.109), (295.470, 70.618), (295.830, 71.473), (296.190, 71.655), (296.550, 71.727),
(296.910, 71.745), (297.270, 71.455), (297.630, 71.691), (297.980, 71.600), (298.340, 71.600),
(298.700, 70.873), (299.060, 69.618), (299.420, 69.055), (299.780, 70.327), (300.130, 75.745),
(300.490, 85.491), (300.850, 96.327), (301.210, 104.636), (301.570, 108.655), (301.930, 110.073),
(302.280, 110.255), (302.640, 110.309), (303.000, 110.018), (303.360, 109.382), (303.720,
107.545), (304.070, 102.455), (304.430, 93.018), (304.790, 82.764), (305.150, 75.018), (305.510,
71.400), (305.860, 70.582), (306.220, 70.655), (306.580, 70.909), (306.940, 71.382), (307.290,
71.636), (307.650, 72.055), (308.010, 72.382), (308.370, 72.964), (308.720, 73.491), (309.080,
74.473), (309.440, 75.873), (309.800, 79.273), (310.150, 89.927), (310.510, 121.436), (310.870,
190.655), (311.220, 304.073), (311.580, 439.364), (311.940, 557.545), (312.300, 634.055),
(312.650, 667.945), (313.010, 678.855), (313.370, 681.600), (313.720, 680.455), (314.080,
671.509), (314.440, 641.636), (314.790, 573.691), (315.150, 461.691), (315.510, 327.855),
(315.860, 210.945), (316.220, 135.564), (316.580, 102.745), (316.930, 92.364), (317.290,
89.745), (317.650, 89.236), (318.000, 89.073), (318.360, 88.982), (318.720, 89.345), (319.070,
89.600), (319.430, 89.982), (319.780, 90.255), (320.140, 90.891), (320.500, 91.782), (320.850,
150
92.527), (321.210, 93.400), (321.560, 94.345), (321.920, 94.873), (322.280, 95.945), (322.630,
96.745), (322.990, 97.964), (323.340, 98.891), (323.700, 100.091), (324.060, 101.036), (324.410,
101.764), (324.770, 103.164), (325.120, 104.455), (325.480, 105.491), (325.830, 107.200),
(326.190, 108.600), (326.540, 110.091), (326.900, 111.491), (327.250, 112.873), (327.610,
114.200), (327.970, 115.418), (328.320, 116.673), (328.680, 117.727), (329.030, 118.855),
(329.390, 119.855), (329.740, 121.091), (330.100, 122.055), (330.450, 123.382), (330.810,
124.618), (331.160, 126.491), (331.520, 130.836), (331.870, 143.436), (332.230, 177.764),
(332.580, 240.109), (332.940, 313.727), (333.290, 377.473), (333.640, 413.418), (334.000,
430.909), (334.350, 437.164), (334.710, 439.545), (335.060, 440.255), (335.420, 437.873),
(335.770, 427.636), (336.130, 395.291), (336.480, 334.745), (336.830, 262.636), (337.190,
201.127), (337.540, 167.655), (337.900, 152.982), (338.250, 149.109), (338.610, 148.309),
(338.960, 148.055), (339.310, 148.073), (339.670, 147.855), (340.020, 147.727), (340.380,
147.745), (340.730, 147.400), (341.080, 146.382), (341.440, 145.055), (341.790, 143.855),
(342.140, 142.309), (342.500, 141.255), (342.850, 140.727), (343.210, 140.545), (343.560,
140.145), (343.910, 139.691), (344.270, 139.018), (344.620, 138.745), (344.970, 138.527),
(345.330, 137.655), (345.680, 137.127), (346.030, 136.600), (346.390, 136.291), (346.740,
135.891), (347.090, 135.473), (347.450, 135.145), (347.800, 134.764), (348.150, 134.382),
(348.500, 133.927), (348.860, 133.545), (349.210, 133.473), (349.560, 132.491), (349.920,
131.655), (350.270, 131.018), (350.620, 130.673), (350.970, 130.200), (351.330, 130.036),
(351.680, 129.873), (352.030, 129.964), (352.390, 130.727), (352.740, 132.236), (353.090,
134.491), (353.440, 137.200), (353.800, 138.945), (354.150, 139.655), (354.500, 140.218),
(354.850, 140.691), (355.200, 140.964), (355.560, 140.836), (355.910, 140.127), (356.260,
138.927), (356.610, 136.509), (356.960, 134.182), (357.320, 132.236), (357.670, 132.345),
(358.020, 134.127), (358.370, 135.545), (358.720, 136.727), (359.080, 137.436), (359.430,
138.745), (359.780, 139.855), (360.130, 141.527), (360.480, 143.764), (360.830, 146.400),
(361.190, 149.582), (361.540, 154.382), (361.890, 162.727), (362.240, 184.418), (362.590,
275.509), (362.940, 495.691), (363.300, 733.091), (363.650, 989.418), (364.000, 1253.764),
(364.350, 1514.273), (364.700, 1763.036), (365.050, 1999.491), (365.400, 2224.091), (365.750,
2436.746), (366.100, 2631.564), (366.460, 2688.873), (366.810, 2503.800), (367.160, 2281.546),
(367.510, 2034.255), (367.860, 1775.709), (368.210, 1519.418), (368.560, 1273.400), (368.910,
1039.982), (369.260, 816.291), (369.610, 598.982), (369.960, 383.927), (370.310, 235.309),
(370.660, 199.055), (371.010, 181.545), (371.370, 170.109), (371.720, 160.927), (372.070,
152.982), (372.420, 145.255), (372.770, 135.982), (373.120, 126.727), (373.470, 119.200),
(373.820, 114.764), (374.170, 112.564), (374.520, 111.491), (374.870, 110.800), (375.220,
110.545), (375.570, 110.745), (375.920, 110.855), (376.270, 110.909), (376.620, 111.418),
(376.970, 112.382), (377.320, 114.436), (377.670, 117.673), (378.020, 121.255), (378.370,
124.527), (378.720, 127.309), (379.070, 129.055), (379.410, 130.182), (379.760, 130.764),
(380.110, 131.800), (380.460, 132.764), (380.810, 133.891), (381.160, 133.091), (381.510,
129.945), (381.860, 125.255), (382.210, 120.636), (382.560, 116.782), (382.910, 113.964),
151
(383.260, 111.691), (383.610, 110.327), (383.950, 108.727), (384.300, 107.764), (384.650,
106.345), (385.000, 105.309), (385.350, 104.836), (385.700, 104.691), (386.050, 104.473),
(386.400, 104.600), (386.750, 104.727), (387.090, 104.964), (387.440, 104.964), (387.790,
105.709), (388.140, 107.055), (388.490, 111.964), (388.840, 124.273), (389.190, 145.909),
(389.530, 172.273), (389.880, 196.073), (390.230, 212.600), (390.580, 222.036), (390.930,
226.709), (391.280, 228.400), (391.620, 228.473), (391.970, 226.091), (392.320, 219.909),
(392.670, 205.909), (393.020, 182.382), (393.360, 154.909), (393.710, 130.527), (394.060,
113.345), (394.410, 103.655), (394.760, 98.473), (395.100, 96.727), (395.450, 95.818), (395.800,
95.582), (396.150, 95.636), (396.490, 96.727), (396.840, 99.527), (397.190, 103.745), (397.540,
108.855), (397.880, 113.418), (398.230, 116.473), (398.580, 118.782), (398.930, 119.927),
(399.270, 120.818), (399.620, 122.291), (399.970, 123.127), (400.310, 123.164), (400.660,
122.618), (401.010, 121.382), (401.360, 120.327), (401.700, 122.818), (402.050, 136.182),
(402.400, 213.055), (402.740, 440.873), (403.090, 694.745), (403.440, 964.309), (403.780,
1234.055), (404.130, 1489.636), (404.480, 1729.091), (404.820, 1955.927), (405.170, 2141.964),
(405.520, 2191.291), (405.860, 2227.091), (406.210, 2240.709), (406.560, 2137.327), (406.900,
2008.327), (407.250, 1832.654), (407.600, 1620.218), (407.940, 1393.673), (408.290, 1166.382),
(408.630, 942.927), (408.980, 757.400), (409.330, 706.200), (409.670, 659.491), (410.020,
571.909), (410.370, 449.800), (410.710, 326.509), (411.060, 232.636), (411.400, 174.891),
(411.750, 145.018), (412.090, 131.109), (412.440, 125.036), (412.790, 120.818), (413.130,
115.873), (413.480, 110.600), (413.820, 105.764), (414.170, 102.345), (414.510, 99.655),
(414.860, 98.236), (415.210, 97.545), (415.550, 97.018), (415.900, 96.309), (416.240, 95.655),
(416.590, 95.091), (416.930, 94.964), (417.280, 94.564), (417.620, 94.382), (417.970, 93.764),
(418.310, 93.545), (418.660, 93.327), (419.000, 92.709), (419.350, 92.709), (419.690, 92.709),
(420.040, 92.727), (420.380, 92.545), (420.730, 92.291), (421.070, 92.073), (421.420, 91.873),
(421.760, 91.945), (422.110, 91.818), (422.450, 91.727), (422.800, 91.636), (423.140, 91.200),
(423.490, 91.327), (423.830, 91.382), (424.170, 91.545), (424.520, 91.673), (424.860, 91.891),
(425.210, 92.127), (425.550, 92.418), (425.900, 92.655), (426.240, 92.945), (426.580, 93.345),
(426.930, 94.036), (427.270, 94.309), (427.620, 95.055), (427.960, 95.727), (428.300, 96.855),
(428.650, 98.327), (428.990, 100.745), (429.340, 103.800), (429.680, 107.400), (430.020,
110.745), (430.370, 114.036), (430.710, 116.855), (431.050, 120.109), (431.400, 124.255),
(431.740, 132.909), (432.090, 154.655), (432.430, 200.691), (432.770, 280.982), (433.120,
410.818), (433.460, 598.145), (433.800, 832.618), (434.150, 1103.400), (434.490, 1411.436),
(434.830, 1728.982), (435.180, 2029.509), (435.520, 2294.854), (435.860, 2523.345), (436.200,
2710.491), (436.550, 2851.182), (436.890, 2828.818), (437.230, 2671.164), (437.580, 2449.909),
(437.920, 2186.418), (438.260, 1882.636), (438.610, 1567.400), (438.950, 1266.164), (439.290,
994.964), (439.630, 746.709), (439.980, 514.455), (440.320, 292.618), (440.660, 183.491),
(441.000, 151.564), (441.350, 135.927), (441.690, 126.273), (442.030, 119.491), (442.370,
114.055), (442.720, 110.073), (443.060, 107.036), (443.400, 104.145), (443.740, 101.873),
(444.080, 100.127), (444.430, 98.745), (444.770, 97.491), (445.110, 96.273), (445.450, 95.200),
152
(445.790, 94.582), (446.140, 94.073), (446.480, 93.673), (446.820, 93.000), (447.160, 92.636),
(447.500, 92.236), (447.850, 92.091), (448.190, 91.636), (448.530, 91.218), (448.870, 91.091),
(449.210, 90.782), (449.550, 90.691), (449.900, 90.273), (450.240, 90.127), (450.580, 89.982),
(450.920, 89.727), (451.260, 89.309), (451.600, 89.073), (451.940, 89.018), (452.280, 88.927),
(452.630, 88.564), (452.970, 88.291), (453.310, 87.655), (453.650, 87.655), (453.990, 87.600),
(454.330, 87.473), (454.670, 87.545), (455.010, 87.855), (455.350, 87.727), (455.690, 87.255),
(456.040, 87.182), (456.380, 87.400), (456.720, 87.473), (457.060, 87.673), (457.400, 87.673),
(457.740, 87.345), (458.080, 87.309), (458.420, 87.164), (458.760, 86.800), (459.100, 86.800),
(459.440, 87.036), (459.780, 86.945), (460.120, 86.909), (460.460, 86.818), (460.800, 86.618),
(461.140, 86.691), (461.480, 86.855), (461.820, 86.782), (462.160, 86.800), (462.500, 87.145),
(462.840, 87.055), (463.180, 86.836), (463.520, 86.764), (463.860, 86.618), (464.200, 86.927),
(464.540, 87.164), (464.880, 87.036), (465.220, 87.255), (465.560, 87.436), (465.900, 87.218),
(466.240, 87.364), (466.580, 87.564), (466.920, 87.782), (467.260, 87.836), (467.600, 88.036),
(467.940, 87.964), (468.280, 87.982), (468.620, 88.218), (468.960, 88.073), (469.300, 88.145),
(469.630, 88.673), (469.970, 88.655), (470.310, 88.818), (470.650, 89.273), (470.990, 89.655),
(471.330, 89.836), (471.670, 90.327), (472.010, 90.527), (472.350, 90.691), (472.690, 91.091),
(473.020, 91.545), (473.360, 91.545), (473.700, 91.655), (474.040, 91.745), (474.380, 91.673),
(474.720, 91.800), (475.060, 92.145), (475.400, 92.073), (475.730, 92.273), (476.070, 92.473),
(476.410, 92.564), (476.750, 92.691), (477.090, 92.909), (477.430, 93.436), (477.760, 93.927),
(478.100, 94.436), (478.440, 94.655), (478.780, 94.873), (479.120, 95.291), (479.450, 95.855),
(479.790, 96.473), (480.130, 97.036), (480.470, 97.545), (480.810, 98.109), (481.140, 98.345),
(481.480, 98.473), (481.820, 98.945), (482.160, 99.436), (482.500, 99.836), (482.830, 100.364),
(483.170, 100.618), (483.510, 101.127), (483.850, 101.745), (484.180, 102.073), (484.520,
102.473), (484.860, 103.309), (485.200, 104.309), (485.530, 104.727), (485.870, 105.582),
(486.210, 106.473), (486.540, 107.145), (486.880, 108.164), (487.220, 109.236), (487.560,
110.291), (487.890, 111.927), (488.230, 113.564), (488.570, 114.436), (488.900, 115.418),
(489.240, 117.782), (489.580, 131.345), (489.920, 166.200), (490.250, 219.600), (490.590,
274.818), (490.930, 316.618), (491.260, 344.764), (491.600, 362.055), (491.940, 371.236),
(492.270, 376.673), (492.610, 379.309), (492.940, 379.073), (493.280, 366.964), (493.620,
333.764), (493.950, 284.891), (494.290, 239.418), (494.630, 211.491), (494.960, 198.691),
(495.300, 193.709), (495.640, 193.273), (495.970, 194.055), (496.310, 194.964), (496.640,
196.582), (496.980, 197.800), (497.320, 198.055), (497.650, 195.145), (497.990, 186.127),
(498.320, 172.745), (498.660, 157.727), (499.000, 145.091), (499.330, 135.982), (499.670,
129.564), (500.000, 125.673), (500.340, 122.655), (500.670, 120.873), (501.010, 120.164),
(501.340, 120.055), (501.680, 120.236), (502.020, 120.036), (502.350, 120.327), (502.690,
120.855), (503.020, 122.345), (503.360, 124.255), (503.690, 126.327), (504.030, 127.800),
(504.360, 127.982), (504.700, 127.545), (505.030, 126.855), (505.370, 126.182), (505.700,
125.691), (506.040, 124.436), (506.370, 123.327), (506.710, 121.218), (507.040, 118.964),
(507.380, 116.782), (507.710, 115.382), (508.050, 114.855), (508.380, 114.455), (508.720,
153
114.418), (509.050, 115.000), (509.380, 115.655), (509.720, 116.873), (510.050, 117.909),
(510.390, 119.782), (510.720, 121.818), (511.060, 123.927), (511.390, 125.509), (511.730,
126.927), (512.060, 127.964), (512.390, 128.564), (512.730, 128.618), (513.060, 128.473),
(513.400, 127.964), (513.730, 127.455), (514.060, 125.945), (514.400, 124.055), (514.730,
121.964), (515.070, 120.382), (515.400, 118.673), (515.730, 117.309), (516.070, 116.273),
(516.400, 115.345), (516.740, 114.655), (517.070, 114.273), (517.400, 113.764), (517.740,
113.927), (518.070, 113.636), (518.400, 113.418), (518.740, 113.036), (519.070, 113.073),
(519.400, 112.818), (519.740, 112.545), (520.070, 112.436), (520.400, 112.164), (520.740,
112.018), (521.070, 112.218), (521.400, 112.182), (521.740, 112.727), (522.070, 113.145),
(522.400, 113.636), (522.730, 114.127), (523.070, 114.818), (523.400, 114.891), (523.730,
115.127), (524.070, 115.745), (524.400, 116.073), (524.730, 115.982), (525.060, 115.964),
(525.400, 115.873), (525.730, 115.691), (526.060, 115.455), (526.390, 115.255), (526.730,
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(528.390, 115.600), (528.720, 115.400), (529.050, 115.418), (529.390, 115.945), (529.720,
116.927), (530.050, 118.073), (530.380, 119.527), (530.710, 120.982), (531.050, 122.109),
(531.380, 122.945), (531.710, 123.727), (532.040, 124.364), (532.370, 125.382), (532.710,
126.345), (533.040, 126.836), (533.370, 127.691), (533.700, 129.545), (534.030, 133.255),
(534.360, 138.055), (534.700, 144.709), (535.030, 152.036), (535.360, 158.600), (535.690,
164.382), (536.020, 168.582), (536.350, 171.564), (536.680, 173.545), (537.020, 174.782),
(537.350, 174.327), (537.680, 171.909), (538.010, 168.164), (538.340, 163.673), (538.670,
158.727), (539.000, 154.691), (539.330, 151.909), (539.660, 151.327), (540.000, 153.418),
(540.330, 158.073), (540.660, 163.200), (540.990, 168.255), (541.320, 172.855), (541.650,
177.527), (541.980, 182.964), (542.310, 189.909), (542.640, 200.309), (542.970, 217.255),
(543.300, 250.564), (543.630, 416.945), (543.960, 707.091), (544.290, 1060.818), (544.620,
1414.945), (544.950, 1769.545), (545.290, 2124.127), (545.620, 2477.291), (545.950, 2820.600),
(546.280, 3080.527), (546.610, 3312.200), (546.940, 3512.891), (547.270, 3568.600), (547.600,
3463.964), (547.930, 3179.418), (548.260, 2855.000), (548.590, 2516.400), (548.920, 2170.891),
(549.250, 1820.855), (549.580, 1476.673), (549.910, 1210.000), (550.230, 963.327), (550.560,
729.709), (550.890, 505.927), (551.220, 316.382), (551.550, 242.291), (551.880, 207.236),
(552.210, 186.218), (552.540, 171.891), (552.870, 162.036), (553.200, 154.491), (553.530,
148.964), (553.860, 144.491), (554.190, 140.691), (554.520, 137.291), (554.850, 134.364),
(555.180, 131.891), (555.500, 129.800), (555.830, 127.636), (556.160, 125.818), (556.490,
124.073), (556.820, 122.509), (557.150, 120.782), (557.480, 119.273), (557.810, 117.982),
(558.140, 116.945), (558.460, 115.909), (558.790, 115.309), (559.120, 114.891), (559.450,
114.764), (559.780, 114.418), (560.110, 113.691), (560.440, 112.873), (560.760, 112.327),
(561.090, 111.836), (561.420, 111.564), (561.750, 111.127), (562.080, 110.745), (562.410,
110.127), (562.730, 109.782), (563.060, 109.418), (563.390, 109.018), (563.720, 108.636),
(564.050, 108.564), (564.380, 108.564), (564.700, 108.455), (565.030, 108.164), (565.360,
108.836), (565.690, 111.091), (566.010, 114.255), (566.340, 117.455), (566.670, 120.109),
154
(567.000, 122.000), (567.330, 123.909), (567.650, 124.982), (567.980, 125.691), (568.310,
126.582), (568.640, 127.473), (568.960, 127.309), (569.290, 125.745), (569.620, 122.855),
(569.950, 120.400), (570.270, 118.709), (570.600, 118.527), (570.930, 119.218), (571.250,
121.309), (571.580, 124.509), (571.910, 128.218), (572.240, 131.982), (572.560, 136.291),
(572.890, 142.509), (573.220, 151.691), (573.540, 164.073), (573.870, 183.309), (574.200,
218.691), (574.520, 390.036), (574.850, 620.709), (575.180, 864.782), (575.500, 1117.055),
(575.830, 1364.727), (576.160, 1603.600), (576.480, 1833.491), (576.810, 2064.400), (577.140,
2301.873), (577.460, 2540.327), (577.790, 2762.182), (578.120, 2836.873), (578.440, 2842.036),
(578.770, 2825.691), (579.100, 2795.364), (579.420, 2766.110), (579.750, 2743.037), (580.070,
2691.036), (580.400, 2557.636), (580.730, 2355.054), (581.050, 2123.691), (581.380, 1882.091),
(581.700, 1645.782), (582.030, 1415.036), (582.360, 1189.945), (582.680, 968.673), (583.010,
749.509), (583.330, 531.600), (583.660, 349.418), (583.980, 244.927), (584.310, 199.636),
(584.640, 174.800), (584.960, 159.636), (585.290, 149.418), (585.610, 142.891), (585.940,
137.418), (586.260, 133.200), (586.590, 130.382), (586.910, 129.255), (587.240, 129.218),
(587.560, 129.109), (587.890, 128.636), (588.210, 127.636), (588.540, 126.436), (588.860,
125.255), (589.190, 123.527), (589.510, 122.073), (589.840, 120.455), (590.160, 118.782),
(590.490, 116.073), (590.810, 112.655), (591.140, 109.127), (591.460, 105.509), (591.790,
102.800), (592.110, 100.764), (592.440, 98.673), (592.760, 97.382), (593.090, 96.509), (593.410,
95.891), (593.740, 95.382), (594.060, 95.218), (594.380, 94.709), (594.710, 94.400), (595.030,
94.400), (595.360, 94.018), (595.680, 93.364), (596.010, 93.364), (596.330, 93.309), (596.650,
93.164), (596.980, 92.855), (597.300, 92.455), (597.630, 91.927), (597.950, 91.818), (598.270,
91.636), (598.600, 91.382), (598.920, 91.345), (599.250, 91.636), (599.570, 91.600), (599.890,
91.236), (600.220, 91.018), (600.540, 90.945), (600.860, 91.036), (601.190, 91.055), (601.510,
90.927), (601.830, 90.873), (602.160, 90.818), (602.480, 90.745), (602.800, 90.473), (603.130,
90.418), (603.450, 90.255), (603.770, 89.982), (604.100, 89.836), (604.420, 89.673), (604.740,
89.818), (605.070, 90.291), (605.390, 91.255), (605.710, 92.327), (606.040, 93.327), (606.360,
94.509), (606.680, 95.200), (607.000, 95.836), (607.330, 96.418), (607.650, 96.636), (607.970,
96.436), (608.300, 96.236), (608.620, 95.600), (608.940, 94.455), (609.260, 93.036), (609.590,
91.782), (609.910, 90.927), (610.230, 91.055), (610.550, 92.109), (610.880, 93.091), (611.200,
94.400), (611.520, 95.691), (611.840, 96.273), (612.170, 96.873), (612.490, 97.255), (612.810,
97.545), (613.130, 97.582), (613.450, 97.055), (613.780, 95.891), (614.100, 94.036), (614.420,
92.545), (614.740, 90.836), (615.060, 89.545), (615.390, 88.636), (615.710, 87.927), (616.030,
87.582), (616.350, 87.364), (616.670, 87.182), (616.990, 87.000), (617.320, 86.745), (617.640,
86.927), (617.960, 86.836), (618.280, 86.891), (618.600, 86.691), (618.920, 86.545), (619.240,
86.455), (619.570, 86.255), (619.890, 86.109), (620.210, 86.127), (620.530, 86.327), (620.850,
86.964), (621.170, 89.073), (621.490, 92.691), (621.810, 97.255), (622.130, 101.673), (622.460,
105.073), (622.780, 107.418), (623.100, 109.218), (623.420, 110.164), (623.740, 111.309),
(624.060, 111.855), (624.380, 111.727), (624.700, 109.764), (625.020, 106.164), (625.340,
101.509), (625.660, 97.164), (625.980, 93.673), (626.300, 91.273), (626.620, 89.345), (626.940,
155
88.236), (627.270, 87.073), (627.590, 86.200), (627.910, 85.655), (628.230, 85.364), (628.550,
85.000), (628.870, 84.782), (629.190, 84.655), (629.510, 84.655), (629.830, 84.455), (630.150,
84.327), (630.470, 84.255), (630.790, 84.091), (631.110, 84.164), (631.430, 84.145), (631.750,
84.000), (632.070, 84.218), (632.390, 84.309), (632.710, 84.055), (633.030, 83.927), (633.340,
84.036), (633.660, 83.927), (633.980, 83.836), (634.300, 83.673), (634.620, 83.255), (634.940,
82.909), (635.260, 82.945), (635.580, 82.545), (635.900, 82.382), (636.220, 82.418), (636.540,
82.200), (636.860, 82.109), (637.180, 82.055), (637.500, 82.127), (637.820, 82.164), (638.130,
82.509), (638.450, 82.673), (638.770, 82.291), (639.090, 82.255), (639.410, 82.218), (639.730,
82.091), (640.050, 82.036), (640.370, 81.836), (640.690, 81.927), (641.000, 81.745), (641.320,
81.855), (641.640, 81.600), (641.960, 81.545), (642.280, 81.527), (642.600, 81.418), (642.920,
81.582), (643.230, 81.764), (643.550, 82.091), (643.870, 82.291), (644.190, 82.200), (644.510,
82.327), (644.820, 81.891), (645.140, 81.855), (645.460, 81.855), (645.780, 81.909), (646.100,
81.945), (646.420, 81.782), (646.730, 81.782), (647.050, 81.618), (647.370, 81.545), (647.690,
81.327), (648.000, 80.873), (648.320, 80.982), (648.640, 81.000), (648.960, 81.000), (649.280,
81.236), (649.590, 81.309), (649.910, 81.164), (650.230, 81.164), (650.550, 80.909), (650.860,
80.709), (651.180, 80.855), (651.500, 81.145), (651.820, 81.200), (652.130, 80.891), (652.450,
80.636), (652.770, 80.345), (653.080, 80.200), (653.400, 80.091), (653.720, 79.673), (654.040,
79.618), (654.350, 79.582), (654.670, 79.255), (654.990, 79.273), (655.300, 79.200), (655.620,
79.091), (655.940, 79.418), (656.250, 79.473), (656.570, 79.455), (656.890, 79.582), (657.200,
79.527), (657.520, 79.818), (657.840, 79.800), (658.150, 79.745), (658.470, 79.473), (658.790,
79.473), (659.100, 79.727), (659.420, 79.527), (659.740, 79.291), (660.050, 79.109), (660.370,
79.073), (660.680, 79.164), (661.000, 79.018), (661.320, 78.909), (661.630, 78.927), (661.950,
79.036), (662.270, 79.055), (662.580, 78.927), (662.900, 78.800), (663.210, 78.836), (663.530,
78.982), (663.850, 79.145), (664.160, 79.036), (664.480, 78.982), (664.790, 78.945), (665.110,
78.927), (665.420, 78.655), (665.740, 78.273), (666.050, 78.255), (666.370, 78.073), (666.690,
77.873), (667.000, 77.655), (667.320, 77.436), (667.630, 77.345), (667.950, 77.073), (668.260,
77.109), (668.580, 77.055), (668.890, 77.182), (669.210, 77.909), (669.520, 78.964), (669.840,
80.782), (670.150, 82.764), (670.470, 84.636), (670.780, 85.636), (671.100, 86.855), (671.410,
87.618), (671.730, 87.964), (672.040, 88.491), (672.360, 88.800), (672.670, 88.400), (672.990,
87.436), (673.300, 85.382), (673.620, 83.582), (673.930, 81.727), (674.250, 80.491), (674.560,
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77.127), (676.450, 76.945), (676.760, 77.418), (677.070, 77.491), (677.390, 77.145), (677.700,
77.055), (678.020, 76.745), (678.330, 76.655), (678.650, 76.618), (678.960, 76.273), (679.270,
76.182), (679.590, 75.945), (679.900, 75.636), (680.210, 75.236), (680.530, 75.000), (680.840,
75.182), (681.160, 75.364), (681.470, 75.618), (681.780, 75.582), (682.100, 75.509), (682.410,
75.564), (682.720, 75.618), (683.040, 75.655), (683.350, 75.836), (683.660, 75.945), (683.980,
75.745), (684.290, 75.709), (684.600, 75.618), (684.920, 75.400), (685.230, 75.218), (685.540,
75.127), (685.860, 75.055), (686.170, 74.909), (686.480, 75.073), (686.790, 75.309), (687.110,
75.527), (687.420, 75.782), (687.730, 75.982), (688.050, 76.600), (688.360, 80.600), (688.670,
156
90.382), (688.980, 104.418), (689.300, 120.400), (689.610, 133.582), (689.920, 143.600),
(690.230, 151.527), (690.550, 157.382), (690.860, 161.509), (691.170, 164.655), (691.480,
166.273), (691.800, 163.727), (692.110, 155.127), (692.420, 141.855), (692.730, 126.127),
(693.050, 113.073), (693.360, 103.182), (693.670, 95.000), (693.980, 88.564), (694.290, 84.145),
(694.610, 80.909), (694.920, 78.418), (695.230, 76.982), (695.540, 76.036), (695.850, 75.073),
(696.170, 74.618), (696.480, 74.345), (696.790, 74.000), (697.100, 73.673), (697.410, 73.636),
(697.720, 73.600), (698.040, 73.455), (698.350, 73.618), (698.660, 73.473), (698.970, 73.291),
(699.280, 73.345), (699.590, 73.400), (699.900, 73.382), (700.220, 73.345), (700.530, 73.527),
(700.840, 73.709), (701.150, 73.582), (701.460, 73.636), (701.770, 73.382), (702.080, 73.473),
(702.390, 73.291), (702.700, 73.400), (703.010, 73.636), (703.330, 73.582), (703.640, 73.382),
(703.950, 73.327), (704.260, 73.291), (704.570, 73.436), (704.880, 73.127), (705.190, 73.309),
(705.500, 73.455), (705.810, 74.400), (706.120, 76.127), (706.430, 78.691), (706.740, 82.200),
(707.050, 85.564), (707.360, 88.418), (707.670, 91.036), (707.980, 92.927), (708.290, 94.582),
(708.600, 95.545), (708.910, 96.218), (709.220, 95.927), (709.530, 94.745), (709.840, 92.091),
(710.150, 88.727), (710.460, 85.673), (710.770, 82.836), (711.080, 80.109), (711.390, 77.909),
(711.700, 76.309), (712.010, 75.109), (712.320, 74.073), (712.630, 73.109), (712.940, 72.309),
(713.250, 71.945), (713.560, 71.836), (713.870, 71.636), (714.180, 71.564), (714.490, 71.564),
(714.800, 71.673), (715.110, 71.582), (715.420, 71.400), (715.730, 71.364), (716.040, 71.745),
(716.350, 71.800), (716.660, 71.982), (716.970, 71.855), (717.270, 71.509), (717.580, 71.327),
(717.890, 71.236), (718.200, 71.327), (718.510, 71.218), (718.820, 71.291), (719.130, 71.145),
(719.440, 71.127), (719.750, 70.964), (720.050, 70.800), (720.360, 70.764), (720.670, 70.964),
(720.980, 70.891), (721.290, 70.945), (721.600, 70.673), (721.910, 70.673), (722.210, 70.618),
(722.520, 70.582), (722.830, 70.491), (723.140, 70.491), (723.450, 70.545), (723.760, 70.473),
(724.060, 70.255), (724.370, 70.200), (724.680, 69.982), (724.990, 70.018), (725.300, 70.091),
(725.610, 70.127), (725.910, 70.145), (726.220, 70.127), (726.530, 70.091), (726.840, 69.745),
(727.140, 69.709), (727.450, 69.709), (727.760, 69.782), (728.070, 69.727), (728.380, 69.655),
(728.680, 69.600), (728.990, 69.491), (729.300, 69.418), (729.610, 69.200), (729.910, 69.036),
(730.220, 69.145), (730.530, 69.182), (730.840, 69.291), (731.140, 69.309), (731.450, 69.436),
(731.760, 69.327), (732.060, 69.273), (732.370, 69.309), (732.680, 69.327), (732.990, 69.455),
(733.290, 69.709), (733.600, 69.945), (733.910, 69.964), (734.210, 69.982), (734.520, 69.964),
(734.830, 70.055), (735.130, 70.291), (735.440, 70.164), (735.750, 70.073), (736.050, 70.309),
(736.360, 69.964), (736.670, 69.891), (736.970, 69.527), (737.280, 69.400), (737.590, 69.418),
(737.890, 69.327), (738.200, 69.055), (738.510, 68.727), (738.810, 68.855), (739.120, 68.855),
(739.430, 68.218), (739.730, 68.255), (740.040, 68.109), (740.340, 68.109), (740.650, 68.000),
(740.960, 67.927), (741.260, 67.800), (741.570, 67.909), (741.870, 68.145), (742.180, 68.200),
(742.490, 68.200), (742.790, 68.327), (743.100, 68.582), (743.400, 68.582), (743.710, 68.691),
(744.020, 68.636), (744.320, 68.291), (744.630, 68.182), (744.930, 68.127), (745.240, 67.982),
(745.540, 67.564), (745.850, 67.218), (746.150, 67.036), (746.460, 66.891), (746.760, 66.636),
(747.070, 66.527), (747.370, 66.745), (747.680, 66.764), (747.990, 66.945), (748.290, 66.927),
157
(748.600, 67.055), (748.900, 67.145), (749.210, 67.182), (749.510, 67.418), (749.820, 67.455),
(750.120, 67.527), (750.430, 67.364), (750.730, 67.127), (751.030, 67.073), (751.340, 67.018),
(751.640, 67.000), (751.950, 66.836), (752.250, 66.691), (752.560, 66.836), (752.860, 66.945),
(753.170, 66.836), (753.470, 66.818), (753.780, 66.800), (754.080, 66.836), (754.380, 67.145),
(754.690, 67.036), (754.990, 66.764), (755.300, 66.691), (755.600, 66.636), (755.910, 66.855),
(756.210, 66.455), (756.510, 66.364), (756.820, 66.291), (757.120, 66.564), (757.430, 66.782),
(757.730, 66.545), (758.030, 66.727), (758.340, 66.964), (758.640, 66.836), (758.940, 66.909),
(759.250, 66.855), (759.550, 67.073), (759.860, 67.018), (760.160, 67.127), (760.460, 67.000),
(760.770, 66.782), (761.070, 66.691), (761.370, 66.255), (761.680, 66.382), (761.980, 66.600),
(762.280, 66.727), (762.590, 66.491), (762.890, 66.636), (763.190, 66.600), (763.500, 66.564),
(763.800, 66.491), (764.100, 66.327), (764.410, 66.400), (764.710, 66.673), (765.010, 66.600),
(765.310, 66.509), (765.620, 66.545), (765.920, 66.455), (766.220, 66.182), (766.530, 66.400),
(766.830, 66.455), (767.130, 66.673), (767.430, 66.800), (767.740, 66.782), (768.040, 66.655),
(768.340, 66.455), (768.640, 66.236), (768.950, 66.236), (769.250, 66.109), (769.550, 66.182),
(769.850, 65.964), (770.160, 65.691), (770.460, 65.455), (770.760, 65.636), (771.060, 66.164),
(771.360, 66.964), (771.670, 68.055), (771.970, 69.236), (772.270, 69.836), (772.570, 70.582),
(772.870, 70.873), (773.180, 71.109), (773.480, 71.582), (773.780, 71.436), (774.080, 71.345),
(774.380, 70.873), (774.690, 70.018), (774.990, 68.836), (775.290, 67.564), (775.590, 66.709),
(775.890, 66.164), (776.190, 66.182), (776.490, 65.818), (776.800, 65.455), (777.100, 65.473),
(777.400, 65.327), (777.700, 65.127), (778.000, 65.055), (778.300, 64.873), (778.600, 65.145),
(778.900, 65.145), (779.210, 65.000), (779.510, 64.636), (779.810, 64.745), (780.110, 64.982),
(780.410, 64.964), (780.710, 65.018), (781.010, 65.000), (781.310, 65.145), (781.610, 65.345),
(781.910, 65.327), (782.210, 65.509), (782.520, 65.727), (782.820, 65.873), (783.120, 65.909),
(783.420, 65.509), (783.720, 65.309), (784.020, 65.073), (784.320, 65.036), (784.620, 64.945),
(784.920, 64.582), (785.220, 64.273), (785.520, 63.836), (785.820, 63.691), (786.120, 63.418),
(786.420, 63.236), (786.720, 63.200), (787.020, 63.309), (787.320, 63.455), (787.620, 63.600),
(787.920, 63.455), (788.220, 63.618), (788.520, 63.836), (788.820, 64.036), (789.120, 63.636),
(789.420, 63.527), (789.720, 63.418), (790.020, 63.564), (790.320, 63.436), (790.620, 63.364),
(790.920, 63.309), (791.220, 63.382), (791.520, 63.473), (791.820, 63.255), (792.120, 63.327),
(792.420, 63.818), (792.710, 64.036), (793.010, 64.036), (793.310, 63.891), (793.610, 63.909),
(793.910, 63.727), (794.210, 63.782), (794.510, 63.927), (794.810, 63.855), (795.110, 64.055),
(795.410, 63.782), (795.710, 63.436), (796.000, 63.327), (796.300, 63.636), (796.600, 63.727),
(796.900, 63.745), (797.200, 63.836), (797.500, 63.545), (797.800, 63.400), (798.100, 63.582),
(798.390, 63.309), (798.690, 63.309), (798.990, 63.145), (799.290, 62.964), (799.590, 62.764),
(799.890, 62.782), (800.180, 63.055), (800.480, 63.055), (800.780, 63.164), (801.080, 63.109),
(801.380, 62.964), (801.680, 63.364), (801.970, 63.382), (802.270, 63.491), (802.570, 63.673),
(802.870, 63.564), (803.170, 63.436), (803.460, 63.145), (803.760, 63.109), (804.060, 63.218),
(804.360, 63.273), (804.660, 63.236), (804.950, 62.873), (805.250, 63.182), (805.550, 63.327),
(805.850, 63.255), (806.140, 63.291), (806.440, 63.455), (806.740, 63.455), (807.040, 63.382),
158
(807.330, 63.255), (807.630, 63.218), (807.930, 63.400), (808.230, 63.455), (808.520, 63.145),
(808.820, 63.109), (809.120, 63.091), (809.410, 63.091), (809.710, 63.000), (810.010, 63.000),
(810.310, 62.945), (810.600, 62.982), (810.900, 62.873), (811.200, 62.855), (811.490, 62.964),
(811.790, 62.945), (812.090, 62.891), (812.380, 62.982), (812.680, 62.964), (812.980, 62.855),
(813.270, 63.055), (813.570, 63.127), (813.870, 63.073), (814.160, 63.236), (814.460, 63.109),
(814.760, 63.073), (815.050, 63.182), (815.350, 63.345), (815.640, 63.200), (815.940, 63.236),
(816.240, 63.400), (816.530, 63.218), (816.830, 62.964), (817.130, 62.873), (817.420, 62.745),
(817.720, 62.691), (818.010, 62.745), (818.310, 62.782), (818.610, 62.764), (818.900, 62.800),
(819.200, 62.945), (819.490, 62.782), (819.790, 63.091), (820.080, 63.364), (820.380, 63.636),
(820.680, 64.000), (820.970, 64.055), (821.270, 63.636), (821.560, 63.636), (821.860, 63.364),
(822.150, 63.291), (822.450, 63.291), (822.740, 63.273), (823.040, 63.036), (823.330, 63.000),
(823.630, 62.727), (823.920, 62.418), (824.220, 62.345), (824.520, 62.545), (824.810, 62.436),
(825.110, 62.455), (825.400, 62.564), (825.700, 62.527), (825.990, 62.527), (826.280, 62.582),
(826.580, 62.709), (826.870, 62.818), (827.170, 62.618), (827.460, 62.436), (827.760, 62.545),
(828.050, 62.582), (828.350, 62.636), (828.640, 62.582), (828.940, 62.345), (829.230, 62.273),
(829.530, 62.127), (829.820, 62.055), (830.110, 62.164), (830.410, 62.345), (830.700, 62.491),
(831.000, 62.327), (831.290, 62.145), (831.590, 62.091), (831.880, 61.836), (832.170, 61.782),
(832.470, 61.891), (832.760, 61.745), (833.060, 61.800), (833.350, 61.564), (833.640, 61.491),
(833.940, 61.636), (834.230, 61.564), (834.530, 61.582), (834.820, 61.727), (835.110, 61.782),
(835.410, 61.782), (835.700, 61.727), (835.990, 61.855), (836.290, 61.673), (836.580, 61.691),
(836.870, 61.727), (837.170, 61.418), (837.460, 61.527), (837.750, 61.636), (838.050, 61.509),
(838.340, 61.782), (838.630, 61.909), (838.930, 61.836), (839.220, 61.873), (839.510, 61.618),
(839.810, 61.418), (840.100, 61.145), (840.390, 61.364), (840.690, 61.491), (840.980, 61.473),
(841.270, 61.509), (841.560, 61.582), (841.860, 61.636), (842.150, 61.582), (842.440, 61.527),
(842.730, 61.527), (843.030, 61.836), (843.320, 62.036), (843.610, 61.873), (843.910, 61.764),
(844.200, 61.691), (844.490, 61.582), (844.780, 61.327), (845.070, 61.091), (845.370, 61.455),
(845.660, 61.473), (845.950, 61.691), (846.240, 61.527), (846.540, 61.655), (846.830, 61.600),
(847.120, 61.291), (847.410, 61.327), (847.700, 61.473), (848.000, 61.327), (848.290, 61.491),
(848.580, 61.291), (848.870, 61.055), (849.160, 61.164), (849.460, 61.000), (849.750, 60.745),
(850.040, 60.818), (850.330, 61.091), (850.620, 60.982), (850.910, 60.873), (851.210, 60.927),
(851.500, 61.018), (851.790, 61.109), (852.080, 61.073), (852.370, 60.836), (852.660, 61.000),
(852.950, 61.273), (853.240, 61.164), (853.540, 61.218), (853.830, 61.545), (854.120, 61.509),
(854.410, 61.509), (854.700, 61.127), (854.990, 60.855), (855.280, 60.982), (855.570, 61.218),
(855.860, 61.255), (856.160, 61.164), (856.450, 61.145), (856.740, 60.964), (857.030, 61.000),
(857.320, 61.055), (857.610, 61.091), (857.900, 61.491), (858.190, 61.655), (858.480, 61.636),
(858.770, 61.564), (859.060, 61.545), (859.350, 61.455), (859.640, 61.727), (859.930, 61.727),
(860.220, 61.291), (860.510, 61.418), (860.800, 61.345), (861.090, 61.291), (861.380, 61.218),
(861.670, 61.327), (861.960, 61.200), (862.250, 61.145), (862.540, 61.218), (862.830, 61.036),
159
(863.120, 61.036), (863.410, 61.036), (863.700, 60.673), (863.990, 62.200), (864.280, 61.000),
(864.570, 60.200), (864.860, 59.200), (865.150, 58.800).
Fig. 4.6 up pointing triangles(black) (format: (time (s) (arbitrary origin), V 2,
power): (26785648, 101.112, 1.656e-06),(26793617, 103.249, 2.038e-06),(26801585, 104.252,
2.346e-06),(26809553, 104.839, 3.15e-06),(26817521, 105.002, 4.202e-06),(26825490, 105.251,
1.218e-05),(26833458, 105.339, 1.999e-05),(26841426, 105.426, 3.133e-05),(26849395, 105.508,
4.96e-05),(26857363, 105.675, 0.0001083),(26865331, 105.761, 0.0001384),(26873299, 105.843,
0.0001684),(26881268, 105.928, 0.0002003),(26889236, 106.096, 0.0002622),(26897204, 106.262,
0.0003222),(26905172, 106.430, 0.0003892),(26913141, 106.685, 0.000496),(26921109, 106.933,
0.000591),(26929077, 107.272, 0.0007136),(26937046, 107.610, 0.0008403),(26945014, 108.115,
0.0009778),(26952982, 108.627, 0.001107),(26960950, 109.308, 0.001228),(26968919, 110.158,
0.001298),(26976887, 111.105, 0.001304).
Fig. 4.6 down pointing triangles(red) (time, V 2, power): (26984855, 110.587,
0.001324),(26992823, 109.731, 0.001275),(27000792, 108.958, 0.001146),(27008760, 108.368,
0.0009858),(27016728, 107.859, 0.0008168),(27024697, 107.440, 0.0006479),(27032665, 107.100,
0.0005127),(27040633, 106.766, 0.0003747),(27048601, 106.507, 0.0002684),(27056570, 106.343,
0.0002031),(27064538, 106.176, 0.0001437),(27072506, 106.005, 9.273e-05),(27080475, 105.923,
7.415e-05),(27088443, 105.836, 5.423e-05),(27096411, 105.671, 2.526e-05),(27104379, 105.593,
1.683e-05),(27112348, 105.504, 1.106e-05),(27120316, 105.421, 6.961e-06),(27128284, 105.251,
3.519e-06),(27136252, 105.086, 2.52e-06),(27144221, 104.917, 2.212e-06),(27152189, 104.582,
1.959e-06),(27160157, 103.751, 1.76e-06),(27168126, 102.177, 1.608e-06),(27176094, 94.808,
1.405e-06).
Fig. 4.12 (b) circles: (-128.9, -0.00139), (-128.7, -0.00138), (-128.4, -0.00137), (-128.1,
-0.00136), (-127.9, -0.00135), (-127.6, -0.00134), (-127.4, -0.00133), (-127.1, -0.00133), (-
126.9, -0.00132), (-126.6, -0.00131), (-126.4, -0.00131), (-126.1, -0.0013), (-125.9, -0.0013), (-
125.6, -0.0013), (-125.4, -0.0013), (-125.1, -0.00129), (-124.9, -0.00129), (-124.6, -0.00129), (-
124.4, -0.00129), (-124.1, -0.00128), (-123.9, -0.00128), (-123.6, -0.00128), (-123.4, -0.00128),
(-123.1, -0.00128), (-122.9, -0.00128), (-122.6, -0.00128), (-122.4, -0.00128), (-122.1, -0.00128),
(-121.9, -0.00128), (-121.6, -0.00128), (-121.4, -0.00128), (-121.1, -0.00127), (-120.8, -0.00127),
(-120.6, -0.00127), (-120.3, -0.00127), (-120.1, -0.00127), (-119.8, -0.00127), (-119.6, -0.00127),
(-119.3, -0.00127), (-119.1, -0.00126), (-118.8, -0.00126), (-118.6, -0.00126), (-118.3, -0.00126),
(-118.1, -0.00126), (-117.8, -0.00125), (-117.6, -0.00125), (-117.3, -0.00125), (-117.1, -0.00125),
(-116.8, -0.00124), (-116.6, -0.00124), (-116.3, -0.00123), (-116.1, -0.00123), (-115.8, -0.00122),
(-115.6, -0.00121), (-115.3, -0.0012), (-115.1, -0.0012), (-114.8, -0.00119), (-114.6, -0.00118),
(-114.3, -0.00116), (-114.1, -0.00115), (-113.8, -0.00114), (-113.5, -0.00112), (-113.3, -0.00111),
(-113.0, -0.00109), (-112.8, -0.00108), (-112.5, -0.00106), (-112.3, -0.00104), (-112.0, -0.00102),
(-111.8, -0.000997), (-111.5, -0.000972), (-111.3, -0.000947), (-111.0, -0.000922), (-110.8, -
0.000896), (-110.5, -0.000866), (-110.3, -0.000836), (-110.0, -0.000806), (-109.8, -0.000775),
(-109.5, -0.000741), (-109.3, -0.000706), (-109.0, -0.00067), (-108.8, -0.000634), (-108.5, -
160
0.000593), (-108.3, -0.000551), (-108.0, -0.000509), (-107.8, -0.000466), (-107.5, -0.000417),
(-107.3, -0.000368), (-107.0, -0.000318), (-106.7, -0.000268), (-106.5, -0.000212), (-106.2, -
0.000155), (-106.0, -9.7e-05), (-105.7, -3.9e-05), (-105.5, 2.6e-05), (-105.2, 9.1e-05), (-105.0,
0.000157), (-104.7, 0.000224), (-104.5, 0.000297), (-104.2, 0.000371), (-104.0, 0.000446), (-
103.7, 0.000521), (-103.5, 0.000605), (-103.2, 0.000688), (-103.0, 0.000772), (-102.7, 0.000857),
(-102.5, 0.00095), (-102.2, 0.00104), (-102.0, 0.00114), (-101.7, 0.00123), (-101.5, 0.00134),
(-101.2, 0.00144), (-101.0, 0.00154), (-100.7, 0.00165), (-100.5, 0.00176), (-100.2, 0.00187),
(-100.0, 0.00199), (-99.7, 0.0021), (-99.4, 0.00223), (-99.2, 0.00235), (-98.9, 0.00248), (-
98.7, 0.00261), (-98.4, 0.00274), (-98.2, 0.00288), (-97.9, 0.00302), (-97.7, 0.00316), (-97.4,
0.00331), (-97.2, 0.00346), (-96.9, 0.00361), (-96.7, 0.00376), (-96.4, 0.00392), (-96.2, 0.00408),
(-95.9, 0.00424), (-95.7, 0.00441), (-95.4, 0.00458), (-95.2, 0.00476), (-94.9, 0.00494), (-94.7,
0.00511), (-94.4, 0.0053), (-94.2, 0.00549), (-93.9, 0.00568), (-93.7, 0.00587), (-93.4, 0.00607),
(-93.2, 0.00628), (-92.9, 0.00648), (-92.7, 0.00668), (-92.4, 0.0069), (-92.1, 0.00711), (-91.9,
0.00733), (-91.6, 0.00754), (-91.4, 0.00777), (-91.1, 0.008), (-90.9, 0.00823), (-90.6, 0.00846),
(-90.4, 0.00871), (-90.1, 0.00895), (-89.9, 0.00919), (-89.6, 0.00944), (-89.4, 0.00969), (-89.1,
0.00995), (-88.9, 0.0102), (-88.6, 0.0105), (-88.4, 0.0107), (-88.1, 0.011), (-87.9, 0.0113), (-
87.6, 0.0116), (-87.4, 0.0118), (-87.1, 0.0121), (-86.9, 0.0124), (-86.6, 0.0127), (-86.4, 0.013),
(-86.1, 0.0133), (-85.9, 0.0136), (-85.6, 0.0139), (-85.3, 0.0142), (-85.1, 0.0145), (-84.8,
0.0148), (-84.6, 0.0151), (-84.3, 0.0154), (-84.1, 0.0158), (-83.8, 0.0161), (-83.6, 0.0164),
(-83.3, 0.0167), (-83.1, 0.0171), (-82.8, 0.0174), (-82.6, 0.0177), (-82.3, 0.0181), (-82.1,
0.0184), (-81.8, 0.0188), (-81.6, 0.0191), (-81.3, 0.0195), (-81.1, 0.0198), (-80.8, 0.0202), (-
80.6, 0.0205), (-80.3, 0.0209), (-80.1, 0.0212), (-79.8, 0.0216), (-79.6, 0.022), (-79.3, 0.0223),
(-79.1, 0.0227), (-78.8, 0.0231), (-78.6, 0.0234), (-78.3, 0.0238), (-78.0, 0.0242), (-77.8,
0.0246), (-77.5, 0.025), (-77.3, 0.0253), (-77.0, 0.0257), (-76.8, 0.0261), (-76.5, 0.0265), (-76.3,
0.0269), (-76.0, 0.0273), (-75.8, 0.0277), (-75.5, 0.028), (-75.3, 0.0284), (-75.0, 0.0288), (-74.8,
0.0292), (-74.5, 0.0296), (-74.3, 0.03), (-74.0, 0.0304), (-73.8, 0.0308), (-73.5, 0.0311), (-73.3,
0.0315), (-73.0, 0.0319), (-72.8, 0.0323), (-72.5, 0.0327), (-72.3, 0.0331), (-72.0, 0.0335),
(-71.8, 0.0339), (-71.5, 0.0343), (-71.3, 0.0347), (-71.0, 0.0351), (-70.7, 0.0355), (-70.5,
0.0358), (-70.2, 0.0362), (-70.0, 0.0366), (-69.7, 0.037), (-69.5, 0.0374), (-69.2, 0.0378), (-69.0,
0.0382), (-68.7, 0.0386), (-68.5, 0.039), (-68.2, 0.0393), (-68.0, 0.0397), (-67.7, 0.0401), (-67.5,
0.0405), (-67.2, 0.0409), (-67.0, 0.0412), (-66.7, 0.0416), (-66.5, 0.042), (-66.2, 0.0423), (-66.0,
0.0427), (-65.7, 0.0431), (-65.5, 0.0434), (-65.2, 0.0438), (-65.0, 0.0441), (-64.7, 0.0445), (-
64.5, 0.0449), (-64.2, 0.0452), (-63.9, 0.0456), (-63.7, 0.0459), (-63.4, 0.0463), (-63.2, 0.0466),
(-62.9, 0.047), (-62.7, 0.0473), (-62.4, 0.0477), (-62.2, 0.048), (-61.9, 0.0483), (-61.7, 0.0487),
(-61.4, 0.049), (-61.2, 0.0494), (-60.9, 0.0497), (-60.7, 0.0501), (-60.4, 0.0504), (-60.2, 0.0507),
(-59.9, 0.0511), (-59.7, 0.0514), (-59.4, 0.0518), (-59.2, 0.0521), (-58.9, 0.0524), (-58.7,
0.0527), (-58.4, 0.0531), (-58.2, 0.0534), (-57.9, 0.0537), (-57.7, 0.054), (-57.4, 0.0544), (-
57.2, 0.0547), (-56.9, 0.055), (-56.6, 0.0554), (-56.4, 0.0557), (-56.1, 0.056), (-55.9, 0.0564),
(-55.6, 0.0567), (-55.4, 0.0571), (-55.1, 0.0574), (-54.9, 0.0578), (-54.6, 0.0581), (-54.4,
161
0.0585), (-54.1, 0.0589), (-53.9, 0.0593), (-53.6, 0.0596), (-53.4, 0.06), (-53.1, 0.0604), (-52.9,
0.0609), (-52.6, 0.0613), (-52.4, 0.0617), (-52.1, 0.0621), (-51.9, 0.0626), (-51.6, 0.063), (-51.4,
0.0635), (-51.1, 0.064), (-50.9, 0.0645), (-50.6, 0.065), (-50.4, 0.0655), (-50.1, 0.066), (-49.8,
0.0665), (-49.6, 0.067), (-49.3, 0.0676), (-49.1, 0.0682), (-48.8, 0.0688), (-48.6, 0.0694), (-48.3,
0.07), (-48.1, 0.0706), (-47.8, 0.0713), (-47.6, 0.072), (-47.3, 0.0727), (-47.1, 0.0734), (-46.8,
0.0742), (-46.6, 0.075), (-46.3, 0.0758), (-46.1, 0.0767), (-45.8, 0.0775), (-45.6, 0.0784), (-45.3,
0.0793), (-45.1, 0.0803), (-44.8, 0.0814), (-44.6, 0.0824), (-44.3, 0.0834), (-44.1, 0.0845), (-
43.8, 0.0857), (-43.6, 0.0869), (-43.3, 0.088), (-43.1, 0.0894), (-42.8, 0.0907), (-42.5, 0.092),
(-42.3, 0.0934), (-42.0, 0.0948), (-41.8, 0.0963), (-41.5, 0.0977), (-41.3, 0.0992), (-41.0, 0.101),
(-40.8, 0.102), (-40.5, 0.104), (-40.3, 0.106), (-40.0, 0.108), (-39.8, 0.11), (-39.5, 0.111), (-
39.3, 0.113), (-39.0, 0.115), (-38.8, 0.118), (-38.5, 0.12), (-38.3, 0.122), (-38.0, 0.124), (-37.8,
0.127), (-37.5, 0.129), (-37.3, 0.131), (-37.0, 0.134), (-36.8, 0.137), (-36.5, 0.139), (-36.3,
0.142), (-36.0, 0.145), (-35.8, 0.148), (-35.5, 0.151), (-35.2, 0.154), (-35.0, 0.157), (-34.7,
0.16), (-34.5, 0.164), (-34.2, 0.167), (-34.0, 0.171), (-33.7, 0.174), (-33.5, 0.178), (-33.2,
0.181), (-33.0, 0.185), (-32.7, 0.189), (-32.5, 0.193), (-32.2, 0.197), (-32.0, 0.201), (-31.7,
0.205), (-31.5, 0.209), (-31.2, 0.214), (-31.0, 0.218), (-30.7, 0.223), (-30.5, 0.227), (-30.2,
0.232), (-30.0, 0.237), (-29.7, 0.242), (-29.5, 0.247), (-29.2, 0.252), (-29.0, 0.257), (-28.7,
0.263), (-28.4, 0.268), (-28.2, 0.273), (-27.9, 0.279), (-27.7, 0.285), (-27.4, 0.291), (-27.2,
0.296), (-26.9, 0.302), (-26.7, 0.309), (-26.4, 0.315), (-26.2, 0.321), (-25.9, 0.327), (-25.7,
0.334), (-25.4, 0.34), (-25.2, 0.347), (-24.9, 0.354), (-24.7, 0.36), (-24.4, 0.367), (-24.2, 0.374),
(-23.9, 0.381), (-23.7, 0.388), (-23.4, 0.395), (-23.2, 0.402), (-22.9, 0.409), (-22.7, 0.417), (-
22.4, 0.424), (-22.2, 0.432), (-21.9, 0.439), (-21.7, 0.447), (-21.4, 0.455), (-21.1, 0.462), (-20.9,
0.47), (-20.6, 0.478), (-20.4, 0.486), (-20.1, 0.494), (-19.9, 0.502), (-19.6, 0.511), (-19.4, 0.519),
(-19.1, 0.527), (-18.9, 0.535), (-18.6, 0.544), (-18.4, 0.552), (-18.1, 0.561), (-17.9, 0.569), (-
17.6, 0.578), (-17.4, 0.586), (-17.1, 0.595), (-16.9, 0.604), (-16.6, 0.612), (-16.4, 0.621), (-16.1,
0.629), (-15.9, 0.637), (-15.6, 0.646), (-15.4, 0.654), (-15.1, 0.663), (-14.9, 0.671), (-14.6, 0.68),
(-14.4, 0.688), (-14.1, 0.697), (-13.8, 0.705), (-13.6, 0.713), (-13.3, 0.722), (-13.1, 0.73), (-
12.8, 0.739), (-12.6, 0.747), (-12.3, 0.755), (-12.1, 0.763), (-11.8, 0.771), (-11.6, 0.779), (-11.3,
0.787), (-11.1, 0.795), (-10.8, 0.803), (-10.6, 0.811), (-10.3, 0.819), (-10.1, 0.827), (-9.8, 0.834),
(-9.6, 0.841), (-9.3, 0.849), (-9.1, 0.856), (-8.8, 0.863), (-8.6, 0.87), (-8.3, 0.877), (-8.1, 0.884),
(-7.8, 0.89), (-7.6, 0.896), (-7.3, 0.902), (-7.0, 0.908), (-6.8, 0.914), (-6.5, 0.919), (-6.3, 0.925),
(-6.0, 0.93), (-5.8, 0.935), (-5.5, 0.94), (-5.3, 0.945), (-5.0, 0.95), (-4.8, 0.954), (-4.5, 0.958),
(-4.3, 0.962), (-4.0, 0.966), (-3.8, 0.97), (-3.5, 0.973), (-3.3, 0.976), (-3.0, 0.98), (-2.8, 0.982),
(-2.5, 0.985), (-2.3, 0.987), (-2.0, 0.99), (-1.8, 0.991), (-1.5, 0.993), (-1.3, 0.994), (-1.0, 0.996),
(-0.8, 0.997), (-0.5, 0.998), (-0.3, 0.999), (0.0, 1), (0.3, 0.999), (0.5, 0.999), (0.8, 0.998), (1.0,
0.998), (1.3, 0.996), (1.5, 0.994), (1.8, 0.993), (2.0, 0.991), (2.3, 0.989), (2.5, 0.986), (2.8,
0.983), (3.0, 0.981), (3.3, 0.978), (3.5, 0.974), (3.8, 0.971), (4.0, 0.967), (4.3, 0.963), (4.5,
0.959), (4.8, 0.955), (5.0, 0.951), (5.3, 0.946), (5.5, 0.941), (5.8, 0.936), (6.0, 0.931), (6.3,
0.925), (6.5, 0.92), (6.8, 0.914), (7.0, 0.908), (7.3, 0.902), (7.6, 0.896), (7.8, 0.89), (8.1, 0.884),
162
(8.3, 0.877), (8.6, 0.87), (8.8, 0.863), (9.1, 0.856), (9.3, 0.849), (9.6, 0.841), (9.8, 0.834), (10.1,
0.827), (10.3, 0.819), (10.6, 0.811), (10.8, 0.803), (11.1, 0.796), (11.3, 0.787), (11.6, 0.779),
(11.8, 0.771), (12.1, 0.763), (12.3, 0.755), (12.6, 0.747), (12.8, 0.739), (13.1, 0.73), (13.3,
0.722), (13.6, 0.714), (13.8, 0.705), (14.1, 0.697), (14.4, 0.688), (14.6, 0.68), (14.9, 0.671),
(15.1, 0.663), (15.4, 0.654), (15.6, 0.646), (15.9, 0.637), (16.1, 0.629), (16.4, 0.621), (16.6,
0.612), (16.9, 0.604), (17.1, 0.595), (17.4, 0.586), (17.6, 0.578), (17.9, 0.569), (18.1, 0.561),
(18.4, 0.552), (18.6, 0.544), (18.9, 0.535), (19.1, 0.527), (19.4, 0.519), (19.6, 0.511), (19.9,
0.502), (20.1, 0.494), (20.4, 0.486), (20.6, 0.478), (20.9, 0.47), (21.1, 0.462), (21.4, 0.455),
(21.7, 0.447), (21.9, 0.439), (22.2, 0.432), (22.4, 0.424), (22.7, 0.417), (22.9, 0.409), (23.2,
0.402), (23.4, 0.395), (23.7, 0.388), (23.9, 0.381), (24.2, 0.374), (24.4, 0.367), (24.7, 0.36),
(24.9, 0.354), (25.2, 0.347), (25.4, 0.34), (25.7, 0.334), (25.9, 0.327), (26.2, 0.321), (26.4,
0.315), (26.7, 0.309), (26.9, 0.302), (27.2, 0.296), (27.4, 0.291), (27.7, 0.285), (27.9, 0.279),
(28.2, 0.273), (28.4, 0.268), (28.7, 0.263), (29.0, 0.257), (29.2, 0.252), (29.5, 0.247), (29.7,
0.242), (30.0, 0.237), (30.2, 0.232), (30.5, 0.227), (30.7, 0.223), (31.0, 0.218), (31.2, 0.214),
(31.5, 0.209), (31.7, 0.205), (32.0, 0.201), (32.2, 0.197), (32.5, 0.193), (32.7, 0.189), (33.0,
0.185), (33.2, 0.181), (33.5, 0.178), (33.7, 0.174), (34.0, 0.171), (34.2, 0.167), (34.5, 0.164),
(34.7, 0.16), (35.0, 0.157), (35.2, 0.154), (35.5, 0.151), (35.8, 0.148), (36.0, 0.145), (36.3,
0.142), (36.5, 0.139), (36.8, 0.137), (37.0, 0.134), (37.3, 0.131), (37.5, 0.129), (37.8, 0.127),
(38.0, 0.124), (38.3, 0.122), (38.5, 0.12), (38.8, 0.118), (39.0, 0.115), (39.3, 0.113), (39.5,
0.111), (39.8, 0.11), (40.0, 0.108), (40.3, 0.106), (40.5, 0.104), (40.8, 0.103), (41.0, 0.101),
(41.3, 0.0992), (41.5, 0.0977), (41.8, 0.0963), (42.0, 0.0948), (42.3, 0.0934), (42.5, 0.092),
(42.8, 0.0907), (43.1, 0.0894), (43.3, 0.0881), (43.6, 0.0869), (43.8, 0.0857), (44.1, 0.0846),
(44.3, 0.0834), (44.6, 0.0824), (44.8, 0.0814), (45.1, 0.0803), (45.3, 0.0793), (45.6, 0.0784),
(45.8, 0.0776), (46.1, 0.0767), (46.3, 0.0758), (46.6, 0.075), (46.8, 0.0742), (47.1, 0.0735),
(47.3, 0.0727), (47.6, 0.072), (47.8, 0.0713), (48.1, 0.0707), (48.3, 0.07), (48.6, 0.0694), (48.8,
0.0688), (49.1, 0.0682), (49.3, 0.0676), (49.6, 0.0671), (49.8, 0.0665), (50.1, 0.066), (50.4,
0.0655), (50.6, 0.065), (50.9, 0.0645), (51.1, 0.064), (51.4, 0.0635), (51.6, 0.0631), (51.9,
0.0626), (52.1, 0.0621), (52.4, 0.0617), (52.6, 0.0613), (52.9, 0.0609), (53.1, 0.0605), (53.4,
0.06), (53.6, 0.0597), (53.9, 0.0593), (54.1, 0.0589), (54.4, 0.0585), (54.6, 0.0582), (54.9,
0.0578), (55.1, 0.0574), (55.4, 0.0571), (55.6, 0.0567), (55.9, 0.0564), (56.1, 0.056), (56.4,
0.0557), (56.6, 0.0554), (56.9, 0.055), (57.2, 0.0547), (57.4, 0.0544), (57.7, 0.0541), (57.9,
0.0537), (58.2, 0.0534), (58.4, 0.0531), (58.7, 0.0528), (58.9, 0.0524), (59.2, 0.0521), (59.4,
0.0518), (59.7, 0.0514), (59.9, 0.0511), (60.2, 0.0508), (60.4, 0.0504), (60.7, 0.0501), (60.9,
0.0497), (61.2, 0.0494), (61.4, 0.049), (61.7, 0.0487), (61.9, 0.0484), (62.2, 0.048), (62.4,
0.0477), (62.7, 0.0473), (62.9, 0.047), (63.2, 0.0466), (63.4, 0.0463), (63.7, 0.0459), (63.9,
0.0456), (64.2, 0.0452), (64.5, 0.0449), (64.7, 0.0445), (65.0, 0.0442), (65.2, 0.0438), (65.5,
0.0434), (65.7, 0.0431), (66.0, 0.0427), (66.2, 0.0423), (66.5, 0.042), (66.7, 0.0416), (67.0,
0.0412), (67.2, 0.0409), (67.5, 0.0405), (67.7, 0.0401), (68.0, 0.0397), (68.2, 0.0394), (68.5,
0.039), (68.7, 0.0386), (69.0, 0.0382), (69.2, 0.0378), (69.5, 0.0374), (69.7, 0.037), (70.0,
163
0.0366), (70.2, 0.0362), (70.5, 0.0359), (70.7, 0.0355), (71.0, 0.0351), (71.3, 0.0347), (71.5,
0.0343), (71.8, 0.0339), (72.0, 0.0335), (72.3, 0.0331), (72.5, 0.0327), (72.8, 0.0323), (73.0,
0.0319), (73.3, 0.0315), (73.5, 0.0312), (73.8, 0.0308), (74.0, 0.0304), (74.3, 0.03), (74.5,
0.0296), (74.8, 0.0292), (75.0, 0.0288), (75.3, 0.0284), (75.5, 0.028), (75.8, 0.0277), (76.0,
0.0273), (76.3, 0.0269), (76.5, 0.0265), (76.8, 0.0261), (77.0, 0.0257), (77.3, 0.0253), (77.5,
0.025), (77.8, 0.0246), (78.0, 0.0242), (78.3, 0.0238), (78.6, 0.0235), (78.8, 0.0231), (79.1,
0.0227), (79.3, 0.0223), (79.6, 0.022), (79.8, 0.0216), (80.1, 0.0212), (80.3, 0.0209), (80.6,
0.0205), (80.8, 0.0202), (81.1, 0.0198), (81.3, 0.0195), (81.6, 0.0191), (81.8, 0.0188), (82.1,
0.0184), (82.3, 0.0181), (82.6, 0.0177), (82.8, 0.0174), (83.1, 0.0171), (83.3, 0.0167), (83.6,
0.0164), (83.8, 0.0161), (84.1, 0.0158), (84.3, 0.0155), (84.6, 0.0151), (84.8, 0.0148), (85.1,
0.0145), (85.3, 0.0142), (85.6, 0.0139), (85.9, 0.0136), (86.1, 0.0133), (86.4, 0.013), (86.6,
0.0127), (86.9, 0.0124), (87.1, 0.0121), (87.4, 0.0118), (87.6, 0.0116), (87.9, 0.0113), (88.1,
0.011), (88.4, 0.0107), (88.6, 0.0105), (88.9, 0.0102), (89.1, 0.00996), (89.4, 0.0097), (89.6,
0.00944), (89.9, 0.0092), (90.1, 0.00895), (90.4, 0.00871), (90.6, 0.00847), (90.9, 0.00824),
(91.1, 0.00801), (91.4, 0.00778), (91.6, 0.00755), (91.9, 0.00733), (92.1, 0.00711), (92.4,
0.0069), (92.7, 0.00668), (92.9, 0.00648), (93.2, 0.00628), (93.4, 0.00608), (93.7, 0.00587),
(93.9, 0.00568), (94.2, 0.00549), (94.4, 0.0053), (94.7, 0.00511), (94.9, 0.00494), (95.2,
0.00476), (95.4, 0.00459), (95.7, 0.00441), (95.9, 0.00425), (96.2, 0.00408), (96.4, 0.00392),
(96.7, 0.00376), (96.9, 0.00361), (97.2, 0.00346), (97.4, 0.00331), (97.7, 0.00316), (97.9,
0.00302), (98.2, 0.00288), (98.4, 0.00274), (98.7, 0.00261), (98.9, 0.00248), (99.2, 0.00235),
(99.4, 0.00223), (99.7, 0.0021), (100.0, 0.00199), (100.2, 0.00187), (100.5, 0.00176), (100.7,
0.00165), (101.0, 0.00154), (101.2, 0.00144), (101.5, 0.00134), (101.7, 0.00123), (102.0,
0.00114), (102.2, 0.00104), (102.5, 0.000951), (102.7, 0.000858), (103.0, 0.000773), (103.2,
0.000689), (103.5, 0.000605), (103.7, 0.000521), (104.0, 0.000446), (104.2, 0.000371), (104.5,
0.000297), (104.7, 0.000223), (105.0, 0.000157), (105.2, 9.1e-05), (105.5, 2.5e-05), (105.7, -
3.9e-05), (106.0, -9.7e-05), (106.2, -0.000155), (106.5, -0.000212), (106.7, -0.000268), (107.0,
-0.000318), (107.3, -0.000368), (107.5, -0.000417), (107.8, -0.000466), (108.0, -0.000509),
(108.3, -0.000551), (108.5, -0.000593), (108.8, -0.000634), (109.0, -0.00067), (109.3, -0.000706),
(109.5, -0.000741), (109.8, -0.000775), (110.0, -0.000806), (110.3, -0.000837), (110.5, -0.000867),
(110.8, -0.000897), (111.0, -0.000923), (111.3, -0.000948), (111.5, -0.000973), (111.8, -0.000997),
(112.0, -0.00102), (112.3, -0.00104), (112.5, -0.00106), (112.8, -0.00108), (113.0, -0.00109),
(113.3, -0.00111), (113.5, -0.00112), (113.8, -0.00114), (114.1, -0.00115), (114.3, -0.00117),
(114.6, -0.00118), (114.8, -0.00119), (115.1, -0.0012), (115.3, -0.0012), (115.6, -0.00121),
(115.8, -0.00122), (116.1, -0.00123), (116.3, -0.00123), (116.6, -0.00124), (116.8, -0.00124),
(117.1, -0.00125), (117.3, -0.00125), (117.6, -0.00125), (117.8, -0.00125), (118.1, -0.00126),
(118.3, -0.00126), (118.6, -0.00126), (118.8, -0.00126), (119.1, -0.00127), (119.3, -0.00127),
(119.6, -0.00127), (119.8, -0.00127), (120.1, -0.00127), (120.3, -0.00127), (120.6, -0.00127),
(120.8, -0.00127), (121.1, -0.00128), (121.4, -0.00128), (121.6, -0.00128), (121.9, -0.00128),
(122.1, -0.00128), (122.4, -0.00128), (122.6, -0.00128), (122.9, -0.00128), (123.1, -0.00128),
164
(123.4, -0.00128), (123.6, -0.00128), (123.9, -0.00128), (124.1, -0.00129), (124.4, -0.00129),
(124.6, -0.00129), (124.9, -0.00129), (125.1, -0.00129), (125.4, -0.0013), (125.6, -0.0013),
(125.9, -0.0013), (126.1, -0.00131), (126.4, -0.00131), (126.6, -0.00131), (126.9, -0.00132),
(127.1, -0.00133), (127.4, -0.00133), (127.6, -0.00134), (127.9, -0.00135), (128.1, -0.00136),
(128.4, -0.00137), (128.7, -0.00138).
Fig. 4.12 (b) pluses: (-128.9, -0.0034), (-128.7, -0.00348), (-128.4, -0.00355), (-128.1, -
0.00363), (-127.9, -0.0037), (-127.6, -0.00377), (-127.4, -0.00384), (-127.1, -0.00391), (-126.9, -
0.00398), (-126.6, -0.00405), (-126.4, -0.00412), (-126.1, -0.00418), (-125.9, -0.00425), (-125.6,
-0.00431), (-125.4, -0.00437), (-125.1, -0.00443), (-124.9, -0.00449), (-124.6, -0.00455), (-
124.4, -0.0046), (-124.1, -0.00466), (-123.9, -0.00471), (-123.6, -0.00476), (-123.4, -0.00481), (-
123.1, -0.00485), (-122.9, -0.0049), (-122.6, -0.00494), (-122.4, -0.00498), (-122.1, -0.00502), (-
121.9, -0.00506), (-121.6, -0.00509), (-121.4, -0.00513), (-121.1, -0.00516), (-120.8, -0.00519),
(-120.6, -0.00522), (-120.3, -0.00524), (-120.1, -0.00526), (-119.8, -0.00529), (-119.6, -0.0053),
(-119.3, -0.00532), (-119.1, -0.00533), (-118.8, -0.00535), (-118.6, -0.00535), (-118.3, -0.00536),
(-118.1, -0.00537), (-117.8, -0.00537), (-117.6, -0.00537), (-117.3, -0.00537), (-117.1, -0.00537),
(-116.8, -0.00536), (-116.6, -0.00535), (-116.3, -0.00534), (-116.1, -0.00533), (-115.8, -0.00532),
(-115.6, -0.0053), (-115.3, -0.00529), (-115.1, -0.00527), (-114.8, -0.00525), (-114.6, -0.00522),
(-114.3, -0.0052), (-114.1, -0.00517), (-113.8, -0.00515), (-113.5, -0.00512), (-113.3, -0.00509),
(-113.0, -0.00506), (-112.8, -0.00502), (-112.5, -0.00499), (-112.3, -0.00495), (-112.0, -0.00492),
(-111.8, -0.00488), (-111.5, -0.00484), (-111.3, -0.0048), (-111.0, -0.00476), (-110.8, -0.00471),
(-110.5, -0.00467), (-110.3, -0.00462), (-110.0, -0.00458), (-109.8, -0.00454), (-109.5, -0.00449),
(-109.3, -0.00444), (-109.0, -0.00439), (-108.8, -0.00434), (-108.5, -0.0043), (-108.3, -0.00425),
(-108.0, -0.0042), (-107.8, -0.00415), (-107.5, -0.0041), (-107.3, -0.00405), (-107.0, -0.004), (-
106.7, -0.00395), (-106.5, -0.0039), (-106.2, -0.00385), (-106.0, -0.0038), (-105.7, -0.00375),
(-105.5, -0.0037), (-105.2, -0.00365), (-105.0, -0.0036), (-104.7, -0.00355), (-104.5, -0.0035), (-
104.2, -0.00345), (-104.0, -0.0034), (-103.7, -0.00335), (-103.5, -0.00331), (-103.2, -0.00326), (-
103.0, -0.00321), (-102.7, -0.00316), (-102.5, -0.00311), (-102.2, -0.00307), (-102.0, -0.00302),
(-101.7, -0.00297), (-101.5, -0.00292), (-101.2, -0.00287), (-101.0, -0.00283), (-100.7, -0.00278),
(-100.5, -0.00273), (-100.2, -0.00268), (-100.0, -0.00263), (-99.7, -0.00259), (-99.4, -0.00254),
(-99.2, -0.00249), (-98.9, -0.00244), (-98.7, -0.00239), (-98.4, -0.00234), (-98.2, -0.00229),
(-97.9, -0.00224), (-97.7, -0.00219), (-97.4, -0.00214), (-97.2, -0.00208), (-96.9, -0.00203),
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94.2, -0.0014), (-93.9, -0.00134), (-93.7, -0.00128), (-93.4, -0.00122), (-93.2, -0.00116), (-92.9,
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90.4, -0.000446), (-90.1, -0.000381), (-89.9, -0.000315), (-89.6, -0.000249), (-89.4, -0.000185),
(-89.1, -0.000121), (-88.9, -5.6e-05), (-88.6, 8e-06), (-88.4, 6.9e-05), (-88.1, 0.00013), (-87.9,
0.000192), (-87.6, 0.000253), (-87.4, 0.000309), (-87.1, 0.000365), (-86.9, 0.000421), (-86.6,
165
0.000477), (-86.4, 0.000525), (-86.1, 0.000574), (-85.9, 0.000623), (-85.6, 0.000671), (-85.3,
0.000711), (-85.1, 0.00075), (-84.8, 0.00079), (-84.6, 0.000829), (-84.3, 0.000857), (-84.1,
0.000884), (-83.8, 0.000912), (-83.6, 0.000939), (-83.3, 0.000954), (-83.1, 0.000968), (-82.8,
0.000982), (-82.6, 0.000996), (-82.3, 0.000996), (-82.1, 0.000996), (-81.8, 0.000995), (-81.6,
0.000995), (-81.3, 0.000978), (-81.1, 0.000962), (-80.8, 0.000945), (-80.6, 0.000929), (-80.3,
0.000895), (-80.1, 0.000862), (-79.8, 0.000829), (-79.6, 0.000796), (-79.3, 0.000745), (-79.1,
0.000694), (-78.8, 0.000644), (-78.6, 0.000593), (-78.3, 0.000525), (-78.0, 0.000457), (-77.8,
0.00039), (-77.5, 0.000322), (-77.3, 0.000237), (-77.0, 0.000152), (-76.8, 6.8e-05), (-76.5, -
1.7e-05), (-76.3, -0.000117), (-76.0, -0.000218), (-75.8, -0.000319), (-75.5, -0.00042), (-75.3,
-0.000535), (-75.0, -0.000651), (-74.8, -0.000766), (-74.5, -0.000882), (-74.3, -0.00101), (-74.0,
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0.00181), (-72.5, -0.00195), (-72.3, -0.00209), (-72.0, -0.00224), (-71.8, -0.00239), (-71.5,
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0.00555), (-66.2, -0.00568), (-66.0, -0.00582), (-65.7, -0.00595), (-65.5, -0.00608), (-65.2,
-0.0062), (-65.0, -0.00632), (-64.7, -0.00644), (-64.5, -0.00656), (-64.2, -0.00667), (-63.9, -
0.00677), (-63.7, -0.00687), (-63.4, -0.00698), (-63.2, -0.00706), (-62.9, -0.00714), (-62.7,
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0.00677), (-56.1, -0.0066), (-55.9, -0.00644), (-55.6, -0.00627), (-55.4, -0.00611), (-55.1, -
0.00589), (-54.9, -0.00568), (-54.6, -0.00546), (-54.4, -0.00525), (-54.1, -0.00498), (-53.9,
-0.00472), (-53.6, -0.00445), (-53.4, -0.00418), (-53.1, -0.00386), (-52.9, -0.00354), (-52.6,
-0.00322), (-52.4, -0.0029), (-52.1, -0.00253), (-51.9, -0.00215), (-51.6, -0.00178), (-51.4, -
0.00141), (-51.1, -0.000984), (-50.9, -0.00056), (-50.6, -0.000135), (-50.4, 0.000288), (-50.1,
0.000761), (-49.8, 0.00123), (-49.6, 0.0017), (-49.3, 0.00218), (-49.1, 0.00269), (-48.8, 0.0032),
(-48.6, 0.00372), (-48.3, 0.00423), (-48.1, 0.00478), (-47.8, 0.00533), (-47.6, 0.00587), (-
47.3, 0.00642), (-47.1, 0.00699), (-46.8, 0.00756), (-46.6, 0.00813), (-46.3, 0.0087), (-46.1,
0.00928), (-45.8, 0.00986), (-45.6, 0.0104), (-45.3, 0.011), (-45.1, 0.0116), (-44.8, 0.0122), (-
44.6, 0.0127), (-44.3, 0.0133), (-44.1, 0.0138), (-43.8, 0.0144), (-43.6, 0.0149), (-43.3, 0.0155),
(-43.1, 0.016), (-42.8, 0.0164), (-42.5, 0.0169), (-42.3, 0.0174), (-42.0, 0.0178), (-41.8, 0.0182),
(-41.5, 0.0187), (-41.3, 0.0191), (-41.0, 0.0194), (-40.8, 0.0197), (-40.5, 0.02), (-40.3, 0.0203),
(-40.0, 0.0204), (-39.8, 0.0206), (-39.5, 0.0208), (-39.3, 0.0209), (-39.0, 0.021), (-38.8, 0.021),
(-38.5, 0.021), (-38.3, 0.021), (-38.0, 0.0208), (-37.8, 0.0206), (-37.5, 0.0204), (-37.3, 0.0202),
(-37.0, 0.0198), (-36.8, 0.0194), (-36.5, 0.019), (-36.3, 0.0186), (-36.0, 0.0179), (-35.8, 0.0173),
166
(-35.5, 0.0167), (-35.2, 0.016), (-35.0, 0.0151), (-34.7, 0.0143), (-34.5, 0.0134), (-34.2, 0.0125),
(-34.0, 0.0114), (-33.7, 0.0103), (-33.5, 0.00917), (-33.2, 0.00805), (-33.0, 0.00672), (-32.7,
0.00539), (-32.5, 0.00406), (-32.2, 0.00273), (-32.0, 0.00123), (-31.7, -0.000279), (-31.5, -
0.00179), (-31.2, -0.00329), (-31.0, -0.00492), (-30.7, -0.00655), (-30.5, -0.00817), (-30.2,
-0.0098), (-30.0, -0.0115), (-29.7, -0.0131), (-29.5, -0.0148), (-29.2, -0.0165), (-29.0, -0.0181),
(-28.7, -0.0197), (-28.4, -0.0213), (-28.2, -0.0229), (-27.9, -0.0243), (-27.7, -0.0258), (-27.4,
-0.0272), (-27.2, -0.0286), (-26.9, -0.0297), (-26.7, -0.0309), (-26.4, -0.032), (-26.2, -0.0331),
(-25.9, -0.0337), (-25.7, -0.0343), (-25.4, -0.035), (-25.2, -0.0356), (-24.9, -0.0356), (-24.7,
-0.0356), (-24.4, -0.0355), (-24.2, -0.0355), (-23.9, -0.0347), (-23.7, -0.0339), (-23.4, -0.033),
(-23.2, -0.0322), (-22.9, -0.0304), (-22.7, -0.0285), (-22.4, -0.0267), (-22.2, -0.0248), (-21.9, -
0.0219), (-21.7, -0.0189), (-21.4, -0.0159), (-21.1, -0.0128), (-20.9, -0.00855), (-20.6, -0.00424),
(-20.4, 7.4e-05), (-20.1, 0.0044), (-19.9, 0.0101), (-19.6, 0.0159), (-19.4, 0.0216), (-19.1,
0.0274), (-18.9, 0.0347), (-18.6, 0.042), (-18.4, 0.0493), (-18.1, 0.0566), (-17.9, 0.0655), (-
17.6, 0.0743), (-17.4, 0.0832), (-17.1, 0.0922), (-16.9, 0.103), (-16.6, 0.113), (-16.4, 0.124),
(-16.1, 0.134), (-15.9, 0.146), (-15.6, 0.158), (-15.4, 0.17), (-15.1, 0.182), (-14.9, 0.196), (-
14.6, 0.21), (-14.4, 0.223), (-14.1, 0.237), (-13.8, 0.252), (-13.6, 0.267), (-13.3, 0.281), (-13.1,
0.296), (-12.8, 0.313), (-12.6, 0.329), (-12.3, 0.345), (-12.1, 0.361), (-11.8, 0.378), (-11.6,
0.395), (-11.3, 0.412), (-11.1, 0.429), (-10.8, 0.447), (-10.6, 0.465), (-10.3, 0.482), (-10.1,
0.5), (-9.8, 0.518), (-9.6, 0.536), (-9.3, 0.554), (-9.1, 0.572), (-8.8, 0.59), (-8.6, 0.608), (-8.3,
0.626), (-8.1, 0.644), (-7.8, 0.661), (-7.6, 0.679), (-7.3, 0.696), (-7.0, 0.713), (-6.8, 0.73),
(-6.5, 0.746), (-6.3, 0.763), (-6.0, 0.779), (-5.8, 0.795), (-5.5, 0.81), (-5.3, 0.825), (-5.0,
0.84), (-4.8, 0.853), (-4.5, 0.866), (-4.3, 0.88), (-4.0, 0.893), (-3.8, 0.904), (-3.5, 0.915), (-
3.3, 0.926), (-3.0, 0.937), (-2.8, 0.945), (-2.5, 0.953), (-2.3, 0.962), (-2.0, 0.97), (-1.8, 0.975),
(-1.5, 0.98), (-1.3, 0.986), (-1.0, 0.991), (-0.8, 0.993), (-0.5, 0.995), (-0.3, 0.998), (0.0, 1),
(0.3, 0.998), (0.5, 0.997), (0.8, 0.995), (1.0, 0.993), (1.3, 0.988), (1.5, 0.983), (1.8, 0.978),
(2.0, 0.972), (2.3, 0.964), (2.5, 0.956), (2.8, 0.948), (3.0, 0.939), (3.3, 0.928), (3.5, 0.918),
(3.8, 0.907), (4.0, 0.896), (4.3, 0.882), (4.5, 0.869), (4.8, 0.856), (5.0, 0.843), (5.3, 0.827),
(5.5, 0.812), (5.8, 0.797), (6.0, 0.782), (6.3, 0.766), (6.5, 0.749), (6.8, 0.733), (7.0, 0.716),
(7.3, 0.699), (7.6, 0.681), (7.8, 0.664), (8.1, 0.646), (8.3, 0.628), (8.6, 0.61), (8.8, 0.592),
(9.1, 0.574), (9.3, 0.556), (9.6, 0.538), (9.8, 0.52), (10.1, 0.502), (10.3, 0.484), (10.6, 0.466),
(10.8, 0.449), (11.1, 0.431), (11.3, 0.414), (11.6, 0.397), (11.8, 0.379), (12.1, 0.362), (12.3,
0.346), (12.6, 0.33), (12.8, 0.314), (13.1, 0.298), (13.3, 0.283), (13.6, 0.268), (13.8, 0.253),
(14.1, 0.238), (14.4, 0.224), (14.6, 0.21), (14.9, 0.197), (15.1, 0.183), (15.4, 0.171), (15.6,
0.159), (15.9, 0.147), (16.1, 0.135), (16.4, 0.124), (16.6, 0.114), (16.9, 0.103), (17.1, 0.0925),
(17.4, 0.0836), (17.6, 0.0746), (17.9, 0.0657), (18.1, 0.0568), (18.4, 0.0495), (18.6, 0.0421),
(18.9, 0.0348), (19.1, 0.0275), (19.4, 0.0217), (19.6, 0.0159), (19.9, 0.0102), (20.1, 0.00442),
(20.4, 7.4e-05), (20.6, -0.00426), (20.9, -0.00858), (21.1, -0.0129), (21.4, -0.0159), (21.7, -
0.0189), (21.9, -0.0219), (22.2, -0.0249), (22.4, -0.0268), (22.7, -0.0286), (22.9, -0.0305),
(23.2, -0.0323), (23.4, -0.0332), (23.7, -0.034), (23.9, -0.0348), (24.2, -0.0357), (24.4, -0.0357),
167
(24.7, -0.0357), (24.9, -0.0357), (25.2, -0.0357), (25.4, -0.0351), (25.7, -0.0345), (25.9, -
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(30.5, -0.00821), (30.7, -0.00657), (31.0, -0.00494), (31.2, -0.00331), (31.5, -0.00179), (31.7,
-0.00028), (32.0, 0.00123), (32.2, 0.00274), (32.5, 0.00408), (32.7, 0.00542), (33.0, 0.00675),
(33.2, 0.00808), (33.5, 0.0092), (33.7, 0.0103), (34.0, 0.0114), (34.2, 0.0126), (34.5, 0.0134),
(34.7, 0.0143), (35.0, 0.0152), (35.2, 0.0161), (35.5, 0.0167), (35.8, 0.0174), (36.0, 0.018),
(36.3, 0.0186), (36.5, 0.0191), (36.8, 0.0195), (37.0, 0.0199), (37.3, 0.0203), (37.5, 0.0205),
(37.8, 0.0207), (38.0, 0.0209), (38.3, 0.021), (38.5, 0.021), (38.8, 0.021), (39.0, 0.021), (39.3,
0.021), (39.5, 0.0209), (39.8, 0.0207), (40.0, 0.0205), (40.3, 0.0204), (40.5, 0.0201), (40.8,
0.0198), (41.0, 0.0194), (41.3, 0.0191), (41.5, 0.0187), (41.8, 0.0183), (42.0, 0.0179), (42.3,
0.0175), (42.5, 0.017), (42.8, 0.0165), (43.1, 0.016), (43.3, 0.0155), (43.6, 0.015), (43.8,
0.0144), (44.1, 0.0139), (44.3, 0.0134), (44.6, 0.0128), (44.8, 0.0122), (45.1, 0.0116), (45.3,
0.0111), (45.6, 0.0105), (45.8, 0.0099), (46.1, 0.00932), (46.3, 0.00874), (46.6, 0.00816),
(46.8, 0.00759), (47.1, 0.00702), (47.3, 0.00645), (47.6, 0.0059), (47.8, 0.00535), (48.1,
0.0048), (48.3, 0.00425), (48.6, 0.00373), (48.8, 0.00322), (49.1, 0.0027), (49.3, 0.00219),
(49.6, 0.00171), (49.8, 0.00124), (50.1, 0.000765), (50.4, 0.00029), (50.6, -0.000135), (50.9,
-0.000562), (51.1, -0.000988), (51.4, -0.00142), (51.6, -0.00179), (51.9, -0.00216), (52.1,
-0.00254), (52.4, -0.00292), (52.6, -0.00324), (52.9, -0.00356), (53.1, -0.00388), (53.4, -
0.0042), (53.6, -0.00447), (53.9, -0.00474), (54.1, -0.005), (54.4, -0.00527), (54.6, -0.00549),
(54.9, -0.0057), (55.1, -0.00592), (55.4, -0.00613), (55.6, -0.0063), (55.9, -0.00646), (56.1,
-0.00663), (56.4, -0.00679), (56.6, -0.00691), (56.9, -0.00703), (57.2, -0.00715), (57.4, -
0.00727), (57.7, -0.00734), (57.9, -0.00742), (58.2, -0.00749), (58.4, -0.00756), (58.7, -0.0076),
(58.9, -0.00764), (59.2, -0.00767), (59.4, -0.00771), (59.7, -0.00771), (59.9, -0.00771), (60.2,
-0.00771), (60.4, -0.00771), (60.7, -0.00767), (60.9, -0.00764), (61.2, -0.00761), (61.4, -
0.00758), (61.7, -0.00752), (61.9, -0.00746), (62.2, -0.0074), (62.4, -0.00734), (62.7, -0.00726),
(62.9, -0.00717), (63.2, -0.00709), (63.4, -0.00701), (63.7, -0.0069), (63.9, -0.0068), (64.2, -
0.0067), (64.5, -0.00659), (64.7, -0.00647), (65.0, -0.00635), (65.2, -0.00623), (65.5, -0.00611),
(65.7, -0.00597), (66.0, -0.00584), (66.2, -0.00571), (66.5, -0.00557), (66.7, -0.00543), (67.0,
-0.00528), (67.2, -0.00514), (67.5, -0.00499), (67.7, -0.00484), (68.0, -0.00469), (68.2, -
0.00454), (68.5, -0.00439), (68.7, -0.00424), (69.0, -0.00408), (69.2, -0.00393), (69.5, -0.00377),
(69.7, -0.00362), (70.0, -0.00347), (70.2, -0.00331), (70.5, -0.00315), (70.7, -0.003), (71.0, -
0.00285), (71.3, -0.0027), (71.5, -0.00254), (71.8, -0.0024), (72.0, -0.00225), (72.3, -0.0021),
(72.5, -0.00196), (72.8, -0.00182), (73.0, -0.00168), (73.3, -0.00154), (73.5, -0.0014), (73.8,
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0.000653), (75.3, -0.000538), (75.5, -0.000422), (75.8, -0.00032), (76.0, -0.000219), (76.3, -
0.000118), (76.5, -1.7e-05), (76.8, 6.8e-05), (77.0, 0.000153), (77.3, 0.000238), (77.5, 0.000323),
(77.8, 0.000391), (78.0, 0.000459), (78.3, 0.000528), (78.6, 0.000596), (78.8, 0.000647),
168
(79.1, 0.000697), (79.3, 0.000748), (79.6, 0.000799), (79.8, 0.000832), (80.1, 0.000866),
(80.3, 0.0009), (80.6, 0.000933), (80.8, 0.00095), (81.1, 0.000966), (81.3, 0.000983), (81.6,
0.000999), (81.8, 0.001), (82.1, 0.001), (82.3, 0.001), (82.6, 0.001), (82.8, 0.000987), (83.1,
0.000972), (83.3, 0.000958), (83.6, 0.000943), (83.8, 0.000916), (84.1, 0.000888), (84.3,
0.000861), (84.6, 0.000833), (84.8, 0.000793), (85.1, 0.000754), (85.3, 0.000714), (85.6,
0.000674), (85.9, 0.000625), (86.1, 0.000577), (86.4, 0.000528), (86.6, 0.000479), (86.9,
0.000423), (87.1, 0.000366), (87.4, 0.00031), (87.6, 0.000254), (87.9, 0.000192), (88.1, 0.000131),
(88.4, 6.9e-05), (88.6, 8e-06), (88.9, -5.7e-05), (89.1, -0.000121), (89.4, -0.000186), (89.6, -
0.000251), (89.9, -0.000317), (90.1, -0.000382), (90.4, -0.000448), (90.6, -0.000514), (90.9,
-0.00058), (91.1, -0.000646), (91.4, -0.000712), (91.6, -0.000778), (91.9, -0.000843), (92.1,
-0.000907), (92.4, -0.000972), (92.7, -0.00104), (92.9, -0.0011), (93.2, -0.00116), (93.4, -
0.00123), (93.7, -0.00129), (93.9, -0.00135), (94.2, -0.00141), (94.4, -0.00147), (94.7, -0.00153),
(94.9, -0.00159), (95.2, -0.00165), (95.4, -0.00171), (95.7, -0.00176), (95.9, -0.00182), (96.2,
-0.00187), (96.4, -0.00193), (96.7, -0.00199), (96.9, -0.00204), (97.2, -0.00209), (97.4, -
0.00214), (97.7, -0.0022), (97.9, -0.00225), (98.2, -0.0023), (98.4, -0.00235), (98.7, -0.0024),
(98.9, -0.00245), (99.2, -0.0025), (99.4, -0.00255), (99.7, -0.0026), (100.0, -0.00265), (100.2,
-0.00269), (100.5, -0.00274), (100.7, -0.00279), (101.0, -0.00284), (101.2, -0.00289), (101.5,
-0.00293), (101.7, -0.00298), (102.0, -0.00303), (102.2, -0.00308), (102.5, -0.00313), (102.7,
-0.00318), (103.0, -0.00322), (103.2, -0.00327), (103.5, -0.00332), (103.7, -0.00337), (104.0,
-0.00342), (104.2, -0.00347), (104.5, -0.00352), (104.7, -0.00357), (105.0, -0.00362), (105.2,
-0.00367), (105.5, -0.00372), (105.7, -0.00377), (106.0, -0.00382), (106.2, -0.00387), (106.5,
-0.00392), (106.7, -0.00397), (107.0, -0.00402), (107.3, -0.00407), (107.5, -0.00412), (107.8,
-0.00417), (108.0, -0.00422), (108.3, -0.00426), (108.5, -0.00431), (108.8, -0.00436), (109.0,
-0.00441), (109.3, -0.00446), (109.5, -0.00451), (109.8, -0.00456), (110.0, -0.0046), (110.3,
-0.00465), (110.5, -0.00469), (110.8, -0.00474), (111.0, -0.00478), (111.3, -0.00482), (111.5,
-0.00486), (111.8, -0.0049), (112.0, -0.00494), (112.3, -0.00497), (112.5, -0.00501), (112.8,
-0.00505), (113.0, -0.00508), (113.3, -0.00511), (113.5, -0.00514), (113.8, -0.00517), (114.1,
-0.0052), (114.3, -0.00522), (114.6, -0.00525), (114.8, -0.00527), (115.1, -0.00529), (115.3,
-0.00531), (115.6, -0.00533), (115.8, -0.00535), (116.1, -0.00536), (116.3, -0.00537), (116.6,
-0.00538), (116.8, -0.00539), (117.1, -0.00539), (117.3, -0.00539), (117.6, -0.00539), (117.8,
-0.0054), (118.1, -0.00539), (118.3, -0.00538), (118.6, -0.00538), (118.8, -0.00537), (119.1,
-0.00536), (119.3, -0.00534), (119.6, -0.00533), (119.8, -0.00531), (120.1, -0.00529), (120.3,
-0.00526), (120.6, -0.00524), (120.8, -0.00522), (121.1, -0.00518), (121.4, -0.00515), (121.6,
-0.00512), (121.9, -0.00509), (122.1, -0.00505), (122.4, -0.00501), (122.6, -0.00497), (122.9,
-0.00492), (123.1, -0.00488), (123.4, -0.00483), (123.6, -0.00478), (123.9, -0.00473), (124.1,
-0.00468), (124.4, -0.00462), (124.6, -0.00457), (124.9, -0.00451), (125.1, -0.00445), (125.4,
-0.00439), (125.6, -0.00433), (125.9, -0.00427), (126.1, -0.0042), (126.4, -0.00414), (126.6,
-0.00407), (126.9, -0.004), (127.1, -0.00393), (127.4, -0.00386), (127.6, -0.00379), (127.9,
-0.00372), (128.1, -0.00364), (128.4, -0.00357), (128.7, -0.00349).
169
Fig. 4.13 circles: (0.000298, 11.02), (0.000959, 10.88), (0.00243, 11.8), (0.00314,
12.78), (0.000674, 17.26), (0.00379, 13.04), (0.00161, 17.52), (0.00449, 11.88), (0.00249,
15.38), (0.00597, 12.1), (0.00422, 12.95), (0.00743, 10.62), (0.00592, 11.22), (0.00891, 10.3),
(0.00759, 10.56), (0.0112, 8.8), (0.0101, 9.239), (0.0134, 8.482), (0.0134, 8.469), (0.0165,
8.409), (0.0168, 7.893), (0.0196, 7.963), (0.0208, 7.723), (0.0243, 8.016), (0.0258, 7.013),
(0.029, 7.157), (0.0315, 6.87), (0.0354, 6.354), (0.0389, 5.851), (0.0433, 5.48), (0.0471, 5.372),
(0.0521, 4.983).
Fig. 4.13 squares: (0.00597, 8.593), (0.0092, 8.274), (0.0132, 7.876), (0.0173, 7.56),
(0.0254, 6.903), (0.0334, 6.469), (0.0496, 5.855), (0.0657, 5.42), (0.0819, 5.144), (0.102, 4.9).
Fig. 4.14: (-0.105, 3.04), (-0.0399, 3.13), (-0.035, 3.05), (-0.0197, 3.17), (-0.02, 3.12),
(-0.0103, 3.38), (-0.0121, 3.34), (-0.00482, 3.76), (-0.0088, 3.55), (-0.00339, 4.08), (-0.0071,
3.72), (-0.00542, 4.11), (-0.00114, 5.28), (-0.000415, 5.63), (-0.00371, 5.00), (0.000298, 5.90),
(-0.00281, 5.50), (0.000959, 6.00), (-0.00186, 5.31), (-0.000999, 4.86), (0.00243, 5.79), (0.00314,
4.73), (0.000674, 4.09), (0.00379, 4.47), (0.00161, 4.03), (0.00449, 4.42), (0.00249, 3.93),
(0.00597, 4.01), (0.00422, 4.01), (0.00743, 3.96), (0.00592, 3.85), (0.00891, 3.76), (0.00759,
3.82), (0.0112, 3.87), (0.0101, 3.81), (0.0134, 3.64), (0.0134, 3.76), (0.0165, 3.58), (0.0168,
3.62), (0.0196, 3.83), (0.0208, 3.82), (0.0243, 3.67), (0.0258, 3.79), (0.029, 3.84), (0.0315,
3.90), (0.0354, 4.00), (0.0389, 4.08), (0.0433, 4.11), (0.0471, 4.04), (0.0521, 4.21).
Fig. 4.15: (-0.105, -0.42), (-0.0399, -0.54), (-0.035, -0.30), (-0.0197, -0.35), (-0.02, -0.24),
(-0.0103, 0.03), (-0.0121, 0.01), (-0.00482, 0.13), (-0.0088, 0.11), (-0.00339, 0.10), (-0.0071,
0.12), (-0.00542, 0.11), (-0.00114, 0.13), (-0.000415, 0.17), (-0.00371, 0.10), (0.000298, 0.13),
(-0.00281, 0.07), (0.000959, 0.07), (-0.00186, 0.08), (-0.000999, 0.06), (0.00243, 0.18), (0.00314,
0.04), (0.000674, 0.00), (0.00379, 0.09), (0.00161, 0.08), (0.00449, 0.09), (0.00249, 0.09),
(0.00597, 0.08), (0.00422, 0.13), (0.00743, 0.07), (0.00592, 0.16), (0.00891, 0.10), (0.00759,
0.15), (0.0112, 0.09), (0.0101, 0.16), (0.0134, 0.12), (0.0134, 0.12), (0.0165, 0.09), (0.0168,
0.11), (0.0196, 0.06), (0.0208, 0.21), (0.0243, 0.11), (0.0258, 0.24), (0.029, 0.17), (0.0315,
0.38), (0.0354, 0.30), (0.0389, 0.62), (0.0433, 0.63), (0.0471, 0.79), (0.0521, 0.77).
Fig. 7.3 up pointing triangles: (3.9646, 0.0001035), (7.9375, 0.0001112), (11.9092,
0.0001036), (15.8798, 0.0001164), (19.8615, 0.0001208), (23.8326, 0.0001351), (27.7959, 0.0001743),
(29.3832, 0.0001996), (30.5744, 0.0002218), (31.7710, 0.0002695), (32.5689, 0.0003041), (33.3560,
0.0003543), (34.1562, 0.0004270), (34.9461, 0.0005144), (35.7425, 0.0006684), (36.5318, 0.0008922),
(36.9392, 0.0010661), (37.3293, 0.0012826), (37.7399, 0.0016052), (38.1324, 0.0020406), (38.5295,
0.0026787), (38.7267, 0.0031291), (38.9277, 0.0036724), (39.1280, 0.0043746), (39.3273, 0.0052670),
(39.7099, 0.0078041), (40.1137, 0.0120341), (40.3161, 0.0146882), (40.5192, 0.0168947), (40.7020,
0.0183380), (40.9912, 0.0186513), (41.3175, 0.0161357), (41.7315, 0.0099564), (42.1443, 0.0055032),
(42.5640, 0.0034826), (42.9834, 0.0028508), (43.7431, 0.0023807), (44.5072, 0.0019860), (45.3642,
0.0013198), (46.1474, 0.0008978), (46.9331, 0.0009134), (47.7242, 0.0010101), (48.8841, 0.0010286),
(50.1450, 0.0011126), (51.7021, 0.0010245), (53.6506, 0.0007805), (55.6324, 0.0006801), (57.6626,
0.0004860), (59.6179, 0.0002889), (63.6394, 0.0002180), (67.5784, 0.0002479), (70.7778, 0.0002373),
170
(71.6383, 0.0002489), (75.5914, 0.0002426), (79.5419, 0.0002386), (83.4674, 0.0002342), (87.4917,
0.0002415), (91.4906, 0.0002429), (95.4576, 0.0002376), (103.3603, 0.0002305), (111.3259,
0.0002356), (119.2947, 0.0002311), (123.2447, 0.0002426), (131.3408, 0.0002374), (143.1774,
0.0002391), (159.0813, 0.0002391), (171.1098, 0.0002432), (199.0006, 0.0002205), (214.8533,
0.0002329), (230.9084, 0.0002329), (246.7313, 0.0002335), (262.6878, 0.0002288), (270.6110,
0.0002271), (278.6826, 0.0002270).
Fig. 7.3 down pointing triangles: (270.6123, 0.0002266), (262.6872, 0.0002314),
(246.7307, 0.0002351), (238.7592, 0.0002292), (230.9144, 0.0002259), (214.8579, 0.0002342),
(199.0091, 0.0002282), (179.1218, 0.0002281), (171.1157, 0.0002407), (159.0870, 0.0002505),
(159.0882, 0.0002368), (143.1845, 0.0002265), (139.2296, 0.0002312), (131.3453, 0.0002394),
(119.3002, 0.0002497), (111.3311, 0.0002290), (107.4426, 0.0002370), (103.3626, 0.0002400),
(95.4615, 0.0002449), (87.4961, 0.0002355), (83.4722, 0.0002426), (79.5456, 0.0002266), (75.5956,
0.0002379), (71.6437, 0.0002405), (67.5814, 0.0002467), (63.6417, 0.0002555), (59.6212, 0.0002811),
(57.6643, 0.0004740), (55.6342, 0.0006282), (53.6510, 0.0007709), (51.7035, 0.0008888), (50.1466,
0.0010902), (48.8860, 0.0012081), (47.7255, 0.0012181), (46.9338, 0.0009482), (46.1476, 0.0007001),
(45.3641, 0.0009084), (44.5079, 0.0017286), (43.7432, 0.0024312), (42.9844, 0.0030458), (42.5644,
0.0039440), (42.1448, 0.0064185), (41.7323, 0.0105997), (41.3174, 0.0164891), (40.9909, 0.0190573),
(40.7027, 0.0185432), (40.5204, 0.0169937), (40.3174, 0.0147354), (40.1148, 0.0121047), (39.3282,
0.0053148), (39.1287, 0.0043947), (38.9286, 0.0037052), (38.7285, 0.0031445), (38.1331, 0.0020604),
(37.7403, 0.0016207), (38.9282, 0.0036742), (38.7282, 0.0031256), (38.5303, 0.0027024), (39.1285,
0.0043812), (38.9288, 0.0037770), (38.7280, 0.0032146), (38.5304, 0.0027589), (38.1332, 0.0020774),
(37.7405, 0.0016465), (37.3297, 0.0013126), (36.9393, 0.0010935), (36.5323, 0.0009123), (35.7430,
0.0006753), (34.9465, 0.0005259), (34.1569, 0.0004269), (33.3562, 0.0003566), (32.5697, 0.0003099),
(31.7716, 0.0002682), (30.5755, 0.0002294), (29.3841, 0.0001999), (27.7965, 0.0001664), (23.8336,
0.0001307), (19.8622, 0.0001155), (15.8805, 0.0001105), (11.9099, 0.0001092), (7.9382, 0.0000989),
(3.9650, 0.0001033), (1.9818, 0.0000981).
Fig. 7.4 up pointing triangles (black): (-0.8921, 9.343e-13), (-0.784, 9.343e-13),
(-0.676, 9.339e-13), (-0.5679, 9.328e-13), (-0.4596, 9.321e-13), (-0.3515, 9.313e-13), (-0.2437,
9.307e-13), (-0.2005, 9.301e-13), (-0.1681, 9.297e-13), (-0.1355, 9.293e-13), (-0.1138, 9.291e-
13), (-0.09239, 9.286e-13), (-0.07062, 9.285e-13), (-0.04912, 9.278e-13), (-0.02745, 9.277e-
13), (-0.005978, 9.272e-13), (0.005107, 9.268e-13), (0.01572, 9.265e-13), (0.0269, 9.261e-
13), (0.03757, 9.255e-13), (0.04838, 9.248e-13), (0.05375, 9.245e-13), (0.05921, 9.241e-13),
(0.06467, 9.238e-13), (0.07009, 9.231e-13), (0.0805, 9.218e-13), (0.09149, 9.21e-13), (0.09699,
9.211e-13), (0.1025, 9.212e-13), (0.1075, 9.223e-13), (0.1154, 9.244e-13), (0.1242, 9.244e-13),
(0.1355, 9.261e-13), (0.1467, 9.255e-13), (0.1582, 9.277e-13), (0.1696, 9.296e-13), (0.1902,
9.298e-13), (0.211, 9.339e-13), (0.2344, 9.341e-13), (0.2557, 9.365e-13), (0.277, 9.37e-13),
(0.2986, 9.388e-13), (0.3301, 9.399e-13), (0.3644, 9.391e-13), (0.4068, 9.442e-13), (0.4598,
9.414e-13), (0.5137, 9.391e-13), (0.569, 9.401e-13), (0.6222, 9.463e-13), (0.7316, 9.393e-
13), (0.8388, 9.542e-13), (0.9258, 9.647e-13), (0.9493, 9.78e-13), (1.057, 9.719e-13), (1.164,
171
9.848e-13), (1.271, 9.87e-13), (1.381, 9.89e-13), (1.489, 9.888e-13), (1.597, 9.875e-13), (1.812,
9.969e-13), (2.029, 1.005e-12), (2.246, 1.009e-12), (2.353, 1.013e-12), (2.574, 1.017e-12),
(2.896, 1.025e-12), (3.329, 1.032e-12), (3.656, 1.035e-12), (4.415, 1.045e-12), (4.846, 1.048e-
12), (5.283, 1.053e-12), (5.714, 1.058e-12), (6.148, 1.061e-12), (6.363, 1.063e-12), (6.583,
1.066e-12).
Fig. 7.4 down pointing triangles (red): (6.363, 1.063e-12), (6.148, 1.061e-12),
(5.713, 1.058e-12), (5.497, 1.056e-12), (5.283, 1.053e-12), (4.846, 1.048e-12), (4.415, 1.045e-
12), (3.874, 1.038e-12), (3.656, 1.034e-12), (3.329, 1.031e-12), (3.329, 1.031e-12), (2.896,
1.023e-12), (2.788, 1.021e-12), (2.574, 1.019e-12), (2.246, 1.004e-12), (2.029, 1.001e-12),
(1.923, 9.982e-13), (1.812, 9.935e-13), (1.597, 9.877e-13), (1.381, 9.821e-13), (1.271, 9.72e-
13), (1.164, 9.697e-13), (1.057, 9.63e-13), (0.9494, 9.523e-13), (0.8389, 9.488e-13), (0.7317,
9.382e-13), (0.6223, 9.358e-13), (0.569, 9.331e-13), (0.5138, 9.368e-13), (0.4598, 9.343e-13),
(0.4068, 9.309e-13), (0.3645, 9.304e-13), (0.3302, 9.322e-13), (0.2986, 9.309e-13), (0.2771,
9.283e-13), (0.2557, 9.28e-13), (0.2343, 9.316e-13), (0.2111, 9.31e-13), (0.1902, 9.282e-13),
(0.1696, 9.228e-13), (0.1582, 9.217e-13), (0.1468, 9.218e-13), (0.1355, 9.193e-13), (0.1242,
9.157e-13), (0.1154, 9.17e-13), (0.1075, 9.148e-13), (0.1026, 9.14e-13), (0.09703, 9.138e-
13), (0.09152, 9.138e-13), (0.07011, 9.153e-13), (0.06468, 9.158e-13), (0.05924, 9.159e-13),
(0.05379, 9.164e-13), (0.0484, 9.165e-13), (0.03759, 9.172e-13), (0.02691, 9.176e-13), (0.05923,
9.159e-13), (0.05379, 9.164e-13), (0.0484, 9.165e-13), (0.0376, 9.158e-13), (0.06468, 9.148e-
13), (0.05924, 9.146e-13), (0.05378, 9.148e-13), (0.0484, 9.152e-13), (0.0376, 9.158e-13),
(0.02691, 9.16e-13), (0.01573, 9.167e-13), (0.00511, 9.17e-13), (-0.005963, 9.172e-13), (-
0.02744, 9.176e-13), (-0.04911, 9.181e-13), (-0.0706, 9.184e-13), (-0.09239, 9.187e-13), (-
0.1138, 9.19e-13), (-0.1355, 9.193e-13), (-0.168, 9.197e-13), (-0.2005, 9.197e-13), (-0.2437,
9.202e-13), (-0.3515, 9.21e-13), (-0.4596, 9.212e-13), (-0.5679, 9.218e-13), (-0.6759, 9.221e-
13), (-0.784, 9.229e-13), (-0.8921, 9.231e-13).
Fig. 7.5 up pointing triangles (black): (-0.8933, 3.311e-11), (-0.7863, 3.311e-11), (-
0.6793, 3.311e-11), (-0.5725, 3.311e-11), (-0.4655, 3.311e-11), (-0.3586, 3.311e-11), (-0.2513,
3.311e-11), (-0.2087, 3.311e-11), (-0.1769, 3.311e-11), (-0.1448, 3.311e-11), (-0.1231, 3.311e-
11), (-0.1021, 3.311e-11), (-0.0802, 3.31e-11), (-0.0592, 3.31e-11), (-0.0359, 3.308e-11), (-
0.0147, 3.307e-11), (-0.0050, 3.307e-11), (0.0067, 3.306e-11), (0.0165, 3.306e-11), (0.0283,
3.306e-11), (0.0381, 3.306e-11), (0.0442, 3.306e-11), (0.0481, 3.306e-11), (0.0541, 3.305e-11),
(0.0602, 3.305e-11), (0.0702, 3.302e-11), (0.0802, 3.3e-11), (0.0862, 3.3e-11), (0.0923, 3.3e-
11), (0.0963, 3.3e-11), (0.1025, 3.3e-11), (0.1127, 3.301e-11), (0.1231, 3.303e-11), (0.1355,
3.307e-11), (0.1458, 3.312e-11), (0.1563, 3.316e-11), (0.1773, 3.328e-11), (0.1987, 3.337e-11),
(0.2201, 3.343e-11), (0.2417, 3.351e-11), (0.2635, 3.358e-11), (0.2854, 3.359e-11), (0.3164,
3.369e-11), (0.3501, 3.393e-11), (0.3911, 3.391e-11), (0.4446, 3.397e-11), (0.4990, 3.41e-11),
(0.5518, 3.419e-11), (0.6054, 3.428e-11), (0.7134, 3.45e-11), (0.8194, 3.496e-11), (0.9070,
3.517e-11), (0.9283, 3.514e-11), (1.0351, 3.541e-11), (1.1422, 3.566e-11), (1.2487, 3.581e-
11), (1.3550, 3.6e-11), (1.4637, 3.615e-11), (1.5690, 3.627e-11), (1.7854, 3.656e-11), (1.9974,
172
3.682e-11), (2.2132, 3.707e-11), (2.3188, 3.715e-11), (2.5346, 3.733e-11), (2.8569, 3.762e-11),
(3.2859, 3.784e-11), (3.6060, 3.798e-11), (4.3574, 3.836e-11), (4.7848, 3.854e-11), (5.2133,
3.873e-11), (5.6428, 3.888e-11), (6.0708, 3.904e-11), (6.2846, 3.912e-11), (6.5016, 3.916e-11).
Fig. 7.5 down pointing triangles (red): (6.2841, 3.904e-11), (6.0703, 3.897e-11),
(5.6423, 3.88e-11), (5.4283, 3.866e-11), (5.2132, 3.855e-11), (4.7846, 3.837e-11), (4.3575,
3.817e-11), (3.8218, 3.788e-11), (3.6061, 3.773e-11), (3.2861, 3.753e-11), (3.2862, 3.754e-
11), (2.8572, 3.722e-11), (2.7468, 3.717e-11), (2.5347, 3.7e-11), (2.2131, 3.67e-11), (1.9973,
3.646e-11), (1.8903, 3.635e-11), (1.7854, 3.624e-11), (1.5690, 3.595e-11), (1.3553, 3.556e-11),
(1.2490, 3.541e-11), (1.1426, 3.521e-11), (1.0354, 3.506e-11), (0.9285, 3.481e-11), (0.8193,
3.449e-11), (0.7134, 3.421e-11), (0.6054, 3.383e-11), (0.5518, 3.381e-11), (0.4990, 3.369e-11),
(0.4446, 3.356e-11), (0.3912, 3.346e-11), (0.3503, 3.345e-11), (0.3165, 3.339e-11), (0.2854,
3.326e-11), (0.2635, 3.318e-11), (0.2417, 3.312e-11), (0.2201, 3.299e-11), (0.1988, 3.292e-11),
(0.1775, 3.282e-11), (0.1565, 3.271e-11), (0.1460, 3.263e-11), (0.1357, 3.258e-11), (0.1233,
3.249e-11), (0.1126, 3.244e-11), (0.1024, 3.243e-11), (0.0963, 3.242e-11), (0.0923, 3.242e-11),
(0.0862, 3.242e-11), (0.0801, 3.242e-11), (0.0600, 3.241e-11), (0.0539, 3.241e-11), (0.0479,
3.241e-11), (0.0440, 3.241e-11), (0.0380, 3.241e-11), (0.0281, 3.241e-11), (0.0163, 3.241e-11),
(0.0481, 3.239e-11), (0.0441, 3.239e-11), (0.0381, 3.239e-11), (0.0283, 3.239e-11), (0.0540,
3.238e-11), (0.0479, 3.237e-11), (0.0440, 3.237e-11), (0.0380, 3.237e-11), (0.0281, 3.237e-11),
(0.0162, 3.238e-11), (0.0065, 3.238e-11), (-0.0052, 3.238e-11), (-0.0149, 3.238e-11), (-0.0360,
3.237e-11), (-0.0593, 3.237e-11), (-0.0804, 3.237e-11), (-0.1022, 3.237e-11), (-0.1233, 3.237e-
11), (-0.1449, 3.238e-11), (-0.1770, 3.238e-11), (-0.2088, 3.238e-11), (-0.2514, 3.238e-11), (-
0.3587, 3.238e-11), (-0.4656, 3.238e-11), (-0.5726, 3.238e-11), (-0.6793, 3.238e-11), (-0.7864,
3.238e-11), (-0.8933, 3.237e-11), (-0.9467, 3.234e-11).
Fig. 7.6 contains too many data points to print here. Check the website for
this dissertation[151] for all the data points in this graph.
Fig. 7.7 (0.00, 1.0000), (0.05, 0.9882), (0.10, 0.9595), (0.15, 0.9226), (0.20, 0.8795),
(0.25, 0.8310), (0.30, 0.7800), (0.35, 0.7296), (0.40, 0.6823), (0.45, 0.6402), (0.50, 0.6045),
(0.55, 0.5758), (0.60, 0.5541), (0.65, 0.5387), (0.70, 0.5287), (0.75, 0.5226), (0.80, 0.5191),
(0.85, 0.5168), (0.90, 0.5145), (0.95, 0.5114), (1.00, 0.5069), (1.05, 0.5009), (1.10, 0.4937),
(1.15, 0.4858), (1.20, 0.4779), (1.25, 0.4708), (1.30, 0.4653), (1.35, 0.4619), (1.40, 0.4609),
(1.45, 0.4624), (1.50, 0.4665), (1.55, 0.4727), (1.60, 0.4804), (1.65, 0.4891), (1.70, 0.4979),
(1.75, 0.5061), (1.80, 0.5132), (1.85, 0.5187), (1.90, 0.5221), (1.95, 0.5233), (2.00, 0.5223),
(2.05, 0.5192), (2.10, 0.5142), (2.15, 0.5077), (2.20, 0.4999), (2.25, 0.4913), (2.30, 0.4822),
(2.35, 0.4729), (2.40, 0.4635), (2.45, 0.4543), (2.50, 0.4454), (2.55, 0.4367), (2.60, 0.4285),
(2.65, 0.4208), (2.70, 0.4136), (2.75, 0.4068), (2.80, 0.4006), (2.85, 0.3950), (2.90, 0.3900),
(2.95, 0.3857), (3.00, 0.3819), (3.05, 0.3787), (3.10, 0.3760), (3.15, 0.3738), (3.20, 0.3719),
(3.25, 0.3703), (3.30, 0.3687), (3.35, 0.3672), (3.40, 0.3655), (3.45, 0.3638), (3.50, 0.3619),
(3.55, 0.3598), (3.60, 0.3575), (3.65, 0.3551), (3.70, 0.3525), (3.75, 0.3499), (3.80, 0.3471),
(3.85, 0.3444), (3.90, 0.3416), (3.95, 0.3389), (4.00, 0.3363), (4.05, 0.3336), (4.10, 0.3308),
173
(4.15, 0.3280), (4.20, 0.3250), (4.25, 0.3220), (4.30, 0.3189), (4.35, 0.3158), (4.40, 0.3126),
(4.45, 0.3094), (4.50, 0.3063), (4.55, 0.3033), (4.60, 0.3005), (4.65, 0.2979), (4.70, 0.2956),
(4.75, 0.2935), (4.80, 0.2916), (4.85, 0.2900), (4.90, 0.2885), (4.95, 0.2872), (5.00, 0.2860),
(5.05, 0.2848), (5.10, 0.2836), (5.15, 0.2823), (5.20, 0.2807), (5.25, 0.2790), (5.30, 0.2770),
(5.35, 0.2749), (5.40, 0.2724), (5.45, 0.2698), (5.50, 0.2670), (5.55, 0.2640), (5.60, 0.2610),
(5.65, 0.2580), (5.70, 0.2549), (5.75, 0.2520), (5.80, 0.2491), (5.85, 0.2464), (5.90, 0.2437),
(5.95, 0.2413), (6.00, 0.2390), (6.05, 0.2368), (6.10, 0.2348), (6.15, 0.2330), (6.20, 0.2313),
(6.25, 0.2297), (6.30, 0.2282), (6.35, 0.2267), (6.40, 0.2252), (6.45, 0.2238), (6.50, 0.2224),
(6.55, 0.2210), (6.60, 0.2194), (6.65, 0.2179), (6.70, 0.2163), (6.75, 0.2146), (6.80, 0.2130),
(6.85, 0.2113), (6.90, 0.2098), (6.95, 0.2082), (7.00, 0.2068), (7.05, 0.2054), (7.10, 0.2042),
(7.15, 0.2030), (7.20, 0.2018), (7.25, 0.2007), (7.30, 0.1995), (7.35, 0.1984), (7.40, 0.1973),
(7.45, 0.1961), (7.50, 0.1950), (7.55, 0.1938), (7.60, 0.1926), (7.65, 0.1914), (7.70, 0.1901),
(7.75, 0.1889), (7.80, 0.1877), (7.85, 0.1864), (7.90, 0.1851), (7.95, 0.1838), (8.00, 0.1825),
(8.05, 0.1812), (8.10, 0.1799), (8.15, 0.1786), (8.20, 0.1773), (8.25, 0.1760), (8.30, 0.1748),
(8.35, 0.1736), (8.40, 0.1724), (8.45, 0.1713), (8.50, 0.1701), (8.55, 0.1690), (8.60, 0.1678),
(8.65, 0.1667), (8.70, 0.1656), (8.75, 0.1645), (8.80, 0.1635), (8.85, 0.1624), (8.90, 0.1614),
(8.95, 0.1604), (9.00, 0.1594), (9.05, 0.1586), (9.10, 0.1579), (9.15, 0.1572), (9.20, 0.1565),
(9.25, 0.1559), (9.30, 0.1554), (9.35, 0.1549), (9.40, 0.1545), (9.45, 0.1541), (9.50, 0.1538),
(9.55, 0.1534), (9.60, 0.1529), (9.65, 0.1525), (9.70, 0.1519), (9.75, 0.1514), (9.80, 0.1508),
(9.85, 0.1501), (9.90, 0.1494), (9.95, 0.1486), (10.00, 0.1478), (10.05, 0.1469), (10.10, 0.1460),
(10.15, 0.1450), (10.20, 0.1440), (10.25, 0.1430), (10.30, 0.1420), (10.35, 0.1410), (10.40,
0.1400), (10.45, 0.1390), (10.50, 0.1379), (10.55, 0.1368), (10.60, 0.1357), (10.65, 0.1346),
(10.70, 0.1336), (10.75, 0.1325), (10.80, 0.1315), (10.85, 0.1306), (10.90, 0.1296), (10.95,
0.1288), (11.00, 0.1281), (11.05, 0.1274), (11.10, 0.1269), (11.15, 0.1264), (11.20, 0.1260),
(11.25, 0.1256), (11.30, 0.1252), (11.35, 0.1248), (11.40, 0.1244), (11.45, 0.1239), (11.50,
0.1234), (11.55, 0.1228), (11.60, 0.1223), (11.65, 0.1217), (11.70, 0.1210), (11.75, 0.1203),
(11.80, 0.1195), (11.85, 0.1188), (11.90, 0.1180), (11.95, 0.1172), (12.00, 0.1164), (12.05,
0.1156), (12.10, 0.1148), (12.15, 0.1140), (12.20, 0.1131), (12.25, 0.1121), (12.30, 0.1111),
(12.35, 0.1101), (12.40, 0.1091), (12.45, 0.1080), (12.50, 0.1069), (12.55, 0.1059), (12.60,
0.1049), (12.65, 0.1039), (12.70, 0.1030), (12.75, 0.1022), (12.80, 0.1013), (12.85, 0.1006),
(12.90, 0.0999), (12.95, 0.0992), (13.00, 0.0986), (13.05, 0.0980), (13.10, 0.0976), (13.15,
0.0972), (13.20, 0.0970), (13.25, 0.0967), (13.30, 0.0965), (13.35, 0.0964), (13.40, 0.0963),
(13.45, 0.0961), (13.50, 0.0959), (13.55, 0.0957), (13.60, 0.0954), (13.65, 0.0951), (13.70,
0.0946), (13.75, 0.0941), (13.80, 0.0935), (13.85, 0.0929), (13.90, 0.0923), (13.95, 0.0916),
(14.00, 0.0910), (14.05, 0.0905), (14.10, 0.0900), (14.15, 0.0896), (14.20, 0.0892), (14.25,
0.0888), (14.30, 0.0883), (14.35, 0.0879), (14.40, 0.0874), (14.45, 0.0869), (14.50, 0.0863),
(14.55, 0.0857), (14.60, 0.0850), (14.65, 0.0843), (14.70, 0.0836), (14.75, 0.0828), (14.80,
0.0820), (14.85, 0.0812), (14.90, 0.0804), (14.95, 0.0797), (15.00, 0.0789), (15.05, 0.0783),
(15.10, 0.0779), (15.15, 0.0775), (15.20, 0.0773), (15.25, 0.0771), (15.30, 0.0771), (15.35,
174
0.0771), (15.40, 0.0772), (15.45, 0.0774), (15.50, 0.0775), (15.55, 0.0776), (15.60, 0.0777),
(15.65, 0.0777), (15.70, 0.0775), (15.75, 0.0773), (15.80, 0.0769), (15.85, 0.0764), (15.90,
0.0757), (15.95, 0.0751), (16.00, 0.0744), (16.05, 0.0737), (16.10, 0.0729), (16.15, 0.0723),
(16.20, 0.0717), (16.25, 0.0711), (16.30, 0.0707), (16.35, 0.0702), (16.40, 0.0699), (16.45,
0.0695), (16.50, 0.0692), (16.55, 0.0689), (16.60, 0.0686), (16.65, 0.0684), (16.70, 0.0681),
(16.75, 0.0678), (16.80, 0.0674), (16.85, 0.0671), (16.90, 0.0668), (16.95, 0.0664), (17.00,
0.0660), (17.05, 0.0657), (17.10, 0.0653), (17.15, 0.0649), (17.20, 0.0645), (17.25, 0.0641),
(17.30, 0.0637), (17.35, 0.0634), (17.40, 0.0630), (17.45, 0.0627), (17.50, 0.0624), (17.55,
0.0620), (17.60, 0.0617), (17.65, 0.0614), (17.70, 0.0610), (17.75, 0.0607), (17.80, 0.0604),
(17.85, 0.0601), (17.90, 0.0598), (17.95, 0.0595), (18.00, 0.0591), (18.05, 0.0587), (18.10,
0.0583), (18.15, 0.0579), (18.20, 0.0574), (18.25, 0.0569), (18.30, 0.0563), (18.35, 0.0558),
(18.40, 0.0553), (18.45, 0.0549), (18.50, 0.0544), (18.55, 0.0540), (18.60, 0.0537), (18.65,
0.0533), (18.70, 0.0530), (18.75, 0.0528), (18.80, 0.0525), (18.85, 0.0522), (18.90, 0.0519),
(18.95, 0.0516), (19.00, 0.0514), (19.05, 0.0511), (19.10, 0.0507), (19.15, 0.0504), (19.20,
0.0500), (19.25, 0.0497), (19.30, 0.0495), (19.35, 0.0492), (19.40, 0.0490), (19.45, 0.0489),
(19.50, 0.0488), (19.55, 0.0488), (19.60, 0.0489), (19.65, 0.0489), (19.70, 0.0488), (19.75,
0.0487), (19.80, 0.0484), (19.85, 0.0481), (19.90, 0.0475), (19.95, 0.0468), (20.00, 0.0460),
(20.05, 0.0451), (20.10, 0.0442), (20.15, 0.0433), (20.20, 0.0424), (20.25, 0.0416), (20.30,
0.0410), (20.35, 0.0404), (20.40, 0.0401), (20.45, 0.0399), (20.50, 0.0398), (20.55, 0.0399),
(20.60, 0.0401), (20.65, 0.0403), (20.70, 0.0406), (20.75, 0.0408), (20.80, 0.0410), (20.85,
0.0410), (20.90, 0.0410), (20.95, 0.0410), (21.00, 0.0408), (21.05, 0.0406), (21.10, 0.0403),
(21.15, 0.0401), (21.20, 0.0400), (21.25, 0.0399), (21.30, 0.0399), (21.35, 0.0400), (21.40,
0.0401), (21.45, 0.0402), (21.50, 0.0404), (21.55, 0.0406), (21.60, 0.0408), (21.65, 0.0410),
(21.70, 0.0410), (21.75, 0.0410), (21.80, 0.0409), (21.85, 0.0407), (21.90, 0.0405), (21.95,
0.0401), (22.00, 0.0396), (22.05, 0.0390), (22.10, 0.0384), (22.15, 0.0378), (22.20, 0.0372),
(22.25, 0.0366), (22.30, 0.0359), (22.35, 0.0353), (22.40, 0.0346), (22.45, 0.0340), (22.50,
0.0334), (22.55, 0.0329), (22.60, 0.0324), (22.65, 0.0320), (22.70, 0.0317), (22.75, 0.0313),
(22.80, 0.0310), (22.85, 0.0307), (22.90, 0.0304), (22.95, 0.0301), (23.00, 0.0298), (23.05,
0.0294), (23.10, 0.0290), (23.15, 0.0286), (23.20, 0.0283), (23.25, 0.0279), (23.30, 0.0276),
(23.35, 0.0273), (23.40, 0.0271), (23.45, 0.0269), (23.50, 0.0267), (23.55, 0.0266), (23.60,
0.0266), (23.65, 0.0266), (23.70, 0.0267), (23.75, 0.0268), (23.80, 0.0269), (23.85, 0.0270),
(23.90, 0.0273), (23.95, 0.0275), (24.00, 0.0277), (24.05, 0.0278), (24.10, 0.0279), (24.15,
0.0279), (24.20, 0.0278), (24.25, 0.0277), (24.30, 0.0274), (24.35, 0.0270), (24.40, 0.0265),
(24.45, 0.0259), (24.50, 0.0253), (24.55, 0.0247), (24.60, 0.0241), (24.65, 0.0237), (24.70,
0.0232), (24.75, 0.0230), (24.80, 0.0227), (24.85, 0.0226), (24.90, 0.0225), (24.95, 0.0224),
(25.00, 0.0222), (25.05, 0.0220), (25.10, 0.0217), (25.15, 0.0213), (25.20, 0.0207), (25.25,
0.0201), (25.30, 0.0194), (25.35, 0.0187), (25.40, 0.0179), (25.45, 0.0171), (25.50, 0.0164),
(25.55, 0.0157).
175
Fig. 7.8 (a) circles: (-0.027, 0.00516), (0.080, 0.005), (0.190, 0.0156), (0.407, 0.0193),
(0.926, 0.0274), (1.489, 0.029), (2.353, 0.0261), (3.329, 0.0214), (4.415, 0.017), (6.583,
0.0123), (5.497, 0.0142), (3.874, 0.0191), (2.788, 0.024), (1.923, 0.028), (1.164, 0.0292),
(0.622, 0.0175), (0.299, 0.0201), (0.136, 0.012), (0.027, 0.00577), (0.065, 0.00569).
Fig. 7.8 (a) squares: (-0.113, 0.002), (-0.015, 0.00176), (0.085, 0.0129), (0.281,
0.0147), (0.757, 0.02), (1.269, 0.0227), (2.057, 0.0205), (2.945, 0.0164), (3.934, 0.0127),
(5.910, 0.00901), (4.921, 0.0104), (3.441, 0.0144), (2.945, 0.0164), (2.454, 0.0186), (1.663,
0.0219), (0.974, 0.0222), (0.480, 0.0114), (0.183, 0.0158), (0.035, 0.00137), (-0.064, 0.00131),
(-0.028, 0.00131), (-0.951, 0.00567).
Fig. 7.8 (a) diamonds: (-0.143, 0.000493), (-0.047, 0.000468), (0.049, 0.00275), (0.240,
0.00263), (0.698, 0.0138), (1.194, 0.016), (1.957, 0.0149), (2.818, 0.012), (3.773, 0.0094),
(5.680, 0.00682), (4.726, 0.00781), (3.297, 0.0106), (2.339, 0.0137), (1.575, 0.016), (0.907,
0.0156), (0.431, 0.00663), (0.145, 0.00248), (0.002, 0.00125), (-0.095, 0.000602), (-0.062,
0.000588), (-0.952, 0.00254).
Fig. 7.8 (a) pluses: (-0.036, 0.000192), (0.070, 0.00019), (0.177, 0.00248), (0.391,
0.00424), (0.907, 0.00942), (1.464, 0.0109), (2.319, 0.01), (3.286, 0.00792), (4.357, 0.00602),
(6.502, 0.00427), (5.428, 0.00497), (3.822, 0.00695), (2.747, 0.00921), (1.890, 0.0109), (1.143,
0.0106), (0.605, 0.00476), (0.285, 0.00474), (0.123, 0.000659), (0.016, 0.000208), (0.054,
0.000202), (-0.947, 0.000645).
Fig. 7.8 (a) crosses: (-0.095, 0.000145), (-0.009, 0.000116), (0.006, 0.00013), (0.106,
0.000457), (0.308, 0.00156), (0.792, 0.00489), (1.317, 0.00523), (2.121, 0.00455), (3.029,
0.00343), (4.036, 0.00249), (6.054, 0.00168), (5.044, 0.00199), (3.534, 0.00291), (2.524,
0.00405), (1.719, 0.00504), (1.015, 0.0052), (0.511, 0.00202), (0.208, 0.00115), (0.056, 0.000746),
(-0.045, 0.00012), (-0.009, 0.000116), (-0.950, 0.000546).
Fig. 7.9 (8.603, 0.0000), (8.606, 0.0000), (8.609, 0.0000), (8.612, 0.0000), (8.615, 0.0000),
(8.618, 0.0000), (8.621, 0.0000), (8.624, 0.0000), (8.627, 0.0000), (8.630, 0.0000), (8.633,
0.0000), (8.636, 0.0000), (8.639, 0.0000), (8.642, 0.0000), (8.645, 0.0000), (8.648, 0.0000),
(8.651, 0.0000), (8.654, 0.0000), (8.657, 0.0006), (8.660, 0.0000), (8.663, 0.0000), (8.666,
0.0000), (8.669, 0.0006), (8.672, 0.0000), (8.675, 0.0006), (8.678, 0.0000), (8.681, 0.0006),
(8.684, 0.0000), (8.687, 0.0006), (8.690, 0.0000), (8.693, 0.0000), (8.696, 0.0011), (8.699,
0.0011), (8.702, 0.0000), (8.705, 0.0011), (8.708, 0.0011), (8.711, 0.0011), (8.714, 0.0006),
(8.717, 0.0006), (8.720, 0.0011), (8.723, 0.0011), (8.726, 0.0011), (8.729, 0.0022), (8.732,
0.0033), (8.735, 0.0006), (8.738, 0.0011), (8.741, 0.0022), (8.744, 0.0022), (8.747, 0.0028),
(8.750, 0.0017), (8.753, 0.0033), (8.756, 0.0044), (8.759, 0.0033), (8.762, 0.0022), (8.765,
0.0033), (8.768, 0.0017), (8.771, 0.0044), (8.774, 0.0044), (8.777, 0.0050), (8.780, 0.0078),
(8.783, 0.0078), (8.786, 0.0094), (8.789, 0.0056), (8.792, 0.0067), (8.795, 0.0094), (8.798,
0.0106), (8.801, 0.0094), (8.804, 0.0122), (8.807, 0.0106), (8.810, 0.0156), (8.813, 0.0156),
(8.816, 0.0144), (8.819, 0.0150), (8.822, 0.0150), (8.825, 0.0233), (8.828, 0.0289), (8.831,
0.0156), (8.834, 0.0167), (8.837, 0.0278), (8.840, 0.0300), (8.843, 0.0283), (8.846, 0.0261),
176
(8.849, 0.0300), (8.852, 0.0378), (8.855, 0.0289), (8.858, 0.0422), (8.861, 0.0422), (8.864,
0.0411), (8.867, 0.0417), (8.870, 0.0450), (8.873, 0.0450), (8.876, 0.0533), (8.879, 0.0589),
(8.882, 0.0617), (8.885, 0.0617), (8.888, 0.0700), (8.891, 0.0678), (8.894, 0.0756), (8.897,
0.0739), (8.900, 0.0694), (8.903, 0.0856), (8.906, 0.0889), (8.909, 0.1006), (8.912, 0.0939),
(8.915, 0.1028), (8.918, 0.0972), (8.921, 0.0983), (8.924, 0.1161), (8.927, 0.1139), (8.930,
0.1206), (8.933, 0.1183), (8.936, 0.1483), (8.939, 0.1356), (8.942, 0.1450), (8.945, 0.1561),
(8.948, 0.1633), (8.951, 0.1511), (8.954, 0.1711), (8.957, 0.1778), (8.960, 0.1950), (8.963,
0.1933), (8.966, 0.2033), (8.969, 0.2111), (8.972, 0.2300), (8.975, 0.2328), (8.978, 0.2489),
(8.981, 0.2383), (8.984, 0.2400), (8.987, 0.2517), (8.990, 0.2661), (8.993, 0.2650), (8.996,
0.2939), (8.999, 0.2878), (9.002, 0.3067), (9.005, 0.3122), (9.008, 0.2944), (9.011, 0.3239),
(9.014, 0.3511), (9.017, 0.3678), (9.020, 0.3800), (9.023, 0.3661), (9.026, 0.3761), (9.029,
0.4000), (9.032, 0.4089), (9.035, 0.4311), (9.038, 0.4361), (9.041, 0.4489), (9.044, 0.4817),
(9.047, 0.4856), (9.050, 0.4917), (9.053, 0.4767), (9.056, 0.4933), (9.059, 0.4950), (9.062,
0.5506), (9.065, 0.5550), (9.068, 0.5689), (9.071, 0.5872), (9.074, 0.6000), (9.077, 0.5933),
(9.080, 0.6050), (9.083, 0.6294), (9.086, 0.6694), (9.089, 0.7194), (9.092, 0.7050), (9.095,
0.6872), (9.098, 0.7394), (9.101, 0.6850), (9.104, 0.7544), (9.107, 0.7394), (9.110, 0.7839),
(9.113, 0.7917), (9.116, 0.8511), (9.119, 0.8422), (9.122, 0.8539), (9.125, 0.8567), (9.128,
0.8644), (9.131, 0.8767), (9.134, 0.9200), (9.137, 0.8933), (9.140, 0.8872), (9.143, 0.9639),
(9.146, 0.9778), (9.149, 0.9883), (9.152, 0.9783), (9.155, 1.0000), (9.158, 1.0983), (9.161,
1.0311), (9.164, 1.0750), (9.167, 1.0600), (9.170, 1.1178), (9.173, 1.1150), (9.176, 1.1289),
(9.179, 1.1733), (9.182, 1.1256), (9.185, 1.1806), (9.188, 1.1700), (9.191, 1.2094), (9.194,
1.2250), (9.197, 1.2483), (9.200, 1.2639), (9.203, 1.2956), (9.206, 1.2922), (9.209, 1.2789),
(9.212, 1.3100), (9.215, 1.3994), (9.218, 1.3667), (9.221, 1.3761), (9.224, 1.3350), (9.227,
1.3817), (9.230, 1.3817), (9.233, 1.4672), (9.236, 1.3889), (9.239, 1.3750), (9.242, 1.4606),
(9.245, 1.4761), (9.248, 1.5028), (9.251, 1.4489), (9.254, 1.4972), (9.257, 1.4600), (9.260,
1.5356), (9.263, 1.5617), (9.266, 1.5600), (9.269, 1.5433), (9.272, 1.6144), (9.275, 1.5578),
(9.278, 1.6206), (9.281, 1.5850), (9.284, 1.6528), (9.287, 1.6372), (9.290, 1.6278), (9.293,
1.6094), (9.296, 1.5806), (9.299, 1.7044), (9.302, 1.6206), (9.305, 1.6250), (9.308, 1.6144),
(9.311, 1.6194), (9.314, 1.6361), (9.317, 1.6461), (9.320, 1.6750), (9.323, 1.5756), (9.326,
1.6617), (9.329, 1.6328), (9.332, 1.6744), (9.335, 1.6272), (9.338, 1.6711), (9.341, 1.6328),
(9.344, 1.6411), (9.347, 1.6744), (9.350, 1.5583), (9.353, 1.6256), (9.356, 1.5311), (9.359,
1.6472), (9.362, 1.6361), (9.365, 1.5928), (9.368, 1.7172), (9.371, 1.5517), (9.374, 1.5861),
(9.377, 1.5661), (9.380, 1.5889), (9.383, 1.5328), (9.386, 1.5533), (9.389, 1.5161), (9.392,
1.5122), (9.395, 1.5583), (9.398, 1.5156), (9.401, 1.5050), (9.404, 1.4611), (9.407, 1.4967),
(9.410, 1.4711), (9.413, 1.4528), (9.416, 1.3867), (9.419, 1.4717), (9.422, 1.5133), (9.425,
1.3867), (9.428, 1.4100), (9.431, 1.3683), (9.434, 1.3594), (9.437, 1.3833), (9.440, 1.4161),
(9.443, 1.3406), (9.446, 1.3706), (9.449, 1.3828), (9.452, 1.3694), (9.455, 1.3172), (9.458,
1.3122), (9.461, 1.2828), (9.464, 1.3344), (9.467, 1.2456), (9.470, 1.2661), (9.473, 1.2606),
(9.476, 1.2639), (9.479, 1.3178), (9.482, 1.2539), (9.485, 1.1917), (9.488, 1.1994), (9.491,
177
1.1833), (9.494, 1.2506), (9.497, 1.1911), (9.500, 1.1739), (9.503, 1.1456), (9.506, 1.1867),
(9.509, 1.1600), (9.512, 1.1372), (9.515, 1.1278), (9.518, 1.1122), (9.521, 1.1283), (9.524,
1.0844), (9.527, 1.1228), (9.530, 1.1028), (9.533, 1.0839), (9.536, 1.1233), (9.539, 1.0478),
(9.542, 1.0444), (9.545, 1.0167), (9.548, 1.0500), (9.551, 1.0156), (9.554, 1.0444), (9.557,
1.0533), (9.560, 1.0083), (9.563, 1.0267), (9.566, 1.0511), (9.569, 0.9850), (9.572, 0.9522),
(9.575, 0.9756), (9.578, 0.9561), (9.581, 0.9417), (9.584, 0.9383), (9.587, 0.9261), (9.590,
0.9867), (9.593, 0.8867), (9.596, 0.9511), (9.599, 0.9367), (9.602, 0.9217), (9.605, 0.9611),
(9.608, 0.8661), (9.611, 0.9011), (9.614, 0.9061), (9.617, 0.8800), (9.620, 0.8928), (9.623,
0.8822), (9.626, 0.8500), (9.629, 0.8778), (9.632, 0.9050), (9.635, 0.8606), (9.638, 0.8394),
(9.641, 0.8744), (9.644, 0.8772), (9.647, 0.8811), (9.650, 0.8461), (9.653, 0.8650), (9.656,
0.8294), (9.659, 0.8272), (9.662, 0.8294), (9.665, 0.8339), (9.668, 0.8283), (9.671, 0.7950),
(9.674, 0.8089), (9.677, 0.8133), (9.680, 0.8050), (9.683, 0.7733), (9.686, 0.8250), (9.689,
0.7950), (9.692, 0.7333), (9.695, 0.8122), (9.698, 0.7867), (9.701, 0.7783), (9.704, 0.7444),
(9.707, 0.7500), (9.710, 0.7406), (9.713, 0.7478), (9.716, 0.7817), (9.719, 0.7500), (9.722,
0.7333), (9.725, 0.7233), (9.728, 0.7017), (9.731, 0.7061), (9.734, 0.6872), (9.737, 0.7028),
(9.740, 0.6889), (9.743, 0.7150), (9.746, 0.6656), (9.749, 0.6878), (9.752, 0.6428), (9.755,
0.6578), (9.758, 0.6856), (9.761, 0.6467), (9.764, 0.6617), (9.767, 0.6678), (9.770, 0.6467),
(9.773, 0.6433), (9.776, 0.6833), (9.779, 0.6161), (9.782, 0.6206), (9.785, 0.6394), (9.788,
0.6317), (9.791, 0.6189), (9.794, 0.5817), (9.797, 0.5761), (9.800, 0.6483), (9.803, 0.6361),
(9.806, 0.6050), (9.809, 0.5856), (9.812, 0.6250), (9.815, 0.5567), (9.818, 0.5961), (9.821,
0.5589), (9.824, 0.5467), (9.827, 0.5856), (9.830, 0.5650), (9.833, 0.5372), (9.836, 0.5506),
(9.839, 0.5506), (9.842, 0.5511), (9.845, 0.5561), (9.848, 0.5311), (9.851, 0.5422), (9.854,
0.5150), (9.857, 0.4772), (9.860, 0.5244), (9.863, 0.5006), (9.866, 0.5117), (9.869, 0.5250),
(9.872, 0.4850), (9.875, 0.4972), (9.878, 0.4844), (9.881, 0.4983), (9.884, 0.5122), (9.887,
0.4928), (9.890, 0.4500), (9.893, 0.4561), (9.896, 0.4572), (9.899, 0.4478), (9.902, 0.4256),
(9.905, 0.4467), (9.908, 0.4194), (9.911, 0.4533), (9.914, 0.4511), (9.917, 0.4194), (9.920,
0.4372), (9.923, 0.4317), (9.926, 0.4200), (9.929, 0.4333), (9.932, 0.3983), (9.935, 0.3706),
(9.938, 0.4056), (9.941, 0.4000), (9.944, 0.3944), (9.947, 0.3772), (9.950, 0.4033), (9.953,
0.3728), (9.956, 0.3600), (9.959, 0.3506), (9.962, 0.3328), (9.965, 0.3717), (9.968, 0.3578),
(9.971, 0.3311), (9.974, 0.3500), (9.977, 0.3411), (9.980, 0.3544), (9.983, 0.3328), (9.986,
0.3089), (9.989, 0.3167), (9.992, 0.3183), (9.995, 0.3117), (9.998, 0.3144), (10.001, 0.2794),
(10.004, 0.2694), (10.007, 0.2800), (10.010, 0.3156), (10.013, 0.2833), (10.016, 0.2722),
(10.019, 0.2744), (10.022, 0.2556), (10.025, 0.2461), (10.028, 0.2617), (10.031, 0.2300),
(10.034, 0.2439), (10.037, 0.2489), (10.040, 0.2411), (10.043, 0.2483), (10.046, 0.2272),
(10.049, 0.2206), (10.052, 0.2289), (10.055, 0.2306), (10.058, 0.2067), (10.061, 0.1939),
(10.064, 0.2222), (10.067, 0.2139), (10.070, 0.1961), (10.073, 0.2039), (10.076, 0.1906),
(10.079, 0.1739), (10.082, 0.1672), (10.085, 0.1756), (10.088, 0.1739), (10.091, 0.2006),
(10.094, 0.1728), (10.097, 0.1633), (10.100, 0.1617), (10.103, 0.1561), (10.106, 0.1617),
(10.109, 0.1383), (10.112, 0.1439), (10.115, 0.1372), (10.118, 0.1428), (10.121, 0.1489),
178
(10.124, 0.1306), (10.127, 0.1344), (10.130, 0.1306), (10.133, 0.1417), (10.136, 0.1311),
(10.139, 0.1300), (10.142, 0.1167), (10.145, 0.1194), (10.148, 0.1194), (10.151, 0.1106),
(10.154, 0.1156), (10.157, 0.1022), (10.160, 0.1044), (10.163, 0.0989), (10.166, 0.0994),
(10.169, 0.0978), (10.172, 0.0889), (10.175, 0.0911), (10.178, 0.0817), (10.181, 0.0811),
(10.184, 0.0783), (10.187, 0.0867), (10.190, 0.0861), (10.193, 0.0633), (10.196, 0.0717),
(10.199, 0.0722), (10.202, 0.0700), (10.205, 0.0717), (10.208, 0.0639), (10.211, 0.0683),
(10.214, 0.0667), (10.217, 0.0639), (10.220, 0.0578), (10.223, 0.0500), (10.226, 0.0556),
(10.229, 0.0394), (10.232, 0.0494), (10.235, 0.0500), (10.238, 0.0478), (10.241, 0.0561),
(10.244, 0.0417), (10.247, 0.0406), (10.250, 0.0489), (10.253, 0.0456), (10.256, 0.0367),
(10.259, 0.0350), (10.262, 0.0294), (10.265, 0.0344), (10.268, 0.0378), (10.271, 0.0272),
(10.274, 0.0194), (10.277, 0.0239), (10.280, 0.0311), (10.283, 0.0233), (10.286, 0.0311),
(10.289, 0.0206), (10.292, 0.0183), (10.295, 0.0294), (10.298, 0.0256), (10.301, 0.0228),
(10.304, 0.0150), (10.307, 0.0156), (10.310, 0.0128), (10.313, 0.0161), (10.316, 0.0194),
(10.319, 0.0133), (10.322, 0.0156), (10.325, 0.0106), (10.328, 0.0200), (10.331, 0.0156),
(10.334, 0.0128), (10.337, 0.0122), (10.340, 0.0111), (10.343, 0.0111), (10.346, 0.0100),
(10.349, 0.0078), (10.352, 0.0061), (10.355, 0.0089), (10.358, 0.0089), (10.361, 0.0117),
(10.364, 0.0028), (10.367, 0.0039), (10.370, 0.0067), (10.373, 0.0044), (10.376, 0.0067),
(10.379, 0.0033), (10.382, 0.0072), (10.385, 0.0067), (10.388, 0.0050), (10.391, 0.0022),
(10.394, 0.0061), (10.397, 0.0056), (10.400, 0.0022), (10.403, 0.0039), (10.406, 0.0039),
(10.409, 0.0022), (10.412, 0.0006), (10.415, 0.0039), (10.418, 0.0022), (10.421, 0.0028),
(10.424, 0.0017), (10.427, 0.0006), (10.430, 0.0022), (10.433, 0.0017), (10.436, 0.0011),
(10.439, 0.0006), (10.442, 0.0011), (10.445, 0.0017), (10.448, 0.0000), (10.451, 0.0006),
(10.454, 0.0000), (10.457, 0.0000), (10.460, 0.0006), (10.463, 0.0000), (10.466, 0.0000),
(10.469, 0.0000), (10.472, 0.0000), (10.475, 0.0000), (10.478, 0.0000), (10.481, 0.0000),
(10.484, 0.0006), (10.487, 0.0000), (10.490, 0.0006), (10.493, 0.0000), (10.496, 0.0000),
(10.499, 0.0000), all the other points are zeros.
Fig. 7.10 (328.993, 0.00000), (329.038, 0.00000), (329.083, 0.00004), (329.128, 0.00000),
(329.173, 0.00000), (329.218, 0.00000), (329.263, 0.00000), (329.308, 0.00000), (329.353,
0.00000), (329.398, 0.00000), (329.443, 0.00000), (329.488, 0.00000), (329.533, 0.00000),
(329.578, 0.00000), (329.623, 0.00004), (329.668, 0.00000), (329.713, 0.00000), (329.758,
0.00000), (329.803, 0.00007), (329.848, 0.00000), (329.893, 0.00000), (329.938, 0.00000),
(329.983, 0.00000), (330.028, 0.00000), (330.073, 0.00000), (330.118, 0.00004), (330.163,
0.00007), (330.208, 0.00000), (330.253, 0.00000), (330.298, 0.00000), (330.343, 0.00000),
(330.388, 0.00004), (330.433, 0.00000), (330.478, 0.00000), (330.523, 0.00000), (330.568,
0.00000), (330.613, 0.00000), (330.658, 0.00000), (330.703, 0.00004), (330.748, 0.00004),
(330.793, 0.00004), (330.838, 0.00004), (330.883, 0.00007), (330.928, 0.00000), (330.973,
0.00004), (331.018, 0.00000), (331.063, 0.00000), (331.108, 0.00004), (331.153, 0.00000),
(331.198, 0.00007), (331.243, 0.00007), (331.288, 0.00004), (331.333, 0.00000), (331.378,
0.00007), (331.423, 0.00004), (331.468, 0.00000), (331.513, 0.00007), (331.558, 0.00011),
179
(331.603, 0.00004), (331.648, 0.00000), (331.693, 0.00000), (331.738, 0.00004), (331.783,
0.00007), (331.828, 0.00011), (331.873, 0.00007), (331.918, 0.00011), (331.963, 0.00011),
(332.008, 0.00004), (332.053, 0.00007), (332.098, 0.00007), (332.143, 0.00007), (332.188,
0.00007), (332.233, 0.00019), (332.278, 0.00015), (332.323, 0.00033), (332.368, 0.00004),
(332.413, 0.00030), (332.458, 0.00019), (332.503, 0.00007), (332.548, 0.00004), (332.593,
0.00022), (332.638, 0.00007), (332.683, 0.00015), (332.728, 0.00015), (332.773, 0.00033),
(332.818, 0.00011), (332.863, 0.00019), (332.908, 0.00030), (332.953, 0.00033), (332.998,
0.00026), (333.043, 0.00026), (333.088, 0.00011), (333.133, 0.00030), (333.178, 0.00022),
(333.223, 0.00026), (333.268, 0.00026), (333.313, 0.00048), (333.358, 0.00041), (333.403,
0.00059), (333.448, 0.00048), (333.493, 0.00037), (333.538, 0.00052), (333.583, 0.00037),
(333.628, 0.00063), (333.673, 0.00033), (333.718, 0.00033), (333.763, 0.00052), (333.808,
0.00041), (333.853, 0.00030), (333.898, 0.00030), (333.943, 0.00056), (333.988, 0.00059),
(334.033, 0.00056), (334.078, 0.00037), (334.123, 0.00056), (334.168, 0.00074), (334.213,
0.00052), (334.258, 0.00056), (334.303, 0.00059), (334.348, 0.00089), (334.393, 0.00041),
(334.438, 0.00074), (334.483, 0.00063), (334.528, 0.00078), (334.573, 0.00074), (334.618,
0.00063), (334.663, 0.00100), (334.708, 0.00052), (334.753, 0.00067), (334.798, 0.00081),
(334.843, 0.00096), (334.888, 0.00111), (334.933, 0.00089), (334.978, 0.00104), (335.023,
0.00104), (335.068, 0.00104), (335.113, 0.00111), (335.158, 0.00119), (335.203, 0.00159),
(335.248, 0.00137), (335.293, 0.00115), (335.338, 0.00130), (335.383, 0.00126), (335.428,
0.00148), (335.473, 0.00115), (335.518, 0.00156), (335.563, 0.00137), (335.608, 0.00126),
(335.653, 0.00163), (335.698, 0.00181), (335.743, 0.00167), (335.788, 0.00189), (335.833,
0.00152), (335.878, 0.00156), (335.923, 0.00174), (335.968, 0.00200), (336.013, 0.00174),
(336.058, 0.00163), (336.103, 0.00211), (336.148, 0.00241), (336.193, 0.00237), (336.238,
0.00230), (336.283, 0.00207), (336.328, 0.00244), (336.373, 0.00211), (336.418, 0.00252),
(336.463, 0.00289), (336.508, 0.00241), (336.553, 0.00233), (336.598, 0.00237), (336.643,
0.00263), (336.688, 0.00267), (336.733, 0.00274), (336.778, 0.00267), (336.823, 0.00274),
(336.868, 0.00278), (336.913, 0.00315), (336.958, 0.00270), (337.003, 0.00289), (337.048,
0.00333), (337.093, 0.00367), (337.138, 0.00370), (337.183, 0.00311), (337.228, 0.00330),
(337.273, 0.00396), (337.318, 0.00326), (337.363, 0.00415), (337.408, 0.00407), (337.453,
0.00441), (337.498, 0.00404), (337.543, 0.00415), (337.588, 0.00415), (337.633, 0.00478),
(337.678, 0.00407), (337.723, 0.00452), (337.768, 0.00463), (337.813, 0.00511), (337.858,
0.00478), (337.903, 0.00470), (337.948, 0.00485), (337.993, 0.00422), (338.038, 0.00526),
(338.083, 0.00500), (338.128, 0.00552), (338.173, 0.00559), (338.218, 0.00607), (338.263,
0.00530), (338.308, 0.00563), (338.353, 0.00689), (338.398, 0.00604), (338.443, 0.00570),
(338.488, 0.00644), (338.533, 0.00759), (338.578, 0.00693), (338.623, 0.00678), (338.668,
0.00707), (338.713, 0.00785), (338.758, 0.00685), (338.803, 0.00700), (338.848, 0.00767),
(338.893, 0.00715), (338.938, 0.00885), (338.983, 0.00770), (339.028, 0.00748), (339.073,
0.00796), (339.118, 0.00844), (339.163, 0.00874), (339.208, 0.01026), (339.253, 0.00867),
(339.298, 0.01007), (339.343, 0.00922), (339.388, 0.01033), (339.433, 0.01044), (339.478,
180
0.01015), (339.523, 0.01141), (339.568, 0.01119), (339.613, 0.01167), (339.658, 0.01074),
(339.703, 0.01081), (339.748, 0.01167), (339.793, 0.01215), (339.838, 0.01230), (339.883,
0.01237), (339.928, 0.01237), (339.973, 0.01152), (340.018, 0.01415), (340.063, 0.01444),
(340.108, 0.01311), (340.153, 0.01293), (340.198, 0.01519), (340.243, 0.01559), (340.288,
0.01359), (340.333, 0.01367), (340.378, 0.01700), (340.423, 0.01507), (340.468, 0.01652),
(340.513, 0.01737), (340.558, 0.01578), (340.603, 0.01730), (340.648, 0.01637), (340.693,
0.01711), (340.738, 0.01804), (340.783, 0.01767), (340.828, 0.01748), (340.873, 0.02030),
(340.918, 0.01926), (340.963, 0.01922), (341.008, 0.02026), (341.053, 0.02019), (341.098,
0.02041), (341.143, 0.02100), (341.188, 0.02300), (341.233, 0.02233), (341.278, 0.02074),
(341.323, 0.02104), (341.368, 0.02233), (341.413, 0.02359), (341.458, 0.02411), (341.503,
0.02444), (341.548, 0.02415), (341.593, 0.02463), (341.638, 0.02507), (341.683, 0.02522),
(341.728, 0.02633), (341.773, 0.02719), (341.818, 0.02859), (341.863, 0.02670), (341.908,
0.02837), (341.953, 0.02833), (341.998, 0.03011), (342.043, 0.02878), (342.088, 0.02841),
(342.133, 0.03144), (342.178, 0.03348), (342.223, 0.03211), (342.268, 0.03215), (342.313,
0.03304), (342.358, 0.03385), (342.403, 0.03330), (342.448, 0.03519), (342.493, 0.03511),
(342.538, 0.03619), (342.583, 0.03822), (342.628, 0.03611), (342.673, 0.03641), (342.718,
0.03919), (342.763, 0.03937), (342.808, 0.03778), (342.853, 0.04367), (342.898, 0.04189),
(342.943, 0.04041), (342.988, 0.04270), (343.033, 0.04481), (343.078, 0.04348), (343.123,
0.04348), (343.168, 0.04396), (343.213, 0.04363), (343.258, 0.04419), (343.303, 0.04544),
(343.348, 0.04648), (343.393, 0.04681), (343.438, 0.04656), (343.483, 0.05007), (343.528,
0.05196), (343.573, 0.05211), (343.618, 0.05178), (343.663, 0.05389), (343.708, 0.05119),
(343.753, 0.05281), (343.798, 0.05678), (343.843, 0.05781), (343.888, 0.05693), (343.933,
0.05596), (343.978, 0.05815), (344.023, 0.05793), (344.068, 0.05707), (344.113, 0.06026),
(344.158, 0.05981), (344.203, 0.06033), (344.248, 0.06296), (344.293, 0.06352), (344.338,
0.06500), (344.383, 0.06444), (344.428, 0.06552), (344.473, 0.06489), (344.518, 0.06970),
(344.563, 0.06978), (344.608, 0.06952), (344.653, 0.06978), (344.698, 0.06841), (344.743,
0.07148), (344.788, 0.07281), (344.833, 0.07422), (344.878, 0.07148), (344.923, 0.07481),
(344.968, 0.07704), (345.013, 0.07363), (345.058, 0.08067), (345.103, 0.07830), (345.148,
0.07848), (345.193, 0.07830), (345.238, 0.07922), (345.283, 0.08115), (345.328, 0.08111),
(345.373, 0.08630), (345.418, 0.08567), (345.463, 0.08667), (345.508, 0.08230), (345.553,
0.08611), (345.598, 0.08922), (345.643, 0.08348), (345.688, 0.08956), (345.733, 0.08689),
(345.778, 0.09044), (345.823, 0.08715), (345.868, 0.08930), (345.913, 0.09630), (345.958,
0.08944), (346.003, 0.09285), (346.048, 0.09041), (346.093, 0.09185), (346.138, 0.09719),
(346.183, 0.09259), (346.228, 0.09789), (346.273, 0.09707), (346.318, 0.09622), (346.363,
0.09967), (346.408, 0.09719), (346.453, 0.09904), (346.498, 0.09941), (346.543, 0.09900),
(346.588, 0.09552), (346.633, 0.10237), (346.678, 0.10093), (346.723, 0.09863), (346.768,
0.09889), (346.813, 0.10333), (346.858, 0.10474), (346.903, 0.10378), (346.948, 0.10130),
(346.993, 0.10522), (347.038, 0.10552), (347.083, 0.09993), (347.128, 0.10541), (347.173,
0.10752), (347.218, 0.10448), (347.263, 0.10704), (347.308, 0.10552), (347.353, 0.10741),
181
(347.398, 0.10726), (347.443, 0.10533), (347.488, 0.10511), (347.533, 0.10748), (347.578,
0.10759), (347.623, 0.10904), (347.668, 0.10578), (347.713, 0.10926), (347.758, 0.10715),
(347.803, 0.11037), (347.848, 0.10733), (347.893, 0.11481), (347.938, 0.11152), (347.983,
0.10600), (348.028, 0.10896), (348.073, 0.10904), (348.118, 0.11174), (348.163, 0.10978),
(348.208, 0.10500), (348.253, 0.10885), (348.298, 0.11074), (348.343, 0.10922), (348.388,
0.10648), (348.433, 0.10670), (348.478, 0.11270), (348.523, 0.10422), (348.568, 0.10878),
(348.613, 0.10511), (348.658, 0.11193), (348.703, 0.10696), (348.748, 0.10481), (348.793,
0.10911), (348.838, 0.10981), (348.883, 0.10319), (348.928, 0.10481), (348.973, 0.10596),
(349.018, 0.10404), (349.063, 0.10289), (349.108, 0.10237), (349.153, 0.10622), (349.198,
0.10667), (349.243, 0.10481), (349.288, 0.10426), (349.333, 0.10385), (349.378, 0.10248),
(349.423, 0.10144), (349.468, 0.09663), (349.513, 0.10270), (349.558, 0.10122), (349.603,
0.10022), (349.648, 0.09419), (349.693, 0.10096), (349.738, 0.10141), (349.783, 0.09474),
(349.828, 0.09581), (349.873, 0.09219), (349.918, 0.09559), (349.963, 0.09081), (350.008,
0.09222), (350.053, 0.09274), (350.098, 0.09393), (350.143, 0.09111), (350.188, 0.09630),
(350.233, 0.08974), (350.278, 0.08937), (350.323, 0.08741), (350.368, 0.08811), (350.413,
0.08752), (350.458, 0.08915), (350.503, 0.08619), (350.548, 0.08589), (350.593, 0.08204),
(350.638, 0.08433), (350.683, 0.08374), (350.728, 0.08119), (350.773, 0.07919), (350.818,
0.08163), (350.863, 0.07570), (350.908, 0.07656), (350.953, 0.07663), (350.998, 0.07689),
(351.043, 0.07744), (351.088, 0.07493), (351.133, 0.07333), (351.178, 0.07593), (351.223,
0.07104), (351.268, 0.07437), (351.313, 0.07074), (351.358, 0.07144), (351.403, 0.07052),
(351.448, 0.06715), (351.493, 0.06907), (351.538, 0.07074), (351.583, 0.06678), (351.628,
0.06507), (351.673, 0.06311), (351.718, 0.06359), (351.763, 0.06056), (351.808, 0.06170),
(351.853, 0.05959), (351.898, 0.05922), (351.943, 0.06004), (351.988, 0.05744), (352.033,
0.05693), (352.078, 0.05715), (352.123, 0.05511), (352.168, 0.05444), (352.213, 0.05415),
(352.258, 0.05385), (352.303, 0.05504), (352.348, 0.05048), (352.393, 0.05174), (352.438,
0.05148), (352.483, 0.05041), (352.528, 0.04856), (352.573, 0.04763), (352.618, 0.04919),
(352.663, 0.04811), (352.708, 0.04541), (352.753, 0.04470), (352.798, 0.04396), (352.843,
0.04311), (352.888, 0.04304), (352.933, 0.04293), (352.978, 0.04244), (353.023, 0.04267),
(353.068, 0.03719), (353.113, 0.03815), (353.158, 0.03756), (353.203, 0.03696), (353.248,
0.03852), (353.293, 0.03448), (353.338, 0.03696), (353.383, 0.03285), (353.428, 0.03363),
(353.473, 0.03200), (353.518, 0.03467), (353.563, 0.03270), (353.608, 0.03115), (353.653,
0.03200), (353.698, 0.03096), (353.743, 0.02807), (353.788, 0.02911), (353.833, 0.02748),
(353.878, 0.02678), (353.923, 0.02467), (353.968, 0.02493), (354.013, 0.02630), (354.058,
0.02470), (354.103, 0.02556), (354.148, 0.02367), (354.193, 0.02456), (354.238, 0.02352),
(354.283, 0.02333), (354.328, 0.02263), (354.373, 0.02204), (354.418, 0.02322), (354.463,
0.02141), (354.508, 0.02044), (354.553, 0.02074), (354.598, 0.02033), (354.643, 0.01844),
(354.688, 0.01967), (354.733, 0.01941), (354.778, 0.01893), (354.823, 0.01719), (354.868,
0.01778), (354.913, 0.01756), (354.958, 0.01763), (355.003, 0.01585), (355.048, 0.01663),
(355.093, 0.01559), (355.138, 0.01559), (355.183, 0.01363), (355.228, 0.01337), (355.273,
182
0.01485), (355.318, 0.01467), (355.363, 0.01415), (355.408, 0.01237), (355.453, 0.01304),
(355.498, 0.01204), (355.543, 0.01222), (355.588, 0.01163), (355.633, 0.01193), (355.678,
0.01219), (355.723, 0.01104), (355.768, 0.01000), (355.813, 0.01048), (355.858, 0.01037),
(355.903, 0.00915), (355.948, 0.01000), (355.993, 0.00970), (356.038, 0.00952), (356.083,
0.00926), (356.128, 0.00863), (356.173, 0.00793), (356.218, 0.00726), (356.263, 0.00889),
(356.308, 0.00807), (356.353, 0.00696), (356.398, 0.00789), (356.443, 0.00615), (356.488,
0.00696), (356.533, 0.00593), (356.578, 0.00604), (356.623, 0.00548), (356.668, 0.00589),
(356.713, 0.00522), (356.758, 0.00578), (356.803, 0.00570), (356.848, 0.00522), (356.893,
0.00578), (356.938, 0.00541), (356.983, 0.00470), (357.028, 0.00456), (357.073, 0.00422),
(357.118, 0.00374), (357.163, 0.00422), (357.208, 0.00396), (357.253, 0.00389), (357.298,
0.00374), (357.343, 0.00378), (357.388, 0.00333), (357.433, 0.00341), (357.478, 0.00322),
(357.523, 0.00344), (357.568, 0.00337), (357.613, 0.00207), (357.658, 0.00281), (357.703,
0.00281), (357.748, 0.00293), (357.793, 0.00311), (357.838, 0.00270), (357.883, 0.00259),
(357.928, 0.00200), (357.973, 0.00237), (358.018, 0.00222), (358.063, 0.00226), (358.108,
0.00211), (358.153, 0.00204), (358.198, 0.00211), (358.243, 0.00219), (358.288, 0.00152),
(358.333, 0.00174), (358.378, 0.00200), (358.423, 0.00167), (358.468, 0.00163), (358.513,
0.00174), (358.558, 0.00104), (358.603, 0.00133), (358.648, 0.00130), (358.693, 0.00144),
(358.738, 0.00133), (358.783, 0.00111), (358.828, 0.00089), (358.873, 0.00081), (358.918,
0.00074), (358.963, 0.00100), (359.008, 0.00070), (359.053, 0.00070), (359.098, 0.00089),
(359.143, 0.00078), (359.188, 0.00081), (359.233, 0.00056), (359.278, 0.00089), (359.323,
0.00037), (359.368, 0.00063), (359.413, 0.00093), (359.458, 0.00059), (359.503, 0.00052),
(359.548, 0.00078), (359.593, 0.00056), (359.638, 0.00067), (359.683, 0.00070), (359.728,
0.00044), (359.773, 0.00026), (359.818, 0.00063), (359.863, 0.00030), (359.908, 0.00052),
(359.953, 0.00070), (359.998, 0.00044), (360.043, 0.00044), (360.088, 0.00044), (360.133,
0.00033), (360.178, 0.00056), (360.223, 0.00037), (360.268, 0.00037), (360.313, 0.00026),
(360.358, 0.00015), (360.403, 0.00022), (360.448, 0.00037), (360.493, 0.00037), (360.538,
0.00030), (360.583, 0.00030), (360.628, 0.00026), (360.673, 0.00030), (360.718, 0.00019),
(360.763, 0.00026), (360.808, 0.00022), (360.853, 0.00019), (360.898, 0.00019), (360.943,
0.00037), (360.988, 0.00022), (361.033, 0.00022), (361.078, 0.00015), (361.123, 0.00015),
(361.168, 0.00011), (361.213, 0.00033), (361.258, 0.00011), (361.303, 0.00015), (361.348,
0.00011), (361.393, 0.00007), (361.438, 0.00007), (361.483, 0.00004), (361.528, 0.00015),
(361.573, 0.00011), (361.618, 0.00019), (361.663, 0.00007), (361.708, 0.00019), (361.753,
0.00007), (361.798, 0.00000), (361.843, 0.00011), (361.888, 0.00011), (361.933, 0.00004),
(361.978, 0.00019), (362.023, 0.00011), (362.068, 0.00004), (362.113, 0.00000), (362.158,
0.00011), (362.203, 0.00004), (362.248, 0.00004), (362.293, 0.00004), (362.338, 0.00000),
(362.383, 0.00000), (362.428, 0.00011), (362.473, 0.00007), (362.518, 0.00004), (362.563,
0.00000), (362.608, 0.00004), (362.653, 0.00000), (362.698, 0.00000), (362.743, 0.00000),
(362.788, 0.00000), (362.833, 0.00007), (362.878, 0.00000), (362.923, 0.00000), (362.968,
0.00004), (363.013, 0.00000), (363.058, 0.00000), (363.103, 0.00000), (363.148, 0.00000),
183
(363.193, 0.00000), (363.238, 0.00000), (363.283, 0.00000), (363.328, 0.00000), (363.373,
0.00000), (363.418, 0.00000), (363.463, 0.00000), (363.508, 0.00000), (363.553, 0.00000),
(363.598, 0.00004), (363.643, 0.00004), (363.688, 0.00000), (363.733, 0.00000), (363.778,
0.00000), (363.823, 0.00000), (363.868, 0.00000), (363.913, 0.00000), (363.958, 0.00000),
(364.003, 0.00000), (364.048, 0.00004), (364.093, 0.00000), (364.138, 0.00000), (364.183,
0.00000), (364.228, 0.00004), (364.273, 0.00000), (364.318, 0.00000), (364.363, 0.00000),
(364.408, 0.00000), (364.453, 0.00000), (364.498, 0.00000), (364.543, 0.00000), (364.588,
0.00000), (364.633, 0.00000), (364.678, 0.00000), (364.723, 0.00007), (364.768, 0.00004),
(364.813, 0.00000), (364.858, 0.00000), (364.903, 0.00000), (364.948, 0.00000), (364.993,
0.00000), (365.038, 0.00000), (365.083, 0.00000), (365.128, 0.00004), (365.173, 0.00000), all
the other points are zeros.
Fig. 7.11 See Table. 7.2 and Table. 7.3. Fig. 7.11 Inset: (-0.1319, 0.01), (-0.0355,
0.00), (0.1584, 0.06), (0.5455, 0.58), (1.1247, 0.63), (1.8990, 0.42), (2.8637, 0.34), (3.8330,
0.29), (5.7645, 0.30), (7.6996, 0.27), (9.6278, 0.26), (13.4943, 0.23), (16.3946, 0.20), (11.5624,
0.24), (8.6665, 0.26), (7.7003, 0.26), (6.7316, 0.28), (4.7972, 0.33), (3.3477, 0.31), (2.3818,
0.38), (1.5105, 0.48), (0.9323, 0.73), (0.3526, 0.53), (0.0612, 0.00), (-0.0354, -0.00), (-0.0840,
-0.00).
Fig. 7.12: See Table. 7.2 and Table. 7.3.
Fig. 7.13 circles: (8.0, 0.534), (16.0, 0.487), (32.0, -0.127), (48.0, -0.183), (60.0,
-0.196), (128.0, -0.16), (500.0, -0.0745), (10000.0, 0).
Fig. 7.13 squares: (8.0, 0.334), (16.0, 0.303), (32.0, -0.104), (48.0, -0.147), (60.0,
-0.0723), (128.0, -0.0751), (10000.0, 0).
Fig. 7.14 up pointing triangles (black): same as Fig. 7.4 up pointing triangles
(black).
Fig. 7.14 down pointing triangles (red): same as Fig. 7.4 down pointing triangles
(red)
Fig. 7.14 solid up pointing triangles (black), format:(ε, average current,
up bar length, down bar length). (-0.027, 9.28e-13, 2.1e-14, 2.07e-14),(0.080, 9.22e-
13, 1.96e-14, 1.95e-14),(0.190, 9.3e-13, 4.23e-14, 3.71e-14),(0.407, 9.44e-13, 6.9e-14, 4.4e-
14),(0.926, 9.65e-13, 1.14e-13, 6.21e-14),(1.489, 9.89e-13, 1.07e-13, 8.07e-14),(2.353, 1.01e-
12, 1.01e-13, 8.25e-14),(3.329, 1.03e-12, 8.5e-14, 7.23e-14),(4.415, 1.04e-12, 7.84e-14, 6.47e-
14),(6.583, 1.07e-12, 5.85e-14, 4.99e-14).
Fig. 7.14 solid down pointing triangles (red): (5.497, 1.06e-12, 6.46e-14, 5.69e-
14),(3.874, 1.04e-12, 8.31e-14, 6.85e-14),(2.788, 1.02e-12, 9.36e-14, 7.96e-14),(1.923, 9.98e-
13, 1.02e-13, 8.54e-14),(1.164, 9.7e-13, 1.1e-13, 7.42e-14),(0.622, 9.36e-13, 8.76e-14, 4.44e-
14),(0.299, 9.31e-13, 5.04e-14, 3.67e-14),(0.136, 9.19e-13, 3.6e-14, 3.26e-14),(0.027, 9.18e-13,
2.19e-14, 2.18e-14),(0.065, 9.15e-13, 2.16e-14, 2.69e-14).
Fig. 7.15 up pointing triangles (black): same as Fig. 7.5 up pointing triangles
(black).
184
Fig. 7.15 down pointing triangles (red): same as Fig. 7.5 down pointing triangles
(red)
Fig. 7.15 solid down pointing triangles (black): (-0.036, 3.31e-11, 2.8e-14, -2.68e-
14),(0.070, 3.3e-11, 2.77e-14, -2.65e-14),(0.177, 3.33e-11, 2.62e-13, -2.47e-13),(0.391, 3.39e-
11, 8.05e-13, -3.04e-13),(0.907, 3.52e-11, 1.41e-12, -1.39e-12),(1.464, 3.62e-11, 1.69e-12, -
1.84e-12),(2.319, 3.71e-11, 1.53e-12, -1.99e-12),(3.286, 3.78e-11, 1.24e-12, -1.59e-12),(4.357,
3.84e-11, 9.78e-13, -1.2e-12),(6.502, 3.92e-11, 7.26e-13, -7.9e-13).
Fig. 7.15 solid down pointing triangles (red): (5.428, 3.87e-11, 8.9e-13, -9.15e-
13),(3.822, 3.79e-11, 1.22e-12, -1.35e-12),(2.747, 3.72e-11, 1.48e-12, -1.77e-12),(1.890, 3.64e-
11, 1.53e-12, -1.9e-12),(1.143, 3.52e-11, 1.55e-12, -1.59e-12),(0.605, 3.38e-11, 1.2e-12, -4.31e-
13),(0.285, 3.33e-11, 4.83e-13, -4.33e-13),(0.123, 3.25e-11, 6.36e-14, -5.75e-14),(0.016, 3.24e-
11, 2.8e-14, -2.99e-14),(0.054, 3.24e-11, 3.03e-14, -2.96e-14),(-0.947, 3.23e-11, 9.27e-14, -
9.99e-14).
Fig. 7.16: (11.19, -2.912e-12), (inf, -2.873e-12), (11.17, -2.835e-12), (11.16, -2.797e-12),
(10.04, -2.758e-12), (10.02, -2.72e-12), (inf, -2.681e-12), (9.698, -2.643e-12), (10.37, -2.605e-
12), (10.36, -2.566e-12), (9.932, -2.528e-12), (9.625, -2.489e-12), (9.597, -2.451e-12), (8.77,
-2.413e-12), (8.645, -2.374e-12), (9.138, -2.336e-12), (8.268, -2.297e-12), (7.998, -2.259e-12),
(7.92, -2.22e-12), (7.948, -2.182e-12), (8.047, -2.144e-12), (7.455, -2.105e-12), (7.279, -2.067e-
12), (7.303, -2.028e-12), (7.034, -1.99e-12), (6.912, -1.952e-12), (6.563, -1.913e-12), (6.615,
-1.875e-12), (6.589, -1.836e-12), (6.265, -1.798e-12), (6.066, -1.759e-12), (6.116, -1.721e-12),
(5.912, -1.683e-12), (5.635, -1.644e-12), (5.369, -1.606e-12), (5.169, -1.567e-12), (5.151, -
1.529e-12), (5.061, -1.491e-12), (4.901, -1.452e-12), (4.785, -1.414e-12), (4.568, -1.375e-12),
(4.482, -1.337e-12), (4.228, -1.298e-12), (4.166, -1.26e-12), (4.078, -1.222e-12), (3.898, -
1.183e-12), (3.699, -1.145e-12), (3.585, -1.106e-12), (3.464, -1.068e-12), (3.278, -1.03e-12),
(3.241, -9.912e-13), (3.021, -9.528e-13), (2.942, -9.144e-13), (2.768, -8.759e-13), (2.624, -
8.375e-13), (2.57, -7.991e-13), (2.364, -7.607e-13), (2.321, -7.223e-13), (2.138, -6.839e-13),
(2.027, -6.454e-13), (1.949, -6.07e-13), (1.762, -5.686e-13), (1.726, -5.302e-13), (1.512, -
4.918e-13), (1.418, -4.534e-13), (1.331, -4.15e-13), (1.178, -3.765e-13), (1.086, -3.381e-13),
(0.9394, -2.997e-13), (0.8163, -2.613e-13), (0.7007, -2.229e-13), (0.5754, -1.845e-13), (0.4467,
-1.461e-13), (0.3883, -1.076e-13), (0.2057, -6.923e-14), (0.1358, -3.082e-14), (0, 7.596e-15), (-
0.1358, 4.601e-14), (-0.2057, 8.443e-14), (-0.3883, 1.228e-13), (-0.4467, 1.613e-13), (-0.5754,
1.997e-13), (-0.7007, 2.381e-13), (-0.8163, 2.765e-13), (-0.9394, 3.149e-13), (-1.086, 3.533e-
13), (-1.178, 3.917e-13), (-1.331, 4.302e-13), (-1.418, 4.686e-13), (-1.512, 5.07e-13), (-1.726,
5.454e-13), (-1.762, 5.838e-13), (-1.949, 6.222e-13), (-2.027, 6.606e-13), (-2.138, 6.991e-13), (-
2.321, 7.375e-13), (-2.364, 7.759e-13), (-2.57, 8.143e-13), (-2.624, 8.527e-13), (-2.768, 8.911e-
13), (-2.942, 9.295e-13), (-3.021, 9.68e-13), (-3.241, 1.006e-12), (-3.278, 1.045e-12), (-3.464,
1.083e-12), (-3.585, 1.122e-12), (-3.699, 1.16e-12), (-3.898, 1.198e-12), (-4.078, 1.237e-12),
(-4.166, 1.275e-12), (-4.228, 1.314e-12), (-4.482, 1.352e-12), (-4.568, 1.391e-12), (-4.785,
1.429e-12), (-4.901, 1.467e-12), (-5.061, 1.506e-12), (-5.151, 1.544e-12), (-5.169, 1.583e-
185
12), (-5.369, 1.621e-12), (-5.635, 1.659e-12), (-5.912, 1.698e-12), (-6.116, 1.736e-12), (-6.066,
1.775e-12), (-6.265, 1.813e-12), (-6.589, 1.851e-12), (-6.615, 1.89e-12), (-6.563, 1.928e-12), (-
6.912, 1.967e-12), (-7.034, 2.005e-12), (-7.303, 2.044e-12), (-7.279, 2.082e-12), (-7.455, 2.12e-
12), (-8.047, 2.159e-12), (-7.948, 2.197e-12), (-7.92, 2.236e-12), (-7.998, 2.274e-12), (-8.268,
2.312e-12), (-9.138, 2.351e-12), (-8.645, 2.389e-12), (-8.77, 2.428e-12), (-9.597, 2.466e-12), (-
9.625, 2.505e-12), (-9.932, 2.543e-12), (-10.36, 2.581e-12), (-10.37, 2.62e-12), (-9.698, 2.658e-
12), (-inf, 2.697e-12), (-10.02, 2.735e-12), (-10.04, 2.773e-12), (-11.16, 2.812e-12), (-11.17,
2.85e-12), (-inf, 2.889e-12), (-11.19, 2.927e-12).
186
Appendix B
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[151] http://www.nls.physics.ucsb.edu/xiaochao/thesis.html . I will
try to maintain this site. In case I have to move the site, please search
for my thesis over the internet.
200
