Space charge effects in high pressure mass spectrometry sources
by Mark Busman
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Chemistry
Montana State University
© Copyright by Mark Busman (1991)
Abstract:
Ion transport in unipolar mass spectrometry ion sources like corona API and electrospray has been
studied theoretically. Unlike previous work in this area, the study has been done with the consideration
of space charge effects. Analytic solutions of the space charge problem have been obtained for simple
ion geometries, including the cases of infinite parallel planes, concentric cylinders of infinite length,
and concentric spheres. These analytic solutions allow, for their respective geometries, the calculation
of electric field, potential, ion density distributions, and ion residence times. It is shown that for typical
operating conditions, the minimum potential required to overcome the space charge effect in corona
API, or electrospray ion sources, constitutes a dominant or significant fraction of the total applied
voltage. Further, the electric field, in the region of the ion sampling orifice and the ion residence time
in the ion source are determined mainly by the space charge. Extending the approach to more general
geometries, absolute sensitivities of corona API ion sources were calculated using a geometry
independent treatment of space charge. Also, general geometries were modelled by a simulation
calculation. The calculation was based on a computer program written to model ion flow in various ion
sources having different geometries. Finally, the space charge influenced ion drift in a drift tube-type
apparatus was modelled, as a function of time. SPACE CHARGE EFFECTS IN HIGH PRESSURE
MASS SPECTROMETRY SOURCES
by
Mark Busman
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Chemistry
MONTANA STATE UNIVERSITY
Bozeman, Montana
July 1991
ii
APPROVAL
of a thesis submitted by
Mark Busman
This thesis has been read by each member of the thesis
committee and has been found to be satisfactory regarding
content, English usage, format, citations, bibliographic
style, and consistency, and is ready for submission to the
College of Graduate Studies.
________ Ju ne 2 4 .
1991
Date
Approved for the Major Department
June 2 4 .
1991
Date
Head, Major Department
Approved for the College of Graduate Studies
/«2, / ? < ? /
Date q
Graduate' Dean
iii
STATEMENT OF PERMISSION TO USE
In presenting this thesis in partial fulfillment of the
requirements for a doctoral degree at Montana State
University, I agree that the Library shall make it available
to borrowers under rules of the Library. I further agree
that copying of this thesis is allowable only for scholarly
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U.S Copyright Law. Requests for extensive copying or
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Microfilms International, 300 North Zeeb Road, Ann Arbor,
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Signat
Date
iv
TABLE OF CONTENTS
Page
H 03 OO CO
I N T R O D U C T I O N ........................ ................... ...
Ion T r a n s p o r t ............. ................... ........................
Convective Transport
Diffusive Transport ..........................
Electrostatic Transport ......................
3
MATHEMATICAL TREATMENT.OF ION TRANSPORT ...............
Drift E q u a t i o n .....................................
Laplacian Field
...................................
Space Charge Field ...........
Historical Background
............................
Mathematical Background
..........................
Analytic Solutions .................................
Planar
. ....................
C y l i n d r i c a l ...............
S p h e r i c a l .....................................
4
4
5
5
6
8
.10
10
12
18
APPLICATIONS OF THE ANALYTIC SOLUTIONS
...............
Corona API in the Spherical G e o m e t r y .............
Electrospray in the Spherical Model
.............
24
27
33
........................
GENERAL GEOMETRIES
Unipolar Charge Drift Formula
.
Calculation of Absolute Sensitivities
...........
Numeric Solutions
.................................
Simulation Method
.................................
Simulation in General Geometries .................
Point to Plane G e o m e t r y ..........................
Electrospray Geometry
...............
API G e o m e t r y .............................. . . . . .
38
38
45
51
55
56
62
62
67
TIME DEPENDENT C A S E .....................................
Simulation With One Type of I o n .................
Simulation With Two Types of I o n s ...............
72
74
76
S U M M A R Y ..................................................
80
REFERENCES CITED
82
.......................................
V
TABLE OF CONTENTS-Continued
Page
A P P E N D I C E S .......................
APPENDIX A: ANALYTIC SOLUTIONS ...................
Planar geometry ..............................
Cylindrical Geometry
........................
Spherical geometry
................. . . . .
APPENDIX B : DERIVATIONS FOR LIMITING CONDITIONS
.
Ion residence time, tres
..........
Sampling ion density, p
Applied potential, V0 . . .... ...............
APPENDIX C: COMPUTER PROGRAMS
. . ................
86
87
88
90
92
95
96
97
98
99
LIST OF FIGURES
Figure
Page
1. Planar Geometry.
n
2. Non-SCD Residence Time and Space Charge..........
13
3. SCD Residence Time and Space Charge..............
14
4. Non-SCD Potential and Fields.....................
15
5. SCD Potential and Fields.........................
16
6 . Cylindrical Geometry.............................
17
7. SCD and Non-SCD Fields...........................
19
8 . SCD and Non-SCD Space Charge Densities..........
20
9. Spherical Geometry...........
21
10. SCD Conditions in the Spherical Geometry. . . .
23
11. Schematic of the Corona-API. . ................
28
12. Conditions in the Corona-API....................
30
13. Residence Time, Potential and Ion Density . . .
34
14. Schematic of the Electrospray..................
35
15. Ion Density vs. Drift T i m e ............. ..
40
16. Field strength map for the nonfocussed case.
.
42
17. Field strength map for the focussed case. . . .
42
18. Ion density contour for the unfocussed case.
.
43
19. Ion density contour for the focussed case.
. .
43
20. Current densities for the Case I and Case 2.
.
44
21. Ion densities for the Case I and Case 2.
...
44
22. Kinetically Influenced API Currents.
50
52
23. Calculated API Currents....................
24. Experimental API Currents.
....................
53
25. Point to Plane Geometry....................
57
26. "Needle in Can" Geometry..........
58
27. Parabolic Gr i d ..............................
59
28. Point to Plane Plate Current Distribution.
. .
63
29. Electrospray Plate Current Distribution.
...
64
30. Spray Pattern from a Taylor Cone..........
65
31. Spray Pattern from the "Umbrella" M o d e ....
66
3 2 . Calculated Electrospray Current Distribution. . 68
33. Field and Space Charge Density Contour....
69
34. Calculated Corona-API Current Distribution.
.
70
35. Corona-API Field, Space Charge Contour....
71
36. Ion Tube in the Fenn Electrospray Source.
...
73
37. Drift Tube Schematic.......................
75
38. One Component Ion Drift Current vs. Time.
...
77
39. Two Component Ion Drift Current vs. Time.
...
78
40. SPACE. B A S .................................
100
41. SPREAD. B A S ............................
119
<
vii
LIST OF FIGURES-Continued
Figure
42. TUBE.M . .■ ........ .. . ................
43,. TUB2.M..............................
44. VELOCITY.M .
................................ ■
45. DISPLACEMENT. M ...................................
Page
124
128
129
viii
ABSTRACT '
Ion transport in unipolar mass spectrometry ion sources,
like corona API and electrospray has been studied
theoretically. Unlike previous work in this area, the study
has been done with the consideration of space charge
effects. Analytic solutions of the space charge problem have
been obtained for simple ion geometries, including the cases
of infinite parallel planes, concentric cylinders of
infinite length, and concentric spheres. These analytic
solutions allow, for their respective geometries, the
calculation of electric field, potential, ion density
distributions, and ion residence times. It is shown that for
typical operating conditions, the minimum potential required
to overcome the space charge effect in corona API, or
electrospray ion sources, constitutes a dominant or
significant fraction of the total applied voltage. Further,
the electric field, in the region of the ion sampling
orifice and the ion residence time in the ion source are
determined mainly by the space charge. Extending the
approach to more general geometries, absolute sensitivities
of corona API ion sources were calculated using a geometry
independent treatment of space charge. Also, general
geometries were modelled by a simulation calculation. The
calculation was based on a computer program written to model
ion flow in various ion sources having different geometries.
Finally, the space charge influenced ion drift in a drift
tube-type apparatus was modelled, as a function of time.
I
INTRODUCTION
In the last half century, mass spectrometry has
gradually evolved from being an interesting tool for the
mass measurement of relatively small ions, into a vital area
of analytical chemistry. This evolution has followed the'
development of creative techniques for the introduction of
molecules into an ion source for mass analysis, and
achieving their ionization.
The ion sources used in mass spectrometry can be
classified according to source pressures. Sources operated
at pressures I torr and below can be considered to be low
pressure ion sources. Low pressure sources include sources
for many of the more commonly used mass spectrometry
techniques: electron impact ionization (EI),1*2 chemical
ionization (Cl),3 and fast atom bombardment (FAB).4 High
pressure ion sources can be thought to be those that operate
in the pressure range between I torr and atmospheric
pressure, and perhaps beyond. High pressure sources include
the sources used for atmospheric pressure ionization
(API),5 thermospray ionization (TSP),6 and electrospray
ionization (ES) .7,8 The ion sources in both of these
categories have been studied to establish appropriate
parameters for their operation. Much has been written with
2
respect to the mechanisms pertinent to the Operation of
these sources.
In mass spectroscopy, the understanding of the
production of ions in the ion source is an important aspect
of understanding the technique. Further, understanding the
processes behind introducing the ions into the mass
resolution part of the instrument is vital for insight into
the identities and quantities of ions available for
detection. The goal of this work is to evaluate certain
aspects of ion transport in high pressure mass spectrometry
sources with regard to the performance of these sources.
Ion Transport
In the ion source of the mass spectrometer ions are
produced from existing molecules. These ions are then driven
toward an aperture separating the high-pressure ion source
from the low-pressure mass analyzer section of the mass
spectrometer, where they can travel towards the detector.
The processes causing the ions to leave the ion source can
be of the following types of transport: convective,
diffusive, and electrostatic.
Convective Transport
The ion source is often at much higher pressure than
the mass analyzer region. The ions will be carried along
with the gas stream down the pressure gradient. The study of
3
how ions can flow along with a moving gas is an important
area in fluid dynamics.9
Diffusive Transport
In many ion sources,
ions are not formed throughout the
source, but, instead, are formed in a small volume in the
ion source. Here, the concentration of ions may be very
high. The ions tend to diffuse out of the areas of high
concentration into areas of lower concentration. The
mathematical treatment of the diffusion of ions in gases is
well developed.10,11
Electrostatic Transport
Ions, as charged species, will drift in existing
fields. An ion will drift in the direction of existing
electrical fields. The drift velocity depends on the field's
intensity. Again, the mathematical treatment, of ionic drift
has already been developed.9,11 Ion drift will now be
discussed in more detail.
4
MATHEMATICAL TREATMENT OF ION TRANSPORT
The application of electrostatics to charged particle
flow and, in particular, ion flow has been important to
physics for many years. The analysis ion flow has been based
on Coulomb's and Maxwell's equations.12 It is these
equations that make the analysis of ion drift in the mass
spectrometer ion source possible.
Drift Equation
Using appropriate laws of electrostatics, one can
predict the magnitude and direction of the forces acting
upon ions at specific locations in the source. Moreover,
assuming certain characteristics of both the ions, and the
neutral gas inside the source, one can predict a velocity
for an ion under the influence of a given field. This
velocity, the drift velocity, v, can be given as a function
of electric field strength, E, by the drift equation,
V=K-E,
(I)
where K is the ion mobility, a constant, dependent on both
the characteristics of the ion and its surrounding gas, and
generally assumed to be independent of field strength.10,13
!
5
The drift equation is essentially an empirical equation
that summarizes a large amount of experimental data. It has
been shown to be applicable at high pressures
(above I
torr). The mobility, k , is the empirical constant in the
equation. The mobility constant's magnitude is dependent on
the charged particle's response to an electric field and to
the collisions with the neutral bath gas that the particle
will experience. Compilations of mobility data have been
published, most notably by McDaniel.13
Laolacian Field
An ion source often has several electrically insulated
components that have different applied potentials. From the
geometry of the ion source, it is possible, in theory, to
map out the electric fields in the source, that result from
the application of the potentials to the various source
components.14 Generally, it is assumed that the field
strength at any point in the source is dependent on only the
external applied field. The field strength at any point is
determined by the source geometry and the respective applied
voltages on various source components. This external field
is commonly called the Laplacian field.11
Space Charge Field
As ion densities in the source increase, the coulombic
forces between the ions will increasingly modify the
i
6
Laplacian electrical fields experienced by ions. In the
source the conduction of current will be unipolar. By saying
that the current is unipolar,
it is meant that particles of
only one polarity exist in the source, and that the ion flow
can be considered accordingly. The ions in the ion source,
traveling at finite velocities, experience a mutual
coulombic repulsion during the time that they transit the
gap between electrodes, thus forming their own electric
field. The Laplacian field is combined with the field caused
by the moving charge, the space charge field. This
combination is the actual field experienced by the ions. In
the extreme, the space charge field, from the unipolar
current, can be so large that it completely dominates any
applied field.
Historical Background
The modification of Laplacian fields by space charge
was.noticed many years ago by workers studying electrical
currents between charged plates. As they attempted to
increase the current between these charged plates by
changing the conditions between the plates, they found that
it was not possible to increase the current beyond a certain
limiting current. This was explained by the modification of
the fields between the plates, by the current flow.15,16,12
The mathematical picture of space charge was initially
developed by Child, in 1908, to model the evaporation of
7
calcium ions from a heated plate.
Langmuir,
Unaware of Child's work,
in 1913, independently developed a similar model
to describe the behavior of thermionic currents from
filaments.16 In 1914, Townsend described the space charge
influenced currents in terms of the mobilities of the charge
carriers. It became common for the maximum currents, as
allowed by the space charge, to be called space charge
limited.12'11'10
The concept of space charge has been used for a wide
range of application. From the initial use for currents
between charged plates, the space charge concept has been
used in electronics for the design of tube-type electronic
components,17 in meteorology to model lightning strikes,
in
aircraft design to evaluate charge buildup on various
airplane structural components,9 in industry for the design
and maintenance of electrostatic precipitators,18 and in
the modelling of fields surrounding high voltage, direct
current power transmission lines.
As the field of mass spectroscopy developed,
it was
natural for the concept of space charge to be applied to the
emerging technique.
However, early workers correctly
assumed that they were working with ion currents and ion
densities that were too low for space charge to be
important. The combinations of source.conditions and
ionization techniques being utilized made the modifications
8
of the Laplacian fields in the sources, by the space charge,
insignificant.19
Recent years have seen a continual evolution of. the
mass spectroscopy field. Hew techniques for ionization have
changed the conditions in the ionization source
considerably. The atmospheric pressure ionization (API)5 and
the electrospray ionization (ESP)7 techniques have utilized
sources that have particularly deviated from the
conventionally expected conditions of high vacuum and low
ion densities. So, it can be argued that an assessment of
the importance of space charge in the sources of some of
these newer techniques is due. Notable in displaying the use
of space charge analysis in high pressure mass spectrometry,
is the ECD model by Cobby, Grimsrud and Warden.20
Mathematica] Background
To evaluate space charge effects in these newer
sources, use of the previously developed methodology for
determining space charge influenced fields must be made.
However, early work with space charge was restricted to the
analysis of simple geometries. Due
to
the difficulty of the
mathematics involved, workers would idealize their
experimental apparatus as either infinite parallel planes,
concentric cylinders of infinite length, or concentric
spheres.'7 Thesej geometries were amenable to mathematical
analysis, yielding data appropriate to the individual
9
worker's needs. The computation of space charge influences
in these geometries was based on the Poisson equation,
-V-E=E1
(2)
where V« E is the gradient of the electric field, p is the
space charge density in terms of charge per unit volume, and
e is electrical permitivity; and the continuity equation.
'dp i-V-J=Or
Ft
(3)
where t is time. When considering stable ion source
conditions, the steady state form of the continuity
equation.
V-J=Or
(4)
can be used by assuming (dp/dt) =0.
Here, J is the ion current density. As was discussed
earlier, the ion current will have contributions from
diffusion, convection, and electrostatic transport. This
dependence of J can be mathematically stated as
J=Kp E-DVp+F r
(5)
where D is the diffusion constant and F is the gas flow
vector. The first term relates to the ion drift
contribution to the current density vector, while the second
and third terms indicate the effects of diffusion and
convection, respectively.10 Diffusion and convection terms
10
are important only in some very special situations, and are
not treated in this work.
Beyond this point the mathematics will be expressed in
scalar notation, instead of the previously used vector
notation.
Analytic Solutions
For this work, the Poisson and continuity equations
were utilized to predict electric fields, space charge
densities, potentials, ion transit times, etc. for ion
sources with certain assumed operating parameters. Given the
wide variation in the geometries of real ion sources, all
three simple geometries, amenable to giving analytical
solutions, were considered. Similar analyses have previously
been described, for use in discharge physics, by Townsend12
and Chapman.21 Descriptions of these analyses, for each of
the three geometries, follow.
Planar
In this geometry, shown in Figure I, two planes of
infinite area are placed parallel to each other at some
finite distance, I. A potential difference, V 0, exists
between the two plates. A current, i, is assumed to flow
between the two plates. Furthermore, all of this current is
carried by one type of charged particle, with a mobility, k .
The unipolar current originates at one surface, and
terminates at the other.
11
Figure I.
Planar Geometry, infinite parallel planes.
The mathematical analysis of the planar geometry gives
electric field strength and ion density as a function of
distance, x, from the source plate. A detailed derivation of
the equations shown here is presented in Appendix A. The
equations for field strength and space charge density are
-X+Er
?T/2
12
(7)
Here E0 is the field strength at the source plate (x=0).
Graphs of the functions for certain sets of initial
conditions are given in Figures 2, 3 , 4 ,
and 5. The Figures
show the remarkable modification of the conditions in the
source, in the space charge dominated (SCD) case. Especially
noteworthy is the decrease in the electric field strength
and the increase in the ion density in the area near the
source plate.
Cylindrical
In this geometry, shown in Figure 6, a cylinder of
radius, r0, is enclosed by a concentric cylinder of radius,
r1. The surface of the inner cylinder acts to supply a
current of charged particles that flow to the outer
cylinder. These concentric cylinders are assumed to be of
infinite length. Again, a potential difference exists
between the two cylinders and the current is completely
carried by charged particles of mobility, k . The field
strength is E0 at the surface of the inner electrode (r=r0) .
The mathematical analysis of the cylindrical geometry
gives electric field strength and ion density in terms of
distance, r, from the axis at the center of the two
8 x IO7
0.003
Time(s)
Ion Density(Ions/cc)
Ion Density
Drift Time
0
1.0
Figure 2.
2.0
3.0
Distance(cm)
4.0
5.0
Non-space charge dominated residence times and
space charge densities, in the planar geometry, as
a function of distance. The current density is 1.0
x IO'8 A/cm2; the total applied potential, V0=IO ,000
V; and the ion mobility. x=l x i o '4 m2/V*s.
0.012
1.8 x IO8
Ion Density
Time(s)
Ion Density(Ions/cc)
Drift Time
0
0
2.0
3.C
Distance(cm)
Figure 3.
Space charge dominated residence times and space
charge densities, in the planar geometry, as a
function of position. The current density is 1.0 x
10'8 A/cm2; the applied potential, V.=3,600 V; and
the ion mobility, K=I x io*4 m2/V-s.
4.28
-n 10
Potential
Z
Laplacian Field
Potential (kV)
Field(kV/cm)
\.
/ Field
O
1.0
Figure 4.
2.0
3.0
Distance(cm)
4.0
5.0
Non-space charge dominated potentials and fields,
in the planar geometry, as a function of distance!
Applied
potential,
current
density,
and
ion
mobility are as specified for Figure 2.
3.6
Potential(kV)
Field(kV/cm)
Potential
Laplacian Field
2.0
3.C
Distance(cm)
Figure 5.
Space charge dominated potentials and fields, in
the planar geometry, as a function of distance.
Applied potential, ion current, and ion mobility
are as specified for Figure 3.
17
Figure 6.
Cylindrical
geometry,
infinite length.
concentric
cylinders
of
cylinders. Mathematically, the functions apply to the region
between the concentric cylinders. A detailed derivation of
the equations shown here is presented in Appendix A. The
equations for field strength and space charge density are
T/2
(r2-r02)+S02
S=—
r
2 itk e0r 02
(8 )
18
1 - 1/2
"
--- i-- - (r2-r02)+E02
2 Trr0ZC 2h k e0r02
.
(9)
A graph of the functions for a certain set of initial
conditions is given in Figures 7 and 8. Interestingly, in
the space charge dominated case, the space charge density is
independent of position. The cylindrical analysis was used
by Shahin to model his cylindrical ion source. However, the
dimensions of his source, as well as his low ion currents,
made space charge influences negligible.19
Spherical
Analogous to the cylindrical case, this geometry, shown
in Figure 9, involves concentric spheres. The inner sphere,
which acts as an ion supply, is of radius, r0. The radius of
the outer sphere is r1. The two spheres are separated by a
potential difference, and current flows from the inner
sphere to the outer sphere by the drift of charged particles
of mobility, k . The field strength is E0 at the inner sphere
surface (r=r0) .
The mathematical analysis of the spherical geometry
gives field strength and ion density in terms of distance,
r , from a point at the center of the two spheres.
Mathematically, the functions are valid in the region
between the concentric spheres. A detailed derivation of the
XlO4
non-SCD
Radius (m)
Figure 7.
Fields with the consideration of space charge and
without the consideration of space charge in the
cylindrical geometry. The radii of the inner and
outer
cylinders
are
0.001
and
0.005
m,
respectively. The field at the inner cylinder
surface, E0= 2 .5 x IO4 V/m; the ion mobility, K=I x
10'4 m2/ V •s ; and the ion current, i=l x lO"*’ A/m.
Ion Density (C/mm3)
XIo-5
non-SCD
Radius (m)
Figure 8.
Space
charge
dominated
and
non-space
charge
dominated space charge densities in the cylindrical
geometry. The conditions are the same as indicated
for Figure 7.
21
Figure 9.
Spherical geometry,
concentric spheres.
equations shown here is presented in Appendix A. The
equations for field strength and space charge density are
E(r) = h i
r
i
1/2
(r3-r0
3)+£02
(1 0 )
6Kne0r0
-
r 3-r03W 02
47re0r02x Qne0Kro'
1/2
(H)
22
A graph of the functions for a certain set of initial
conditions is given in Figure 10. From the Figure it is
evident that any analysis of such a space charge dominated
source, without the consideration of space charge, would
dramatically underestimate the actual field strength at
distances far from the inner sphere surface.
Experimental work has been done, in the different
geometries, to verify the equations. The cylindrical and
planar geometries were experimentally studied by Child,
Langmuir and Townsend at the time that the mathematical
formulations were originally presented.15,16,12 Much later,
Chapman produced similar experimental data to verify the
description of the spherical model. Chapman measured the
magnitude of space charge limited currents, and showed these
currents to be in agreement with the analytic solutions22.
7.5 x IO9
1.8
Potential
Y---'
U
IZl
C
O
U
2.
0.9 d
in
U
Q
c
O
O
3
0.5
Figure 10.
r>
-zX D rift
1.0
1.5
2.0
Radius(cm)
2.5
3.0
Space charge dominated conditions in the spherical
geometry: Electric field potential, ion density
and ion drift time for the space charge dominated
case. The applied voltage, V0=7,693 V; the supply
surface radius, r0=O.l cm; the outer sphere radius,
r0ut=3 cm; the ion mobility, *=i x io"4 m2/V-s; and
the ion current, i=6/LiA.
24
APPLICATIONS OF THE ANALYTIC SOLUTIONS
The analytical solutions for the simple geometries
presented in the previous section give at least a
qualitative picture of what is happening in a actual ion
source. There are two reasons why the models may not
correspond directly to physical reality. First, they do not
correspond precisely to the actual physical geometry of most
sources. Second, they rely on a rather artificial
distinction between the ionization and drift regions of the
source, as will be discussed below. However, they do serve
some instructive purpose. These models can be viewed as some
crude approximation of actual source geometries. These
models can give important insight into the perturbation of
the Laplacian fields by space charge.
The spherical, source, especially, can be used to model
many ion sources. Many sources can be viewed as having a
central ion source, and having walls, at least approximately
equidistant, in all directions. This view gives the
opportunity to evaluate a particular ion source with respect
to some of its known operating parameters
(i.e., potential
and supply current). The data from this evaluation can give
interesting information about the conditions inside the
source. Especially important will be the way these
0#
25
conditions will be influenced by the space charge density in
the source. The data must then be assessed with respect to
its dependence on the geometric approximations made in
constructing the model.
Particularly useful pieces of information for an ion
source are the ion residence time in the source,
ion
concentration at the sampling aperture and ion drift
velocity at various locations in the source. Such
information is readily available from this method of
analysis, and is very interesting for a variety of
applications, especially when the reaction kinetics for the
drifting ions is important.
The ion source can be thought to be comprised of two
rather arbitrarily divided regions; the ionization region
and the drift region. When modelling the ion source by this
method, the entire volume of the source is not being
modelled. Instead, only the drift region is being
considered. The ionization region will contain mechanisms
that will produce ions, at its surface, at a given rate.
Regardless of the ion source in question, processes of
complexity beyond the scope of this study will occur in the
ionization region. In the drift region, however,
ions merely
drift to the counter electrode. This reduces the source
modelling into a much more workable problem.
The somewhat arbitrary division between the ionization
region and the drift region has precedent.9 Much of
26
application of space charge theory is rooted in the study of
corona discharges. A corona is used in the corona API source
and is discussed below in that context. A corona discharge
can be stimulated by biasing a needle with a large voltage.
Events centered around the tip of the needle arise from the
effect of the applied voltage on the air surrounding the
needle. Many researchers observed light being emitted from
the area immediately around the tip of the needle, at the
center of a corona. This light emitting region was assumed
to contain a plasma region, whereas the drift region was
assumed to be on the outside of the light emitting region.
The shape and size of the light emitting region was taken to
be an indication of the shape and size of the plasma. This
visual observation gave early researchers the justification
for the division between the two regions.9*10'11 Shahin used a
similarly arbitrary division for the analysis of his
cylindrically symmetric corona mass spectrometer source.19
With mathematical models of the space charge influence
in the simple geometries developed, attention can be turned
to mass spectroscopic techniques utilizing sources in which
space charge may have an important effect. Two of these
techniques are Atmospheric Pressure Ionization (API) and
Electrospray Ionization (ESP). Both of these techniques
operate with relatively high source pressures and ion
currents.11*7*6 Moreover, the drift region of each source is
essentially unipolar.
27
Corona API in the Spherical Geometry
In the positive corona API source, shown in Figure 11,
the ionization is caused by biasing a needle at a positive
voltage high enough to initiate a corona. A detailed
description of the processes involved in the corona has been
provided by Loeb,10 Rees23 and Sigmond.11 Equations have been
formulated to empirically model the behavior of the corona,
immediately around the needle. An electron travelling in the
corona will strike a number of neutral molecules causing an
avalanche of electrons and, as a result,
leaves positive
ions. This primary ionization process corresponds to
Townsend's primary ionization coefficient, a .
A secondary
ionization process, where ions are produced by collisions
between molecules and the electrons produced in the
avalanche,
is described by a secondary Townsend ionization
coefficient, y. Through this complex regime of electronmolecule reactions, positive ions charge are driven outward,
away from the needle tip. Similarly, negatively charged
particles are attracted to the needle, and collected
there.10,11
Outside of a certain distance from the needle tip, only
particles of a charge being the same as the needle's will
exist. This region, away from the needle tip, can be
considered to be the drift region. In this drift region, the
only process occurring is the motion of the ions, through
ion drift.
28
-I
H
I
mi
r
-i
Figure 11.
Schematic of the corona API
Using the spherical geometry as a model for the API
source, one can model the ion motion in the drift region.
First, the ionization region of the source must be made to
conform to the developed model. The ionization region can be
reduced simply to a spherical ion supply surface, the inner
sphere in the spherical model. The idea of the spherically
symmetric ion supply surface has been supported by analysis
of photographs of the plasma regions surrounding corona
needles.9 This ion supply surface will produce an ion
current uniformly over its entire surface are a . The ion
29
current will consist of ions having a mobility,
k
.
According
to tabulated data of ion mobilities, most ions in an API
source will have a mobility of about I x IO"4 m 2/V- s .13
Here, a problem with the entire modelling process must
be faced. The solution to the derived equations for the
spherical model is dependent on boundary conditions. These
boundary conditions are values of important conditions on
the boundaries of the region to be modelled. Especially
important are the conditions on the ion supply surface.
Since the modelling of the ionization processes in the
corona, or in terms of the model, the conditions at the
supply surface, are beyond the scope of this study, the
problem of deciding appropriate parameters for this model
becomes especially difficult. It was decided to model the
API source with the ability to specify E0, the field
strength at the ion supply surface. E0 was varied over a
wide range of values and then the entire range of results
was evaluated. Figure 12 shows results,
in terms of E, p, t
and V, of modelling the API over a range of E0 values. As is
demonstrated in the Figure, the choice of E at the inner
boundary of the drift region does not greatly influence
conditions, except for potential, further out in the drift
region.
The parameters used in the model are
k
,
r0, I0, and E0.
For the Figure, K - I x IO'4 m2/V, rQ = 0.03 m and I0 = 6 pA
were used. Six different choices for E0 were used: 40, 35,
Fields
Potentials
Potential(kV)
Laplacian Field
Radius(cm)
Figure 12.
Conditions
in
the
corona-API
source:
Field
strengths
and
potential
distributions
versus
radius
for
6
different
values
of
applied
potential.
31
30,
25,. 20, and 0 kV/cm. As is apparent in the figure, only
in the area near the ion supply surface is the effect of the
choice of E0 important. The comparison of the Laplacian
field curve and the other curves, for actual electric
fields, makes obvious the important effect that space charge
plays in this system. The space charge domination serves to
coalesce the field strengths in the outer area of the
spherical source.
In the corona API source, the analyte molecules are
usually not directly ionized by the corona. Instead, the
ions supplied from the corona are largely reagent ions.
These reagent ions result from the ionization of some of the
neutral gas molecules present in the source. Often the ions
in the source are hydrated hydronium ions, H^O+ (H2O)n. The
analyte molecules are of much lower concentration than those
of the reagent gas. The reagent ions will,
in turn, react
with the traces of analyte molecules present in the source,
often in a reaction of the type:
H3O+ (H2O) n + B ^ B H +(H2O) n+1,
(12)
where B is an analyte molecule. The extent of analyte
ionization, and thus sensitivity, will depend on the time
that it takes for the reagent ions to transit the gap
between the ion supply surface and the sampling orifice on
the outer wall of the source. Hence,
it becomes important to
32
know the residence time, tres for the reagent ions in the
source. It is necessary to know the residence time in order
to properly evaluate the kinetic data acquired in corona API
experiments.
Approximate expressions for ion transit times, along
with other interesting source parameters, have been derived
from the spherical geometry solutions. For a space charge
limited source with the ion supply surface radius, rQ, being
much smaller than the outside surface radius, r 1,
expressions for ion residence time, t res; sample ion density,
Psamp; and applied potential, Vd can be written.
7/2
STrer1
3k i
3ei
(13)
7/2
samp
(14)
STrzer13
2Ir1 7/2
(15)
3 Trzee
Derivations of these equations are shown in Appendix B . The
33
equations are graphed, over a certain range of time, in
Figure 13.
Electrosnrav in the Spherical Model
Likewise, the Electrospray (ESP) source, shown in
Figure 14, can be analyzed by the spherical space charge
model. Similar problems with choices of parameters, relating
to boundary conditions, exist in this model. In the ESP
source the analyte is introduced into the ion source by
injecting a stream of a volatile liquid, containing a small
concentration of analyte molecules, through a hypodermic
needle. The needle is put at high voltage with respect to
the rest of the ion source. As the stream of liquid exits
the tip of the needle, it becomes charged and starts to
break into droplets. The droplets each carry a charge of the
same sign as the charge on the needle. Even though the
dynamics of charged droplets, such as these, have been
studied since the time of Lord Rayleigh, the exact
mechanisms involved in the behavior of these droplets are
still not well understood. However, through the work of
Vestal,6 Dole,7 Iribarne and Thomson,24 a general scheme for
ion production has been developed. As a droplet travels
further away from the needle, it is observed that both ions
and solvent molecules evaporate from the droplet's surface.
Several effects combine to determine the course of the
droplet's reduction in size. Surface tension will tend to
Ion Density(Ions/cc)
Voltage(kV)
Drift Time
Drift
Voltage
• ^ —Ion Density
Current(jaA)
Figure 13. Variation
of
residence
time,
potential
distribution
and
ion density with discharge
current in the spherical ion source geometry. The
conditions are as specified in Figure 10.
35
Figure 14.
Schematic of the Electrospray.
keep the droplet's surface intact. This will restrict the
droplet's contained liquid to a certain volume. Coulombic
repulsions will tend to keep the carriers of the droplet's
charge separated as far as possible, and therefore, near the
droplet's surface. The energy in the system will cause
neutrals to evaporate from the surface of the droplet. As
the droplet's size reduces, the coulombic repulsive forces
will overcome the surface tension and cause the droplet to
split into two or more droplets thus reducing the crucial
parameter for the coulombic forces, charge per surface area.
36
Further solvent evaporation will cause the cycle of
evaporation and droplet splitting for the droplets to
continue, until the droplets become very small. At some time
during this cycle of events, ions may evaporate from the
surface of a charged droplet taking some or all of the
droplet's charge with th e m .6 The production of a wide range
of both sizes and charges of ions has been noticed. Charges
as high as 1000, as well as masses up to 2,000,000 daltons
have been reported.7
It is apparent that droplets of varying radii and
charge are involved in the production of an ion current at a
sampling orifice. The range of charges and masses involved
spans a wide range of ion mobilities. To predict the
response of the larger droplets to the convective, as well
as electrostatic, forces acting in the source would be a
difficult task. However, making the simplification of
equating the area immediately around the capillary tip with
a spherical ion supply surface and assuming all of the ions
in the drift region to be of equal charge and mobility,
makes the analysis easily possible. Although much
mechanistic detail is sacrificed with the model, the
information available from the simple analysis gives some
interesting and useful information.
Using equation 14, the degree of space charge influence
can be calculated. Using a needle to orifice distance of 5
cm, a ion mobility value of I x 10"4 m 2/V s, and an ion
37
current of IO'6 A, the space charge potential is about 3500
V. This potential is in the, range of the applied voltage
used in the operation of the electrospray source. Therefore
the electrospray source is strongly space charge influenced,
if not dominated.
By this construction of an electrospray model, the
model becomes essentially identical to the previously
discussed API model. The operating parameters used in the
API model are appropriate for the normal range of ESP
operating parameters. From Figure 12, the illustration of
the API modelling,
modelled,
it is apparent that the ESP source, as
is also strongly space charge influenced. The
areas distant from the hypodermic needle tip are relatively
unaffected by the changes of E0 at the ion supply surface.
However,
in a recent paper ion drift times were calculated
using the Laplacian electric field.25 Clearly, such
analyses will be substantially in error.
38
GENERAL GEOMETRIES
The extension of the mobility model of ion transport to
more general geometries requires a mathematical approach
that can be applied to these general geometries. One
possible approach to generalizing the technique is to use
numerical methods to solve the same differential equations
that were solved analytically for the simple geometries.
Another approach would be to take some observations from the
simple solutions, and to generalize them without explicitly
calculating the electric fields at every point in an ion
source. Both approaches will be discussed here, with the
latter generalization being pursued first.
Unipolar Charge Drift Formula
The solution to the cylindrical model has been extended
into a form that can be applied to more general geometries.
The unipolar charge drift formula,
I
P (t)
I
eO
(t-to) '
provides space charge densities, for unipolar currents,
without requiring any explicit knowledge of the source
(16)
39
geometry. Here p(t) is space charge density as a function of
time, t. pQ and t0 are the initial space charge.and time,
respectively. This equation is useful for. estimating the
space charge effect in almost any system. The natural
tendency of ions is to spread out, by nature of their
coulombic repulsions. It must be cautioned that, although
the formula is geometry independent, it is dependent on the
ion flow being unipolar.
All that is needed to apply this formula is an initial
space charge and the time required to travel from the
initial point to the point in question. Regardless of the
application, the formula will describe the gradual
broadening of a cloud of charge, as it drifts.11 The
application of the formula to three different initial space
charge densities is shown in Figure 15. Interestingly, the
ion densities converge at the long drift times.
An important consequence of the unipolar drift formula
is that it is impossible to "squeeze" space charge closer
together. A source cannot be constructed that will increase
the concentration of ions in a dilute volume of space
charge. The futility of efforts to "focus" ions is apparent
in the simplicity of this equation. The equation is totally
independent of applied fields. According to the formula, the
only way to increase the sampled concentration of ions is to
reduce the time between the creation of the ion density and
their sampling. This situation has been reinforced by the
Timc(s)
Figure 15.
Ion density versus time in a unipolar ion source
for three different values of p0, the ion density
at t =0, calculated for the unipolar drift formula.
41
use of a simulation.26'27 In this simulation, a high
potential was applied to the walls of a source, in an
attempt to "squeeze" ions together. Figures 16, 17, 18 and
19 show the field and ion density contours for the
simulation with, and without, the application of the
focussing potential. While the ion density contour lines
seem to be concentrated toward the sampling aperture, the
sampled ion density actually goes down. The applied
potential has actually slowed ion flow, and consequently
increased the time for the ion density to spread. Further
indication of this reduction in sampled ion density is given
in Figures 20 and 21, where the focussed and unfocussed ion
densities and ion currents are given.
42
=5 0.02
5000 V
l * 1 0 - ‘ C/m
N e e d le
Figure 16. Case I: Potential and field contour.
5000 v
5000 V
0.045
5 0 0 0 V.
0.035
"
0.03
0.025
■4 0.02
5000 V
1 * 1 0 * C /m ’ O1
z axil (meters)
Figure 17.
Case 2: Potential and field contour.
43
~
0.02
Figure 18.
Case I: Charge density contour.
H 0.03
~
0.02
z axis (meters)
Figure 19.
Case 2: Charge density contour.
44
lrIO4
Figure 20.
Current density vs. radius.
1IO4
C a se I
C aee 2
Figure 21.
Charge density vs. radius.
45
Calculation of Absolute Sensitivities
The spherical ion source model is adequate for rough
estimates of residence times for ions that would allow
kinetically controlled processes to occur during ion
transport. An example of this utility is the use of the
unipolar formula for the evaluation of kinetically
controlled ion molecule reactions.
One can assume a model for the kinetically controlled
ion source. The ion supply surface supplies only one type of
ion, a reagent ion, R+. This ion can react with a molecule,
P. The reaction between these two species is second order.
The molecule, P, is the analyte molecule of interest, and
therefore, the ion current of the analyte ion, P+, can be
modelled on the basis of the transport and reaction
processes occurring in the source. The second order reaction
is,
R ++P -> R+ P +
(I?)
with the reaction having a second order rate constant k 1.
Because the source is at high pressure, the ion
densities will be small, as compared to the neutral
densities. This condition allows the kinetics to be
evaluated as being pseudo-first order. Putting the kinetics
in terms of the concentration of the reagent ion
concentration gives
46
dt
7Ci[i?+] [P]#
(18)
or in terms of pseudo-first order kinetics
d [ R +]
* ZilX] ,
dt
(19)
where the pseudo rate constant is
JrVk1CP] .
- (2 0 )
Rearranging the pseudo-first order rate equation into
an integratable form gives
(2 1 )
[£ ]
which upon integration gives
l n [ R +] - l n [ R +]0= - k \ ( t - t 0) .
(2 2 )
Assuming t0=O gives
ln[R+]=-k/1t+ln[R+]0
(23)
or
[R+]=e(
(24)
47
Upon substituting for the pseudo-first order rate constant,
the reagent ion concentration is given as
[iT] = [i?+]0e ^ klCP]t).
(25)
The current, I, exiting an orifice in the source would
be the product of the space charge density, p, and the
ventilation rate, S , at the orifice aperture
I=P-S.
(26)
The current, I, is composed of charged species passing
through the orifice. In this case it would be composed of
ions R+ and P+. To produce an expression for the analyte ion
current would require a ratio of analyte ion density to
total ion density. This ratio, multiplied by the total
current at the orifice, would give the analyte current at
the orifice. The analyte current, Ip+, would be:
I +=P-S- analY te lon concentration
p
total ion concentration
Here, the total ion concentration would be the sum of the
reagent and analyte ion concentrations, which would,
in
turn, be equal to the initial ion concentration:
total ion concentration= [R+]+ [P+]= [R+^0.
The analyte ion concentration would be the reagent ion
(28)
48
concentration subtracted from the initial reagent ion
concentration:
analyte ion concentration= [P+]= [R+]0-[R+] .
(29)
From the above kinetic argument, the ratio of analyte ion
concentration to total ion concentration would be:
analyte ion concentration=
[P+]
^on concentration
[p+]+ [p+]
[P+]0-[P+]
(31)
[i?+]0
= l-e
-k,[P]t
[R+Io-LR+10e~k'mt
LR+Io
Substitution of this expression into the equation for
analyte ion concentration gives:
(32)
(33)
49
I p*=P ' S ’
(34)
This equation can be used to model the analyte ion current
in a spherical ion source, that is operated with conditions
similar to typical corona-API sources. The implications of
this relationship are illustrated in Figure 22, where
analyte ion current is plotted against ion drift time. The
curves correspond to a variety of initial reagent ion and
analyte molecule concentrations. The maxima in the curves
result from two competing processes. First, the ion-molecule
reaction will produce analyte ion concentrations that will
increase with time. Second, the space charge expansion will
cause the total ion concentration, and thus the analyte ion
concentration, to decrease. The canceling effect, ,that these
processes have on each other, produces the broad maxima
found in Figure 22. For curve e, in Figure 22, a 4mm depth
of a curtain gas was included. The other conditions were the
same as in curve c. The curtain gas is a current of gas
flowing in front of the sampling aperture. It is often
composed of dry nitrogen, and serves to decluster the
analyte molecules as they exit the ion source. Here, the
curtain gas serves to end the region where reactivity is
possible, but the unipolar expansion will continue as the
ions continue to drift toward the orifice. The x-coordinate
for distance was calculated from equation 13, the equation
50
r l (cm)
Io n
C u rre n t
(C o u n ts /s )
0.01
10-'
10"'
IO"'
T i m e (s)
Figure 22.
Kinetically influenced corona API ion currents:
analyte ion intensities in a corona API source
versus ion drift t i m e . The upper x-axis shows
needle to orifice distance for a discharge current
of 3(.lA. The curves correspond to: a.) p0=l Oiz
ions/cm3, c„ (analyte concentration)=100 ppb; b .)
P0=IO10 ions/cm3, c.= 100 ppb; c.) po=10lz ions/cm3,
C1=I ppb; d .) Po=IO10 ions/cm3, C1=IOO ppb; and e .)
same as c.) but with 4mm of curtain gas in front
of the orifice.
51
for ion residence time in the limiting condition of space
charge domination. A current of 3AtA was used for the
calculation.
This methodology can be compared to experimental
results for such a system. Figure 23 shows calculated
analyte intensities as a function of distance between the
source needle and the sampling orifice. The analyte,
pyridine, was assumed to have a rate constant, k=2 x IO"9
cm3/molec/s in ambient air. The pyridine was present at a
concentration of 0.1 ppb. Included in the model was a 4 mm
depth of a curtain gas. Figure 24 shows the corresponding
experimental result.28
This model gives insight into the factors determining
the absolute ion current at a sampling orifice. Prior
to this work there was little understanding of these
factors.
Numeric Solutions
In mathematics, alternatives to analytic methods for
solving differential equations have been developed. The
field of numerical methods of differential equations has
provided methodologies for solving different classes of
differential equations. These methods, in general, involve a
discretization of a mathematical region. The differential
equations are then mapped over the discretized region. This
necessitates the transformation of the differential
needle-orifice distance (mm)
Figure 23.
Calculated corona API ion currents: Based upon 0.1
ppb analyte
(pyridine)
concentration,
4mm of
curtain gas and P0=IO12 ions/cm3.
H 3O+(H2O)
f
PyrH
needle-orifice distance (mm)
Figure 24.
Experimental corona API ion currents: detailed in
Reference (28).
54
equations into difference equations. It is the difference
equations that are mapped over the grid. A variety of
methods, including the finite difference and the finite
element methods, have been developed to obtain a solution,
based on the mapped difference equations. The obtained
solution can be used to remap the difference equations on
the grid. Typically, an iterative scheme is used to provide
a "best" solution to the differential equations, based on
the repeatedly remapped difference equations.29
The rigorous numerical solution of a space charge
system requires the use of such techniques. These methods
require much computational time. The general concept of this
application to space charge dominated systems has been
discussed by Felici and Atten.17'30 Specific use of the
technique to m o d e l .space charge dominated systems has been
made by Weber,31 McDonald,18 Leutert and Bohlen.32 Much of
the work done in this area was done to model the operation
of electrostatic precipitators. A good general work in this
area, describing the space charge problems in electrostatic
precipitators and how they relate to' the general body of
knowledge in electrostatics, has been written by Hinds.33
In general, the work done to model precipitators involves
geometries that are either cylindrical or planar. Work in
the cylindrical geometries has been done mostly by using the
analytical solution to the concentric cylindrical geometry.
The planar geometry has required the use of finite
55
difference analysis of the parallel planar system. In the
finite difference analysis, the geometry has been reduced to
two dimensions by assuming the planar system to possess
certain symmetry properties.32 Application of these
numerical methods, to the modelling of space charge
dominated mass spectrometry sources, has been made by
Vogel.25,26
Simulation Method
In addition to the actual numerical solving of the
differential equations, another approach was used in the
study of these systems. The previously discussed numerical
methods for solving the differential equations,
in addition
to requiring large amounts of computer time, necessitate the
stipulation of boundary conditions that are not available
from the analysis of experimental conditions used for ion
sources. An alternate simulation method was used to model
these systems. In this method, the flow of ions is simulated
by modelling the flow of ions through a grid of discrete
cells.
The computer program written to allow a user to perform
the simulation uses simple equations, previously described,
to model the ion's movement as a response to the fields in
the ion source. The coulombic forces between the source
components and the drifting ions, as well as between the
drifting ions and their own space charge field, are
56
considered. An initial field is set up, over the source, and
an initial space charge in each of the source's discretized
cells is assumed. Then, through a series of "time" steps, a
current is introduced into the ion supply area of the
source. The amount of charge introduced is appropriate to
the length of the time step and the assumed current. As the
simulation progresses, the conditions in each cell converge
upon a steady state. For a source under simulation the
steady state is assumed to give the desired information.
Simulation in General Geometries
With a simulation strategy in hand, attention can be
turned to geometries out of the realm of the analytic
solutions. These geometries include sources that are
actually of more than one dimension. Interesting geometries
that could be thought of as being mathematically two
dimensional include the point to plane geometry and the
finite length cylindrical geometry. These geometries are
illustrated in Figures 25 and 26.
To discretize the source into a grid, the source must
be divided so that the source is covered with a net of
cells. The cells could be regular, such as a set of
identically sized squares. However, one could attempt to
concentrate the larger number of cells in the areas where
the largest changes in the field conditions would be
expected to occur. For the simulations, both approaches were
57
I
Figure 25. Point to plane geometry.
used. For some sources, with little need for a variable grid
or little prior knowledge about source conditions, the grid
was mapped with square or rectangular cells. Other sources,
because of prior knowledge about the source conditions, were
mapped with a parabolic grid. Figure 27 illustrates the
geometry of the parabolic grid. The two dimensional grid,
when rotated around the y-axis,
geometry.
forms a cylindrical
Each cell then becomes a ring, centered around the
y-axis. In this manner, the simulation gives the opportunity
to model a variety of sources, providing that they possess
Figure
26. "Needle in can" geometry.
59
Figure 27.
Parabolic grid.
symmetry around the axis of the cylindrical geometry. It is
assumed that the axis of the geometry is perpendicular to
the collector plate.
A separate mechanism for mapping an ion supply surface
is needed for each geometry. The program user is given the
opportunity to specify the current and details concerning
the geometry of the supply surface.
In the parabolic
geometry, the user is asked to describe the width of the ion
supply surface in terms of angular width. In the rectangular
60
grid geometry, the user is asked to describe a ring shaped
supply surface by giving the rings inner and outer radii.
An intercellular distance is required for each cell to
cell interaction. These distances are stored in an array.
Based on the intercellular distances, fields can be
calculated for each cell in the grid. These fields are a sum
of both the Laplacian and spade charge fields. The Laplacian
fields result from fields calculated from a point charge
that is placed at a specified distance behind the ion supply
surface. The fields that each cell experiences are resolved
I
into x- and y-components.
Upon the calculation of the fields for each cell, the
program can allow for the movement of charge density
throughout the grid. This movement is based on the
components of the calculated ion velocities and the
intercell distances. The fraction of the charge leaving a
cell is based on a proportion of the distance travelled by
ions in a cell to the size of the cell in which the ions are
contained. A mechanism to allow for the movement of ions,
based on space charge expansion in a single cell is
provided.
The ion movement process also allows charge density to
exit the entire grid, either by moving charge density to the
collector plate or by hitting other defined source walls.
The locations at which ion density exits the source are
recorded.
61
The charge movement process repeats for a number of
cycles. At the end of this set of cycles, the program
recalculates the field strengths on the basis of the changed,
charge distribution. At certain intervals a charge grid and
two field grids, describing, respectively, charge contained
in each cell the x- and y- components of field strength in
each cell, are saved by the computer. The progress of the
simulation can be monitored by the examination of the
calculated grids.
The "pseudo-time" intervals for charge movement, field
recalculation and the total time of program operation are
selected upon starting the simulation program. The times can
be varied to be appropriate to the initial source
parameters. Along with the time parameters, geometry type,
needle charge and various source dimensions can be specified
at the program's execution.
To enhance the applicability of the program, it was
assumed that the collector plate would be constructed of a
conducting material. A conductor, having a constant
potential at every point, requires a special treatment for
the calculation of electric fields. In this simulation the
"method of mirrors." is used. This method places an
imaginary mirror charge, on the other side of the conductor
s u r f a c e , f o r each charge. Therefore, each source component
has a corresponding "mirror" component, and each grid cell
has a corresponding "mirror" grid cell.
62
Point to Plane Geometry
For the point to plane geometry, the current intensity
on the plate is of interest. Using the two dimensional
simulation, a plot of current versus the distance from the
axis of the geometry, was drawn. This plot is shown in
Figure 28. This current distribution has previously been
observed. An empirical modelling of this distribution has
also been used by various workers.11
i
'
Electrosorav Geometry
In electrospray sources it has been noted that an
interesting current distribution exists for a geometry that,
at least at first glance, appears to look like a point to
■
plane geometry. The current distribution is shown in Figure
29.24 The contrast with the above modelled distribution is
striking. This discrepancy can be explained by an important
:
detail at the tip of the ESP needle. Instead of proceeding
;
out of the needle in a straight jet, the liquid in the ESP
can spread out in the form of droplets from the surface of a
Taylor cone. However, often this Taylor cone is distorted
into an umbrella shaped pattern. The droplets then seem to
separate from the "rim" of the umbrella. The result is a
I
ring shaped source of charged droplets, that are caused to
j
drift towards the plate. The two forms of the electrospray
flow are shown in Figures 30 and 31. The simulation of the
'I
;
"ring to plate" geometry gives an entirely different current
\
■I
3.5
E
O
o
X
<
>>
C
Z
)
C
<D
Ch
w
C
O
Ui
O
U
Figure 28.
Calculated point to plane current distribution as
a function of radius. For the calculation, the
radius of the source is 4 cm, the length of the
source is 5 cm, the current is I x i o ‘8 A, the
needle charge is I x i o '6 C.
1.8
E
E
o
x
<
03
(U
W
m
C
(U
u*
U
Radius (cm)
Figure 29.
Experimental electrospray collector plate current
distribution as a function of radius: detailed in
Reference (25).
Figure 30. Spray pattern from a Taylor cone
Figure
31. Spray pattern from the "umbrella" mod e .
I
67
distribution. This distribution, shown in Figure 32, is seen
to closely correspond to the experimentally observed
distribution.
The contrast between the two current distributions is a
direct consequence of the space charge effects, and cannot
be explained in terms of Laplacian fields alone. The
distribution can also be, at least qualitatively, described
by using the unipolar space charge formula. The field and
space charge density distributions for the electrospray
source are shown in Figure 33.
API Geometry
Another interesting geometry, accessible with the two
dimensional simulation,
is the finite length cylindrical
geometry. This geometry can be used to model the typical API
source. This simulation is distinguished from the point to
plane geometry by having defined walls, in addition to
having a collector plate. Using standard operating
parameters, a simulation can be used to model such things as
current distributions on the source walls. Perhaps more
importantly, it can be used to determine the degree of
approximation that is made, when the spherical geometry,
with its analytic solution,
is used to model the API source.
Results of the API simulation are shown in Figure 34 and 35.
4.8 X !O'9
Z--S
CO
£
£
U
CZD
C
CD
"O
CD
bfi
u
03
-O
CO
U
Radius (cm)
Figure 32. Calculated electrospray current distribution, at
the collector plate, based upon ion densities at
the perimeter of the source. For the calculation,
the radius of the source is 10 cm, the length of
the source is 2.5 cm, the electrospray current is
I x IO'5 A, the electrospray needle charge is I x
IO'6 C, and the electrospray current is injected
into the source in a ring with inner and outer
radii being 1.0 and 2.0 cm, respectively.
Figure 33.
Field and space charge density contour
for the
electrospray
simulation.
The
calculation
parameters are the same as specified in Figure 32.
E
c
U
C
<D
Q
C
(L)
o
U
Radius (cm)
Figure 34.
Calculated corona API current distribution at the
collector plate. For the calculation, the radius
of the source is 10 cm, the length of the source
is 2.5 cm, the corona-API current is I x io'9 A
the corona needle charge is I x io'8 C, and the
corona current is injected into the source in a
circle with a radius of 2.0 cm.
Figure 35.
Field and space charge density contour for the
corona API simulation. The calculation parameters
are the same as specified in Figure 34.
TIME DEPENDENT CASE
The space charge model can be extended beyond steady
state systems. The unipolar flow of ions in a dynamic system
could be strongly influenced by space charge effects. An
example of this type of system should demonstrate the
utility of such an approach.
For this example, a cylindrical apparatus could be
used. This cylinder would be constructed with conditions so
that the Laplacian field strength would be constant
throughout its entire length. This apparatus could be
thought to represent, in a crude way, the ion tubes that are
now used in many electrospray sources. The electrospray
sources introduced by Fenn and Chait utilize ion tubes to
transport ions from one region of the ion source to another.
Figure 36 shows a schematic of such a source. A variety of
materials have been successfully utilized for the ion
tubes.7,34 Moreover, this apparatus can model drift tubes
that are in use for ion mobility spectrometry (IMS) .35,36
To
model ion transport in these ion tubes,- it is
convenient to imagine operating the system in pulse mode. To
introduce ions into this system, ions would be filled behind
a "gate" that would be a distance, a, from the source end of
the tube. From the source plate to the gate, the ion
r
Electrostatic
len ses
Cylindrical
Quadru pole
m ass spectrometer
electrode
Capillary
N eedle
Liquid
sam ple
Skimmer
First pumping
stage
Figure 36.
Second pumping
stage
Ion tube in the Fenn electrospray source.
74
density, yO0, would be constant. All ions in this region
would be of the same mobility and charge. Between the gate
and the collector plate, the ion density would be zero. The
model system is illustrated in Figure 37.
At some time, t0, the gate would be 'opened7 allowing
the drift of the ions toward the collector plate. The
velocity of the ion drift would be dependent on the field
strength that each respective ion would experience. This
field, as before, would be dependent on both the Laplacian
field and the space charge field. Whether the space charge
field would be important would depend on the initial space
charge behind the gate.
Simulation With One Type of Ion
The ion tube is treated here by using the infinite
parallel plate geometry. This treatment is based on the
assumption that both systems have constant Laplacian field
strength throughout their entire lengths. A computer model
of this system was constructed. In this model, the length of
the drift tube was divided into cells of equal length. The
number of cells could be chosen oh the basis of the desired
accuracy of the model. The cells were each filled with ion
density appropriate to the initial conditions. At the each
end of the tube was a plate of a certain assigned charge.
The model then went through a series of time steps. At each
time step the actual field strength in each cell was
<-------C
a
— "H
-N
Figure 37. Drift tube schematic.
76
calculated. This field strength, in turn, gave a drift
velocity for the ions in each cell. The length of the
timestep multiplied by the length of the timestep would give
an indication of the displacement of ion density out of each
cell. At the end of each timestep, a new ion density
distribution, appropriate to the ion displacements, was
constructed. To record the simulation of the ion flow, the
ion collection current, at the collector plate, was recorded
as a function of time. This collection current is shown in
Figure 38.
Simulation With Two Types of Ions
As a further example of the utility of this approach, a
similar program was written to allow the presence of two
types of ions in the tube. The construction of the model was
the same as in the preceding example, except for the need to
account for two separate ion densities in each cell. The two
ion types were assumed to have different ion mobilities. For
the simulation, one ion type, ion A, was given an ionic
mobility, K = A m2/V-s. The second ion type, ion B, was given
an ion mobility, Kg=2 m2/V-s.
The collection current for this simulation is shown in
Figure 39. The current profile, as a function of time,
displays the modification of chromatography, based on ionic
mobilities, by space charge. This chromatography,
ion
mobility spectrometry (IMS), is based on transport in the
CoUector Current as a function of time after pulse
time(sec.)
Figure 38.
One component ion drift current versus time. The
ion mobility for the drifting ion is 2 x io'4
m2/V*s. The field strength in the drift region is
0.5 V/m. The pulse contains 5 x io*7 ions.
Collector Current as a function of time after pulse
time(sec.)
Figure 39.
Two component ion drift current versus time. The
field strength in the drift region is 0.05 V/m.
The pulse contains 5 x io-6 ions of each drifting
species.
y
79
ion tube being a function of ion mobility.34
In the case of the ion tube in the electrospray source,
the actual retarding effects in the ion flow are not well
understood. The importance, in the ion transport, of
adsorption of the ions on the ion tubes walls, the degree of
hydration of the ions and fluid flow considerations are not
well understood and are presently the subject of
experimental study. Furthermore,
in typical uses of these
ion tubes, the effect of the ion transport through the tube
may often be overwhelmed by the events occurring at both the
entrance and the exit to the tube. The fluid and coulombic
influences on ion flow would seem to be difficult to
monitor, or model,
in these regions. Clearly, additional
work will be necessary in this area.
80
SUMMARY
In this work, ion transport in unipolar mass
spectrometry ion sources like corona API and electrospray
has been studied theoretically. Unlike previous work in this
area, the study has been done with the consideration of
space charge effects. Analytic solutions of the space charge
problem have been obtained for simple ion geometries,
including the cases of infinite parallel planes, concentric
cylinders of infinite length, and concentric spheres. These
analytic solutions allow, for their respective geometries,
the calculation of electric field, potential,
ion density
distributions, and ion residence times. It is shown that for
typical operating conditions, the minimum potential required
to overcome the space charge effect in corona API, or
i
'
electrospray ion sources, constitutes a dominant or
significant fraction of the total applied voltage. Further,
the electric field, in the region of the ion sampling
'i
orifice and the ion residence time in the ion source are
determined mainly by the space charge. Extending the
approach to more general geometries, use of the unipolar
formula, numerical methods of differential equations and a
simulation approach was made. With the unipolar formula,
absolute sensitivities of corona API ion sources were
81
calculated. Also, general geometries were modelled by a
simulation calculation. The calculation was based on a
computer program written to model ion flow in various ion
sources having different geometries. Finally, the space
charge influenced ion drift in a drift tube-type apparatus
was modelled, as a function of time.
The analyses, for the first time, provide methods for
evaluating the ion sampling efficiency in high pressure,
high current ion sources such as corona API and
electrospray. Also, presented for the first time is a method
for evaluating absolute sensitivities in the corona API
source.
REFERENCES CITED
83
1.
Dempster, W., "A New Method of Positive Ray Analysis,"
Ph v s . R e v . . 1 1 . 1918, 316-325.
2.
Bleakney, W. "A New Method of Positive Ray Analysis and
its Application to the Measurement of Ionization Potentials in
Mercury Vapor," Ph v s . R e v . . 34, 1929, 157-160.
3.
Munson, M.S.B., Field, F .H ., "Chemical Ionization Mass
Spectrometry," J. Am. Chem. So c . . 88., 1966, 2621-2630.
4.
Barber, M.N., Bordoli, R.S., Sedgick, R . D . , Tyler, A . N . ,
"Fast Atom Bombardment of Solids (F.A.B.): A New Ion Source
for Mass Spectrometry," J. Chem. Soc.,, Chem. Commun. . 7., 1981,
325-327.
5.
Mitchum, R . K . , Korfmacher, W . A . , "Atmospheric Pressure
Ionization Mass Spectrometry," Anal. Chem. . 55., 1983, 1485A1496A.
6.
Vestal, M.L., "Studies of Ionization Mechanisms Involved
in Thermospray LC-MS," J. Mass Spectrom.. 4 6 . 1983, 193-196.
7. . Fenn, J.B., Mann, M., Men g , C.K., Wong, S.F., Whitehouse,
C.M., "Electrospray Ionization for Mass Spectrometry of Large
Biomolecules," Science. 246. 1989, 64-71.
8.
Chapman,
S.,
"Carrier
Mobility
Spectra
of
Electrified Liquids," Phvs. Rev. . 52., 1939, 184-190.
Spray
9.
Chapman, S., "The Magnitude of the Corona Point Discharge
Current," Journal of Atmospheric Sciences. 34., 1977, 18011809.
10. Loeb, L.B. Basic Processes of Gaseous Electronics.
University of California Press, Berkeley, 1955.
11.
Sigmond, R . S . , "Electrical Coronas," Electrical Breakdown
of Gases. Meek, J.M. and Craggs, J.D. (ed.) Wiley, New York,
1978.
12.
Townsend, J.S., "The Potentials Required to Maintain
Currents Between Coaxial Cylinders," Phil. M a g . . 2 8 . 1914, 8390.
13. McDaniel, E.W., Mason, E .A ., The Mobility and Diffusion
of Ions in Gases. John Wiley and Sons, New York, 1973.
14. Abou-Seada,
M.S.,
Nasser,
E.,
"Digital
Computer
Calculation of the Electric Potential and Field of a Rod Gap,"
Proceedings of the IEEE. 5 6 . 1968, 813-820.
84
15.
Child, C.D. , "Discharge from Hot CaO," P h v s . Rev. . 32.,
1908, 492-511.
16.
Langmuir, I., "The Effect of Space Charge and Residual
Gases on Thermionic Currents in High Vacuum," P h v s . Rev. . 2.,
1913, 450-486.
17.
Felici, N., "Recent Advances in the Analysis of D.C.
Ionised Electric Fields," Dir. Curr. . 8., 1963, 252-260, 278287.
18. McDonald, J.R, Smith, W . B . , Spencer, H. W . , Sparks, L.E.,
"A Mathematical Model for Calculating Electrical Conditions in
Wire-Duct Precipitation Devices," J. A p p I . Phv s . , 4 8 . 1977,
2231-2243.
19.
Shahin, M.M.,
"Mass-Spectrometric Studies of Corona
Discharges in Air at Atmospheric Pressures." J. Che m . Phvs.,
45, 1966, 2600-2605.
20.
Gobby, P.L., Grimsrud, E.P., Warden, S.W., "Improved
Model of the Pulsed Electron Capture Detector," A n a l . Chem..
52, 1980, 473-482.
21.
Chapman, S ., Discharge of Corona Current from Points on
Aircraft or on the Ground. Cornell Aeronautical Laboratory,
Buffalo, 1955.
22.
Chapman, S., Corona Discharge form an Isolated Point in
Wind, Rep. No. 161, Cornell Aeronautical Laboratory, Buffalo,
1967.
23. Rees, J.A., "Fundamental Processes in the Electrical
Breakdown of Gases," Electrical Breakdown of G a s e s . Meek, J.M.
and Craggs, J.D. (e d .) , Wiley, New York, 1978..
24.
Iribarne, J.B, Thomson, B.A., "On the Evaporation of
Small Ions from Charged Droplets," J . Chem♦ P h v s .. 6 4 . 1976,
2287-2294.
25.
Ikonomou,
M.G.,
Blades,
A.T.,
Kebarle,
P.,
"Investigations of the Electrospray Interface for Liquid
Chromatography/Mass Spectrometry, " Anal. Chem., 62 , 1990, 957967 .
26.
Vogel, C., Busman, M., Sunner, J., in preparation.
27.
Busman,
M.,
Sunner, J.,
Vogel,
C.,
"Space-Charge
Dominated Mass Spectrometry Ion Sources:
Modelling and
Sensitivity," J. Am. Soc. Mass Spectrom. . 2., 1991, 1-10.
85
28.
Sunner, J., Nicol, G., Kebarle, P., "Factors Determining
the Relative Sensitivity of Analytes in Positive Mode
Atmospheric Pressure Ionization Mass Spectometry," A n a l .
Chem.. 6 0 . 1988, 1300-1307.
29.
Botha, J.F., Pinder, G.F., Fundamental Concepts in the
Numerical
Solution
of
Differential
Equations.
WileyInterscience, New York, 1983.
30. Atten, P. , "Methode Generale de Resolution du Problems du
Champ Electrique Modifie par Une Charge D zEspaCe Unipolaire
Injectee." Rev. Gen, de Electr.. 8 3 . 1974, 143-153.
31. Weber, C., "Numerical Solution of Laplace's and Poisson's
Equations and the Calculation of Electron Trajectories and
Electron Beams," Chapter 1.2, Focusing of Charged Particles.
Vol. I . Septier, A. (ed.), Academic Press, New York, 1967.
32.
Leutert, G., Bohlen, B., "Der Raumliche
Elektrischer Feldstarke und Raumladungsdichte
Elektrodefilter," Staub. 3 2 . 1972, 297-301.
Verlauf von
im Platten-
33. Hinds, W.C., Aerosol Technology. Properties. Behavior,
and Measurement of Airborne Particles. John Wiley and Sons,
New York, 1982.
34.
Chowdhury, S.K., Katta, V., Chait, B.T., "An Electrospray
Ionization MS with New Features for Protein Analysis,"
Presented at the 3 8th ASMS Conference on Mass Spectrometry and
Allied Topics, June 3-8, 1990.
35. Revercomb, H.E.,
Mason,
E.A.,
"Theory
of
Plasma
Chromatography / Gaseous Electrophoresis - A Review," Anal.
Chem.. 47, 1975, 970-983.
36.
Cohen, M.J., Karasek, F.W., "Plasma Chromatography-A New
Dimension for Gas Chromatography and Mass Spectrometry," J.
Chromatogr. Sci., 8, 1970, 330-337.
86
APPENDICES
87
APPENDIX A
ANALYTIC SOLUTIONS
88
Derivations of field strength expressions for planar,
cylindrical and spherical geometries are shown.
Planar geometry
The ion current .density, J, is
J = P ’V,
(35)
where v is the ion density. Also,
i=J'A,
(36)
where i is the ion current and A is the unit area. So,
i=J.
(37)
V=K 'E.
(38)
The ion velocity, v, is
The space charge density, p, becomes
p =- =-±- .
v K 'E
(39)
The Poisson equation in the planar geometry is
VzV=-L
(40)
eo
d dV
~dx ~dx
(41)
_ dE
~dx *
(42)
89
This equation, with the expression for space charge.
p0, can be integrated to give E .
dE= R- dx= — -— dx
e0
E0X-E
(43)
E
(44)
TI/2
E= (2i X+E0Z
G0K
(45)
E 2= — X+E02
G0K
(46)
Using the relation
P = K -E
/
the space charge density is given by
(47)
90
2i
HI/2
^ +-E02
(48)
Cylindrical Geometry
The ion current density, J, is
J=p*v
(49)
and the ion current, i, is
(50)
where A is the surface are of a unit length of the cylinder,
with a radius, r. The surface area, A, is given by
A=2nr.
(51)
V=K-E
(52)
Using,
for ion velocity, the space charge density, p, becomes
P=
2nrKE
(S3)
The Poisson equation for a cylindrically symmetric geometry
is
V 2V = J L
eo
(54)
91
i a sr.
r dr
(55)
(56)
Using the above expression for space charge density, p, one
obtains
— — (rE) = ___ ____,
r dr
2TrrzcSen
(57)
or upon rearrangement
(rS)£
(rS)= 2 ^ w r ,
(58)
This equation can be integrated to give an expression for
field strength, E.
I.
(rE)d(rE)
(rE)
i
rdr
2nKe - J
(59)
1 .rz
(60)
T0E0 STTZfe0
(r2-r02)
(61)
(r2-r02)+T02S02
(62)
ITTKec
r 2E 2=
STTZfe0
r
92
i
r>2_ rO^
r‘
(63)
2TZVCe0 rQ2 2TTVCe0
7/2
r2-r02+E02
(64)
2TTKe0r0d
Using the relation
(65)
2nrKE'
the space charge density is then given by
> 1/2
2 Trr0ZC 2 Trzce0T 02
(r2-r02)+E02
(6 6 )
Spherical geometry
The ion current density, J, is
and the ion current,
J = P 'V,
(67)
I = J ttA l
(68)
i, is
where the area, A, is the surface area of a sphere of
radius, r,
93
A=Airr2
(69)
V=K -E.
(70)
The ion velocity, v, is
The space charge density, p, can then be given as
(71)
Anr2KE
The Poisson equation for a spherically symmetric
geometry is
V 2V=JL
I
d .2 dV
dr
(72)
(73)
With the use of the above expression for space charge
density, p, one obtains
<75)
This equation can be integrated to give an expression
for E.
d (r2E)
Ane0K E c
(76)
94
(r2E) d (T2E ) = __i__r 2dr
47re0zc
(77)
T2E
J (r2S) d (r2S) =
,1
—
fr2c?z-
(78)
472'E0Zf J
ro
1 (T2E )2 r I=
2
P02E0
— I r 3r|
^ n e 0K
3
(79)
r0
T ltE 2- T ^ E 2= - J - (T 3- T 03)
(80)
Z-4S
4rf 6ii ^ (r3-r "3)+r»<£»2
(81)
Sne0K
-
Z 3 _ ^O3
fro I
k 2+
1
Z
0
S tte 0ZC Z 4 Z 4
(82)
Il/2
(83)
T 3- T 03 W E 02
Sne0Kr011
Then using
(84)
Znr2KE
the space charge density can be given as
I
i
4 TTE0Z 02ZC
STre0Zer0
-
(Z3-Z03)+S02
1/2
(85)
95
APPENDIX B
DERIVATIONS FOR LIMITING CONDITIONS.
96
Derivations of expressions for ion residence time, t
sampling ion density, Psamp, and applied potential, V0, at the
limit of space charge domination and r0« r
in a spherically
symmetric system are shown.
Ion residence time, t
The expression for field strength, E , in spherical
geometry is
1/2
i
E=
r
r 3 _ r 03
? 2+ ^
0 Sne^K r O4 r04
(8 6 )
Upon application of the limit E0>0,
T1/2
i
1 _ r O3 I
Sne0K r r 4
(87)
Now, if r0« r , the field strength, E, becomes
i
111/2
Sne0K r
(88 )
The velocity, v, of ions in the source is given by the drift
equation.
/•7>•
V=K-E=^E1.
dt
Substituting the above relation for E , into the drift
equation gives
(89)
97
dr
K ' i
dt
Sne0Kr
67re0rJ
(90)
or
'67repT/2
Ki
r'1 dr.
(91)
Upon integration, one obtains
(6ne0y/z rI
Jr172CJr
Ki
I
(92)
'67re0W 2 2
Cr1372- ^ 372)
Ki
I"
(93)
'Srre0TH/2 2 „ 3/2
Ki
3r1
(94)
or, if r » r n.
tres
STretJr13
3/ci
(95)
Sampling ion density, o.
An expression for ion current density is
J=KpE.
(96)
At the exterior boundary of the spherical ion source, the
ion density, p^, is
U1
P r /CE1 /
(97)
98
where J1 and E1 are, respectively, the ion current density
and field strength at the exterior surface of the ion
source.
The ion current density, J 1, is given by
(98)
47rr1
So, the ion density, P1, will become
1(1/2
'Gne0K
(99)
4TTT12
-j—
3ie0
r1
1/2
(100)
STrzcr13
Applied potential. V0
Again, the field strength, E, is
I T/2 dV
Gne0K r
dr
(101)
Upon integration, an expression for applied potential is
produced:
I
1 T /2
Gne0K r
I 7/2
~2 (rS/z-r0y2)
Gne0K
(102 )
(103)
APPENDIX C
COMPUTER PROGRAMS
100
Figure 40. SPACE.BAS: A program, written in VAX BASIC
Version 3.3, used for the two dimensional modelling of ion
sources.
i*
I*
!*
!*
!*
I*
!*
!*
!*
I*
I*
!*
!*
S P A C E .BAS
A s i m u l a t i o n to m o d e l a t w o - d i m e n s i o n a l
ap a c e charge p r o b l e m w i t h c y l i n d r i c a l geometry.
W R I T T E N BY DR. J A N S U N N E R
DEPT. OF C H E M I S T R Y
MONTANA STATE UNIVERSTIY
AND
MARK BUSMAN
DEPT. OF C H E M I S T R Y
MONTANA STATE UNIVERSTIY
1 *****»*****»**»*«*******»**»»»«*****************,
1*************»*****************»*****»**»***«**»*******»«**«»**»**»***«*»*»«,»»
t*
I*
!*
!*
!*
!*
I*
I*
I*
I*
!*
!*
!*
!*
I*
!*
I*
I*
I*
I*
I*
!*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
I*
L I S T OF V A R I A B L E S
CONSTANTS:
Z K A P P A : Ion m o b i l i t y .
EPSO:
Permitivity
C O U L C : I / <4*pi*EPS0)
TIME:
( s e conds)
TTOT:
M a x i m u m for total e l a p s e d time
T S T E P : Movement timestep
T S N A P : T i m e s t e p b e t w e e n " s napshots"
TE:
Tim e step b e t w e e n f i e l d calculations
T:
T i m e cou n t e r for e l a p s e d t ime
T2:
T i m e cou n t e r for "snap s h o t s "
TC:
T i m e cou n t e r for f i e l d r e c a l c .
T S T A R T : Ini t i a l t ime in p r o g r a m r u n .
CURRENT:
ZJO :
CURR:
QDC:
(amps)
C u r r e n t at source
G r i d of current m e a s u r e d at surface
C u r r e n t i n j e c t e d in a g i v e n cell
*
*
*
»
»
»
*
*
*
*
*
*
*
*
*
*
*
*
*
GEOMETRY:
GRID?
NCX
NCY
Q
CHARGE:
QN
QD
QNUM
QS U M N U M
I n d i c a t i o n of g e o m e t r y t y p e
No. of in c r e m e n t s in x - d i r e c t i o n
No. of i n c r ements in y - d i r e c t i o n
T a n g e n t of angle in p a r a b o l i c geo.
(coulombs)
N e e d l e charge
I n j e c t i o n char g e
Charge grid
Total charge of e n t i r e g r i d
q g a p n u m Total charge of r e d u c e d (gap) g rid
Charge grid
Q
QCH
Charge transfer matrix
DISTANCES:
(meters)
FP:
D i s t a n c e of n e e d l e c h a r g e b e h i n d tip
D:
D i s t a n c e b e t w e e n p l a n e s (SQ)
D:
P l a n e - need l e t i p d i s t a n c e (PARA)
DP:
D i s t a n c e b e t w e e n cells p a r . to plates
Y N E E C : D i s t a n c e b e t w e e n p l a t e a n d n e e d l e tip
RDIS:
M a x i m u m charge input r a d i u s
R O D I S : M i n i m u m charge in p u t radi u s
R A D M I N : Min. input rad. at r e s o l . of g r i d
R A D M A X : Max. input rad. at r e s o l . of g r i d
RD:
D i s t a n c e b e t w e e n a d j a c e n t cells
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
101
F i g u r e 40.
S P A C E .B A S - c o n t inued
!*
I*
I*
!*
I*
I*
I*
I*
!*
I*
!*
I*
I*
!•
!*
I*
I*
I*
I*
AD:
G:
COUNTERS:
NNX:
NY:
MISC:
ADIS
RDEG
FI D I S
SN:
FINPS:
TlS:
T2S:
T3S:
T4$:
T5S:
Cl:
02:
ASTEP:
D i s t a n c e b e t w e e n a d j a c e n t cells
D i s t a n c e f r o m focal p o i n t to plate
U s e d to count loops w o r k i n g in x-dir
U s e d to count loops w o r k i n g in y - d i r
M a x i m u m an g l e for c u r r e n t i njection
A n g u l a r ra n g e in p a r a b o l i c g e o m e t r y
# of f i d t i m e a t e p s p e r m o v e t i m e s t e p
R e s t a r t status i d e n t i f i e r
D u m m y v a r i a b l e s t o in p u t n o n e s s e n t i a l
s t r i n g s f rom dis k f i l e s .
T e r m s in s o l u t i o n t o q u a d r a t i c form,
u s e d for s e t t i n g p a r a b o l i c grid.
A n g u l a r i ncrement in t o r u s i n t e g r .
CNm
!Set u p arrays, constants, etc.
D I M X (15,15), Y (15,15), E X (15,15), E Y ( 1 5 , 1 5 ) , D O U B L E Q N U M (15,15)
D I M A ( I S ) , B (15) , A N G (15) , ANGR(IS)
D I M D O U B L E Q C H (15,15), D O U B L E Q D (15), D O U B L E Q D C (15)
4
D I M SING L E Q (15)
5
D I M R D (15 , 1 5 , 2 , 1 ), A D ( 15,15,2,1 ) , D R ( 1 5 , 15), DA(15,15)
D I M D O U B L E Q S U M N U M (I) , D O U B L E Q G A P N U M (I)
I
I
I N T E G E R I-N
D O U B L E A - H , O-Z
Z K A P P A - IE-4
E P S O - 8.85E-12
COULC - 1/EPS0/4/PI
10
Iinput dat a - s e c t i o n
I N P U T "DO Y O U W A N T TO R E A D F R O M
IF (FZ$-"y" O R FZ$-"n")
T HEN
P R I N T "Set 'Caps L o c k ' "
G O T O 10
E N D IF
IN P U T " F I L E N A M E ", FILES
IF FZ $ « " Y "
THEN
INPUT " S T A R T T I M E / m s ", TS
TSTART-TS*lE-3
INPUT "TOT A L T I M E / m s ", TS
TT0T-TS*lE-3
T2-0
TC-O
G O S U B 1000 ISNAPRI
ITIMESTEPS
P R I N T "TSTEP
", TSTEP
P R I N T "TSNAP
", TSNAP
P R I N T "T,E - f i e l d ", TE
P R I N T " D i s c h . c u r r . ", ZJO
I N P U T " Change (Y/N) ", AS
IF A $ - " Y "
THE N
TSTEP2-TSTEP
INPUT "TIME S T E P , C U R R / u s
"
T S T E P - T S * I E -6
I N P U T "TIME S T E P ,E - F I E L D / u s
F I L E (Y/N) ", FZS
!ASSURE SHIFT L O C K
F I N D A F I L E F O R R E S T A R T OF P R O G R A M
!MOVEMENT T I M ESTEP
!"SNAPSHOT" T I M ESTEP
!FIELD C A L C U L A T I O N T I M E S T E P
!CURRENT
!CHANGE A N Y TIM E P A R A M E T E R S ?
TS
!TIMESTEP M O D I F I C A T I O N S
", TS
102
Fi g u r e 40.
S P A C E . B A S - con t i n u e d
TE-TS*lE-6
I N P U T "TIME S N A P / u s
T S N A P asTS* IE- 6
E N D IF
G O S U B 1100
PRINT T
I
I
!
ISN A P R
", TS
READ THE SAVED FILE F O R FIELD DISTRIBUTIONS
IF T < ( T S T A R T - 1 E - 6 ) T H E N G O T O 100 E N D IF
C L O S E #1
O P E N F I L E S F O R O U T P U T AS #1
FINF$-"Y"
T2-0
T C-O
P R I N T " T S T A R T / s ", T
F O R N Y = O TO N C Y + 1
C U RR-O
F O R N N X - O TO N C X + I
Q (NNX) - C S N G (QNUM (NNX, NY) )
Q ( N N X ) = Q N U M ( N N X fNY)
P R I N T NNX, NY, Q(NNX)
NEXT NNX
F O R N N X = I TO N C X
C U R R - C U R R + E Y (NNXrN Y ) * Z K A P P A * Q (NNX)/ (D/NCY)
NEXT NNX
P R I N T " C u rrent at N Y NY; " is "; C U R R
INPUT "Pause ", P
N E X T NY
ELSE
FINP$-"N"
I
I
INPUT "Sq u a r e or P a r a b o l i c gri d s y s t e m (SQ/PARA) ", GRIDS
I DETERMIN THE G E O M E T R Y OF THE G R I D S Y S T E M
IF G R I D S-"PARA"
!PARABOLIC G E O M E T R Y
THE N
G O S U B 2 8 0 0 !GRID
ELSE
G O S U B 275 0 IF G R I D S — "SQ" ! G R I D S Q ---R E C T A N G U L A R G E O M E T R Y
E N D IF
IN P U T " P r i n t - o u t of c o o r d i n a t e s (Y/N), p r i n t e r on? ", AS
IF A S - "Y" T H E N G O S U B 2850 E N D IF GROUT
I N P U T "TIME S T E P , C U R R / u s
", TS
IInput t i m e s t e p p a r a m e t e r s
T S T E P = T S * I E -6
IN P U T "TIME S T E P ,E - F I E L D / u s ", TS
T E - TS*lE-6
INPUT "TIME S N A P / u s
", TS
T S N A P — TS* l E - 6
IN P U T " T O T A L T I M E / u S ", TS
TTO T - TS*lE-6
IN P U T " D I S C H A R G E C U R R E N T / u A
", ZJ
ZJ0=le-6*ZJ
ItO A
T-O
T2- 0
TC-O
F I N P S - "N"
G O S U B 1400 ICHINP
E N D IF
IF GRIDS = nS Q n
IINPUT P A R A M E T E R S
PERTINENT TO RECTANGULAR
THEN
!GEOMETRY
IN P U T "NE E D L E C H A R G E ", QN
I N P U T " D I S T A N C E B E H I N D P L A T E /mm
", FPO
FP-FPO/IOOO
YNEEC-D+FP
ELSE
IF G R I D S - nP A R A n
IINPUT P A R A M E T E R S
PERTINENT TO PARABOLIC
103
Figure 40. SPACE.BA S -continued
IG E O M E T R Y
I N P U T " NEEDLE C H A R G E ", QN
I N P U T "DIS T A N C E B E H I N D P L A T E /mm
FP-FP0/1000
YNEEC-D+FP
E N D IF
E N D IF
G O S U B 1500 ICHIN P 2
DNSUM=O
F O R N N X = I TO N C X
D N S U M = D N S U M + Q D (NNX)
NEXT NNX
P R I N T "ND S U M = ", D N S U M
then
", FPO
IF F I N P $ = " Y "
THEN
IF G R I D S “ "PARA" T H E N G O S U B 2800 E N D IF !GRID
IF G R I D S = nSQ" T H E N G O S U B 2750 E N D IF !GRIDSQ
G O S U B 2820 !DIST
G O S U B 2100 IEFLD
FIRSTS="!"
LT=I
G O T O 300
E N D IF
F I R S T S - nN"
G O S U B 2820 !DIST
SN-TE/TSTEP
T-TE
G O S U B 1600 !CHINJ
G O S U B 1200 ISNAPINI
G O S U B 2100 !EFLD
300
!time s t e p p i n g
T-T+TSTEP
T2 —T2+TSTEP
TC=TCtTSTEP
P R I N T "T= ", T
IF T> (TTOT+1E-10)
THEN
G O S U B 226 0 !SNAP
P R I N T "THIS IS THE END ! ! ! "
PRINT
G O T O 3000
ELSE
SN=I
G O S U B 1600 ICHINJ
IF T O (TE- (IE-10) )
THEN
TC=TC-TE
P R I N T " Q - e u m ", Q S U M M t*M
P R I N T " Q - g a p ", Q G A P N U M
IF F I R S T S - nN" T H E N GO S U B 210 0 E N D IF
F I R S T S - nN"
LT=I
E N D IF
IF T 2 > (T S N A P - (IE - 1 0 ) )
THE N
T2-T2-TSNAP
TMU=T*1E6
P R I N T "Snap output to file"
P R I N T "T/us- ", T M U
G O S U B 226 0 !SNAP
E N D IF
GO S U B 200 0 ICURR
GO S U B 1900
INEWCH
IEFLD
104
Figure 40. S P A C E .B AS-continued
E N D IF
G O T O 300
!End of M a i n P r o g r a m
SUBROUTINE SECTION
********
********
L I S T OF S U B R O U T I N E S
********
CHCPOUT
L O C A T E D B Y U S I N G G O S U B 1800
CH I N I
C h a r g e initiation
L O C A T E D B Y U S I N G G O S U B 1300
CHINJ
Inj e c t s c h a r g e s int o first cell line
L O C A T E D B Y U S I N G G O S U B 1600
CHINP
C a l c u l a t e i n j e c t e d ch a r g e s int o first cell line
L O C A T E D BY U S I N G G O S U B 1400
CHIN P 2
C o n v e r t s i n j e c t i o n cur r e n t t o i n j e c t e d c h arge
L O C A T E D B Y U S I N G G O S U B 100
CURR
Calculates charge transfer matrix
L O C A T E D B Y U S I N G G O S U B 2000
DIST
C a l c u l a t e s inte r c e l l d i s t a n c e s
L O C A T E D BY U S I N G G O S U B 2820
EFLD
Electric field calculation
L O C A T E D B Y U S I N G G O S U B 2100
GRID
G e o m e t r y input and cell c o o r d i n a t e c a l c u l a t i o n
L O C A T E D B Y U S I N G G O S U B 2800
GRIDSQ
Sam e as G R I D but for s q u a r e g r i d s y s t e m
L O C A T E D B Y U S I N G G O S U B 2750
GOUT
O u t p u t t o s c r e e n of g e o m e t r y a n d c o o r d i n a t e s
L O C A T E D B Y U S I N G G O S U B 2900
NEWCH
C a l c u l a t e s n e w charge d i s t r i b u t i o n
L O C A T E D B Y U S I N G G O S U B 1900
RINGINT
I n t e g r a t i o n of t h e c h a r g e
G O S U B 250 0
in a n y r i n g
R I N G S A M E I n t e g r a t i o n of the c h a r g e in the same r ing
as a s e l e c t e d cell
L O C A T E D BY U S I N G G O S U B 2400
RINTX
L O C A T E D B Y U S I N G G O S U B 2700
RINTY
L O C A T E D B Y U S I N G G O S U B 2600
SNAP
W r i t e s s n a p - i n f o r m a t i o n t o file
L O C A T E D BY U S I N G G O S U B 2260
105
Figure 40. SPACE.BA S -continued
S NAPINI I n i t i a t e s S N A P - f ile for output,
L O C A T E D B Y U S I N G G O S U B 1200
w r i t e s input p a r a m e t e r s
SNAPR
R e a d s e l e c t r i c f i e l d a n d c h a r g e d i s t r i b u t i o n f r o m SNAP file
L O C A T E D B Y U S I N G G O S U B 1100
SNAPRI
R e a d s d a t a - h e a d fro m S N A P - f ile
L O C A T E D B Y U S I N G G O S U B 1000
1000
* * » * * » * * » • • * * * * * * * * * * * * * * * * * * , , * * * * *
SNAPRI - s u b r o u t i n e
r e a d d a t a - h e a d fro m S N A P - f i l e
O P E N FILES F O R IN P U T A S #1
INPUT #1, GRIDS
INPUT #1, TlS
IF GRIDS- "SQ"
T HEN
P R I N T " Square c o o r d i n a
INPUT *1, D, DP, N C Y , i
ELS E
IF G R I D S = nP A R A n
THE N
INPUT #1, D, F P , N C Y , I
END Il
END IF
INPUT #1, T S N A P , TE, TSTEP
INPUT #1, ZJ0
F O R NNX-I TO N C X
INPUT #1, QDC(NNX)
NEXT NNX
RETURN
,,,,,,,
*****
!to disk
Ig e o m e t r y type
!" p o i n t - t o - p l a n e corona'
Is q u a r e g e o m e t r y
Iinput s q u a r e p a r a m e t e r s
Iinput p a r a b o l i c p a r a m e t e r s
1100
|************#************************************e**eeeeeeeeeeeeeeileielleilll
IS N A P R - su b r o u t i n e
Iinput of e l e c t r i c f i e l d d i s t r i b u t i o n f r o m SNAP file
|**********************»*********»»***********«***************»**»*************
!ERROR TRAP
O N E R R O R G O T O 1150
W H I L E 0-0
INPUT #1, T2$, T
!"T- "
P R I N T "T ", NS, T
INPUT #1, T3S
I"ELE C T R I C F I E L D A N D C H A R G E D I S T R I B U T I O N S "
I N P U T #1, T4$, Q S U M N U M
I"Q,Sum- "
INPUT #1, T5S, Q G A P N U M
I"Q,gap- "
Q S (NS)- C S N G (QSUMNUM)
Q S G (NS)- C S N G (QGAPNUM)
F O R NY- I TO N C Y
F O R NNX-I T O N C X
INPUT #1, I, J, E X ( N N X 1N Y ) , E Y ( N N X 1N Y ) , QT I
I
I
I
I
Q N U M (NNX1N Y ) - C D B L (QT)
Q N U M ( N N X 1N Y ) - Q T
P R I N T I 1J, E X ( N N X 1N Y ) , Q N U M ( N N x 1NY)
NEXT NNX
N E X T NY
NNX=O
106
Figure 40. S P A C E .BAS-c ontinued
I
I
I
I
1150
1160
F O R N Y - O TO N C Y + I
IN P U T #1, I, J, QT
QNXJM (NNX1NY) - C D B L (QT)
QNXJM (NNX1N Y ) - Q T
N E X T NY
NNX-NCX+1
FOR NY=-O TO N C Y + 1
IN P U T #1, N N X , NY, QT
QNXJM (NNX1NY) - C D B L (QT)
Q N U M ( N N X 1N Y ) - Q T
N E X T NY
N Y-O
F O R NNX-I TO N C X
I N P U T #1, N N X , NY, QT
Q N U M ( N N X 1NY)- C D B L ( Q T )
QNXJM (NNXfNY) - Q T
NEXT NNX
N Y —N C Y + I
F O R NN X = I T O N C X
IN P U T #1, N N X , NY, QT
QNXJM(NNXfN Y ) - C DBL(QT)
QNXJM(NNXfN Y ) = Q T
NEXT NNX
N E X T !WHILE
IF ERR= I l T H E N R E S U M E 1160 E L S E ON E R R O R G O T O 0
RETURN
1200
!SNAPINI - su b r o u t i n e
!i n i t i a l i z e S N A P - f ile
I********************
O P E N FI L E S F O R O U T P U T AS #1
IF GRIDS = 11PARA"
THEN
P R I N T #1, GR I D S
!Geometry t ype
P R I N T #1, " P o i n t - t o - p l a n e corona"
P R I N T #1, D;
FP;
NCY;
NCX;
RDEG
a r a m e t e rs
ELSE
IF G R I D S — "SQ"
THEN
P R I N T #1, G R I D S
P R I N T #1, " P l a n e - t o - p l a n e c orona"
P R I N T #1, D;
DP;
NCY;
NCX;
ADIS
ameters
E N D IF
E N D IF
P R I N T #1, T S N A P ;
TE;
T S T E P ; ","; T T O T
P R I N T #1, ZJ0
!Current
F O R N N X = I TO N C X
P R I N T #1, QDC(NNX)
NEXT NNX
I
C L O S E #1
RETURN
!PARA geome t r y p
ISQ g e o metry par
ITim e parameters
1300
**********
*************
ICHINI - s u b r o u t i n e
Icharge d i s t r i b u t i o n i n i t i a t e d
j******************************************************************************
INPUT "Charge on needle
QN
107
Figure 40
S P A C E .BAS-continued
P R I N T "Charge i n i t i a t i o n "
!FOR NY- I TO N C Y
!FOR NNX-=I TO N C X
!Q N U M (1,5)— ID-12
! Q N U M (3,5)- O D - I2
IQ N U M (4,5)— OD-12
!NEXT N N X
!NEXT NY
RETURN
1400
I*******************************************************,********,*.,*,,+*,,,,,
ICHINP - s u b r o u t i n e
!c a l c u l a te i n j e c t e d c h a r g e s i nto first cell line
!******************************************************************************
!
I
P R I N T " C a l c u l a t e s i n j e c t i n g c h a rges"
N=O
NY=NCY
**
CHARGE INJECTION FOR PARABOLIC GEOMETRY
**
IF G R I D S = "PARA"
THEN
INPUT " C U R R E N T I N J E C T I O N A N G L E / d e g ", A D I S I
A D I S = A D I S I / 1 80 * P I
!max an g l e for d i s c h a r g e cu r r e n t
F I D I S = I - C o s (ADIS)
I s t eradia n s /2/pi
F O R NN X = I TO N C X
IF N=I
T HEN
QDC(NNX)-O
ELSE
IF N N X - I
THEN
Al = O
ELSE
A l - A N G R ( N N X ) -AST E P R / 2
E N D IF
A 2 -ANGR(NNX)+ASTEPR/2
IF A 2 > A D I S
THEN
A 2 —A D I S
N-I
ELSE
QDC(NNX) - (COS ( A l ) - C O S (A2))/ F I D IS*ZJ0
E N D IF
E N D IF
NEXT NNX
ELSE
•*
CHARGE INJECTION F O R SQUARE GEOMETRY
**
IF G R I D S — "SQ"
THEN
INPUT " M A X I M U M C H A R G E INPUT R A D I U S / m m ", R D I S O
!Define r i n g for
R D I S - R D I S 0/1000
Ii n j e c t i o n of charge
INPUT " M I N I M U M C H A R G E INPUT R A D I U S / m m ", R 0 D I S 0
RODIS-RODISO/IOOO
T O T S = P I * (RDIS*R D I S - R 0 D I S * R 0 D I S )
F O R NNX-I TO N C X
IF N=I
THEN
QDC(NNX)-O
ELSE
IF ( N N X - 0 .5)* D P <RDIS !R a d i u s
THEN
!Total are a of i n j e c t i o n ring
insi d e of ring ' s ou t e r b oundary?
108
Figure 40. SPACE.BAS-continued
R A D M A X - (NNX-0.5) *DP
ELSE
RADMAX-RDIS
N-I
E N D IF
IF NN X - I
THEN
RADMIN-O
ELSE
R A D M I N - ( N N X - 1 .5)*DP
E N D IF
IF R A D M I N < R O D I S
!R a dius
inside of r i n g 's in n e r b o u n d a r y ?
IF R A D M A X < R 0 D I S
THEN
RADMIN-RADMAX
ELSE
RADMIN-RODIS
E N D IF
E N D IF
O D C ( N N X ) - P I * (RADMAX* R A D M A X - R A D M I N *R A D M I N )/ T O T S * Z J O
IC u r r e n t in a cell is the total i n j e c t i o n current
!multi p l i e d by the r a t i o of t h e cell's area to the
!total area of t h e i n j e c t i o n r i n g .
E N D IF
NEXT NNX
E N D IF
E N D IF
F O R N N X = I TO NCX
P R I N T QDC(NNX)
N E X T NN X
RETURN
1500
!CHINP2 - s u b r o u t i n e
!Converts i n j e c t i o n c u r r e n t t o i n j e c t e d c h arge
**********
F O R N N X - I TO N C X
Q D ( N N X ) - Q D C ( N N X ) * T STEP
!Charge i n j e c t e d in a cell is c u r r e n t
! m ultiplied b y l e n g t h of m o v e m e n t timestep.
NEXT NNX
RETURN
1600
I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * , , , , , , , , , , , , , , , , , , , , , , , , , ,
I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * , * * « * » * « * * » , * , , , „ , , , , ttt
!CHINJ - s u b r o u t i n e
Id i s c h a r g e c h arge input
F O R NNX-I TO N C X
Q N U M ( N N X 1N C Y ) - Q N U M ( N N X fN C Y ) + S N * Q D (NNX)
!Puts i n j e c t e d c h a r g e into
Ichar g e grid
NEXT NNX
RETURN
1900
I*.*
t****************
INEWCH - s u b r o u t i n e
!c a l c u l a tes n e w char g e d i s t r i b u t i o n
IAA****************************************************************************
109
Figure 40. S P A C E .BAS - continued
I
QSUMtTUM-O
QGAPNUM-0
P R I N T "New c h a r g e d i s t r i b u t i o n calcu l a t e d "
F O R N Y - O TO N C Y t l
F O R NNX-O TO N C X + I
Q N U M (NNX.NY) - Q N U M (NNX1NY) + Q C H (NNX1NY)
!Add m o v e m e n t c h a r g e to e x i s t i n g char g e
Q S U M N U M - Q S U M N U M + Q N U M (NNX, NY)
!Total ch ar g e of e n t i r e g r i d
P R I N T Q N U M ( N N X fN Y ) , Q C H ( N N X 1NY)
NEXT NNX
NEXT NY
F O R NY=I TO N C Y
F O R NN X = I TO N C X
Q G A P N U M - Q G A P N U M + Q N U M (NNXfNY)
ITotal c h a r g e of "gap" gri d
NEXT NNX
N E X T NY
!PRINT " T OTAL C H A R G E S
RETURN
", Q G A P N U M , Q S U M N U M
2000
1»** *
# »»****«*
CURR -subroutine
c a l c u l a t i o n of c h a r g e t r a n s f e r b e t w e e n cells
"*********************************«*+*****************************************
** L O C A L V A R I A B L E D E S C R I P T I O N S **
CHARGE:
QCH:
C h a r g e t r a n s f e r matr i x
QL:
C h a r g e m o v i n g to left
CR:
C h a r g e m o v i n g t o right
QU:
C h a r g e m o v i n g up
CD:
C h a r g e m o v i n g dow n
QUL:
C h a r g e m o v i n g up and to left
QUR:
C h a r g e m o v i n g up and t o right
QDL:
C h a r g e m o v i n g down and
t o left
QDR:
C h a r g e m o v i n g d o w n and
to left
C L E A V E : Total c h arge l e a v i n g a cell
Q:
C h a r g e of a cell fro m w h i c h c h a r g e is being
moved.
VELOCITY:
VR:
Radial velocity
VA:
Angular velocity
DIST A N C E :
DR:
G rid for r a dial d i s t a n c e t r a v e l l e d by a cell.
RW:
R a d i a l "width" of a cell
RC:
R a d i a l d i s t a n c e t r a v e l l e d by acell
RU:
R a d i a l d i s t a n c e t r a v e l l e d b y t h e cel l "up"
in t h e grid.
RD:
R a d i a l d i s t a n c e t r a v e l l e d b y t h e cell "down"
in the grid.
RDB:
A v e r a g e d i s t a n c e t r a v e l l e d b y a cell and
the cell "down" in the grid.
RUB:
A v e r a g e d i s t a n c e t r a v e l l e d b y a cell and
the cell "up" in the grid.
RWN:
R a d i a l d i s t a n c e t r a v e l l e d b y a cell with
c o m p e n s a t i o n for " e x p a n s i o n ."
DA:
G r i d for axial d i s t a n c e t r a v e l l e d by a cell
AW:
Ax i a l "width" of a cell
AC:
A x i a l d i s t a n c e t r a v e l l e d b y a cell
AR:
A x i a l d i s t a n c e t r a v e l l e d by t h e cell to the
HO
Figure 40. S P A C E .BAS-continued
AL:
ALB:
ARB:
AWN:
right.
A x i a l d i s t a n c e t r a v e l l e d by t h e cell to the
left.
A v e r a g e d i s t a n c e t r a v e l l e d b y a cell and
t h e cell to t h e l e f t .
A v e r a g e d i s t a n c e t r a v e l l e d b y a cell and
t h e cell t o the right.
A x i a l d i s t a n c e t r a v e l l e d b y a cell wit h
compensation for "expansion."
FIELD:
ER:
R a d i a l c o m p o n e n t of f i e l d
EA:
A n g u l a r c o m p o n e n t of field
CHARGE/FRACTIO N :
QFU:
F r a c t i o n of char g e m o v i n g up.
QFU:
F r a c t i o n of char g e m o v i n g down.
QFU:
F r a c t i o n of char g e m o v i n g to t h e r i g h t .
QFU:
F r a c t i o n of c h a r g e m o v i n g to t h e left.
QFU:
M a x i m u m f r a c t i o n of c h arge m o v i n g .
******************************************************************************
******************************************************************************
P R I N T "Charge t r a n s f e r / c u r r e n t c a l c u l a t e d "
E R A S E QCH
F O R N Y-O T O NCY+1
F O R N N X = O TO NCX+1
Q C H ( N N X fNY)-O
NEXT NNX
NEXT NY
QMAX=O
F O R NY-I TO N C Y
!choose cell
F O R NNX=I TO N C X
!c a l c u l a t e r a dial e l e c t r i c fi e l d
E R - ( - R D (NNXfN Y , 0 , 0 ) * E X (NNXfN Y ) - R D ( N N X fN Y f I , 0 ) * E Y (NNXfN Y ) ) / R D (NNXfN Y f2,0)
!calculate axial e l e c t r i c f i e l d
E A - (-RD (NNXfN Y f I, 0) *EX (NNXfN Y ) + R D (NNXfN Y f 0, 0) *EY (NNXfNY) ) /RD (NNXfN Y f2,0)
IF NNX=I T H E N E A - O E N D IF
!radial v e l o c i t y
VR=ZKAPPA*ER
Iaxial v e l o c i t y
VA-ZKAPP A*EA
Id i s t a n c e of radial t r a v e l for cell center
D R ( N N X fNY) —V R + T S T E P
Id i s t a n c e of ax i a l trav e l for cell cent e r
D A ( N N X fN Y ) - V A * TS T E P
NEXT NNX
N E X T NY
F O R NNX-I T O N C X
F O R NY-I T O N C Y I
If r a c t i o n of c h a r g e e x i t i n g left a n d right
IF NN X - I
!On edg e of g r i d
T HEN
R I - A D ( N N X fN Y f2 , I ) /2
R F - R I + D A ( N N X + 1 ,N Y ) /2
Q F R - (RF"RF/ R I / R I - 1 )
QFL-O
ELSE
!M e c h a n i s m for the axial "expa n s i o n " of a cell in response
Ito m o v e m e n t of charge in n e i g h b o r i n g cells .
A W - A D ( N N X fN Y f2 , I)
A C - D A ( N N X fNY)
IAx i a l v e l o c i t i e s
A L = D A ( N N X - I fNY)
IF N N X - N C X
IOn ou t e r e dge
THEN
Ill
Figure 40. S P A C E .BAS-continued
AR-2*AD-AL
E LSE
A R - D A (NNX+1, NY)
E N D IF
A L B - (AL+AC)/2
A R B - (AR+AC) /2
RWIT-AW-AT,B+APB
IF A L B < 0 T H E N Q F L — A L B / A W N E L S E Q F L - O E N D IF
IF A R B > 0 T H E N Q F R - A R B /A W N E L S E QF R = O E N D IF
E N D IF
!f r a c t i o n of c h a r g e e x i t i n g u p a n d down
R W - R D (NNX,N Y , 2,0)
!Distance
R C - D R ( N N X 1NY)
!Radial distance.
IF N Y = N C Y
ISame 'slice' of grid?
T HEN
!Me c h a n i s m for the radial "expansion" of a c ell in response
!to m o v e m e n t of c h arge in n e i g h b o r i n g c e l l s .
RUB-O
ELSE
R U - D R (NNX,N Y + I)
!Radial distance.
R U B - ( R U + R C ) /2
E N D IF
IF NY=I
THE N
RDB=2*RC-RU
ELSE
R D = D R (NNX,N Y - I )
!Radial distance.
R D B = (RD+RC)/2
E N D IF
RWN-RW+RUB-RDB
IF RU B > 0 T H E N Q F U = R U B / R W N E L S E QF U = O E N D IF
!Charge up
IF R D B C O T H E N Q F D — R D B / R W N E L S E Q F D - O E N D IF
!Charge d own
!testing for Q - m a x
IF Q F U > Q M A X
!Monitor f r a c t i o n of m o v e m e n t - - u p
THEN
QMAX=QFU
NID-I
II n d i c a t e t h e d i r e c t i o n of m a x i m u m m o v e m e n t
NXM=NNX
NYM-NY
E N D IF
IF Q F R > Q M A X
!Monitor f r a c t i o n of m o v e m e n t - - r i g h t
THEN
QMAX=QFR
NID=2
NXM=NNX
NYM=NY
E N D IF
IF Q F D > Q M A X
!Monitor f r a c t i o n of m o v e m e n t --down
THEN
QMAX=QFD
NID-3
NXM-NNX
NYM-NY
E N D IF
IF Q F L > Q M A X
!Monitor f r a c t i o n of m o v e m e n t - - I e ft
THEN
QMAX-QFL
NID-4
NXM=NNX
NYM-NY
E N D IF I
I
Q = C S N G ( Q N U M (NNX,N Y ) )
O = Q N U M (NNX,NY)
112
Figure 40. SPACE.BAS-continued
O L - Q F L * ( 1 -QFU-0FD)*Q
!Charge g o i n g left from cell
Q R - Q F R * ( I - Q F U - Q F D ) *Q
IC h arge g o i n g right f rom cell
Q U - Q F U * ( I - Q F L - Q F R ) *Q
IC h arge g o i n g up from cell
Q D - Q F D * ( I - Q F L - Q F R ) *Q
IChar g e g o i n g dow n from cell
QUL-QFU*QFL*Q
!Charge g o i n g up a n d left f r o m cell
QUR-QFU*QFR«Q
!Charge g o i n g up and right f r o m cell
OnL-QFD*QFL*Q
IChar g e g o i n g d own and left f r o m cell
Q D R - Q F D *QFR*Q
!Charge g o i n g u p a n d left f r o m cell
Q L E A V E (QL+QR+QU+Q D + Q U L + Q U R + Q D L + Q D R )
!Charge l e a v i n g cell
IT r a n a f e r m o v e m e n t i n f o r m a t i o n t o a " m o v e m e n t " m a t r i x
Q C H (NNX,N Y ) - Q C H ( N N X 1N Y ) - Q L E A V E
Q C H ( N N X t l 1N Y ) - Q C H ( N N X t l 1N Y ) t Q R
Q C H ( N N X - I 1N Y ) - Q C H ( N N X - I 1N Y ) tQL
Q C H (NNX,N Y t I )- Q C H (NNX, N YtlJtQU
Q C H (NNX,N Y - I )- Q C H ( N N X 1N Y - I ) tQD
Q C H ( N N X - I 1N Y - I ) - Q C H ( N N X - I 1N Y - I ) tQD L
Q C H ( N N X t I 1N Y - I ) - Q C H (N N X t l , N Y - 1 ) t Q D R
Q C H (NNX- I 1N Y t l ) - Q C H (NNX-I 1N Y t l ) tQU L
Q C H ( N N X t l 1N Y t l ) - Q C H ( N N X t l 1N Y t l ) t Q U R
N E X T NY
NEXT NNX
IF LT=I
THEN
P R I N T "Q-max ", Q M A X 1 N I D 1 N X M 1 N Y M
F O R I-I TO 100
N-I
NEXT I
LT=O
E N D IF
IF Q M A X > 0 .05
!Ia m a x i m u m m o v e m e n t over t h e aet t h r e s h o l d ?
THEN
P R I N T "WARNING, Q 1m a x - ", Q M A X 1 N I D
!Display m o v e m e n t
P R I N T "TIME ", T 1 N N X 1 NY
!warning
W A R N S - "Y"
E N D IF
RETURN
2100
|******************************************$************************,*,,,*„,**,
|A**************************************,******,******************,,,***,,*,,,,,
!EFLD - s u b r o u t i n e
!for c a l c u l a t i o n of e l e c t r i c f i e l d
I******************************************************************************
I
I
I
!returns e l e c t r i c f i e l d v e c t o r in e a c h cell
P R I N T " C a l c u l a t i o n of E-field"
F O R NY= I TO N C Y
!cell for w h i c h e l e c t r i c f i e l d is t o be calc.
F O R N N X - I TO N C X
!NY=2: N N X — 2
!PRINT "NNX, N Y ", N N X 1 N Y
EX=O
EY-O
D S - X (NNX,NY)
Ix-coordinate of cell
Y l - Y ( N N X 1NY)
! y - coordinate of cell
special t r e a t m e n t for axial c e l l s ? ? ?
F O R K Y-I TO N C Y
!ch a r g e d t o r u s t o be i n t e r a c t e d w i t h cell
F O R K X-I TO N C X
L-O
Q N U M ( K X 1K Y ) - I E - S I
IF (KX=I A N D N N X = I
D R = X ( K X 1KY)
Y 2 = Y (KX1KY)
Q = Q N U M ( K X 1KY)
DL=Y2-Y1
A N D N Y - K Y ) T H E N G O T O 2150 E N D IF
Ix-coordinate for torus
Iy - c o o r d i n a t e for torus
Icharge for torus
!Distance b e t w e e n cell and t o r u s (y-comp.)
113
Figure 40. S P A C E .B AS-continued
I
DL2-Y2+Y1
!Distance b e t w e e n cell a n d " m i r r o r torus"(v-cp.
NSTEP-20
IF (K X - N N X A N D K Y - N Y ) IIs t h e cell c o n t a n i n e d in t h e t o rus?
THEN
!same t o r u s i n t e r a c t i o n
P R I N T KX, N N X , KY, NY
IF N N X - I
T HEN
EXT-O
!Define E - f l d for "same t o r u s " a n d first row
EYT-O
EXM-O
E Y M - -C O U L C *Q /DL2/DL2
ELSE
G O S U B 2 400 IRINGSAME
D e f i n e E - f l d for "same torus"
L=I
GOSUB 2500
IRINGINT
L-O
E N D IF
ELSE
G O S U B 250 0
!R I N G I N T
i n t e g r a t e s c o u l o m b o ver c h a r g e d torus
Iret u r n s ey2 a n d ex2
E N D IF
EX-EX+EXT+EXM
EY-EY+EYT+EYM
!x - c o m p o n e n t of E - f l d in a cell is the
!s u m of the p r e v i o u s v a l u e of E-fld, the
Ix - c o m p o n e n t f r o m the to r u s a n d t h e
!x - c o m p o n e n t f r o m t h e "mi r r o r t o rus"
!y-component of E - f l d in a cell is the
Isum of the p r e v i o u s va l u e of E-fld, the
Iy - c o m p o n e n t f r o m the to r u s a n d the
Iy - c o m p o n e n t f r o m t h e " mirror toru s "
!IF IKX-I A N D KY=4)
•THEN
!PRINT N N X , NY
!PRINT " E 1T - C H A R G E ", KX, KY, EXT, EYT
!PRINT " E ,M I R R O R - C H " , KX, KY, E X M , E Y M
!END IF
2 150
NEXT KX
IC o n t i n u e w i t h an i n t e r a c t i o n w i t h a n o t h e r t o r u s
N E X T KY
!electric f i e l d f r o m n e e d l e c h arge
D Y N - Y N E E C -Y (N N X ,N Y )
Iy - c o m p o n e n t of d i s t a n c e f r o m needle
D X N = X ( N N X fNY)
Ix - c o m p o n e n t of d i s t a n c e f r o m needle
DN2«DYN*DYN+DXN*DXN
D N - S Q R (DN2)
!Distance f r o m n e e d l e
E X N - C O U L C * Q N /DN2 * D X N /DN
Ix - c o m p o n e n t of E - f l d f r o m n e edle
E Y N - - C O U L C * Q N /DN2 *D Y N /DN
Iy - c o m p o n e n t of E - f l d f r o m n e edle
!electric f i e l d f r o m n e e d l e m i r r o r charge
D Y N M = Y N E E C t Y ( N N X fNY)
Iy - c o m p o n e n t of d i s t a n c e
!from "mirror nee d l e "
D N M 2 — D Y N M 4D Y N M + D X N * D X N
D N M = S Q R (DNM2)
!distance f r o m "m i r r o r n e e d l e "
E X N M - -C O U L C * Q N / D N M 2 1D X N /DNM
Ix - c o m p o n e n t of E - f l d from
I"mirror n e e d l e "
EYNM-CouLC 1QNZDNlti1DYNMZDNM
!y-component of E - f l d from
I"mirror n e e d l e "
I
P R I N T N N X fNY
I
P R I N T "NEEDLE ", E X N , E Y N
I
P R I N T " M I R R O R ", EXNM, E Y N M
E X ( N N X fN Y ) - E X + E X N + E X N M
E Y (NNX,N Y ) —E Y + E Y N + E Y N M
!x-component of E - f l d in a cell is the
!sum of t h e p r e v i o u s va l u e of E-fld, the
Ix - c o m p o n e n t fro m the n e e d l e a n d the
!x-component f r o m the "mirror nee d l e "
!y-component of E - f l d in a cell is the
Isum of the p r e v i o u s v a l u e of E-fld, the
Iy - c o m p o n e n t f r o m the n e e d l e a n d the
!y-component f r o m the "mirror nee d l e "
114
Figure 40. SPACE.BA S -continued
I
Q T - C S N G (QNtTM (NNX ,NY) )
Q T - Q N U M ( N N X 1NX)
P R I N T NNX, NY, E X ( N N X 1N Y ) , E Y ( N N X 1N Y ) , QT
NEXT NNX
N E X T NY
RETURN
!next cell
2260
I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * it***,,*********,,,**
I******************************************,,*,*,,,*,,*,,***,,*,*,
ISNAP - s u b r o u t i n e
!output of e l e c t r i c f i e l d d i s t r i b u t i o n t o h a r d disc
I*********************************,,*,,**,,,,,,,,,,,,,,,,*,,,,,,,,,,,,,,,,,,,,,
I
I
O P E N FI L E S F O R O U T P U T AS I I ,O R G A N I Z A T I O N S E Q U E N T I A L F I X E D !&
ACCESS APPEND
!Reopen e x i s t i n g file
P R I N T #1, "T—
T
!Sen d the c u r r e n t t i m e
P R I N T #1, " E L E C T R I C F I E L D A N D C H A R G E D I S T R I B U T I O N S "
P R I N T #1, "QsumQSUMNUM
!Sum of c h a r g e s in g e o m e t r y
P R I N T #1, "Q g a p — ";
QGAPNUM
ISum of c h a r g e s in "reduced geom
etry"
I
I
I
I
I
I
F O R NY-I TO N C Y
F O R N N X = I TO N C X
Q T - C S N G ( Q N U M (NNX,N Y ) )
Q T - Q N U M ( N N X fNY)
P R I N T *1, NNX;
NY;
E X ( N N X fN Y ) ;
E Y ( N N X fN Y ) ;
QT
IStore cell " x - i d e n t i f i e r ," cell " y - i d e n t i f i e r , "
Ix - c o m p o n e n t of E-fld, y - c o m p o n e n t of E-fld,
!a n d the char g e in t h e c e l l .
NEXT NNX
N E X T NY
NNX=O
F O R N Y = O TO NCY+1
Q T - C S N G ( Q N U M (NNXfN Y ) )
Q T - Q N U M ( N N X fNY)
P R I N T #1,
NNX;
NY;
QT
IS t o r e cell " x - i d e n t i f i e r , " cell " y - i d e n t i f i e r , "
land the c h arge in t h e cell.
N E X T NY
N N X —NCX+1
F O R N Y - O TO NCY+1
Q T - C S N G ( Q N U M I N N X fN Y ) )
Q T = Q N U M ( N N X fNY)
P R I N T #1,
NNX;
NY;
QT
N E X T NY
NY-O
F O R NNX-I TO NCX
Q T - C S N G ( Q N U M (NNXfN Y ) )
Q T - Q N U M ( N N X fNY)
P R I N T #1,
NNX;
NY;
QT
NEXT NNX
NY-NCY+l
F O R N N X - I TO N C X
Q T - C S N G ( Q N U M (NNXfN Y ) )
Q T - Q N U M ( N N X fNY)
P R I N T #1,
NNX;
NY;
QT
NEXT NNX
C L O S E #1
RETURN
2400
I**************************************************************************** + *
I******************************************************************************
IRINGSAME - s u b r o u t i n e
115
Figure 40. SPACE.BA S -continued
!numerical i n t e g r a t i o n f o r sam e rin g
I********************************************************************,
Z - ( N N X - I ) *2
ALIMO-ATN(1/SQR(Z*Z-1))
II n t e g r a t i o n l i m i t s .
ALIMl-PI
A S T E P - (AT,TMI - AT T M O ) /NSTEP
!Increment for i n t e g r a t i o n .
ALMO-ALIMO+ASTEP/2
!Middle of first i n c r e m e n t .
SUM-O
IZero the " p r o j ection" counter.
F O R I-I TO N S T E P
IL o o p for integration.
A L F A - A L M O 4 (I-I)-ASTEP
IA n g l e for i n t e g r a t i o n .
F U N C - I / S Q R (I- C O S ( A L F A ) )
I" P r o j ection" f u n c t i o n .
SUM-SUM4FUNC-ASTEP
!S u m of " p rojection."
NEXT I
E X T — C O U L C *Q / P I / D R / D R / 2 . 8 2 8 4 2 7 - S U M
!x - c o m p o n e n t of field.
EY T - O
!Field has no y - c o m p o n e n t .
RETURN
2500
I******************************************************************************
!********************************-*********************************************
!RINGIHT - s u b r o u t i n e
!numerical i n t e g r a t i o n of t h e i n t e r a c t i o n w i t h c h a r g e d ring
2 550
ALIMO-O
!Initial a n g l e for i n t e g r a t i o n
ALIMl-PI
!F i n a l angle for i n t e g r a t i o n
A S T E P = (A L I M l - A L I M O )/NSTEP
IA n g l e incr e m e n t for i n t e g r a t i o n
A L M O -AL I M O +AS T E P /2
!Middle of first i n c r e m e n t
TR=DR
!x - c o o r d i n a t e of cell g i v i n g the field,
ROFF=DS
!x - c o o r d i n a t e of t h e cell w h i c h the f i e l d acts on.
IF L=I T H E N G O T O 255 0 E N D IF
!Mirror c h arge only?
Y =DL
Iy - c o m p o n e n t of d i s t n a c e b e t w e e n cells.
G O S U B 2600 !RINTY
EYT=-COULC-Q*Y/PI-SUM
Iy - c o m p o n e n t of f i e l d f r o m t o r u s .
G O S U B 270 0 IRINTX
E X T — C O U L C - Q / P I - SUM
!x-component of f i e l d f r o m t o r u s .
Y-DL2
G O S U B 260 0 !RINTY
EYM=-COULC*Q«Y/PI-SUM
Iy - c o m p o n e n t of fi e l d f r o m m i r r o r c h a r g e .
G O S U B 270 0 !RINTX
Ix - c o m p o n e n t of fi e l d f r o m m i r r o r charge.
E x M =COULC*Q / P I-SUM
RETURN
2600
j******************************************************************************
J******************************************************************************
IRINTY - s u b r o u t i n e
I
I********************************************--********************************
SUM-O
F O R I=I T O N S T E P
!Loop for angle i n c r e m e n t i n g
A L F A - A L M O 4 (I-I)-ASTEP
!Angle for i n t e g r a t i o n
F U N C = I / ( Y - Y + R O F F - R O F F + T R - T R - 2 - T R - R O F F - C O S ( A L F A ) )*(3/2)
ly-proj e c t i o n for incr e m e n t
SUM=SUMtFUNC-ASTEP
!Total y - p r o j e c t i o n
NEXT I
RETURN
2700
l***********************************e»*******************«**-«*************»***
I******************************************************************************
!RINTX - s u b r o u t i n e
I
f******************************************************************************
116
Figure 40. S P A C E .BAS-continued
SUM-O
F O R I-I TO N S T E P
A L F A - A L M O + (I-I)1A S T E P
!Loop for angle i n c r e m e n t i n g
IA n g l e for i n t e g r a t i o n
F U N C - (TR1C O S (ALFA) -ROFF)/ (VlYtRoFFlRoFFtTRlTR-Z1TR1ROFF1C O S (ALF A ) )A (3/2)
Ix - p r o j e c t i o n for incr e m e n t
S U M - S U M + F U N C 1A S T E P
!Total x - p r o j e c t i o n
A d e g -A l f a z p i 1ISO
NEXT I
RETURN
2750
******************
******************
!G R I D S Q - su b r o u t i n e
!square g r i d s y s t e m
**********
IF F I N P $ — "N"
THEN
INPUT "Plane t o p l a n e d i s t a n c e / m m ", DO
D-D0/1000
INPUT " Number of cell lines p a r a l l e l to p l a n e s ", N C Y
IN P U T " Number of cell lines p e r p e n d . to p l anes ” , N C X
INPUT " D i s t a n c e b e t w e e n t h e s e cell l i n e s / m m
", DPO
DP-DP0/1000
GRID$="SQ"
E N D IF
N C T - N C Y 1N C X
P R I N T "Total n u m b e r of cells
", NCT
F O R N Y-O TO NCYtl
F O R N N X = O TO NC X t l
X ( N n X i N Y ) - D P 1 (NNX-I)
Ix - c o o r d i n a t e for eac h cell
Y (NNX,N Y ) - (D/NCY)1 ( N Y - 0 .5)
!y-coordinate for e a c h cell
NEXT NNX
N E X T NY
RETURN
2800
I*********************************************************************+**,,**,*
IGRID subroutine
Ito set up p a r a b o l i c g r i d syst e m
I********************************************************************,,,„,,*,,*
I
IF F I N P $ — "N"
!Restart status
THEN
I N P U T "Tip t o p l a n e d i s t a n c e / m m
D-DO/IOOO
INPUT "Focal p o i n t to t i p Zmm
FPO
FP-FP0/1000
INPUT "Number of p a r a b o l a cell lines
NCY
I N P U T "Nu m b e r of radial g r i d cells
NCX
IN P U T " A n g u l a r range, fro m 0 d e g to
", R D E G
ERASE QNUM
E N D IF
P - I Z (4*FP)
YNEEC=DtFP
N C T - N C Y 1N C X
P R I N T "Total n u m b e r of cells
NCT
ASTEP-RDEGZ(NCX-I)
IA n g l e i n c r e m e n t in d e g r e e s .
ASTEPR=ASTEP/180*PI
IA n g l e i n c r e m e n t in r a d i a n s .
!Find p a r a b o l a c o n s t a n t s for an e q u a t i o n y
a x A2 + b.
F O R NY=I TO N C Y
B ( N Y ) - D / N C Y 1 (NY-0.5)
117
Figure 40. S P A C E .B A S -continued
A ( N Y ) - 1 / 4 / (D+FP-B(NY))
NEXT NY
I
!
A(O)-O
B(O)-O
B (NCY + 1 )— D
A ( N C Y + 1 )— 1 / 4 /FP
F O R N N X = O T O NCX-I I
A N G (NNX) - (NNX-I)*ASTEP
!Angle in degrees.
A N G R ( N N X ) - A N G ( N N X ) /180*PI
!Angle in radians.
NEXT NNX
G O S U B 285 0 G R O U T
!cell c o o r d i n a t e s
F O R N N X - O TO N C X + 1
IIncr e m e n t a n g l e .
O-TAN(ANGR(NNX))
F O R NY- I TO N C Y + 1
!Incr e m e n t p a r a b o l a b e i n g described.
IF A B S ( Q ) C l E - I O
THEN
Y ( N N X 1N Y ) - B (NY) !y - c o o r d i n a t e for e a c h cell
X ( N N X 1N Y ) - O
I x - coordinate for e a c h cell
ELSE
C - A ( N Y ) *Q*Q
G-D+FP
ITotal d i s t a n c e f r o m focal p t . t o plane.
Q l “ '(1/C+2*G)
11st c o n s t a n t in q u a d r a t i c e q u a t i o n .
Q 2 — G * G + B ( N Y ) /C
12 n d c o n s t a n t in q u a d r a t i c e q u a t i o n .
!y-coo r d i n a t e is d e t e r m i n e d by s o l v i n g a q u a d r a t i c equation.
Y - - Q l / 2 - S Q R (01*01/4-Q2) !y-coordinate for e a c h cell
!x-coo r d i n a t e is d e t e r m i n e d b y s u b s t i t u t i n g the y - c o o r d i n a t e
!into the e q u a t i o n y - ax'2 + b.
X = S Q R ( ( Y - B ( N Y ) )/ A (NY))
!x-coordinate for e a c h cell
Y ( N N X 1N Y ) - Y
X ( N N X 1N Y ) = X
E N D IF
N E X T NY
NEXT NNX
F O R NY=I TO N C Y + 1
Y ( O fN Y ) - Y (2,NY)
X (0fNY) — -X (2, NY)
N E X T NY
F O R N N X - O TO N C X + I
Y ( N N X f0)-0
X (NNX, 0) - T A N (ANGR (NNX) ) * (D+FP)
NEXT NNX
G R I D S - nPARA"
RETURN
2820
I******************************************************************************
I******************************************************************************
IDIST - s u b r o u t i n e
!"CAt.UI.ATES I N T E R - C E L L D I S T A N C E S "
PRINT "CALCULATES INTER-CELL DISTANCES"
F O R NY=I TO N C Y
F O R NNX-I T O N C X
R D ( N N X fN Y f0 , O i - X ( N N X fN Y - I ) - X ( N N X fNY)
R D ( N N X fN Y f0 , I)- X ( N N X fN Y + ! ) - X ( N N X fNY)
R D ( N N X fN Y f I , 0)- Y ( N N X fN Y - I ) - Y ( N N X fNY)
R D ( N N X fN Y fI fI ) - Y (N N X fN Y + ! ) - Y ( N N X fNY)
RD (NNX, N Y f 2, 0)- S Q R (RD (NNX, N Y f 0,0) "2+RD (NNXfN Y f I, 0) '2)
RD (NNX, N Y f2, I) - S Q R (RD (NNXfN Y fO fI) A2 + R D (NNX, N Y f I, I) A2)
A D (NNX, N Y f 0 , 0 ) - X (NNX - I fNY) -X (NNX, NY)
A D (NNXfN Y f O fD - X (NNX + 1 ,NY) - X ( N N X 1NY)
A D (NNX, N Y f I, 0) -Y (NNX - I 1NY) -Y (NNXfNY)
A D ( N N X fN Y fI f I ) - Y ( N N X +1,NY)- Y ( N N X fNY)
A D (NNX, N Y f2,0) - S Q R (AD (NNX, N Y f 0,0) A2+AD (NNXfN Y fI f 0) A2)
118
Figure 40. S P A C E .BAS-continued
I
A D (NNX, NY, 2, I)- S Q R (AD (NNX, N Y ,0,1) * 2 +AD (NNX, NY, I , I ) ''2)
NEXT NNX
N E X T NY
NNX-3: N Y-3
RETURN
2900
|****************************************e***lllwlllllllllllllllllllllllJtl1t1tlltl
!GOUT - a u b r o u t i n e
Ioutp u t of g r i d - p o i n t s
PRINT "PARABOLIC CONSTANTS"
P R I N T " N U M B E R OF C E L L S ", N C X , " TI M E S
", N C Y
PRINT "
I
A(I)
B (I) "
F O R NY- I T O N C Y
P R I N T NY, A(NY) , B (NY)
N E X T NY
P R I N T "J
A N G (J)
T A N (ANG (J) ) "
F O R NNX=I TO N C X
P R I N T NNX, A N G (NNX) , T A N (A N G R (NNX) )
NEXT NNX
PRINT
P R I N T " P A R A B O L I C G R I D P O INTS "
PRINT " J
X
L
F O R NNX-I TO N C X
F O R NY=I TO NCY
Y M = Y ( N N X fN Y ) *1000
X M = X ( N N X 1N Y ) *1000
P R I N T NNX, XM, NY, YM
N E X T NY
NEXT NNX
RETURN
3000
I
3001
IF W A R N S - "Y"
T HEN
P R I N T #1 "WARNING:
E N D IF
C L O S E #1
END
Q M A X T O O HIGH"
Y"
119
Figure 41. SPREAD.BAS: A program, written in VAX BASIC
Version 3.3, used to read the data files created by
SPACE.BAS.
1001
*
*
*
*
*
*
SPREAD. BAS
READS A POINT-TO-PLANE PILE
LIST
OF
VARIABLES
*
TIKE:
T : Total t i m e of calculation.
T S S : S e l e c t e d t i m e for e n a p - a h o t .
TSNAP:
TE:
TSTEP:
TTOT:
VOLTAGE:
V O L T : Integrated Potential
CURRENT:
CURRENT: C u r r e n t p e r m m A2
CURR:
C: Total cu r r e n t
CHARGE:
0:
QNUM:
QSUMNUM:
QGAPNUM:
QT:
POSITION:
XMIN:
XMAX:
D:
DP:
FIELD:
EX: F i e l d s t r e n g t h
E Y : Field strength
(x-component).
(y-component).
AREA:
COUNTERS:
N N X : C o u n t e r in x-direction.
NY: C o u n t e r in y - d i r e c t i o n .
N C X : N u m b e r of x - g r i d s .
N C Y : N u m b e r of y - g r i d s .
MISC:
ADIS:
RDEG:
PK$: C a l c u l a t i o n t y p e (PROF or SMALL)
F I L E $ : F i l e to be read.
G R I D S : S e l e c t e d g e o m e t r y (PARA or SO).
C O N S TANTS:
120
Figure 41. S P R E A D .BAS-c ontinued
Z K A P P A : Ion m o b i l i t y
**************
I
D I M X (15,15), y (15,15), E X (15,15) , E Y (15,15), QT (15, 15)
D I M T ( I O O ) , V O L T ( I O O ) , C U R R ( I O O ) , QS(IOO), Q S G ( I O O ) , 0(15)
D I M S U R F (15), C U R R E N T (15)
200
IN P U T "Name of f i l e ", FI L E S
IN P U T "Cell p r i n t - o u t
(Y/N) ", P2S
IF (P25-"y" or P2$-"n")
!ASSURE SH I F T L O C K
THE N
P R I N T "Set 'Caps L o c k ' "
G O T O 200
E N D IF
INPUT "Current p r o f i l e (Y/N) ", P3S
INPUT "What c a l c u l a t i o n s ? (PROF/SMALL) ", PK$
ZK A P P A - I . O E -4
!Open a file for input. D i s p l a y i n t r o d u c t o r y information.
G O S U B 2 800 !SNAPINI
P R I N T "2-D space c h arge c a l c u l a t i o n "
P R I N T "File nam e ", FILES
P R I N T " G r i d - t y p e - ", GRIDS
P R I N T "D= ", D
P R I N T "DP= ", DP
P R I N T "NCX ",- N C X , "NCY "; N C Y
P R I N T "ADIS ", ADI S
P R I N T "T S N A P - "; T S N A P , "TE- "; TE
P R I N T "T S T E P T S T E P , "T T O T - "; TTOT
NS = O
IT=O
!N E W S N 2 :
400
IN P U T "NEXT S N A P - S H O T ? ( Y / N ) ", TSS
IF T SS="Y"
THE N
G O T O 500
ELSE
G O T O 700
E N D IF
!N E W S N 3 :
500
NS=NStl
!Trap errors.
ON E R R O R G O T O 600
G O S U B 290 0 !SNAP
R e a d d a t a f r o m file.
IF T(NS) < (TSS-1E-5) T HEN G O T O 500 E N D IF 1NEWSN3
G O S U B 2400 IWCALC
D i r e c t t h e d e s i r e d p r o c e s s i n g of i n f o r m a t i o n .
G O T O 400 !NEWSN2
600
650
690
700
IF E R R=Il T H E N R E S U M E 650
ID eal w i t h t r a p p e d E O F error.
E L S E R E S U M E 690
!Deal w i t h other e r r o r s .
E N D IF
P R I N T "END OF F I L E (EOF) R E A C H E D "
G O T O 700
PRINT "UNEXPECTED ERROR";ERR
C L O S E #1
G O T O 3100
I***************************************,*,**,,,,,,,,,*,,,,,,.,,,,,,,.,,..,.,,,
S U B R O U T I N E SEC T I O N
121
Figure 41. SPREAD.B A S -continued
W C A L C : I n q u i r e s about a c h oice of a d e s i r e d calculation.
A c c e s s e d b y G O S U B 2 400
PROF:
C r e a t e s a c u r r e n t p r o f i l e on t h e c o u n t e r - e l e c t r o d e .
A c c e s s e d b y G O S U B 2500
P S M A L L : D o e s a "small p r i n t - o u t " of the
A c c e s s e d b y G O S U B 2600
"snap" result.
S N A P I N I : R e a d s t h e d a t a - h e a d f r o m t h e SNAP-file.
A c c e s s e d b y G O S U B 2600
SNAP:
Inputs t h e e l e c t r i c f i e l d d i s t r i b u t i o n f r o m t h e
A c c e s s e d b y G O S U B 2900
SNAP f i l e .
C A L C : P e r f o r m s c u r r e n t c a l c u l a t i o n s b a s e d on f i e l d a n d c h a r g e d a t a .
A c c e s s e d b y G O S U B 3000
*******************************************************************************
*******************************************************************************
S u b r o u t i n e W C A L C : Inquires about c h o i c e of a d e s i r e d calculation.
A c c e s s e d by G O S U B 2400
*******************************************************************************
2400
!which c a l c u l a t i o n ?
IF P K $ « " P R 0 F "
THEN
G O S U B 2500 IPROF
ELSE
G O S U B 2600 !PSMALL
E N D IF
RETURN
!Display
c u r r e n t p r o f i l e data.
!Display
"small output" of S N A P - f i l e data.
j*******************************************************************************
I
!
S u b r o u t i n e P R O F : D i s p l a y s c urrent p r o f i l e on c o u n t e r - e l e c t r o d e .
A c c e s s e d b y G O S U B 250 0
2500
NY= I
PRINT T
P R I N T "S n a p n u m b e r ", NS
P R I N T "Time
", T(NS)
P R I N T "NNX", "AREA", "AMPS", " A M P S / m m A2"
F O R NNX=I TO NCX
IF NNX=I
THEN
XMIN=O
ELSE
X M I N = ( N N X - 1 .5)"DP
E N D IF
X M A X = ( N N X - 0 .5)"DP
S U R F (NNX)= 3 . 1 4 1 5 * (XMAX*XMAX-XMIN«XMIN)
C — Z K A P P A ' E Y (NNX1N Y ) / ( D / N C Y ) * Q T (NNXfNY)
C U R R E N T ( N N X ) = C Z S U R F ( N N X ) "IE-6
!current p e r mm2
P R I N T NNX, SUR F (NNX), C, C U R R E N T (NNX)
NEXT NNX
RETURN
I
I
S u b r o u t i n e P S M A L L : D i s p l a y a small p r i n t - o u t of the s nap r e s u l t .
A c c e s s e d by G O S U B 2600
I*******************************************************************************
*******************
l******«****»**«»»»»**«i>«*«**»*********«»*****»****«********»»»"""»*****
122
Figure 41. S P R E A D .BAS - continued
2 600
PRINT
P R I N T "Snap n u m b e r ", NS
P R I N T "Time
", T(NS)
P R I N T "NNX", "NNY", "QT", "EX", "EY"
F O R N Y - O TO N C Y + 1
CURR-O
IF P2$ - " Y "
THEN
F O R N N X - O TO NC X + 1
IQ (NNX)-CSNG (QNUM (NNX,NY) )
IQ (NNX) - Q N U M (NNXfNY)
P R I N T NNX, NY, Q T (NNX, NY), E X ( N N X fN Y ) , E Y ( N N X 1NY)
NEXT NNX
E N D IF
F O R NNX=I T O N C X
C U R R - C U R R + E Y (NNXfN Y ) * Z K A P P A * Q T (NNX, N Y ) / (D/NCY)
NEXT NNX
IF P3S-"Y"
T HEN
P R I N T " C u rrent at N Y - "; NY; " is "; C U R R
E N D IF
IF P 2 $ - " Y " T H E N IN P U T "Pause ", P E N D IF
N E X T NY
G O S U B 3000 !CALC
PRINT "V,centerV O L T (NS), " ! , p l a t e "; CURR(NS)
P R I N T "Q-a u m ";
Q S (NS); " Q - g a p
QSG(NS)
RETURN
**********************
S u b r o u t i n e S N A P I N I : Re a d s the d a t a - h e a d f r o m the SNAP-file.
A c c e s s e d b y G O S U B 2800
***********
2800
O P E N FILES F O R IN P U T AS #1
P R I N T "SN A P I N I subro u t i n e "
IN P U T #1, GRIDS
IN P U T #1, TlS
!" p o i n t - t o - p l a n e corona"
IF G R I D S — "SQ"
THEN
P R I N T "Sq u a r e c o o r d i n a t e system"
I N P U T #1, D, DP, NCY, N C X , A D I S
ELSE
IF G R I D S — "PARA"
THEN
P R I N T " P a r a b o l i c c o o r d i n a t e system"
I N P U T #1, D, DP, NCY, NCX, R D E G
E N D IF
E N D IF
I N P U T #1, T S N A P , TE, T S T E P , T T O T
IN P U T *1, ZJO
F O R N N X - I TO N C X
IN P U T *1, QD(NNX)
NEXT NNX
IS N A P I l :
RETURN
I
S u b r o u t i n e SNAP; Inpu t s t h e e l e c t r i c fi e l d d i s t r i b u t i o n
I
f r o m the SNA P file.
I
A c c e s s e d by G O S U B 2900
I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
2900
INPUT #1, T2S,
P R I N T T2S, NS,
T (NS)
T (NS)
!"T- "
123
Figure 41. S P R E A D .BAS-continued
I N P U T #1, T3$
I" E L E C T R I C F I E L D A N D C H A R G E D I S T R I B U T I O N S "
I N P U T #1, T4S, Q S U M N U M
!"Q,eum- ”
INPUT #1, T5S, Q G A P N U M
I"Q,gap- "
OS(NS)-QSUMNUM
Q S G (NS)- Q G A P N U M
P R I N T "I", "J", "E x", "QT"
F O R NY-I T O N C Y
F O R NN X - I T O N C X
I N P U T #1, I, J, E X ( N N X fNY) , E Y (NNXfNY) , Q T ( N N X fNY)
P R I N T I fJ, E X ( N N X fN Y ) , Q T ( N N X fNY)
NEXT NNX
N E X T NY
NNX-O
P R I N T "I", "J", "QT"
F O R NY-O TO NCY-H
IN P U T *1, I f J f Q T ( N N X fNY)
P R I N T I, J f Q T ( N N X fNY)
N E X T NY
NNX-NCX+I
F O R N Y-O TO N C Y + I
INPUT #1, N N X f N Y f Q T ( N N X fNY)
P R I N T N N X f N Y f Q T ( N N X fNY)
N E X T NY
N Y-O
F O R NN X = I TO N C X
INPUT #1, N N X f N Y f Q T ( N N X fNY)
P R I N T N N X f N Y f Q T ( N N X fNY)
NEXT NNX
N Y —N C Y + I
F O R NN X - I TO N C X
IN P U T *1, N N X f N Y f Q T ( N N X fNY)
P R I N T N N X f N Y f Q T ( N N X fNY)
NEXT NNX
RETURN
I*******************************************************************************
I
I
Subroutine C A L C : Performs calculations
A c c e s s e d b y G O S U B 3000
of p o t e n t i a l
and c u r r e n t .
3000
NNX-I
ESUM-O
F O R NY=I T O N C Y
E S U M - E S U M + E Y (NNXfNY)
N E X T NY
V O L T (NS)- E S U M * D / N C Y
N Y-I
CURR(NS)-O
F O R NN X = I T O N C X
C U R R (NS)- C U R R (NS)- E Y ( N N X fN Y ) * Z K A P P A * Q T (NNXfN Y ) / (D/NCY)
NEXT NNX
RETURN
3100
lend p r o g r a m
P R I N T "THIS IS T H E END"
(*******************************************************************************
END
124
Figure 42. TUBE.M : A program, written for the command
language of the MATLAB program, used to simulate ion flow
through a drift tube.
%
% TUBE: A p r o g r a m t o s i m u l a t e the f l o w of Iona,
t a d r ift tube.
of one apeciesm,
thr o u g h
’
%
4
% Input i n f o r m a t i o n ab o u t t h e drift t u b e a n d t h e ions p r o d u c e d in it.
4 Also, i n i t i a l i z e some of t h e variables, constants, etc.
N - i n p ut ('no. of g r i d p o i n t s - ');
steps - input('no. of t i m e s t e p s - ');
t i m e (I)-0;
p o « i n p u t ( ' s p a c e c h a r g e - ');
rho - d i a g (zeros (N));
I - i n p u t ('l e n g t h ( m e t e r s )- ');
c hg - i n p u t ( ' p l a t e c h a r g e - ');
t h i c k - 1/N;
a - i n p u t ('d i s t a n c e f r o m r e p e l l e r to
m - a/thick;
4Tube is d i v i d e d int o N slices.
4N u m b e r of t i m e steps.
4 l n i t i a l time.
4Space c h arge p e r "slice."
4 S pace c h arge v e c t o r .
4L e n g t h of d r i f t t u b e .
4 D i f ference b e t w e e n c h a r g e s on plates.
4 T h i c k n e s s of a s l i c e .
e (meters)- ');
4Number of slic e s on t h e source side
4of t h e g a t e .
4
% Set u p t h e i n i t i a l c o n d i t i o n s for the simulation.
4
for x = l : m
rho (x) - po;
end
vo - v e l o c i t y (rho,m,N,chg);
4Fill t h e r e g i o n w i t h sp a c e c h a r g e .
4Call "velocity" to c a l c u l a t e the
4 v e l o c i t y of ions in t h e fastest slice,
S D e t e r m i n a t i m e s t e p t o g ive m a n a geable
4d r i f t distance.
t i m e s t e p - thic k / v o ;
4
4 M ove t h e ions a c c o r d i n g to fi e l d condtions.
for t i m e _ i n c = 2 :steps
r h o _ n e w = d i a g ( z e r o s (Ntl));
if s u m (rho) > p o / l e l O
%"New" space c h a r g e vector.
S C o n d i t i o n for t h e e n d of the
S s i m u l a t i o n is the e xit of mos t of the
S s p a c e c h a r g e to t h e c o l l e c t o r plate.
for c=l:N
if rho(c)
v(c)
- v e l o c i t y (rho,c , N , c h g ) ;
S C a l c u l a t e t h e a v e r a g e v e l o c i t y of ion
Sdri f t in a "slice," c .
r h o _ a d d - d i s p l a c e m e n t ( v (c),r h o ( c ) ,t h i c k , timeste
P,C,N);
S T r a n s l a t e ion drift v e l o c t i y into
S d i s t a n c e of m o v e m e n t d u r i n g a timestep.
r h o _ n e w - r h o _ n e w + r h o _ a d d (I: N + 1 ) ;
S Report t h e ion d i s p l a c e m e n t in terms
Sof a space c h arge vector.
end
end
4
4 I n c r e m e n t time.
4
time( t i m e _ i n c )
t i m e ( t i m e _ i n c - l ) +timestep;
S A c c u m u l a t e d time af t e r the gate is
Sopened.
125
Figure 42. T U B E .M - continued
%
% Co u n t t h e c o l l e c t e d i o n s .
%
collect (time m e )
- rho n ew(N+l);
♦ C o l l e c t e d ions are ion s tha t mak e it to
♦the last cell of t h e dr i f t tube.
r h o - r h o _ n e w (I: N ) ;
♦Th e n e w space c h a r g e vector, r esulting
♦ f r o m the s u m m i n g of t h e displacements,
%is u s e d for t h e n ext t i m e step.
tend
en d
clt ■= c o l l e c t ./timestep;
♦Collector c u r r e n t :
% (number of ions c o l l e c t e d /
♦ l e n g t h of c o l l e c t i o n period.
%
% P l o t t h e c a l c u l a t e d data.
%
plot (time, clt)
t i t l e ('C o l l e c t o r C u r r e n t as a f u n c t i o n
y I a b e I ('C u r r e n t ( i o n s / s e c ) ')
x l a b e l ( ' t i m e ( s e c .)')
t i m e after p u l s e ' )
126
Figure 43. TUB2.M: A program, written for the command
language of the MATLAB program, used to simulated the flow
of a group of ions, having two different ion mobilities,
through a drift tube.
%
% TUB2: A p r o g r a m to s i m u l a t e the m o v e m e n t of t w o d i f f e r e n t k i n d s of ions,
* differing
m o b i l i t i e s , t h r o u g h a drift tube.
of
%
% Input i n f o r m a t i o n about t h e drift tub e and the ions p r o d u c e d in it.
% Also, i n i t i a l i z e some of the variables, constants, etc.
N - in p ut('no. of g r i d p o i n t s - ');
!Tube is d i v i d e d into N slices.
steps - input ('no. of t i m e s t e p s - ' );
% N u m b e r of tim e steps.
t i m e (I)-O;
! I n i t i a l time.
p o a - i n p u t ( ' s p a c e c h a r g e of a- ');
!Spa c e charge p e r "slice."
p o b - input ('space c h a r g e of b - ');
rho - d i a g (zeros (N));
! S p a c e c h arge v e c t o r s .
r hoa - d i a g (zeros (N));
r hob - d i a g (zeros (N));
ka « i n p u t ( ' m o b i l i t y of a- ');
!Ion m o bilities.
kb - i n p u t ('m o b i l i t y of b - ');
I - i n p u t ('l e n g t h ( m e t e r s ) - ');
! L e n g t h of drift t u b e .
chg - i n p u t ( ' p l a t e c h a r g e = ');
! D i f f e r e n c e b e t w e e n cha r g e s on plates.
th i c k « 1/N;
! T h i c k n e s s of a slice,
a - i n p u t ( ' d i s t a n c e f r o m r e p e l l e r to g a t e ( m e t e r s )- ');
m = a/thick;
!Nu m b e r of slices on t h e s o urce side
!of t h e g a t e .
%
! Set u p t h e ini t i a l c o n d i t i o n s
%
for t h e
for x = l : m
• r h o (x) - poa+pob;
r h o a (x) - poa;
r h o b (x) - pob;
end
!vo - v e l o c i t y (rho,m,N,chg);
simulation.
!Fill t h e r e gion w i t h space charge.
!Call "velocity" to c a l c u l a t e the
! v e l o c i t y of ions in t h e fastest s l i c e .
v oa - v e l _ i o n ( r h o a , m , N , c h g , k a ) ;
v ob - v e l _ i o n ( r h o b , m , N , c h g , k b ) ;
if v o a < v o b
t i m e s t e p - th i c k / v o b ;
else
t i m e s t e p - th i c k / v o a ;
end
! D e t e r m i n a t i m e s t e p t o g i v e m a n a geable
! d rift d i s t a n c e .
!
! M o v e t h e ions a c c o r d i n g to t h e f i e l d condi t i o n s .
!
for t i m e _ i n c - 2 :steps
r h o _ n e w - d i a g ( z e r o s ( N + l ) );
r h o a _ n e w - d i a g ( z e r o s ( N + l ) );
r h o b _ n e w - d i a g ( z e r o s ( N + l )) ;
if s u m (rho) > p o a / l e l O
!"New" space c h a r g e vector.
! C o n d i t i o n for t h e e n d of the
! s i m u l a t i o n is the exit of mos t of the
! s p a c e c h arge to the c o l l e c t o r plate.
for c - 1 :N
if rho(c) —
0
vb(c) - v e l _ i c n (rho, c, N, chg, kb) ;
va(c) - vel i o n (rho,c,N,chg,ka);
!CaTculate the average velocity of ion
!drift in a "slice," c .
rhoa_add - displacement(va(c),rhoa(c),thick,time
step,c,N);
127
Figure 43. T U B 2 .M - continued
r h o b _ a d d - d i e p l a c e m e n t ( v b (c),r h o b (c),thick,time
step, c,N) ;
! T r a n s l a t e ion drift v e l o c t i y into
! d i s t a n c e of m o v e m e n t d u r i n g a timestep
r h o a n e w - r h o a _ n e w + r h o a _ a d d (I: K + 1 ) ;
r h o b _ n e w - r h o b _ n e w + r h o b _ a d d (I :N + 1 ) ;
!Re p o r t the ion d i s p l a c e m e n t in terms
!of a space char g e v e c t o r .
end
end
end
I
I Increment time.
!
time(time_inc)
- t i m e < t i m e _ i n c - l )+timestep;
! A c c u m u l a t e d t i m e a f t e r t h e gate is
!opened.
!
% Count the ions as t h e y are collected,
collect(time_inc)
- rho a n e w ( N + l ) + r h o b n e w ( N + l );
! C o l l e c t e d ions are ions t h a t mak e it to
!the last cell of t h e dr i f t tube.
rhoa - r h o a _ n e w (I: N ) ;
!Th e n e w space c h a r g e vector, r esulting
r h o b - r h o b _ n e w (I: N ) ;
! f r o m t h e s u m m i n g of t h e displacements,
rho - rho a + rhob;
!is u s e d for the n ext t i m e step.
!end
end
clt = c o l l e c t ./timestep;
!Collector cu r r e n t :
! (number of ions c o l l e c t e d /
! l e n g t h of c o l l e c t i o n period.
I
% P lot t he c a l c u l a t e d d a t a .
%
p l o t (time, clt)
t i t l e ( ' C o l l e c t o r C u r r e n t as a f u n c t i o n of t i m e a f t e r p u l s e ' I
y l a b e l ( ' C u r r e n t ( i o n s / s e c ) ')
x l a b e l ( ' t i m e ( s e c . ) ')
128
Figure 44. VELOCITY.M : A function, called by both TUBE.M and
TUB2.M, used to calculate ion velocities.
function[z]-velocity(rho,b,N,chg,m o b i l i t y )
!velocity
F u n c t i o n c a l c u l a t e s v e l o c i t i e s of ion drift in a cell b,
%
b a s e d on p l a t e charges, ap a c e char g e and ion mobility.
r h o _ r t - 0;
r h o _ l f - 0;
eO - 8 . 85e-12;
fo r k - b :N
rho_rt - rho_rt + rho(k);
! S u m of apace char g e to t h e right
!of a "slice."
end
for k - I :b
r h o _ l f - r h o _ l f + rho(k) ;
of
! S u m of ap a c e c h arge to t h e left of
%a "slice."
end
E -
(1/eO)* ( r h o _ l f - r h o _ r t + c h g ) ;
z - mobility/E;
! F i e l d s t r e n g t h in an i n f i n i t e paralell
!pla n e system.
!Drift v e l o c i t y in a s l i c e as a result
!of ion m o b i l i t y and f i e l d strength.
129
Figure 45. DISPLACEMENT.M : A function, called by both TUBE.M
and TUB2.M, used to calculate ion movements from cell to
ce l l .
f u n c t i o n [ d _ p l a c e ]- d i s p l a c e m e n t ( v e l o c i t y , r h o , t h i c k , timestep,c,N)
%d i s p l a c e m e n t
C a l c u l a t e s the m o v e m e n t of s p a c e charge from, cell to cell,
%
b a s e d on gi v e n space c h a r g e densities, d e t e r m i n e d drift
%
v e l o c i t i e s and a g i v e n t i m e s t e p .
d _ p l a c e - d i a g (zeros (Ntl));
% S p a c e c h a r g e v e c t o r as r e s u l t i n g from the
%dr i f t f r o m a " s l i c e , " c .
d i s t a n c e - v e l o c i t y * t i mestep; »The resu l t f r o m "velocity" is t r a n s l a t e d into
♦ d i s t a n c e of m o v e m e n t .
disp - distance/thick;
♦ " d i s t a n c e " is r e p o r t e d in t e r m s of n u mber of
♦"slices" m o v e d .
i f rho > 0
♦If #0
if abs(disp) > -2
♦If #1
♦ V e r i f i c a t i o n is m a d e that " r e a s o n a b l e " amounts
♦ of m o v e m e n t p e r t i m e - s t e p a r e b e i n g m a d e .
if d i s p < -I
♦ I f #2
d _ p l a c e (c-2) - r h o * ( d i s p - 1 );
d_ p l a c e ( c - 1 ) - rho - d _ p l a c e ( c - 2 ) ;
else
♦ E l s e for if #2
if d i s p < 0
♦If #3
d_place(c-1) - rho* d i s p ;
d _ p l a c e (c) - r h o - d _ p l a c e ( c - 1 ) ;
e lse
♦Else for if #3
if dis p < I
♦If 44
d _ p l a c e ( c + 1 ) - d i s p * rho;
d _ p l a c e (c) - r h o - d _ p l a c e ( c + 1 );
else
♦ E l s e for if t4
if d i s p < 2
♦If #5
d _ p l a c e (c+2) - (disp-1) » rho;
d _ p l a c e ( c + 1 ) - rho - d _ p l a c e ( c + 2 ) ;
end
♦End for if # 5 (No else for 5)
end
♦ E n d f o r if #4
end
♦En d for if #3
end
♦En d for if #2
♦ E l s e f o r if #1
e lse
d i s p ('displacement of space c h a r g e t o o g r e a t ' )
♦ w a r n i n g of too large
♦a m o v e m e n t from a c e l l .
break
♦ E n d f o r if #1
end
♦ E n d for if #0
end
♦ R e t u r n v a l u e of d i s p l a c e m e n t v e c t o r
d _ p l a c e - d _ p l a c e (I: N + 1 ) ;
♦ w i t h c o r r e c t l e n g t h stipulated.
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