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Michigan State
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This is to certify that the

dissertation entitled

IMPROVEMENTS TO RESOLVING POWER
IN TIME-OF-FLIGHT MASS SPECTROMETRY

presented by

George Edward Yefchak

has been accepted towards fulfillment
of the requirements for

Ph. D. degreein Chemistry

 

 

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cmmwd

IMPROVEMENTS TO RESOLVING POWER
IN TIME-OF-FLIGHT MASS SPECTROMETRY

By

George Edward Yefchak

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1990

ABSTRACT

IMPROVEMENTS TO RESOLVING POWER
IN TIME-OF-FLIGHT MASS SPECTROMETRY

BY
George Edward Yefchak

As the separation efficiency available. from modern gas chromatog-
raphy (GC) continues to improve, the increasingly short elution time
observed for each mixture component presents a difficulty for mass spec—
trometric detection and analysis of these components. Time-of-flight mass
spectrometry (TOFMS) offers the potential for hundreds of spectra to be
acquired per second, hence TOFMS has been identified as the mass spec-
trometric technique of choice for high-speed GC. In order to accomplish
high-resolution capillary column GC-TOFMS, researchers at Michigan
State University have developed data acquisition equipment capable of
obtaining and recording spectra at a rate of 100 Hz or more. In addition to
this detection equipment, improvements to the mass resolving power
achievable with time-of-flight mass analysis are required in order for GO-
TOFMS to become a viable technique. Methods by which such improve-

ments can be made are examined in this dissertation.

The causes for the poor resolving power generally obtained in
conventional TOFMS arise from the distributions of initial velocities and

positions that ions have at the time of formation. They are carefully exam-

ined, and conclusions drawn from this analysis are used to develop and
evaluate new TOFMS techniques. One new technique, dynamic-field
focusing (DFF), relies on a time-varying electric field to preferentially
accelerate trailing ions within each isomass packet so that nearly simul-
taneous arrival of all isomass ions is obtained.1 This technique is presented
and analyzed in detail; the study shows that although DFF can provide
better mass resolving power than the previously-developed techniques
which used time-invariant fields, DFF is not the panacea sought. Two
superior techniques for improving the resolving power in TOFMS are then
presented and discussed. In the first of these, a continuous beam of ions is
modulated by means of a deflection field in order to form ion packets for
time-of-flight mass analysis.2 In the second, a novel pulsed electron-
impact ion source having ion storage capabilities is used in combination
with a grid-free ion mirror.3 These two techniques are examined, and both
are found to provide mass resolving power adequate for GC-TOFMS.
Finally, the relative merits of the two techniques, including sensitivity and

detection limit considerations, are discussed.

 

1 G. E. Yefchak, C. G. Enke, and J. F. Holland, Int. J. Mass Spectrom. Ion
Processes, 1989, 87, 313.

2 T. K. Fowler and W. M. Good, Nucl. Instrum. Math. 1980, 7, 245.

3 R. Grix, R. Kutscher, G. Li, U. Grfiner, and H. Wollnik, Rapid Commun. Mass
Spectrom, 1988, 2, 83.

Copyright © by
GEORGE EDWARD YEFCHAK
1990

 

ACKNOWLEDGEMENTS

I am grateful for the help provided to me by a large number of people.
In particular, I would like to thank my advisers Dr. Christie G. Enke and
Dr. Jack Throck Watson, as well as Dr. John F. Holland, who helped to
guide me throughout my years in the graduate program.

Kevin Hart and Chris Evans provided lots of assistance and lots of
friendship. Eric Erickson and Brian Eckenrode, two past participants in
the time-of-flight project, provided much assistance as well. Thanks go
also to Dr. John Allison and the present members of the time-of-flight team,
namely Gary Schultz, Ron Lopshire, Mary Puzycki, Dr. Ron Tecklenburg,
and Dr. Damaschin Ioanoviciu. Dr. Stan Crouch and Dr. Adrian Wade
provided many helpful discussions, and the computers were kept running
with much assistance from Dr. Tom Atkinson, Mel Micke, and Mike
Davenport. Helpful programming advice was provided by Drake Diedrich;
Drake also provided some graphics routines for the beam deflection project.
Kevin McNitt programmed and maintained the vast integrating transient
recorder software system. I would like to thank Professor Hermann
Wollnik and his coworkers at Universitat Giessen for all their help in our
collaborative project. Special thanks go to Dr. Raimund Grix who came to
MSU from Giessen. I am also grateful for all the assistance provided by
past Enke-group members Norm Penix and Mark Victor, as well Bruce
N ewcome, who made time-array detection a reality. And I would like to
express my appreciation to the entire Enke and Watson groups, past and
present.

Keith Caserta deserves special recognition for advising me to enter
the graduate chemistry program at Michigan State University.

finally, I am most appreciative of tIie guidance, support, and [ave of
my wonderful wife and fiest friend, Wendy fileen Teifl'er, to wfiom
tliis dissertation is Jedicaterfi

TABLE OF CONTENTS

List of Tables ...................................................................................... ix

List of Figures ..................................................................................... x

Chapter 1: Introduction ................................................................. 1

1.1. The Problem ........................................................................... 1

1.2. A Solution—The Michigan State University Plan ................ 7

1.2.1. Time-Array Detection ........................................................ 8

1.2.2. Improved Mass Resolving Power ......................................... 9
Chapter 2: Mass Resolving Power in 'l‘ime-of-Flight Mass

Spectrometry .......................................................................... 11

2.1. Introduction to Time-of-Flight Mass Spectrometry ............. 11

2.1.1. Principle of Mass Analysis ................................................ 12

2.1.2. Definition of Resolving Power ............................................. 13

2.2. Ion Behavior Patterns Which Degrade Resolving Power ...14

2.3. Previous Approaches to Improving Resolving Power ......... 16

2.3.1. Wiley-McLaren Developments ............................................ 18

2.3.2. Early Techniques for Space and/or Energy Focusing ....... . ..... 19

2.3.2.1. Change of initial ion behavior ................................. 19

2.3.2.2. Change of ion behavior after extraction .................... 22

2.3.3. The Reflectron .................................................................. 24

2.4. Requirement of Non-Static Fields for Complete Focusing . .27

2.5. Methods Which Incorporate Dynamic Fields ...................... m

2.6. Summary ............................................................................... a)

Chapter 3: Dynamic-Field Focusing ............................................ 34

3.1. Models for Space and Energy Focusing ............................... 34

3.1.1. Four Post-Source Focusing Models ..................................... 35

3.1.2. Velocity Adjustment: Best Suited to Dynamic Fields ............ :£

3.2. Investigation of Post-Source Velocity Adjustment ........... :D

3.2.1. Study of Muga’s Work ....................................................... {D

3.2.1.1 Theory of Velocity Compaction ................................ 40

3.2.1.2 Digital Simulation of Velocity Compaction ............... 42

3.2.2 Empirical Studies of Dynamic Acceleration ......................... 46
3.2.2.1 Cases VC, A, and B ............................................... 47

3.2.2.2 Cases S and E ....................................................... 52

3.2.3. The Infinitely-Thin DFF Region ......................................... 57

3.2.4. The DFF Equations ........................................................... 59

3.2.5. Selection of Instrument Parameters for DFF-TOFMS .......... 65

3.2.6. Simulated DFF-TOFMS Performance ................................. Q

3.2.7. The DFF Animation Program ............................................ 74

3.2.8. Predicted Metastable Ion Behavior ...................................... 77

3.2.9. Practical Considerations ................................................... 82
Chapter4: BeamDeflection .......................................................... as
4.1. Background ........................................................................... 87
4.2. Theory .................................................................................... 89
4.2.1. Models for Beam Modulation Techniques ............................ $

4.2.2. Ion Packet Quality ............................................................ 93

4.3. Simulations ........................................................................... w
4.4. Experimental Results ......................................................... 102
4.5. Practical Considerations .................................................... 103
4.5.1 Mass Resolving Power .................................................... 103

4.5.2 Signal-to-Noise Considerations ........................................ 104

4.6. Conclusions ......................................................................... 105

Chapter5: CombinationdPuldelectronImpactSource

with Ian Mirror ................................................................... 107
5.1. Background ......................................................................... 108
5.2. Preliminary Studies of Ion Behavior in the

Wollnik Source .................................................................... 112
5.3. A Model for Ion Behavior in the Wollnik Source ............... 114

5.3.1. The Revised Model Simulation ......................................... 114

5.3.2. The Hypothesis ............................................................... 119

5.3.3. Experimental Studies in Giessen ...................................... 122
5.4. Studies Performed with the MSU M-TOF .......................... 127
Chapter 6: Collaborations ........................................................... 136
6.1. Studies on the Mass-Dependence of Time-Lag Focusing. 136

6.2. Studies of Two-Stage Homogeneous-Field Ion Mirrors 140

6.2.1 Background Information .................................................. 140

6.2.2 Ion Mirror Transfer Matrix .............................................. 143

6.2.3 Digital Simulation ........................................................... 144

6.2.4 A Convenient Simulation Method ...................................... 147

Chapter 7: Conclusions ............................................................... 152

Appendix: Program Listings ...................................................... 156
viii

 

 

3- 1.
3-2.
3-3.
3-4.
3-5.

4-1.

4-2.

4-3.
5-1.

LIST OF TABLES

Values used by program VCTOF ................................................. 46
Effect of voltage rise on final velocities .......................................... 49
Effect of voltage rise on arrival times ............................................ 50
Final expressions for k ............................................................... 64
Values used by program SIMTOF. .............................................. 71
Specific k values for 1000 u ions (V0 = 614.6 V). .............................. 73
Parameters used for simulated beam deflection peak shape. ........ 100
Comparison of peak variances obtained from calculation using

Eqs. 1-3 and from digital peak shape simulation ......................... 101
Comparison of measured and simulated peak widths .................. 163
M-TOF source voltages used to obtain data in Figure 5-8 ............... 1:!)

ix

LIST OF FIGURES

Two problems in GC-MS caused by slow spectral generation
rates. ........................................................................................ 6

Overall mass analysis process in time-of-flight mass
spectrometry ............................................................................. 12

Two methods of quantitating mass resolving power. The
quantity reported as “% Valley” (Percent Valley) is the height of
the overlap formed between two equally intense peaks,
expressed as a percentage of the peak height. The FWHM
(Full Width at Half Maximum) value is the width of the peak at
its half-intensity height, expressed in mass units .......................... 14

Flight time variations in TOFMS. Lengths of the arrows
indicate relative velocities of isomass ions extracted with
energy U, from a single-field source. Relative positions and
velocities are shown at three different times: ion formation (to),
arrival at the space focus plane (t1), and later in the field-free
region “2). ................................................................................ 17

Three representations of resolving power are plotted against
mass: a: two methods of Sanzone, “R- 1” and “Unit Resolu-
tion” b: the percent valley method ................................................ Z)

One-stage ion mirror. Motion of two isomass ions having
different initial kinetic energies is illustrated with relative
positions and velocities at four different times ............................... 5

Two-stage ion mirror (“reflectron”). Motion of two isomass
ions having different initial kinetic energies is illustrated with
relative positions and velocities at six difi‘erent times. .................... 3

a: Ions starting at rest from difi‘erent positions within the
source (case S) behave identically to case E+ ions starting from
the space-focus plane. b: Similar behavior occurs for ions
having some initial kinetic energy, but the location of the
space-focus plane is shifted ......................................................... 23

3-2.

3—4.

3-5.

3-6.

Illustration of ion motion in a conventional TOFMS instru—
ment. Two case E+ ions for each of two masses m1 and m2 are
illustrated. Ions begin at the origin (time and distance both
zero) in the source. They are accelerated out of the source and
move with constant velocity through a field-free region. The
shaded lines represent motion of ions with non-zero initial
energies; these ions achieve higher final velocities and thus
reach the detector before the reference ions (represented by
solid lines). Motion stops after ions traverse a certain path

length and strike the detector ...................................................... 35

Four models for focusing by post-source modification of ion
behavior. For each case, motion of two case E+ ions for each of

two masses m1 and m2 is illustrated as in Figure 3-1 ..................... 37

Schematic diagram and voltage profile for the velocity com-

paction TOFMS instrument (diagram adapted from ref. 4).
Ions are created in the source, accelerated by a small electric
field into the first field-free region (FF1), experience post-
source acceleration in the dynamic field region, and finally

travel through a second field-free region (FF2) to the detector .......... 40

Selection of values corresponding to a distribution function
p[x]. The curve represents a cumulative distribution function

P[x]=l:.p[xldx. Equally spaced values of P[x] (i.e., a uniform

distribution) correspond to a ptxl distribution of x .......................... 44

Simulated peaks for m/z 600 and 601 from a conventional
linear TOFMS instrument (a) and from the velocity compaction

instrument (b). In each case, 0., corresponds to 500 K and

0’. = 0.01 cm. .............................................................................. 47

Models of ion behavior used for initial studies. In case VC, two
isomass ions enter the dynamic field region simultaneously
with different velocities but leave with equal velocities. In case
A, the ions enter the region as in case VC, but the initially
slower ion is accelerated to a higher velocity so that it catchs up
to the other ion at the detector. In case B, two ions enter the
dynamic field region separated in time but with equal
velocities; as in case A, the trailing ion is accelerated and it

catches up to the leading ion at the detector. ................................. 48

Relation of k (dV/dt) values to initial voltage for cases VC, A,

and B. Parameters other than V,- are those in Table 3-1 .................. 51

Two new models of ion behavior used for empirical studies. In
case S, two ions begin at rest from different positions within
the source. They travel through FF1 (which may extend

xi

beyond the space-focus plane, as pictured here) and enter the
DFF region at different times and with different velocities.
They are accelerated in this region and travel with new
velocities through FF2, finally arriving simultaneously at the
detector. Case E is similar except that the ions start from the
same position but with positive and negative initial velocities .......... 52

3-9. Relation of k values to initial voltage for cases S and E. For the
data labled “S (short)” and “S Gong)”, d = 0.02 m and 0.04 m,
respectively. Identical points were obtained with both values of
d for case E. Parameters other than V,- and d are those in
Table 3-1. .................................................................................. 54

3-10. Dependence of It on FF1 length, d, for cases S and E and with
V; = 100 V and 221 V. Values for case S approach infinity at
d = 0.02 m, the location of the space-focus plane. Values for
case S and case E appear to converge at d = 0.16 m ......................... 55

3-11. The two ions in case E have the same kinetic energy so the plot
of energy vs separation distance is a horizontal line. In case S,
the two ions have different energies, thus the plot is slanted.
As the ions move further along the flight path, however, their
separation increases. This reduces the slope making it more .
similar to that for case E ............................................................. 56

3-12. Peak shape sketches for m = 600 u based on data from the sim-
ulation program. Peaks in the first row were generated using
only an initial energy distribution (i.e., a planar source); those
in the second row arise from an initial spatial distribution.
Peaks simulated for a real gaseous source are shown in the
third row. Peaks were generated using 12 values for both case E
and case S and also using the mean of these two. Velocity
compaction results (from VCTOF) are shown in the first
column for comparison. ............................................................. 57

313. Model for dynamic-field focusing within an isomass group. a:
Illustration of ion behavior. b: Relationships between ion
velocity and ion position at regions of the flight path. c:
Voltage profile throughout the flight path ..................................... 6)

3-14. Two arbitary isomass ions used in deriving the general focus-
ing equation. a: time of entry into FF2, b: time of arrival at

detector. ........................... _ ........................................................ 61
3-15. The DFF voltage function ............................................................ Q
316. Simulated DFF-TOFMS peaks for ions of mass 1000 and 1001 u
for parameters shown in Table 3-5 ......................................... . ...... 71
xii

3-17.

3-18.
3-19.

4-2.

4—3.

5-2.

5-3.

Simulated % Valley resolving power of DFF- TOFMS as a

function of mass ........................................................................ 72
Deconvoluted DFF-TOFMS peaks at masses 1000 and 1001 u. .......... 73
Eight snapshots of the DFF Animation program (parts c-h

continue through next three pages). ....................................... 78—81

Ideal ion behavior in beam modulation; the process essentially
yields a slice from the continuous beam. Arrow lengths
indicate relative ion velocities; only isomass ions are
considered in this illustration. .................................................... $

Three models for beam modulation techniques. Isomass ions
are shown to pass through a pair of deflection plates and move
toward an aperture. Relative duration and polarity of the
deflection voltage pulse is indicated for each case. The darker
shading indicates ions which will pass through the aperture
and form the ion packet. a: Gate model; b: Impulse Sweep
model; c: Differential Impulse Sweep model. ............................... 92

Two ions entering the deflection plate region with different
trajectory angles. Field reversal occurs when the lower ion is

at point a and the upper ion is at point b. ...................................... %
Illustration of ion behavior modeled by the simulation program
(see text). .................................................................................. 99
Simulated peak shape for m/z 1000. — Bakker effect
only,m Bakker effect with energy dispersion, --- Bakker effect
with angular dispersion, --— — all efi'ects combined ..................... 101

a: Schematic diagram of the Wollnik-type instruments
showing ion source, mirror, detector, and nominal ion path.
b: three-dimensional view of the ion source showing circular
filament and “electron pusher” electrodes. c: Block diagram of
the M-TOF. ............................................................................. 109

Applied potential profile within the Wollnik ion source.
a: storage mode, b: extraction mode .......................................... 110

Simulated peak shapes for m/z 866 from various initial kinetic
energy distributions (temperatures) calculated for parameters
matching the 4 m Wollnik instrument in Giessen. The peak
corresponding to 50 K is most similar to the experimentally
obtained peak. ......................................................................... 111

Geometry (above) and potential profile (below) of the Wollnik
source. In the illustration of potential between G4 and G2, the

xiii

shaded lines represent the storage condition; black lines
represent the potential during extraction. .................................. 112

5-5. Plot of ion packet width (analogous to peak base width) as a
function of the electric field strength between grids G2 and G3
using a temperature of 500 K. The abscissa, “Voltage Factor”,
is the quantity f defined in the text. For f values above 1.06,
A widths at source exit; 0 widths at space-focus plane (see
text). ...................................................................................... 116

5-6. Plot of peak width at half-height as a function of initial ion
temperature calculated for parameters matching the 4 m
Wollnik instrument in Giessen with a two-stage acceleration
field. The horizontal line indicates the experimentally-

measured peak width ............................................................... 118
5-7. Plot of peak width (FWHM) and intensity for m/z 18 as a
function of extraction pulse frequency ........................................ 1%

5-8. Two segments from the mass spectrum of tris(heptafluoro-
propyl)-S-triazine obtained from the M-TOF using time-array
detection. a. m/z 28—32 region (residual air ions), showing
resolution of 920 (FWHM = 12 ns); b. m/z 466—467 region,
showing resolution of 1500 (FWHM = 30 ns). ............................... 12;

5-9. Reconstructed total-ion chromatogram; the nine peaks are:
1. cyclohexanol, 2. bromohexane, 3. 2-cyclohexene-1-one,
4. methoxy-ethyl ether, 5. 1,3-dichlorobenzene, 6. benzyl
chloride, 7. 1,2-dichlorobenzene, 8. acetophenone, 9. benzyl

bromide. Column: DB-5, 5 m x 100 um. Temperature
program: 50°C-150°C, 20°C/min. Sample size: 1 ng of each
component injected on column .................................................. 132

5-10. Mass spectrum of 2—cyclohexene-1-one obtained with the M-
TOF (top); reference spectrum (bottom). ..................................... 133

5-11. Mass spectrum of 1-bromohexane obtained with the M-TOF
(top); reference spectrum (bottom). ............................................ 134

6-1. Illustration of the CVC 2000 mass spectrometer showing
nomenclature for parameters used in the modeling program
by Erickson. Voltages are represented by the V terms and
geometry by the as terms. The distance from an ion’s initial
position to the first grid is 31, and the rise time of the voltage
pulses applied to the first and second grids is t, ........................... 134

62. Resolution and intensity boundaries shown on lag time-m/z
space. The shaded area represents the locus of points at which

6-3.

adequate (see text) peak shapes are obtained. The shaded
curve shows the optimum-focus lag time (proportional to \fnT). ..... 135

Simulated peak shapes for m/z 1000 and 1001 from planar and
gas-phase (electron impact) ion sources. The widths at half-
height are 2.7 and 72 ns, respectively. ........................................ 142

Chapter 1

Introduction

“We live in the age of the sorcerer’s apprentice,
in that we create data faster than we can sweep it away.”
Raymund E. Dessy

1.1 The Problem

Chemical analyses generally involve several steps, including
sampling, separation, measurement, and data processing. The separation
step is often critical, since many measurement techniques respond in some
way to many species present rather than only to the species of interest. In a
conductometric analysis, for example, all ionic species present in the
sample contribute to the total measured current; it is the responsibility of
the analyst to make certain that only charge carried by this species is
significant. One separation method of great importance in many chemical
analyses is gas chromatography (GC). In this technique, sample materials
are vaporized and carried through a column by the flow of a carrier gas.
During this motion, referred to as elation, chemically distinct species in the
sample are separated due to differences in their partitioning between the

2

mobile phase (the gas) and the stationary phase (the column wall or
packing material).

Afier components of a mixture have been separated by GC, some
measurement step is generally taken in which the quantity and perhaps
identity of each species that elutes from the column is determined. Mass
spectrometry (MS) is one measurement tool that can provide both
qualitative and quantitative information about the chromatographically
separated sample components. In an MS technique, the sample molecules
are ionized by one of several methods, of which the most common is
bombardment with fairly high-energy electrons. After ionization, some of
the molecular ions fragment into smaller ions and neutral species; then
the abundances and mass-to-charge (m/z) ratios of all these ions are
determined. The 5_ resulting fragmentation pattern, represented as a
histogram of ion abundance vs. mass-to-charge ratio (termed a mass
spectrum), often serves as a “fingerprint” for identification of the original
compound. In addition, the overall quantity of ions detected may be used to
find the abundance of this species.

The combination of GC with MS (GC-MS) was described recently in
the following manner.

Among the various possible combinations of chromatog-
raphy with different analytical techniques, that of gas
chromatography with mass spectrometry has been the most
successful, useful, and routinely applicable. The reasons
for this are not hard to see. Both techniques use very small
quantities (ng or pg range) of the sample, both require the
sample to be volatile, and the mass spectrometer can respond

asfastastheGCcanelute.1

Indeed, GC-MS is a widely-used analytical methodology. Its applications
span virtually the entire range of chemical analyses, from study of drug
metabolism2 and quantitative metabolic profiling3 to trace analyses of
dioxins in the environment.4 A problem with the combination of G0 with
MS has arisen, however, since the continuing improvements in GC
separation efficiency have not been accompanied by an increase in the
spectral generation rates of MS instrumentation. In particular, the
statement that “the mass spectrometer can respond as fast as the GC can
elute” quoted above is no longer always true, especially if a mass spectrum

is desired for each eluant.

Individual species can often elute from very high-resolution GC
columns, called capillary columns because of their small diameter, in
times on the order of one second. (This refers to the span of time over which
the species actually exits the column, rather than the total elution time
between injection of the sample onto the column and the elution of a species
ofi‘ of it.) It is thus necessary to be able to acquire a mass spectrum in about
one second. 5Moreover, since the exact elution times of all species present
are not known in advance, spectra must be acquired at rates substantially
higher than 1 Hz in order to assure that at least one complete spectrum is
obtained from each of the eluting species. As will be shown below, even this
rate may not be adequate to provide true mass spectra or to sufficiently

determine retention time.

If a device which continuously records the quantity of analyte eluting
from the column is used for detection, a plot of analyte signal intensity vs.
time may be obtained. Each species present is then represented by a peak in
this plot, which is called a chromatogram. Much interest in the field of

4

chromatography is placed on the shapes of these peaks. This can be
complicated somewhat when discrete measurements are used to represent
peak shapes, as is the case when total ion abundances from successive
mass spectra are used to reconstruct a chromatogram. It has been shown
that at least 40 points must be obtained across the elution profile in order to
adequately represent the true peak shape.5 A convenient rule-of-thumb is
that 10 points per standard deviation are needed, assuming the peak shape
approximates a normal distribution. As a result of this, it is desirable to
achieve mass spectral generation rates on the order of 50 Hz for GC-MS
with currently-available capillary columns. As GC technology continues to
improve, the data-generation rates required from MS will become even

more stringent.

The ever-increasing incompatibility between GC and MS time scales
was fully recognized in the literature with the article “Mass Spectrometry
on the Chromatographic Time Scale: Realistic Expectations” which
appeared in 1983.6 In this work, carried out by several researchers at
Michigan State University (MSU), two difficulties associated with an
inadequate mass spectral generation rate were described. The first of these
problems, illustrated in Figure 1-1a, is the inability to fully represent a
chromatographic separation performed by the GC. An idealized “true”
chromatogram representing a plot of concentration vs time is shown on the
left, together with the times at which mass spectral acquisitions have been
made. The reconstructed chromatogram, formed simply by connecting the
discretely measured points, is shown on the right. Since the data
generation rate in this example is too low, the adjacent peaks from

separafid species have been misrepresented as a single peak.

5

The second problem, illustrated in Figure 1-1b, is caused not directly
by the low spectral generation rate, but by the accompanying slow speed at
which each individual spectrum is acquired. Most mass spectrometers
operate in a scanning mode. That is, a signal is obtained from ions of one
m/z ratio at a time; this m/z value is incremented, or scanned, in order to
obtain a complete spectrum. If the scan rate is slow compared to the
elution of the peak, the intensity measurements for individual m/z values
can be skewed relative to those from an ideal spectrum taken very rapidly.
This results from the fact that the partial pressure of the analyte molecules
(essentially the analyte concentration) within the ion source changes

during the scan.

Many mass spectrometers can be operated in a selected-ion
monitoring (SIM) mode in addition to the normal scanning mode. In SIM,
only the abundance of a single, selected m/z (or perhaps a few m/z values)
is monitored. Since no scanning is required to monitor ions of only one
m/z, the abundance can be recorded continuously; thus SIM can be used
obtain true chromatographic profiles. Moreover, the continuous nature of
these measurements leads to a much higher signal-to-noise (S/N) ratio in
SIM than in the scanning mode of data collection. The ideal MS detector for
G0 would be able to achieve the high S/N ratios and hence low
detectabilities found with SIM as well as the complete spectral acquisition

provided by scanning.

 

True .
Chromatogram

Reconstructed
Chromatogram
l |

l l
K Spectral

 

 

 

(a) Acquisitions
Chromatographic
Peak acquisition time Mass SpeCtru-m

H for ideal scan

Ideallllll

\ u -|. o

(b) for one real scan

 

 

Figure 1-1. Two problems in GC-MS caused by slow spectral generation
rates.

 

 

1.2 A Solution—The Michigan State University Plan

After describing the problems of chromatographic detection by MS in
the 1983 publication, the MSU workers went on to examine the various
mass spectrometer technologies and discuss the relative merits of each of
these toward achieving the goal of a 100 Hz spectral generation rate
covering the range of m/z 50—500. Time-of-flight mass spectrometry
(TOFMS) was selected over magnetic sector, quadrupole, and Fourier
transform techniques. It is interesting to note that in the 1950s, the earliest
achievement of GC-MS also made use of a time-of-flight mass
spectrometer;7 TOFMS was chosen because of its relatively high spectral
generation rates at that time as well. As a result of this analysis,
researchers at MSU developed a two-part plan to accomplish high-
resolution capillary column GC-MS using TOFMS. This plan addresses
three major goals: (1) the need for data acquisition equipment capable of
obtaining and recording spectra at the desired spectral. generation rate of
100 Hz, (2) the need to collect all the information made potentially available
from the sample molecules by the time-of-flight mass spectrometer so that
sufficient sensitivity and detectability are achieved, and finally (3) the need
to improve the quality (i.e., mass resolving power, discussed below) of time-
of-flight mass “spectra. As described below, the first two of these goals have
been achieved with the development of time-array detection. The third goal,
improved mass resolving power in TOFMS, is the subject of this

dissertation.

 

 

 

1.2.1 Time-Array Detection

At first, a time-of-flight spectrum exists in the form of a transient
electronic signal generated by ions striking a detector. (As described in
Chapter 2, mass analysis in TOFMS is accomplished by measuring the
arrival times of ions after their “flight” through the instrument; these
measurements take the form of a brief signal from the detector.) The
duration of this signal, called a transient, can be as short as 100 us.
Furthermore, the signal portion representing a single m/z value might be
as short as 10 ns. Mass spectra of this type can be displayed fairly easily on
an oscilloscope screen—this, in fact, was the data acquisition mode used in
the first GC-MS applications—but the digitization and storage of all the
data required to represent 500 m/z values at spectral generation rate of
100 Hz requires novel instrumentation. Since the detection of complete
transients is essentially the acquisition of an entire array of time-encoded
data, the term time-array detection (TAD) is used for this detection scheme.
The first phase of the MSU plan was the design and building of new data
acquisition equipment for TAD. Work on this project has resulted in the
design and construction of an integrating transient recorder (ITR) which
can acquire 10,000 transients each second.8 Several consecutive transients
are added together (hence “integrated”) to yield a scan file analogous to a
conventional mass spectrum. This integration serves both to reduce the
spectral generation rate from 10 kHz to a more manageable level and to
improve the signal-to-noise ratio of the acquired data. For example, if each
1000 successive transients are summed, scan files are produced at a rate of

100 Hz; this meets the desired mass spectral generation rate for GC-MS.

9

Furthermore, each of these scan files will have a S/N ratio improved by a
factor of \l 1000 over that of each individual transient signal.

1.2.2 Improved Mass Resolving Power

The need for improved TOFMS spectral quality is addressed in the
second phase of the MSU plan. As described in Chapter 2, time-of-flight
spectra generated from gaseous samples typically suffer from poor mass
resolving power relative to other MS techniques. Mass resolving power,
which is essentially the capacity to differentiate ions of one m/z value from
those of another, tends to become more and more important at higher
m/z values and thus often limits the effective mass range of a spectrometer.
When TOFMS was first developed, a mass range of a few hundred daltons
could be achieved; this was fairly acceptable in comparison to other MS
technologies at that time. Later, however, achievable mass ranges
increased dramatically for magnetic sector, quadrupole, and Fourier
transform instruments, etc., while the mass range of TOFMS continued to
be limited by its low mass resolving power. The reasons for the poor
resolving power in TOFMS are examined carefully in Chapter 2, and a
large variety of methods which have been used to improve resolving power
are described as well. Unfortunately, these conventional techniques are not
compatible with the analysis of gaseous samples, such as those provided by
GC, or are unsuitable in some other way for TAD. This dissertation
continues from Chapter 3, then, to describe novel improvements to

resolving power in TOFMS for use with GC-MS.

REFERENCES

B. Ravindranath, Principles and Practice of Chromatography, Ellis
Horwood Limited: Chichester, 1989, 175.

W. A. Garland and M. P. Barbalas, in Mass Spectrometry in Biomedical
Research, Ed. S. J. Gaskell, Wiley: Chichester, 1986, 379—403.

J. F. Holland, J. L. Leary, and C. C. Sweeley, J. Chmmatogr., 1988, 397,
3.

L. M. Smith, D. L. Stalling, and J. L. Johnson, Anal. Chem., 1984, 56,
1830.

S. N. Chesler and S. P. Cram, Anal. Chem., 1971, 43, 1922.

J. F. Holland, C. G. Enke, J. Allison, J. T. Stults, J. D. Pinkston, B.
Newcome, and J. T. Watson, Anal. Chem., 1983, 55, 997A.

R. S. Gohlke, Anal. Chem., 1959, 31, 535.

R. Tecklenburg, B. Newcome, M. Davenport, C. G. Enke, and J. F.
Holland, in preparation for submission to Rev. Sci. Instrum.

Chapter 2

Mass Resolving Power in
Time-of-Flight Mass Spectrometry

“Time wasted is existence;
used is life.”
Edward Young

2.1 Introduction to Time-of-Flight Mass Spectrometry

Time-of-flight mass spectrometry (TOFMS) was conceived by
Cameron and Eggers1 in 1948 and developed into a practical technique by
Wiley and McLaren2 in 1955. The technique evolved only gradually for
many years after that time, but it remained in use due to its simplicity and
relatively low cost. Eventually, time-of-flight mass spectrometers fell out of
favor as other types of instruments capable of much higher resolving power
were developed. Prior to the development of TAD, the method by which
computerized data acquisition was performed led to poor sensitivity; this
also contributed to the decline of TOFMS. More recently, however, the
advent of naturally pulsed ion sources (e.g., radioactive decay) which are
well suited to the Weof-flight mass spectrometer, as well as the potential

for TOFMS instruments to provide spectra over a large mass range at

12

acquisition rates far higher than other mass spectrometers, led to a

renewed interest in time-of-flight mass spectrometry.

2.1.1 Principle of Mass Analysis

Ions which have been accelerated to equal energies move with
velocities related to their mass-to-charge ratios; these characteristic
velocities are used for mass analysis in TOFMS. Ions simultaneously

accelerated out of an ion source separate into groups according to their

Acceleration Grids

:5

Ion Source Flight Tube
I Detector

\
52>

® 0 O
.

    
 

 

l

 

 

 

 

 

. h

Spectrum

 

 

Figure 2-1. Overall mass analysis process in time-of-flight mass
spectrometry.

velocities as they travel through an evacuated, field-free tube, as shown in
Figure 2-1. The time elapsed between the extraction of an ion from the
source and its detection at the end of the tube is measured and used to
calculate mass. (Strictly, this calculation yields the mass-to-charge ratio,
m/z.) In a typical commercial TOFMS instrument,3 the energy applied for

extraction is sufficient to cause ions up to about m/z 1000 to arrive at the

13

detector within 100 us of the extraction pulse. The instrument is therefore
capable of producing a signal representing 104 complete mass spectra each
second. With high-speed data acquisition devices such as the ITR, these
spectra can now be recorded with no loss of information. Unfortunately,
the spectra produced by time-of-flight instruments have until recently

suffered from poor mass resolving power.

2L2 Definitionof‘fleeolvingPower”

The degree to which ions with different m/z values may be
difi’erentiated by a mass spectrometer is referred to as the mass resolving
power, or resolution. Several methods of quantifying resolving power are
commonly used;4 the choice of method is often based on personal preference
or local custom. In this work, resolving power will generally be reported in
one of the two ways illustrated in Figure 2-2. The percent valley definition
is most useful for describing digitally simulated peaks, since it relies on the
“valley” formed between two peaks of adjacent integral m/z values and
equal intensity.5 The full width at half maximum definition (FWHM) is
useful for experimentally obtained peaks, and this is the quantity most often
given in the literature. In addition, the term unit mass resolution will be
used. This refers to any resolving power suficient to permit difi‘erentiation
between two peaks of adjacent integral m/z values. It is a less quantitative
term unless some additional information is given, such as “unit mass
resolution with <10% valley”. Regardless of the quantity used to represent
resolving power, the ability to distinguish adjacent masses always

decreases with increasing mass; this results in an upper mass limit for

14

which useful mass resolution can be attained with any particular
instrument.

4—D—>

 

For m2— m1 = 1,

C ¢ %Valley = 100 B/A
FWHM = mD/C

\B
T

<—:>—>

m m
1 2

Figure 2-2. Two methods of quantitating mass resolving power. The
quantity reported as “% Valley” (Percent Valley) is the height of the overlap
formed between two equally intense peaks, expressed as a percentage of the
peak height. The FWHM (Full Width at Half Maximum) value is the .
width of the peak at its half-intensity height, expressed in mass units.

2.2 Ion Behavior Patterns Which Degrade Resolving Power

Ideally, all ions in each isomass group should arrive simultaneously
at the detector. Such coincidence for the detection of isomass ions can be
achieved if all ions start from the same point, have no initial motion, and
are accelerated to the same final energy. Any deviations from these
conditions cause perturbations in the flight times which, unless
compensated for, degrade the capacity to resolve adjacent masses. For ions
created from gaseous samples, the distribution of initial positions in the
ionization region and the distribution of initial kinetic energies due to
thermal motion give rise to a distribution of arrival times within each
isomass group. The processes by which these perturbations occur,
described in detail below, are often referred to as the space and energy

effects, respectively. In addition, the half of the ions having some

15

component of thermal motion directed away from the detector must turn
around under the influence of the extraction field before moving out of the
source. This delay, referred to as the turn-around time, results in a further

spread of arrival times.

Models of the three patterns of ion behavior which lead to decreased
resolving power are illustrated in Figure 2-3. (Presentation of ion behavior
in this manner was inspired by a similar figure of Poschenriederfi) One of
these patterns, case S, is caused by the distribution of initial positions (the
“space effect”). The other two, cases E4» and E-, result from the initial
kinetic energy distribution (the “energy effects”). All ions shown in
Figure 2-3 have the same mass. In each case, a reference ion, located
initially at position so with no kinetic energy, leaves the source with kinetic
energy U, after extraction. Ions which start either at other positions,
so + 63, or with initial energies, 6U, are described by the perturbation

parameters a and 5 defined by

a = (so + GEO/so (2-1)

and

p= w/ o, (2-2)

In case 8, ions which start at some nonideal position aso are considered.
An ion starting farther from the detector than so (a > 1) experiences a
larger potential drop, and thus receives a higher final kinetic energy, than
the reference ion. It catches up to the reference ion at the spacefocus plane
and eventually reaches the detector first. The ions of case E-i- have some
initial motion directed toward the detector. Such an ion having initial
kinetic energy 5U. leaves the source with a final energy (1 + fl)U., so it

16

arrives at the detector before the reference ion. In case E-, the initial
motion is directed away from the detector. The extraction field decelerates
the ion as it travels farther back into the source so that the ion stops, turns
around, and eventually returns to its initial position. At that time, it has
regained its initial velocity but is then moving toward the detector. From
there on, it behaves identically to the case E+ ion, except that the total flight
time has been increased by the turn-around time.

2.3 Previous Approaches to Improving Resolving Power

In 1955, Wiley and McLaren reported the first and, until the past
several years, most important improvements to TOFMS resolving power.
They developed two techniques which, when combined, raised the upper
limit for unit mass resolution to several hundred daltons.

 

 

Source Field-Free Region
F—W A
\
U U
C1139- o o—'> (>4)
Ideal U. U,
0 O——> H

Space-Focus aU

Plane \ ' .—aU—'>
280 Us Us
H

{xx}

 

 

(1 + [3X], (1 + EU,

U. U.

O——> O-—>
.......................... ,q. =4... .4 -...

(1 4» EU, (1 4» EU,

U. U.
O—-> O—>

\_7_J \_T__J
1 2

Figure 2-3. Flight time variations in TOFMS. Lengths of the arrows
indicate relative velocities of isomass ions extracted with energy U. from a
single-field source. Relative positions and velocities are shown at three
different times: ion formation (to), arrival at the space focus plane (t1). and
later in the field-free region (t2).

18

2.3.1 Wiley-McLaren Developments2

In this early work, equations were given for exact flight times in the
source as well as in the flight tube, and the mathematical basis for
existence of a space-focus effect, shown above in Figure 2-3, was explained.
Ions starting at various positions throughout the ionization region “catch
up” to each other at locations which fall on a plane termed the space-focus
plane; in the simple one-field source illustrated here, this action occurs at a
distance 280 past the source exit. One of the Wiley-McLaren developments
was a two-field ion source which caused the space-focus effect to occur at
the detector surface rather than at the original location. In this
arrangement, the second acceleration reduces the relative difference in
velocity between ions formed at different locations; the faster, trailing ions
therefore catch up to the leading ones at the end of the flight path (the
detector surface) rather than at the original space-focus location.
Unfortunately, the location of the space-focus plane is perturbed by initial
kinetic energies.

The second development, time-lag focusing, concerns the initial
energy distribution. In this technique, an adjustable delay is introduced
between the time of ion formation and the extraction of ions from the
source. During this delay period, thermal motion carries the ions to new
positions within the source in such a way that resolving power is increased
for a small, selectable range of masses. (The fact that this focusing is
mass-dependent results in a need for superior mass-independent focusing

techniques such as those described in this dissertation. A study on the size

19

of mass windows which do receive adequate focusing by time lag is reported
in Chapter 6.)

2.3.2 EarhTechniquestorSpaceand/orEnergyFocusing

After Wiley and McLaren’s work, a variety of other approaches were
taken toward decreasing the spread of isomass ion arrival times and thus
increasing the resolving power in TOFMS. These methods, as well as those
of Wiley and McLaren, fall into two distinct categories. Some techniques
are based on altering the initial spatial and/or energy distributions of ions
in the source prior to their extraction into the flight tube. Other methods
are based on compensating for the deleterious effects of the spatial or

energy distributions after the ions have left the extraction region.

2.3.2.1 Changeofinitialion behavior

Those techniques in which the ion behavior is altered prior to
extraction include Wiley and McLaren time-lag focusing, the impulse-field
focusing technique, and ionization methods in which ions are formed in
essentially coplanar positions, such as from solid samples or in molecular
beams. Time-lag focusing is considered here because the positions of the
ions change during the lag time between their formation and their

extraction.

Marable and Sanzone developed the technique of impulse-field
focusing (IFF) in which an electric field pulse of high amplitude and short
duration is used to alter the initial kinetic energy distribution of ions in the
source.7 This short pulse is followed by a longer extraction pulse; ions

ideally remain in the source for the duration of the first pulse and leave

2)

only under the influence of the longer extraction pulse. The first pulse
serves to bring all the sample ions to essentially the same kinetic energy,
relative to their thermal energies. In addition, this pulse causes all the
ions to have velocity components in the direction of the flight path. Thus the
loss of resolving power due to the case E-l- effect is reduced, and the turn-
around problem is eliminated. The remaining problem (Case S) is
compensated for by Wiley-McLaren space focusing. Implementation of [FF
has been impeded by the inability to produce pulses of sufficiently large
amplitude and short duration.8

Fi-1

 

 

 

 

 

 

 

 

ca 300 s~
. UR I
5- L
. .700 6_
4-
,_ . r600 52‘
l 3- r =4-
‘1 ‘ +500: g
2- , °\o
a 2.
1'1 '400
1 .
(3)0 'l'l—fl'l' 300 “90- ,',Tr'
0100200300400500 0100200300400500
Mass (u)' Mass (U)

Figure 24. Three representations of resolving power are plotted against
mass: a: two methods of Sanzone, “R — 1” and “Unit Resolution” b: the
percent valley method.

The methods by which Marable and Sanzone presented resolving
power data were found to be somewhat confusing by this author. This led,
in part, to the development of the “percent valley” expression of resolving
power described above. (Sanzone’s term impulse-field focusing and its
acronym IFF served, however, as inspiration for the name of the dynamic-
field focusing (DFF) technique described in Chap. 3.) Comparison of these

various presentations, using the data of Sanzone and coworkers,9 is shown

21

in Figure 2-4. A common use for plots such as these is to identify the
highest mass value which may be distinguished according to some

predetermined degree of resolution. The “R — 1 method of Sanzone
(Figure 2-4a) is not well suited to this application since the ordinate
asymptotically approaches zero for the masses of interest. The “Unit
Resolution” method is better, but the ordinate value corresponding to the
minimum acceptable resolving power is not readily determinable. The
percent valley method (Figure 2-4b) directly provides the maximum mass
resolvable to any particular degree of overlap—for example, masses up to
450 u can be resolved with a less than a 5% valley between adjacent peaks of

equal intensity.

Liquid and solid ionization techniques such as 2“Cf-plasma
desorption,10 secondary ion mass spectrometry (SIMS),11 and laser
microprobe mass analysis (LAMMA)12 essentially produce ions from a
plane. Resolving power is therefore increased because the initial spatial
distribution of ions in the source is eliminated. The turn-around effect is
also eliminated because initial motion in the wrong direction is prevented
by the liquid or solid phase. Placement of these techniques in this category
occurs because the ionization region has been changed fi'om that of the gas-
phase situation to this new, planar, configuration. Impressive resolving
powers have been reported, but these are due to the use of energy focusing
devices such as the reflectron (described below) in addition to the planar
ionization region. For example, Standing13 reported that resolving power of
3200 for a linear (i.e. , without mirror) instrument incorporating SIMS

increased to about 10,000 with the addition of a single-stage mirror.

22

Another technique in which the overall ion state is radically changed
from that in the gas-phase situation involves the use of molecular beams.
In such a beam, molecules travel with a limited span of trajectories. The
distribution of velocity components perpendicular to the beam axis is
therefore much narrower than the Maxwell-Boltzman distribution of
velocities present in a gas at thermal equilibrium. If ions are formed in a
molecular beam by laser ionization and then extracted along a flight path
perpendicular to the beam axis, high resolving powers are obtained. For
example, Reilly and coworkers reported a resolving power of 4650 for CoHo+
(m/z 78).14 In this work, the authors present a detailed theoretical model of
TOFMS peak shapes based on probability theory both for conventional
electron impact ionization and molecular beam/laser ionization;
experimentally obtained peaks were found to be in excellent agreement with
the theoretical predictions.

2.3.2.2 Changeofionbehaviorafla‘ech‘action

A second classification covers those methods which are based on
alteration of ion behavior after ions have left the extraction region. These
include filtering methods, in which ions having significant deviations from
the desired energies are removed, and focusing methods, in which some
correction in the velocity or flight path of each ion is made to promote
simultaneous arrival of all isomass ions. Focusing techniques which
compensate for either the initial spatial or initial energy distribution are
said to be space focusing or energy focusing techniques, respectively.
Energy focusing techniques generally do not compensate for the turn-

around problem.

23

Energy filtering has been reported with the use of an electric sector. 15
A continuous ion source (i.e., one that is not pulsed) originally part of a
magnetic mass spectrometer was used to form an ion beam and direct it
into the electric sector. The energy-filtered beam emerging from the
electric sector was “chopped” into ion packets by a beam modulation
technique. (This technique alone can be used as a way to obtain improved
resolving power over conventional TOFMS; in this capacity, however, it
does not fit easily into either of the two categories considered here. Beam
modulation is the subject of Chapter 4 below.) These ion packets were
mass-analyzed by time of flight; resolving powers as high as 900 were
reported.

Electric sectors also have been used to accomplish energy focusing.
Several instrument designs have been proposed to provide energy focusing
with the use of magnetic sectors16 and electric sectors.17"19 Benninghoven
has implemented one of these, incorporating a toroidal electric sector and
two field-free regions, for SIMS applications.20 Matsuda has developed a
more complex design which incorporates four tandem electric sectors.21
Implementation of this design has yielded a resolving power of 730 for Xe
ions (m/z 32) from an electron impact ionization source with an ion

transmission of about 90%.22

Either energy focusing or space focusing (but not both
simultaneously) can be achieved by another arrangement of electric fields

termed a reflectron or ion mirror, treated below.

2.3.3TheReflectron

A simplified illustration of reflectron energy focusing is shown in
Figure 2-5. (Reflectron space focusing will be discussed in Section 2.4.) The
device is simply a region having a retarding electric field of sufficient
potential to reflect the ions. Ions starting from the same plane with
difi‘erent initial kinetic energies travel to different depths within the mirror
before turning around. When they exit the mirror, their velocities are the
same as the initial velocities upon entering the mirror. The longer paths
taken within the mirror by ions having larger initial energies compensate
for their shorter flight times in the field-free region. If the mirror field is
correctly adjusted, both the faster and slower ions arrive simultaneously at
the detector. This figure illustrates a “one-stage homogeneous field” ion
mirror—the simplest possible design. As the description indicates, only
one electric field stage is used (in addition, of course, to the field-free region
outside the mirror), and this field is homogeneous. Such a system can
accomplish only first-order focusing. The term first order refers to a
mathematical description of the focusing ability. Expressions for flight
time, which are functions of initial kinetic energy, mass, acceleration
potential, etc., can often be written as series expansions; a system
accomplishing nth-order focusing will focus ions for which all n + 1 and
higher order terms are insignificant. Second order focusing can be

achieved with a two-stage mirror.

Whatever distribution of ion positions and velocities (in the direction
of the ion path) exists at the source is essentially reproduced by the mirror

at the detector . (When discussing ion mirrors, it is often convenient to

%

refer to the first focal plane as the source and the second as the detector,
even though these objects may not actually exist precisely at these
locations.) This explains why the tum-around effect is not eliminated by
mirrors; ions traveling at identical velocities but separated in time are
affected identically by the mirror, so they arrive at the detector with the

same separation they had at the source.

 

Figure 2-5. One-stage ion mirror. Motion of two isomass ions having
different initial kinetic energies is illustrated with relative positions and
velocities at four different times.

Mamyrin and coworkers pioneered the use of ion mirrors in TOFMS
in the early 1970s.23:24 These first experiments used electron impact
ionization and a two-stage mirror design similar to that shown in Figure 2-
6. This instrument achieved a resolving power of 2800 for 199Hg12712+
(m/z 453) and resolving powers greater than 3000 for ReBra clusters having
m/z 1266—1290. Mamyrin later built a linear mirror instrument in which
ions were deflected 180° by the mirror and thus returned essentially anti-
parallel and colinear to their initial trajectory.25 The ions were collected by
an annular detector which surrounded the ion source. This instrument

achieved a resolving power of 1200 for 199Hg127I” (m/z 326) with a smaller

%

flight path length. In 1983, Wollnik and coworkers reported a mirror
instrument similar in design to the first Mamyrin instrument.26
Resolving power of 2000 was reported from this instrument for N2"'

(m/z 28).

 

Figure 2-8. Two-stage ion mirror (“reflectron”). Motion of two isomass
ions having different initial kinetic energies is illustrated with relative
positions and velocities at six different times.

In 1982, Schlag and coworkers reported the first use of ion mirror
TOFMS with multiphoton ionization (MIPI).27 Benzene was introduced into
the instrument in an effusive molecular beam, ionized by MP1, and mass
analyzed by a two-stage ion mirror. Resolving power of 3900 was achieved
for CoHo+ (m/z 78). Two years later, Della Negra and LeBeyec first used an
ion mirror with 2520f ionization TOFMS, or plasma desorption mass
spectrometry (PDMS).28 Resolving power of about 2500 was achieved for
various molecules including nigericin (m/z 724). Two years after that,
Schlag and coworkers increased the MPI-TOFMS resolving power to 10,000
in a collaboration with Bruker-Franzen Analytik.29 Yang and Reilly have
reported mass resolving power of 11,000 for aniline (m/z 93) with a novel
instrument which incorporates laser-induced desorption ionization by

prism internal reflection.30 In this technique, a sample is placed on a

 

27

transparent medium and illuminated “from behind” by the laser pulse.
Probably the highest resolving power yet achieved with TOFMS was
reported recently by Bergman and coworkers.31 Cesium ions (m/z 134)
formed by pulsed laser ionization were focused by a pair of quadrupoles and
mass analyzed using a two-stage ion mirror with a mass resolving power of

35,000.

In almost all implementations except the original Mamyrin designs,
mirrors have been used with ionization methods which produce ions in a
tiny, well-defined region of space. The laser ionization methods form ions
only in the region where the beam is most focused; in PDMS ions are
formed essentially on the surface of the sample foil. With these modes of
ionization, space focusing is essentially unnecessary. There is a
substantial distribution of kinetic energies, of course, but these are
adequately compensated for by the mirror. Gas-phase ionization, such as
electron impact ionization in particular, requires both space and energy
focusing because ions are formed throughout a distribution of locations as
well as with a distribution of kinetic energies. The reflectron alone cannot
accomplish this complete focusing—in fact, no technique described above is
capable of complete focusing—for reasons described in the following

section.

2.4 Requirement of Non-Static Fields for Complete Focusing

In Figures 2-5 and 2-6, the starting positions of the ions in the mirror
instruments are coplanar, and this plane (indicated by the dashed line) is
perpendicular to the flight path. This situation is identical to case E 4-
described above. Figure 2-7a illustrates the transformation of case S into

%

case E-I- at the space-focus plane. Because of this transformation, an ion
mirror can accomplish space focusing, rather than energy focusing, for
ions which start from various initial positions; in such a case, one can
think of the space-focus plane as a “virtual source” for the mirror. As
illustrated by Figure 2-7b, ions having a given initial kinetic energy also
come into spatial focus at a space-focus plane, but the location of this plane
differs from that for ions of different energy. Any real distribution of initial
energies would result in a distribution of space-focus planes; since the
mirror simply reproduces its source (or virtual source) image, a
distribution of arrival times corresponding to this distribution of planes
would be obtained. This leads to the unfortunate conclusion that an ion
mirror can accomplish either space or energy focusing, but not both

concurrently. (Perhaps a compromise between the two, however, can be

achieved.)

0 H
(a) . II
. I
Ob EM
0» :M
(b) 0» EH

Figure 2-7. a: lens starting at rest from different positions within the
source (case 8) behave identically to case E-i- ions starting from the space-
focus plane. 1): Similar behavior occurs for ions having some initial
kinetic energy, but the location of the space-focus plane is shifted.

This conclusion is a specific case of a more general result developed
by Stein.32 This work showed that achievement of simultaneous space and

energy focusing is impossible through the use of static (time-invariant)

29

fields. This applies to any conservative force field, including electric and
magnetic fields. The time-lag technique does result in some degree of both
space and energy focusing, but only over a limited mass range for each
value of the lag time. (Moreover, the delayed application of the extraction
voltage probably prohibits consideration of time-lag as a static-field
technique.) Certainly concurrent space and energy focusing cannot be

achieved in a mass-independent manner by the use of static fields.

For many applications which incorporate planar sources, Stein’s
conclusion is not a serious problem because space-focusing is not needed.
For gas-phase ionization methods, however, it implies the need to explore
the possibility that dynamic (time-variant) fields could provide

simultaneous space and energy focusing. ‘

2.5 Methods Which Incorporate Dynamic Fields

Exploration into the use of dynamic fields has so far resulted in
techniques which incorporate time-variant acceleration either in the source
or in a region past the source. Within-source dynamic acceleration has
been pursued through the IFF technique described above. Post-source
acceleration has been implemented in several ways—the velocity
compaction technique of Muga, the post-source pulse focusing method of
Kinsel and Johnston, and the dynamic-field focusing technique described
in this dissertation.

In velocity compaction, a dynamic acceleration field is used to
equalize ion velocities within each isomass packet (hence “compaction” of
the velocity distribution) in order to reduce the spread of arrival times.33

Velocity compaction was carefully studied prior to development of the

so

dynamic-field focusing technique, and a detailed description of it appears in
the next chapter. Kinsel and Johnston have recently reported a new
technique termed post-source pulse focusing in which a large section of the
flight tube is held field-free, while a large range of isomass packets travel
into it, followed by application of an accelerating electric field throughout

this same region.34

2.6 Summary

This Chapter describes techniques for improvement of mass
resolving power in TOFMS which have been used since its inception in 1948
until mid-1989. Though not exhaustive, this review does include all the
major approaches that have been taken throughout this period. Despite
these considerable efforts by many investigators, the resolving power of
TOFMS has not yet reached the levels more easily attained by several other
types of mass spectrometers—Fourier transform and sector instruments in
particular—except under special circumstances such as those of laser
ionization. The research goal for this dissertation has not been to elevate
TOFMS to the level of these “high resolution” instruments but rather, as
described in Chapter 1, to achieve a level of resolving power adequate for
GC-MS.

14

REFERENCES

A. E. Cameron and D. F. Eggers, Jr., Rev. Sci. Instrum., 1948, 19, 605.
W. C. Wiley and I. H. McLaren, Rev. Sci. Instrum., 1955,26, 1150.
CVC 2000, CVC Product Inc., Rochester, New York.

S. T. Mullen and A. G. Marshall, Anal. Chim. Acta., 1985, 178, 17.

This method for quantification of resolving power was developed by
Wendy Peiffer (the author’s wife).

W. P. Poschenrieder, Int. J. Mass Spectrom. Ion Phys, 1972, 9, 357,
Fig. 1.

N. L. Marable and G. Sanzone, Int. J. Mass Spectrom. Ion Phys, 1974,
13, 185.

J. A. Browder, R. L. Miller, W. A. Thomas, and G. Sanzone, Int. J.
Mass Spectrom. Ion Phys, 1981, 37, 99.

J. A. Browder, R. L. Miller, W. A. Thomas, and G. Sanzone, Int. J.
Mass Spectrom. Ion Phys, 1981, 37, 99, from Figs. 6 and 7.

R. D. Macfarlane and D. F. Torgerson, Int. J. Mass Spectrom. Ion
Phys, 1976, 21, 81.

B. T. Chait and K. G. Standing, Int. J. Mass Spectrom. Ion Phys, 1981,
40, 185.

J. Rosmarinowski, M. Karas, and F. Hillenkamp, Int. J. Mass
Spectrom. Ion Processes, 1985, 67, 109.

X. Tang, R. Beavis, W. Ens, F. Lafortune, B. Schueler, and K. G.
Standing, Int. J. Mass Spectrom. Ion Processes, 1988, 85, 43.

R. B. Opsal, K. G. Owens, and J. P. Reilly, Anal. Chem., 1985, 57, 1884.

J. D. Pinkston, M. Rabb, J. T. Watson, and J. Allison, Rev. Sci.
Instrum., 1986, 57, 583.

16

17

18

19

Z)

32

W. P. Poschenrieder, Int. J. Mass Spectrom. Ion Phys, 1971, 6, 413.
J. M. B. Bakker, Int. J. Mass Spectrom. Ion Phys, 1971, 6, 291.
W. P. Poschenrieder, Int. J. Mass Spectrom. Ion Phys, 1972, 9, 357.

L. R. Kilius, E. L. Hallin, K. H. Chang, and A. E. Litherland, Nucl.
Instrum., Meth., 1981, 191, 27.

P. Steffens, E. Niehuis, T. Friese, D. Greifendorf, and A. Benninghoven,
J. Vac. Sci. Technol. A, 1985, 3, 1322.

T. Sakurai, T. Matsuo, and H. Matsuda, Int. J. Mass Spectrom. Ion
Processes, 1985, 63, 273.

T Sakurai, Y. Fujita, T. Matsuo, and H. Matsuda, Int. J. Mass
Spectrom. Ion Processes, 1985, 66, 283.

V. I. Karataev, B. A. Mamyrin, and D. V. Shmikk, Sov. Phys. Tech.
Phys, 1972, 16, 1177.

B. A. Mamyrin, V. I. Karataev, D. V. Shmikk, and V. A. Zagulin, Sov.
Phys. JETP, 1973, 37, 45.

B. A. Mamyrin and D. V. Shmikk, Sov. Phys. JETP, 1979, 49, 762.

W. Gohl, R. Kutscher, H. J. Laue, and H. Wollnik, Int. J. Mass
Spectrom. Ion Phys, 1983, 48, 411.

U. Boesl, H. J. Neusser, R. Weinkauf, and E. W. Schlag, J. Phys.
Chem., 1982, 86, 4857.

S. Della N egra and Y. LeBeyec, Int. J. Mass Spectrom. Ion Processes,
1984, 61, 21.

K. Walter, U. Boesl, and E. W. Schlag, Int. J. Mass Spectrom. Ion
Processes, 1986, 71, 309.

M. Yang and J. P. Reilly, Int. J. Mass Spectrom. Ion Processes, 1987,
75, 209.

T. Bergmann, T. P. Martin, and H. Schaber, Rev. Sci. Instrum.,1989, 60,
792.

R. Stein, Int. J. Mass Spectrom. Ion Phys, 1974, 14, 205.
M. L. Muga, Anal. Instrum., 1987 , 16, 31.

33

34 G. R. Kinsel and M. V. Johnston, Int. J. Mass Spectrom. Ion Processes,
1989, 91, 157.

Chapter 3

Dynamic-Field Focusing

“T he algorithms are only
human.”
John F. Holland

3.1 Models for Space and Energy Focusing

In the broadest sense, there are at least four approaches to the post-
source modification of ion behavior for improvement of resolving power in
TOFMS. Each of these involves a change in some aspect of distance, time,
velocity, or acceleration. (For simplicity, higher derivatives have not been
considered.) For example, an ion having excess kinetic energy may be
allowed to travel over a longer distance or for a longer time, etc., in order
for it and its reference ion to arrive at the detector simultaneously. Each of
the techniques categorized in Chapter 2 as a “change of ion behavior after

extraction” may be described by one or more of these models.

 

 

 

 

 

35

3.1.1 Four Post-Source Focusing Models

Ion behavior in a TOFMS instrument employing no focusing
elements and a single-field source is illustrated in Figure 3-1. Two case E+

ions are considered here, but this type of presentation is applicable to case

 

 

5’
ii
§

Field-

Region

Path Length
3

t t

m m
1 2 .

Time

Figure 3-1. Illustration of ion motion in a conventional TOFMS instrument.
Two case E+ ions for each of two masses m1 and m2 are illustrated. Ions begin at
the origin (time and distance both zero) in the source. They are accelerated out of
the source and move with constant velocity through a field-free region. The
shaded lines represent motion of ions with non-zero initial energies; these ions
achieve higher final velocities and thus reach the detector before the reference
ions (represented by solid lines). Motion stops after ions traverse a certain path
length and strike the detector.

121- and S ions as well. Progress along the flight path (Simply a straight line
in this case) is plotted against time. The two ions become more and more
separated in space as time increases; they finally strike the detector—
reaching the same coordinate along the flight path—at different times. At
any point along the graph, the slope of the path length vs time curve repre-

$

sents the instantaneous velocity of the ion. For example, the ion motion is
represented by straight lines past the source; velocity remains constant in

this region.

Conceptual models of the four approaches to post-source ion behavior
modification are illustrated in Figure 3-2. The path length adjustment
model (Figure 3-2a) provides a shorter path length for slower moving ions,
and vice versa, within each isomass group. Focusing with electric fields
falls in this category. (Electric sector focusing also involves preferential
acceleration and deceleration or slower and faster moving ions, respec-

tively, in addition to path-length adjustment.1)

In the remaining three models, all ions travel through the same
distance. The time-delay adjustment model (Figure 3-2b) involves the
temporary removal of ions from the flight path, followed by their reinsertion
at the appropriate times. The ion mirror can be described in terms of this
model if the time spent within the mirror is considered to be the delay time;
faster ions are delayed by the mirror longer than slower ions within each

isomass group.

In the velocity adjustment model (Figure 3-2c), a time-dependent
acceleration field increases the velocity of each ion. Within each isomass
group, trailing ions receive slightly more acceleration than leading ions, so
these trailing ions catch up to the leading ones as they reach the detector
surface. Velocity compaction is closely related to this model. Focusing
could also be achieved by deceleration; trailing ions would receive a smaller
decrease in velocity than leading ions. The fourth model, acceleration

adjustment (Figure 3-2d), would be similar to velocity adjustment except

37

that the ions would experience non-zero acceleration at all times. The
magnitude of this acceleration would decrease (or increase) during the

flight to focus the ions.

Path Length

 

 

 

 

 

 

 

"'2 (d) t": 5.2

1
Time Time

Figure 3-2. Four models for focusing by post-source modification of ion
behavior. For each case, motion of two case E+ ions for each of two masses
m1 and m2 is illustrated as in Figure 3-1.

38

3.1.2 VelocityAdiusiment: BestSuiwdtoDynamicFields

The key to achievement of concurrent space and energy focusing in a
mass-independent manner, rather than just for a small range of masses,
is the use of non-static fields. In addition, the ability to effect some degree of
mass separation prior to any compensation for nonideal ion behavior would
permit the appropriate degree of compensation to be applied specifically to
ions of each mass. Path length adjustment with electric sectors requires a
constant electric field, so mass-independent space and energy focusing
cannot be achieved using this approach. Time delay adjustment using an
ion mirror seems to be incompatible with the use of non-static fields, since
ions having a wide range of masses could reside inside the mirror at any
given time. Dynamic fields appear to be inapplicable to acceleration
adjustment as well, again because the dynamic acceleration could affect a
large mass range. In addition, this method would require complicated
instrumentation. The velocity adjustment method is compatible, however,
both with the use of non-static fields and with mass separation prior to ion

behavior modification.

In an instrument incorporating post-source velocity adjustment,
isomass groups would be separated in a field-free region prior to post-
source acceleration. They would then pass through a short acceleration
region and continue into a second field-free region. A time-dependent
acceleration function would be designed to bring about focusing for ions of
each mass individually, since only a small range of masses would reside in
the acceleration region simultaneously. This method of ion behavior modi-
fication was chosen for detailed study, and a model for dynamic post-source

$

acceleration has been developed. Mathematical analysis and computer
simulations based on equations from this model indicate that the method
can provide substantial mass-independent focusing for space, energy, and

turn-around effects. A complete description of the study is presented below.

3.2 Investigation of Post-Source Velocity Adjustment

This investigation of post-source velocity adjustment began with a
detailed study of Muga’s velocity compaction technique.2 The goal, at first,
was to thoroughly understand the technique in order to improve upon it, or
at least be able to implement it at MSU. In addition to the mathematical
analysis of velocity compaction, a computer simulation was developed
which verified the velocity compaction effect. A superior approach to post-
source velocity adjustment was discovered and pursued, however; mathe-
matical analysis and computer simulations then led to development of a

new technique termed dynamic-field focusing (DFF).3

3.2.1 Study of Muga’s work

The essential components of the velocity compaction time-of-flight
mass spectrometer are shown in Figure 3-3. Ions are created by electron
impact in the source and are drawn into the first field-free region by a
single extraction field. The ions begin to separate into isomass groups as
they pass through this region, just as in conventional TOFMS. Ions within
each isomass group enter the dynamic field region at a distribution of times
and with a distribution of velocities due to the case S, E+, and E- effects.

The monotonically increasing acceleration field serves to equalize the ion

4D

velocities by preferentially accelerating slower ions within each isomass
group.
Source FF1 Dynamic Field FF2 Detector

/ ,_/1 r“ .I\.

 

 

 

 

 

 

 

 

Figure 3-8. Schematic diagram and voltage profile for the velocity com-
paction TOFMS instrument (diagram adapted from ref. 4). lens are
created in the source, accelerated by a small electric field into the first
field-free region (FF1), experience post-source acceleration in the dynamic
field region, and finally travel through a second field-free region (FF2) to
the detector.

3.2.1.1 Theory of Velocity Compaction

Consider an isomass ion packet traveling through the first field-free
region, FF1. The packet begins to separate from other isomass packets just
as in conventional TOFMS. The packet begins to spread out in space and
time since the ions within it have a distribution of velocities. Muga derived
an expression to show that the distribution of ion velocities could be reduced
if the ions experience a particular time-dependent force in the dynamic
field region given by a voltage (applied between FF1 and FF2) having the
rate of change

41
fl = 2mvl
0" m2

where m and v are the mass and mean velocity, respectively, of ions in the

(3-1)

isomass packet, and r is the flight time through the dynamic field region
having length I. This rate-of-change dV/dt is designed to have the following
efi‘ect: two ions which enter the dynamic field region simultaneously but
with different velocities will be accelerated to identical velocities as they
leave. Their exit from the dynamic field region, however, will take place at
different times despite their simultaneous entrance into the region. For

ions extracted from a single-field ion source, Eq. 3-1 becomes

8V
9% = {,- x/ 3"" (3-2)

where V, is the extraction voltage. The variable 1' was eliminated by an

 

unpublished approximation to yield

g—=i-K 2er

dt 1 m (3-3)

 

For ions starting at a distance so from the source em’t and for an FF1 region
of length d, the mean arrival time of the isomass packet is given by

t = (280+d) 72an (34)

Thus mass may be eliminated from Eq. 3-3 to yield the final differential

equation
d V I V _

This equation describes a family of voltage functions which could bring
about velocity compaction. Using the boundary condition that V(t) = Vo at

42

the time ions of mass 1 u enter the dynamic field region, Eq. 3-5 is
integrated to yield the velocity compaction voltage function
V

0 tr

V(t) = 7(— t (3'6)

r

where K = [(230 +d)/\/2er/ 1 am] , r = [(230 + d) / l], and 1 man is the

 

value of the atomic mass unit, 11. Although V(t) is not linear, it is assumed
that during the short time any particular isomass packet spends within the
dynamic field region, V(t) may be approximated with a linear function
having the slope given in Eq. 3-1. The complete derivation of Eq. 3-6 from
first principles, which is rather long, remains the property of M. L. Muga.

3.2.1.2 Digital Simulation of Velocity Compaction

Little experimental data had been presented to demonstrate velocity
compaction at the time this study began, so a digital simulation was
developed to verify Muga’s results, most of which were simulated as well.
A complete listing of the program, which was written in PDP-ll
FORTRAN-7 7 , appears on pages VC 1—VC 13 of the appendix. The heart of
the program is the function FTIME(M, Q, s, U, DIRECT), described
below, which returns the flight time through the instrument for an ion of
mass m, charge q, initial position 8, and initial kinetic energy U. The
argument DIRECT is set to +1 or -1 to indicate initial motion toward or away
from the detector, respectively. The user specifies means and standard
deviations for the initial spatial and velocity distributions, which are
assumed to be normal (Gaussian), as well as the ion mass and limits for
the time axis of the output peak shape. The assumption of a normal velocity
distribution is acceptable if the ionization process does not greatly disturb

43

the Maxwell-Boltzmann distribution of velocities which, in the case of

motion in one dimension only, is a normal distribution having a mean of

zero and a standard deviation 0,, of /kBT/m . The choice of a normal spatial

distribution has less fundamental support, but it seems reasonable as well
as convenient. The program constructs cumulative distribution functions
(CDF’s) for these normal distributions and stores them in two arrays.
Various initial positions and velocities are then chosen by iteration along
the ordinate of these CDF’s followed by calculation of the corresponding
spatial or velocity values, as shown in Figure 3-4. This has the effect of
selecting initial conditions which follow the desired distributions; e.g.,
more values are chosen near the mean than far from it. Each chosen
velocity is used to generate an initial energy as well as to select a value for
DIRECT. Flight times corresponding to (position, velocity) pairs are calcu-
lated and tabulated in an array to form a histogram representing the peak
shape. If desired, standard deviations of zero may be specified for either the
energy or space distribution. This would permit the study of efl‘ects due to
either the spatial or energy distribution alone, respectively. For example,

using a standard deviation 0. = 0 for the spatial distribution causes the

program to simulate a planar source.

The function FTIME () calculates flight time through the instrument
as the sum of flight times in the source, the two field-free regions, and the
dynamic field region. Flight time in the source is

t _ s(2m)1l2
‘ ' [ 1’2 (3-7)

8
‘8‘;er + Ui] i ((4)112

 

 
  

 
 
  

pm 222225252?

    

 

 
 

Figure 8-4. Selection of values corresponding to a distribution function
plx]. The curve represents a cumulative distribution function

PIX] = Dam. Equally spaced values of PM (i.e., a uniform distribution)
correspond to a p[x] distribution of x.

and that in FF1 is
t _ d(m)1/2
d

 

' 1/2

{3% er. + U2] (3.8)
After calculating t. and t4 from Eqs. 3-7 and 3-8, the function FTIME () calls
a subroutine LTIME () which calculates t1 and the final velocity Of. These
are returned to FTIME () which calculates tf as f/vf and returns the total
flight time tm = t. + ta + t: + tftO the calling program. The function LT IME ()
could calculate flight time through the dynamic field region in two ways.
Runge-Kutta integration could be used to simulate ion motion through

many consecutive steps of duration At. This method was used at first. A

faster (though still iterative) method was later derived as follows;

The dynamic voltage V(t) is proportional to t", or 1‘10 when r = 1/2, as
is usually the case. Letting 1' represent the time the ion has traveled within
the dynamic field region and letting t“; = t. 4» t4, so that t = 1' + tug, we have

2
1/2
Sit—g = d“. tad) (3'9)

where x is coordinate representing distance into the dynamic field region

eV
0 . .
and c = m l K' Integrating Eq. 3-9 yields

 

d_x (310)

-k 3’2
(fi-3(T+t‘d) +1:

3/2
To evaluate x, 1is set to zero yielding K = v0 - %td , where 00 is the velocity

into the dynamic field region. Integrating Eq. 3-10 yields

-22 5’2 341
x-15(1'+t.d) +rt+11 ( )

5/2
Again setting 1'to zero reveals thatl=-£1§-t.d.

completely through the dynamic field region (:1 = l), 1' represents the total

When the ion has passed

flight time tz. Thus t1 may be found by setting an in Eq. 3-11 to l and solving

for 1'. Then the final velocity,vf, is calculated from Eq. 3-10 as [%J

1’- it

Several variables in the program are set to the values given in
Table 3-1, and values for dynamic voltage function are calculated from
Eq. 3-6. Upon completion, the program outputs the histogram to an ASCII
file as a listing of (time, intensity) pairs which may be plotted to yield the
peak shape.

46
TI‘lalle 3-1. Values used by program VC‘IfiF.

 

so Center of ionization 0.01 m
d FF1 length 0.02 m
l Dynamic-field length 0.08 m
f FF2 length 1.30 m
V; Extraction voltage 10.0 V
V0 Reference voltage 70.0 V

 

Sample output from program VCTOF is shown in Figure 3-5. Here
peaks simulated for m/z 600 and 601 from the velocity compaction instru-
ment are compared to simulated peaks for a conventional Wiley-McLaren
instrument without time lag focusing. These peak shapes were generated
by a program similar to VCTOF. Comparable results were obtained for
other masses. This demonstrated that velocity compaction should, indeed,

significantly improve resolving power over a large mass range.

3.2.2 Empirical Studies of Dynamic Acceleration

Encouraged by the validation of the velocity compaction effect, a
series of empirical studies on ion behavior with dynamic acceleration was
begun. In each of these, various “cases” of ion behavior were studied. As
will be described below, these classifications eventually led to the model

cases S, 13+, and E-.

47

 

 

 

 

2 , J. .

' I
71.07 71.17 71.27 147.6 147.7 147.8 147.9

(a) Flight'l‘ime (#860) (b) Flight'rime (usec)

 

Figure 3-5. Simulated peaks for m/z 600 and 601 from a conventional
linear TOFMS instrument (a) and from the velocity compaction instru-
ment (b). In each case, 0., corresponds to 500 K and a. = 0.01 cm. .

3.2.2.1 Cases VC, A, and B

The first of these studies examined the three cases illustrated in
Figure 3-6. Case VC represents the velocity compaction efi'ect. Cases A and
B represent other effects which seemed to be superior since, in each of
these, the ions under consideration reach the detector simultaneously.
Programs were written, based on the function FTIME, which would report
final velocities, positions, etc, for ions entering the dynamic field region at
specified times with specified velocities. With these programs, empirical
searches were carried out to find voltage functions which would cause any
of the case VC, A, or B efi‘ects to occur. Only linear voltage functions were
considered, since as above it was assumed that the isomass packets would
pass through the dynamic field region rapidly compared to any nonlinear-
ity in the rate of voltage change. These studies therefore required only the

48

search for appropriate values of V; and k to yield a voltage function
V(t) = V,- + kt ‘which would bring about the desired efi'ect.

Dynamic-Field

Region
”-1 .——> v1
0 t f v0 = v1
VC ”3 .—-D a? f f
Detector
01 v1
A 0 i H f
0,- r—bv;
0
vi v1
B ' ’ of
H”!

 

Figure 8-6. Models of ion behavior used for initial studies. In case VC, two
isomass ions enter the dynamic field region simultaneously with different
velocities but leave with equal velocities. In case A, the ions enter the
region as in case VC, but the initially slower ion is accelerated to a higher
velocity so that it catchs up to the other ion at the detector. In case B, two ions
enter the dynamic field region separated in time but with equal velocities;
as in case A, the trailing ion is accelerated and it catches up to the leading
ion at the detector.

The velocity into the dynamic field region for an ion of m/z 100 and a
value for V, of 10 V is 4393 m/s. The instantaneous value of the velocity
compaction voltage when the ion enters this region (using the same
parameters as in Table 3-1) is 221 V, and dV/dt at this time is, from Eq. 3-5,
12.1 V/us. The first test in this study was to obtain this dV/dt, now repre-
sented as k, empirically. Dynamic voltage functions V(t) = V.- + kt with

V.- = 221 V and various values of I: were used to calculate final velocities v:

49

and v; for two ions entering the region with kinetic energies of 10.0 and

10.1 eV, respectively. The relative difference in final velocities was calcu-

lated for each 11; results are shown in Table 3-2.
Table 3-2. Efiei ct of voltage rise on final velocities.

 

Case VC
v1 — v°
_L_£
v°
k f
0. Was 219.x10-6
10. 82.
20. —6.
30. -51.
40. -102.
50. —133.

 

The relative velocity difi'erence passes from positive to negative between
I: = 10 and k = 20 V/us, indicating that a value of zero (corresponding to the
desired 11) should occur in this range. More detailed study identified the
desired It to be 19.9 :I: 0.1 Was. The large difference between this empiri-
cally obtained k and the value obtained from Eq. 3-5 (12.1 V/us) was eventu-
ally found to be due to the approximation for 1' made in the derivation of
Eq. 3-5. (This part of Muga’s derivation is not in the public domain.)

The program used to study case VC was modified slightly to report
arrival times at a specified point along the flight path (representing the
detector location). Again using parameters from Table 3-1 (including, in
particular, V,- = 221 V and f = 1.3 m), the following relative arrival time

difi‘erences were obtained.

50
iT‘ahle 8-3 Efie' ct ofE voltage rise on arrival times.

 

 

Case A
t1 ' to
k to
0. V/us -293.x10-6
10. -168.
20. —86.
30. —41.
40. 7.
50. 36.

 

These data indicate that the case A effect (which results in a relative arrival
time difference of zero) can be made to occur, but at a higher voltage sweep
rate (30—40 V/us) than needed for case VC. Further study identified It for
this case to be 40.3 i 0.3 Was. The study was repeated using a lower initial
voltage of V,- = 100 V. For this case, 12 was found to be 20.8 :t 0.4 V/us. A
series of such studies were then undertaken to obtain values of k as a func-
tion of V; for all three cases VC, A, and B. The results are plotted in Figure
3-7. For any particular initial voltage V;, a lower sweep rate is needed to
bring about case B behavior, a somewhat higher rate to equalize the veloci-
ties, and a much higher rate to achieve case A. Note that for any V,-, the
sweep rate corresponding to case V0 is very nearly the mean of the the case

A and case B sweep rates.

51

 

 

50 "
0 VC
. A i
40 " ‘ B I
3 3° ' I
\ ..
B i
M 20 d I .
I I
10 " i ! V ' I
.4
. I
o I T U I ' I j l 1 I
0 50 100 150 200 250
V-initial (Volts)

Figure 3-7. Relation of k (dV/dt) values to initial voltage for cases VC, A,
and B. Parameters other than V,- are those in Table 3-1.

From these results it would first appear that the velocity compaction

voltage function brings about an improvement to resolution not only due to

the intended case VC efi‘ect alone, but also because this voltage is a
“compromise” between the case A and case B effects. Recall, however, that
the case VC values obtained empirically do not correspond to any actual
velocity compaction voltage function due to Muga’s imprecise approxima-
tion for 1'. The disparity between the different k’s needed to bring about each
Of the three efi'ects decreases with decreasing Vi, indicating that some
optimization of the various instrumental parameters might yield superior

re sults. Before attempting to achieve such an improvement, however, it

52

was decided to study ion behavior through the entire instrument (including
the source) rather than only from the dynamic field region onward.

3.2.2.2 CasesSandE

Two new cases of ion behavior were conceived and investigated in

this new study. These are case S and case E effects (referring to space and

 

Source FF1 Region DFF Region FF2 Region Detector
O H
S - O
4—d —><—z———~ f—u—>
O» ' H
E d-O O—>

Figure 3-8. Two new models of ion behavior used for empirical studies. In
case S, two ions begin at rest from different positions within the source. They
travel through FF1 (which may extend beyond the space-focus plane, as
pictured here) and enter the DFF region at different times and with different
velocities. They are accelerated in this region and travel with new
velocities through FF2, finally arriving simultaneously at the detector.
Case E is similar except that the ions start from the same position but with
positive and negative initial velocities.

energy, respectively) illustrated in Figure 3-8. The patterns of ion behavior
seen in each of these cases were ascertained with the help of the DFF
Animation Program described below. The dynamic field region was given
the name “DFF region” in anticipation of the “dynamic-field focusing"
technique.

Empirical searches for appropriate k and V,- values were carried out

fOr cases S and E using values of so = 0.01 m, d = 0.02 m, f: 0.08 m,

53

l = 1.3 m, and m = 100 u. The study was repeated with d = 0.04 m. As

shown in Figure 3-9, alteration of d changed the k-V} relationship for case
S but not for case E. This occurs because the case E effect is essentially a
compensation for the turn-around effect. The ions having initially opposite
directions of motion travel through FF1 with identical velocities. The
spatial separation they acquire during extraction remains constant
throughout this region, therefore the voltage function appropriate to cause
focusing for these ions is independent of the FF1 length. The fact that the k
value for case S changes as a function of d, but the value for case E does not,
indicates that perhaps a d value exists which would make the k values
equal for both cases (and for a specific Vi). If this were true, then a single
voltage function having the appropriate form could completely focus the
isOlfnass packet. Ideally, this d would be independent of mass so that all

masses could be focused.

 

 

4o-
. S(short) i
' S(long)
ao- ‘ E
‘3 q
E i
a
10 i i
I I
'1 I ‘
0 v J I r I 1 l ' l
o 50 100 150 200 250
V-initialWolts)

Figure 3-9. Relation of k values to initial voltage for cases S and E. For the
data labled “S (short)” and “S Gong)”, d = 0.02 m and 0.04 m, respectively.
Identical points were obtained with both values of d for case E. Parameters
other than V; and d are those in Table 3-1.

Using values of so = 0.01 m, f: 0.08 m, l = 1.3 m, and m = 100 11,
values for It were calculated with V; = 100 V and 221 V for both case S and
case E. The results, plotted in Figure 3-10, indicate that with both V,- values,

k ’8 for the two cases become equal (within the precision of the calculations)
V"hen d is approximately 0.16 m. Moreover, the functions seem to converge
as d increases beyond this value. This convergence can be explained as
t‘Ollows: Consider the plots of kinetic energy vs separation distance for case
S and case E ions shown in Figure 3-11. For case E, this plot is always a
h~<Jrizontal line, since both ions have the same kinetic energy. For case S,

55

the plot is a slanted line since the leading ion has more kinetic energy than
the trailing one. As the ions travel through FF1, however, the separation
between them increases; this reduces the slope of the energy vs distance
plot making it become more similar to the case E plot. After traveling an
infinite distance in FF1, the case S plot would become completely horizontal
and thus k values for case S and case E approach each other as d

approaches infinity.

0 3(100)
A 3(221)
— E (100)
I --- 5(221)

10-

(IV/d!

 

 

v v I ' I V I ' i
0.00 0.04 0.03 0.12 0.16 0.20
FF1 Length (m)

Figure .3-10. Dependence of It on FF1 length, d, for cases S and E and with
V; = 100 V and 221 V. Values for case S approach infinity at d = 0.02 m, the
location of the space-focus plane. Values for case S and case E appear to

converge at d = 0.16 m.

56

 

 

 

 

 

 

Case E Case S Case S
Early in flight Later in flight
.53 Q 9. a 2 B
E E .———C E 5:3 / E g /
it? re E! rx: :2 a:
separation separation separation

Figure 3-11. The two ions in case E have the same kinetic energy so the plot
of energy vs separation distance is a horizontal line. In case S, the two ions
have different energies, thus the plot is slanted. As the ions move further
along the flight path, however, their separation increases. This reduces the
slope making it more similar to that for case E.

Since the two k values become quite similar at d = 0.16 m, it would
seem that using this FF1 length would enable complete focusing for 100 u
ions, This would be true, however, only if focusing of the somewhat
academic case E and case S ions happens to focus ions having all other
Possible initial conditions as well. In order to ascertain whether this was
true, a program was written to simulate peak shapes using the newly
discovered instrumental parameters. Indeed, complete focusing is
obtained by a single voltage function when the FF1 length is set to 0.16 m.
Furthermore, this same value is appropriate for a large range of masses.
Slietches of several peak shapes generated by the program with m = 600 u
are shown in Figure 3-12. In the best case, a resolving power of about 7500
was indicated for this mass.

5'7

FF1 = 0.02 m FF1 = 0.16 m
VC E mid S E mid S

Energy_l__L_/—_—~_;=_L_L_L
Space AAA_L_1__L_L
Both AAAQJLLJL

Figure 3-12. Peak shape sketches for m = 600 11 based on data from the sim-
ulation program. Peaks in the first row were generated using only an
initial energy distribution (i.e., a planar source); those in the second row
arise from an initial spatial distribution. Peaks simulated for a real
gaseous source are shown in the third row. Peaks were generated using 1:
values for both case E and case S and also using the mean of these two.
Velocity compaction results (from VCTOF) are shown in the first column
for comparison.

 

 

3.2.3 The Infinitely-Thin DFF Region

The results obtained up to this point were quite promising, in that
better than unit mass resolution could apparently be achieved for various
masses with the use of appropriate instrumental parameters and the
DI‘Oper voltage function. These results were obtained by empirical selection
01' the parameters, however, with the consequence that little theoretical
understanding of the processes involved had been achieved. The somewhat
iIituitive understanding of dynamic-field focusing which was attained
t=arne more from the use of the DFF Animation Program than from the

58

numerical simulations described above. Clearly a more fundamental and

theoretical understanding of the focusing technique was desired.

During a conversation regarding patterns of ion behavior elucidated
with the DFF Animation Program, Dr. J. F. Holland and this author jointly
proposed that there was no need for the DFF region to be as long as the
0.08 m used previously. In fact, except for practical concerns, there was no
need for it to have any macroscopic length at all. If the DFF region were
considered to be infinitely thin, then the flight time t; through it would be
zero and the acceleration would occur merely as an instantaneous increase
in velocity. This situation would have the advantage that the total flight
time through the instrument could be found directly as t,“ = t. + ta + tf with
110 need to evaluate t1 through either of the iterative methods used above. A
relatively simple expression for the total flight time could therefore be
Written; inspection of this expression could lead to an increased under-

8handing of the ion behavior and perhaps of the focusing process as well.

Ideal ion behavior in an instrument incorporating an infinitely-thin
DFF region is shown in Figure 3-13. (This behavior was again determined
by use of the DFF Animation Program). Ions are extracted out of a single-
field source and enter the first field-free region. As the ions approach the
8Dace-focus plane, there is no simple relationship between their positions
and velocities. Further along the flight path, however, velocity becomes
"elated to position in such a way that the velocity adjustment necessary to
Qchieve simultaneous arrival at the detector (within the isomass group) is a
Monotonic function of arrival time at the DFF region. Ions are accelerated
as they pass through the DFF region and enter the second field-free region.
The magnitude of this acceleration is related to the time of arrival at the

59

end of FF1; ions arriving later receive more energy than leading ions.
Thus, in FF2, the relationship between position and velocity is inverted (i.e.,
within any isomass group, the leading ions are the slowest). The applied
acceleration is tailored at each instant to make the trailing ions just catch

up to the leading ones at the detector.

3.2.4TheDFFEquations

Using this new approach, a formalism was developed to directly
calculate the value of It needed to focus two arbitrary isomass ions. In order
to obtain simultaneous arrival, the deviations in arrival times at the end of
FF1 for these two ions must be compensated by an exactly complementary
difi‘erence in flight times through FF2. (Note that the DFF region length is
now assumed to be short enough that flight times through it are negligible.)
This focus condition may be described mathematically using the two ions of
mass m and kinetic energies Uo and U1 shown in Figure 3-14. The values
of Uo and U1 need not be specified at this point in the formalism; specific
energies related to various ion cases will be addresed later. Ion 1 enters
FF2 At seconds before ion 0, as pictured in Figure 3-14a. In Figure 3-14b
where simultaneous arrival at the detector has been achieved, the flight
time through FF2 for ion 1, which has been accelerated to new energy Ul’ ,

must have been equal to the time for ion 0 to reach the end of FF1 (traveling
with energy U0) plus its flight time through FF2 with new energ Uo’. Thus

the general focus condition is written

Source Field-Free Region 1 Field-Free Region 2 D ctor

    

 

   

 

 

n A {-DFF 'on
oi : jm——> o~—->'
OE ' 9—) H
E I q»—> o-r—>
so: o—> o—>
40: (3+9 0*)
‘96“ d x f —v
(a)
U v
+i— 3
x x

 

 

 

 

 

Figure 3-13. Model for dynamic-field focusing within an isomass group.
a: Illustration of ion behavior. b: Relationships between ion velocity and
ion position at regions of the flight path. c: Voltage profile throughout the
flight path.

ewe”

2U1’ W;

The acceleration is brought about by an electric field placed across the DFF
region by the voltage function V(t). The final energies U0’ and UI’ are there-
fore related to the initial energies as

Uo’ = U0 - eV(to) (3-13)

and

UI’ = U1 — eV(to-At) (314)

where to is defined to be the time ion 0 crosses the DFF region. (The voltage
V(t) is always negative, so U0’ and UI’ are greater than Uo and U1, respec-

tively, for positive ions.)

FF1 FF2
IU1 |
.p
U
e—L _f__

 

 

 

 

 

\ m... x m...
(a, .....,/ (b, .:.,/

Figure 3-14. Two arbitary isomass ions used in deriving the general focus-
ing equation. a: time of entry into FF2, b: time of arrival at detector

62

The electric field strength is again assumed to vary linearly while
ions within a particular isomass group pass through the DFF region, so the
voltage function may be approximated by

V(t) = Vo - kAt (3-15)
during this time. Letting Vo = V(to) and t = to - At, Eqs. 3-12—3-15 may be
solved for k to yield

11 = [Vo + (U1’- U1)/e] /At (316)
where

-2
j. i "2
. _ m m .

Thus, for any two arbitrary ions having mass m and known values of At,
Uo, and U1, Eqs. 3-16 and 3-17 specify k as a function of Vo. That is,

k = F(Vo, m) (318)
Thus Eq. 3-18 describes a family of V(t) functions which can bring about

simultaneous arrival for these two ions.

Since mass separation begins to occur as ions travel through FF1,
ions having different masses pass through the acceleration region at differ-
ent times. All masses may be focused, therefore, by constructing a
synchronized voltage function, the “DFF voltage function”, that behaves
similarly to the appropriate linear ftmction for each mass, given by Eq. 3-18,
during the time that ions of that mass pass through the DFF region. That

is, at any time t, the instantaneous rate-of-change dV/dt of the DFF voltage
fimction is given by

dV/dt = k = F(V(t), m) (319)

63

if At, Uo, and U1 are known. Since m can be related to arrival time at the

DFF region, this may be written as

dV/dt = k = F’(V(t), t) (3-20)
which may be integrated to yield
V(t) = G(V(0), t) (321)

Eq. 3-21 represents a family of DFF voltage functions, each of which has a
unique value for the arbitrary constant V(O). Any of these functions will
apply to all masses.

In order to apply the formalism described above to a practical case, it
is necessary to know values for the parameters of Eqs. 3-16 and 3-17 that
depend upon the initial positions and kinetic energies of the ions, namdy
At, Uo, and U1. Values for each of these will difl'er in form for each case of
ion behavior. By the time this formalism had been developed, cases S and E
had matured into the cases S, E4», and E- described in Chapter 2. If ion 0 in
Figure 3-14 is defined to be the reference ion so that a :1 and fl = 0 (recall
Eqs. 21 and 2-2), three expressions for U1 and At may be derived, one for

each of these cases:

U1. = “U. (322)
”1,, = (“M’- (3-23)
U12- = (1 + DU. (324)
2m W 1/2 '
423 = [77:] {ell-(a) ] + %(1 — 5%} (3.25)
1/2

AtE+ = [%J {so[1 4» (mm - (1+mm] + %[1 - (1+flfm]} (3-26)

64

mE_=[‘2t%)m{so[1-(fi)m —<1+m"2]+ [—141.41 ”2]}<3~27)

These are substitued into Eqs. 3-16 and 3-17 to yield the three expressions for
k given in Table 3-4.

 

m expressionrfor 11.
v0 +1-{L-1—d—(1-H-a'10-1-—;°7.-(1-a")+(U.-Voer"]4 - aU.}

 

 

 

 

ks: viva—[30(1- -a'-) +d(1- a“)

ten. v0 dngyh- (1+p>"')+—UL(1+p"- -(1+p)”)+(v.- voerl-I' -(1+fl)U.}
l l VE[ro(1+p"—(1+p)")+ 4141.417]:

1 Va JWh- -(1+m"')+ 33:0- 5"— (1+m”)+(v.- woof-mom}

3..

 

 

W' 1+5)"- (1+m‘)+ 41- (14%]

 

If ions were created that sufi‘ered from only one of the efi'ects S, E+, or
E—, and if the related perturbation parameter (a or 0) were restricted to a
single value, then the appropriate expression for It could be integrated
immediately to give a family of DFF voltage functions. Unfortunately, a
gaseous source will produce ions having a distribution of a and fl values.
In addition, these ions will sufi‘er from not only the efi'ects S, E+, and E—,
but also from combinations of these efi'ects. In order to obtain a DFF voltage
function that can adequately focus all ions of interest, the following

assumptions were made:

1. An average ’1 calculated from a practical range of
a or 5 values will provide adequate focusing for all

ions of interest within any of the three cases.

65

2. Instrument parameters (lengths of the field-free
regions, etc.) can be found that make 123, km, and
leg. similary enough that a single voltage function

can adequately focus ions from all three cases.

3. Any process which simultaneously focuses ions of
cases S, E+, and E- will focus all ions, including
those which are not described by any of the three

cases alone.

The success of the simulations described above, as well as Muga’s reported
success for velocity compaction, suggested that these assumptions were
valid. A search for instrument parameters which would satisfy the condi-

tions needed for dynamic-field focusing was therefore begun.

3.2.5 Selection of Imtrument Parameters for DFF-TOFMS

The equations in Table 3-4 express each h as a function of so, (1, f, U,,
a, fi, Vo, and m for each of the cases S, E+, and E-. In order to focus all
ions of interest, values must be found for so, d, f, and U. that will make
113 ~ leg, ~ 113. for all practical values of a, [3, Vo, and m. In addition, these
parameters should make each 12 approximately constant with respect to a
and [3. Using these parameters, It can be treated as a function of Vo, and m

only, as in Eq. 3-18, so that integration will yield the desired V(t).

Inspection of Eq. 2-1 indicates that so should be made large relative to
typical values of 6s, for any given extraction energy, so that any perturba-
tion in initial position would result in only a small change to a and hence
only a small change in kg. For initial positions obeying a normal distribu-

$

tion with a standard deviation of 0,, the choice of so = 1000, seems reason-
able. Estimating 0'. to be 0.1 mm for the CV C 2000 instrument, so was set to
10 mm. (Blueprints provided with the instrument indicated an electron
beam slit width of 0.007 inches, or 0.178 mm. Thus 0', was set to be approx-
imately half of this slit width.)

Similarly, inspection of Eq. 22 indicates that U, should be made large
relative to typical values of 5U. Initial kinetic energies are on the order of
kBT/2, or about 0.02 eV at 500K. Thus U, should be at least 100 times larger,
2 eV, in-order to assure that km. and kg. do not depend strongly on EU. In
addition, increasing U. causes 8 to approach zero. Inspection of the latter
two equations in Table 3-4 indicates that km and kg. approach each other as
)3 approaches zero. Unfortunately, kg., and kg- have a complicated depen-
dence on U, which causes them to diverge from each other as U, increases,

even though this causes [3 to decrease, so a final choice for U, was not made

by inspection of these equations alone.

As in any TOFMS instrument, differences in the arrival times of
adjacent mass reference ions increase as the total flight path length is
increased. Because this total length is essentially the instrument size, it
cannot be enlarged without limit. The DFF technique was expected to be
implemented first in the CV C 2000 instrument which had a path length of
2 m, so this length was chosen for the simulated instrument as well. The
question of how to partition the total path length between d and f was not
resolved, however, by inspection of the equations in Table 3-4.

Values for U,, d, and f were chosen by simplex optimization. This
technique searches for an optimum response from a system controlled by

several independent parameters.5 In order to use simplex, it is necessary

67

to construct a response function which will be minimized or maximized.
In this case, the response must be a function of two parameters, U, and d.
(Since the sum of d and 1“ must equal the total length of 2 m, f is not an
independent parameter.) The first response function used was a measure
of the degree to which kg, kg." and kg. remain approximately equal over all
practical values of a, fi, Vo, and m. Simplex optimization of this function,
using a routine based on that of Press, et al.6, produced the values d = 34 cm
and U = 5.75 eV which resulted in a rapidly-increasing “superlinear” DFF
voltage function. (This occurred because d was so large.) From this first
response function, the resulting DFF voltage function was guaranteed to
produce small spreads in arrival time, but not necessarily high resolving
power, since nothing about the separation between adjacent mass arrival
times was specified. The superlinear voltage function would, in fact, cause
ions of any given mass m + 1 to arrive very near to, or even before, those of

mass m.

Since the goal of dynamic-field focusing is to obtain good space and
energy focusing over a large range of masses, a response function was
designed to yield a measure of this overall resolving power directly. For a
particular set of a, [3, Vo, m, d, and U, values, each of the three k values
were calculated from the equations in Table 3-4 and then averaged to yield a
single value, k“, These values of km... a, p, Vo, m, d, and U, were used to
calculate a flight time tm. In addition, reference ion flight times if“ and t1“
were calculated by setting a to 1 and b to 0. The absolute difi‘erence between
tn, and ti“, relative to the peak separation (ti;+1 - ti), was taken to be a

represenation of the resolving power for mass m. This quantity was

68

summed over a range of a, [3, Vo, and m values to yield the new response

function as

o 0 V 1 V 2 'tm - ti,
rein = if” f 3’: i“ ——,—
5%

0
“03. 412.. 6U, - 001, 002... Vo .. 100,200.. m . 100, 200.. tm+1 - tm

(3-28)
Simplex optimization of R(d, U.) yielded values of 0.010 m for cl and 10 eV for
U,. The program SIMPLX which accomplished this is listed in the

appendix.

Setting so, d, f, and U, to the selected values enabled the three k
values to be treated as functions of Vo, m, and a or 13.

ks = F1(V0. m. a) (3-29)
kE-o- = F2(V09 ms p) (3‘30)
kn- = F 3(V0. m, 5) (3-31)

‘ From these, F(V(t), m) was approximated as

F (V(t), m) =
[F1(V(t), m, 1.01) + F1(V(t), m, 1.001) + F1(V(t), m, 1.0001) +
F2(V(t), m, 0.01) + F2(V(t), m, 0.001) + F2(V(t), m, 0.0001) +
FsCV (t), m, 0.01) + F3(V(t), m, 0.001) + F3(V(t), m, 0.0001)] / 9
(3-32)

Substituting [2U,t2 / (280 + d)2] for m then yielded F’(V(t), t). The initial
value problem dV/dt = F’(V(t), t) was integrated numerically, using the
boundary condition that V(t) = 140 V as ions of mass 20 u pass through the
DFF region, to yield the DFF voltage function shown in Figure 3- 15. The
choice of the boundary condition affects the range and slope of V(t). The
value of 140 V was chosen to keep V(t) < 1000 V and dV/dt < 10 V/us for
masses up to 2000 u. The program SOLVE which accomplished the
integration to calculate V(t) is listed in the appendix.

m

3.2.6 Simulated DFF-TOFMS Performance

A simulation program similar to VCTOF was written to generate
peak shapes for a DFF-TOFMS instrument. This program, SIMTOF, is
listed in the appendix. Although the DFF region length was assumed to be
zero in the theoretical development, a finite length, l, was used for the
simulation so that exact flight times could be calculated. No serious devia-
tions from the expected focusing effects were seen for 1 values of up to

5m.

1000 ‘
800 "

600‘

\hflflagai

400 ‘

200 ‘1

 

 

O - I ' I ' I fl I

o 20 40 60 so 100 120
{Dhaus(us)

Figure 3-15. The DFF voltage function.

Flight times are again calculated as the sum tm = t, + ta + t; + tf.
Eqs. 3-7 and 3-9 yield t, and t4, then it is found through solution of a cubic

70

equation as in VCTOF. Motion in the DFF region is described by the
function

x = p3(t + 1)3 + p2t2 «1» mt + D0 (333)

where the ion position, x, ranges from 0 to l as t increases from 0 to t1. Here

0
1' -'-(tm - in), so the DFF voltage is given by Vo + (t + 1)k. The coeficients in

Eq. 3-33 are
p3 = ek/6ml (334)
p2 = eVol 2ml (335)
p1 = d/t .1 - ek12/2ml (3.35)
120 = —ek13/6ml (3-36)

To find i), x is set to l in Eq. 3-33 and the appropriate root of the equation
pg? + <3p31+p2xf + (ti/1d», - z = o (237)

is found by Newton’s method. In the final field-free region, the velocity of

the ion, Uf, is given by

vf = 311301 + 192 + Zpgtz + p1 (3-38)
so tf is found as
tf = [71],? (3-39)

Using the parameter values shown in Table 3-5, simulated peaks
were produced for a variety of masses. The results for masses 1000 and
1001 u, shown in Figure 3-16, indicate a resolving power of 3% valley, or
2900 using the FWHM method. Unit resolving power (5 50% valley) was

achieved at all masses up to 2000 u, as shown in Figure 3-17.

71
TaBle 3-5. Values used by program SIM‘l‘CF.

 

 

 

 

T Temperature 500. K
a, spatial distribution std. dev. 0.0001 m
so Center of ionization 0.0100 m
d FF1 length 0.1000 m
l DFF length 0.0020 m
f FF2 length 1.9000 m
U, Extraction energy 10.0 eV
12000 ' .
1000 1001
:3
E .
E
.5
o- ,4..fi....,.fi..,.
259.30 259.35 259.40 259.45 259.50 259.55

Flight Time (1.18)

Figure 3-16. Simulated DFF-TOFMS peaks for ions of mass 1000 and
1001 u for parameters shown in Table 3-5.

% Valley

 

 

 

 

' I I I ' T 1
0 400 800 1200 1600 2000

Mass (u)

Figure 3-17. Simulated % Valley resolving power of DFF-TOFMS as a
function of mass.

The space and energy focusing effects were investigated separately by
simulating peaks with either a, or a, set to zero. Keeping all other parame-
ters at the values listed in Table 3-5, such deconvoluted peaks were gener-
ated for masses 1000 and 1001 u; these are shown in Figure 3-18. It would
appear from this figure that DFF energy focusing is more efi‘ective than

space focusing. The reference ion of mass 1000 u enters the DFF region at
time ‘2000 = 86.38 us; the value of the DFF voltage at this time is 614.6 V and

the instantaneous slope is -7.25 V/us. Comparison of this slope to the k
values calculated from the equations in Table 3-4, shown in Table 3-6, indi-
cates that dV/dt is closer to both kg, and kg_ than it is to ks. This explains
the superior energy focusing in this case. Certainly the weighting of the
individual k values used to calculate dV/dt (Eq. 3-32) could be altered to
equalize the degree of space and energy focusing, but this adjustment was

73

not studied. Such weighting could also be used to accentuate either space

or energy focusing for cases where either a, or 0,, is dominant.

-— NO spatial distribution
--------- No energy distflbution
8000 "

Intensity (arb. units)

 

 

 

 

 

  

 

0 . '

259.30 259.35 259.40 259.45 259.50 259.55
FlightTimeQIS)

Figure 3-18. Deconvoluted DFF-TOFMS peaks at masses 1000 and 1001 u.

 

Table 3-6. m 07: values for 1000 11 ions

(V0 = 614.6 V)

a 5 ks 1311+ ks—
(V/us) (V/us) (V/us)

1.01 0.01 -7.55 —7.17 -7.20

1.001 0.001 —7.56 -7.21 —7.22

74

3.2.7 TheDFFAnimationProgram

An important tool used in the theoretical development of the DFF
technique was the program AN IMATE, which illustrates motion of ions
through a TOFMS instrument. Ions of user-specified mass, initial velocity
and initial positions are displayed in any of eight user-specified colors.

These ions move through an instrument described by user-specified

parameters for the various region lengths as well as the Vo and k for the
DFF voltage function. The program was written to run under IBM PC-DOS
with IBM Professional Fortran, and it uses the IBM Graphics Development
Toolkit device drivers for graphics output; a listing appears in the

appendix.

The screen is divided into two major windows; the upper window
portrays the entire instrument while the lower one represents a close-up
view of the ions. The upper view is divided into regions representing the ion
source, FF1, and FF2 (shown at the left, middle, and right, respectively).
The DFF region is considered to have negligible length in the version of the
program listed in the appendix, but another version of ANIMATE
presented motion through finite-width DFF regions as well—in that
version, Runge-Kutta integration was used to advance the ions step-by-step
through the DFF region. FF2 is scaled to extend through most of the right-
hand half of the screen; the remaining portions of the instrument are
scaled to fit the rest of the screen. In this upper window, ions move
through the instrument leaving trails of the appropriate color behind. The
lower view represents a “moving window” which travels along with the

ions; motion of this window is matched to that of a reference ion so that the

75

location of that ion appears to remain fixed. Elapsed time is shown at the
top center of the screen. The velocity of each ion is shown and constantly
updated at the left edge of the upper window; current position values for

each ion are shown at the left edge of the lower window.

A sample run of the program is illustrated in Figure 3-19, where
eight “snap-shot” views of the screen have been reproduced to represent the
animation. This example shows motion of seven m/z 800 ions, as well as
reference ions of m/z 799 and 801, traveling through an instrument with
so = 0.01 m, d = 0.05 m, and l = 1.95 m. The center ion, shown in black, is
considered to be the reference ion of m/z 800. The three upper-most m/z 800
ions, shown in grey, are initially located slightly farther back in the source;
the remaining three m/z 800 ions, also shown in grey, start slightly forward
in the source. The m/z 799 and 801 ions, shown in white above and below
the other ions, respectively, have the same starting position as the m/z 800
reference ion. Within each group of three non-ideal m/z 800 ions, the
upper-most ion has a positive initial velocity, and lower-most has a negative
initial velocity, and the center ion starts at rest. These initial conditions are
shown in Figure 3-19a. In Figure 3-19b, 11.25 us have elapsed (as indicated
at the top of the figure) and the ions are nearing the source exit. The ions
have moved into FF1 in Figure 3-19c and continue with constant velocities
through this region in Figures 3-19d—e. Note that the ions have just passed
the space-focus plane in Figure 3-19d. (The space-focus plane is located
20 mm past the source exit; the ions are all approximately 21 mm past the
source,as indicated at the left of the figure.) In Figure 3-19f, the ions have
been accelerated by the DFF region and have entered FF2. Though widely
separated by this acceleration, the m/z 800 ions begin to come together while

76

the m/z 799 and 801 ions begin to separate, as shown in Figure 3-19g.
Finally, all the m/z 800 ions have become focused (between the m/z 799 and
801 reference ions) at the end FF2, shown in Figure 3-19h.

The input data file used by program ANIMATE in this example is

shown below:

9 Numberofions

799., -0.0100, 0., 1 Mass,position,velocity,andcolorofion1
800., -0.0105, 50., 2 Mass,position,velocity,andcolorofion2
800., -0.0105, 0., 2 Mass,position,velocity,andcolorofion3
800., -0.0105, -50., 2 Mass,position,velocity,andcolorofion4
800., -0.0100, 0., 3 Mass,position,velocity,andcolorofion5
800., -0.0095, 50., 2 Mass,position,velocity,andcolorofion6
800., -0.0095, 0., 2 Mass,position,velocity,andcolorofion7
800 . , -0 . 0095, -50 . , 2 Mass, position, velocity, andcolorofion8
801., -0.0100, 0., 1 Mass,position,velocity,andcolorofion9
5, 0.020 Ref.ionis#5,halfwindowsizeis0.020m
58-8, 0. aisEOnafimelag-O

654.67, 8.9E6, 20., 0.05, 0.050001, 2.0 VakV,d,d+l,d+l-1f
5, 10 Screen-updatecpeeds

In a typical study involving program ANIMATE, an input data file such as
this is created, the program is run, a modification is made to the input data
file, the program is run again, and so forth. Since the input data are
contained in a plain ASCII file, this frequent alteration of the input data file
is easily made with any of several editors or word processors. Though not
as “user-friendly” as a menu or icon-based user interface, this method of
data entry was very easily implemented as well as being satisfactory in

performance.

Ion motion can be stopped at any time by pressing the BEIUBN key.
This enables the user to closely examine or record the exact positions or
velocities of the ions. In addition, the key sequence E BELLE]! causes a
hard-copy “snap-shot” of the screen to be generated. (The IBM Graphics

77

Development Toolkit provides a device driver for the Hewlett-Packard 7475A
six-pen plotter.)

3.2.8 PredictedMetastableIonBehavior

In conventional TOFMS, metastable ions (that is, ions which
fragment after leaving the ion source) are detected at the same flight time
as their parent ions. This is because the fragmentation process affects the
ion velocity only slightly, and this perturbation takes place isotropically (in
the center-of-mass reference frame). The result of a metastable decomposi-
tion is therefore the detection of a somewhat broadened peak superimposed
over the parent ion peak. In a DFF-TOFMS instrument, however, a “more
complicated situation will occur. Due to the relatively low extraction energy
used, and hence the relatively long flight times through FF1, most
metastable decompositions will occur prior to acceleration by the DFF

voltage. Consider ions of mass mp traveling through FF1. If some of these
ions undergo fragmentation along the way, the resulting fragment ions of

mass mf will arrive at the DFF region concurrent with the parent ions.

These fragment ions will therefore experience a dynamic acceleration field

adjusted to focus ions of mass m rather than mf. Because mf< mp, the

P

instantaneous rate of voltage change will be too low to properly focus the
fragment ions. In addition, these ions will be accelerated to a higher veloc-
ity than the parent ions, due to their lower mass, and will therefore arrive
prior to the parent ions. The effect is most severe for fragment ions having

vastly decreased mass relative to their parent ions.

 

Source

78

FF1

FF2

Detector

 

 

0.0000E+00

 

 

-0.01000
-0.01050
-0.01050
-0.01050
-0.01000
-0.00950
-0.00950
-0.00950
-0.01000

 

 

 

(a)

 

'''''''''''''''''''

 

0.11253-04

 

 

 

 

(b)

Figure 3-19. Eight snapshots of the DFF Animation program (parts c-h
continue through next three pages).

79

 

1522
1588

oooooooooooooooooooooooooo

 

1586

 

1590

 

1550

 

1509

 

1514

 

1512

 

1548 ............... gm”

'''''''''''''''''''''''''''''''''''''''''''''''''''

 

0.20258-04

 

 

 

0.01145
0.01185
0.01119
0.01055
0.01143
0.01223
0.01165
0.01100
0.01140

 

 

''''''''''''''''''''''''''''''''''''''''

 

 

 

...............

 

 

0.2650E-04

 

 

 

0.02115
0.02177
0.02110
0.02049
0.02111
0.02167
0.02111
0.02145
0.02107

 

 

 

80

1552 0.3450E-04

1588
150
159
1550
1509
151
15
154

0.03356
0.03447
0.03379
.03321
.03351
.03374
.03322
.03254
.03346

12553 0.5600E-04

12592
1263
126
126
126
12702 -
12735
12656 .....

0.18879
0.19871
0.19381
0.18907
0.18840
0.18736
0.18233
0.17754
0.18795

 

81

0.14353-03

1.29721
1.30052
1.29895
1.29722
1.29648
1.29577
1.29374
1.29188
1.29534

 

2.00027
1.99938
1.99992
2.00011
1.99933
1.99881
1.99869
1.99870
1.99774

 

82

A variety of methods could be used to eliminate the interference
caused by metastable fragmentations. For example, a potential barrier
nearly equal in magnitude to U, could be placed (by means of grids) just
prior to the DFF region. This would prevent fragment ions, which have
lower kinetic energy than the parent ions, continuing through the instru-
ment. Those fragment ions that had lost only a small fraction of their
parent mass would pass through this barrier, but these same ions would be

fairly well-focused as well.

In addition to the interference caused by fragment ions, the neutral
species formed by metastable decomposition will be detected as well. These
species are not accelerated by the DFF voltage and thus will travel through
the entire instrument with their original velocity. Elimination of this inter-
ference requires some part of the ion path to be curved (by electrostatic

deflection, for example) so that the neutral species do not strike the detector.

The studies described above demonstrate that dynamic-field focusing
should be a viable technique for improvement of resolving power in TOFMS.
It should provide focusing over the mass range of 0—2000 u for ions formed
from from gaseous samples in an electron impact ion source. Even if this
mass range is overly-optimistic (due to nonideal ion behavior not modeled
in this study) by as much as a factor of 4, DFF would provide focusing
adequate for routine GC-MS analyses based on TOFMS with time-array
detection. Unfortunately, however, the predicted resolving power cannot be
increased greatly over that described above due to two factors. First, the

boundaries for voltage range and sweep rate of V(t) < 1000 V and

83

dV/dt < 10 V/us used to obtain the DFF voltage function approach the prac-
tical limits with current electronics technology. Moreover, simulated
studies of a variety of instrumental and sample parameters have indicated
that the initial spatial distribution of ions present with gas-phase ionization
remains, in spite of the dynamic-field approach, a serious impediment to

achievement of adequate mass-independent focusing in TOFMS.

In section 3.2.5, it was noted that the first attempt to choose DFF
instrument parameters (by Simplex optimization of the mutual equality
between ks, kg.,, and kg.) was unsuccessful. The parameters chosen in this
first attempt were those that produce the smallest possible peak widths
rather than the best mass resolving power. Though reduction of peak
widths is a goal common to virtually all resolution-enhancement tech-
niques in TOFMS, the failure to achieve good resolving power by merely
minimizing peak widths is not entirely unexpected. For example, consider

the most basic TOFMS experiment in which flight time through a distance

of length l is given by the relation t =% , where m is the mass and U is

the acceleration energy. The separation in arrival time between two ions of
adjacent mass is filfiflm + 1 — m) and is thus inversely proportional to U.

The turn-around time effect (and hence the peak width) is reduced,
however, as U is increased. A compromise between the lowering of U to
increase adjacent peak separation and the raising of U to decrease peak
width is therefore required. This is analogous to the compromise between
peak width and peak separation in the DFF situation. In fact, simplex
optimization of an overall resolving power function (analogous to Eq. 3-28)
could be used to pick the most appropriate value for U, as well as other

84

source parameters, for conventional TOFMS just as it was in the DFF

studies.

Notwithstanding these similarities between DFF and conventional
TOFMS, the fact that efforts to improve resolving power by reducing peak
widths can actually make it worse (not only by reducing the separation
between peaks but, in the case of DFF, by actually reversing their order!)
indicates that these approaches to the improvement of resolving power may
have serious flaws. Recall that the study of techniques incorporating
dynamic fields came about as a result of Stein’s conclusion (see section 2-4)
that complete space and energy focusing could not be achieved with time-
independent fields alone. Even the use of dynamic-fields, however, has not
yet resulted in a truly attractive focusing technique—DFF has the problems
described above, these must be shared by velocity compaction (due to its
similarity to DFF), and post source pulse focusing (section 2-5) does not
achieve truly mass-independent focusing despite its use of dynamic fields.
Though these results do not preclude success for all dynamic-field
techniques in the definitive way Stein’s conclusion proved concurrent space
and energy focusing could not be achieved solely with static fields, they
strongly suggest that DFF is unlikely to provide the panacea sought. Two
alternative strategies which are quite different were pursued after
completion of the DFF analysis; these are the subjects of the following two
chapters. '

REFERENCES

W. P. Poschenrieder, Int. J. Mass Spectrom. Ion Phys, 1972, 9, 357.
M. L. Muga, Anal. Instrum., 1987 , 16, 31.

G. E. Yefchak, C. G. Enke, and J. F. Holland, Int. J. Mass Spectrom.
Ion Processes, 1989, 87, 313.

M. L. Muga, U. S. Patent 4 458 149, 19842.
S. N. Deming and S. L. Morgan, Anal. Chem., 1973, 45, 278A.
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,

Numerical Recipes, Cambridge University Press, New York, 1986, pp.
292—293.

Chapter 4

Beam Deflection in TOFMS

“That is the curse of statistics, that it can never prove things. only
disprove them! At best, you can substantiate a hypothesis by
ruling out, statistically, a whole long list of competing hypotheses,
every one that has ever been proposed. After a while your adver-
saries and competitors will give up trying to think of alternative
hypotheses, or else they will grow old and die, and then your
hypothesis will become accepted. Sounds crazy, we know, but
that’s how science works!”

From Numerical Recipes, by William H. Press, Brian P.
Flannery, Saul A. Teukolsky, and William T. Vettening

All varieties of time-of-flight (TOF) mass spectrometers share a need
for sychronized ion packet production; the departure of ion packets from the
ion source serves to indicate the starting times for ion velocity measure-
ments and the dimension of the ion packet determines mass resolving
power. Though most TOF mass spectrometers make use of so-called
“pulsed ion sources” (i.e., ion sources which generate ion packets directly),
there are a number of circumstances in which it is desirable to use sources
which produce ions continuously. Beams produced by these “continuous
ion sources”, which are similar to sources used in magnetic sector and
quadrupole mass spectrometers, are spatially modulated as described in

this Chapter to create the desired ion packets for TOF mass analysis. The

$

87

study of the resolving power aspects of beam deflection TOFMS presented
here is part of a manuscript prepared by this author together with Gary
Schultz and Drs. J. Allison, J. F. Holland, and C. G. Enke.1

4.1 Background

A theoretical treatment of beam modulation techniques in nuclear
physics appeared in 1960,2 and in 1973 many theoretical considerations for
beam modulation in time-of-flight mass spectrometry (TOFMS) were
elegantly described by Bakker.3 Shortly thereafter, Bakker reported the
construction of a beam modulation TOF mass spectrometer and an experi-
mental confirmation of the theory.4. Various other TOF instruments
incorporating beam modulation have since been reported, including those
of Pinkston,5 Glish,6 and Eckenrode.7 A beam modulation ion source was
also developed for the early Bendix TOF spectrometers by Futrell.3v9 In
addition, a similar technique has been used in a tandem TOF mass spec-
trometer to select parent ions prior to a second stage of mass analysis.10
The “orthogonal-acceleration” TOFMS instrument recently described by
Dawson and Guilhaus11 incorporates beam-deflection in the broadest
sense, but the operating principle of orthogonal acceleration differs sum-
ciently from conventional beam-modulation techniques to exclude its

discussion in this work.

Ion packet production by modulation of a continuous beam elimi-
nates some of the difficulties encountered with more conventional pulsed-
source techniques, particularly when gaseous samples such as those
encountered in GC-MS are ionized. With pulsed ion sources, three patterns

of ion behavior can lead to decreased mass resolving power.12»13 The distri-

88

bution of initial positions of ions within the source, especially important for
gaseous samples, and the distribution of initial ion velocities both lead to a
spread in final kinetic energy. This spread in kinetic energy leads to a
distribution of arrival times for isomass ions and hence to the loss of resolv-
ing power. Furthermore, the thermal motion of ions moving with equal
speeds toward and away from the source exit prior to the application of the
extraction field results in the “turn-around” separation between isomass
ions ultimately having identical on-axis energies. The deterioration of
resolving power due to this turn-around problem has normally been the
most severe, necessitating the use of high extraction field gradients which
minimize the turn-around time in pulsed ion sources. Unfortunately, the
use of large field gradients exacerbates the spatial problem; this incompati-
bility between space focusing and energy focusing with gaseous samples
has been described in the literature. 12» 14 Production of ion packets by modu-
lation of continuous beams eliminates the turn-around problem since all
ions in the accelerated beam are moving in the same direction.
Furthermore, this elimination of the turn-around efi‘ect permits low extrac-
tion gradients to be used which significantly decrease the energy spread
caused by the initial spatial distribution. For these reasons, studies of beam
modulation have been undertaken in an effort to achieve adequate mass
resolving power in GC-MS with TOF instrumentation. Both the theoretical
and experimental results reported here demonstrate that beam deflection
TOFMS is a viable technique for GC-MS when used in combination with
time-array detection.

4.2Theory

4.2.1 ModelsforBeamModulation Techniques

The ideal, desired behavior of all beam modulation techniques is
illustrated in Figure 4-1; a packet having an arbitrarily short temporal
width is “chopped” out of a continuous beam. Note that the packet retains
the kinetic energy distribution present in the original beam and is finite in

 

. .—) . ) . ’
Continuous r 3 ’ O—* H

H H O—> H
beam H. ’90 I >

H

Ion packet ..__,'

H

 

 

 

)

Figure 4-1. Ideal ion behavior in beam modulation; the process
essentially yields a slice from the continuous beam. Arrow lengths
indicate relative ion velocities; only isomass ions are considered in
this illustration.

spatial and temporal width; these are the only nonidealities. All ions
depicted in Figure 4-1 and throughout this work have the same mass-to-
charge ratio. In actual operation, of course, an instrument would produce
ions of various masses. Ion trajectories in a beam deflection instrument
are independent of mass, though the velocities of the different isomass

packets are mass-dependent.

90

Modulation techniques have so far all relied upon deflection of the
beam by an electric field, generated by some voltage function V(t), applied
across a pair of deflection plates between which the ions pass. Some portion
of the beam is deflected across an aperture, causing an ion packet to
emerge from the opposite side. Various V(t) functions may be used to
generate ion packets via at least three modes of ion behavior. The ion
behavior patterns corresponding to these modes are illustrated in Figure 4-
2. These include a pattern referred to here as the Gate mode, as well as the
Impulse Sweeping and Differential Impulse Sweeping modes that were
identified by Fowler and Good.2 In each of these modes, the “default”
condition, which exists before and after application of the packet-forming
voltage pulse, is the deflection of the ion beam away from the aperture. In
the Gate mode (Figure 4-2a), the field changes to allow the beam to pass
directly through the aperture for some period of time; at the end of this time
the field returns to its original state. In this mode, V(t) serves most closely
as a “gate”, either preventing or allowing passage of ions through the aper-
ture. The Impulse Sweep mode (Figure 4-2b) is similar to the Gate mode, in
that the field is pulsed “on” for a period of time and then returned to its
original value. In Impulse Sweeping, however, the deflection voltage pulse
is short in duration, compared to the flight time through the plates. The
pulse serves to add a velocity component perpendicular to the direction of
ion motion; the resulting angular trajectories cause part of the beam to be
swept into the aperture. Here the width of the emerging ion packet is
related, but not equal, to the duration of the deflection pulse. The width and
ion trajectories are also mass-dependent in this mode. Impulse Sweeping
can be dificult to accomplish, since it requires precise geometrical designs
well as a voltage impulse of precisely-defined amplitude. Furthermore, it

91

does not appear to have any advantage over the other two modes, so it will
not be discussed further here. In the Differential Impulse Sweep mode
(Figure 4-20), the deflection voltage switches polarity causing a “kink” to
form in the beam. The portion of this kink which continues to move
parallel to the original beam passes through the aperture to form the
packet.

The Gate mode is the simplest case. Here the width of the ion packet
is essentially equal to the duration of the deflection pulse (neglecting the
intermediate ion behavior which occurs when the voltage change takes
place). Ions which pass through the plates while the deflection field is off
continue onward and pass through the aperture; those that pass through
before or after this pulse do not. In the description of this mode, the tran-
sient ion behavior caused by the voltage change itself is ignored; only the
steady-state trajectories either through or away from the aperture are
considered. Ion packets produced in the Gate mode are therefore always
longer in duration than the flight time of the ions through the deflection
plate region. This disadvantage may perhaps be offset by the fact that the
Gate mode yields ion packets for which the temporal width is independent

of mass.

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4-2. Three models for beam modulation techniques. Isomass ions
are shown to pass through a pair of deflection plates and move toward an
aperture. Relative duration and polarity of the deflection voltage pulse is
indicated for each case. The darker shading indicates ions which will pass
through the aperture and form the ion packet. a: Gate model; b: Impulse
Sweep model; c: Differential Impulse Sweep model.

93

The transient ion behavior which was ignored in the Gate mode is in
fact responsible for packet formation in the Differential Impulse Sweep
mode. This mode is therefore capable of producing much shorter ion pack-
ets. Consider ions which enter the deflection plate region just before the
switch in the field direction. (The time under consideration here is some-
what earlier than that of the “snap shot” shown in Figure 4-2c.) These ions
are deflected downward as they travel through the region. When the field
changes polarity, however, they begin to experience an upward force.
Those ions that have traveled half-way through the plate region when the
pulse occurs will experience equal durations of downward and upward
forces and thus exit the region moving parallel to their original trajectory.
The aperture is assumed here to be located in line with the initial beam axis
and the deflection plates. Some ions very near to the exact half-way point of
the deflection plate region at the time of the field reversal will also obtain
final trajectories which allow for their passage through the aperture. The
extent to which this occurs contributes to the temporal width of the packet

and will be discussed in the following section.

4.2.2 IonPacketQuality

As derived by Bakker,3 the temporal base width of an ion packet
resulting from the Differential Impulse Sweep process is given by the

expression

w, .9229. an (4.1)
LVd 2e

where B is the beam width, S is the aperture width, D is the deflection plate
separation, L is the distance between the plates and the slit, and V, and Vd

94

are the acceleration and deflection voltages respectively. (That is, the
deflection voltage switches from —Vd to +Vd.) The subscript in w 1 is meant
to represent “ideal” since this is the width which would be obtained if the
ion beam were ideal in terms of kinetic energy and angular dispersion.
The principal assumptions for Eq. 4-1 are that the rise time for the voltage
change is negligible and, more importantly, that the distance from the
plates to the aperture, L, is much greater than the length of the plates.
(Note, however, that the actual plate lengths do not appear in the expres-
sion.) This latter assumption implies that the aperture should not be
placed near the deflection plates but rather at the detector. In fact, it is
possible for the detector surface itself to serve as the aperture.

In addition to the ideal width w I derived by Bakker, the packet is
broadened by the kinetic energy distribution of ions within the packet. Two
ions leaving an initial position x = 0 simultaneously, but with differing
kinetic energies U and (1 + [3)U, will arrive at a location 1: = L separated in
time by At ~ flL(2m/U)1’2/4. Thus an ion packet having negligible initial

width but with a normal kinetic energy distribution having standard devia-
tion O'U would be broadened, after traveling distance L, to have a half-width

given approximately by
- 22%. PI:
wU - Vae 4 (4‘2)

where [3 = 1.670y/Uo.

%

A third cause for packet broadening was identified in this study for
ion beams having nonzero angular divergence. Consider the two ions
shown in Figure 4-3. The lower ion enters with no off-axis energy and is
initially accelerated downward by the deflection field. As it reaches point a,
half-way through the plate region, the field reverses and accelerates the ion
in the upward direction. Since downward and upward force are experi-
enced for equal periods of time, the ion leaves the region with no off-axis

energy and thus is included in the final packet. The upper ion enters the

 

Figure 4-8. Two ions entering the deflection plate region with
different trajectory angles. Field reversal occurs when the
lower ion is at point a and the upper ion is at point b.

region At seconds prior to the lower ion and at an angle 9 from the beam
axis. Due to its off-axis energy, this ion continues to move upward for a
period of time but is eventually deflected downward by the electric field.
This ion experiences the downward force longer than the the lower ion
does, since the upper ion travels to point b before the field reverses. This
longer duration of the applied downward force, followed by a shorter dura-
tion of upward force (experienced from point b to the end of the deflection

region), compensates for the initial off-axis energy. The ion therefore

%

leaves the region parallel to the beam axis and is included in the final

packet, despite being offset from the ideal ion by a time At.

The value of At which satisfies these conditions may be related to off-
axis angle,9, as follows. Consider an “ideal” (i.e. initially on-axis) ion trav-
eling with the most-probable velocity vo = (2Vae/m)1’2. An ion entering the
deflection plate region with an off-axis angle 9 will have a velocity vector
perpendicular to the beam axis given by vy = v0 cos 9. When leaving the
region, however, the perpendicular velocity vy’ is zero. If the ideal ion
passes half-way through the plate region in time tm, then the off-axis ion

experiences downward and upward forces for times tm + At and “/2 — At,

respectively. The final perpendicular velocity of the off-axis ion is related to
the initial value by the expression

V dc V dc
v’=0=vsin0-—(t +At)+—-—(t -At)
f 0 mD 1’2 mD 1’2 which rearranges to
At = % vosin 0 . This represents the broadening of the peak to lower

flight times due to ions with trajectories that diverge in the upward direc-
tion. (Only two-dimensional motion in the plane of Figure 4-3 is considered
here.) For a true beam having a divergence half-angle of 9, those ions
having trajectory angles in the downward direction will cause similar
broadening to higher flight times as well. Thus the quantity 2At could be
used to approximate the base width of a packet broadened only by this angu-
lar efi‘ect, yielding

2V m
we = 1%- ——e° sin 9 (4.3)

97

Equation 4-3 yields an underestimate of peak broadening due to the angular
effect, however, since it is not necessary for ions to be moving exactly paral-
lel to the beam axis as they leave the deflection region in order to pass
through the aperture. Consider, for example, an ion which enters the
deflection region with the same trajectory as the upper ion discussed above
but slightly ahead of it in time. This ion would experience even more net
downward force than the upper ion pictured in the figure, and would thus
leave the region with a small downward velocity component. This final
trajectory could, however, still carry the ion through the aperture (at a posi-
tion somewhat below the point at which the pictured ion passes through).
Ions acting in this way, as well as the analogous ions having initial down-
ward motion, serve to futher increase the final packet width. Though Eq. 4-
3 cannot be used to obtain the complete magnitude of the peak broadening
due to angular divergence, the digital simulation described below can be

used to measure this efi‘ect.

The broadening caused by ions with positive and negative initial
trajectory angles need not be symmetric, since in both cases the forward
velocity (in the direction of the beam axis) of these off-axis ions is reduced
from the ideal velocity by a factor of cos 0 for ions having the same kinetic
energy. The flight time for ions having an initial downward velocity
component is increased over that of the ideal ion both by their lower velocity
in the beam axis as well as the angular efi‘ect described above. For the ions
having some initial upward motion, however, flight times tend to be
decreased by the angular effect but increased by their reduced velocity. The
net result of this is that the peak is skewed toward higher flight times.

%

It is difficult to combine the expressions for w 1, wt}, and we directly to
obtain an expression for true peak width, but any of the expressions can be
used alone to approximate peak widths for cases in which one effect
(geometry, energy distribution, or angular divergence) predominates. The
combination of the three effects was explored by means of digital simula-
tions as described below; the individual expression were useful for. checking

the validity of these simulations as well.

4.3 Simulations

In order to explore the combination of the energy spread and angular
divergence effects with the ideal packet width given by Eq. 4-1, computer
programs were developed to model ion packet behavior. First, an anima-
tion program was developed which illustrates ion behavior in beam modu-
lation by moving colored dots, representing individual ions, across the
screen of an IBM AT-compatible computer. (This program was used to
ascertain the pattern of ion motion shown in Figure 4-3.) Then a
simulation program was written to generate peak shapes for beam
modulation TOFMS. The method used for this simulation will be explained
with the aid of Figure 4-4. The program models the motion of various ions,
chosen systematically from the region indicated, by means of Runge-Kutta
integration through the deflection plate region and onward to the aperture.
Flight times for those ions found to pass through the aperture are tabulated
in the form of a histogram to generate the peak shape. The three grey
regions in Figure 4-4 are meant to indicate the locus of points occupied by

these ions at three instants in time: prior to entering the deflection region,

Q

part-way along the flight path, and just after passage through the aperture.
(See ref. 3 for a complete description of the shapes of these regions.)

Packet prior Packet while Final
to deflection in beam packet

/ /

       
 

h—d
Search
area

 

 

Figure 4-4. Illustration of ion behavior modeled by the simulation program
(see text).

An example of the output from this program is given in Figure 4-5 for
the parameters shown in Table 4-1. The peak widths obtained by this
simulation method may be approximated by means of Eqs. 4-1—3 by using
calculated values of wt, wt], and we to obtain estimates of peak variances.
The variance of the overall peak is then given by the sum of the contributing
variances. Examples of this process and comparisons with the simulated
peaks are summarized in Table 4-2. Values for variances shown in paren-
theses were calculated as the difference between the other simulated values
under the additivity-of-variances assumption. The peak shape obtained by
the Baker effect alone may be approximated as - a triangle. (This was found
to yield more accurate results than a rectangular peak shape.) The
variance of an isosceles triangle having a base width w I is given by w12/24.

Similarly, the variance contributed by the angular efl‘ect may be approxi-

100

mated by w92/12; a rectangular peak shape is assumed here to model cases
in which all values of 0 within the limits given are equally likely. If the

peak shape corresponding to the energy efl‘ect is assumed to be normal with
a standard deviation of wU/2.35, the variance contributed by the energy

effect is then (w U/2.35)2. The accuracy of the simulated peak variances
cannot be well-characterized without, for example, detailed Monte Carlo
analysis, but these widths are believed to fall within a few percent of the
true values. Note that the prediction of peak widths from variance calcula-
tions is unreliable, in particular for the angular divergence effect. This is
due to the variety of peak shapes (nearly rectangular to nearly Gaussian)
which can occur as the various parameters are altered. For cases in which
the angular divergence is very small (e.g. 9 < 0.01°), the predictions can be
quite accurate; this was the case in the experimental study described below.

 

Table 4-1. FEameters used for simulated
beam deflection peak shape.
1000 u

500. V

100. V

1.500
0.005
0.005
0.003

or
=. 1.67(kBT/2) = 0.0215 eV

to CD wMUh§°<3
8 BB 8

 

101

 

fible 4-2. (Ymparison of peak variances obtainedfrbm calcula-
tion using Eq. 1—3 and from digital peak shape simulation.

Source of calculated calculated simulation
peak broadening base width variance variance
x10‘9 8 x10"18 8 x10'18 8
Bakker effect 13.6 8 9
Energy 7.7 11 (11)
Angle 8.9 7 (16)
Bakker 4» Energy — 19 2)
Baker + Angle — 15 %
Bakker + Energy + Angle — as 35

 

 

0.1'

 

Intensi

 

 

 

 

 

0.0 e - . - - - .
150.62 150.64 150.66 150.68

Flight Time (us)

Figure 4-5. Simulated peak shape for m/z 1000. —Bakker effect

only,--° Bakker effect with energy dispersion, --- Bakker effect with
angular dispersion, -- — all effects combined.

102

4.4 Experimental Results

A spectrum of perfluorotributylamine (PFTBA) was obtained by
Schultz with direct inlet introduction with an ion source temperature of
150°C; 5000 transients were collected in 1 s and summed by the ITR. A
comparison of peak base-widths for major ions in this spectrum to those
produced by digital simulation is shown in Table 4-3. For the simulated
data, parameters matching the physical dimensions and voltages of the
instrument were used, namely L = 2.1 m, S = 6.35 mm, B = 0.13 mm,
D = 13 mm, Va = 3000 V, and Vd = 63 V. The standard deviation of the
kinetic energy distribution, on, was approximated as 0.5 eV based on the
potential distribution within the source; the angular divergence of the beam
was neglected. The slope of base-width vs. W for the experimental
values is calculated to be 3.7 :1: 0.2 ns, and that for the simulated data is
4.0 i 0.1 ns, demonstrating that the instrument performance follows the
theory described above. An “expected” slope can be obtained by summing
Eqs. 4-1 and 4-2, then factoring out m. This procedure yields a value of
3.9 us if 5 is assigned a value of 6oU/Uo to represent base-widths. This
simple method is not appropriate when the angular divergence is large, but

it is quite adequate for this case.

103
Table 4-3. Comparison of measured and

 

simulated peak widths. -
ma Measured Simulated
(:5 ns) (:3 ns)

m 40. 38.

100 36. 42.

131 52. 46.

219 64. 61.

%4 67. 69.

502 88. 92.

 

45 Practical Considerations

4.5.1 Mass Resolving Power

The PFTBA spectrum described above was obtained with unit or
better mass resolution up to at least m/z 502. An estimate of the upper
mass limit for unit mass resolving power with this instrument can be
found as follows. The flight time separation between ions of two adjacent

m/z values is given by

-161.
At-WUm/z-l-l—Vm/z) (44)

where u is the atomic mass unit. Eq. 4-4 may be appron'mated by the first

term of the Taylor series as

At ~ _IdiI__
2%? (4-5)
In the previous section, the peak base width was found to be given approxi-
mately by 3.942751... (Either the experimental slope or the value found via
simulation could also be used.) For simplicity, the width at half-height may
be approximated as half of this value, thus

104
1017/2 '3 2.0 X 10—9 m/z 8 (4'6)
Setting At ~ 501,2 yields the m/z value at which peak separation just equals

peak width at half-height, and hence the approximate upper m/z limit for
unit mass resolving power. For the parameters in this example, this upper

limit is an m/z value of 708.

An improvement could be made if an ion mirror were incorporated
into the instrument. The ion mirror would remove the efl‘ect of the energy
dispersion and, in addition, increase the effective flight path length by a
factor of four. (The decreased flight time in the mirror contributes a factor
of two, while the travel in the forward and reverse directions through the
flight path contributes another factor of two.) In this case, the expected
slope is obtained from Eq. 4-1 alone and has the value 2.5% ns. Using
this value, and replacing L by 4L in Eq. 4-5, the upper m/z limit increases to
4400. This is certainly adequate for most GC-MS applications.

4.5.2 Signal-toNoise Considerations

It would appear that beam modulation in TOFMS would experience
greatly reduced detectability and sensitivity due to the very small portion of
the ion beam that is selected for detection. In addition, it would appear that
the beam dimensions required for beam deflection, which are much
smaller than those in conventional TOFMS, would further contribute to
poor ion statistics. It is important to realize, however, that during the time
an ion packet is striking the detector, the measured ion flux is the same as
that seen from a continuous beam. The consequences of this have been
analyzed in detail by Enke,15a1 who showed that TOFMS with TAD enjoys a

significant SIN advatage over scanning mass spectrometers.

105

4.6 Conclusions

Beam deflection is becoming a common technique in TOFMS, yet
beam deflection systems are often designed with the assumption that the
ion behavior so produced will follow the Gate model. For optimum time-
resolution, however, the more complex Differential Impulse Sweep mecha-
nism is required. This study has shown that accurate predictions of ion
behavior under this model can be obtained either by simple inspection of the
equations given or by digital simulation of ion motion. Such predictions can
be of assistance in the design of beam deflection TOFMS instruments. The
study has also demonstrated that a beam deflection TOFMS instrument
incorporating an ion mirror and an ITR would have mass resolution and

sensitivity suitable for modern GC-MS applications.

REFERENCES

1 G. E. Yefchak, G. A. Schultz, J. Allison, C. G. Enke, and J. F. Holland,
submitted for submission in J.A.S.M.S.

T. K. Fowler and W. M. Good, Nucl. Instrum. Meth. 1960, 7, 245.
J. M. B. Bakker, J. Phys. E: Sci. Instrum., 1973, 6, 785.
J. M. B. Bakker, J. Phys. E: Sci. Instrum., 1974, 7, 364.

J. D. Pinkston, M. Rabb, J. T. Watson, and J. Allison, Rev. Sci.
Instrum., 1986, 57(4), 583.

6 G. L. Glish and D. E. Goeringer, Anal. Chem., 1984, 56, 2291.

7 B. A. Eckenrode, J. T. Watson, C. G. Enke, and J. F. Holland, Int. J.
Mass Spectrom. Ion Processes, 1988, 83, 177.

8 J. H. Futrell, T. D. Tiernan, F. P. Abramson, and c. D. Miller, Rev. Sci.
Instrum., 1968, 39, 340.

9 C. D. Miller, T. D. Tiernan, and J. H. Futrell, Rev. Sci. Instrum., 1969,
40, 503.

10 K. Schey, R. G. Cooks, R. Grix, and H. Wollnik, Int. J. Mass Spectrom.
Ion Processes, 1987, 77, 49.

memento

11 J. H. J. Dawson and M. Guilhaus, Rapid Commun. in Mass Spectrom.,
1989, 3, 155.

12 G. E. Yefchak, C. G. Enke, and J. F. Holland, Int. J. Mass Spectrom.
Ion Processes, 1989, 87, 313.

13 W. C. Wiley and I. H. McLaren, Rev. Sci. Instrum., 1955, 26, 1150.
14 R. Stein, Int. J. Mass Spectrom. Ion Physics, 1974, 14, 205.

15 C. G. Enke, Proceedings of the 37th ASMS Conference on Mass
Spectrometry and Allied Topics, Miami Beach, Florida, May 21-26, 1989.

Chapter 5

Combination of a Pulsed Electron Impact Source
with an Ion Mirror

“I have yet to see any problem, however complicated,
which when you looked at it the right way did not become
_still more complicated.”

P. Anderson

As the DFF project was nearing its completion, Dr. Enke delivered a
presentation at an American Society for Mass Spectrometry meeting1 in
which he pointed out the requirement of non-static fields for complete
focusing, as well as the other issues discussed in section 24. The specific
point illustrated in Figure 2-7 (that initially-moving ions would be spatially
focused at some location distinct from the nominal space-focus plane, and
hence ion mirrors could not be used in this situation) caught the attention
of Professor Hermann Wollnik; after the presentation, Wollnik told Enke of
a gas-phase electron impact ion source which could be used together with
an ion mirror without serious complications from this efi'ect. Preliminary
results from Wollnik’s laboratory in Giessen, Federal Republic of
Germany, indicated that resolving powers as high as 20,000 (FWHM) had

107

108

been achieved with this source. With encouragement fiom these results,
together with Wollnik’s desire to import the ITR technology to his
laboratory, a collaboration between MSU and the Giessen laboratory was
begun. This collaboration has so far resulted in the construction of an
instrument similar to those in Giessen, with the novel ion source and a

grid-free ion mirror, for use at MSU.

5.1 Background

A schematic diagram of the Wollnik-type instruments is shown in
Figure 5-1. Instruments in this family have been termed M-TOF
instruments, where “M” represents “mirror”. This name was chosen for
consistency with other TOFMS instruments at MSU (e.g., the B-TOF and E-
TOF spectrometers), and so it is generally used to refer to the particular
Wollnik-type instrument now located at MSU, rather than to those in
Giessen. The source consists of a three-grid storage or trap region, in
which ions are formed by the emissions of a nearly-circular filament, and
an acceleration region, through which ions exit after extraction. The ions
are formed continuously by the filament current, and some of these ions are
confined to the trap region by a potential well formed by voltages applied to
the three grids. (A more detailed description of the trapping process is the
subject of studies described below in §5.3.2.) At the time of extraction, the
potential applied to the center grid is raised (see Figure 5-2), so that the
electric field between the first and second grids matches that between the
first grid and the aperture at the end of the acceleration region. This
electric field accelerates the ions out of the source and toward the ion

mirror. The mirror is somewhat more complex than the homogeneous-

109

field mirrors illustrated in Figures 2-5 and 2-6, since it incorporates no
grids. The field distribution is therefore inhomogeneous (i.e., it varies in
two dimensions, rather than just one) so that the device acts as a lens as
well as a mirror. An Einsel lens, placed between the source and the
mirror, acts in combination with the lens function of the mirror to focus the
reflected ion packets spatially (as well as temporally, of course) on the

detector.

 

 

 

 

 

 

Einsel
Source Lens Mirror
lnilll Ill llllllllll
3W
(a) I“; ................................................................................. “lulu”
Detector
Electron Pushers

Filament

   

(c)

Figure 5-1. a: Schematic diagram of the Wollnik-type instruments showing
ion source, mirror, detector, and nominal ion path. 1): three-dimensional
view of the ion source showing circular filament and “electron pusher”
electrodes. c: Block diagram of the M-TOF.

110

 

 

 

 

I I I I I Tj I

G4 GB G2 G1 G4 G3 G2 G1

(a) (b)

Figure 5-2. Applied potential profile within the Wollnik ion source.
a: storage mode, b: extraction mode.

Preliminary results fiom an instrument similar to that pictured in
Figure 5-1 demonstrated a resolving power of about 20,000 (FWHM) for
m/z 866.2 (This instrument had an overall length of approximately 4 m and
a total flight-path length of about 7 m; the accelerating potential used when
these data were obtained was 500 V.) The FWHM value for the m/z 866 peak
was about 20 ns. An attempt was made by this author to compare these
experimental results with those of a digital simulation. Since the ion
motion within the grid-free mirror is quite complex, and since the tools
needed to simulate such trajectories were not available, ion motion was at
first simulated only up to the nominal space-focus plane of the source. This
is equivalent to simulating a perfect ion mirror, since such a mirror would
simply reproduce, at its object plane (the detector surface), the ion behavior
which occurs at its image plane (the space-focus plane). The results of this
study are presented in Figure 5-3. Since a perfect mirror was assumed in
this study, it would seem that the simulated peak should have a width
somewhat smaller (or at least not larger) than the peak obtained

experimentally with a real mirror. In fact, however, only peaks simulated

111

for extremely low ion temperatures showed widths less than or equal to that

of the experimentally-obtained peak; simulations for more “realistic”

temperatures produced much broader peaks.

source which performs “better than it should” was both serendipitous and
puzzling. While plans for applying this source technology were proceeding
rapidly, therefore, a parallel effort was taken by this author towards

elucidating the ion behavior mechanism responsible for the unexpectedly

high resolving power.

 

 

.3; III:
.6 .-
. l J“ ‘a
g 1 ' I “\t -
’5 4
O MM/‘I I I '
7.55 7.56 7.57 7.58 7.59
Flight Time (us)

 

 

 

7.60

The discovery of an ion

Experimentally-
obtalned peak

7.61

Figure 53. Simulated peak shapes for m/z 866 from various initial kinetic
energy distributions (temperatures) calculated for parameters matching
the 4 m Wollnik instrument in Giessen. The peak corresponding to 50 K is

most similar to the experimentally obtained peak.

112

5.2 PreliminaryStudies oronBehaviorintheWollnik Source

The first modeling program used in this study of the Wollnik source
simulated ion behavior from the time of extraction until the time of arrival
at the nominal space-focus plane. The geometry and potential profile of this
situation, as reported by Grix, et al.2, is shown in Figure 54. Prior to the
time of extraction, the potential applied to grid 3 is held slightly lower than
that of grids 2 and 4. (The field shape resulting fi‘om these grid potentials

i “1+2az i

 

ament
G4 G3 G2

I?"

Hun I ll /

 

 

 

 

 

 

Electron Space-Focus
“Pushers” Plane
V
I I I I
G4 G3 G2 G1

Figure 5-4. Geometry (above) and potential profile (below) of the
Wollnik source. In the illustration of potential between G4 and G2, the
shaded lines represent the storage condition; black lines represent the
potential during extraction.

113

forms, or at least affects, the potential well in which ions are trapped.) At
extraction, however, the voltage applied to grid 3 is raised to the value
needed to make a linear acceleration field across the region of grid 3 to
grid 1. The ions are accelerated out of the source and achieve good
temporal focusing at the space-focus plane, located at the distance a1 + 2a2
from grid 1. In order to generate peak widths similar to those of the
experimentally-produced peaks, the temperature parameter used by the
program had to be set to approximately 50K (recall Figure 5-3). The value
for this parameter, which essentially determines the initial energy
distribution of ions in the source, has typically been set to 500 K in other
studies; this higher value is thought to represent the true kinetic energy

distribution in a typical TOFMS electron impact ionization source.

To verify that the assumption of a perfect mirror, and hence the
simulation of ion motion only from the source to the space-focus plane, did
not in some way yield erroneously-high peak widths, motion through a
complete mirror instrument was simulated as well. The resources
necessary to model the grid-free Wollnik mirror were not available, so the
two-stage homogeneous-field mirror simulation developed with Damaschin
Ioanoviciu (described in §6.2.3) was used to model ion extraction out of the
Wollnik source, motion through the field-free regions, and motion within
the mirror. This more complex study produced peak shapes almost
identical to those of the perfect-mirror model. These results clearly
indicated that some unexplained behavior within the ion source—
something not modeled by the simulation programs—was causing the

reduced peak widths.

114

5.3 A Model for Ion Behavior in the Wollnik Source

The search for one or more mechanisms which could explain the
surprisingly narrow peak shapes discussed above lasted for over a year.
During this time, the collaboration between MSU and the Giessen
laboratory began to develop, first with a short visit by Enke and Holland to
Giessen. Experiments performed at that time, which revealed primarily
that GC-MS should be possible with a Wollnik-type TOFMS instrument, led
to a three-month visit to Giessen by Dr. Ron Tecklenburg, as well as several
trips to MSU by Wollnik. Tecklenburg succeeded in performing the first
GC-TOFMS experiments using a Wollnik-type instrument. Meanwhile,
Enke, Holland, and this author formed a hypothesis for the ion behavior
which could account for the high mass resolving power seen in the Giessen
experiments. More recently, Mary Puzycki (from the Enke laboratory)
spent three months in Giessen working, in part, to carry out experiments
designed by this author to test this hypothesis. Finally, Dr. Raimund Grix,
who was heavily involved in the development of the Wollnik-type

instruments, came to MSU to participate in the M-TOF project here.

5.3.1 The Revised Model Simulation

Grix indicated to this author that calculations of peak widths made
in Giessen did not yield results which differed as severely from the
experimentally-measured widths as those shown in Figure 5-3. The
Giessen calculations were not as sophisticated as those described above, in
that they did not produce entire peak shapes, but they were thought to be

sufficiently accurate to warrant concern over the validity of this author’s

115

results. A search for some modification to the model which would yield
narrower peak widths was therefore carried out. It seemed possible that
the optimum focus might occur at some location other than the space-focus
plane. Consider, for example, case E+ (recall §2.2). The ideal and non-
ideal ion are initially located at the same positions and become continuously
more separated as they travel along the flight path. Though spatial focus
(i.e., compensation for the case S effect) occurs at the space-focus plane,
therefore, dispersion due to case E-l- must be more severe at that location
than anywhere prior to it. A “compromise” between space and energy
focusing might be reached at some intermediate location. Simulations
were performed to calculate peak widths at a variety of locations on both
sides of the space-focus plane, but'the minimum width was found to occur

precisely at this plane.

An hypothesis for ion behavior was eventually developed which
accounts for a significant degree of peak narrowing (see the following
section), but this ion behavior still did not account for the extraordinarily-
narrow peaks seen in Giessen. Eventually, however, it became clear that
the compromise between space and energy focusing sought above could be
reached by increasing the extraction field strength between G2 and G3
relative to that between G1 and G2. This would produce a two-stage
extraction field which would behave in the opposite manner to the Wiley-
McLaren two-stage source (see §2.3.1). That is, the space-focus plane would
be moved closer to the source exit, where the energy dispersion would be
reduced. Discussion with Grix indicated that, indeed, the optimum
performance was achieved with an extraction pulse applied to G3 having a
greater magnitude than that indicated by Figure 5—4.

116

The position of the space-focus plane may be found from Eq. 5 of Wiley

and McLaren3, shown here as Eq. 6-1.

 

1ng = 211r3’2 (1— LL )

11(r + r1/2) (6-1)

In this expression, lsfp is the distance from the source exit to the space-
focus plane, 11 is the distance from the center of ionization to G2, 12 is the
separation between G2 and G1, and r is a quantity given by the expression

r = (llEo +12Eb)/11Ea, (6-2)
where Ea and E5 are the field strengths in the regions G3 to G2 and G2 to

120}I

 

 

11o-
. i
’37 100- I
E 90- i
k . i l
g 80-
. l i
70- i
60 . . . . . . - 1 . u
1.00 1.02 1.04 1.06 1.08 1.10
VoltageFactor,f

Figure 5-5. Plot of ion packet width (analogous to peak base width) as a
function of the electric field strength between grids G2 and G3 using a
temperature of 500 K. The abscissa, “Voltage Factor”, is the quantity f
defined in the text. For f values above 1.06, A widths at source exit;

0 widths at space-focus plane (see text).

117

G1, respectively. To test the two-stage extraction field theory, ion packet
widths were simulated for a range of field distributions established by the
potentials 0 V, 450 V, and 500f V applied to G1, G2, and G3, respectively.
(For a completely linear field, f would be unity.) The ratio 12/11 was
approximated with a value of 0.9. For each value of f, the position of the
space-focus plane was calculated from Eq. 6-1, and packet widths were
simulated for travel to that position. The results of this study, using a
temperature of 500 K, are shown in Figure 5-5. As the G3 voltage is
increased, and hence the field between G2 and G3 becomes greater with
respect to that between G1 and G2, the packet widths (analogous to peak
base widths) decrease. The field strength between G2 and G3 can be
increased only to a certain limit, however, before the space-focus plane
becomes located behind the source exit (i.e., lafp becomes negative). The
value off corresponding to this limit may be found by setting 14,, = 0 in Eq. 6-
1 and solving for r, then substituting this value into the expression
r = 500f/ (500f- 450), which is equivalent to Eq. 62 for the values under
consideration. This yields a result of f = 1.065. Packet width values for
voltage factors greater than this value are shown in Figure 5-5 both as the
“ideal” width at the calculated space-focus plane position and at the source
exit. Note that no practical problem is caused by the space-focus plane
being located behind the source exit, or even entirely behind the source.
The situation is somewhat analogous to that of a virtual image in light
optics; the mirror (i.e., mirror voltages) must be tuned to view the space-
focus plane as a virtual source, but the position of this plane relative to the

actual ion source is irrelevant.

118

The temperature study shown in Figure 5-3 was repeated for the two-
stage extraction field case with the parameter f set to 1.06. This value was
chosen, after discussion with Grix, to correspond approximately to the

conditions used in the Wollnik source to obtain the experimental data in

i
l

 

 

 

10 f I ' I ' I v 1 v '
0 100 200 300 400 500

Temperature(K)

Figure 5-6. Plot of peak width at half-height as a function of initial ion
temperature calculated for parameters matching the 4 m Wollnik
instrument in Giessen with a two-stage acceleration field. The
horizontal line indicates the experimentally-measured peak width.

question. (Recall that this author’s suggestion of the two-stage acceleration
field was made on purely theoretical grounds; only later did dialogue with
Grix provide confirmation of the theory.) The results of this study are
shown in Figure 5-6 as a plot of peak width at half-height (i.e., FWHM)

119

versus ion temperature. In this revised model, the temperature
corresponding to the measured peak width is approximately 150 K, rather
than the 50 K or less indicated by the initial study. This new value seems
somewhat more reasonable than the initial estimate, but a temperature of
150 K still would seem rather low for an electron-impact source. Thus, it
seems that an additional mechanism responsible for resolving power
enhancement must be at work. The hypothesis developed prior to discovery
of the two-stage acceleration field appears to account for the remaining

effect; this subject is addressed in the following section.

5.3.2 The “Velocity-Ines” Hypothesis

Prior to discovery of the two-stage extraction field described above,
Enke, Holland, and this author attempted to explain the high mass
resolving power of the Wollnik-type instruments in the following way.
Studies made in Giessen strongly suggested that a significant number of
ions are stored in the source prior to application of the extraction field, so it
seemed likely that a “plasma-like” ensemble of ions and electrons must
exist in the ionization region (and hence the “storage region”) of the source
during this “storage time”. Such an ensemble, consisting of electrons and
ions (positively charged ions, in particular) would be, in effect, a conductor.
Except under the passage of current, the electric field within a conductor is
uniform; the potential profile through the storage regi0n was therefore
hypothesized to be nearly flat, in contrast to the profile suggested by the
voltages applied to the grids illustrated in Figure 5-4. Under the
assumption that this was the case, it was first thought that ions having

significant kinetic energies would escape from the storage region during

12)

the storage time. Because only ions having very low kinetic energies would
remain at the time of extraction, the ion behavior (i.e., peak widths)
observed would correlate with that of a very low temperature sample. The
peak width for the temperature of 50 K in Figure 5-3, for example, would
correspond to a 2x10“3 eV barrier. (This is obtained from the relationship
U = 1/2kBT, where U is the energy for a system with one degree of freedom;
the one-dimensional treatment is used here because only motion along the
flight-path dimension is of consequence.) Similarly, the value of 150 K
indicated by Figure 5-6 would correspond to a barrier of 6x10“3 eV. Such
low values for the trapping field depth seemed plausible under the
assumption that the plasma-like ensemble of particles should act nearly

like a conductor.

The assumption of a nearly-flat potential field contradicts somewhat
the study of Baker and Hasted,4 however, who derived an expression for the
potential distribution within an electron beam used to trap positive ions.
This potential distribution function is mathematically complex, but it can
be described in general as a radially-symmetric function which decreases
to a minimum value at a certain radius from the beam center and then
increases from there outward. Positive ions having hnetic energies less
than the depth of the well would be expected to occupy the radial trap
region. The shape of the trap is a consequence of two opposing
contributions: the potential caused by space charge of the electrons alone
would have a minimum at the center of the beam and increase
monotonically outward, while the positive ions alone would yield a
maximum potential at the center and would decrease monotonically
outward. Except for the shape of the trapping potential, however, the ion

behavior described in this work seemed fairly similar to that described

121

above. No clear procedure for calculation of the well depth could be found
for the geometrical configuration of the Wollnik source. A sample
calculation in the article indicated that trap depths of more than 0.1 eV
could be “readily obtained” for the case of a cylindrical electron beam
confined by a magnetic field, and this author’s attempt to apply the Baker
and Hasted equations to the Wollnik source geometry also resulted in a
value of nearly 0.1 eV for the well depth. These calculations may have little
relevance to the Wollnik source, however, since there are at least two
potentially important differences; the Wollnik source does not incorporate a
magnetic field, and the electron current emitted from the circular filament

is quite different fi'om that in a linear beam.

The potential well depths suggested by the Baker and Hasted study
differ by two orders of magnitude from those indicated by the simulations
described above. This author maintained, however, that the local
hypothesis for Wollnik-source ion behavior was viable enough to warrant
experimental investigation. A complete statement of the proposed

hypothesis is given below.

 

The superposition of the electric fields applied to G2, G3, and G4
with the electron space charge yields a region of nearly-constant
potential between G2 and G3. (The potential is slightly lower near the
middle of this region, along the plane of the filament.) The formation
of positively charged ions in this region contributes to near-flatness of
the potential in this region as well. This near-flat potential is
established fairly rapidly, relative to the storage time. Ions formed in
the trap region after establishment of the near-flat potential acquire
energies given by the sum of their initial kinetic energies and the
potential energies associated with their point of formation on the
electric field potential surface.

 

 

122

The newly-formed ions move about in the trap region according to
their energies. Those with sufficient energies and directions of
motion escape from the region and are lost. Others oscillate in the
region; some of these strike wires of G3 and are also lost. Eventually,
only those ions which do not acquire kinetic energies sufficient for
escape from the trap region are left when the extraction field is
applied. At the time of extraction, these ions gain final kinetic
energies which are the sum of their kinetic energies at that time and
their initial location on the extraction—field potential surface (not the
same as their initial location on the trap potential surface).

As the storage time is increased, an increasing fraction of the
generated ions are able to leave the trap. Those ions having greater
kinetic energies are most able to leave; thus, the spread of final
kinetic energies decreases as the storage time is increased. (In an
alternate model, the trap region becomes more nearly flat as more
positive ions are formed; this would result in a similar relationship
between final kinetic energy spread and storage time.)

 

 

 

Experimental work performed in order to test the validity of this hypothesis

is described below.

5.3.3 ExperimentalStudiesinGiessen

The studies of ion behavior in the Wollnik-type source were begun
several months before the MSU M-TOF was available, so this author
suggested a number of experiments to be peformed by Puzycki on a 1 m
Wollnik-type instrument in Giessen. The first of these was to measure true
ion-packet widths near the nominal space-focus plane by placing the
detector at various positions shortly past the ion source exit. If the ions
were to act as expected, a minimum packet width should be observed for a
certain flight path (i.e., when the detector is placed at the true space-focus

123

plane). If some other behavior were found, such as packet widths which
increase monotonically with increasing path length, this would indicate a
serious incompatability between the Wollnik source and the Wiley/
McLaren-like treatment used to model it. Puzycki found that this
experiment had already been done by workers in Giessen. As expected, the
packet widths reached a minimum when the detector was located precisely
at the nominal space-focus plane. (A completely linear field, i.e., f = 1, was
used for this study.) Thus, it would seem that the most general aspects of
the model are correct. The remaining studies described here were made

with the detector located at its normal position.

The next experiment suggested by this author was to record the effect
of the voltage applied to- G3 on peak width and intensity. It was reasoned
that if the trapping process depended strongly the electron space charge,
little effect on peak shape would be seen for small changes in the grid
voltages. Adjustment of the G3 voltage would shift the overall potential of
the entire electron cloud rather than changing the potential distribution
within the cloud. If, however, a precise relationship between the G2 and G3
voltages were required, changes in the G3 voltage in the millivolt range
(i.e., comparable to the proposed trap depth) should strongly affect the peak
shape. Puzycki found that the G3 voltage could be varied over a range of
about 1 V without significantly affecting peak shape or intensity. It seems,
therefore, that the trap shape is determined primarily by the electron space

charge.

Under this author’s proposal that the more energetic ions
preferentially escape from the trap, the observed peak widths should

decrease as storage time is increased (i.e., as the extraction frequency is

124

decreased). This is because a greater proportion of the “faster” ions would
escape, compared to the “slower” ions, per unit storage time. Puzycki
therefore measured peak widths and intensities as a fimction of extraction
frequency. In this experiment, as well as those described below, the m/z
values chosen for observation are those for major “background” ions; no
sample introduction was used. Results from this study are shown in
Figure 5-7. As the storage time is increased (right-to-left on the frequency
axis), the peak width (FWHM) decreases from 14.0 us to about 10.5 ns. The
sigmoidal shape of this curve is consistant with that of curves observed for
other m/z values. From the peak-width data alone, it would seem that the
preferential loss of high-energy ions occurs as predicted. The situation
becomes more complex, however, when the intensity data are considered.
For the frequency range in which peak width decreases with increasing
storage time, the peak intensity exibits a corresponding increase. This
would indicate that though the proportion of “faster” ions decreased with
storage time, the total ion population increases with storage time. This is
explained by the fact that ionization continues throughout the storage time
in this experiment, and hence the more energetic ion can be replaced by
nacent, less-energetic ones. An attempt was made to pulse the filament
voltage in such a way that a constant ionization period was maintained as
the storage time was varied, but experimental dificulties made the results
from this experiment inconclusive. Note that both the decrease in peak
width and the increase in peak intensity cease as the storage time is
increased to 500 ps (i.e., as the extraction frequency is reduced to about
2 kHz). It would seem that the ions remaining in the trap beyond this time
do not have suflicient kinetic energy to escape. As the storage time is
increased beyond this value, however, the intensity begins to decrease. This

125

is unexpected, since ionization continues to occur throughout the storage

time.

According to the Velocity-Loss hypothesis under consideration, more
and more ions should be able to escape from the trap as the storage time is
increased. A continuous supply of new ions was provided during the
trapping time, however, since the electrons continued to be emitted by the
filament throughout the storage time. Consider, then, a model of ion
storage in which the number of trapped ions, n, behaves according to the
expression dn/dt = 121 — kgn. Here 161 represents the constant formation
rate, and Inn represents a loss rate proportional to the number of trapped
ions. (This is a reasonable approximation for the loss process, but this
treatment does not attempt to model the velocity-dependent loss postulated
in the Velocity-Loss hypothesis.) The solution to this equation yields the

k

function n(t) =%1-;1e4¢; this is simply a growth curve which rises
2

. 2
exponentially from zero, at t = 0, to the limit k1/k2 at infinite time. That is,
the intensity (related to the number of trapped ions at the time of extraction)
would increase and eventually level off with increasing storage time under
this model. This steady-state condition (in which the intensity would level
off after the formation and loss rates became equal) was expected to occur,
since it was believed that the trap region would become “full” in the sense
that the electron space chage would repell all positive ions beyond a certain
population. The eventual decrease in intensity observed with increasing

storage time suggests, however, that steady-state was not achieved.

1500- -15

/‘

. Intensity " 13

1000 -

a FWHM
' 12
500 u ‘

Intensity (counts/shot)
FWHM (ns)

1 ‘M
e" A

 

 

 

o I __'-‘ I I I ' I ' I ‘ I 10
0 2000 4000 6000 8000 1 0000
Extraction Pulse Frequency (Hz)

Figure 5-7. Plot of peak width (FWHM) and intensity for m/z 18 as a
function of extraction pulse frequency.

It appears that some mechanism in which the loss rate eventually
exceeds the formation rate must be used to account for the observed data.
Perhaps this mechanism would include a decrease in the ion formation
rate (assumed up to this point to remain constant), but no evidence for this
has been developed to date. On the other hand, some evidence suggests that
the eventual loss in intensity observed for m/z 18 ions discussed above might
be caused by their preferential loss relative to higher-mass ions. Plots
similar to those in Figure 5-7 obtained by Puzycki for other masses suggests
that the storage time corresponding to the maximum intensity varies as a
function of m/z. The curve for m/z 4 reached a maximum at about 4 kHz,

while the curve for m/z 220 seemed to remain fairly constant as the

127

frequency was varied. This seems to indicate that as storage time is
increased, ions having very light mass (i.e., m/z 4) are depleted and
replaced by higher-mass ions (i.e., m/z 18). As storage time continues to
increase, these ions are replaced by even heavier ones. No decrease in
intensity was observed for the m/z 220 ions, which would be expected since
these were the heaviest ions in the experiment having significant
abundance. Since ion velocity in the source is proportional to the square
root of m/z, this preferential loss of low-mass ions is in accord with the

Velocity-Loss hypothesis.

Regardless of the mechanistic details, the change in peak width
caused by variation of the extraction frequency must point to the
“unexplained behavior” responsible, together with the two-stage extraction
field, for the extrordinarily high resolving power in the Wollnik-type

instruments.

5.4 Studies Performed with the MSU M-TOF

After the M-TOF arrived at MSU, a period of several months passed
before fairly reliable operation was achieved. During this time, the
occasional periods of M-TOF operation often occurred concurrently with
periods in which the ITR was inoperative. When all components of the
system were working, most of the work done was aimed at implementing
and demonstrating the performance of GC-MS. Future studies are planned
to further explore the effect of extraction frequency on peak shape. In
particular, plans are underway to permit separate control of the ionization
and extraction periods so that effects of storage time alone, rather than
combined storage and ionization time, can be studied. Results from these

128

future studies will be presented at the upcoming American Society for Mass

Spectrometry conference.5

One interesting discovery was made by accident with the M-TOF
when it was found that the instrument had been running for several days
with no voltage applied to G4. This author had previously thought that
floating G4 might eliminate the trapping behavior, so that the source would
operate similarly to a conventional TOFMS pulsed source. The apparent
irrelevance of the G4 voltage, however, further indicates that the space
charge located between G2 and G3, rather than the fields defined directly by
the grids, has the dominate effect on ion behavior.

An example of the mass resolving power achievable with the M-TOF
is shown in Figure 58 The compound tris(heptafluoropropyl)-S-triazine,
often used in mass calibration, was introduced into the M-TOF by drawing
about 0.2 ul of the pure liquid into a syringe and then inserting the syringe
into the GC inlet without advancing the syringe plunger. This procedure,
which is the only convenient method of direct sample introduction
available, other than chromatographic introduction, provides a steady
supply of the sample material which lasts for ten minutes or more. (This
method of sample introduction is not suitable, however, for studies of
sensitivity or detection limits. Such studies have not been carried out
directly by this author, but preliminary studies made by other workers at
MSU have indicated that detection limits in the low picogram range will be
feasible. Sensitivity and detection limit values will be reported in the
future.6) The voltages applied to the various source elements are listed in
Table 5-1. The ITR was used to sum 2500 successive transients with a 5 kHz

extraction frequency, therefore the summed spectrum was collected in

129

500 ms. (That is, scan files representing summed spectra were generated
at a rate of 2 Hz.) Two regions of the observed spectrum are shown. In
Figure 5-8a, a resolving power of 920 is seen for m/z 28. In Figure 5-8b, the
m/z 466 peak shows a resolving power of 1500. In fact, much higher
resolving power was observed via oscilloscope; data obtained with the ITR
suffer from peak broadening of several nanoseconds. The lower resolution
observed for the low-mass region occurs because peak broadening by the

ITR afl‘ects narrow, low-mass peaks more severely than the wider, high-

 

 

 

 

 

mass peaks.
120001 4000
1 as ,
100001 . 45°
4 am:
so 6000< .
5 60001 E 20004
40001 ‘
20001 . I ,
0----.----.fifi ....... f---.----.
22.5 23.0 23.5 24.0 24.5 91.0 91.5 920
. Flight Time (118) Flight Time (us)

(a) (b)

Figure 5-8. Two segments from the mass spectrum of tris(heptafluoro-
propyl).S-triazine obtained from the M-TOF using time-array detection.
a. m/z 28-32 region (residual air ions), showing resolution of 920
(FWHM = 12 ns); b. m/z 466-467 region, showing resolution of 1500
(FWHM = 30 ns).

It was not possible to compare the experimentally-observed resolution
to that of the simulation program since the field shape within the M-TOF

source is quite different from that in the modeled source and not accurately

130

known. (This difficulty was caused by the failure of various high-vacuum
resistors, etc. within the source.) Resolving power predicted by the
simulation (assuming the intended source configuration and using the
voltages listed in Table 5-1) is nearly a factor of two better than the observed
values, due to the altered source details as well as the ITR peak broadening.
(In fact, M-TOF resolving powers up to nearly 3000 have been seen on the
oscilloscope, but such data were unfortunately not collected.) Though
perhaps disappointing, the prediction of better resolving power than that
seen in practice seems much less disturbing than the opposite situation

seen earlier in this work.

 

Table 5-1. M-TOF source voltages used to obtain
data in Figure 5-8.

G4 827 V
G3-extract 979
G3-store :M
G2 817
Filament 777
Pusher #1 773
Pusher #2 779

 

An example of the complete GC-TOFMS-ITR system performance is
shown in Figures 5-9—11. A sample of a nine-component mixture
containing approximately 1 ng of each component was introduced into the
GC column; a temperature program was selected such that all these
components eluted within a two-minute time span. The M-TOF was
operated at an extraction frequency of 5 kHz so that successive summation
of 250 transients by the ITR yielded mass spectral scan files at a rate of
20 Hz. The reconstructed total-ion chromatogram, excluding the solvent
peak, is shown in Figure 5-9. Each of the 2400 dots represents the time at

131

which a complete scan file was generated. Since each chromatographic
peak has a width of at least 1 s, more than 20 mass spectral scan files were
generated for each of the nine components. Peaks 2 and 3 overlap some-
what; if mass spectra were generated at a rate of 1 Hz, for example, a
reconstructed chromatogram would not adequately represent the chro-
matographic separation actually obtained (see Figure 1-1a). Sample mass
spectra from peaks 2 and 3 (scan files #668 and #704, respectively) are
shown in Figures 510 and 5-11. Reference spectra are taken from the Atlas
of Mass Spectral Data.7

Relative Intensity

 

6.034

 

 

 

 

 

 

 

 

 

2 7 '
3 4 20 Scans/Sec

ll
’ 5
l

4.0E4 -~ 1
, s

1 . 6

‘ 9

2.034 "P ' 1

l
0.0 ‘ A r H . -~ 4 - : ,
0 600 1200 1600 2400

Sean Number

Figure 5-9. Reconstructed total-ion chromatogram; the nine peaks are:
1. cyclohexanol, 2. bromohexane, 3. 2-cyclohexene-1-one, 4. methoxy-ethyl
ether, 5. 1,3-dichlorobenzene, 6. benzyl chloride, 7. 1,2-dichlorobenzene,
8. acetophenone, 9. benzyl bromide. Column: DB-5, 5 m x 100 um. Tem-
perature program: 50°C-150°C, 20°C/min. Sample size: 1 ng of each

component injected on column.

Scan 668 2-Cyclohexen-1-one

 

100

Intensity
0| \1
O 01
I I

to
(It
I

 

 

 

III I ll .1I I .

20 60 so 100

-l

 

 

1000 _
800 ..
600 _
400 - .

200 ..

 

 

 

o IllIlllI Illll! I I II I I I I fil I

30 40 50 60 70 80 90 1 00

Figure 5-10. Mass spectrum of 2-cyclohexene-1—one obtained with the M-
TOF (top); reference spectrum (bottom).

Scan 704 1-Bromohexane

 

 

 

100
,Q‘ 75'
U)
5 50~
.E
25
1 l u

 

 

50' 717 9'0 '110j150'150'1io
m/z

1000 -
800 -
600 ..
400 ..

200 -

 

 

 

 

 

0-
10 30 50 70 90 110 130 150 170

Figure 5-11. Mass spectrum of 1-bromohexane obtained with the M-TOF
(top); reference spectrum (bottom).

REFERENCES

C. G. Enke, “Time-of-Flight Mass Spectrometry: Meeting Chromatog-
raphy’s Challenge”, Presented at the 36th ASMS Conference on Mass
Spectrometry and Allied Topics, San Francisco CA, June 5—10, 1988.

R. Grix, R. Kutscher, G. Li, U. Griiner, and H. Wollnik, Rapid
Commun. Mass Spectrom., 1988, 2, 83.

W. C. Wiley and I. H. McLaren, Rev. Sci. Instrum., 1955, 26, 1150.

F. A. Baker and J. B. Hasted, Phil. Trans. Roy. Soc. London, 1966, 261,
33.

G. E. Yefchak, M. A. Puzycki, C. G. Enke, R. Grix, J. F. Holland, G. Li,
Y. Wang, and H. Wollnik, to be presented at the 38th ASMS Conference,
Tucson, AZ, June 3—6, 1990.

R. Tecklenburg, Jr., R. Grix, J. Allison, J. F. Holland, C. G. Enke, C. C.
Sweeley, J. T. Watson, U. Gruner, H. Glitz, and H. Wollnik, to be
presented at the 38th ASMS Conference, Tucson, AZ, June 3—6, 1990.

Ed. E. Stenhagen, S. Abrahamsson, and F. W. McLafl‘erty, Wiley
(Interscience), New York, 1969.

Chapter 6

Models Developed for Collaborative Work

“Scientific research is becoming more and
more a cooperative enterprise... We know
Edison invented the phonograph, but who is
the inventor of television?”

Martin Gardner

The potential for TOFMS to provide unsurpassed mass spectral
generation rates and superior sensitivity has led to a number of research
projects at MSU both prior to and concurrent with the studies described
above. This author has participated in some of these as well, and two of
these collaborative projects are described in this Chapter. The accounts
have been made fairly brief since complete reports on these works have been

(or will be) made by the collaborators.

6.1 Studies on the Mass-Dependence of Time-Lag Focusing

The Wiley-McLaren time-lag focusing/space-focusing technique
(§2.3.1) was for many years the standard method by which fairly reasonable
mass resolving power could be achieved in TOFMS. Time-lag focusing was

136

 

 

137

initially considered inappropriate for TAD, however, because ions from
only a small range of m/z values could be focused by any particular set of
instrument parameters. Yet during the development of the DFF technique,
it became clear that the question of just how small such m/z ranges were
had not been addressed. Conventional TOFMS instrumentation
incorporating time-slice detection generally provides a lag time appropriate
to focus whatever m/z value the detection electronics are set to observe;
when the observed m/z value is incremented, the lag time is also adjusted to
insure focusing for the new m/z value. When only one m/z value is
sampled for each transient (time-slice detection), this adjustment schedule
is appropriate. If, however, TAD is used and a particular lag time could
result in adequate focus for ten m/z values instead of one (for example), an
entire spectrum could be collected with one tenth the number of transients
required for time-slice detection. In this approach, the portions of each
transient that received sumcient focusing would be appended to each other
to yield a complete spectrum. Eric Erickson studied the “mass windows”
over which adequate focusing could be achieved, as well as the effects
which various instrumental parameters have on the size and location of
these windows.1 The results of this study indicate that as few as three
distinct lag-time settings could provide completely focused spectra over the
mass range 50—700 u. Though spectra produced in this way would not have
mass resolving power as high as those produced by, for example, the M-
TOF, this technique does permit TAD to be applied over a fairly wide mass
range by the use of conventional TOFMS instrumentation. The approach
used in this study and the results obtained are summarized below.

 

W {H

V0 V1 2 V3 V4 V4

 

t t

7‘ I‘

Figure 6-1. Illustration of the CVC 2000 mass spectrometer showing
nomenclature for parameters used in the modeling program by Erickson.
Voltages are represented by the V terms and geometry by the x terms. The
distance from an ion’s initial position to the first grid is 31, and the rise
time of the voltage pulses applied to the first and second grids is t,.

The heart of this study was the digital simulation of peak shapes as a
function of the various instrumental parameters, most notably the acceler-
ation grid voltages and the lag time, for various masses. This simulation
was based on the CV C 2000 time-of-flight mass spectrometer, which incor-
porates the four-region ion source pictured in Figure 6-1. Ions are formed

by electron impact in the first region. Then, after the lag time occurs, the
voltages applied to grids 1 and 2 are pulsed from V1.- to Vlf and 0 to sz,

respectively. Grids 3 and 4 are always held at potentials V3 and V4, respec-
tively. Erickson wrote a FORTRAN function to calculate the flight time of

139

an ion from the'source to the detector as a function of its mass, its initial
velocity in the flight direction, its distance s1 from grid 1, the lag time, and
the voltages V1.3 Vlf , V2,- , V3 and V4. (During the lag time, the simulated
ion moves only according to its initial velocity and the field applied by volt-
age Vlg.) Using this function, Erickson and this author wrote a program
(similar to the SIMTOF programs described in Chapter 3) to simulate peak
shapes for pairs of adjacent m/z values. From these data, the efi‘ects of the
lag time and the various voltages were analyzed. (In addition, the efi‘ect of a
linear voltage change between the initial and final values of the voltages
applied to grids 1 and 2 was studied. The linear function was meant to
approximate the behavior of imperfect pulse shapes which occur in prac-

tice.)

 

 

 

  

 

 

 

5 ‘ Optimum Focus
4 w ‘k
’i - 90% Intensity
j; 3 . Boundaries
IE
g 2 'll
3° , .
0 - 1 . . V4
0 200 400 600 800 1000

m/z

Figure 6-2. Resolution and intensity boundaries shown on lag time—m/z
space. The shaded area represents the locus of points at which adequate (see
text) peak shapes are obtained. The shaded curve shows the optimum-focus

lag time (proportional to V; ) .

The most significant results of this study are summarized in

Figure 6-2. As an example, consider the behavior for m/z 200. The figure

140

indicates that the optimum lag time for m/z 200 is approximately 1.3 us.
With lower or higher lag times, the peak becomes somewhat broadened and
reduced in height. If the lag time is lowered to 1.2 as or raised to 1.7 119, the
peak height is reduced to 90% of its maximum value. If the lag time is
increased to 4.0 as, the resolving power deteriorates to the 10% valley level.
Consider now the mass dependence of focusing for, as an example, a lag
time of 1.5 ps. The figure indicates that the mass window 150 < m/z $250 is
adequately focused with this lag time.I This study indicates that the three
lag time values of 1.0, 1.5, and 2.4 as can be used to build a completely
focused mass spectrum from m/z 50 to 700. The three partial spectra would
comprise m/z ranges of 50 to 150, 150 to 250, and 250 to 700, respectively.

The results of this study indicate that the mass spectra of reasonable
quality should be obtainable by TAD without the use of the novel resolution-
enhancement techniques described elsewhere in this thesis. Erickson
experimentally demonstrated the feasibility of this technique using the
CV C 2000 instrument and the ITR.2 The practical implications of the study
are, however, somewhat limited since the technology required to carry out
TAD (i.e. an ITR) is currently much more complex and expensive than

either M-TOF or beam deflection instrumentation.

6.2 Studies of Two-Stage Homogeneous-Field Ion Mirrors

6.2.1 Backgxmnd Information

This second collaboration involved derivation of the second-order
transfer matrix for TOFMS spectrometers incorporating two-stage homo-

geneous-field ion mirrors (i.e. mirrors such as the one illustrated in

141

Figure 2-6). A complete discussion of transfer matrices is beyond the scope
of this work, but a simplified explanation will be given here.3 Consider an
ion initially located at coordinates (x0, yo) and moving at angles x’o = dx/dz
and y’o = dy/dz to the y—z and x—z planes, respectively. This situation may

be expressed by the vector A0 = . If the ion is moving through a field-

 

 

3",,
L. .—

free region, its inertia will carry it a certain distance dz along the z axis

after time dt; this new state can be described by the vector A1 =

y’,

 

 

Since the direction of motion remains constant in a field-free region,
x’1 = x’o and y’1 = y’o. The new positions are thus given by the relations
x1 = xo + x’odz and y1 = yo + y’odz. The transfer matrix for this field-free
region is the matrix M which satisfies the relationship A1 = AM and is

thus given by M = . Each element of M may be identified

OHOC
HNOOI

 

 

OOOH
COD-IN

142

by a symbol of the form a lb, where a and b represent rows and columns of
the matrix, respectively. Thus the second element of the first row of M is
symbolized by x lx’ and can be thought of as the effect x’ has on x for a given
value of dz. That is, since x1 = xo(x Ix) + x’o(x lx’), x Ix and x Ix’ represent
the “contributions” to at; from x0 and x’o, respectively. As presented in this
example so far, transfer matrices simply provide a concise way to represent
the solutions to the equations of motion for a particular optical element.
Their true utility, however, results from the fact that matrices for consecu-
tive optical elements may be multiplied to yield a transfer matrix which

describes the combined efi‘ect of the two elements.

Other optical elements more complex than a field-free region can be
described by transfer matrices as well. Furthermore, additional elements
can be added to describe other quantities of interest, such as energy,
momentum, or mass. By applying the appropriate relationships between
energy, mass, and velocity, flight-time information can be conveyed by
transfer matrices as well. These additional quantities are often expressed
in the form of energy difl‘erences (63), mass differences (6",), and time dif-
ferences relative to some reference ion having a specified kinetic energy,
mass, and arrival time. The final ion coordinates, angles, and arrival time
differences at some reference plane are found from the initial parameters
at some previous reference plane through equations of the following form.

961 = fa(xo, x’o.yo.y’o. 5K. 5m)

x 1 = fxixo. x’o.yo.y’o. 5K. 5.1.)
y1 = $060. x’o.yo.y’o. 6x. 6".)

y’]. "' 34x01 x’OI 3’0» y’O, 6K: 5m)
to + ft(x0: x’09 yo: y’OI 6K: 8m)

N
H
II

143

If terms from the series developments of these equations are available, a
first, second, or higher-order description of the ion motion may be obtained,
depending on the desired accuracy.

6.2.2 Ion MirrorTransferMatrix

The second-order transfer matrix for two-stage homogeneous-field
mirror TOFMS instruments is much more complex than that for a field-
free region. As above, however, the matrix could be used to yield a vector
A1, describing the state variables of an ion which has arrived at the detec-
tor, from a vector A0 which represents the initial conditions of the ion leav-
ing the source. The additional complexity is due in part to the number of
state variables required to describe the situation (there are three position
variables x, y, and 2, two angles a and 6, an energy term dx, and a mass
term 6",) and also because second-order terms (such as zl a8) have been
included. As derived by Damaschin Ioanoviciu, et al.4 the complete list of
non-zero elements from the portion of the transfer matrix which carries

flight-time information is given below:

tlx=—tan9/v tla=tan9(A-L)/v tl6,,,=—tan9’(2v)
t I the = (A —-L)/(2v) t Ixa = tan26/v 11x5, = -tan9’(2v)
t Ixsg = tanO/(Zv) t I are: = (L + B/2) tan29/v + (L -A)/(2v)
t I aim = tan6 (A — L)/(2v) t I mix = tano (A + B+L)/(2v)
t I 8min = -(C + L)/(8v) t I smog = (A — L)/(4v)

tl5x53=(3L+B-A)/(80) tIfifl=-t|53

144

(The definitions for the symbols are not essential to this discussion.) Using
the transfer matrix, Ioanoviciu then derived the following expression for

resolving power of the two-stage homogeneous field mirror instrument

62]+,u plus -1

K
6-1
&—;—-_ :1) < )

5R; Lu A+stan0[1+ 20ltan9+
Here file the resolving power (FWHM definition). L is the total field-free
region length, A is the initial ion packet width at the source exit, 9 is the
source slit width, and [.11 is the ratio of the accelerating potential to the
retarding potential applied to the first mirror stage, respectively. As an

example, a value of 25,000 was calculated for K using the parameters
m = 1000 u, s = 1mm,L= 1m, a: 15°, 0:: 1°, 5K: 104,111 =2/3, andil=0.
Similarly, an expression for the resolving power of a single-stage mirror
instrument was found as

-1

6K ‘ 52
RF [%4- L —tan9 [1+ 2atan9+?] + aztanza + 2ao‘K time + '5' (6‘2)

6.2.3 Digital Simulation

Values for resolving power may also be found by numerical simula-
tion of peak shapes. This author used the transfer matrix elements derived
by Ioanoviciu to write a function FTIME ( ) , similar to that discussed in
Chapter 3, which calculates flight times given values for ion mass, initial
energy, position, direction (angles) of motion, etc. This function was then
used with a modified version of program SIMTOF, listed in the appendix, to

generate peak shapes. Using the same parameters as the example above,

145

the simulated peak shapes indicate a resolving power of 25,300 which is in
close agreement with the calculated value. Numerical simulation enables
ion behavior in the source to be easily modeled as well. For example, the
simulation was repeated for a similar instrument incorporating a typical
electron-impact ion source. Values for a were chosen from a normal
distribution function of 025°, while values for x were selected from a
uniform distribution corresponding to a 1 mm object slit. Ions were
assumed to have a Boltzmann distribution of initial kinetic energies at
500 K and a normal initial spatial distribution having a standard deviation
of 0.2 mm. The source employed a single extraction field of 500 V applied
across a 20 mm region. The peak shapes from both simulations are shown
in Figure 6-3. The vastly reduced resolving power of 1300 obtained for the
electron-impact source illustrates the reason that mirrors have been used
almost exclusively with desorption or laser ionization techniques in which
the ionization volume is nearly planar and thus the initial spatial distribu-
tion is nearly eliminated. (The Wollnik source discussed in Chapter 5 is, of

course, a recent exception.)

The function FTIME () actually exists in three versions. The main
version, listed in the appendix, calculates flight times for the gas-phase
ionization situation as the sum t = twum + Inf-1' tum. Here twang , the time
required for the ion to leave the source and travel to the space-focus plane,

is calculated as

 

tm = “80% + 280 ___m__
V (“trifle l}? i [fie—K— ,\/ ias'l'pslx (6.3)

146

1000 1001

Gas-Phase

K

Planar
/

,.. L_1

143.75 ' 149.95 145.95 ' 144.05
Flight Time (usec)

Intensity (arb. units)

 

 

 

 

 

Figure 6-3. Simulated peak shapes for m/z 1000 and 1001 from planar and
gas-phase (electron impact) ion sources. The widths at half-height are 2.7
and 72 ns, respectively.

where a, and )3, are the perturbation parameters defined in Eqs. 21 and 2-2,
K is the ion acceleration energy, and so is the distance from the center of
ionization to the source exit. The quantity tnf is the flight time for an ideal
ion (that is, one formed the center of ionization with no initial energy and
moving exactly along the beam axis) from the space focus plane to the detec-
tor given by the expression

t,¢=4ll§(%v£) (6-4)

147

where v is the ideal ion velocity (for the mass under consideration) and d2
and V2 are the length of and voltage applied to the second mirror stage,
respectively. (Note that parameters related to the first mirror stage are not
required for this calculation when the mirror potentials appropriate for
energy focusing are used.) The final quantity, tmt, is the deviation from the
ideal flight time calculated from the transfer matrix as

_ it row,- x col.-

‘M‘ " .-.1 (6-5)
U

 

In order to simulate the planar-ionization case, a second version of
FT IMB () was written in which the quantity twum, calculated from Eq. 6-3,
was neglected. A third version of FITIME () , which does not make use of the
transfer matrix, was written in order to check the validity of the main
version. Fortunately this version produced results identical to the transfer
matrix method. The calculation procedure used in this test version is
particularly convenient and has general utility for a variety of applications,
so it will be discussed in detail below.

6.2.4 A Convenient Simulation Method

Once an ion has been accelerated out of the source, its motion along
the beam axis (i.e., in one dimension) can be modeled as a series of move-
ments within various regions—these may be field-free regions, or regions
in which accelerating or decelerating fields are applied. Considering only
static fields, two possibilities exist. The ion may enter a region with suffi-
cient kinetic energy to pass completely through it (this is the case, of course,

for field-free or accelerating regions), or it may enter a decelerating region

148

with only enough energy to slow down and stop, he reaccelerated in the
opposite direction, and finally to exit the region at its entrance point. These
situations may be described mathematically as follows. Let Ugnmaz = 1/2mv2,
where m is the ion mass and v is the velocity, and let Ufinal = Uinmaz + qV,
where q is the ion charge and V is the voltage applied to the region of length
I under consideration. If Ufinal > 0, the ion will pass completely through the

region, and the flight time for this will be given by

[[275

”mm at...

(6-6)

Furthermore, the ion will leave the region with a new velocity

v' = W (no
If Ufiml < 0, the ion will pass only part-way into the region and will exit it
after time i '
212112 (6—8)

with the same velocity (except, of course, for the sign) as it had when enter-

ing the region.

A function tregion () was written in C which calculates flight times
for an ion (of specified mass and initial velocity) through a region using the
equations above. The function is called as

tregion (l, V, ion)
where l and V are variables representing the length of the region and the
voltage applied to it, and ion is a pointer to a structure in the form

struct ionstrt {
double m;
double vel;
double time;
} *ion;

149

Once the structure ion has been initialized (that is, ion .m is set to the
appropriate mass, ion . vel is set to the initial velocity, and ion . time is set
to zero), the ion motion can be modeled by repeated calls to tregion () with
the appropriate I and V values. Upon each call, the function increments
ion . time by an appropriate value calculated from Eq. 66 or 6-8, and it
replaces ion . vel with a new value (either —ion . vel or a value calculated
from Eq. 6-7, as appropriate). The call sequence used to simulate the two-
stage mirror instrument described above is shown below. 7

ion .m = (double) m;

ion .vel == sqrt (2 .0 * (alpha_s + beta_s) * K *

COUELE / (m * KGAMUH;
ion.time = 0.0;

tregion(L, 0. 0, &ion); Flightthroughfield-freeregions
tregion(d_ refl _1, -V_ refl _1, &ion); mwdmmfiml
tregion(d_ refl_ _,2 -V_ refl_ _2, &ion); reflection
tregion(d_ refl— __1, v:refl_1, &ion); re-acceleration

The variables d_ref1_1, V_refl_1 , d_refl_2, and v_ref1_2 represent the
lengths and voltages of the first and second mirror stages, respectively.
Note that it is necessary to provide a value for the first mirror region length
(d_ref1__1) when modeling the instrument in this way, even though this
parameter was not needed for the transfer matrix calculation. (The voltage
applied to this region is determined by the focusing equations even though
the length is not.) In the simulation, this region was given an almost
negligible length (1 pm) in order to simplify the comparison of the results
with those of the matrix calculation. In an actual instrument, of course,
there would be a measurable length for this region and a significant flight
time through it; this would slightly alter the length of the field-fies regions
necessary to achieve temporal focusing, but this efl‘ect was not studied in

detail.

150

The repeated calls to function tregi on () make this algorithm fairly
slow, in comparison to a more direct approach, but the tregion () method
has the great advantages of convenience and broad applicability. Many
TOFMS configurations of interest can be easily modeled by an appropriate
set of function calls. In fact, it is not necessary for one to be more than
casually acquainted with C programming in order to build a complete
instrument simulation in this way. One need only to write an FTIME ()
function with an appropriate set of tregion () calls and inserting this func-
tion into an existing program such as SIMTOF () listed in the appendix.
(Some modifications to SIMTOF () would be necessary in order to account for
some parameters specific to the instrument under consideration, of

course.)

151

REFERENCES

E. D. Erickson, G. E. Yefchak, and C. G. Enke, Presented at the 37th
ASMS Conference on Mass Spectrometry and Allied Topics, Miami
Beach, FL, May 21-26, 1989.

E. D. Erickson, G. E. Yefchak, C. G. Enke, and J. F. Holland, Int. J.
Mass Spectrom. Ion Processes, in press.

The tutorial on transfer matrices presented here ls based loosely on that
of J. H. Boerboom, Adv. Mass Spectrom., 1979, 5, 269.

D. Ioanoviciu, G. E. Yefchak, and C. G. Enke, Int. J. Mass Spectrom.
Ion Processes, 1989, 94, 281.

Chapter 7

Conclusions and Future Work

“Cool, calm dellberatlon
disentangles every knot”
Anonymous

Time-of-flight mass analysis compares favorably in many respects to
other techniques used in mass spectrometry. TOFMS can be accomplished
in an instrument having great mechanical simplicity, and ion
transmission is quite efflcient since few apertures are needed along the
flight path. Mass spectra can be obtained very quickly in TOFMS, but the
complexity of the ITR required to do this overshadows, to some extent, the
simplicity of the mechanical components. As advances in available
electronic technology continue to be made, however, the design and
construction of data acquisition equipment similar to the ITR should
become more practical. The need for improved resolving power in TOFMS
has been addressed in this work, which has shown that both the beam
deflection technique and the combination of a pulsed storage source with an
ion mirror (i.e., the M-TOF) can provide resolving powers adequate for a

variety of mass spectrometric applications.

152

153

The M-TOF can provide higher resolving power than that obtained
from a beam deflection instrument having similar length. Moreover, the
increased sensitivity expected from the storage of ions between extractions
makes the M-TOF approach even more attractive. On the other hand, the
beam deflection technique can be readily applied to any ionization process,
while the M-TOF is currently only capable of electron impact ionization.
For most GC-MS applications in which electron impact ionization is used,
the mass resolving power provided by both techniques is more than
adequate. Current plans for continued development of high-speed GC-MS
at MSU involve mainly the M-TOF, since the sensitivity and detection limits
expected from a mature version of this instrument should match or

surpass those of commercial GC-MS instruments such as the ion trap.

The new technique of orthogonal acceleration TOFMS1 may compete
strongly with the M-TOF in the filture. In this technique, ions are formed
continuously, as in beam deflection; TOF mass analysis begins when an
acceleration field is applied orthogonally to the beam axis. The energy
provided by this orthogonal field greatly exceeds the original ion
acceleration energy, so the ion packets produced by the orthogonal
acceleration travel nearly perpendicularly to the beam axis. Flight times
are recorded after these packets travel through a field-free space along their
new trajectory and strike a detector. The advantage of orthogonal
acceleration TOFMS lies in the fact that Stein’s conclusion (§2.4) does not
apply to cases where ion extraction and flight time analysis occur in
separate dimensions. Collimating the ion beam so that velocity components
perpendicular to the beam axis are reduced will reduce the effective initial
kinetic energy distribution, and hence increase mass resolving power, even

though the total energy distribution within the beam is not reduced. Thus

154

energy focusing can be accomplished here with static fields, and space
focusing can be simultaneously accomplished by either a Wiley/McLaren-
like two-stage orthogonal acceleration field or by an ion mirror. The
orthogonal acceleration technique in its present conception, however, does
not provide any means of ion storage, and hence it may not provide the high

sensitivity expected from the M-TOF.

It should be noted here that time-of-flight mass analysis can be
applied to tandem mass spectrometry. For example, Glish has combined
TOF with a quadrupole mass filter;2 parent ions are selected by the
quadrupole, then fragment ion mass analysis is performed by beam
deflection TOF analysis. At Michigan State University, Stults measured
flight times through a magnetic mass spectrometer3 so that parent and
metastable or collisionally-induced fragment ion masses could be
determined through a combination of magnetic field strength and flight
time data. It seems certain that the improvements to resolving power for
single-stage TOFMS discussed in this work will find application in MS/MS
techniques as well.

REFERENCES

1 J. H. J. Dawson and M. Guilhaus, Rapid Commun. in Mass Spectrom.,
1989, 3, 155.

2 G. L. Glish, S. A. McLuckey, and H. S. McKown, Anal. Instrum., 1987,
16, 191.

3 J. T. Stults, C. G. Enke, and J. F. Holland, Anal. Chem., 1983, 55, 1323.

APPENDIX

Program Listings

Weinberg’s Second Law: If builders built buildings
the way programmers wrote programs, then the first
woodpecker that came along would destroy civilization.

From Murphy '6 Law and other reasons why things go
wrong, by Arthur Bloch

Velocity Compaction Project .......................... 157
Dynamic-Field Focusing Project ..................... 168
Beam Deflection Project ................................. 225
Transfer Matrix Project ................................ 268

157
Simulation Program for Velocity Compaciton

This is a listing of the program VCTOF which simulates peak shapes
from Muga’s Velocity Compaction TOFMS instrument.

VCTOF ................................................................................... 16)
GETX ..................................................................................... 1.
GENCDF ................................................................................ 163
ERF ....................................................................................... 164
TAB ....................................................................................... 164
FTIME ................................................................................... 165

LTIME . .' ................................................................................. 1%

VCTOF

VCTOF is a program which simulates time-of-flight peak shapes.
Only two non-instrumental parameters affect the flight time of an ion
having a given mass-to-charge ratio—the initial location of the ion in the
source and the initial velocity of the ion in the direction of the flight path. .
The program calculates flight times for ions having a specified mass (one
positive charge is assumed) and randomly chosen initial positions and
velocities corresponding to normal distribution functions. These flight
times are tabulated as a histogram (an array of (time, intensity} values)
which represents the peak shape and output to a file.

The user supplies values for the ion mass, the mean and standard
deviation of the initial velocity distribution, the mean and standard
deviation of the initial spatial distribution, and step sizes related to the
number of desired iterations through the two distributions. In addition, the
user supplies the range of the desired histogram (usually in the form of
mass values from which to calculate low and high time limits) and the
desired name for the output file.

Description of Routines:

PROGRAM VCTOF
The main program obtains the required input from the user and
controls the iterations through the space and energy distributions.

FUNCTION GETX
This function returns an abscissa corresponding to a specified
cumulative distribution value.

SUBROUTINB GENCDF
This routine generates the cumulative distribution function for a
normal distribution having specified mean and standard deviation.

FUNCTION ERF
This function, from Press, Flannery, Teukolsky, and Vetterling,

Numerical Recipies, p. 164, evaluates the error function erf(x).

SUBROUTINE TAB

159

In this routine, flight times are tabulated in a histogram. The flight
times are mapped into bin numbers (array positions, that is); then the
corresponding array value is incremented.

FUNCTION FTIME
This function calculates and returns flight time through the instru-

ment for an ion of specified mass fiom its initial position and velocity.

SUBROUTINE LTIME
This routine calculates flight time through the dynamic field region

of the velocity compaction TOFMS instrument.

GOO

000

10

000

PROGRAM VCTOF
Simulate velocity compaction time-of-flight peak shapes

PARAMETER (SIZE - 64)
PARAMETER (NCDF - 128)

REAL TMIN, TMAX

REAL VCDF (NCDF) , SCDF (NCDF)

REAL VMIN, VMAX, SMIN, SMAX

REAL VSIGMA, SSIGMA, v0, so

REAL PV, PS, VSTEP, SSTEP, v, E, 5, TIME
REAL TEMP

INTEGER SPECT(SIZE)

INTEGER DIRECT, MASS

CHARACTER*4O FNAME

Init spectrum

DO 10 I I 1, SIZE
SPECT(I) ‘ 0
CONTINUE

TYPE *, 'Enter STD. DEV. and MEAN for velocity distribution'
ACCEPT *, VSIGMA, vo

TYPE *, 'Enter STD. DEV. and MEAN for space distribution'
ACCEPT *, SSIGMA, SO

IF (VSIGMA .NE. 0.0) THEN
CALL GENCDF(VCDF, VMIN, VMAX, VSIGMA, V0)
TYPE *, 'Enter PV step size'
ACCEPT *, VSTEP

ENDIF

IF (SSIGMA .NE. 0.0) THEN
CALL GENCDF(SCDF, SMIN, SMAX, SSIGMA, SO)
TYPE *, 'Enter PS step size'
ACCEPT *, SSTEP

ENDIF

TYPE *, 'Enter mass'
ACCEPT *, MASS

TYPE *, 'Calculate min & max from what two masses?‘
TYPE *, '(Enter equal numbers for manual input)‘
ACCEPT *, MINMAS, MAXMAS

Find min and.max times of spectrum

IF (MINMAS .EQ. MAXMAS) THEN
TYPE *, 'Enter TMIN and TMAX'
ACCEPT *, TMIN, TMAX
ELSE
TMIN - FTIME(MINMAS, 1, 0.2, 0.0, 1)
TMAX - PTIMEtMAXMAS, 1, 0.2, 0.0, 1)
ENDIF

100

200

300

000

999

1000
900

161
TYPE *, 'TMIN, TMAx - ', TMIN, TMAx

IF (VSIGMA .NE. 0.0 .AND. SSIGMA .NE. 0.0) THEN
DO 100 PV = 0.0, 1.0, VSTEP
v a GETX(PV, VCDF, VMIN, VMAX)
E - 0.5 * V * V I Note absence of MASS (mv‘Z/Z)
DIRECT - 1
IF (V .LT. 0.0) DIRECT - -1

DO 100 PS - 0.0, 1.0, SSTEP
S ' GETX(PS, SCDF, SMIN, SMAX)
TIME - FTIME(MASS, 1, S, E, DIRECT)
CALL TAB(TIME, SPECT, TMIN, TMAX)
CONTINUE

ELSE IF (VSIGMA .EQ. 0.0 .AND. SSIGMA .NE. 0.0) THEN
E ' 0.5 * V0 * V0
DIRECT - 1
IF (V0 .LT. 0.0) DIRECT 3 -1

DO 200 PS = 0.0, 1.0, SSTEP
s = GETX(PS, SCDF, SMIN, SMAX)
TIME - FTIME(MASS, 1, S, E, DIRECT)
CALL TAB(TIME, SPECT, TMIN, TMAX)
CONTINUE

ELSE IF (VSIGMA .NE. 0.0 .AND. SSIGMA .EQ. 0.0) THEN

S - SO

DO 300 PV - 0.0, 1.0, VSTEP
V - GETX(PV, VCDF, VMIN, VMAX)
E - 0.5 * v * V ! Note absence of MASS (mv‘2/2)
DIRECT - 1
IF (V .LT. 0.0) DIRECT = -1
TIME - FTIME(MASS, 1, S, E, DIRECT)
CALL TAB(TIME, SPECT, TMIN, TMAX)

CONTINUE
ELSE

DIRECT - 1

IF (V0 .LT. 0.0) DIRECT - -1

TYPE *, 'TIME - ', FTIME(MASS, 1, SO, E0, DIRECT)
ENDIF

Spectrum complete, now output to file

TYPE *, 'Enter output filename'
ACCEPT 999, FNAME

FORMAT(A)

OPEN(UNIT-1, FILE-FNAME, TYPE-'NEW')
TEMP = (TMAx - TMIN) / SIZE

Do 900 I - 1, SIZE .
TIME - (FLOAT(I) - 0.5) * TEMP + TMIN
WRITE(1, 1000) TIME, SPECT(I)
FORMATth, F10.6, ',', Is, '.')
CONTINUE

CLOSE (UNIT=1)
CALL EXIT
END

162

0000

000

10

20

000

100

163
FUNCTION GETX(P, CDF, XMIN, XMAX)

Find and return X value having cumulative distribution function
(CDP) value I P

PARAMETER (NCDF I 128)

REAL P, CDF(NCDF), XMIN, XMAX
REAL DELTAX, DELTAY
INTEGER I

IF (P .LT. 0.0) STOP 'INVALID P VALUE IN GETX()'
IF (P .GT. 1.0) P I 1.0

DELTAX I (XMAX - XMIN) / FLOAT(NCDF - 1)

This should use a binary search instead...

DO 10 I I 1, NCDF
IF (CDF(I) .GE. P) GO TO 20
CONTINUE
GETX I XMAX
RETURN

DELTAY I CDF(I) - CDF(I - 1)

GETX I (P - CDF(I - 1)) * DELTAX / DELTAY +
1 (I - 2) * DELTAX + XMIN

RETURN

END

SUBROUTINE GENCDF(CDF, XMIN, XMAX, SIGMA, X0)
Generate gaussian cumulative distribution function
PARAMETER (NCDF I 128)

REAL CDF(NCDF), XMIN, XMAX, SIGMA, X0
REAL FAC

XMIN I X0 - 3.0 * SIGMA
XMAX I X0 + 3.0 * SIGMA
RANGE I XMAX - XMIN

FAC - RANGE / FLOAT(NCDF - 1)
$2 - SIGMA * SIGMA

Do 100 I - 1, NCDF
x - XMIN + PAC * FLOAT(I - 1)
CDF(I) I 0.5 *
1 (1.0 + ERF((x - x0) / SQRT(2.0 * 52)))
CONTINUE
RETURN
END

000

000

FUNCTION ERF(X)
From Numerical Recipies, p. 164.

Z I ABS(X)

T I 1./(1. + 0.5*Z)

ERFC I T * EXP(-Z*Z-1.26551223+T*(1.00002368+T*(0.37409196+

1 T*(0.09678418+T*(-0.18628806+T*(0.27886807+T*(-1.13520398+
2 T*(1.48851587+T*(-0.82215223+T*0.17087277)))))))))

IF (X .LT. 0.0) ERFC I 2.0 - ERFC

ERF I 1.0 - ERFC

RETURN

END

SUBROUTINE TAB(TIME, SPECT, TMIN, TMAX)
Tabulate flight times in a histogram

PARAMETER (SIZE I 64)

REAL TIME, TMIN, TMAX

INTEGER SPECT(SIZE)

INTEGER INDEX

RANGE I TMAX - TMIN

INDEX I INT((TIME - TMIN) * SIZE / RANGE) + 1

IF (INDEX .GT. SIZE + 1 .OR. INDEX .LT. 1) RETURN

IF (INDEX .EQ. SIZE + 1) INDEX I SIZE

SPECT(INDEX) I SPECT(INDEX) + 1

RETURN
END

000

0000000

165
REAL*4 FUNCTION FTIME(M, Q, R4SS, R4U0, DIRECT)
Return flight time through velocity compaction TOFMS

IMPLICIT REAL*8 (A-H, O-Z)
INTEGER M, Q, DIRECT
REAL*4 R4SS, R400

M I mass (amu)

Q - charge (elementary charges)

ss - initial position (cm)

uO I initial energy (eV)

direct I initial direction (+1 or —1)

REAL*8 SS, 00

REAL*8 TIME

REAL*8 CONVER

REAL*8 L, TS, TD, TL, TF
REAL*8 TOP, TEMP

DATA CONVER /101.805/ I SQRT(amu / elementary charge) * 1E6
DATA vx, so, D, L, F /10.0, 0.01, 0.02, 0.08, 1.3/

SS = R488
U0 I R4U0

S I SS + 0.8
S - S / 100. I Conver cm to meters

TOP I CONVER * SQRT(2.0 * M)

TEMP - SQRT(Q * s * vx / so + U0) + DIRECT * SQRT(UO)
Ts - TOP * s / TEMP

TEMP I 2.0 * SQRT(Q * S * VX / $0 + UO)
TD I TOP * D / TEMP

V0 - D / TD I Velocity into velocity compaction region
TSD I TS + TD
CALL LTIME(M, Q, SO, D, L, V0, VX, TSD, TL, VF)

TF I F / VF
FTIME I TS + TD + TL + TF

RETURN
END

000

000

20

166

SUBROUTINE LTIME(M, Q, S, D, L, V0, VX, TSD, TL, VF)
IMPLICIT REAL*8 (A-H, O-Z)

COMMON /CCR/ CC, R

INTEGER M, Q

REAL*8 L, K, V0, TL, VF
REAL*8 KGAMU, COUELE
REAL*8 YY(2)

DATA VREF /70./ I VOLTAGE AT M I 1

DATA CONVER /101.805/

DATA KGAMU /1.66057E-27/ I kg per amu

DATA COUELE /1.60219E-19/ I elementary charge in Coulombs

DELTAT I 0.1

Conver time to seconds

TSD I TSD / 1.E6
DELTAT I DELTAT / 1.E6
V0 I V0 * 1.E6

R - 0.5

K = (CONVER/1.E6*((2. * S + D) / SQRT(2. * VX))) ** R
CC = VREF / x

VINIT - CC * SQRT(TSD)

CC - CC * COUELE / (L * M * KGAMU)

Start TL at Muga's approximation

TL I 2. * M * KGAMU * L * L / (COUELE * VINIT)
TL I SQRT(TL)

P I V0 - CC * (TSD**(R + 1.)
QQ I -CC * (TSD**(R + 2.)) /
XX I CC * ((TSD+TL)**(R + 2.)
1 + P * TL + QQ - L
IF (XX .GT. 0.0) THEN

SGN I I1.

) / ( + 1.)
( 1.) * (R + 2.))
R + 1 ) * (R + 2.))

ELSE
SGN I 1.
ENDIF

DELTAT - DELTAT / 10.
IF (ABS(XX) .LT. 1.E-8) Go To 30
OLDXX - xx
TL - TL + SGN * DELTAT
xx - CC * ((TSD + TL)**(R + 2.)) / ((R + 1.) * (R + 2.))
1 + P * TL + QQ — L
IF (SIGN(1.0Do, XX) .NE. SIGN(1.0D0, OLDXX)) THEN
SGN - —SGN
DELTAT - DELTAT / 2.0
ENDIF
Go To 20

30

000

167

VF I CC * ((TSD + TL)**(R + 1.))
XX I CC * ((TSD + TL)**(R 2.))
1 + P * TL + 00

Back to meters per mdcrosecond

VF - VF / 1.E6
TL I TL * 1.E6
RETURN

END

/ (
/ (

R+1.)+P
(R+1.) * (R+2.))

168
' Programs for the Dynamic-Field Focusing Project

The following are listings of programs developed over the period from

1986 to 1988 which were used for the Dynamic-Field Focusing (DFF) project.

Program SIMTOF: ........................................................................... 170
main ...................................................................................... 172
gasdev .................................................................................... 177
ran ........................................................................................ 177
tab ......................................................................................... 178
dfitime ................................................................................... 179
gettvk ..................................................................................... 181
find ........................................................................................ 182

Program SOLVE: ............................................................................. 183
main ...................................................................................... 184
rk4 ......................................................................................... 185
derivs ..................................................................................... 186
ave ......................................................................................... 187
kep ......................................................................................... 187
kem ....................................................................................... 188
ks .......................................................................................... 188

Program SIMPLX: ........................................................................... 189
SIMPLX ................................................................................. 191
FUNK .................................................................................... 192
AMOEBA ............................................................................... 193
respon .................................................................................... 1%
kep ......................................................................................... 197
kem ....................................................................................... 197
ks .......................................................................................... 1%
dfltime ................................................................................... 1%

gettvk ..................................................................................... 201

169

Program VALLEY: ....................................................................... ...202
main .............................................................. ' ........................ 203
gauss ..................................................................................... 206
finds ...................................................................................... 216

Program AN IMATE: ........................................................................ 207
AN IMATE .............................................................................. 209 ~
DONE ..................................................................................... 213
DRAW .................................................................................... 214
FAST ..................................................................................... 215
FINDXV ................................................................................. 216
GCLOSE ................................................................................. 217
GENY .................................................................................... 217
GOPEN .................................................................................. 218
MOVE .................................................................................... 22)
NUMBERS .............................................................................. 221
XFORM .................................................................................. 223
XMAP .................................................................................... 223
YMAPl .................................................................................. 224

YMAP2 .................................................................................. 224

170
SIMTOF

SIMTOF is a program which simulates time-of—flight peak shapes
using a Monte Carlo method. Only two non-instrumental parameters
affect the flight time of an ion having a given mass-to-charge ratio—the
initial location of the ion in the source and the initial velocity of the ion in
the direction of the flight path. The program calculates flight times for ions
having a specified mass (one positive charge is assumed) and randomly
chosen initial positions and velocities corresponding to normal distribution
functions. These flight times are tabulated as a histogram (an array of
(time, intensity} values) which represents the peak shape and output to a
file.

The program uses the file “FUNCTION.DAT” produced by program
SOLVE to obtain the DFF voltage and thus calculate flight times. The user
supplies values for the ion mass, the temperature (which determines the
width of the initial velocity distribution), the mean and standard deviation
of the initial spatial distribution, and the number of desired iterations
through the two distributions. In addition, the user supplies the range of
the desired histogram (usually in the form of mass values from which to
calculate low and high time limits) and the desired name for the output file.

Description of Routines:

main

The main program obtains the required input from the user, opens
access to the DFF voltage file function . dat, and controls the iterations
through the space and energy distributions.

gasdev

This routine, based on Press, Flannery, Teukolsky, and Vetterling,
Numerical Recipes, p. 203, generates random numbers drawn from a
normal distribution with zero mean and unit variance.

ran
This routine returns a random number between 0.0 and 1.0. The

built-in library routine rand supplied by the VAX-11 C compiler generates
integer random numbers from the range 0 to (231 - 1). These integers are

171

converted to floating point and divided by (231 — 1) to yield the desired
uniform deviates.

. tab

In this routine, flight times are tabulated in a histogram. The flight
times are mapped into bin numbers (array positions, that is); then the
corresponding array value is incremented.

dfftime
This routine calculates the flight time for an ion having specified

mass, charge, initial position, and initial velocity. The velocity information
is actually specified as a combination of kinetic energy (a scalar quantity,
unlike velocity) and a direction. The integer variable direct, one of the
arguments sent to this routine, is set by the calling program to be either +1
or —1 to indicate motion either towards or away from the detector,
respectively.

gettvk
This routine calculates and returns the flight time for the reference

ion, the initial DFF voltage, and the DFF voltage rate of change for an ion of
specified mass.

find .
This routine executes a binary search through the DFF voltage
function array.

/*
*‘k
**
**
*‘k
**
*‘k
*‘k
**
*‘k
**
**
**

*/

172

simtof.c
Monte-Carlo simulation of time-of-flight peak shapes.

Means and standard deviations are input for generation of
gaussian distribution functions representing the spatial and
velocity spreads of ions in the source. Ions are randomly
picked from.these distributions, and the flight times of these
ions are calculated. The flight times are tabulated in a
histogram to generate the peak-shape.

George Yefchak

#include <stdio.h>
#include <math.h>

#define SIZE 64 /* Spectrum array size */
#define NCDF 128 /* Cumulative distribution function array size
*/
#define MAXV 500 /* Maximum array size for FUNCTION.DAT */
#define KBOLT 1.38007e-23 /* Boltzmann constant */
#define COUELE 1.60219e-19 /* electron charge */
#define KGAMU 1.66057e-27 /* kg / amu */
#define CHARGE 1 /* One electron charge on ions */
/*
** External declarations for DFF paramters
*/
double are I 0.01;
double dffreg - 0.100;
double dffend - 0.102;
double tofend - 2.000;
double Vext I 10.0:
float tfile[MAXV], vfile[MAXV];
int mref, fsize:
main()
I
double tmin, tmax; /* Spectrum extrema */
double u0, so, vsigma, ssigma; /* Means and standard
deviations */
double v, e, s, time; /* Velocity, energy, location,
time */
double temp; /* Temperature */
double dev; /* Deviation from.mean */
double range; /* Range of spectrum */
double mkg; /* Mass (kg) */
int spect[SIZE]; /* Histrogram for spectrum */
int nviter, nsiter; /* I of velocty and space
iterations */
int iveloc, ispace, i; /* Loop variables */
int mass, direct; /* Ion mass, ion direction */
int mdnmass, maxmass; /* Spectrum.extrema (Da) */
int peak; /* Index for peak 1 or 2 */

char string[80];

/*

*‘k

*/

/*

**

*/

/*

*‘k

*/

173

FILE *fp;
double gasdev(); /* Gassian deviate function */
double dfftime(); /* Time-of—flight function */

Open DFF Voltage Function file

if ((fp I fopen("FUNCTION.DAT", "r")) II NULL)
{
printf("Unable to open FUNCTION.DAT\n");
exit(0);
}
for (i I 0; i < MAXV; ++i)
{
if ((fscanf(fp, "%f, %f, %*f", &tfile[i], &
vfile[i])) II EOF) break;

}
fclose(fp);
fsize I i;

Initialize spectrum to zeros

for (i = 0; i < SIZE; ++i)
spect[i] I 0;

Input mass, distribution function parameters, etc.

printf("Enter mass (amu)\n");
mass I atoi(gets(string));
mkg I (double) mass * KGAMU;
mmef I mass;

printf("Enter temperature (K)\n");
temp I atof(gets(string));

if (temp II 0.0)
{
printf("Enter energy (eV) and direction (+l- 1)\n");
scanf("%f, %d", &u0, 5direct);
gets(string); /* Clean up buffer from scanf() call */
vsigma I 0.0;
}
else
{
no I 0.0;
vsigma I sqrt((KBOLT * temp) / mkg);
}

printf("Enter mean for spatial distribution\n”);

so I atof(gets(string));

printf("Enter standard deviation for spatial distribution\n");
ssigma I atof(gets(string));

if (vsigma :- 0.0)

/*

*‘k

*/

174

{
printf("Enter number of energy iterations\n");

nviter I atoi(gets(string));
}
if (ssigma !I 0.0)
{
printf("Enter number of space iterations\n");
nsiter I atoi(gets(string));

}

if (vsigma !I 0.0 II ssigma !I 0.0)
{
printf("Calculate min 5 max from what two masses?\n“);
printf(" (Enter two equal numbers for manual input.)\n");
scanf("%d, %d", aminmass, &maxmass);
gets(string); /* Clean up buffer from scanf() call */
if (minmass II maxmass)
{
printf("Enter tmin\n");
tmdn I atof(gets(string));
printf("Enter tmax\n");
tmax I atof(gets(string));
}
else
{
tmin I dfftime(minmass, CHARGE, s0, u0, 1);
tmax I dfftime(maxmass, CHARGE, s0, no, 1);
}
printf("Tmin I %e, Tmax I %e\n", tmin, tmax);

srand(1); /* Initialize random number generator */
}

Now calculate spectrum

for (peak I 1; peak <- 2; ++peak) /*** START OF PEAK LOOP ***I
{
if (peak II 2)

(

mass I mass + 1;

mkg I (double) mass * KGAMU;

mref I mass;

)

if (vsigma !I 0.0 as ssigma !I 0.0) /* REAL CASE... */
for (iveloc I l; iveloc <I nviter; ++iveloc)
{
dev I gasdev();
v I vsigma * dev + sqrt((2. * u0) / mkg);
eI0.5*mkg*v*v/COUELE;
direct I (v >I 0.0) ? l : I1:
for (ispace I 1; ispace <I nsiter; ++ispace)
{
dev I gasdev();
s I ssigma * dev + 30;
time I dfftime(mass, CHARGE, s, e, direct);
tab(time, spect, tmin, tmax, SIZE);

/*

**

*/

175

}
}

else if (vsigma II 0.0 && ssigma !I 0.0) /* MONO-ENERGETIC */

I
e I uO;
v I sqrt((2. * uO) / mkg):
for (ispace I l; ispace <I nsiter; ++ispace)
{
dev I gasdev();
s I ssigma * dev + 30;
time I dfftime(mass, CHARGE, s, e, direct);
tab(time, spect, tmin, tmax, SIZE);

}
}

else if (vsigma !I 0.0 && ssigma II 0.0) /* PLANAR */

{
s I s0;
printf("s I %f\n”, 3):
for (iveloc I 1; iveloc <I nviter; ++iveloc)
(
dev I gasdev();
v I vsigma * dev + sqrt((2. * u0) / mkg);
e I 0.5 * mkg * v * v / COUELE;
direct I (v >= 0.0) ? 1 : -1;
time I dfftime(mass, CHARGE, s, e, direct);
tab(time, spect, tmin, tmax, SIZE);
} .
}

{

s I 30;

v I sqrt((2. * uO) / mkg);

time I dfftime(mass, CHARGE, so, no, direct);
printf("Time I %e\n", time);

exit(0);

}

else /* SINGLE ION */

} /*** END OF PEAK LOOP ***/

Spectrum complete; now output to file

printf("Enter output filename\n");
gets(string);
if ((fp I fopen(string, "w")) II NULL)

{
printf("Error opening fi1e\n");

exit(0);
}

range I (tmax - tmin) / SIZE;

for (i I 0; i < SIZE; ++i)

{
time I ((float) i - 0.5) * range + tmin;

fprintf(fp, "%12.8e, %8d.0\n", time, spect[i]);
}

fclose(fp);

176

177

double gasdev()

/*
**
*‘k

**

*/

Return gaussian deviate
See Numerical Recipes, p. 203.

{

double v1, v2, r, fac;
static double gset;
static int iset I 0;

double ran();

if (iset II 0)

{

do
{
v1 I 2. * ran() - 1.
v2 I 2. * ran() - l.
r I vl*vl + v2*v2;
}
while (r >- 1.):

fac I sqrt(-2. * log(r)/r);

gset I v1 * fac;

iset I 1;

return(v2 * fac);

}

o
I
o
I

else
{
iset I 0;
return(gset);
}

double ran()

/*
*‘k
**
*‘k
*‘k

*/

Return random number between 0.0 5 1.0

Library routine rand() returns an int between 0 and 2‘31 - 1.
We divide this by 2‘31 to return a double >- 0. and < 1.

{
double temp;
int rand();

temp I(double) rand() / 2147483648.;
return(temp):
}

178

tab(time, spect, tmin, tmax, spsize)
double time, tmin, tmax;
int spect[l, spsize;

/*

**

*/

Tabulate ion‘s time-of-flight in histogram

{
double range;
int index;

range I tmax - tmin;
index I (int) ((time - tmin) * (double) spsize / range);

if (index >- spsize ll index < 0)
return; /* ...out of range */

if (index II spsize)
--index;

spect[index] +- 1;
}

179

/*
** dfftime.c

**

** Return flight time through DFF TOFMS instrument
*/

#include <stdio.h>
#include <math.h>

#define KGAMU 1.66057E-27
#define COUELE 1.60219E-19
#define CONVER 1.018056E-4 /* sqrt(KGAMU/COUELE) */

#define MAXNM 100 /* Maximum iterations for Newton's Method
*/
#define MAXV 500 /* Maximum size of FUNCTION.DAT */

double dfftime(m, q, s, u0, direct)
double 3, uO;
int m, q, direct;
{
double v;
double top, tmp, tsrc, tffl, time, tf, tref;
double tdff, old;
double tp, dfflen;
double pp, qq, rr, 33, aa, bb, cc, dd;
double f, fprime;
double dffVO, dffk;
int i;

extern double src, Vext, dffreg, dffend, tofend;

/*
** Get DFF voltage function parameters:
**
** Time to OFF region for reference ion I tref
** Initial DFF voltage I dffvo
** Rate-of—change of DFF voltage I dffk
*/
gettvk(m, &tref, EdffVO, &dffk);
/*
** Now find flight time
*/
top I CONVER * sqrt(2.*m);
tmp I sqrt(q * s*Vext/src + u0) + direct*sqrt(u0);
tsrc I top * s / tmp;
tmp - 2, * sqrt(q * s*Vext/src + u0);
tffl I top * dffreg / tmp;
v I dffreg / tffl;
time I tsrc + tffl;
tp I time - tref;
dfflen I dffend - dffreg:
/*

** We need to solve

**
**
**
**
*‘k
**
**
**

*/

180

dfflén I pp*(tdff+tp)**3 + qq*tdff**2 + rr*tdff + 88

This is expressed as

aa*tdff**3 + bb*tdff**2 + cc*tdff + dd I 0
which we solve using Newton's Method.

tmp I COUELE / (m * KGAMU * dfflen);
pp I tmp * dffk / 6.;

qq I tmp * dffVO / 2.;

rr I v - tmp * dffk * tp*tp / 2.;

ss I -tmp * dffk * tp*tp*tp / 6.;

aa I pp;

bb I (3.*pp*tp + qq):

00""

dd I Idfflen;

tdff I 1.e-4; /* Initial point for Newton's Method */

for (i I 1; i <- MAXNM; ++i)
{
f I aa*tdff*tdff*tdff + bb*tdff*tdff + cc*tdff + dd;
fprime I 3.*aa*tdff*tdff + 2.*bb*tdff + co;
tdff II f/fprime;
if (tdff < 0.0)
tdff I 0.;
if (fabs(f) <I 1.e-10)
break;
}
if (i >- MAXNM)
printf("Newton failed: f I %e, f' I %e, t I %e\n",
f, fprime, tdff);

tmp I COUELE / (m * KGAMU * dfflen);
v - 3.*pp*(tdff+tp)*(tdff+tp) + 2.*qq*tdff + rr;
tf I (tofend - dffend) / v;

return(time + tdff + tf);
}

181

gettvk(m, t, v, k)
double *t, *v, *k;

int m;

/*

*‘k

*/

/*

**

*/

{

double top, tmp, tsrc, tffl;
double tmref;

int i, index;

extern double src, dffVO, dffk, Vext, dffreg, dffend,
extern float tfile[MAXV], vfile[MAXV];
extern int mref, fsize;

Find tref for reference mass ion

top I CONVER * sqrt(2.*mref);
tmp I sqrt(Vext);

tsrc I top * src / tmp;

tmp I 2.*sqrt(Vext);

tffl I top * dffreg / tmp;
tmref I tsrc + tffl;

Find tref for current ion

top I CONVER * sqrt(2.*m);
tmp I sqrt(Vext);

tsrc I top * src / tmp;
tmp I 2.*sqrt(Vext);

tffl I top * dffreg / tmp;
*t I tsrc + tffl;

index I find(*t, tfile, fsize);

*k I (vfilelindex+1] - vfile[index]) / (tfile[index+1]

tfilelindexJ);

*v I vfile[index] + *k * (*t - tfileIindexl):

return;

}

tofend;

/*
**
**
kt
**

*/

find.c

Binary search through array

find(x, array, asize)
double x;

float array[];

int asize;

(
float xhi, xlow, xmid;

int hi, low, mid;

low I 0;

hi I asize - 1;
xlow I array[low];
xhi I array[hi];

if (x < xlow II x > xhi)
{
printf("%f is out of range!\n", x);
exit(0):
}
if (x II xhi)
return(hi);-

mid I (hi + low) / 2;
xmid I array[mid];
while (mid !I low 85 x !I xmid)
{
if (x < xmid)
hi I mid;
else
low I mid:
mid I (hi + low) / 2;
xmid I array[mid];
}
return(mid):

}

SOLVE

SOLVE is the program which numerically integrates the average 12
function to yield the final DFF voltage function. Routines ks(), kep(), and
kemO return values for leg, km, and kE_, respectively; these are averaged by
routine ave(). The initial value problem dV/dt = k(V, t) is solved by fourth-
order Runge-Kutta integration. The actual integration is performed by
routine rk4(). The user must supply the boundary condition as a voltage to
be applied at the time ions of a specified mass pass through the DFF region.
For the published work, the value of 140 V was assigned to mass 20 u.

Description of Routines:

main

The main program accepts the required boundary condition from the
user, specified as values for a mass and a voltage, open the output file, and
controls the integration loop.

rk4
This routine, based on Press, Flannery, Teukolsky, and Vetterling,
Numerical Recipes, p. 553, performs fourth-order Runge-Kutta integration.

derivs
This routine calculates derivatives (dV/dt values) for rk4. These
values are found simply by calling ave.

ave
This routine averages individual ks, leg-1, and leg. values to yield an

overall k, which is dV/dt . The values of 0.0001, 0.001, and 0.01 for fl and
1.0001, 1.001, and 1.01 for a were picked in a fairly arbitrary manner.

kep

This calculates 133+.
kem

This calculates k 3..
ks

This calculates ks.

/*
** solve.c
*1:

** Find V(t) from known dV/dt I f(t, V).

*‘k

** Runge-Kutta integration is used to calculate V(t) as
** V(t) I integration { f(t, V)}.

**

*/

#include <stdio.h>
#include <math.h>

#define S 0.01
#define D 0.100
#define Us 10.

#define AMU 1.66057e-27
#define E 1.60219e-19

main()
{
float t, tmax, dt, y[1], dydx[1];
float vold:
char string[80];
FILE *fp;
FILE *fopen();
/*
** Initialize t to arrival time (at DEF) for 20 amu ion.
**
*/
printf("Enter initial mass\n“);
t I atof(gets(string)); /* AMU */
t I sqrt(2.*AMU*t) * (S/sqrt(Us*E) + D/(2.*sqrt(Us*E)));
tmax I 2100.0; /* AMU */
tmax I sqrt(2.*AMU*tmax) * (S/sqrt(Us*E) + D/(2.*sqrt(Us*E)));
/*
** Input starting voltage and time increment
*/
printf("Enter voltage for initial ion\n");
y[0] I atof(gets(string));
printf(“Enter delta t (seconds)\n");
dt I atof(gets(string));
/*
** Open output file
*/

printf("Enter output filename\n");
gets(string):
if ((fp I fopen(string, "w")) II NULL)
{
printf(”Error opening file\n");
exit(0);
}

185

/*
** Now do Runge—Kutta integration to find V(t)
*/
while (t (I tmax && y[0] <I 1100.)
(
vold I y[0];
rk4(y, 1, t, dt);
fprintf(fp, "%15.8f, %15.8e", t, y[0]);
if (vold !I y[0])
fprintf(fp, ", %f\n", y[0] - vold);
else
fprintf(fp, "\n"):
t +- dt;
}
/*
** Close file and exit
*/
fclose(fp);
}
/*
** rk4.c
**
** Runge-Kutta integration routine.
** See Numerical Recipes, p. 553
*/

rk4 (y, n, x, h)
float y[], x, h;

int n;

{

float ytil], dy0[1], dyt[1], dellli
float hh, h6, xh;

int i:

hh I h * 0.5;
h6 I h / 6.;
xh I x + hh;

derivs(x, y, dyO):
for (i I 0; i < n; ++i)

ytlil ' Yii] + hh * dy0[i];

derivs(xh, yt, dyt);
for (i I 0; i < n; ++i)

yt[i] ' Yli] + hh * dyt[i];

derivs(xh, yt, dym);
for (i I 0; i < n; ++i)

(

ytti] - xiii] + h * dymli];
dymti] .. dytli] + dymti];
)

derivs(x+h, yt, dyt);

for (i I 0; i < n; ++i)
Hi] .. viii + 116 * (dyotil + dytli] + 2.*dymli]);
return;

}

/*
** derivs.c

**

** Return dV/dt to Runge-Kutta routine
**

*/

#include <math.h>

#define S 0.01

#define D 0.100

#define L 1.900

#define Us 10.0

#define AMU 1.66057e-27
#define E 1.60219e-19

derivs(t, v, dvdt)
float t, v[1], dvdt[1]; /* Array usage is for compatability with

rk4, which can solve sets of

simultaneous

/*
**
*‘k
**
**

*/

(2

/*
**
*‘k

*/

equations. */

{
float dvdtlo, dvdthi, tlo, thi;
int mlo, mhi;

double ave():

.*AMU));

Find mlo & mhi such that:

tlo I ffltime(mlo), thi I ffltime(mhi)

tlo (I t <- thi
mlo I (int) (pow(t/(S/sqrt(Us*E)+D/(2.*sqrt(Us*E))),2.) /
mhi I mlo + 1;
tlo I sqrt(2.*AMU*mdo) * (S/sqrt(Us*E) + D/(2.*sqrt(Us*E))),
thi I sqrt(2.*AMU*mhi) * (S/sqrt(Us*E) + D/(2.*sqrt(Us*E)));
Find dV/dt for mlo & mhi for voltage v. Then interpolate
linearly to find dV/dt for time t.

dvdtlo I ave((double)mlo, v[0]);

dvdthi I ave((double)mhi, v[0]):

dvdt[O] I (t-tlo)*(dvdthi-dvdtlo)/(thi-tlo) + dvdtlo;
return;

}

double ave(m, v)
double m, v; '
{

double sum;

double kep(), kem(), ks():

sum I 0;

sum += kep(s, D, L, Us, v, m, 0.0001);
sum +I kep(S, D, L, Us, v, m, 0.001);
sum +I kep(S, D, L, Us, v, m, 0.01);

sum +I kem(S, D, L, Us, v, m, 0.0001);
sum +I kem(S, D, L, Us, v, m, 0.001);
sum +I kem(S, D, L, Us, v, m, 0.01);

sum.+- ks(S, D, L, Us, v, m, 1.0001);
sum +I ks(S, D, L, Us, v, m, 1.001);
sum.+I ks(S, D, L, Us, v, m, 1.01);

sum /I 9.0:

return(sum);

}

#define Us us .
double kep(s, d, 1, Us, 00p, m, b)
double 3, d, l, Us,_UOp, m, b;
/*
** kE+
*/
{
double q, r;

if (b II 0.0) return(0.0);

m *I AMU;
Us *I E;
UOp *- E;

q I d*(1.-l./sqrt(l.+b))lsqrt(Us);

q +- 2.*s*(1.+sqrt(b)-sqrt(1.+b))/sqrt(Us);
q /I l;

q += 1./sqrt(Us+UOp);

q I UOp - 1./(q*q) + Us*(1.+b);

I s*(1.+sqrt(b)-sqrt(1.+b))/sqrt(Us);
+I d*(1.-1./sqrt(1.+b))/(2.*sqrt(Us));
*I sqrt(2.*m);

HHH

return(q/(r*E));
}

188

double kem(s, d, 1, Us, UOp, m, b)
double 3, d, 1, Us, UOp, m, b;
/*
** kg-
*/
{
double q, r;

if (b II 0.0) return(0.0);

m *I AMU;
Us *I E;
UOp *I E;

q I d*(1.-1./sqrt(1.+b))/sqrt(Us);

q +- 2.*s*(1.-sqrt(b)-sqrt(1.+b))/sqrt(Us);
q /I l;

q +I 1./sqrt(Us+U0p);

q - UOp - 1./(q*q) + Us*(1.+b);

" .

I s*(1.-sqrt(b)-sqrt(1.+b))/sqrt(Us);
+I d*(1.-1./sqrt(1.+b))/(2.*sqrt (Us) );
*I sqrt(2.*m);

NH

return(q/(r*E)):
}

double ks(s, d, 1, Us, UOp, m, a)
double 3, d, 1, Us, UOp, m, a;

{

double q, r;

if (a II 1.0) return(0.0);

m *I AMU:
Us *I E;
U0p *I E;

q I d*(l.-1./sqrt(a))/sqrt(Us);
q +- 2.*s*(1.-sqrt(a))/sqrt(Us);
q /I l;
q +I 1./sqrt(Us+UOp);

q I U0p - 1./(q*q) + Us*a;

I s*(1.-sqrt(a))/sqrt(US);
+- d*(1.-1./sqrt(a))/(2.*sqrt(Us)):
*- sqrt(2.*m);

HHH

return(q/(r*E));
}

SIMPLX

SIMPLX is a program which performs downhill simplex method
optimization to find the best DFF-TOFMS instrument parameters. The
main program and two subroutines are written in VAX FORTRAN; the
remaining routines are written in VAX C.

Description of Routines:

S IMP LX
The main program accepts the starting coordinites of the simplex as
well as the tolorance condition for termination of the search.

FUNK
This little function calls the C function respon() which evaluates the
actual response function.

AMOEBA

This routine, based on Press, Flannery, Teukolsky, and Vetterling,
Numerical Recipes, pp. 292-293, carries out the multidimensional
minimization of FUNKO by means of the downhill simplex method.

respon

This C function calculates the response function.

kep

This calculates 133+.
kem

This calculates 1:13..
ks

This calculates ks.
dfftime

This routine calculates the flight time for an ion having specified
mass, charge, initial position, and initial velocity. The velocity information
is actually specified as a combination of kinetic energy (a scalar quantity,
unlike velocity) and a direction. The integer variable direct, one of the
arguments sent to this routine, is set by the calling program to be either +1

190

or —1 to indicate motion either towards or away from the detector,
respectively. ‘

gettvk

This routine calculates and returns the flight time for the reference
ion, the initial DFF voltage, and the DFF voltage rate of change for an ion of
specified mass.

00000000

10

20

191
PROGRAM SIMPLX

Run~simplex optimdzation routine AMOEBA to find best
parameters for DFF.

Response function is respon(s, d, 1, Us), written in C.
This program is linked to AMOEBA and a version of respon()
callable by FORTRAN.

PARAMETER (NDIM I 2)
PARAMETER (MP I 3)
PARAMETER (TOTAL I 2.0)

REAL F(MP, NDIM), Y<MP), X(NDIM)
REAL FTOL

REAL D, L, U

INTEGER I, J, ITER

REAL FUNK

TYPE *, 'Enter center point (X1, Y1)‘
ACCEPT *, x1, Y1
TYPE *, 'Enter other values (X2, Y2)‘
ACCEPT *, x2, Y2

P(1, 1) - x1
F(l, 2) I Y1

P(2, 1) I X2
P(2, 2) I Y1

P‘3r l) ' X1
9‘31 2) " Y2

no 20 J - 1, MP
no 10 I - 1, NDIM
X(I) - P(J, I)
CONTINUE
Y(J) - FUNK(X)
CONTINUE

TYPE *, 'ENTER FTOL'

ACCEPT *, FTOL

CALL AMOEBA(P, Y, MP, NDIM, NDIM, FTOL, FUNK, ITER)
TYPE * ITER, ' ITERATIONS'

I
TYPE *, '8 set to 0.01'
TYPE *, 'Us- ', P(1,1)
TYPE *, 'D I ', P(l,2)
TYPE *, 'L I ', TOTAL I P(1,2)
CALL EXIT

END

FUNCTION FUNK(X)
PARAMETER (NDIM I 2)
PARAMETER (TOTAL I 2.0)

REAL X(NDIM)
REAL*8 S, D, L, U

S I 0.01

U = DBLE(X(1))
D I DBLE(X(2))
L I TOTAL - D

FUNK I RESPON(S, D, L, U)
RETURN
END

000000000

0000

12

13

14

15

SUBROUTINE AMOEBA

From Numerical Recipes, p. 292.

This is the Nelder & Mead Downhill Simplex Method
Note that this routine finds a MINIMUM, not a maximum.

SUBROUTINE AMOEBA(P, Y, MP, NP, NDIM, FTOL, FUNK, ITER)
PARAMETER (NMAX I 20, ALPHA I 1.0, BETA I 0.5, GAMMA I 2.0)
PARAMETER (ITMAX I 500)

DIMENSION P(MP, NP), Y(MP), PR(NMAX), PRR(NMAX), PBAR(NMAX)

MPTS I NDIM + 1
ITER I 0
ILO I 1
IF (Y(1) .GT. Y(2)) THEN
IHI I 1
INHI I 2
ELSE
IHI I 2
INHI I 1
ENDIF
DO 11 II1, MPTS
IF (Y(I) .LT. Y(ILO)) ILO I I
IF (Y(I) .GT. Y(IHI)) THEN
INHI I IHI
IHI I I
ELSE IF (Y(I) .GT. Y(INHI)) THEN
IF (I .NE. IHI) INHI I I
ENDIF
CONTINUE

NOTE!!! This convergence criterion will not work for functions
who's true minimum is zero!

RTOL - 2. * ABS(Y(IHI) - Y<ILO)) / (ABS(Y(IHI)) + ABS(Y(ILO)))
IF (RTOL .LT. FTOL) RETURN

IF (ITER .EQ. ITMAX) PAUSE 'EXCEEDING MAXIMUM ITERATIONS'

ITER - ITER + 1

no 12 J - 1, NnIM
PsAR(J) - o.
CONTINUE
no 14 I - 1, MPTs
IF (I .NE. IHI) THEN
no 13 J - 1, NnIM
PsAR(J) - PBAR(J) + P(I, J)
CONTINUE
ENDIF
CONTINUE‘
no 15 J - 1, NnIM
PBAR(J) - PBAR(J) / NnIM
PR(J) - (1. + ALPHA) * PBAR(J) - ALPHA * P(IHI, J)
CONTINUE
YPR - FUNK(PR)

16

17

18

19

21

22

23

24

25

194

IF (YPR .LE. Y(ILO)) THEN
DO 16 J I 1, NDIM

PRR(J) I GAMMA * PR(J) + (1. - GAMMA) * PBAR(J)

CONTINUE
YPRR I FUNK(PRR)
IF (YPRR .LT. Y(ILO)) THEN
no 17 J I 1, NDIM
P(IHI, J) I PRR(J)
CONTINUE
Y(IHI) I YPRR
ELSE
no 18 J I 1, NnIM
P(IHI, J) I PR(J)
CONTINUE
Y(IHI) I YPR
ENDIF
ELSE IF (YPR .GE. Y(INHI)) THEN
IF (YPR .LT. Y(IHI)) THEN
no 19 J I 1, NDIM
P(IHI, J) I PR(J)
CONTINUE
Y(IHI) I YPR
ENDIF
no 21 J I 1, NDIM

PRR(J) I BETA * P(IHI, J) + (1. - BETA) * PBAR(J)

CONTINUE
YPRR I FUNK(PRR)
IF (YPRR .LT. Y(IHI)) THEN
DO 22 J I 1, NDIM
P(IHI, J) I PRR(J)
CONTINUE
Y(IHI) I YPRR

ELSE
DO 24 I I 1, MPTS
IF (I .NE. ILO) THEN
no 23 J I 1, NnIM
PR(J) I 0.5 * (P(I, J)
P(I, J) I PR(J)
CONTINUE
Y(I) I FUNK(PR)
ENDIF
CONTINUE
ENDIF
ELSE
no 25 J I 1, NDIM
P(IHI, J) I PR(J)
CONTINUE
Y(IRI) I YPR
ENDIF
GO TO 1

END

/*

** respon.c

*1:

** Find response function for simplex Optimization.

**

** THIS VERSION OF RESPON.C CAN BE CALLED BY FORTRAN
**
*/
#include <stdio.h>
#include <math.h>

#define amu 1.66057e-27
#define e 1.60219e-19

double respon(sp, dp, 1p, Usp)
double *sp, *dp, *lp, *Usp;
{
double 3, d, 1, Us;
double a, b; /* alpha I s' / 3, beta I dU/Us */
double ds, d0; /* Position & energy perturbations
*/
double UOp, m; /* UOprime, mass */
double k1, k2, k3, kave;
double trefO, trefl, timel, timez;
double respon:

extern double src, Vext, dffreg, dffend, tofend;
extern double dffVO, dffk;
extern int mref;

double kep(), kem(), ks(); /* dV/dt functions for 8+, E-, s S
*/
double dfftime(): /* flight time function */

3 I *sp;
d I *dp;
l I *lp;
Us I *Usp;

src I s;
Vext I Us;

dffreg I d;

dffend I d + 0.001:

tofend I d + 1 + 0.001;

respon I 0.0;

printf("s I %f, d I %f, 1 I %f, Us I %f\n", s, d, 1, Us);

for (ds I -3e-4, a-(s+ds)/s; ds <I 3.e-4; ds +I 1e-4, a-(s+ds)/s)

{ /* Space loop */
for (dUI0.01, b-dU/Us; dU <- 0.06; dU+-0.01, b-dU/Us)
{ /* Energy loop */
for (00p I 100.; UOp <- 1100.; UOp +- 250.)
{ /* Voltage loop */

dffVO I UOp; /* Extern dffVO */
for (m.I 100.; m,<I 2100.: m +I 400.)
{ /* Mass loop */

/~k
**
**
**
*‘k

*/

}

printf("respon I %e\n",
return(respon);
}

196

mref I (int) m: /* Extern mref */
k1 I kep(s, d, 1, Us, UOp, m, b);
k2 I kem(s, d, 1, Us, UOp, m, b);
k3 I ks(s, d, 1, Us, UOp, m, a):
kave = (k1 + k2 + k3) / 3.0:

dffk = kave; /* Extern dffk */
timel I

dfftime((int) m, 1, s+ds, d0, 1);
trefO I ‘

dfftime((int) m, l, s, 0.0, l):
trefl I

dfftime((int) (m+1.), 1, s, 0.0, 1);

respon is the reciprocal of the
resolution, since we need to
MINimize respon. That is,
respon I width/separation.

respon +-
fabs(timel - trefO) / (trefl - trefO);

respon);

197

double kep(s, d, 1, Us, UOp, m, b)
double 3, d, 1, Us, UOp, m, b;
/*
** kE+
*/
{
double q, r;

if (b II 0.0) return(0.0);

m *I amu;
Us *I e;
UOp *I e;

q I d*(1.-1./sqrt(1.+b))/8qrt(US):

q +- 2.*s*(l.+sqrt(b)-sqrt(1.+b))/sqrt(Us);
q /- l;

q +- 1./sqrt(Us+UOp);

q - UOp - 1./(q*q) + Us*(1.+b);

H

I s*(1.+sqrt(b)-sqrt(1.+b))lsqrt(Us);
+I d*(l.-l./sqrt(l.+b))/(2.*sqrt(Us));
*I sqrt(2.*m);

HH

return(q/(r*e));

}

double kem(s, d, 1, Us, UOp, m, b)
double 3, d, 1, Us, UOp, m, b;
/*
** kg-
*/
{
double q, r;

if (b II 0.0) return(0.0);

m *I amu;
Us *I e;
UOp *I e;

q I d*(1.-1./sqrt(1.+b))/sqrt(Us):

q +I 2.*s*(1.-sqrt(b)-sqrt(1.+b))/sqrt(Us);
q /= l;

q +- 1./sqrt(Us+U0p);

q I UOp - 1./(q*q) + Us*(1.+b);

I s*(1.-sqrt(b)-sqrt(1.+b))/sqrt(Us):
+I d*(1.-1./sqrt(1.+b))/(2.*sqrt(Us));
*- sqrt(2.*m);

HHH

return(q/(r*e)):
}

double ks(s, d, 1, Us, UOp, m, a)
double 3, d, 1, Us, UOp, m, a;

{ .

double q, r;

if (a II 1.0) return(0.0):

m *I amu:

q I d*(1.-1./sqrt(a))/sqrt(Us);

q +I 2.*s*(1.-sqrt(a))/sqrt(Us):
q /I l;
q +- 1./sqrt(Us+U0p);

q I UOp - 1./(q*q) + Us*a:

N

I s*(1.-sqrt(a))/sqrt(Us);
+I d*(1.-1./sqrt(a))/(2.*sqrt(Us));
*- sqrt(2.*m);

HH

return(q/(r*e)):
}

/*
** dfftime.c

**

** Return flight time through DFF TOFMS instrument
** October, 1987

** George Yefchak

*/

#define KGAMU 1.660578-27

#define COUELE 1.602198-19
#define CONVER 1.018056E-4 /* sqrt(KGAMU/COUELE) */
#define MAXNM 100 /* Maximum iterations for Newton's Method

*/

double dfftime(m, q, s, uO, direct)
double 3, uO:
int m, q, direct:

/*

/*
**
**
**
**
*‘k

*/

/*

**

*/

/*

**

{

double v:

double dffk, dffVO; */

extern double dffk, dffVO;

double top, tmp, tsrc, tffl, time, tf, tref:
double tdff, Old;

double tp, dfflen;

double pp, qq, rr, ss, aa, bb, cc, dd;
double f, fprime;

int i;

extern double src, Vext, dffreg, dffend, tofend;

Get DFF voltage function parameters:

Time to DFF region for reference ion I tref
Initial DFF voltage I dffVO
Rate-Of-change of DFF voltage I dffk

gettvk(m, atref, &dffV0, 8dffk):

Now find flight time
top I CONVER * sqrt(2.*m);

tmp I sqrt(q * s*Vext/src + u0) + direct*sqrt(u0):
tsrc I top * s / tmp;

tmp I 2. * sqrt(q * s*Vext/src + u0):
tffl I top * dffreg / tmp:

v I dffreg / tffl;

time I tsrc + tffl:

tp I time - tref;

dfflen I dffend - dffreg;

We need to solve

**
*‘k
**
**
**
*‘k

**

*/

zoo

dfflen I pp*(tdff+tp)**3 + qq*tdff**2 + rr*tdff + as
This is expressed as

aa*tdff**3 + bb*tdff**2 + cc*tdff + dd I 0
which we solve using Newton's Method.

tmp I COUELE / (m * KGAMU * dfflen);

pp I tmp * dffk / 6.:

qq I tmp * dffVO / 2.;

rr I v - tmp * dffk * tp*tp / 2.;

as I -tmp * dffk * tp*tp*tp / 6.:

an - pp:

bb I (3.*pp*tp + qq):

cc I v:

dd I Idfflen;

tdff I 1.e-4; /* Initial point for Newton's Method */

for (i I 1; i <- MAXNM; ++i)
{
f I aa*tdff*tdff*tdff + bb*tdff*tdff + cc*tdff + dd;
fprime I 3.*aa*tdff*tdff + 2.*bb*tdff + cc:
tdff II f/fprime:
if (tdff < 0.0)
tdff I 0.;
if (fabs(f) <I 1.e-10)
break;
}
if (i >I MAXNM)
printf("Newton failed: f I %e, f' I %e, t I %e\n",
f, fprime, tdff);

tmp I COUELE / (m * KGAMU * dfflen);
v I 3.*pp*(tdff+tp)*(tdff+tp) + 2.*qq*tdff + rr;
tf I (tofend - dffend) / v;

return(time + tdff + tf);
I

201

gettvk(m, t, v, k)
double *t, *v, *k;
int m;
/*
** This dummy routine returns the external values for dffVO * dffk.
*/
{
double top, tmp, tsrc, tffl;
double tmref:

extern double src, dffvo, dffk, Vext, dffreg, dffend, tofend;
extern int mref:

/*
** Find tref for reference mass ion
*/

top I CONVER * sqrt(2.*mref);

tmp I sqrt(Vext);

tsrc I top * src / tmp:

tmp I 2.*sqrt(Vext);

tffl I top * dffreg / tmp:

tmref I tsrc + tffl;
/*
** Find tref for current ion
*/

top I CONVER * sqrt(2.*m);
tmp I sqrt(Vext);

tsrc I top * src / tmp:
tmp I 2.*sqrt(Vext);

tffl I top * dffreg / tmp;
*t I tsrc + tffl;

*v I dffVO + (*t - tmref) * dffk;
*k I dffk:
return;

}

VALLEY

This program generates a % Valley plot (“Wendy Plot”) for the DFF-
TOFMS instrument.

Description of Routines:

main

The main program accepts the required input from the user and
then loops through various user-specified masses calculating the
appropriate percent valley value. For each of these, intensity at the valley
and peak locations are calculated (rather the generating the entire peak
silape). The ratio of these two (multiplied by 100%) yields the percent valley
v ue.

gauss
This returns the value at location x of the a gaussian distribution
function having mean mu and standard deviation 8 igma.

finds
This function returns the value of the spatial coordinate s which

yields a flight time t for an ion of energy e and direction direct.

/*
** valley.c
**
** Calculate % valley height.
**
**
** George Yefchak
*/
#include <stdio.h>
#include <math.h>
#define MAXV 500 /* Maximum array size for FUNCTION.DAT */
#define KBOLT 1.38007e-23 /* Boltzmann constant */
#define COUELE 1.60219e-19 /* electron charge */
#define KGAMU 1.66057e-27 /* kg / amu */
#define CHARGE 1 /* One electron charge on ions */
/*
** External declarations for DFF paramters
*/
double src I 0.01:
double dffreg I 0.100:
double dffend I 0.102;
double tofend I 2.000:
double Vext I 10.0:
float tfile[MAXV], vfilelMAXV];
int mmef, fsize;
main()
{
double tpeakl, tpeak2, tvalley; /* Flight times */
double Ppeak, Pvalley, Pe, Ps; /* Probabilities */
double vmin, vmax, smin, smax; /* Distrib. function extrema
*/
double vinc; /* Velocity increment */
double uO, so, vsigma, ssigma; /* Means and standard
deviations */
double v, e, s, time; /* Velocity, energy, location, time
*/
double temp; /* Temperature */
double dev; /* Deviation from mean */
double mkg; /* Mass (kg) */
int nviter; /* # of velocty iterations */
int iveloc, i; /* Loop variables */
int mass, direct; /* Ion mass, ion direction */
char string[80];
FILE *fp;
double dfftime(); /* Time-of-flight function */
double find8(): /* Returns value for s */
double gauss(); /* Gaussian distribution */
/*

**

*/

Open DFF Voltage Function file

/*

**

*/

/*

**

*/

/*

**

*/

204

if ((fp I fopen("FUNCTION.DAT", "r")) II NULL)
{
printf("Unable to open FUNCTION.DAT\n");
exit(0);
}

for (i I 0; i < MAXV; ++i)
{
if ((fscanf(fp, "%f, %f, %*f", &tfile[i], &
vfile[i])) II EOF) break;
}

fclose(fp);
fsize I i;

for(;:)
{

Input mass, distribution function parameters, etc.

printf("Enter mass (Da) or negative number to stop\n");
mass I atoi(gets(string));
if (mass <= 0)
exit(0);
mkg I (double) mass * KGAMU;

printf("Enter temperature (K)\n");
temp I atof(gets(string));

no I 0.0;

vsigma I sqrt((KBOLT * temp) / mkg):

printf("Enter mean for spatial distribution\n");

so I atof(gets(string));

printf("Enter standard deviation for spatial distribution\n");
ssigma I atof(gets(string));

printf("Enter number of energy iterations\n");
nviter I atoi(gets(string));

Get flight times:

mref I mass:

tpeakl I dfftime(mass, CHARGE, so, uO, 1);
mass +- 1;

mref I mess;

tpeak2 I dfftime(mass, CHARGE, so, uO, 1);
tvalley I (tpeakl + tpeak2) / 2.0;

mass II 1;

mref I mass;

Now calculate maximum.and valley intensities.

Ppeak I 0.0;

Pvalley I 0.0;

vmax I 3.0 * vsigma;
vinc I vmax / nviter;

205

for (v I 0.0, iveloc I 1; iveloc <I nviter; ++iveloc, v += vinc)
{ .
e I 0.5 * mkg * v * v / COUELE; /* P[E] */
Pe I gauss(v, 0.0, vsigma);

s I finds(tpeakl, e, l); /* P[S] for peak max */
Ps I gauss(s, so, ssigma):

Ppeak +I Pe*Ps:

s I finds(tpeakl, e, I1):

Ps I gauss(s, so, ssigma):

Ppeak +I Pe*Ps:

s I finds(tvalley, e, 1); /* P[S] for valley from left

Ps I gauss(s, so, ssigma):
Pvalley +I Pe*Ps:
s I finds(tvalley, e, I1):
Ps I gauss(s, so, ssigma):
Pvalley +I Pe*Ps:

mass +- 1; /* P[S] for valley from right */

mref I mass:

s I finds(tvalley, e, 1):

Ps I gauss(s, s0, ssigma):

Pvalley +I Pe*Ps:

s I finds(tvalley, e, I1):

Ps I gauss(s, sO, ssigma):

Pvalley +I Pe*Ps:

mass II 1:

mref I mass:

)
printf("Mass I %d, % valley I %f\n", mass, 100.0*Pva11ey / Ppeak);
}

double gauss(x, mu, sigma)
double x, mu, sigma;
#define PI 3.141592654
{
return((exp(-(x-mu)*(x—mu)/(2.0*sigma*sigma)))
/ (sqrt(2.0*PI)*sigme));
}

double finds(t, e, direct)
double t, e:
int direct:
{
double thi, tlo, tmid;
double shi, slo, smid:

extern int mref;
double dfftime();

shi I 0.009; /* (Large 3 yields low time, and vice
versa */

510 I 0.011;

thi I dfftime(mref, CHARGE, shi, e, direct);

tlo I dfftime(mref, CHARGE, slo, e, direct):

smid I (shi + slo) / 2.0;

while (fabs((shi I slo) / slo) > 1e-5)
{
tmid I dfftime(mref, CHARGE, smid, e, direct):
if (t < tmid)
{
shi I smid:
thi I dfftime(mref, CHARGE, shi, e, direct):
. }
else
{
slo I smid: .
tlo dfftime(mref, CHARGE, slo, e, direct):
}
smid I (shi + slo) / 2.0;
}
return(smid):
}

ANIMATE

This is the DFF Animation. The program displays ions in user-
specified colors on the screen and moves them (hence “animation”) through
a DFF-TOFMS instrument.

Description of Routines:

ANIMATE

This is the main program. It accepts the required user input and
calls routines to open the graphics window, move the ions, and finally close
the window.

DONE
This function test whether or not any of the ions have reached the end
of the flight path; if so, the animation is “done”.

DRAW
This routine draws the ions on the screen.

FAST

This function tests to see if all the ions have entered the final field-
free region; if so, the animation is speeded up to account for the much
longer length usually assigned to this region.

FINDXV

This routine calculates new positions and velocities for the ions to
move them forward one step. The routine, which is called by MOVE () . is
written to handel the case of an infinitely-thin DFF region. A previous
version of AN IMATE used Runge-Kutte integration in routine MOVE ()
rather than FINDXV () to deal with finite-length DFF regions.

GCLOSE
This routine closes the graphics window when done.

GENY
This routine generates the initial vertical coordinites for the ions,

which remain constant throughout the animation.

GOPEN

$8

This routine opens the graphics window and draws the basic screen
layout. '

MOVE
This routine moves the ions through one step.

NUMBERS
This routine outputs the numerical portions of the display.

XFORM
This routine transforms instrument coordinates to generic screen
coordinates.

XMAP
This function maps horizontal screen coordinates to device
coordinates.

YMAP 1
This function maps vertical screen coordinates to device coordinates

for the upper window.

YMAP 2
This function maps vertical screen coordinates to device coordinates

for the lower window.

0000000

000

209
PROGRAM ANIMATE

Show motion of ions through DFF TOFMS.

George Yefchak
January, 1987

PARAMETER (NMAX I 9)

REAL*8 X(NMAX), XINIT(NMAX), V(NMAX), V0(NMAX)

REAL*8 MASS(NMAX), Y(NMAX), XMARK(NMAX)

REAL*8 XREF, VREF, MREF

REAL*8 DELTAT, TIME, WINSIZ, A, TEMP, TLAG

REAL*8 KGAMU, COUELE

REAL*8 DFFVO, DFFK, TEREE, VEXT, DFFREG, DEFEND, TOFEND
REAL*8 TOP, TMP, 5, U0

INTEGER NIONS, I, UPnAT1, UPnATz, USAVEl, USAVEZ, IREF
LOGICAL FAST, DONE, CHECK

CHARACTER*1 ANSWER

CHARACTER*40 FNAME

COMMON /DFF/ DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND

VDI declarations:

INTEGER*2 DEVHDL, DEVPL, DUMMY(2), COLOR(NMAX), PEN(NMAX)
INTEGER*2 WKINSC(19), WKINPL(19)
INTEGER*2 VSMSTR, VQERR

DATA KGAMU, COUELE /1.6606E-27, 1.60219E-19/
nATA PEN /NMAX*1/
nATA WKINSC /0, 1, 1, 3, 1, 1, 1, o, o, 1, 1,

* 68, 73, 83, 80, 76, 65, 89, 32/

DATA WKINPL /0, 1, 1, 3, 1, 1, 1, 0, 0, 1, 1,

* 80, 76, 79, 84, 84, 69, 82, 32/

CHECK I .TRUE.

PRINT *, 'Do you want to use a command file? (Y/N) '
READ(*, 999) ANSWER
IF (ANSWER .EQ. 'Y' .OR. ANSWER .EQ. 'y') THEN
PRINT *, 'Enter filename: '
READ (*, 999) FNAME
OPEN (UNITIl, FILE-FNAME, STATUSI'OLD')
READ(1, *) NIONS
IF (NIONS .GT. NMAX) STOP 'TOO MANY IONS'
DO 5 I I 1, NIONS
READ(1, *) MASS(I), X(I), V(I), COLOR(I)
CONTINUE
READ(1, *) IREF, WINSIZ
READ(1, *) DELTAT, TLAG
READ(1, *) DFFVO, DFFK, VEXT, DFFREG, DFFEND, TOFEND
READ(1, *) UPDATl, UPDATZ
CLOSE(UNITI1)

10

210

ELSE
PRINT *, 'Enter number of ions, or 0 to STOP: '
READ (*, *) NIONS
IF (NIONS .LT. 1) STOP
IF (NIONS .GT. NMAX) THEN
PRINT *, 'TOO MANY IONS'
GO TO 2
END IF
DO 10 I I 1, NIONS
PRINT *, 'Enter mass, x, v, and color for ion', I,

READ(*, *) MASS(I), X(I), V(I), COLOR(I)
CONTINUE
PRINT *, 'Enter reference ion number and window half-size:

READ(*, *) IREF, WINSIZ

PRINT *, 'Enter time step size and lag time: '

READ (*, *) DELTAT, TLAG

PRINT *, 'Enter DFFVO, DFFK, VEXT, DFFREG, DFFEND, and

TOFEND: '

000M

PRINT *
READ(*, *) DFFVO, DFFK, VEXT, DFFREG, DFFEND, TOFEND
PRINT *, 'Enter screen-update parameters: '
READ(*, *) UPDATl, UPDATZ
END IF

USAVEI I UPnATl

USAVEZ I UPnATz

no 20 I I 1, NIONS
XINIT(I) I X(I)
XMARK(I) I X(I)
V0(I) I V(I)

CONTINUE

Initialization complete, now Open graphics.

CALL GENY(NIONS, I)

CALL GOPEN(DEVHDL, WKINSC)

Calculate time of arrival at the DFF region for the reference

IF (X(IREF) .GT. DFFREG) STOP 'Can"t calculate Tfree'
IF (X(IREF) .EQ. DFFREG) THEN
TFREE I 0.0
ELSE
S I IXINIT(IREF)
TOP I SQRT(Z. * MASS(IREF) * KGAMU / COUELE)
U0 I 0.5 * MASS(IREF) * KGAMU * V0(IREF) * V0(IREF)
TMP I SQRT(S * VEXT / 0.02 + 00)
IF (V0(IREF) .GE. 0.) THEN
TMP I TMP + SQRT(UO)
ELSE
TMP I TMP I SQRT(UO)
ENDIF

211

TFREE I TOP * s / TMP
TMP I 2.0 * SQRT(S * VEXT / 0.02 + U0)
TFREE I TFREE + TOP * DFFREG / TMP

ENDIF
C
C Set time I 0 and draw ions in their initial positions.
C
TIME I 0.0
CALL DRAW(NIONS, XINIT, X, Y, V, XMARK, 1, IREF, WINSIZ,
* DEVHDL, COLOR, TIME)
C
C Start animation when <RETURN> is pressed.
C If "P<RETURN>" is pressed, copy screen to plotter.
C
99 READ(*, 999) ANSWER

IF (ANSWER .EQ. 'P' .OR. ANSWER .EQ. 'p') THEN

CALL GOPEN(nEVPL, WKINPL)

CALL nRAW(NIONS, XINIT, x, Y, v, XMARK, 1, IREF,
* WINSIZ, nEVPL, COLOR, TIME)

CALL GCLOSE(DEVPL)

PRINT *, 'PLOTTING DONE'

GO TO 99
ENDIF

100 IF (DONE(NIONS, X, TOFEND)) GO TO 149
C
C Check for a key; pause if found.
C
STATUS I VSMSTR(DEVHDL, 1, 0, DUMMY, ANSWER)
IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
IF (STATUS .GT. 0) THEN
CALL DRAW(NIONS, XINIT, X, Y, V, XMARK, 1, IREF,
* WINSIZ, DEVHDL, COLOR, TIME)
IF (ANSWER .EQ. 'P' .OR. ANSWER .EQ. 'p') THEN
CALL GOPEN(DEVPL, WKINPL)
CALL DRAW(NIONS, XINIT, X, Y, V, XMARK, 1, IREF,
* WINSIZ, DEVPL, COLOR, TIME)
CALL GCLOSE(DEVPL)
PRINT *, 'PLOTTING DONE'
PRINT *
ENDIF

105 STATUS I VSMSTR(DEVHDL, 1, 0, DUMMY, ANSWER)
IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
IF (STATUS .EQ. 0) GO TO 105
ENDIF

IF (TIME .LT. TLAG) THEN
TIME I TIME + 10. * DELTAT
CALL MOVE(NIONS, X, XINIT, V, V0, 10.*DELTAT,
TIME, MASS, TLAG)
ELSE
TIME I TIME + DELTAT
CALL MOVE(NIONS, X, XINIT, V, V0, DELTAT, TIME,
MASS, TLAG)
ENDIF

252

IF (UPDATl .EQ. 1) THEN

CALL
* DRAW(NIONS, XINIT, X, Y, V, XMARK, UPDAT2,
* IREF, WINSIZ, DEVHDL, COLOR, TIME)
END IF

IF (UPDATl .GT. 1) THEN
UPDATl I UPDATl - 1
ELSE
UPDATl I USAVEl
END IF
IF (UPDATZ .GT. 1) THEN
UPDATZ I UPDATZ - 1
ELSE
UPDAT2 I USAVE2
END IF

IF (CHECK) THEN
IF (FAST(NIONS, x, DFFEND)) THEN
DELTAT I DELTAT * 10.
CHECK I .FALSE.
END IF
END IF
GO TO 100
149 CALL DRAW(NIONS, XINIT, x, Y, v, XMARK, 1, IREF, WINSIZ,
* DEVHDL, COLOR, TIME)

150 READ(*, 999) ANSWER
IF (ANSWER .EQ. 'P' .OR. ANSWER .EQ. 'p') THEN
CALL GOPEN(nEVPL, WKINPL)
CALL nRAW(NIONs, XINIT, x, Y, v, XMARK, 1, IREF,
* WINSIZ, nEVPL, COLOR, TIME)
CALL GCLOSE(nEVPL)
PRINT *, 'PLOTTING DONE'

ELSE
CALL GCLOSE(DEVHDL)
GO TO 1

ENDIF

GO TO 150

999 FORMAT (A)
END

253
LOGICAL FUNCTION DONE(N, x, TOFEND)

Return true if any ions have reached the end of the
flight path.

0000

REAL*8 X(N), TOFEND
INTEGER I

DONE I .FALSE.
DO 10 I I 1, N
IF (X(I) .GE. TOFEND) DONE I .TRUE.
10 CONTINUE
RETURN
END

000

000

000

O(7()F'

214

SUBROUTINE DRAW(NIONS, XINIT, x, Y, v, XMARK, UPDATE,
IREF, WINSIZ, DEVHDL, COLOR, TIME)

Draw line from Old point to new point for each ion.

REAL*8 TOFSTA, L0, R1
PARAMETER (TOFSTA I -0.02)
PARAMETER (L0 I 0.0)
PARAMETER (R1 I 1.0)

REAL*8 XINIT(NIONS), X(NIONS), Y(NIONS), V(NIONS),
XMARK(NIONS), WINSIZ, TIME

REAL*8 XLINE(2), YLINE(2)

REAL*8 RANGE, LEFT, RIGHT

REAL*8 OFFSET

REAL*8 DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND

REAL * 8 XFORM

INTEGER UPDATE, IREF
INTEGER I, K

COMMON /DFF/ DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND
VDI declaration:

INTEGER*2 DEVHDL,- STATUS, ZERO

INTEGER*2 XMAP, YMAPl, YMAP2

INTEGER*2 XYVDI (20) , COLOR(NIONS)

INTEGER*2 VQERR, VPLINE, VPMARK, VSMCOL, VSLCOL

DATA OFFSET /o.05/
DATA ZERO /0/

Draw ion trajactories in upper window.

no 10 I I 1, NIONS
STATUS I VSLCOL(DEVHDL, COLOR(I))
IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
XLINE(l) I XFORM(XINIT(I))

XLINE(Z) I XFORM(X(I))

YLINE(1) I Y(I)

YLINE(2) I Y(I)

XYVDI(1) I XMAP(XLINE(1), XFORM(TOFSTA), XFORM(TOFEND))
XYVDI(3) I XMAP(XLINE(2), XFORM(TOFSTA), XFORM(TOFEND))
XYVDI (2) I YMAP1(YLINE(1))

XYVDI(4) I YMAP1(YLINE(2))

STATUS I VPLINE (DEVHDL, 2, XYVDI)
IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
CONTINUE

Now draw ions in lower window.

LEFT I X(IREF) _ WINSIZ

RIGHT I X(IREF) + WINSIZ

RANGE I RIGHT - LEFT

IF (RANGE .GT. 0.0 .AND. UPDATE .EQ. 1) THEN

000

100

000

105

110

0000

10

215

Erase Old ions...

STATUS I VSMCOL(DEVHDL, ZERO)
IF (STATUS .LT. 0) PRINT *, 'ERROR IN VSMCOL', VQERR()
K I 0
no 100 I I 1, NIONS
IF (XMARK(I) .GE. 0.0 .AND. XMARK(I) .LE. 1.0) THEN
XYVDI(K+1) I XMAP(XMARK(I), Lo, R1)
XYVDI(K+2) I YMAP2(Y(I) + OFFSET)
K I K + 2
ENDIF
CONTINUE
STATUS I VPMARK(DEVHDL, K/2, KYvnI)
IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()

.and draw new ones.

DO 105 I I 1, NIONS
XMARK(I) I (X(I) - LEFT) / RANGE
CONTINUE
DO 110 I I 1, NIONS
IF (X(I) .GE. LEFT .AND. X(I) .LE. RIGHT) THEN
STATUS I VSMCOL(DEVHDL, COLOR(I))
IF (STATUS .LT. 0) PRINT *,
'ERROR IN VSMCOL', VQERR()
XYVDI(l) I XMAP(X(I), LEFT, RIGHT)
XYVDI(2) I YMAP2(Y(I) + OFFSET)
STATUS I VPMARK(DEVHDL, 1, XYVDI)
IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
ENDIF
CONTINUE
CALL NUMBERS(NIONS, V, X, Y, DEVHDL, TIME, COLOR)
END IF
RETURN
END

LOGICAL FUNCTION FAST(N, X, DFFEND)

Return true if all ions have passed into the final
flight path.

REAL*8 X(N), DFFEND
INTEGER I

FAST I .TRUE.
DO 10 I I 1, N
IF (X(I) .LT. DFFEND) FAST I .FALSE.
CONTINUE
RETURN
END

000

000

000

000

000

216
SUBROUTINE FINDXV(MASS, x0, vo, TIME, x, V)
Return new X and V using zero-length DFF region.

IMPLICIT REAL*8 (AII, OIZ)
REAL*8 MASS, X0, V0, TIME, X, V

REAL*8 ACCEL, TS, TD, KGAMU, COUELE
REAL*8 TOP, TMP, U0
REAL*8 DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND

COMMON /DFF/ DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND
DATA KGAMU, COUELE/ 1.6606EI27, 1.60219E-19/

ACCEL I COUELE * VEXT / (MASS * KGAMU * 0.02)
TOP I SQRT(2. * MASS * KGAMU / COUELE)
U0 I 0.5 * MASS * KGAMU * vo * v0 / COUELE
TMP I SQRT(IXO * VEXT / 0.02 + U0)
IF (v0 .GE. 0.) THEN
TMP I TMP + SQRT(UO)
ELSE
TMP I TMP - SQRT(UO)
ENDIF
TS I -xo * TOP / TMP
TMP I 2.0 * SQRT(IXO * VEXT / 0.02 + U0)
Tn I TS + TOP * DFFREG / TMP

In source...
IF (TIME .LT. TS) THEN
X I ACCEL * TIME * TIME / 2.0 + V0 * TIME + X0
V I ACCEL * TIME + V0
ELSE IF (TIME .GE. TS .AND. TIME .LT. TD) THEN
In FF1...
X I V * (TIME - TS)
ELSE IF (V .GT. 0.) THEN
After FF1, but not yet accelerated by DFF region...
TMP I 0.5 * MASS * KGAMU * V * V
TMP I TMP + COUELE * (DFFVO + DFFK * (TIME I TFREE))
V I ISQRT(2. * TMP / (MASS * KGAMU))
X I DFFEND
ELSE
In FF2... (V set to IV as a flag)
X I IV * (TIME - TD) + DFFEND
ENDIF

RETURN
END

000

10

217
SUBROUTINE GCLOSE(DEVHDL)
INTEGER*2 DEVHDL, STATUS, VQERR
STATUS I VCLRWK(DEVHDL)
STATUS I VENCUR(DEVHDL)
STATUS I VCLSWK(DEVHDL)
RETURN
END
SUBROUTINE GENY(NIONS, Y)

Generate ordinate values for plotting ions on the screen.

REAL*8 Y(NIONS), DELY
INTEGER I

IF (NIONS .EQ. 1) THEN

Y(l) I 0.5
ELSE
DELY I 0.8 / FLOAT(NIONS - 1)
no 10 I I 1, NIONS
Y(I) I 0.85 - DELY * FLOAT(I - 1)
CONTINUE
END IF
RETURN

END

218

SUBROUTINE GOPEN(DEVHDL, WORKIN)

C
C Open graphics window and draw TOFMS regions.
C
REAL*8 ZERO, ONE, BACK
PARAMETER (ZERO I 0.0)
PARAMETER (ONE I 1.0)
PARAMETER (BACK I I0.02)
REAL*8 X0, X1, X2, X3, X4, Y0, Y1
REAL*8 XBOX(5), YBOX(5), XZBOX(5), YZBOX(5)
REAL*8 XLINE(2), YLINE(2)
REAL*8 DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND
REAL*8 XFORM
INTEGER I, K
COMMON /DFF/ DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND
C
C VDI Graphics declarations here:
C

IMPLICIT INTEGER*2 (V)
INTEGER*2 WORKIN(19), WORKOT(66)
INTEGER*2 STATUS, DEVHDL
INTEGER*2 XYVDI(lO)
INTEGER*2 SPEED

C Function calls:
INTEGER*2 XMAP, YMAPI, YMAP2

DATA Yo, Y1/o.o, 1.0/

DATA YBOX /0.0, 0.0, 1.0, 1.0, 0.0/
DATA xzsox /0.0, 1.0, 1.0, 0.0, 0.0/
DATA stox /0.0, 0.0, 1.0, 1.0, 0.0/
DATA YLINE /0.0, 1.0/

DATA SPEED /25/

X0 I XFORM(BACK)

X1 I XFORM(0.0D0)
X2 I XFORM(DFFREG)
X3 I XFORM(DFFEND)

X4 = XFORM(TOFEND)
XBOX(1) I XFORM(BACK)
XBOX(2) I XFORM(TOFEND)
XBOX(3) I XFORM(TOFEND)
XBOX(4) I XFORM(BACK)
XBOX(5) I XFORM(BACK)

C

C********** OPEN WORKSTATION

C
STATUS I VOPNWK(WORKIN, DEVHDL, WORKOT)
IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
STATUS I VPNSPD(DEVHDL, SPEED)

C

C********** DRAW LOWER BOX
C

10

C

219

K I 1

DO 10 I I'l, 5
XYVDI(K) I XMAP(X2BOX(I), ZERO, ONE)
XYVDI(K+1) I YMAP2 (Y2BOX(I))
K I K + 2

CONTINUE

STATUS I VPLINE(DEVHDL, 5, XYVDI)

IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()

C********* DRAW UPPER BOX AND TOFMS REGIONS

C

20

K I 1

DO 20 I I 1, 5
XYVDI (K) I XMAP(XBOX(I), XFORM(BACK), XFORM(TOFEND))
XYVDI(K+1) I YMAP1(YBOX(I))
K I K + 2

CONTINUE

STATUS I VPLINE (DEVHDL, 5, XYVDI)

IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()

XYVDI (2) I YMAP1(YLINE(1))

XYVDI (4) I YMAP1(YLINE(2))

XLINE (1) XFORM (X1)

XLINE (2) XFORM (X1)

XYVDI(l) I XMAP(XLINE(1), XFORM(BACK), XFORM(TOFEND))
XYVDI(3) I XMAP(XLINE(Z), XFORM(BACK), XFORM(TOFEND))
STATUS I VPLINE (DEVHDL, 2, XYVDI)

IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
XLINE(l) I XFORM(XZ)

XLINE(Z) I XFORM(XZ)

XYVDI(l) I XMAP(XLINE(1), XFORM(BACK), XFORM(TOFEND))
XYVDI(3) I XMAP(XLINE(Z), XFORM(BACK), XFORM(TOFEND))
STATUS I VPLINE (DEVHDL, 2, XYVDI)

IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
XLINE(l) I XFORM(X3)

XLINE(Z) I XFORM(X3)

XYVDI(l) I XMAP(XLINE(1), XFORM(BACK), XFORM(TOFEND))
XYVDI(3) I XMAP(XLINE(2), XFORM(BACK), XFORM(TOFEND))
STATUS I VPLINE(DEVHDL, 2, XYVDI)

IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()

RETURN

END

0000

000

100

220
SUBROUTINE MOVE(NIONS, X, X0, V, V0, DELTAT, TIME, MASS, TLAG)

Replace each ion's position with the position it
should have at TIME + DELTAT.

REAL*8 X(NIONS), X0(NIONS), V(NIONS), v0(NIONS)

REAL*8 DELTAT, TIME, MASS(NIONS), TLAG

REAL*8 DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND
REAL*8 KGAMU, COUELE

REAL*8 CC, R, TSD, L

REAL*8 YY(2), XX, nT

INTEGER NIONS

INTEGER I

COMMON /DFF/ DFFVO, DFFK, TFREE, VEXT, DFFREG, DFFEND, TOFEND
COMMON /CCR/ CC, R, TSD

DATA KGAMU, COUELE /1.6606n-27, 1.60219n-19/
TSD I TFREE

L I DFFEND I DFFREG
VX I VEXT
IF (TLAG .NE. 0.) PRINT *, 'TIME LAG NOT ALLOWED'

Here we go...

DO 100, I I 1, NIONS
CALL FINDXV(MASS(I), X0(I), V0(I), TIME, X(I), V(I))
CONTINUE
RETURN
END

000

000

000

999

000

1000

10

0(7()

2000

221

'SUBROUTINE NUMBERS(NIONS, V, X, Y, DEVHDL, TIME, COLOR)

Print ion velocities, etc. on graphics screen.

PARAMETER (NMAX I 9)

REAL*8 V(NIONS), X(NIONS), Y(NIONS), TIME
REAL*8 OFFSET

INTEGER I

CHARACTER*7 TEXT

CHARACTER*9 TEXT9

VDI Declarations:

INTEGER*2 DEVHDL, STATUS
INTEGER*2 XVDI, YVDI, XTVDI
INTEGER*2 YMAPl, YMAP2
INTEGER*2 VQERR, VGTEXT
INTEGER*2 COLOR(NIONS), WHITE

DATA OFFSET /0.04/
DATA xvnI /1838/
DATA XTVDI /15000/
DATA WHITE /1/

Put time in upper window.

WRITE(TEXT9, 999) TIME

FORMAT(E9.4)

YVDI I YMAP1(Y(1) + OFFSET)

STATUS I VSTCOL(DEVHDL, WHITE)

STATUS I VGTEXT (DEVHDL, XTVDI, YVDI, TEXT9)

Put velocities in upper window.

DO 10 I I 1, NIONS
IF (X(I) .LE. 0.0) THEN
WRITE(TEXT, 1000) V(I)
ELSE
WRITE(TEXT, 1000) ABS(V(I))
ENDIF
FORMAT(F7.0)
YVDI I YMAP1(Y(I) + OFFSET)
STATUS I VSTCOL(DEVHDL, COLOR(I))
STATUS I VGTEXT(DEVHDL, XVDI, YVDI, TEXT)
IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
CONTINUE

Put positions in lower window.

DO 20 I I 1, NIONS

WRITE(TEXT, 2000) X(I)
FORMAT(F7.5)

YVDI I YMAP2(Y(I) + OFFSET)

222

STATUS I VSTCOL (DEVHDL, COLOR(I))

STATUS I VGTEXT(DEVHDL, XVDI, YVDI, TEXT.)

IF (STATUS .LT. 0) PRINT *, 'ERROR ', VQERR()
20 CONTINUE

RETURN
END

000000

000

223
REAL*8 FUNCTION XFORM(X)
Transform world coordinates to screen coordinates.

This squeezes the last field-free region into the
second half of the screen.

REAL*8 X
REAL*8 DFFK, DFFVO, TFREE, VEXT, DFFREG, DFFEND, TOFEND
COMMON /DFF/ DFFK, DFFVO, TFREE, VEXT, DFFREG, DFFEND, TOFEND

IF (X .LT. DFFEND .OR. DFFEND .GE. TOFEND) THEN
XFORM I X
ELSE
XFORM I (X - DFFEND) * DFFEND / (TOFEND - DFFEND) + DFFEND
ENDIF
RETURN
END

INTEGER*2 FUNCTION XMAP(X, XL, XR)
Transform x to device coordinates.
PARAMETER (MAXINT I 32767)
REAL*8 X, XL, XR
IF (X .LT. XL) THEN
XMAP I 0

ELSE IF (X .GT. XR) THEN
XMAP I MAXINT

ELSE
XMAP I INT(MAXINT * (X - XL) / (XR I XL))
ENDIF
XMAP I 1638 + INT(0.9 * FLOAT(XMAP))
RETURN

END

000

000

224

INTEGER*2 FUNCTION YMAP1(Y)

Map Y from unit coordinates to upper window
PARAMETER (MAXINT I 32767)
PARAMETER (HLFMAX I 14000)
PARAMETER (TOPHLF I MAXINT I HLFMAX)
REAL*8 Y
YMAPl I TOPHLF * Y + HLFMAX
YMAPl I 1638 + INT(0.9 * FLOAT(YMAP1))
RETURN
END
INTEGER*2 FUNCTION YMAP2(Y)

Map Y from unit coordinates to lower window

PARAMETER (MAXINT I 32767)
PARAMETER (HLFMAX I 14000)

REAL*8 Y

YMAP2 I HLFMAX * Y

YMAP2 I 1638 + INT(0.9 * FLOAT(YMAP2))
RETURN

END

225
Simulation Programs for the Beam Deflection Project

This following are listings of the programs BBB and BDTOF, which
were used in the beam deflection studies.

PROGRAM BDR ............................................................................... 2%
main ...................................................................................... 28
gopen ..................................................................................... 234
xmap ..................................................................................... 2%
ymap ...................................................................................... 2%
rk4 ......................................................................................... 237
draw ...................................................................................... 238
move ...................................................................................... 240
derivs ..................................................................................... 241
done ....................................................................................... 241
collide .................................................................................... 242
data_table ............................................................................... 244
Misc. utility functions .............................................................. 246
DIG routines ........................................................................... 247

PROGRAM BDTOF ........................................................................... 257
main ...................................................................................... 269
rk4 ......................................................................................... %2
derivs ..................................................................................... 263
test ......................................................................................... 261
tab ......................................................................................... %
gasdev .................................................................................... %6
ran ........................................................................................ $7

Misc. utility functions .............................................................. %7

This is the animation program which models ion motion through a
pair of deflection plates. (The name “BDR” is an acronym for Beam
Deflection/Reflectron; it was originally intended for the program to model
motion through both the beam deflection plate and a reflectron.) This
program is written in Microsoft C and is to be run on an IBM (or
compatible) computer. In a manner similar to program ANIMATE, from
the dynamic-field focusing project, it displays colored dots on the screen
which represent ions, and it moves these dots to represent ion motion.
Graphics output to the screen is accomplished via the Microsoft C graphics
library (included with the compiler). Output to an HPGL-language plotter
and to a Postscript-language printer are accomplished routines written by
Drake Diedrich using the dig . h header file.

Description of Routines:

main

The main pregram obtains the required input from the user and
controls the iterations through the routines which calculate ion motion,
display ions on the screen, etc. Compilation of this and other modules
requires the header files bdr .h and dig . h.

gopen
Open graphics window, draw box, deflection plates, slit, and ions at
their initial positions.

xmap
Map x coordinate in (O, 1) space to physical device units.

ymap
Map y coordinate in (0, 1) space to physical device units.
rk4
This routine, based on Press, Flannery, Teukolsky, and Vetterling,

Numerical Recipes, p. 553, performs fourth-order Runge-Kutta integration.

derivs
This routine calculates derivatives (dV/dt values) for rk4 .

draw

227

Draw ions in new positions (and remove old ones) to simulation ion
motion.

move
Calculate new positions of ions after one time step.

done
Check to see if program is done (all velocities equal zero).

collide
Stop ions which hit the plates, etc.

data_tab 1e
Output position and velocity of moving ions, as well as the packet
width (max — min of moving ions) to the screen.

Misc. utility functions
These are functions from the C version of Numerical Recipies which
allocate array space, etc.

DIG routines
This is Drake Diedrich’s library of Device-Independent Graphics
routines for screen, HPGL, and Postscript graphics output.

/*
** bdr.h
*/

#define TRUE -1

#define FALSE 0

#define OR ll

#define AND &&

#define MAXIONS 50

#define KGAMU 1.66063-27
#define COUELE 1.60219E-l9
#define PI 3.14159265359
#define RperD (2.0 * PI / 360.0)

#define ROWl (short) 1

#define COLl (short) 7

#define ROWZ (short) 2

#define COLZ (short) 7

#define ROW3 (short) 3

#define COL3 (short) 7

#define ROW4 (short) 1

#define COL4 (short) 40
#define Rows (short) 2

#define COLS (short) 40
/*

** dig.h

*/

#include <graph.h>
#include <stdio.h>

extern FILE *gcom;

#define SCREEN 0
#define HP 1
#define POST 2

int setvideomode(int,int);

struct videoconfig *getvideoconfig(struct videoconfig *,int);
int setcolor(int,int);

int ellipse(int,int,int,int,int,int);

void clearscreen(int,int);

int rectangle(int,int,int,int,int,int)z

int lineto(int,int,int);

struct xycoord moveto(int,int,int);

int setpixel(int,int,int);

void outtext(char *,int);

struct rccoord settextposition(int,int,int);
int settextcolor(int,int);

int arc(int,int,int,int,int,int,int,int,int);

/*

**

*/

bdr.c

#include <stdio.h>
#include <conio.h>
#includeo<stdlib.h>
#include <string.h>
#include <math.h>
#include <graph.h>
#include "bdr.h"
#include "dig.h"

/*
**
**
**
**
**
*‘k
**

*/
/*
*1:

*/

*/

BDR
Animation of beam-deflection reflectron instrument.

Deflection plates located from plate_x1 to plate_x2 in the
x-direction, and at plate_y1 and plate_y2 in the y-direction.

External variables:

double window_x, window_y: /* View window extrema */
double x_slope, y_slope, x_intcp, y_intcp; /* Screen scale factors

double plate_xl, plate_x2, plate_y1, plate_y2; /* Dev. plate */
double slit_x, slit_y1, slit_y2;/* Slit parameters */

double def_volt; /* Deflection plate voltage */
double def_v1, def_v2, def_v3; /* Initial, mid, and final

voltages */

double def_v_tl, def_v_t2, def_v_t3, def_v_t4;
/* Voltage held at def_v1 till
time dev_v_t1, then ramped to
def_v2, reaching it at time
dev_v_t2, held there till def_v_t3,
then ramped to def_v3, reaching it
at time def_v_t4. */

char *quittext I "Hit <ESC> to quit ";

char *datatext I "Hit <SPACE> for data table “:

char *pausetext I "Hit any other key to pause ('P' to Plot)

char *conttext "Hit any key to continue ";

char *donetext "DONE: Hit any key to finish ".
main(int argc, char *argv[])

{
/*
** Variable declarations:
*/

double massIMAXIONS], x[MAXIONS], VXIMAXIONS], y[MAXIONS],
vy[MAXIONS];

double time, delta_t, rampl, ramp2; /* t, dt, def. volt. ramp
times */

double vel, ang, swap; /* Temporary storage */

int color[MAXIONS]; /* Ion colors */

steps

int n_ions; /* Number of ions */

int draw_freq; /* Redraw after this # of
*/ .

int pltdev; /* Plot device code */
int 1:

char string[80], ch;
struct videoconfig vc;

FILE *fp;
/*
** Function declarations:
*/

void gopen(int); /* Open graphics
*/

void draw(int, double I], double [1, int [1, int); /* Draw ions
*/

void move(int, double *, double *, double *, double *,

double *, double, double); /* Move ions */

int done(int, double *, double *, double *, double *):/* See if
done */

void collide(int, double *, double *, double *,

double *): /* Stop ions which

hit plates, etc. */
void data_table(int, double *, double *, double *, double *):
/* Output data */

/*
** Program:
**
** INPUT DATA
*/

if (argc > 1) /* Open file: */

{
if ((fp I fopen(argv[1], "r")) II NULL)

{

printf("Error opening %s\n", argv[1]);

exit(0);

}

}
else
do {

printf("Enter name of data file: "):

} while ((fp I fopen(gets(string), "r")) II NULL);
n_ions I atoi(fgets(string, 80, fp)); /* Get number of ions */
if (n_ions > MAXIONS)

{
printf("ERROR: Too many ions!\n");
exit(0);
l
for (i I 0; i < n_ions; ++i)
{
fgets(string, 80, fp); /* Read a line... */
sscanf(string, "%lf, %lf, %lf, %lf, %lf, %d\n",
mass+i, x+i, y+i, &vel, sang, color+i); /* ... and get
initial

position, velocity, angle,
and ion symbol color */

 

231

vx[i] I vel * cos(RperD * ang): /* Calculate velocities */
vy[i] I vel * sin(RperD * ang); /* in x and y directions */
}
delta_t I atof(fgets(string, 80, fp)): /* Get time step-size */
fgets(string, 80, fp); /* Read a line and ... */
sscanf(string, "%lf, %lf, %lf, %lf\n", /* ... get plate
dimensions */
&plate_x1, splate_x2, aplate_yl, &plate_y2):
fgets(string, 80, fp); /* Read a line and ... */
sscanf(string, "%lf, %lf, %lf, %lf, %lf, %lf, %lf, %lf\n",
/* ... get voltage parameters */
adef_vl, &def_v2, &def_v3, &def_v_t1, &def_v_t2, &def_v_t3,
&def_v_t4);
fgets(string, 80, fp); /* Read a line and ... */
sscanf(string, "%lf, %lf, %lf\n", /* ... get slit parameters
*/
Sslit_x, &slit_yl, sslit_y2);
fgets(string, 80, fp): /* Read a line and ... */
sscanf(string, "%lf, %lf\n", /* ... get window extrema
*/
&window_x, Swindow_y);
draw_freq I atoi(fgets(string, 80, fp));/* Get redraw frequency */

fclose(fp);
/*
** Make sure plate and slit dimensions are ordered properly:
*/
if (plate_x1 > plate_x2)
{
swap I plate_x1;
plate_x1 I plate_x2;
plate_x2 I swap;
}
if (plate_yl > plate_y2)
{

swap I plate_y1:
plate_y1 I plate_y2;
plate_y2 I swap;
)

if (slit_y2 < slit_yl)
{
swap I slit_yl;
slit_y1 I slit_y2;
slit_y2 I swap:
}

/*
** Calculate deflection voltage ramps:
*/
if (def_v_t1 lI de£_v_t2)
rampl I (def_v2 - def_vl) / (def_v_t2 - def_v_tl);
if (def_v_t3 !I def_v_t4)
ramp2 I (def_v3 - def_v2) / (def_v;t4 - def_v_t3);

/*
** OPEN GRAPHICS
*/

gopen(SCREEN);

232

if (argc > 2)

t

if (stricmp(argv[2], "HP") II 0)
pltdev I HP:

else if (stricmp(argv[2], "POST") II 0)
pltdev I POST:

else
pltdev I SCREEN:

if (pltdev !I SCREEN)
if (argc > 3)
{
gcom I fopen(argv[3], "w");
}
gopen(pltdev):
}
draw(n _ions, x, y, color, SCREEN);
if (pltdev lI SCREEN)
draw(n _ions, x, y, color, pltdev);

time I 0. 0; /* Initialize time */
_settextposition(ROW4, COL4): /* Output time ... */
printf("Time I %10. 5e", time): /* ... on screen */
def_volt I def_v1;/* Initialize voltage */
_settextposition(ROW5, COLS); /* Output voltage on

screen */
printf("Deflection Voltage I %+10. 5f", def _volt);

while (lkbhit() ) /* Pause until keystroke
*/
if ((ch I getch()) II 27) /* Clean up char, */
goto EXIT: /* if ESC, quit. */
else if (ch II ' ')/* see if <SPACE> for data */
{
data_table(n_ions, x, vx, y, vy);
_settextposition(ROWl, COLl);
_outtext(quittext):
_settextposition(ROW2, COLZ);
_outtext(datatext);
}
_settextposition(ROW3, COL3);
_outtext(pausetext):
/*
** START ANIMATION
*/

i I 0;
while (! done(n_ions, x, y, vx, vy))
{
time +- delta_t:
if (time >- def_v_t1 AND time < def_v_t2)
def_volt I def_vl + (time I def_v_tl) * rampl;
else if (time >- def_v_t2 AND time < def_v_t3)
def_ volt I def _v2:
else if (time >- def_ v _t3 AND time < def_ v _t4)
def_ volt I def_ _v2 + (time I def_ v _t3) * ramp2;
else if (time >- def_v_t4)
def_volt I def_v3:

/*

**

*/

EXIT:

233

move(n_ions, mass, x, vx, y, vy, time, delta_t):
if (kbhit() ) /* If keystroke... */
{
if ((ch I getch()) II 27) /* Clean up last char, */
goto EXIT; /* if <ESC>, quit, else */
else if (ch II ' ')/* see if <SPACE> for data */
data_table(n_ions, x, vx, y, vy);
else
{ /* see if <P> for print */
if (toupper(ch) II 'P')
draw(n_ions, x, y, color, pltdev);
‘_settextposition(ROW3, COL3):
_outtext(conttext);
while(!kbhit()) /* pause, and wait for */
; /* next keystroke, */
getch(): /* then clean it up */
}

Restore text:

_settextposition(ROW1, COLl);
_outtext(quittext);
_settextposition(ROW2, COL2);
_outtext(datatext);
_settextposition(ROW3, COL3):
_outtext(pausetext);

}

collide(n_ion3, x, vx, y, vy);
‘ /* Stop ions which have hit plates */

_settextposition(ROW4, COL4); /* Update time on screen */
printf("Time I %10.Se", time);

if (1 II draw_freq)
{
‘_settextposition(ROW5, COLS):
printf("Deflection Voltage I %+10.5£", def_volt);
draw(n_ions, x, y, color, SCREEN):
i I 0:
}
else
++i:

DONE, NOW PAUSE TO VIEW SCREEN, THEN EXIT

_settextposition(ROW3, COL3);
_outtext(donetext);

while(!kbhit() ) /* Wait for keystroke */
getch(); /* Clean up char */
._setvideomode(_DEFAULTMODE); /* Return to text screen */

if (pltdev !I SCREEN):
setvideomode(_DEFAULTMODE, pltdev):

}

#include <stdio.h>
#include <math.h>
#include <graph.h>
#include "bdr.h"
#include "dig.h"

void gopen(int pltdev)

{
/*

**

*/

/*

**

*/

/~k

**

*/

/*

**

*/

/*

**

*/

Variable declarations:
struct videoconfig vc:

extern double x_slope, y_slope, x_intcp, y;intcp;

extern double plate_x1, plate_x2, plate_y1, p1ate_y2:
extern double slit_x, slit_y1, slit_y2;

extern double window_x, window_y;

extern char *quittext, *datatext, *pausetext, *conttext:

Function declarations:

int xmap(doub1e, int): Ymap(double, int):

Program:

setvideomode(_ERESCOLOR, pltdev):
if (pltdev !I SCREEN)
setcolor(l, pltdev);

if (pltdev II SCREEN)
{
_settextwindow(1, 1, 3, 80):
_clearscreen(_CCLEARSCREEN);
_settextcolor(7);
}

getvideoconfig(&vc, pltdev);

x_slope I (0.9 * (float) vc.numxpixels);
x_intcp I (0.05 * (float) vc.numxpixels);
y_slope I (I0.8 * (float) vc.numypixels);
y_intcp I (0.95 * (float) vc.numypixels);

Draw box around screen.

moveto(xmap(0., 1), ymap(0., 1), pltdev);
lineto(xmap(l., 1): ymap(0., 1), pltdev);
1ineto(xmap(1., 1), ymap(1., 1), pltdev):
lineto(xmap(0., 1), ymap(1., 1), pltdev);
lineto(xmap(0., 1), ymap(0., l), pltdev);

Output instruction text

if (pltdev II SCREEN)
{

/*
**
**

*‘k

*/

/*

**

*/

/*

**

*/

235

_settextposition(ROW1, COLl);
_outtext(quittext):
_settextposition(ROW2, COLZ);
_outtext(datatext):
_settextposition(ROW3, COL3):
‘_outtext(conttext);

)

Now set up proper transformation to map window onto screen.
(Window goes from 0 to window_x in x-direction and 0 to
window_y in y direction.)

x_slope /I window_x;
y_slope /I window_y:

Draw deflection plates.

moveto(xmap(plate_x1, 1), ymap(plate_y1, l), pltdev);
lineto(xmap(plate_x2, l), ymap(plate_y1, 1), pltdev);
moveto(xmap(plate_x1, 1), ymap(plate_y2, 1), pltdev);
lineto(xmap(plate_x2, l), ymap(plate_y2, l), pltdev);

Draw slit.
moveto(xmap(slit_x, 1), ymap(slit_yl, 1), pltdev);
lineto(xmap(slit_x, 1), ymap(O, 1), pltdev);
moveto(xmap(slit_x, l), ymap(slit_y2, 1), pltdev);
lineto(xmap(slit_x, 1), ymap(window_y, 1), pltdev);

return:

/*
**
**
**
*‘k

*/

xmap.c

Convert x-coordinate in (0, 1) space to physical device units
for the window specified by mode.

extern double x_slope, y_slope, x_intcp, y_intcp:

int xmap(double x, int mode)

{

/*
**
**
**

**

*/

switch (mode)

{

case 1: return((int) (x * x_slope + x_intcp));
break:

default:
printf("Invalid mode (%d) in xmap\n", mode);
exit(0);

ymap.c

Convert y-coordinate in (0, 1) space to physical device units
for the window specified by mode.

extern double x_slope, y_slope, x_intcp, y_intcp:

int ymap(double y, int mode)

{

switch (mode)

{

case 1: return((int) (y * y_slope + y_intcp));
break:

default:
printf("Invalid mode (%d) in ymap\n", mode);
exit(0);

 

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <malloc.h>

/*
**
*t
**
*a:

*/

rk4.c

This is a Runge-Kutta routine based on that in "Numerical

Recipies in C", p. 572.

void rk4(double y[J, int n, double x, double h,
void (*derivs)(double, double *, double *))

{
/*

**

*/

/*

**

*/

Variable declarations:

int i:
double xh, hh, h6, *dyO, *dym, *dyt, *yt:

Function declarations:

double *vector(int, int):
void free_vector(double *, int, int);

dyO I vector(0, n I 1);
dym I vector(0, n I 1);
dyt I vector(0, n I 1);
yt I vector(0, n I 1);
hh I h * 0.5:
h6 I h / 6.0:
xh I x + hh;
(*derivs) (x, y, dyO):
for (i I O; i < n; ++i)
ytlil ' Ylil + hh * dy0[i];
(*derivs)(xh, yt, dyt):
for (i I 0; i < n; ++i)
ytti] - Yli] + hh * dytli]:
(*derivs)(xh, yt, dym):
for (i I 0; i < n; ++i)
{
ytli] - y[i] + h * dymti];
dymli] +- dytli]:
}
(*derivs)(x+h, yt, dyt):
for (i I 0; i < n; ++i)

y[i] +- 116 * (dyOliJ + dyt[i] + 2.0 * dym[i]);

free_vector(yt, 0, n I 1):
free_vector(dyt, 0, n I 1):
free_vector(dym, 0, n I l):
free_vector(dy0, 0, n I 1):
return;

 

 

/*
** draw.c
**
** Draw ions.
*1):
*/
void draw(int n_ions, double x[], double y[], int color[], int pltdev)
{
/*
** Variable declarations:
*/
int i:
int x1, y1, x2, y2;
static double xold[MAXIONSl, YoldtMAXIONS]:
static int first I TRUE:
/*
** Function declarations:
*/
int xmap(double, int), ymap(double, int);
/*
** Erase old ions.
*/
if (! first AND pltdev II SCREEN)
{
for (i I 0: i < n_ions: ++i)
{
if (xoldli) II x[i] AND yold[i] II y[iJ)
continue:
x1 I xmap(xold[i], 1) I 2;
yl I ymap(yold[i], 1) - 2:
x2 I x1 + 4;
y2 I yl + 4;
_setcolor(0):
_ellipse(_CFILLINTERIOR, x1, y1, x2, y2);
_setcolor(color[i]);
_ellipse(_CBORDER, x1, y1, x2, y2);
}
}
else if (pltdev II SCREEN)
{
for (i I 0; i < n_ions; ++i)
I
xoldIi] I x[i]:
YOldli] - y[i];
}
}
/*
** Draw ions in thein new locations.
*/

for (i I 0; i < n_ions: ++i)
{
if ((l first) AND (pltdev II SCREEN) AND
(xold[il II x[i] AND yold[i] II y[i]))
continue:
setcolor(color[i], pltdev);

239

x1 I xmap(x[i], 1) I 2,
yl I ymap(yli], 1) - 2:
x2 I x1 + 4:
y2 I yl + 4:

ellipse(_CFILLINTERIOR, x1, yl, x2, y2, pltdev):
}

if (first)
first I FALSE:

for (i I 0: i < n_ions: ++i)
(
xold[i] I XIi]:
YOldli] - y[iJ;
)

if (pltdev !I SCREEN) .
grestore(): -
}

 

#include <stdio.h>
#include <math.h>
#include "bdr.h"

/*
** move.c
**
** Move ions one step.
*/
void move(int n_ions, double mass[], double x[], double vx[], double
Y”!
double vy[], double time, double delta_t)
{
/*
** Variable declarations:
*/
double xxISJ:
int i:
/*
** Function declarations:
*/
void rk4(double *, int, double, double, void (*)()):
void derivs(double, double *, double *):
/*
** Move ions:
*/

for (i I 0; i < n_ions; ++i)
{ .

um] I Hi]:

xx[1] I vx[i];

xxl2] I Hi]:

xx[3] I vyli]:

xx[4] I mass[i]:

rk4(xx, 5, time, delta_t, derivs):

x[i] I xx{0]:

vx[i] I xxlll:

y[i] I xxl2]:

vy[i] I xxl3]:

)

 

/*

*‘k

*/

241

Here is the derivative function:

void derivs(double t, double x[], double dx[l)

{

/*
*i'
**

*/

/*

*‘k

*/

extern double plate_x1, plate_x2, plate_y1, plate_y2:
extern double def_volt:

First, do x-direction:
(Just move at constant velocity.)

dx[O] I x[l]: /* dx/dt I V */
dx[l] 0.0: /* dv/dt I'a I 0.0 */

Now y-direction:

region: */

if (x[0] < plate_x1 OR x[0] > plate_x2 OR /* Field-free: */
x[2] < plate_y1 OR x[2] > plate_y2)
I
dx[2] I x[3]; /* dy/dt I v */
dx[3] I 0.0: /* dv/dt I a I 0.0 */
}
else /* In deflection
{
dx[2] I x[3]; /* dy/dt I v */

dx[3] I COUELE * def_volt /
(KGAMU * x[4] * fabs(plate_y2 I p1ate_y1));
}
dx[4] I 0.0; /* dx[4] is not used */

#include <stdio.h>
#include "bdr.h"

/*
**
**

**

*/

done.c

Return TRUE if all velocities are zero.

int done(int n_ions, double x[], double y[], double vx[], double vy[])

{

int i:
extern double window_x, window_y:
for (i I 0: i < n_ions: ++i)
if (vx[i] > 0.0 OR vy[i] > 0.0)
return(FALSE):

return(TRUE);

 

#include <stdio.h>
#include "bdr.h"

/*

** collide.c

**

** Stop ions which have hit a deflection plate or slit, or leave
window.

**

** Note: this code may not function properly if slit_x II 0.

*/

void collide(int n_ions, double x[], double vx[], double y[], double
VY[])
{

int i:

static double xold[MAXIONS];

extern double plate_x1, plate_x2, plate_y1, plate_y2:

extern double slit_x, slit_y1, slit_y2:
extern double window_x, window_y;

/*
** Check for those that have hit a deflection plate:
*/
for (i I 0: i < n_ions; ++i)
{ .
if ((xti] >I plate_xl AND x[i] <I plate_x2) AND
(y[i] <I plate_yl OR y[i] >I plate_y2))
{
vx[i] I 0.0:
VYli] I 0.0:
if (y[i] <I plate_y1)
y[i] I 0.999 * p1ate_y1:
else if (y[i] >I plate_y2)
y[i] I 1.001 * plate_y2:
}
}
/*
** Check for those that don't pass throught slit
*/
for (i I 0; i < n_ions; ++i)
{
if ((xoldli] <I slit_x AND x[i] > slit_x) AND
(y[i] < slit_y1 OR y[i] > slit_y2))
{
vx[i] I 0.0:
vy[i] I 0.0:
}
xoldli] I x[i]:
}
/* '
** Check for those that have left window:
*/

for (i I 0; i < n_ions: ++i)
{

 

 

243

if (x[i] < 0.0 OR x[i] > window_x OR

y[i] < 0.0 OR y[i] > window_y)
{

vx[i] I 0.0:
vy[i] I 0.0:
}

 

#include <stdio.h>
#include <conio.h>
#include <graph.h>
#include "bdr.h"

/*
** data_table.c

**

** Output position and velocity for moving ions.

:: Also output packet width ([xmax-xmin] of moving ions).

void data_table(int n_ions, double x[], double vx[], double y[], double

VYIJ)
{
double min, max, width:
int i_min, i_max;
int line, last, 1:
char string[80];

/*
** POSITIONS AND VELOCITIES:
*/
line I 1;
last I 0:
‘_settextposition(l, 1):
_outtext("\n\n\n\n\n"); /* Clear text window */
for (i I 0; i < n_ions; ++i)
{
if (vx[i] !- 0.0 OR vy[i] !I 0.0)
{
sprintf(string,
"\nIon %2d: x I %10.5f, vx I %10.4f, y I %10.5f, vy I
%10.4f",

i+1l XIi], vx[i]: Y[i]r VYULJ);
_settextposition(3, 1);
_outtext(string);

last I line:

if (line < 3)

++line:

else
{
while (! kbhit()) /* Pause to view */
getch(); /* Clean up char */
line I 1:

l
}

if (last !I 3)
{
for (i I l; i <- (3 - last); ++i)
_outtext("\n"); /* Blank lines */
while (! kbhit()) /* Pause to view */

getch(); /* Clean up char */
}

 

/~k
** PACKET WIDTH:
*/

_outtext("\n\n\n\n\n"): /* Clear text window */

min I 1.e32;
max I I1.e32:
for (i I 0; i < n_ions; ++i)
I
if (vx[i] II 0.0 OR vy[i] II 0.0)
I
if (x[i] < min)
I
min I x[i]:
i_min I i:
}
if (x[i] > max)
I
max I Klil:
i_max I i:
l
}
}
width I max I mdn:
sprintf(string, "Packet Width I %10.5f m", width):
_settextposition(1, 1):
_outtext(string);
sprintf(string, " II> %15.8e sec",
width * 2.0 / va[i_max] + vx[i_min]));
_settextposition(2, 1):
_outtext(string);

while (! kbhit()) /* Pause to view */

getch(); /* Clean up char */
_settextposition(1, 1):

}

‘_outtext("\n\n\n\n\n"); /* Clear text window */

246

/*
** Here are the utility routines:
*/

void nerror(char error_text[])

I
void exitI):

fprintf(stderr, "Numerical Recipies run-time error...\n");
fprintf(stderr, "%s\n", error_text);

exit(l);

}

double *vector(int nl, int nh)
I

double *v:

v I (double *) malloc((unsigned) (nh I n1 + 1) * sizeof(double));
if (!v)
nerror("allocation failure in vectorI)"):
return(v I nl):
}

void free_vector(double *v, int nl, int nh)

I
free((char *) (v + nl)):

}

 

247

#include <graph.h>
#include <stdioih>
#include <math.h>
#include "dig.h"

#define PI 3.1415926535

FILE *gcom-NULL;

int hp_x,hp_y,hp_highI0,hp_wideI0,hp_pen-0;
float post_scale:

int post_wide,post_high,post_x,post_y:

#define SCREEN 0
#define HP 1
#define POST 2

int
setvideomode(mode,dev)
int mode;
int dev:
I
switch (dev) I
case SCREEN:
return (_setvideomode(mode));
case HP:
switch (mode) {
case _DEFAULTMODE:
if (gcom) {
fprintf(gcom,"SP 0;\n"):
fclose(gcom):
}
gcomINULL:
return (_DEFAULTMODE);
case‘_TEXTBW40:
case _TEXTC40:
hp_wideI40:
hp_high-25:
break:
case _TEXTBWBO:
case _TEXTCBO:
case _TEXTMONO:
hp_wideI80:
hp_highI25:
break:
case _MRESACOLOR:
case _MRESNOCOLOR:
case';MRE8256COLOR:
hp_wideI320:
hp_highI200:
break:
case _RRESBW:
case _RRESIGCOLOR:
hp_wideI640:
hp_high-200:
break:
case _ERESNOCOLOR:
case _ERESCOLOR:
hp_wideI640:

I .

hp_high-350:
break;

case _VRESZCOLOR:

case _VRESlGCOLOR:
hp_wideI640:
hp_high=480:
break;

}

if (gcomIINULL) gcomIfopen("com1","w");
if (gcomIINULL) return (0);
fprintf(gcom,"IN; SC 0,%d,0,%d;\n“,hp_wideIl,hp_high-l);
return (mode);
case POST:
switch (mode) {
case _DEFAULTMODE:
if (gcom) {
fprintf(gcom,"showpage\ngrestore\r\n");
fclose(gcom):
}
gcomINULL;
return (_DEFAULTMODE);
case _TEXTBW40:
case _TEXTCdO:
post_wideI40:
post_high-25:
break;
case _TEXTBW80:
case _TEXTC80:
case _TEXTMONO:
post_wideI80:
post_highI25:
break:
case _MRESdCOLOR:
case _MRESNOCOLOR:
case _MRESZSGCOLOR:
post_wideI320:
post_high-200:
break:
case _RRESBW:
case _RRESlGCOLOR:
post_wideI640:
post_high-200:
break:
case _ERESNOCOLOR:
case _ERESCOLOR:
post_wideI640:
post_high-350:
break:
case _VRESZCOLOR:
case _VRESlGCOLOR:
post_wideI640:
post_high-480:
break;
}

if (gcomIINULL) gcomIfopen("com1","w");
if (gcomIINULL) return (0);

249

fprintf(gcom,"gsave erasepage %f %f scale 90 rotate\n",
(float)8*72./post_high,(float)10*72./post_wide);

fprintf(gcom,"%d %d translate\n",post_wide/25,I
(post_high*26/25)):

fprintf(gcom,"/Times-Roman findfont %d scalefont
setfont\n",post_high*4/5/25);

return (mode);

}

return (0):

}

struct videoconfig *
getvideoconfig(config,dev)
struct videoconfig *config:
int dev;
I
switch (dev) {
case SCREEN:
return ((struct videoconfig *)_getvideoconfig(config));
case HP: 5
config->numxpixels=hp_wide: ‘
configI>numypixelsIhp_high:
configI>numtextcolsI25:
configI>numtextrowsI80;
configI>numcolorsI7;
configI>bitsperpixelI3;
configI>numvideopagesI1:
return (config);
case POST:
config->numxpixelstost_wide:
configI>numypixelstost_high:
configI>numtextcolsI25:
configI>numtextrowsI80:
configI>numcolors-2:
configI>bitsperpixelI1:
configI>numvideopagesI1:
return (config);

I

}
return (NULL):

}

int
setcolor(color,dev)
int color:
int dev:
I
int last:
switch (dev) I
case SCREEN:
return (_setcolor(color));
case HP:
if (gcom-INULL) return (0);
fprintf(gcomr”SP %d;\n",color);
last-hp_pen:
hprenIcolor:
return (last);

return (0):

}

int
ellipse(control,x1,y1,x2,y2,dev)
int control,xl,y1,x2,y2;
int dev;
I

int cx,cy,x,y,scale;

switch (dev) I
case SCREEN:
return (_ellipse(control,x1,y1,x2,y2));
case HP:
if (gcomIINULL) return (0);
cx=(x1+x2)/2;
cyI(y1+y2)/2:
fprintf(gcom,"PU %d,%d; ",cx,hp_high-1Icy);
scale-hp_high/(y2-cy)*(x2-cx);
fprintf(gcom,"SC O,%d,0,%d; ",hp_wide-l,scale);
fprintf(gcom,"CI %d; ",x2-cx);
fprintf(gcom,"SC 0,%d,0,%d;\n",hp_wide-1,hp_high-1);
return (1);
case POST:
if (gcomIINULL) return (0);
cxI(x1+x2)/2:
cy-(yl+y2)/2:
/* don't ask me why I couldn't cx-xl cnyl scale, it didn't
work */
fprintf(gcom,"gsave %d %d translate 1 %f scale newpath 0 0
%d 0 360 arc closepath ",
cx,post_high-cy-1,(float)(cy-yl)/(cx-x1),cx-x1);
switch (control) {
case _GBORDER:
fprintf(gcom,"stroke grestore\n"):
return (1);
case _GFILLINTERIOR:
fprintf(gcom,"fill grestore\n");
return (1):
default:
fprintf(gcom,"grestore\n");
return (0):
}
fprintf(gcom,"grestore\n”):
}
return (0):

}

void
clearscreen(area,dev)
int area:
int dev;
{
switch (dev) {
case SCREEN:
_clearscreen(area);
return;
case POST:

251

if (gcomIINULL) break;
fprintf(gcom,"erasepage\n");
return:

}

int
rectangle(control,x1,y1,x2,y2,dev)
int control,xl,yl,x2,y2:
int dev;
I
switch (dev) (
case SCREEN:
return (_rectangle(control,x1,y1,x2,y2));
case HP:
if (gcom-INULL) return (0);
fprintf(gcom,"PU %d,%d; EA %d,%d;\n",x1,hp_high-y1-
1,x2,hp_high-y2-1);
hp_xIx1:
th'yl:
return (1):
case POST:
if (gcomIINULL) return (0);
fprintf(gcom,"newpath %d %d moveto %d %d lineto %d %d lineto
%d %d lineto %d %d lineto closepath stroke\n",
x1,post_high-y1-1,x2,post_high-y1-1,
x2,post_high-y2-l,x1,post_high-y2-1,
x1,post_high-yl-l);
post_XIx1;
post_y=y1:
return (1):
}
return (0);

}

int
lineto(x,y,dev)
int x,y;
{
switch (dev) {
case SCREEN:
return (_lineto(x,y));
case HP:
if (gcomIINULL) return (0);
fprintf(gcom,"PD %d,%d;\n",x,hp_high-1Iy);
return (1):
case POST:
if (gcom-INULL) return (0);
fprintf(gcom,"newpath %d %d.moveto %d %d lineto stroke\n",
post_x,post_high-post_y-1,x,post_high-y-1);
post_xIx:
post_ny:
return (1):
}
return (0):
}

struct xycoord

 

'5.

252

moveto (x,y, dev)
int x,y:
int dev;
I
struct xycoord xy:

switch (dev) {
case SCREEN:
return (_moveto(x,y));
case HP:
xy.xcoord=hp_x:
xy.ycoord=hp_y:
if (gcom-INULL) break;
fprintf(gcom,”PU %d,%d;\n",x,hp_highI1-y);
hp_xIx:
th-y:
return (xy);
case POST:
xy.xcoordeost_x:
xy.ycoordeost_y:
post_x-x;
P°3t_Y'Y:
return (xy);
}
return (xy);
}

int
setpixel(x,y,dev)
int x,y;
int dev:
I
switch (dev) I
case SCREEN:
return (_setpixel(x,y));
case HP:
if (gcomIINULL) return (0);
fprintf(gcom,"PU %d,%d; PD; PU;\n",x,hp_high-1Iy);
return (1):
1
return (0):
}
void

outtext(text,dev)
char *text:
int dev;
I
int i:
switch (dev) {
case SCREEN:
_outtext(text);
return;
case HP:
if (gcomIINULL) return:
fprintf(gcom,'PU %d,%d; LB %s\003\n",hp_x,hp_high-hp_y-
1,text):
return;

253

case POST:
if (gcomIINULL) return;
fprintf(gcom,"%d %d moveto (",post_x,post_high-post_y+2);
for (iI0;text[i]!I0;i++) {
switch (text[i]) {
case '\n': fprintf(gcom,"\\n"); break;
case '\t': fprintf(gcom,"\\t"); break;
case '\r': fprintf(gcom,"\\r"); break;
case '\b': fprintf(gcom,”\\b"); break;
case '\f': fprintf(gcom,"\\f"); break;
case '(': fprintf(gcom,"\("); break;
case ')': fprintf(gcom,"\)”): break:
default: if (text[i])31 && text[i]<127)
fputc(text[i],gcom); break;
}
)
fprintf(gcom,") show\n");
return;
}

return:

 

}

struct rccoord
settextposition(row,column,dev)

I
struct rccoord rc:

switch (dev) {
case SCREEN:
return (_settextposition(row,column));
case RP:
if (gcomIINULL) break;
rc.col-hp_x*80/hp_wide+1:
rc.row-25Ihp_y*25/hp_high:
hp_xI(columnI1)*hp_wide/80:
hp;yIrow*hp_high/25:
return (rc):
case POST:
if (gcomIINULL) break;
rc.col-post_x*80/post_wide+1:
rc.rowI25-post_y*25/post_high;
post_xI(columnI1)*post_wide/80;
post_yIrow*post_high/25:
return Irc):
l
return (rc);

}

int
settextcolor(pixel,dev)
int pixel;
int dev:
I
int pen;
switch (dev) (
case SCREEN:
return (_settextcolor(pixel));
case HP:

254

if (gcomIINULL) return (0);
fprintf(gcom,"SP %d:\n",pixel):
hp_penIpixel;

return (pen);

 

}

return (0):

}

extern FILE *gcom;

int
arc(x1,y1,x2,y2,x3,y3,x4,y4,dev)
int x1,y1,x2,y2,x3,y3,x4,y4;
int dev:
I
int cx,cy,x,y,dx,dy;
float start,sweep,r,scale:
double atan(),sqrt()psin(),cos();

switch (dev) {
case SCREEN:
return (_arc(x1,y1,x2,y2,x3,y3,x4,y4));
case HP:
if (gcomIINULL) return (0);
cx-(x1+x2)/2:
cyI(y1+y2)/2:

 

scale-(float)(x2-cx)/(y2-cy);

start-atan( (double)Iy3-cy)*scale/(x3-cx) )*180/PI;
if (x3>cx) start+I180:

sweep-atan( (double)(y4-cy)*scale/(x4-cx) )*180/PI;
if (x4>cx) sweep+-180:

sweep-startIsweep:

while (sweep<0) sweep+I360:

while (sweep>-360) sweep-I360;

rIx2ch:

xch+(x3ch)*r/sqrt( (double) (x3-cx)*(x3-cx)+(y3-cy)*(y3-
CY) ):

yIcy+(y3-cy)*r/sqrt( (double) (x3-cx)*(x3-cx)+(y3-cy)*(y3-
CY) ):

fprintf(gcom,"PU %d,%d; PD;",x,hp_high-1Iy);

fprintf(gcom,”SC 0,%d,0,%3.0f; ",hp_wide-1,scale*(hp_high-
1)):

fprintf(gcom,”AA %d,%3.0f,%3.0f; ",cx,(hp_high-cy-
1)*scale,sweep);

fprintf(gcom,"SC 0,%d,0,%d;\n",hp_wideI1,hp_highI1);

return (1):

case POST:
if (gcomIINULL) return (0);
cx-(x1+x2)/2;
cyI(y1+y2)/2i
scale-(float)(x2ch)/(Y2-cy);
start=atan( (double)(cy-y3)*scale/(x3-cx) )*180/PI;
if (x3<cx) start+I18O:

$5

while (start<0) start+I360:

while (start>I360) start-I360;

sweep-atan( (double)(cy-y4)*scale/(x4-cx) )*180/PI:

if (x4<cx) sweep+-180:

while (sweep<0) sweep+I360;

while (sweep>-360) sweep-I360;

fprintf(gcom,"gsave %d %d translate 1 %f scale newpath 0 0

%d %f %f arc stroke grestore\n",

cx,post_high-cy-1,(float)(cy-yl)/(cx-x1),cx-x1,
start,sweep):

return (1);

 

}

return (0):

 

 

fiinclude
#include
#include
#include
#include

/*
**
**

*‘k

*/

void gre
I
/*
**
*/
St

<stdio.h>
<math.h>
<graph.h>
"bdr.h"
"dig.h"
grestore.c

Restore global scale factors to those of the screen.

store(void)

Variable declarations:

ruct videoconfig vc:

extern double x_slope, y_slope, x_intcp, y_intcp:
extern double window_x, window_y:

/*
*‘k
*/

in
/*

**

*/

Function declarations:

t xmap(double, int), ymap(double, int):

Program:

_getvideoconfig(&vc):

x_slope I (0.9 * (float) vc.numxpixels);
x_intcp I (0.05 * (float) vc.numxpixels);
y_slope I (IO.8 * (float) vc.numypixels);
y_intcp I (0.95 * (float) vc.numypixels);

x_slope /I window_x;
y;slope /I window_y:

return:

 

k...

257
817101"

This program, which is similar to the SIMTOF programs presented
in this appendix, generates peak shapes for a beam deflection TOFMS
instrument.

main
The main program obtains the required input from the user and
controls the iterations through the space and energy distributions, etc..

rk4
This routine, based on Press, Flannery, Teukolsky, and Vetterling,

Numerical Recipes, p. 553, performs fourth-order Runge-Kutta integration.

derivs

This routine calculates derivatives (dV/dt values) for rk4 .

test

This routine checks to see whether the given ion (specified by initial
location within the beam, velocity, and mass, etc.) will pass through the
aperture and thus be part of the ion packet.

tab
In this routine, flight times are tabulated in a histogram. The flight

times are mapped into bin numbers (array positions, that is); then the
corresponding array value is incremented.

gasdev
This routine, based on Press, Flannery, Teukolsky, and Vetterling,

Numerical Recipes, p. 203, generates random numbers drawn from a
normal distribution with zero mean and unit variance.

ran

This routine returns a random number between 0.0 and 1.0. The
built-in library routine rand supplied by the VAX-11 C compiler generates
integer random numbers from the range 0 to (231 - 1). These integers are
converted to floating point and divided by (231 — 1) to yield the desired
uniform deviates.

 

"/

 

258

Misc. utility functions
These are functions from the C version of Numerical Recipies which
allocate array space, etc.

 

#include <stdio.h>
#include <math.h>

/*

** bdtof.c

**

** Simulate Beam-Deflection Time-of-Flight Mass Spectrometer
** peak shapes.

*1:

*/

#define TRUE I1
#define FALSE 0
#define AND as

 

#define OR ||

#define PI 3.14159265359

#define RperD (2.0 * PI / 360.0)
#define KGAMU 1.6606EI27
#define COUELE 1.60219EI19

 

#define SPECSIZE 64

/*
** External declarations:
*/
double plate_xl, plate_x2, plate_y1, plate_y2:
double plate_l:
double slit_x, slit_y1, slit_y2;
double def_volt;
double mass:
double def_vl, def_v2, del_v;
double def_v_t1, rise_time, rise_end;
double delta_t;
double xx[5];
FILE *fp;

main()
I
/*
** Variable declarations:
*/
double xmin, xmax, ymin, ymax, thetamax, uO, u_stdev:
double x, y, theta, u;
double ycen, yp, ktheta:
double dev:
float tstep, ustep:
float tmin, tmax, leo, mLhi, time, step:
float generic_l;
int spect[SPECSIZE];
int nx, ny, nt, nu:
int 1:
char string[80];

 

/*
** Function declarations:
*/

int derivsI):

/*

**

*/

/*

*‘k

*/

double gasdev();
double test();

Input parameters:

printf("Enter plate_x1, plate_x2: ");
scanf("%lf, %lf", aplate_xl, &plate_x2);
printf("Enter plate_y1, plate_y2: ")i
scanf("%lf, %lf", aplate_yl, &plate_y2):
plate_l = fabs(plate_y2 - plate_y1);
printf("Enter slit_x, slit_y1, slit_y2: ")3
scanf("%lf, %lf, %lf", &slit _x, aslit_y1, sslit_y2);
printf("Enter def _vl, def_ v2: "):
scanf("%lf, %lf", &def_v1, &def_y2):
del_ v = def_ v2 - def_ v1;
printf("Enter def_ v_t1: ");
scanf(" %lf", &def_ v_t1):
printf("Enter rise time: ");
scanf("%lf", &rise_time);
rise_end - rise_time + def_y_t1;
printf("Enter mass: ");
scanf("%lf", &mass);
printf("Enter delta_t: ");
scanf("%lf", 8delta_t);
printf("Enter grid xmin, xmax: ")3
scanf("%lf, %lf", £xmin, &xmax);
printf("Enter nx: ");
scanf("%d", anx):
printf("Enter grid ymin, ymax: "):
scanf("%lf, %lf", 8mmin, symax);
printf("Enter ny: ");
scanf("%d", &ny);
printf("Enter maximum angle: ");
scanf("%lf", &thetamax);
printf("Enter 00 and standard deviation (eV): ");
scanf("%lf, %lf", &u0, &u_stdev);
if (n_stdev !- 0.0)

{

printf("Enter nu: ");

scanf("%d", anu);

}

printf("Enter m _lo and m.hi to calculate\n");
printf("bounds of plot: ");

scanf("%f, %f" , 8m _lo, 8m.hi);

generic_ l - slit_ x - (xmin + (xmax - xmin) / 2. );
tmin - generic_ l / sqrt((2. *uO*COUELE)/(m.lo*KGAMU));
tmax - generic_l / sqrt((2. *u0*COUELE)/(m:hi*KGAMU));

gets(string): /* Clean up input buffer */

Initialize spectrum and random number generator

for (i - 0; i < SPECSIZE; ++i)
spect[i] - 0;
srand(1);

 

 

 

/*

**

*/

/*

*‘k

*/

%1

Calculate spectrum

ycen - (ymax + ymin) / 2.0;
ktheta - thetamax / (ymax - ycen);

for (x - xmin; x <- xmax; x +. (xmax - xmin)/(nx-l))
{
for (yp - ymin; yp <- ymax; yp +- (ymax - ymin)/(ny-l))
{
theta - ktheta * (yp - ycen);
y - yp + (x - xmin) * tan(RperD * theta);
if (u_stdev -- 0.0 OR nu <- l)
{
u = uO;
if (test(x, y, theta, u) !- 0.0)
{ .
tab(x, u, theta, tmin, tmax, spect, SPECSIZE);
}
}
else
{
for (i - 1; i <- nu; ++i)
{
dev - gasdev();
u - n_stdev * dev + u0;
if (test(x, y, theta, u) !- 0.0)
{
tab(x, u, theta, tmin, tmax, spect, SPECSIZE);

}

} /* u loop */
}
l /* y loop */
} /* x loop */

Output spectrum and exit:

printf("Enter output filename: ");

fp = fopen(gets(string), "w");

step - (tmax - tmin) / (float) (SPECSIZE - 1);

for (i - O; i < SPECSIZE; ++i)
{
time - (float) i * step + tmin;
fprintf(fp, "%e, %d.\n", time, spect[i]);
}

fclose(fp);

}

 

/*
** rk4.c

**

** This is a Runge-Kutta routine based on that in "Numerical
** Recipies in C", p. 572.

*/

rk4(y, n, x, h, derivs)
double y[], x, h;
double (*derivs)();

 

int n;
{
/*
** Variable declarations:
*/
int i;
double xh, hh, h6, *dyO, *dym, *dyt, *yt;
/*
** Function declarations:
*/

double *vector():
int free_yector();

dyO - vector(0, n - 1);
dym - vector(0, n - 1);
dyt - vector(0, n - 1):
yt - vector(0, n - 1);
hh = h * 0.5:
h6 - h / 6.0;
xh = x + hh:
(*derivs) (x, y, dyO);
for (i = o; i < n; ++i)
yt[i] - y[i] + hh * dyOIi]:
(*derivs)(xh, yt, dyt);
for (i - 0; i < n; ++i)
ytm - y[i] + hh * dytm;
(*derivs)(xh, yt, dym);
for (i - 0; i < n; ++i)

( .

ytli] ' Yli] + h * deIiJ:

dymti] +- dytli];

}

(*derivs) (x-I-h, yt, dyt);
for (i - 0; i < n; ++i)

y[i] +- he * (dy0[i] + dyt[i] + 2.0 * dymtil);
free_yector(yt, 0, n - l);
free_vector(dyt, 0, n - l):

.free_yector(dym, 0, n - l);
free_yector(dy0, 0, n - 1):
return:

/*
** derivs()

*/

 

derivs(t, x, dx)
double t, x[], dxl];
{
extern double plate_xl, plate_x2, plate_yl, plate_y2;
extern double plate_l:
extern double def_yolt:

dx[O] = XII];
dx[l] = 0.0;

if (x[0] < plate_xl || x[0] > plate_x2 ll
x[2] < plate_y1 ll x[2] > plate_y2)

{

dXIZJ - 8(3):
dx[3] - 0.0:
}

{
dx[2] ' x[3];
dx[3] - COUELE * def_volt / (KGAMU * x[4] * plate_l);
}
dx[4] - 0.0;
}

 

else

double test(x, y, theta, u)
double x, y, theta, u:

/*
*‘k
0.0.
*/

/*

**

*/

/*

**

*/

/*

**

*/

See if ion passes through slit. Return 1.0 if so, else return

{
double vel;
double time;

extern double plate_xl, plate_32, plate_yl, plate_y2;
extern double plate_l;

extern double slit_x, slit_yl, slit_y2;

extern double def_volt:

extern double mass:

extern double def_vl, def_v2, del_v;

extern double def_v_tl, rise_time, rise_end:

extern double delta_t;

extern double xx[5];

vel - sqrt(2. * u * COUELE / (mass * KGAMU));
time - 0.0;

def_volt - def_vl;

xxlO] - x;

xx[1] - vel * cos(RperD * theta);
xx[2] - y;

xx[3] - vel * sin(RperD * theta);
xx[4] - mesa;

First, move to the beginning of the plate region:

time - (plate_xl - xx[0]) / xx[1]: /* t - dx / (dx/dt) */
xx[0] - plate_x1; /* x - edge of plates */
xx[2] +- time * xx[3]; /* y +' t * (dy/dt) */

Now, do Runge-Kutte integration through the plate region:

do {
time +- delta_t:
if (time >- def_v_t1)
{
if (time < rise_end)
def_volt - def_vl +
del_v * (time-def_v_t1) / rise_time;
else
def_volt - def_v2;
}
rk4(xx, 5, time, delta_t, derivs):
if ((xxlO] >- plate_xl && xxlO] <- plate_32) &&
(xx[2] <- plate_yl ll xx[2] >- plate_y2))
return(0.0);
} while (xxlOJ < plate_x2);

Finally, move to the detector:

 

 

*/

265
time = (slit_x - plate_32) / xx[l]; /* time in FF region - l / vx

xx[2] += (time * xx[3]); /* move t*vy in y direction */
if (xx[2] > slit_yl as xx[2] < slit_y2)

return(1.0):
else

return(0.0);

}

 

 

 

/*
** should be void tab()
*/
int tab(x, u, theta, tmin, tmax, spect, size)
double x, u, theta;
float tmin, tmax;
int size, spect[J;
{

float vel, time, range;
int index;

extern double plate_x2, slit_x, mass;

vel = sqrt(2. * u * COUELE / (mass * KGAMU));
vel *- cos(RperD * theta);
time - (slit_x - x) / vel;

range = tmax - tmdn;
index = (int) ((time - tmin) * (float) size / range) + 1;
if (index >- 0 AND index < size)
spect[index] +- 1;
}

double gasdev()

/*
** Return gaussian deviate

**

** See Numerical Recipies, p. 203.
*/

{
double v1, v2, r, fac;

static double gset;
static int iset - 0;

double ran();

if (iset -- 0)
{
do
{
v1 = 2. * ran() - 1.,
v2 - 2. * ran() - 1.;
r - v1*vl + v2*v2;
}
while (r >- 1.);
fac - sqrt(-2. * log(r)/r);
gset - v1 * fac;
iset - 1;
return(v2 * fac);
}
else
{
iset - 0;
return(gset);
}

 

 

%7

double ran()

/*
** Return random number between 0.0 & l.Q

**

** Library routine rand() return an int between 0 and 2‘31 - 1.

** We divide this by 2‘31 to return a double >= 0. and < 1.
*/

{

double temp;

int rand();

temp -(double) rand() / 2147483647.;
return(temp);

}

/*
** Here are the utility routines:

*/

nerror(error_text)
char error_text[];
{
fprintf(stderr, "Numerical Recipies run-time error...\n");
fprintf(stderr, "%s\n", error_text);
exit(l);
}

double *vector(nl, nh)
int nl, nh;

{

double *v;

v - (double *) malloc((unsigned) (nh - nl + 1) * sizeof(double));
if (!v)

nerror("allocation failure in vector()”);
return(v - n1);

}

free_vector(v, nl, nh)

double *v;

int nl, nh;
{
free((char *) (v + nl));
}

 

$8
Simulation Program for Transfer Matrix Calculations

This is a listing of the program SIMTOF which simulates peak
shapes using Damaschin Ioanoviciu’s transfer matrix for two-stage homo-
geneous field mirror TOFMS instruments.

 

PROGRAM SIMTOF ......................................................................... 29
simtof ..................................................................................... 271
gasdev .................................................................................... 276
ran ........................................................................................ 2'77
tab ......................................................................................... 277
ftime ...................................................................................... 278

 

 

SIMTOF

This program is similar to that of the same name used in the DFF
project. As in that SIMTOF program, the user supplies values for the ion
mass, the mean and standard deviation of the initial velocity distribution,
the mean and standard deviation of the initial spatial distribution, and step
sizes related to the number of desired iterations through the two
distributions. In this version, the user also supplies a value for the exit slit
width and the standard deviations of the a and ,8 angle distributions. In
addition, the user supplies the range of the desired histogram (usually in
the form of mass values from which to calculate low and high time limits)
and the desired name for the output file.

Description of Routines:

simtof

The main program obtains the required input from the user and
controls the iterations through the space, energy, angle, and position
distributions.

gasdev
This routine, based on Press, Flannery, Teukolsky, and Vetterling,

Numerical Recipes, p. 203, generates random numbers drawn from a
normal distribution with zero mean and unit variance.

ran

This routine returns a random number between 0.0 and 1.0. The
built-in library routine rand supplied by the VAX-11 C compiler generates
integer random numbers from the range 0 to (231 - 1). These integers are
converted to floating point and divided by (231 - 1) to yield the desired
uniform deviates.

tab
In this routine, flight times are tabulated in a histogram. The flight

times are mapped into bin numbers (array positions, that is); then the
corresponding array value is incremented.

ftime

270

This routine calculates the flight time for an ion having specified
mass, charge, slit-exit position, angle 0:, angle ,8, initial position, and initial
velocity. The velocity information is actually specified as a combination of
kinetic energy (a scalar quantity, unlike velocity) and a direction. The
integer variable direct, one of the arguments sent to this routine, is set by
the calling program to be either +1 or —1 to indicate motion either towards
or away from the detector, respectively.

f t
The actual matrix multiplication is done by this routine.

 

/*
**
*‘k
**
**
**
**

*/

#include <stdio.h>

simtof.c

Monte-Carlo

271

simulation of time-of-flight peak shapes using

Ioanoviciu's transfer matrix.

#include <math.h>

#define SIZE
#define NCDF

*/

#define KBOLT
#define COUELE
#define KGAMU

George Yefchak

128 /* Spectrum array size */
128 /* Cumulative distribution function array size

1.38007e-23 /* Boltzmann constant */
1.60219e—19 /* electron charge */
1.66057e-27 /* kg / amu */

main()
{

double tmin, tmax; /* Spectrum extrema */

double vdcf[NCDF], scdf[NCDF]; /* Cum. distrib. function arrays
*/

double vmin, vmax, smin, smax; /* Distrib. function extrema */

double uO, so, vsigma, ssigma; /* Means and standard deviations
*/

double v0, v, e, s, time; /* Velocity, , energy, location,
time */

double width; /* For D. I.'s calculations */

double x, alpha, beta; /* For D. I.'s calculations */

double theta, A, B, C, L; /* For D. I.'s calculations */

double V_accel; /* Acceleration voltage */

double xinc, asigma, bsigma; /* Ditto... */

double temp; /* Temperature */

double dev; /* Deviation from.mean */

double range; /* Range of spectrum */

double mkg; /* Mass (kg) */

int spect[SIZE]; /* Histrogram.for spectrum.*/

int nviter, nsiter; /* # of velocty and space iterations
*/

/*

int nxiter, naiter, nbiter; /* t of x, alpha, and beta iter's */

int iveloc, ispace, i; /* Loop variables */

int ix, ialpha, ibeta: /* Ditto */

int mass; /* Ion mass */

int minmass, maxmass; /* Spectrum extrema (Da) */
int dir; /* Direction (1 or -1) of motion */
int peak; /* Index for peak 1 or 2 */

char string[80];

FILE *fp;

double gasdev(); /* Gassian deviate function */
double ftime(); /* Flight time function */

**

*/

/*

**

*/

272

Initialize spectrum to zeros
for (i‘- 0; i < SIZE; ++i)
spect[i] - 0;
Input mass, distribution function parameters, etc.
printf("Enter mass (Da)\n");
mass - atoi(gets(string));
mkg - (double) mass * KGAMU;

printf("Enter acceleration voltage\n");
v_accel - atof(gets(string));

printf("Enter total field-free length\n");
L - atof(gets(string));

A-L;
B - -2.0 * L;
C - L / 3.0;

printf("Enter theta\n"):
theta - atof(gets(string));

printf("Enter temperature (K)\n");
temp - atof(gets(string));

if (temp - 0.0)
{
printf("Enter energy (eV) and direction (+/- 1)\n");
scanf("%f, %d", &u0, 5dir);
gets(string); /* Clean up buffer from scanf() call */
vsigma - 0.0;
l

{
uO - 0.0;
vsigma - sqrt((KBOLT * temp) / mkg);

}

else

printf("Enter mean for initial spatial distribution\n");
so - atof(gets(string));

printf("Enter standard deviation for spatial distribution\n");
ssigma - atof(gets(string));

printf("Enter slit half-width\n");
width - atof(gets(string));

printf("Enter standard deviation for alpha distribution\n");
asigma - atof(gets(string));

printf("Enter standard deviation for beta distribution\n");
bsigma - atof(gets(string));

if (vsigma !- 0.0)
{

/*

*‘k

*/

273

printf("Enter number of energy iterations\n");
nviter - atoi(gets(string));
}
if (ssigma !- 0.0)
{
printf("Enter number of space iterations\n");
nsiter - atoi(gets(string));
}
if (width !- 0.0)
{
printf("Enter number of x iterations\n");
nxiter - atoi(gets(string));
}
else
nxiter - 1;
if (asigma !- 0.0)
{
printf("Enter number of alpha iterations\n");
naiter - atoi(gets(string));
}
else
naiter - 1;
if (bsigma !- 0.0)
{
printf("Enter number of beta iterations\n");
nbiter - atoi(gets(string));
}
else
nbiter - 1;

printf("Calculate min 8 max from what two masses?\n");
printf(" (Enter two equal numbers for manual input.)\n");
scanf("%d, %d", &minmass, &maxmass);
gets(string); /* Clean up buffer from.scanf() call */
if (minmass -- maxmass)

{

printf("Enter tmin\n");

tmin - atof(gets(string));

printf("Enter tmax\n");

tmax - atof(gets(string));

}

else

{

tmin - ftime(minmass, v_acce1, theta, 0.0,
0.0, 0.0, s0, s0, 0.0, 1, A, B, C, L);
tmax - ftime(maxmass, Y_accel, theta, 0.0,
0.0, 0.0, s0, s0, 0.0, 1, A, B, C, L);

}
printf("Tmin - %e, Tmax - %e\n", tmin, tmax):

srand(1); /* Initialize random number generator */

Now calculate spectrum

for (peak - 1; peak <- 2; ++peak) /*** START or PEAK LOOP ***/
{

***/

274

if (peak - 2)
{
mass - mass + l;
mkg - (double) mass * KGAMU;

}

if (nxiter > 1)
xinc - 2.0 * width / ((double) nxiter - 1.0);

else
xinc - 0.0;
for (ix - 0; ix < nxiter; ++ix) /*** X LOOP ***/
{
x - -width + xinc * (double) ix;
for (ialpha - 0; ialpha < naiter; ++ia1pha) /*** ALPHA LOOP

{
if (naiter -- 1)
alpha - 0.0;
else
{
dev - gasdev();
alpha - asigma * dev;
}
for (ibeta - o; ibeta < nbiter; ++ibeta); /*** BETA LOOP ***/

{
if (nbiter -- 1)

beta - 0.0;
else
{
dev - gasdev():
beta - bsigma * dev:
}
if (vsigma !- 0.0 && ssigma !- 0.0) /* REAL CASE... */

for (iveloc - 1; iveloc <- nviter; ++iveloc)
{
dev - gasdev();
v0 - vsigma * dev + sqrt((2. * COUELE * u0) / mkg);
dir - (v0 > 0.0) ? 1 : -1;
e = 0.5 * mkg * v0 * v0 / COUELE;
for (ispace - 1; ispace <- nsiter; ++ispace)
{
v - v0:
dev - gasdev():
s - ssigma * dev + s0;
time - ftime(mass, v_accel, theta, alpha,
beta, x, s, 30, e, dir, A, B, C, L);
tab(time, spect, tmin, tmax, SIZE);
}
}
else if (vsigma -- 0.0 as ssigma !- 0.0) /* MONO-ENERGETIC */
{
v0 - sqrt((2. * COUELE * uO) / mkg);
dir - (v0 > 0.0) ? 1 : -1;
for (ispace - 1; ispace <- nsiter; ++ispace)
{
e - 0.5 * mkg * v0 * v0 / COUELE;
dev - gasdev();

/*

*‘k

*/

275

s - ssigma * dev + 30;
time - ftime(mass, V_accel, theta, alpha,
beta, 1:, s, 30, e, dir, A, B, C, L);
tab(time, spect, tmin, tmax, SIZE);
1
}
else if (vsigma !- 0.0 && ssigma -- 0.0) /* PLANAR */
{

for (iveloc - 1; iveloc <- nviter; ++iveloc)
{
dev - gasdev();
v - vsigma * dev + sqrt((2. * uO) / mkg);
dir - (v > 0.0) ? 1 : -1:
e - 0.5 * mkg * v * v / COUELE;
time - ftime(mass, v_acce1, theta, alpha,
beta, x, s0, s0, e, dir, A, B, C, L);
tab(time, spect, tmin, tmax, SIZE);
} A
}
else /* SINGLE ION */
{
s - 30;
v - sqrt((2. * COUELE * uO) / mkg);
e - 0.5 * mkg * v * v / COUELE;
time - ftime(mass, V_accel, theta, alpha,
beta, x, s, 30, e, dir, A, B, C, L);
tab(time, spect, tmin, tmax, SIZE);
} .

} , /*** END OF BETA LOOP ***/
} /*** END OF ALPfiA LOOP ***/
} /*** END or x LOOP ***/

1 /*** END or PEAK LOOP ***/

Spectrum complete; now output to file

printf("Enter output filename\n");
gets(string);
if ((fp - fopen(string, "w”)) -- NULL)

{
printf("Error opening file\n");

exit(0);
)

range - (tmax - tmin) / SIZE;

for (i - 0; i < SIZE; ++i)

{
time - ((float) i - 0.5) * range + tmdn;

fprintf(fp, ”%12.8e, %8d.0\n", time, spect[i]);
}

fclose(fp);

 

276

double gasdev()

/*
*‘k
*‘k

**

*/

Return gaussian deviate
See Numerical Recipies, p. 203.

{
double v1, v2, r, fac;

static double gset;
static int iset - 0;

double ran();

if (iset -- 0)
{
do
{
v1 - 2. * ran() - 1.;
v2 - 2. * ran() - 1.;
r - vl*vl + v2*v2;
}
while (r >- 1.);
fac - sqrt(-2. * log(r)/r):
gset - v1 * fac;
iset - 1;
return(v2 * fac);
}
else
{
iset - 0;
return(gset);
}

277

double'ran()
/* .
** Return random number between 0.0 & 1.0
**
** Library routine rand() return an int between 0 and 2‘31 - 1.
** We divide this by 2‘31 to return a double >- 0. and < 1.
*/
{
double temp;
int rand();

temp -(double) rand() / 2147483647.;
return(temp);
}

tab(time, spect, tmdn, tmax, spsize)
double time, tmin, tmax;
int spect[], spsize;
/*
** Tabulate ion's time-of—flight in histogram
*/
{
double range;
int index:

range - tmax - tmin:
index - (int) ((time - tmin) * (double) spsize / range);

if (index >- spsize ll index < 0)
return; /* ...out of range */

if (index -- spsize)
--index;

spect[index] +- 1;
}

 

278

#include <stdio.h>
#include <math.h>

#define KGAMU 1.66057E-27

#define COUELE 1.60219E-19

#define P1180 0.01745392 /* Pi / 180 */
/* ftime.c

**
**

*/

Flight time calculation from.Damaschin Ioanoviciu's transfer matrix.

double ftime (m, K, theta, alpha, beta, x, s, so, no, dir, A, B, C, L)
double K, theta, alpha, beta, x, s, so, no, A, B, C, L;
int m, dir;

/*
**
**

*/

/*
**
**
*‘k
*‘k

*/

*/

/*
*‘k
**

*/

{

double row[14];

double d;ref1, V_refl;

double t_ref, t_mat, t_source;

double v_ref, v, delta_k;

double alpha_s, beta_s; /* alpha and beta from Yefchak, et. al */

double ft();

First find t_source, the flight time out of the source and to
the space-focus plane:

alpha_s - s / s0;
beta_s - uO / K;

t_source - alpha_s * s0 * sqrt(2.0 * (double) m.* KGAMU);
t_source /- (sqrt((alpha_s + beta_s) * K * COUELE) +
(double)dir * sqrt(beta_s * K * COUELE));

t_source +- 30 * 2. * sqrt((double) m * KGAMU) /
sqrt(2.0 * (alpha_s + beta_s) * K * COUELE);

Now find t_ref, the flight time of the reference ion:

Let d_refl - L * 0.1 m. (This is the length of the second,
reflecting, stage of the mirror.)

v_ref - sqrt(2.0 * K * COUELE / (m * KGAMU));

v - sqrt(2.0 * (alpha_s + beta_s) * K * COUELE / (m * KGAMU));
d_refl - L * 0.1:

V_refl a 6.92820323 * d_refl * K / L; /* 6.928... - 4*sqrt(3)

t_ref - 4.0 * d_refl * K / (v_refl * v_ref);
t_ref *- 2.309401; /* 2.309... - 4/sqrt(3) */

Finally, use transfer matrix calculations to find tJmat, which
is the flight time relative to that of the reference ion:

theta *- PI180;
delta_k - 0.5 * m * KGAMU * v * v / (K * COUELE):
row[0] - x;

279

row[l] - alpha;

row[2] - 0.0; /* delta_m - 0 */

row[3] - delta_k:

row[4] - x * alpha;

row[S] - 0.0; /* x * delta_m = 0 */
row[6] - x * delta_k;

row[7] - alpha * alpha;

row[8] - 0.0; /* alpha * delta_m - 0 */
row[9] - alpha * delta_k;

row[lO] - 0.0; /* delta_m * delta_m.- 0 */
row[ll] - 0.0; /* delta_m * delta_k - 0 */
row[12] - delta_k * delta_k;

row[l3] - beta * beta;

t_mat - ft (m, K, theta, v_ref, A, B, C, L, row);

return(t_source + t_ref + t_mat):
}

$0

double ft(m, K, theta, v, A, B, C, L, row)
double K, theta, v, A, B, C, L, row[14];

int m;

/*

**

*‘k
**
**
**
**
**
**
**
**
*‘k

**

*/

{

double col[14];

double
double
int i;

tn;
t_mat;
row[0] x
1 alpha
2 delta_m.
3 delta_K
4 x alpha
5 x delta_m
6 x delta_k
7 alpha alpha
8 alpha delta_m
9 alpha delta_k
10 deltaJm delta_m
ll delta_m.delta_k
12 delta_k delta_k
13 beta beta

tn = tan(theta);

col[O]
col[l]
col[2]
col[3]
col[4]
col[S]
col[6]
col[7]
col[8]
col[9]

-tn;

tn * (A - L):

-tn / 2.0;,

(A - L) / 2.0;

tn * tn;

-tn / 2.0:

tn / 2.0;

(L + B/2.0) * tn * tn +
tn * (A - L) / 2.0;

tn * (A + B + L) / 2.0;

(L - A) / 2.0;

col[lO] - -(C + L) / 8.0;

col[ll] - (A — L) / 4.0;

col[lZ] - (3.0 * L + B - A) / 8.0;
col[13] - -col[3];

t_mat - 0.0;

for (i - 0; i < 14; ++i)

t_mat +- row[i] * col[i];
t_mat /- v;

return (t_mat) ;

}