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EXPERIMENTAL AND THEORETICAL MODELING STUDIES OF

THE SIEGFRIED MASS IDENTIFICATION SYSTEM

By

Marcello Michael DiStasio

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

1980

ABSTRACT

EXPERIMENTAL AND THEORETICAL MODELING STUDIES OF

THE SIEGFRIED MASS IDENTIFICATION SYSTEM

By

Marcello Michael DiStasio

The electrostatic mass identification system, SIEGFRIED
was used to identify short-lived nuclear reaction products.
A combination of Helium Jet Recoil Transport and Time-of-
Flight methods was employed for Al, KCl, Ti, Sm and NaF
targets. Both B-mass and y-mass coincidence experiments
were performed.

All measurements were made using the 70—MeV 3He beam
from the Michigan State University Sector-Focused Cyclo-
tron.

Theoretical modeling of the Time-of-Flight system was
accomplished by numerical methods. This modeling was
used to explain observed mass peak broadening in terms of
the recoil nucleus initial kinetic energy which results
from B decay. The possibility of measuring B-decay Q values

from observed peak broadening is demonstrated.

ACKNOWLEDGMENTS

Thank you to Professor William C. McHarris for serving
as my thesis adviser and for all his support through my
graduate career. I would also like to thank Dr. William
Kelly for all the time and trouble he spent in fulfilling
his role as my second reader. To Ray warner and Richard
Firestone I want to express my appreciation for much help
and guidance.

Thanks to Dr. D. O. Riska for invaluable discussions
and suggestions concerning theoretical matters. I would
also like to thank Richard Au and Barb Woodward.

Special thanks to Wayne Bentley for countless hours of
help on the experimental work for this thesis.

Thank you to Peri-anne Warstler for an excellent Job
of typing this work.

Thanks to Roy Gall and Sandy Koeplin for the special
friendship they extended to me.

My sincerest thanks to my mother and father for con-
tinuing faith and love, all that I accomplish is but a small
acknowledgment of their devotion.

To my brother and sisters, thank you for the love and
support you have always given.

Finally, a continuing and loving thanks to my wife,

Ernie, for her love and understanding over all these years.

ii

Chapter

LIST OF

TABLE OF CONTENTS

TABLES. . . . .

LIST OF FIGURES

I. INTRODUCTION .
II. DETAILS OF TOF SYSTEM.
III. EXPERIMENTAL RESULTS .
IV. RECOIL ENERGY FITTING.
V. MODELING THE TOF SYSTEM. . .
VI. THEORETICAL MODELING RESULTS
VII. CONCLUSIONS.
APPENDICES
I. SIMPLE TOF MODEL .
II. RELATION BETWEEN B-DECAY AND
RECOIL ENERGY. . . . . . . .
III. DOUBLE 3-POINT INTERPOLATION .
IV. FINITE DIFFERENCE APPROXI-
MATIONS. . . . . . . . . .
BIBLIOGRAPHY.

iii

Page

iv

vi

16
75
101
1142

175

177

18L!
187

. 191

195

LIST OF TABLES

Table Page
III-l LSQ Fit to Al TOF Spectrum. . . . . . . . 25
III-2 Predicted Cross-Sections and

Integrated Areas for Al Peaks . . . . . . 28
III-3 Results of Kinetics Study

of Al Spectrum. . . . . . . . . . . . . . 37

III-A(a) Properties of Reaction Products
from 23Na + 70-MeV 3He. . . . . . . . . . Al

III-4(b) Properties of Reaction Products

from 19F + 70-MeV 3He . . . . . . . . . . Al
III-5 LSQ Fit for KCl Spectrum. . . . . . . . . A9
III-6 Properties of Products of “5T1

+ 70—MeV 3H8. . . . . . . . . . . . . . . 55
IV-l Results of SAMPO Fitting of

KCl Spectra . . . . . . . . . . . . . . . 8l
IV-2 Calculated TOF'S for KCl

Spectrum (in nsec). . . . . . . . . . . . 8A
IV-3 Fit to KCl Spectra Recoil

Energies (no F-GT Correction) . . . . . . 9O
Ivsu Fit for Al Spectrum ~- No

F-GT Correction . . . . . . . . . . . . . 93

iv

Table

VI-l

VI-2

Fermi-Gamow-Teller Parameters
for Peaks in KCl Spectrum
Recoil Fit for KCl Spectrum
(With F-GT Correction).

Fit to Al Spectrum Recoil
Energies (With F-GT Correction)
Comparison of Relaxation and
SOR Results for Infinite Coaxial
Conductors. . . . .

Comparison of Speeds for Runge-
Kutta vs. Predictor-Corrector
Routines.

Comparison of Bound Values. . .
Comparison of Peak Widths for

Various Distributions

Page

99

100

100

116

luO

15A

168

LIST OF FIGURES

Figure Page
2-1 Schematic Diagram of SIEGFRIED. . . . . . 7
2-2 Electronics for TOF mass measure-

ment. . . . . . . . . . . . . . . . . . . 10
2-3 SIEGFRIED'S vacuum system . . . . . . . . l2
2-A Schematic for CEMA failsafe . . . . . . . 15
3-1 TOF spectrum for products of

27Al + 70-MeV 3He . . . . . . . . . . . . 2A
3-2 ALICE predictions for 27Al + 3He. . . . . 26
3-3 ALICE predictions for 23Na + 3He. . . . . 39
3-u ALICE predictions for 17F + 3He . . . . . no
3-5 TOF spectrum for 70-MeV 3He on

NaF target. . . . . . . . . . . . . . . . A3
3-6 ALICE predictions for 39K + 3He . . . . . as
3-7 ALICE predictions for 35C1 + 3He. . . . . A6
3-8 TOF spectrum for 70-MeV 3He on

KCl target. . . . . . . . . . . . . . . . A8
3-9 ALICE predictions for I"STi + 3He. . . . . 5A
3-10 ALICE predictions for ”7T1 + 3He. . . . . 57

3-11 TOF spectrum for products of 70-MeV
3He on Ti target. . . . . . . . . . . . . 58
3-12 ALICE predictions for 58Ni + 3He. . . . . 61

vi

Figure Page

3-13 TOF spectrum for products of AS-MeV

3He on Ni target. . . . . . . . . . . . . 62
3-1A TOF spectrum for products of 70-MeV

3He on Ni target. . . . . . . . . . . . . 65
3-15 TOF spectrum for 27Al + 70—MeV

3He reaction products (using NaI

as start detector). . . . . . . . . . . . 67
3-16 NaI y-spectrum in coincidence with

spectrum of Figure 3-15 . . . . . . . . . 70
3-17 ALICE predictions for 3He + 1""Sm . . . . 71

3-18 TOF spectrum for products of 70-MeV

3He on 1""Sm target . . . . . . . . . . . 73
A-l Comparison of peak width for KCl

spectrum. . . . . . . . . . . . . . . . . 76
A-2 Decay of 28Al . . . . . . . . . . . . . . 91
A-3 Fermi-Gamow—Teller recoil energy

distributions . . . . . . . . . . . . . . 96
5-1 Two-dimensional x-y grid with dij's

at free points after 1 iterations.

The grid spacing is the same in the

x and y directions. x = 1A, y = 5A,

where 1,3 are integers. . . . . . . . .‘. . 111
5-2 Cross section of infinitely long

coaxial cylindrical conductors. The

inner conductor has radius a = 1.0 and

vii

Figure

5-3

544

5-5

5-6

547

5-8

Page

is held at 83 = 100; the outer

conductor has radius b = 10.0 and

is held at ob = 0.0. The analytic

solution is given in the text. A¢=

¢a - ob . . . . . . . . . . . . . . . . . 115
Finite cylindrical conductor held

at ¢ = 100, enclosed in larger grounded
cylinder. . . . . . . . . . . . . . . . . 118
Equipotentials resulting from SOR

solution to problem geometry of

Figure 5-3. . . . . . . . . . . . . . . . 119
Finite cylindrical conductor (with

o = 100) enclosed in grounded outer

conductor that abruptly expands radius

by a factor of two. . . . . . . . . . . . 120
Equipotentials resulting from SOR

solution to problem geometry of

Figure 5-5. . . . . . . . . . . . . . . . 121
Equipotentials in the acceleration

zone calculated by overrelaxation

method. . . . . . . . . . . . . . . . . . 125
Example of Z and p components of

the calculated electric field in the
acceleration zone. For Ez curve x = z,

for ED curve x = p. . . . . . . . . . . . 126

viii

Figure Page

5-9 Equipotentials in the drift zone

calculated by overrelaxation method . . . 131
6-1 Approximate model of the TOF

system. . . . . . . . . . . . . . . . . . 1A8
6-2 To - T¢ bound curves for several

initial positions . . . . . . . . . . . . 155
6-3 To — T¢ hit efficiency curves . . . . . . 157
6—A Graphs of final 0 position as

function of T¢ (for fixed initial

9 velocities) . . . . . . . . . . . . . . 159
6—5 Graphs of final p position as func-

tion of T (for fixed initial p
velocities) . . . . . . . . . . . . . . . 160
6-6 Recoil energy distributions for

pure Fermi and pure Gamow-Teller

decays. . . . . . . . . . . . . . . . . . 162
6-7 Recoil energy distribution for mixed

decay . . . . . . . . . . . . . . . . . . 163
6-8 Theoretical TOF peak predicted

from flat momentum distribution . . . . . 167
6-9 Theoretical TOF peak predicted from

Fermi distribution. . . . . . . . . . . . 170

6-10 Theoretical TOF peak predicted from

Gamow-Teller distribution . . . . . . . . 171

ix

Figure Page

6-11 Observed TOF spectrum of products

of 27A1 + 70-MeV 3He reaction . . . . . . 172

CHAPTER I

INTRODUCTION

The development and refinement of many new and ingenious
instruments for investigations of nuclei from the region
of B-stability has made such endeavors challenging and re-
warding. Experiments that were very difficult ten years
ago are fast becoming relatively easy and almost routine.
Among the new techniques are the Helium Jet Recoil Trans-
port system (HeJRT), the Rabbit system, and a number of
new time-of-flight (TOF) mass identification systems. The
HeJRT and Rabbit systems were developed to provide a means
of fast transport of short-lived products of nuclear re-
actions to a low background counting area. This is a very
important function when one is dealing with reaction
products that have half-lives in the tenths of second range.
Besides their short half-lives, another difficulty one
must deal with is interfering products. Generally the
result of the interaction of a projectile beam with a
chosen target is never exclusively the product of interest.
One is always faced with the problem of differentiating
the species of interest from a crowded field of interfering

reaction products. As a means of separating out and studying

a particular isotope of interest, on-line mass separation
systems have been developed. The time—of-flight spectrom-
eter is a conceptually simple and increasingly popular
instrument for isotope identification. Our system, SIEG-
FRIED, was constructed toward this end.

The basic principle of SIEGFRIED, as of all TOF systems,
is that nuclei of varying masses may be differentiated ac-
cording to the time each mass requires to traverse a given
flight path length. In our system each mass is initially
accelerated to a known velocity and then allowed to drift
a known length. Measurement of the time-of-flight yields
the associated mass. In contrast to many other TOF systems,
SIEGFRIED was developed as a means of labeling a B-decay
event with the associated mass resulting from the decay;
it was not originally intended to provide highly precise
mass measurements. In this respect SIEGFRIED is analogous
to the fast chemical separation techniques used for the
study of short-lived nuclei. These techniques were in-
valuable in dealing with complex mixtures of radioactive
reaction products. Serving a similar function SIEGFRIED is
a fast, mass identification system.

Our mass identification system is actually a result
of coupling a HeJRT system, for transport, to the TOF system,
for mass identification. Some of the original work along
these lines was accomplished by a group at Texas A & M

(Ju7l). They used their MAGGIE system to analyze recoils

from a-decay. To demonstrate that recoils from B-decays
could be mass analyzed by TOF methods the MSU SIEGFRIED
project was initiated (Ed76). As with so many supposedly
simple instruments unexpected results often manifest them-
selves. Closer inspection of the details of our TOF
spectra leads one to suspect that a great deal of interest-
ing information about the nuclear decay associated with an
observed recoil mass, is being preserved in the observed
mass peaks. In order to confirm or refute such suspicions,
it is necessary to gain a clear understanding of precisely
how the SIEGFRIED TOF system works. In a large measure,
this will be a main thrust of our discussion. In this thesis
we will attempt to dovetail experimental results in various
mass regions with a rigorous theoretical modeling study of
the system. Of course the model will not be an exact simu-
lation of the system yet we will strive for the most real-
istic representation possible while still practical.

Before proceeding to the bulk of our presentation I
would like to point out that our system has resulted directly
and indirectly from the work of many researchers in the area
of TOF measurements. As is the case with many experimental
systems, SIEGFRIED has provided us with a number of suc-
cesses, a fair share of frustrations, and a plethora of
intriguing potentialities. Among the successes we should
first count the fact that SIEGFRIED has proven that B-recoil

ions can be routinely mass analyzed. This is something that

only a few years ago was an uncertain hypothesis. Second,
as will be shown in later chapters, we have employed
SIEGFRIED for mass identification over a range of dif-
ferent mass regions with fair success. These experiments
have been performed, in part, to investigate the utility.
of, and difficulties associated with our system when applied
to a variety of cases that one would expect to obtain reason-
able results. Third, we will demonstrate in this work

that a prime bit of information that can be obtained from
our mass peaks is the initial recoil energy of the observed
B-decay product. This is a quantity that is directly

related to the B-decay energy.

CHAPTER II
DETAILS OF TOF SYSTEM

Since a detailed analysis of our Time—of-Flight (TOF)
system will comprise a significant portion of subsequent
discussions it is worthwhile at this point to provide a
brief exposition concerning the design and use of the
SIEGFRIED spectrometer. Many aspects of our instrument and
the TOF measurement process are not unique to SIEGFRIED
but are common to many TOF systems. In the following
presentation the only details relating to SIEGFRIED that
will be stressed are those that represent changes from the
original construction. They are modifications effected
by the author that should be noted by future users of
SIEGFRIED. For a full discussion of the design and construc-
tion of SIEGFRIED, see reference (Ed76).

In a "typical" experiment for TOF measurements, nuclear
reaction products are generated with the MSU Sector-Focused
Cyclotron and transported to SIEGFRIED by means of a Helium
Jet Recoil Transport System (HeJRT). In this HeJRT system
a gaseous mixture of He, impurities, and reaction products
flow through a capillary tube and into an evacuated chamber

of SIEGFRIED that is called the skimmer chamber. The

gaseous mixture is directed at a conical skimmer which
acts to remove most of the helium gas (See Figure 2-1).
The reaction products are attached to large molecular
"clusters" which have a smaller divergence as they flow
from the end of the capillary and so have a higher prob-
ability of passing through the skimmer assembly and striking
the stainless steel collecting plate in the collection
chamber. These molecular clusters stick to the collecting
plate and provide a very thin source. The metallic collect-
ing plate is held at a static, positive high voltage, usually
+6 kV with respect to SIEGFRIED ground potential.

When one of the radioactive species undergoes a B-decay,
a simple sequence of events occurs that is the basis for
our time-of-flight measurements. As the nucleus of interest
emits the 8 particle, it will recoil to conserve momentum
and leave the surface of the collector. Also, when a
nucleus emits a 8 particle, there is a resulting sudden
change in the nuclear charge from Z to Z i 1. This very
fast change causes a pertubation on the electrostatic po-
tential in which the atomic electrons move and often
results in the ejection of an atomic electron. Studies
(Ca63) have shown that in the majority of ionizing events
a +1 charge state results.

The positive ions so created are then accelerated
across a region of nearly uniform electric field, trans-

forming the 6 kV of potential into kinetic energy. The

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acceleration zone is in effect a circular plate capacitor
with the collector at 6 kV as one plate and a disc formed
of fine wire mesh (90% transmission mesh) acting as the
ground plate.

After passing through the wire mesh disc, which ef-
fectively completes the ion acceleration, the recoiling
ions enter a weak field region we will refer to as the
"flight tube". In this portion of the instrument there is
a very small diameter (0.005 cm) wire held at a small nega-
tive voltage (-110 V) concentric with a relatively large
radius (5.25 cm) pipe that is held at ground. The result
of this configuration is a region of logarithmic potential:
the wire, commonly called an Electrostatic Particle Guide
(ESPG), acts to focus the recoil ions in the radial direc-
tion and onto a set of Channel ELectron Multiplier Arrays
(CEMA) that serves as the ion detector. A set of two
CEMA's is often called a Chevron detector. The CEMA's
are positioned close to the end of the flight tube, roughly
1 m away from the collector plate and the source.

The electronics necessary for a mass measurement is
made up of standard NIM modules that are widely employed
by nuclear experimentalists. The basis of the measurement
is as follows: (1) a radioactive species emits a 3 particle
that strikes a plastic scintillator, which provides a start

pulse for a Time to Pulse Height Converter (TAC); (2) the

recoil ion, after acceleration and drift down the length

of the flight tube, strikes the CEMA detector and generates
a pulse that stops the TAC; (3) the TAC then gives a voltage
output that is proportional to the recoil ion time of flight.

In Figure 2—2 is shown a minimum electronics block
diagram for a mass TOF measurement. The function of this
set-up is to perform a simple delayed coincidence measure-
ment. The anode output from the plastic scintillator
photomultiplier tube is fed to timing filter amplifier (TFA)
for shaping and amplification. The TFA output is then pro-
cessed by a Constant Fraction Timing Discriminator to pro-
vide a very sharp, fast start pulse for the TAC. The output
from the CEMA preamp is sent through an identical set of
NIM modules and so generates the stop signal for the TAC.
The output of the TAC is sent to an analog-to-digital con-
verter (ADC); the digitized output of the ADC is then
stored in a computer as the TOF spectrum of interest.

The basic operation of SIEGFRIED has remained virtually
unchanged since the original construction was completed in
1976 by M. Edmiston. Nevertheless, we have made a few
modifications related to the system that are necessary to
document for future use of SIEGFRIED. The two most im-
portant changes are: (1) design and inclusion of a safety
control unit for the power to the CEMA's and (2) a new set
of CEMA detectors.

The CEMA's are a rather delicate set of detectors, and

lO

 

 

 

SCIn‘IiIlO‘IOl’ I CEMA
PHOTOMULTIPLIE
ANOOE OUTPUT PreO mp

 

 

 

 

 

 

I I

 

 

 

 

 

 

 

 

 

TIMING TIMING

FILTER FILTER

AMPLIFIER AMPLIFIER
cms'mm CONSTANT
FRACTION FRACTION
TIMING TIMING
DISCRIMINATOR - DISCRIMINATOR

 

 

 

 

 

 

 

TIMING TO PULSE
HEIGHT

310”. Stop
CONVERTER

 

 

 

 

 

ADC

 

 

 

Figure 2-2. Electronics for TOF mass measurement.

11

the manufacturer (Bendix Corporation) warns that they

should not be powered in any vacuum worse than 10-5 torr at
the risk of their destruction. In the original operation
.mode of SIEGFRIED the only safeguard against the destruc-
tion of the CEMA detectors during an experiment was through
interlocks between two vacuum valves, VA and V6 (see Figure
2-3), and the power supply for the detectors. Actually,
these interlocks were not meant to serve as a safeguard
system against problems arising during the length of an
experiment. Originally the interlock system was intended to
ensure only that the CEMA detectors were not powered before
the vacuum in SIEGFRIED was below a level safe for operation
of the detectors. Well, true to form this interlock system
did not function as the failsafe it was never intended to
be, as the old CEMA detectors met with a rather quick and
complete demise during the course of an early experiment.
This sudden, distasteful development is what led to the
second of the aforementioned modifications related to
SIEGFRIED -- acquisition of a new set of CEMA detectors.

The new channel plates were purchased from the Bendix
Corporation, product number 302-B-005-MA. In many respects
the new CEMA's are much the same as the old ones, both sets
functioning according to the same ingenious theory of opera-
tion which is well described in the Technical Application
Note provided by the manufacturer. The new CEMA's are

thinner and draw less current than the old set, but a most

l2

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95 ”S X = Vacuum Valve
BLOWER
[_'_'> = Vacuum Gauge
- - LN2 Trap

 

 

 

f 7 ll:
VANE PUMP

I

 

Figure 2-3. SIEGFRIED's vacuum system.

l3

important difference between the old set and the new that
cannot be overemphasized is that the new CEMA detectors
require -2000 V bias. The old set required -2700 V for
operatiOn. This particular difference is especially im-
portant to note carefully since using the wrong high voltage
setting can easily lead to serious damage of the new CEMA's
-- if not their complete destruction as functional detectors.

Another difference between the old and new sets of CEMA
detectors is of much less importance than the power require-
ments but interesting nonetheless. The bias angle of a CEMA
is the angle that the channels make with the normal to the
detector face. Greater bias angles alleviate a problem that
is known as ion feedback. The new CEMA's have bias angles
of 8° as compared to 0° and 5° of the old CEMA chevron set,
this means that the new set will have a better signal—to-
noise ratio than the old set.

As a result of our experience of destroying the CEMA's,
it was decided that a fast and reliable means of safeguard-
ing the new set of detectors was a rather obvious necessity.
A safety-control unit was designed, constructed, and put
into operation as a reliable means of quickly shutting down
power to the CEMA detectors in the event of a sudden bad
vacuum and/or a large current flow through the CEMA's -- both
of which can damage or destroy the multiplier array detectors
that are vital for the operation of SIEGFRIED and are very

expensive.

1A

In Figure 2-A is shown the circuit schematic for the
Safety control unit for the CEMA detectors. Power is from
an ORTEC A59 power supply that has a remote shutdown feature
and so can be effectively turned off by the control unit.

The 10 uA meter shown monitors the current drawn by the
CEMA's; if this current exceeds the setpoint (usually =3 uA),
the meter will cut off the power to the CEMA's. In many
tests of this system, intentional or not, the failsafe has
been highly successful. The vacuum interlock system was

not so useful as the current interlock portion. There were

‘ two reasons for this. First, the only region of vacuum that
is of value to the safeguard system is around 10-5 torr.

The only reliable means to monitor the vacuum in that range
is utilizing vacuum gauges that generate ions in such quanti-
ties as to create severe noise problems in the CEMA's.
Second, the vacuum interlock system only provides a redun-
dancy of the overcurrent safeguard that, in view of the noise

problem, is of reduced value.

l5

ORTEC ‘59
0'5 KV

 

HV
OUT TO CEMA

2
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ORTEC I
459 I
INTIRLOCK I
I
‘V
I I
I 24 vac
I
rJ
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I
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Figure 2-A. Schematic for CEMA failsafe.

CHAPTER III

EXPERIMENTAL RESULTS

An important aspect of any experimental instrument is
its range of utility. In order to make the greatest use of
any apparatus it is necessary to investigate and define the
types of systems that are most suited to the capabilities
and limitations of the method of measurement. This is es—
pecially true of the SIEGFRIED, and the following discussion
will attempt to deal with these aspects in a manner that will
aid in further use of this time-of-flight system.

The results of a number of TOF measurements for several
different targets will be presented and discussed in a manner
that will display the reasoning and logic required to design,
run, and interpret a mass-identification experiment. In
addition, these measurements will present a survey on the
use of SIEGFRIED in a number of different mass regions.

The coupling of the HeJRT system to SIEGFRIED presents
the experimentalist with a number of advantages that makes
such a combination very well suited to the study of short-
lived nuclei that fulfill certain criteria. Naturally,
species that do not have the appropriate characteristics

yield poor or null results. The aforementioned criteria

16

17

are unfortunately not quantitative conditions, but the
following discussion should serve as a guideline for deter-
mining whether a particular TOF measurement is feasible.

The obvious first step in designing an experiment with
SIEGFRIED is to choose an appropriate target. Because of the
total efficiency of the HeJRT-SIEGFRIED system it is im-
portant to select cases with high reaction cross-sections
and targets that can withstand maximum beam on target.

Foil targets are well suited to such conditions. In a

number of runs with chloride and oxide targets it was found
that a large quantity of powdery substance had been generated
in the target area and had clogged the input end of the

HeJRT capillary, which effectively reduced the transport
efficiency to zero. The nature of this powder and the
mechanism for generating it are both unknown but do consti-
tute a real problem with oxide and chloride targets.

Having decided on the target most suited to the reaction
of interest, it is worthwhile to estimate the products of
the chosen reaction and their relative cross-sections. This
is a task that is best accomplished by using the computer
routine called ALICE (B170). ALICE is Fortran code written
by M. Blann and F. Plasil and is available on the MSU 27
computer as a user's code. The results of ALICE computa—
tions give the reaction cross-sections as a function of
energy for a number of different reaction products. A

number of figures in this section are the result of such

18

computations which provide a convenient tool for estimat—
ing the most probable products from the reaction of interest.
After determining the probable reaction products, one must
consider the half-lives of these products. The time re-
quired for transport of a product from the target area up

to the collection plate of SIEGFRIED has been estimated to
be on the order of hundreds of milliseconds, so species
with half-lives less than a few hundred milliseconds will
decay away before reaching SIEGFRIED. Using the HeJRT
system at MSU, Edmiston (Ed76) was able to measure the
half—lives of ”7Cr, “6V and 55Ni. He found these to be

A60 i 1.5 msec, A20 i 3 msec and 219 i 6 msec, respectively.
These measurements can be used to gain a rough estimate on
the upper limit of the transport time. Species with half-
lives greater than a few minutes are also difficult to
measure with SIEGFRIED. In such cases one must optimize
counting and collection intervals as is discussed in this
chapter.

The chemical properties of the reaction products and
the associated molecular clusters are an important aspect-
of the transport and measurement process. Though less under-
stood than many details Of the system, these properties
should be considered in designing an experiment. For the
sake of this discussion I will divide the relevant chemical
properties into two categories: (1) transport efficiency

and (2) sticking efficiency. (This division may well be

19

partially artificial but will aid in understanding two
chemically related aspects of our system.)

Kosanke (K073) studied the efficiency of transporting
a number of different reaction products with the HeJRT
system. It was found that the transport efficiencies varied
from 2A% to 60% and was also a function of the type and
concentration of impurity added to the pure He. The situa-
tion now is that the actual nature of the recoil-cluster
association has not been elucidated to the point where it
is possible to predict the transport efficiency for a given
species, but it remains an important point.

Once a species has been transported from the target
area to the counting area, it is sprayed onto a surface and
if the sample adheres to the surface, a thin film source
results. However, it is by no means certain in all cases
that the nuclei of interest will stick well enough to remain
on the surface until they decay. This "sticking efficiency"
was also looked into by Kosanke (K073) but only with regard
to varying capillary-collecting surface angles and distances.
No studies thus far have dealt with this efficiency with
respect to differing chemical species, since it has been
assumed that the "clusters" stick to the surface and the
nuclei of interest somehow adhere to the clusters.

Whatever the actual chemical nature of the transport and
sticking process, a few practical observations have been

made. In no instances have any noble gases (i.e., He, Ne,

20

Ar, Xe) been observed even when the appropriate reaction
cross-section is fairly high. This seems to be a rather
obvious chemical effect, since the noble gases are known to be
relatively chemically inert. In a number of experiments per-
formed it has been noted that chlorine and fluorine products
had very reasonable cross-sections but were not observed.
This may suggest a poor transport and/or sticking efficiency
for halide ions.

Once a set of reasonable reaction products have been
predicted, one should calculate the time of flight for the
heaviest mass expected. This is necessary in order to choose
the appropriate TAC scale to be used in the measurement. The
‘time of flight for a nucleus of mass number A for our system

can be calculated from the formula
TOF(usec) = 2.536 /A7HV (3-1)

vvhere HV, the voltage applied to the collector plate is in

lciloVolts. For a more complete discussion of this equation,

see Appendix I.

Having chosen the appropriate time scale for the TAC,

:lrt is very important to calibrate the TAC. A convenient
method of calibration is to use the Tennelec Model 1030
UDILme Calibrator, a NIM module that generates start and stop
$3Zlgnals suited for most TAC's. A second means of calibration

1;53 to use some standard target with a well-known TOF spectrum.

21

In all of our experiments an Al foil target was used for
calibration, since it gives high statistics, well separated
peaks, and is simple to construct. This latter method is
actually preferable to the former, since it gives a calibra-
tion for the system -- not just the TAC. It has been found
that a combination of the two calibration procedures is
preferred in most cases. (The Time Calibrator allows deter-
mination of a precise number Of nsec per channel.)

After completion of the data acquisition, one uses the
time calibration to extract the measured time of flight;
that is, transform channel number to time units. This trans-
formation is a simple linear one. Once the TOF's for each
peak are known, the mass number can be extracted by the
simple relationship

A 0.155 (HV)(T0F)2 (3-2)

After determining the mass number corresponding to each
Lbeak, one must determine the particular isotope that is
Eissociated with the known mass number. If a Y-mass coinci-
<ience has been run, it may be possible to associate a par-
tzicular y ray with a known mass, thus specifying the

IDearticular decay. Unfortunately, in our experiments it
fleas been found that the majority of y rays is annihilation
ir‘eadiation, which, of course, is of little use in character-

ii.2:ing a decay -- except to show it occurs via positron

22

emission.

For most of our runs we have used B-mass coincidence,
recording the TOP and the 8 energy. For decays with
highly different B-endpoint energies one can determine the
correct species straightforwardly. However, in many cases
the B energies are not greatly different, and because of
the poor energy resolution of plastic scintillator no
unique characterization is possible.

In the absence of direct measurements uniquely de-
termining Z, one must rely on reasonable deductions. Com-
parison of the observed masses with the predictions from
.ALICE calculations often narrows the possible isotopes to
one or two choices. After choosing a number of potential
candidates, one must consider the relative half-lives and
:reaction cross-sections to characterize finally the decay

‘that gives rise to the observed TOF peak.

2 7A1 + 3He

Al foils were the first and most frequent targets used
I70r our experiments. The material is readily available in
‘Vrarying thicknesses and the 27A1 isotope comprises 100% of
t:lne material, so there are no interfering contaminants.
A33r1 addition, the 27Al (p,xnyp) and 27Al (3He,xnyp) reactions
Egzlve products that are very well suited for measurements

EDJV SIEGFRIED. In the developmental stages of our TOF

23

system, these reactions were used almost exclusively to
provide simple and familiar spectra. Another advantage
of these reactions was that an identifiable mass TOF spec-
trum would result from an hour's worth of counting time with
a beam current of roughly 1/2 uA of 70-MeV 3He on target.
Also, the decays that give rise to the Observed TOF spectra
are straightforward ground-state to ground-state decays in
most cases. The resultant peaks have a shape that is
prototypical of the mass peaks we obtain with SIEGFRIED.

In light of the aforementioned advantages of using Al
foil targets, it is natural that this target has come to
be used as a source for a standard spectrum for many of
our runs. This is especially true in the mass region
20-50 amu.

Using a TAC calibration from the Tennelec Time cali-
‘brator, the first peak in Figure 3-1 is found to occur at
£1 TOF of A.915 usec. Then, using Equation (3-2) the mass
Iiumber is found to be 23. Mass 2A does not appear in the
spectrum, but, following the same procedure, we assign
J; = 25-29 successively. As a double check, a linear least
squares fit to the TOF's as a function of centroid was
Iatarformed. The results are given in Table III-l; x2 for
tslne fit was 8.0 x 10'6.
In Figure 3-2 the theoretical excitation functions for
2 7A1 + 3He are plotted. These curves are the result of

Eit'lALICE calculation. The plots do not include stable

2A

003v

 

.omm >mzlo> + H<h~ mo mu0300LQ Low

sapwoodm hoe

 

A035 “.05.
003 00.0 come
- — p — b
. _
.43 f
.1 .
M“ M \
ii __ i
.55 r ._
I M
.48 i _
awn

 

00h?

.Hum ocsmfig

cone
+0

 

 

Lag:

25

Table III-1. LSQ Fit to A1 TOF Spectrum.

 

 

Centroid T

A TOF (Channel LSQ A A
(AMU) (usec) number) (nsec) (nsec) LSQ

23 A.9l5 5087 A.91A 0.001 22.99

25 5.127 5288 5.126 0.001 25.00

26 5.229 5387 5.231 0.002 26.02

27 5.330 5A8A 5.33A 0.00A 27.0A

28 5.A28 5569 5.A2A 0.00A 27.95

 

lCKD-

5C)-

0'
(mb)

 

F igure 3-2. ALICE

26

25A]

30 4O 50 60 7O

EsHe (MGV)

predictions for 27Al + 3He.

27

products or species with half-lives greater than a few hours,
since these products will not be measurable with our system.

The spectrum shown in Figure 3-1 is the result of 70-MeV

3He on 27A1. For this experiment we collected and counted

for roughly two hours with 1 HA of beam target. In Table

III-2 are the integrated areas and the theoretical cross-

sections for 70-MeV 3He.

Now that we have the mass number for each of the peaks
we must assign an atomic number Z for complete characteriza-

tion. According to the ALICE calculations (see Figure 3-2)

there are two possible sources of the A = 23 peak. They

are 23Mg and 23Ne, with 23Mg having about six times as large

a reaction cross-section. In addition, 23Ne is a noble gas

and thus known to have very poor transport and sticking

efficiency. 23Mg is well-suited for SIEGFRIED; studies by

Kosanke (K073) showed it has good transport and sticking

jproperties. On this basis we have assigned the mass 23

‘peak to 23Mg. For mass 25, the 25Al is the only possibility

‘with a reasonable half-life (7.2 sec) and reaction cross-

section (11.5 mb). '25Na has a suitable half-life (60 sec)

taut very small cross-section (1.5 mb). The mass 25 peak

vvas assigned to 25Al. In the case of the A = 26 peak, we

Ilave two good candidates, 26A1 and 26Si. The 26A1 has a

‘V'ery large cross-section (117 mb) according to the ALICE
'czalculation, while the 26Si has a cross-section of 3.8 mb.

Piowever, the difficulty with the 26A1 species is that its

28

Table III-2. Predicted Cross-sections and Integrated
_ Areas for A1 Peaks.

 

 

 

0
(mb) Area
23Mg A1 68683
25A1 11 6A23
26A1 120 5A897
2’31 28 21139

28Al 3 6605

 

29

ground state (5") has a very long half-life (7.2 x 105y)
ibut its first excited state (0+) has a half-life of 6.A.
sec. Obviously SIEGFRIED will "see" only the decay of the

0+

state. The problem lies in deciding what fraction of
tkie cross-section results in populating the 0+ state. Yet
eaven if only 10% of the cross section leads to the 0+ state,
tflae "effective" cross-section for the 0+ state of 26Al is
stigll roughly A times that of the 26Si. In light of these
considerations, we consider the A = 26 peak to be 26Al
preciominantly, with some contribution from 2581.

The case of A = 27 is fairly straightforward, since
27Ek1. is the only real possibility. It has a reaction cross-
sect::ion at 70-MeV 3He of 28 mb and a half-life of A.l
sec.. The mass-28 peak has been assigned to the 28A1 species.
The strength of this assignment lies in the absence of
reassonable alternatives. The cross-section for producing
28A]. is only 2.6 mb, which is small compared to the cross-
Sectiions from other members of the spectrum. The half-
1if63 for 28A1 is 2.3 min -- much longer than one might
exDfiect to observe strongly with SIEGFRIED'S counting mode
(trtis counting arrangement will be discussed later in this
ChaIDter). However, among the members of A = 28 isobars
thelce are no other reasonable candidates. The 28P has
neEligible cross-section and a half-life (270 msec) far too

Shorw to be feasible for measurement by SIEGFRIED. In View

Of the relative strength of the A = 28 peak, it may be that

30

the ALICE calculation has underestimated Orx for ”Al.
For the small A = 29 peak, the ALICE calculation sug-
gests that nothing should be there -- an obvious under-
estimate. 29P has a half-life of A.l sec and a moderately
larger 8+ energy (A.9A MeV). This is also the result of a

f:.41(3He,n)‘:':P reaction, which, while not highly favorable

with 70-MeV 3He, is the only reasonable means of obtaining

a mass 29 from 27A1 + 3He.

SIEGFRIED Kinet ic 5

During the collection interval we spray activity onto
a collection plate continuously so that the amount of fresh
source accumulated would be linearly proportional to the
collection time if the species were stable. The deposited
activity is radioactive; it has a characteristic half-
life and associated decay constant. Therefore, during the
COllection interval some of the source decays. We will
aSSLune that the kinetic equation describing the change in

the number of species per unit time is

R=-AN+S (3‘3)

3 is the source term, assumed to be constant during the col-

lection interval. The solution to this equation is

N(t) = NO e‘At + § (l-e-At) (3—u)

31

where t is the time measured from the start of the interval
and NO is the number of atoms at t = 0. During the counting

interval there is no source term and we have the simple ex-

ponent ial relation ,

N = N e‘At (3-5)

where N1 is the number of atoms present at the time t1.
In effect, the source term turns on and off at regular inter-
val s, as shown schematically below. This corresponds to a

slightly different kinetic behavior for successive "bins."

 

 

 

Collection Count Collection Count
C l) (l) (2) (2)
dN
33 == -AN+S %% = -AN g% = -AN+S %% = -AN
t
0 t1 t2 1;3 t“

Since we have regular intervals, by taking t0 = C, we have

t1 = iAt. We are interested in the numbers present at the

endpoints of the intervals.

At the start of the first collection interval, N0 = 0,

there are no species present. Defining g(At) = § (l-e’xAt),

We have the following sequences:

32

t = t1 = At N(At) = g(At)
t = t2 = 2At N(2At) = N(At)e'Mt
= g(At)e"Mt
t = t3 = 3At N(3At) = (I\I(2At)e‘)”3t)e‘Mt + g(At)
N(3At) = g(At) (1 + e'2AAt)
t = tu = AAt N(AAt) = N(3At)e'Mt
N(AAt) = g<At> (e‘Mt + e'3AAt)
t = (2j-l)At N((2j-1)At) = g(At) :E: e-k(2xAt)
t: = 2jAt N(2jAt) = N((2j-l) t) e‘Mt

We count the number of disintegrations in each count

int erval ,
D(2jAt) = N<<2I-1)At)(1-e‘*At> <3—6)

The total number of disintegrations counted after M counting

int ervals is D:

M M AAt
DM = z D(2jAt) = z (1-e‘ )N(2<J-1)At)
J=l 3:1
-AAt M 3‘1 -k(2AAt)
DM = g(At)(l-e ) Z 2 e

j=l k=0

33

So we see that the total number of possible disintegra-
tsions counted is a double sum of decreasing exponentials --
2a rather inconvenient form to evaluate, since both sums
gare finite. Normally one might choose some sufficiently
snail number to be the truncation limit and use that trun-
Icated sum. But careful inspection shows that the sum is
euztually a geometric progression, which turns out to reduce

tc> much simpler form: (Gr65)

 

 

 

321 e-kX g -(k-1)x 1-e-JX
k=0 k=l 1-e"x
1M = z (1‘e x ) 1_x { z 1 - z e'Jx}
3:1 1-ep l-e j=l 3:1
A“ M M ’Mx
z 1 = M 2 e-JX = e-X. 2 e-(J-l)x e-x {l-e }
J=1 3:1 j=l 1-e“x
-M
I = l {M e-x(l'e x)}
M -x -x
l-e l-e

So we have succeeded in collapsing a finite double sum

 

3A

 

 

to a simple algegraic form. Taking x = 2At, we have
_ § _ ->.At 2 _ .
DM - A (l e ) IM - s FM(A,At)
- S (l-e'AAt)? e-2AAt(l_e-2AAt)
D - X M

(l-e'2AAt) (i-e"ZAAtI

A much more concise form than the double sum.
Turning to the mechanistic side, we see that the total
number of disintegrations is linearly proportional to the

source term S, which is intuitively reassuring. If we wish

to compare calculations with observed results, a model of

S my st be hypothesized. This is very difficult, since the

source term S, representing the mechanism for adding fresh

sample to the collection plate, is an unknown but certainly

Cornplicated quantity! Realistically, it should be function

015' the relevant reaction cross-section, transport ef-
f1C31ency, transport time, and sticking efficiency. We

can use the ALICE calculations to provide reaction cross-

Sections. In order to take the transport time into account,

we- suppose that the radioactive species with a decay constant
A will decay during the total transport time 1:; thus, a

factor exp(-AT) reduces the source term from the start of

tr‘ansport. Different species might have different transport

35

times, but if we accept a molecular cluster transport mechan-

ism, we would expect I to be species independent, since the

massive clusters should have roughly the same T. To compare

with the observed peak areas, Ai we will assume:

- i _

where e is the efficiency of SIEGFRIED itself, which should

be independent of species. For source term we use the

form

Yi is the transport efficiency for species 1, Ci is the

reaCtion cross-section, I is the beam current on target,

and N is the number of target atoms. We are actually in-

ter‘ested in relative quantities, so we will normalize to a

spec ific case. For the case of the Al spectrum I used the

laI‘gest peak “Mg for the normalization. If we take a1

Ai/A23 and ”i = Y1/123

(A23-A1)T Fmoi)
—- e -———3-

36

From the last form we can estimate the relative trans-
port efficiency ”i from ai, an experimental quantity, and
”’1’ a calculated parameter. For the spectrum of Figure 3-1
we collected for two hours with A-sec counting intervals,
so At = A.0 and M = 900. The results of this calculation
are given in Table III-3. All parameters have been normalized
to the 23Mg peak, the largest in the spectrum. The last
column gives the relative transport efficiency according to
our model. The values for the 25A1, 26A1, and 27Si cases
app ear reasonable in the absence of any corraborative fig-
ures. The value for the 28A1 is excessively high, perhaps
reflecting an unusually low value for the reaction cross-
section (2.6 mb). This dependence on the calculated cross-
SCCtion may also explain the difference in the transport co-
efficients for the 25Al and 26Al cases. Since both are
aluminum isotopes, and so chemically equivalent, their

transport coefficients should be the same.

N__3~F + 3He

In an attempt to observe some light mass TOF spectra
and to obtain a high multiplicity of peaks it was deemed
WC>3:“thwhile to use targets made up of binary compounds. We
wanted to use compounds whose components both seemed likely

'50 result in products that would be suited to measurement

by SIEGFRIED.

37

 

 

 

Hm.H emo.o moo.o owm.o moo.o cmo.fi NMH Ha.“
m:.o mom.o cmc.o Hem.o no.0 m:m.o m.e em.~
mm.o mme.o mm.m omm.o mm.m mem.o e.c Hemw
:m.o mmo.o msm.o mmm.o mm.o Hem.o m.e Hem“
o.H o.H o.H o.H o.H o.H H.NH mzm~
A: He He gmmce Emmefiev :Hfmmco who

AHAVG

 

seesaw 2 .3 seem ones .3 mass .3: .28

38

Our first choice was a NaF pressed powder target. The
excitation functions resulting from ALICE calculations for
23Na + 3He and 19F + 3He are shown in Figures 3-3 and 3-A,
respectively. In Table III-Aa and III-Ab are given the
expected products, their associated half-lives, calculated
cross-sections, B decay energies, and associated recoil
energies for 23Na + 70-MeV 3He and 19F + 70-MeV 3He, respec-
tively. All the cases tabulated possess a number of attrac-
tive characteristics that make them well suited for TOF
measurements by SIEGFRIED. The majority of half-lives are
on the order of tens of seconds. The decay energies are
large enough to result in large recoil energies. In the
cases of 16N and 2°F, the very large decay energies give
rise to huge recoil energies -- A keV and 1.5 keV, respec-
tively. Such recoil energies are quite appreciable fractions
of the usual 6-kV accelerating voltage employed in our
system.

As will be shown in following discussions, the broaden-
ing effects from such large initial kinetic energies would
be very pronounced in the resulting TOF spectrum. This
would provide us with a highly graphic demonstration of
the dependence of TOF peak widths on the initial recoil
energy. Furthermore, the NaF target should provide us with
a real multiplicity of TOF peaks with very little inter-
ference from recoils with the same A. As a result, we

should obtain a light mass spectrum that is easily

39

 

 

 

 

KDC>-
so — "'- "’F
°F
25
Al 23Mq
0(mb) ' 2 INC
./ 3‘" 2°... ..-N
K3--
5 _ 22Mg
I90
I l l J l I
I0 20 30 4O 50 60 70
E3 (MGV)
He

Figure 3-3. ALICE predictions for 23Na + 3He.

A0

 

 

 

IOO -
20F- I50
so —
2'N0
0’ ‘ IGN
(mb) V» \ I9
N
IO —
I6
5 __ F
L I I l L l
20 30 40 so 60 70
E3 (MeV)
He

Figure 3-A. ALICE predictions for 19F + 3He.

 

Table III-Aa.

A1

Properties of Reaction Products from 23Na
+ 70-MeV 3He.

 

 

 

t E Recoil
1/2 0 B Energy
Isotope (3) (mb) (MeV) (eV)
16N 7.1 7 10.u2 A000.
17F 6A.5 A7 1.7A 152.
190 27 u A.82 795.5
20F 11 A2 7.03 1519.3
21Na 3.9 16 2.52 228.2
22Mg 12.1 A 3.77 AAl.0
23Mg 7.2 37 3.03 280.1
Table III-Ab. Properties of Reaction Products from 19F

+ 70-MeV 3He.

 

 

t E Recoil
1/2 a B Energy
Isotope (5) (mb) (MeV) (eV)
150 122 68 1.7A 172.0
1°N 7.1 23 10.u2 u000.
l7F 6A 22 1.7A 152.0
19Ne l7 19 2.22 203.0
2°F 11 7 7.03 1519.3

 

A2

calibrated and contains some interesting peaks.

For this eXperiment we calibrated the system by first
obtaining an 27Al + 70-MeV 3He spectrum from two hours of
counting. After cleaning the collection plate, we col-
lected data on the NaF spectrum for l-l/2 hours with 10A
of beam current on target. Shown in Figure 3-5 is the
result of the l-l/2 h counting from the NaF target. Ob-
viously we didn't need to worry about overlapping or inter-
fering TOF peaks! The single mass peak occurring in the
spectrum occurs exactly where the 23Mg peak occurs in the
Al calibration spectrum.

The results of this run are slightly disconcerting, as
we mentioned that the pertinent characteristics of the
reaction products seem to be almost tailored to SIEGFRIED.
It goes almost without saying that we must have overlooked
something important! In a few cases it might be that the
real cross-section is much smaller than the ALICE predic-
tions but it is very improbable that all but one cross-
section were overestimated. The characteristic that we
did not take into consideration was the transport prOperties
0f the reaction products. Except for the Mg and Na isotopes,
all the reaction products are potentially volatile species
that would have poor transport efficiencies. In studies
using a system very similar to SIEGFRIED, H. Wolnik (W076)
found that elements like the noble gases, Br, and I are

hardly attached to the molecular clusters but still pass

A3

 

 

 

 

8402-—
23Mg
.J
UJ
Z
2
<1:
I
L)
$\
09
I...
Z
I)
c:
C)

 

 

0-J —~Aee -
5500 000

CHANNEL NUMBER

Figure 3-5. TOF spectrum for 70-MeV 3He 0n NaF target.

AA

through the capillary with reasonable efficiency. Never-
theless, at the collector foil it was found that the total
efficiency is very low. Evidently the same phenomenon is
occurring for the F, N, O, and Ne isotopes that we expected

to observe.

KCl Target

The next binary compound used as target material for
a possible TOF measurement was potassium chloride (KCl).
For these experiments pressed powder targets on a thin Al
backing were used. Figure 3-6 gives the calculated excita-
tion functions for the 39K + 3He reaction for the energy
range 20 - 75 MeV. The excitation functions for 35C1 + 3He
are given in Figure 3-7. As can be seen from these plots
we can expect a fair number of suitable reaction products
especially at the higher energies for the incident 3He.
Another consideration favoring higher energies is the
transport efficiencies of the products. The excitation
functions for the 39K + 3He reaction show that at the lower
energies (E3He < 50 MeV) two chloride isotopes account for
the majority of the reaction cross-section. If our ex-
perience with the NaF target taught us anything, poor
transport efficiency should be expected with potentially
volatile species such as chlorides. In view of these
facts, we ran the KCl experiments with a beam energy of

70 MeV. Since the heaviest possible product would have a

A5

 

 

 

 

H30
34%“
50L
6
(mb) \ 29p
IC)- 3|
/ s 28AI
5— ' 33CI
39to
I I I I I I
20 3O 4O 50 60 7O

E3He (MGV)

Figure 3-6. ALICE predictions for 39K + 3He.

A6

IOO -
50 - “A!
275‘
0'
(mb) V
I0 -
”AI
9,:
5 I...
l l l l l l

 

 

E3 . (MGV)

H

Figure 3-7. ALICE predictions for 35C1 + 3He.

time of flight of about 6.8usec we used an 8 usec TAC
range. This is also a convenient range for the Al Cali-
bration Spectrum.

We collected data on the KCl target for approximately
ten hours. The count rate with luA of 70 MeV 3He on target
was decidedly lower than the A1 calibration. Nonetheless
after an hour of counting, a multiplicity of TOF peaks was
observed. Using the Al calibration we determined the lepe,
K and the intercept, t for the relation:

0

TOF(CN) + K . CN + tO (3.3)

This allows one to calculate the time-of-flight as a

function of channel number (CN). From the TOF(CN) we are

able to determine the corresponding mass number from Equation

3.2, with RV = 6 kV. The accumulated spectrum is shown

in Figure 3-8. Using the K and t0 from the Al spectrum

and Equation 3.3 we determine the correct mass numbers.

The fact that masses 26 and 27 occur at precisely the same

position on both the calibration and KCl spectrum streng-

thens these assignments. Finally as a double check, a

linear least squares fit to the assigned TOF's, as a func-

tion of Channel number, was run. The results of this fit

are given in Table III-5. Thexz for the fit was 2.3 x 10‘5.
Assignment of the correct Z corresponding to each peak

in the KCl spectrum is more difficult than in the previous

.pomgmp Hos co omm >ozlo> pom sapwoodm mos .wIm ogswfim

mumZDZ JMZZ<IQ
OOMn OWOm

 

A8

 

- III 0

kav.mv
M? 2V

 

 

'ISNNVHO / SiNOOO

 

00mm

 

 

 

 

_ _ norm

 

A9

 

 

Table III-5. LSQ Fit for KCl Spectrum.
A TOP 762223;? TLSQ A ALsQ

(amu) (nsec) number) (nsec) (nsec) (amu)
26 5.279 5515 5.278 0.001 25.00
27 5.380 5619 5.385 -0.005 27.00
28 5.A78 570A 5.A37 -0.005 28.00
29 5.575 5809 5.581 —0.006 29.00
30 5.671 5892 5.667 -0.006 30.00
31 5.76A 5983 5.671 -0.003 31.00
39 6.A66 6668 6.A67 -0.001 39.00

 

50

two cases analyzed. However, conspicuous by their ab-
sence are masses 33 and 3A which, according to the ALICE
predictions, would correspond to the chloride isotopes
33Cl and 35Cl. This may be due to the poor transport and/or
sticking efficiencies of chlorides. Recalling the results
of the NaF experiment and those of H. Wolnik (W076) it
appears that F, Cl, Br and I isotopes display similar trans-
port prooerties. This chemical equivalence shown by this
set of elements is not too surprising when one notes that
this group makes up the halogen family. As is known from
elementary chemical principles the halogens all act chemi-
cally similar in many reactions.

The mass 39 peak is fairly simple to assign since
39Ca is the only possibility at that mass. The only other
members, 39C1 and 39Ar, have poor transport efficiency,
little or no cross sections and half-lives that are much
too long (60 m and 269 y respectively). 39Cl has a half-
1ife of 0.86 sec and a calculated cross-section of 5 milli-
barns. However, the 39Ca peak is easily the largest single
feature in the spectrum. The collection-counting inter-
vals used for this run were A sec, which may favor such
half-lives but not to the extent that it would make up for
such a relatively small cross-section. It seems that the
ALICE calculation may well be underestimating the actual
cross-section. Another intriguing point is that in the

KCl spectrum it appears that 39Ca has a very good transport

51

efficiency while in the Al and NaF spectrum 23Mg has ex-
cellent transport properties. Noting that both Mg and Ca
belong to the same chemical family -- the chalogens, it is
intriguing to hypothesize a potential correlation between
transport efficiencies and chemical families.

The peak for mass 31 is also easily identified by
elimination. Among the A = 31 isobars 318 is the only
choice, it has a half-life of 2.7 s and a calculated cross-
section of 27 millibars. The only other unstable mass 31
is 3181 with a half-life of 2.6 hr which immediately
eliminates it as a possible contribution.

The two mass peaks corresponding to A = 26 and 30
represent a bit more difficult task to assign. For the
mass 26 we have already mentioned in the section dealing
with the standard Al target that this mass may arise from
either 26mA1 and/or 2581. The assignment of 26Si is tenta-
tive at this point although evidence presented in the next
chapter seems to lend weight to the assignment. The poten-
tial contribution from 26Al cannot be determined at this
point. A similar problem exists for the mass 30 peak.

The possible candidates for this mass are 3°S or 3°P.

The 3°P has a half-life of 3 m while the 3°S has a half-
life of l.A s which is more favored by our short-time count-
ing arrangement. On this basis the mass 30 is tentatively
assigned to 3°S. 29P has been assigned to the A = 29. It

has a calculated cross-section of 28 millibarns and a

52

half-life of A.A s. The only other A = 29 isobar that is
unstable is 29Al which has a 6.6 m half-life and a very
small reaction cross-section. 29P is a 1/2+ + 1/2+
(ground state to ground state) transition by positron emis-
sion.

As in the 27Al + 3He TOF spectra, we find that 28Al
is the only reasonable contribution for the A = 28 peak.
The 28Al has a fairly large cross section (0 = 50 mb) from
the 35C1 + 3He (70 MeV) reaction and a small contribution
(0 = 6 mb) from the 39K + 3He. Again it should be pointed
out that 28Al differs from the majority of species observed
in that it decays by 3’ decay rather than positron emission.

25 is due to the

The small peak at the position of A
decay of 25A1. This isotope has a half-life of 7.2 s and
a calculated reaction cross-section of 10 millibarns from
the 35C1 + 3He reaction at 70 MeV. Comparison of the cross-
section and the half-life with other members of the spectrum
would lead one to expect this peak to be much larger. The
fault cannot lie with transport efficiency since the 28Al
seems to be transported very well. It may be that the cal-
culated cross-section is too large. In studies of I‘°Ar
on 16°Dy, 1°“Dy and 17"Yb, Y. LeBeyec gt _1. (LeB76) has
found systematic discrepancies in comparisons of ALICE

predictions and experimental results. The origin of these

discrepancies is not well understood.

53

“5T1 Target

Towards the end of the experiment using the KCl target
the count rate drOpped gradually but continuously until
there was no rate at all! After checking the obvious things
such as the cyclotron, we opened the target chamber. A
large quantity of yellowish-white powder was found to coat
much of the thermalizer chamber. There was enough of an
accumulation on the walls of the chamber to plug up the
capillary tube. This served to block the transport of re-
coil products and so the count rate drop-off occurred. The
origin of this powder is not known.

For our next experiment we decided to use a natural Ti
foil target as a possible means of avoiding the aforemen-
tioned problem with the powder generated in the thermalizer
chamber. Although the “eTi isotope accounts for part of
the target material ALICE calculations indicate that this
isotope would not give rise to any species that would be
measurable by SIEGFRIED. This is also true for the HTi
and “9Ti isotopes. Shown in Figure 3-9, are the calculated
cross-sections for the reaction products for “5T1 + 3He.

In Table III-6 are given the associated half-lives and
cross-sections for production with a 70-MeV 3He beam. The
“5T1, with a half-life of 3 hr, is included because the
other mass A5 predicted is l’5‘V which is so far unknown.

If we see a peak corresponding to mass A5 in our TOF

 

5A

 

500-.
47V
45Ti
I00-
50-
425c
0'
(mb)
46
I0— V
5- 45
' 43Ti
I I L I l 1
IO 20 30 4o 50 60 70
E3 (MeV)
He

Figure 3-9. ALICE

predictions for “sTi + 3He.

55

Table III-6. Properties of Products of “6T1 + 70-MeV

 

 

3He.

t 0
1/2 (mb)

“25c 0.7 s 50
“3Ti 0.6 s 1.1

“5T1 3.0 h 110.
“5V Unknown 5.5
“5V O.A s 3.0

“7v 33 m 0.0

 

56

spectrum it must be fairly strong if we want to argue that
we might be making the unknown “5V isotope. A weak peak

in the TOF spectrum would be attributed to the l’sTi isotope
since it has a large production cross-section. The reaction
of l”Ti + 3He gives rise to a few products similar to those
from the “GTi isotope. The calculated cross-sections are
shown in Figure 3-10.

For this experiment we again ran an 27Al target for
calibration and used a Tennelec Time Calibration for a
double check. We then collected statistics on the Ti tar-
get for A hours with 1/2 0A of 70—MeV 3He beam on target.
The TAC range used was 8 nsec. The results of the collection
are shown in Figure 3-11. The first, large peak corresponds
to mass A2. The ALICE predictions show I”Sc to have a very
large production cross-section. Besides the ground state
decay (tl/2 = 0.7 sec) “280 has a metastable state at 0.617
MeV that positron decays with a half-life of 62 sec. On
the shoulder of the “280 peak is a peak corresponding to
A = A3. The ALICE predictions indicate that “3Ti is the
only A = A3 isobar with any cross-section that has a half-
life of less than a few hours. l”Sc, with a half-life of A
hr, has a large cross-section (90 - 100 mb) but seems un-
likely to contribute in view of the relatively long half-
life. Tentatively we identify this peak at “3T1.

At longer TOF, just beyond the “28c and “3T1 peaks is

a broad distribution of TOF events. This feature arises

57

 

H30

50“-

(r36)

l0—

 

 

 

20I 3&3

 

Figure 3-10. ALICE predictions for “7T1 + 3He.

58

 

 

 

  

 

 

ZCHD
.J
UJ
2:
2:
it:
U 4250
‘\
00
’—
2:
:3
O 43
C) 'Ti
1
A.
45+46
I i a
l

   

 

52 . <52 '12
TOF (nsec)

Figure 3-11. TOF spectrum for products of 70-MeV 3He on
Ti target.

59

from a combination of two masses, A5 and A6. The mass AA
should occur in the valley between the “3Ti shoulder and
this broad distribution. A mass A7 recoil would occur at
7.1 s which is slightly above the mass conglomerate. Since
the two masses peaks are so poorly resolved we have no
information on the strengths of the individual peaks but
there are two mass peaks in the wide distribution. “6V

is the most probable source of the mass A6 contribution to
the broad peak. Among the A = A6 isobars it is the only
one with a suitable half-life and ALICE predictions indicate
that it has a 3 mb production cross-section. As for the

A = A5 contribution there are no convincing prospects.
According to the ALICE calculations “5Ti should be produced
in relatively large amounts. However, the 3 hr half-life
of this isotope is very unfavorable. The other potential
candidate, “5V has a cross-section of roughly 6 mb but the
half-life of this isotope is unknown. “5V has so far only

been observed in charged-particle measurements (Mu76).

Nickel + 3He

A natural nickel foil was used as the target for our
next experiment. This represents a step of about ten
to twelve amu from the titanium foil target previously
employed. This allows us to investigate the performance
of our system in a slightly heavier mass region. The

isotopic composition of the target is as follows:

60

58Ni (68%), 6°Ni (20%), 61Ni (1.25%), 62Ni (8.7%) and
°“Ni (1.1%). ALICE calculations were performed for all
the constituents. The main contribution to the reaction
cross-section was from the 58Ni isotope. The results of
the ALICE calculations for 58Ni + 3He are shown in Figure
3-12. Note that the 55Ni isotope has a reasonable pro-
duction cross-section between 35 and 60 MeV. This particu-
lar isotope is as yet poorly known but the ALICE calculations
predict a 7 mb production cross-section at A5 MeV. Also
at this energy we might expect a fairly strong 5"Co peak
to appear in our TOF spectrum.

For our first run with the Ni target, a A5-MeV 3He
beam was used. Since the recoil mass products were in the
range of 50-60 amu, the TAC was run on the 10 nsec range.
For the counting arrangement it was decided to collect
activity and then count for twenty-second intervals since
the predicted products have half-lives in the tens of
seconds to minutes range. We collected statistics for
approximately four hours. This was interrupted by frequent
problems with the cyclotron. For the start detector we used
a NaI(Tl) detector, which for annihilation radiation is
as efficient as the plastic scintillator. Shown in
Figure 3-13 is the TOF spectrum resulting from the
58Ni + A5 MeV 3He run. As was expected, a large peak
occurs for Sl+Co. At slightly earlier TOF's there appears

to be a triplet of peaks centered on A = 53. We have

61

 

 

 

500
”Fe
ICC—-
50 — ”Cu
0'
(mb) “Co
IO-
53
“Cu 55 . CO
NI
I l l I l
20 4O 5O 60 70
E3 (MeV)
He
Figure 3-12.

ALICE predictions for 58Ni + 3He.

 

62

 

 

 

 

 

 

200
..I
m
E 54
4 Co
I
u
\
m
'2 ”Fe
3 ”Ni
0
0
sec”
0 I I l i I I
/6.B 72 7.6 8.0 8.4 8.8
TOF(useC)

Figure 3-13. TOF spectrum for products of A5-MeV 3He on
Ni target.

63

tentatively assigned this group to 53Fe which decays to
states in 53Mn which immediately y-decay to the ground
state. The effect of the y-decay on the recoil TOF may be
reflected in this splitting. Just beyond the 5“Co there
is a fairly large peak at the position of A = 55. We have
tentatively identified this as 55Ni. However, this cannot
be taken as an unequivocal assignment, fromcnu'experience
with the ALICE calculations it would seem rash to place
too much confidence in the predictions. The 55Cr isotope
has a very small production cross-section according to cal-
culations but it has a reasonable half-life (3.5 m) and
may be a possible source of the mass 55 peak. The assign-
ment of 55Ni to this peak should be taken as tentative at
this point.

The next feature in the TOF spectrum of Figure 3-13
is a smaller, broad peak whose centroid corresponds to mass
57.5. The peak probably has two components: A = 57 and
58. However it is not really possible to resolve this peak.
From the ALICE calculations we would expect some produc-
tion of the 58Cu isotope which has a very large decay
energy, (QEC = 8.57 MeV) and positron decays to various
levels in 58Ni with a half-life of 3.2 s. The large decay
energy results in a maximum recoil energy of 600 eV, which
is 10% of the 6 kV acceleration voltage and may explain
the distinct broadening seen in this peak. If there is

an A = 57 component to this peak it is very difficult to

6A

rationalize. The only possibility amoung the A = 57 iso-
bars is 57Mn but there is no reasonable reaction of
58Ni + 3He to produce 57Mn at these energies.

After a day of running we were forced to stop due to
difficulties with the cyclotron. When we came on-line again
it was decided to run at 70-MeV He on the same Ni target.
The TOF spectrum shown in Figure 3-lA is the result of ap-
proximately 8 hrs of counting. As expected from the ALICE
predictions the mass 53 peak is now the largest feature due
to the appreciable increase in cross-section for 53Fe and
the onset of production of 53Co. The 5"C0 peak is cor-
respondingly reduced as suggested by calculations. At
thepositioncfi'A.= 55 there is a small peak on the shoulder
of the 5"Co peak. On the basis of the previous run with
the Ni target we assign ssNi to this weak peak but again
urge caution in this assignment. Rather unexpected is
the smaller, partially resolved doublet at the position of
A = 58 and 59. The ALICE calculation for 58Ni + 3He
indicate both 58Cu and 59Cu to have appreciable cross-
section at much lower energies. However, 62Ni, (A% of
natural) may be a potential source for these species.

In the absence of any other reasonable species we tenta-

tively assign 58Cu and 59Cu to this doublet.

65

 

102-—

COUNTS/CHANNEL

 

0 MINI LII ‘
000

Figure 3-1A.

 

 

I l 1 I ,
.l‘ I I ‘II
1500

CHANNEL NUMBER

 

TOF spectrum for products of 70-MeV 3He on
Ni target.

66

Al y-Mass Coincidence

The use of a y ray detector for the start signal of a
TOF event is an attractive alternative to the plastic
scintillator used heretofore. As is well-known, y transi-
tions are discrete as compared to the continuum resulting
from 8 transitions. It is much easier to determine the
energy of a y transition than it is to measure the end-
point energy of a B decay. As a result it is more useful
to label a mass peak with a given y ray energy.

In order to demonstrate the feasibility of such experi-
ments, we decided to use a NaI (Tl) scintillator as the
start detector for an 27A1 + 70 MeV 3He experiment. As
discussed earlier, this particular reaction and the result-
ing TOF spectrum are familiar and well understood from many
earlier experiments. Since the majority of decay products
we see in the spectrum result from positron emission, an-
nihilation radiation will provide effective start signals
for any ground state to ground state transitions.

The 27Al target was the same one used for the experi-
ments described in Section I, as were the beam energy (70
MeV) and TAC scale (8 nsec). The amount of beam on tar-
get was roughly 0.5 uA and data were collected for 2.5
hrs. The resulting TOF spectrum is shown in Figure 3-15.
The two most striking features of this spectrum are the

background and the peak distribution as compared to the

67

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68

same spectrum using a plastic scintillator for the start
detector. This is especially noticeable in the case of the
23Mg peak which in all spectra resulting from plastic
scintillator start signals is easily the largest peak in
the set by a factor of two. In the y-mass spectrum of
Figure 3-15, this is obviously not the case. Actually the
mass 26 peak is now the larger peak. This is not well
understood at this point. One would expect the only dif-
ferences in distributions to occur as a result of large
differences in positron energies. The NaI(Tl) scintillator
will detect positrons by their annihilation radiation which
occurs once the positrons have been stopped and the scin-
tillator should be more effective stopping lower energy
positrons. Accordingly we would expect NaI(T1) scintillator
to be more sensitive to lower energy positron. However,

in the case of the 23Mg and 26Al, thepositron energies are
about the same. If the 26Si is the main component its
positron energy is 0.8 MeV greater than the 23Mg which is
in conflict with the efficiency argument.

The highest background observed is probably due to the
large number of random start signals resulting from brems-
strahlung and Compton scattering. For a 3 MeV electron
passing through stainless steel, energy losses due to
bremsstrahlung is about 11% of the loss due to ionization
(8166). Since there are always some residual gas mole-

cules in the TOF chamber striking the CEMA's and the

69

NaI(Tl) scintillator is very efficient for low energy radia-
tion such as bremsstrahlung the higher background is ex-
pected.

Figure 3-l6a is the NaI(T1) spectrum associated with the
TOF spectrum of Figure 3-15. Figure 3-l6b is the spectrum
after a 3-channel smoothing was performed to accentuate
details of the spectrum. Since the majority of decays are
positron emitting ground state to ground state transitions
the very large 511 keV peak is an expected feature. The
large "hump" preceding the 511 keV peak is due to a combina-
tion of Compton scattering events and bremsstrahlung. The
small peak at 1780 keV is due to the y-transition following

the decay of 28Al.

l'olfism + 3He

As an example of a heavier mass TOF spectrum, we decided
to use a 1““Sm203 target. This was available in the
separated isotope form (99%). The ALICE calculations
for the products of 1““Sm + 3He are given in Figure 3-17.
The predicted cross-sections are much larger than any of
the previous experiments discussed here. Since we expect
recoils in the region of 150 amu, the minimum TAC range,
with 6 kV acceleration, is 13 nsec. To be safe we used
a 20 nsec TAC range. One difficulty with such a long TAC
range is that we have cut down on our resolution -- there

are more nsec per channel in this case. Also working

7O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) Smoothed Spectrum
5H keV
4
LIJ
Z
Z
q
I
U
\
U)
1...
Z
3
o .
v .I
I
I
I780 keV
o ‘u i‘ -———A—- ;A“-L AL ._
O 500 1000
CHANNEL NUMBER
2'102
NHwV
(0) Raw Data
4
LL]
2
Z
<
I
U
\
U)
*—
2
D
O
U
I780 keV
0 - .
500 I000

Figure 3—16.

 

CHANNEL NUMBER

NaI y-spectrum in coincidence with spectrum

of Figure 3-15.

71

 

ICMDO
500 t- I38Pm
I'Eu
Iaesm
RIO-

 

(mb) so —-.44
Eu

 

IOP

 

 

 

 

 

I I I I L I
20 30 40 50 60
E3He (MEV)

Figure 3-17. ALICE predictions for 3He + 1"“Sm.

I42

 

72

against us in the heavier mass region is the fact that the
differences in TOF's for successive masses are decreasing

so the peaks will tend to bunch together, making it dif-
ficult to resolve adjacent mass peaks. Normally one way

to alleviate this problem is to use a very precise delay

for the start signal for the TAC. This allows one to use

a shorter TAC range and thereby expand the region of
interest. This effectively increases the resolution sub—
stantially. In light of these considerations the Precision
Digital Delay Model 7030 was bought from Berkeley Nucleonics
Corporation. This module is essentially a sophisticated,
highly stable delay line. Unfortunately, the reliability of
the box is significantly lower than its other qualities.

It was malfunctioning before this run and was not used.

It was deemed worthwhile to do the experiment for a number
of reasons. First, in a number of runs the coincidence
rates were very low making complex experiments of interest
impossible. To have a rough idea of the collection rate
would be very helpful for planning further experiments.

Also of interest was how serious is the problem of mass

peak punching; possibly the adjacent peaks "ride" on one
another's tails yet are still resolvable. Lastly, in View
of the importance of the transport and sticking efficiencies
as established earlier it seemed worthwhile to find out if
we would "see" anything in reasonable quantities.

Shown in Figure 3—18 is the TOF spectrum resulting

73

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from two hours of counting of the products from the

1HSm + 3He. At the long TOF end of the spectrum are two
large, very broad peaks. The first of these two peaks
occurs at the position of A = 1A1 but, as expected it is
broad enough to encompass masses 138-1AA. Obviously it is
impossible to resolve meaningfully the individual mass com-
ponents of this peak. The only thing one can surmise from
this feature of the spectrum is that we have measured the
TOF's of a number of species with masses between A = 138
and 1AA. The second large peak shown in the spectrum seems
to allow some resolving of the individual components. The
masses most prominent in the peak correspond to 155, 158
and 159. Yet how can such masses result from a compound
nucleus of maximum mass of 1A7? A number of experimenters
(Ne78), (N879), (W077) have observed very similar phenomena
in their TOF spectra and attribute the heavy mass peaks to
molecular species that are somehow volatilized from the col-
lector surface. The fact that the second large mass peak
corresponds to the earlier masses + 16 amu, which is the
mass of 160, suggests that the components of this peak are
the same as the lighter masses except there is an attached
oxygen (A = 16).

In the remainder of this spectrum can be found a large
number of small TOF peaks that are not explained as simply
as the peak centered around A = 158. They may correspond to
radicals (CnHmOk’ etc.) that have been ionized by collision

with the primary recoils.

CHAPTER IV

RECOIL ENERGY FITTING

A close inspection of the mass time-of-flight peaks
acquired with SIEGFRIED reveals a number of phenomena
that are not exactly what one would expect for simple
measurement of an ion flight time. Referring to Figures
A-l, 3-1, and 3-5, it is easy to notice that the peaks are
all rather wide -- much wider than any electronic timing
contributions might account for, as will be shown. Per-
haps more striking to the eye is the very peculiar shapes
exhibited in the spectra; all have long nearly exponential
tails on the low time side, implying some mechanism that
suppresses higher energy events and favors longer TOF
events.

It seems reasonable to View the source of the recoils
as a thin film deposited on the collecting plate, since
massive clusters provide the means of transport of the
product nuclei from the target area. A simple picture of
recoils passing through a surface layer of cluster deposit
seems to favor a higher probability of escape for high

energy recoils than for low energy recoils. However, higher

75

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77

energy recoils should transform into shorter TOF events,

and this is not what one observes in the accumulated TOF
spectra; in fact, the converse seems to be true. That is,
the low energy events seem to be the more probable events

to be detected. The entire puzzle is especially intriguing,
since in the early days of development of SIEGFRIED and its
forerunners it was not known for sure whether recoils from

B decays would be energetic enough to escape the surface on
which they were deposited (Ed76), (Ma7A). It appears,
however, that not only are the recoils energetic enough to
free themselves, but also the recoil energy is a significant
contribution to the peak broadening.

Just how important are broadening effects such as
electronic timing contributions in our measurements is a
reasonable concern that we will deal with briefly here.

A very rough argument for nanosecond range precision can
be made by consideration of the performance specifications
for the various NIM modules used for the TOF measurements.
This is not quite so convincing as an internal measure or
upper bound on intrinsic broadening. Happily, nature sees
fit to provide us with just such a means to estimate an
order of magnitude figure of merit. Referring to Figure
A-l, it is found that a narrow peak occurs at exactly the
position predicted for an H+ ion. Obviously, a proton is
the result of a neutron undergoing a B decay and it seems

a bit far-fetched to hypothesize neutrons being transported

78 .

and deposited on the collection plate; however, the cluster
molecules are somehow formed from saturated hydrocarbons
(e.g., CGHG), so there is certainly an abundant source of
hydrogen atoms at the collection site. The exact mechanism
for how the H+ ions result from decay-related events is
certainly not clear. However, the recoil energies involved
in a "typical" 8 decay of medium-light nuclei are on the
order of a few hundred electron volts. When one compares
such recoil energies with typical carbon-hydrogen bond
energies, which are roughtly 5 electron volts, the dis-
sociation and ionization of a hydrogen atom from a recoil-
bearing cluster is a plausible occurrence. Furthermore, in
a number of spectra taken by a group at Orsay on a TOF
system much like SIEGFRIED (Be78) a large, prominent spike
is also assigned to the H+ ion.

Accepting this assignment we can use the full width
at half maximum (FWHM) to obtain an estimate of the intrinsic
broadening effects due to the electronics. As shown, the
H+ ion has a FWHM of 8 nsec which represents a real limit
on the peak broadening. In comparison, the mass groups at
longer times of flight have FWHM's of about A-5 times that
of the H+ peak. As we intend to show, the peak widths
result from the spread in recoil energy that the daughter
nucleus has after the B decay.

Another aspect of the TOF Spectra that is especially

intriguing is the overall shape of the peaks. This is

79

most easily seen in the large, isolated peaks correspond-
ing to 39Ca in Figure A-1 and 23Mg in Figure 3-1 and 3-5.
All exhibit a long, exponential-type tailing on the short
TOF side with a very sharp drop-off on the higher channel
side. This distinctive feature seems to just elicit further
investigation.

As a first step in the analysis, we decided to attempt
to fit the TOF spectra with the program SAMPO, since it is
a fairly flexible and easily run data analysis routine.
SAMPO is a Fortran routine written and developed by J. T.
Routti and S. G. Prussin (R069) to perform automatic data-
analysis of y-ray spectra. The routine is familiar to
spectroscopists involved in Ge(Li) work but is probably
rather foreign to mass spectroscopists in general. In
order to elucidate our fitting results, a brief description
of the analysis routine and the associated terminology will
be given here.

In SAMPO the peak shape is approximated by a function
that is basically a Gaussian with possible high— and/or
low-channel tailing that "goes" as exponentials.- The
amount of tailing is determined by the distance in channel
number from the centroid to the points at which the Gaus-
sian is joined to the appropriate exponential. At these
junctions the function and its first derivatives are con-
tinuous; as a result, the complete peak shape, aside from

the normalization, is specified by three parameters:

80

(1) CW, the width of the Gaussian, (2) CL, the slope param-
eter of the low-channel exponential tail and (3) CH, the
parameter of the high-channel tail. The signifiCance of
the magnitudes of these shape parameters is very straight-
forward: (1) a large CW means a wide Gaussign contribu-
tion, (2) a large CL yields a fast rise on the low-channel
side, and (3) a large CH gives a rapid fall-off of the
peak on the high-channel side.

Shown in Table IV-l are the results of a SAMPO fit for
the KCl spectrum of Figure A-l. As indicated by the
X2 for each, the fitting routine has proved to be well
suited to deal with the rather asymmetric shapes. The
CW's for all the identified species except the 28Al case
are very large. (CW as was discussed earlier, is the width
parameter for the Gaussian contribution to the peak shape.)
The magnitudes of the CW's shows a significant broadening
effect in the TOF peaks. The low-channel tail parameter
(CL) for the peaks are generally small and reflect a long,
slow drop-off on the shorter TOF side. Especially striking
is the size of CH for masses 26 through 39, excluding the
28A1. All the CH's for this set are greater than 5; in
fact, all except the 27Si case are greater than 10!
This indicates that the exponential fall-off on the high-
channel side of the peak is very, very rapid. In fact,
the large size of the CH's implies some type of cutoff --

a limit on the maximum time-of-flight allowed. If we

81

 

 

 

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82

equate the time of flight with the total energy of the
recoil ion, as shown in Appendix I, the TOF is inversely
proportional to the kinetic energy of the ion. In light

of this, it is easily seen that the maximum TOF corresponds
to a minimum allowed kinetic energy, that acquired by ac-
celeration across the gap region.

To sum up the results of the fitting briefly, the peak
shapes predicted by SAMPO will display a long tailing for
the shorter TOF's, a wide Gaussian portion and a very fast
fall-off on the high time side. A glance at the KCl
spectrum readily exhibits this predicted behavior. Ob—
viously there must be a reason for these asymmetric peak
shapes, although the causes may not be immediately pin-
pointed. Nonetheless, the simplest hypothesis concerning
the TOF peaks seems to be that the initial recoil energy
of the daughter nucleus is somehow related to the observed
spectral shapes. If this is so, then we can develop a
quantitative relationship between the observed peak shape
parameters and the recoil energy of the daughter nuclei.

Obviously, the broadening of the mass peaks is not an
electronic effect. Is it reasonable to attribute the effect
to the initial recoil energy of the daughter nuclei? As
is shown in Appendix I, using a gross model of the SIEG-
FRIED system, we can estimate the time-of-flight of a
singly charged ion of mass number A and initial recoil

energy R by the expression:

83

 

TOF = ml/2 {[2(E:;?]l/2 + 11%18 [/EIE _ Rl/21} (A-l)

Rigorously R = l/2MV3 where V2 is the initial velocity
in the z direction, that is, in the direction of the
flight tube axis. In Table IV-2areethe results of times
of flight calculated for the mass set observed in the KCl
spectrum. R is the maximum initial recoil energy, T(R)
the corresponding TOF, T(O) the TOF for zero initial recoil,
and AT the difference between T(O) and T(R). The TOF dif-
ferences are all in the neighborhood of a few hundred nano-
.seconds. Considering our timing resolution to be of the
order of nanoseconds, it is imminently plausible that the
recoil energy effect is a significant contribution to the
broadening.

We have argued that the peak shape, especially the
broadening is because of the initial recoil energy of the
measured mass species; if that is so, then it seems reason-
able to attempt to develop a quantitative relationship
between some measured peak shape parameters and the recoil
energy of the ion. First, we need to have some rough form
-of the parameterization. If we consider the ions drifting
along the flight tube, with negligible time of acceleration

(correct to ~3%, see Appendix I) then we can write:

8A

Table IV-2. Calculated TOF's for KCl Spectrum (in nsec).

 

A R TOF(R) TOF(O) AT AT/T

 

mean
26 381 5055 52AA 189 .036
27 362 5160 53AA 18A .03A
28 213 5323 5AA2 119 .022
29 359 53A9 5539 190 .031I
3O A3A 5AO6 5633 227 .OAO
31 A09 5507 5726 219 .038

39

A9A

613A

6A23

289

.OA5

 

85

E = l/2mv2 = l/2m —— (A-2)

E is the ion kinetic energy, v its velocity, m the mass,
A, the flight length and t the time of flight.

Now suppose an ion starts with zero recoil energy,
then B = qV 5 E0 = 1/2mvg, where V is the electrostatic
potential on the collection plate and q is the charge on
the recoil ion. If the ion starts out with the maximum
recoil energy, with zero transverse momentum (pg/2m = R,

R the maximum recoil energy), then E = EO + R = l/2mv2

2 l/2m22/ti.

So we have

_ 2 2
EO - 1/2m2 /t0 (“-3)
= 2 2
El l/2m2 /tl (A-A)
81-80 = R = 1/2m22(;1§ - 52-) (II-5)
l o

If we make the assumption that the measured TOF is linear
in channel number and the spread is proportional to the
FWHM (or equivalently to CW from SAMPO fitting), we can

write:

86

t0 = K(CP + CW) (A-6)
t1 = K(CP - CW) (A—7)
t: = K2(CP2 + 2CP cw + CW2) (u-8)
= K2CP2(1 + 2CW/CP + CW2/CP2) (u-g)
Similarly,
ti = K2CP2(1 - 2CW/CP + CW2/CP2) (A-lO)

Since typical values of CW/CP are on the order of 10-3,
we can drop the (CW/GP)2 terms.

Using the binomial theorem and retaining terms to first

order, we have

 

 

1 = l (l + 2CW/CP) (A-ll)
:5 K2CP2
l
.33 z _§l—§ (1 — 2CW/CP) (A-l2)
t K CP
0
2 2
m 1 1 ml 1 ACW
R = -— E - ——l = ( ) (A-13)
2 g?" t2 2 K201,2 CP
0
R = 0A EHL (A-lA)

87

This last expression is the form that will be used to re-
late the recoil energy to the TOF peaks parameters CW
and CP.

We now have a parametric form to test. Next we need
some experimental cases to apply it to. Since we are
really working in a new area, it is obvious that there
are certain criteria for the spectra we will attempt to
fit. We require fairly high statistics in each peak and
that the decays be fairly simple; also we prefer as high
a multiplicity of peaks in the spectrum as is obtainable.
The reasons for these conditions are fairly obvious; we
are in the position of hypothesizing a relationship that we
believe can be expressed by a simple functional relation-
ship. Now, in order to prove or, at least further, this
view, we must start with the most reasonable set of tests
available. The KCl and Al spectra come closest to ful-
filling our rather loose conditions. The Al target has
been the standard for our runs and is well understood,
while the KCl spectrum has a fair multiplicity of single
species, peaks, good statistics and the "characteristic"
shapes of TOF peaks occurring in a number of instances.

The actual form of the equation used for the fitting

was

y1 = ax + b (A-15)

'88

xi = CW/(CP)3 - A (CP in kilochannels) (A-16)

y1 = Recoil energy of species i

A linear least squares fit was performed on three

known cases, and the resulting parameters were used to

predict the recoil energy of the members of the set not

included in the least squares fitting. The predicted

values were then compared with the known values.

The maximum recoil energy of a daughter nucleus with

mass number A was calculated using the following equation:

where

E

A

R

R = 5&1 (EO + 1.022)) (A-17)

is the maximum kinetic energy of the 8
particle in MeV

is the mass of the recoil in amu

is the maximum energy of the recoil in eV

(For the derivation see Appendix II.)

The first example of the fitting is for the KCl spectrum.

Our procedure was to choose two sets of three peaks as the

"known" energies and predict energies for the remaining

peaks in the spectra. The first set chosen for the

89

calibration corresponds to 26Si, 2781 and 3OS. In Table
lV-3 are the results of a linear least squares fit to the
recoil and decay energies for this set. The predicted
decay energies are calculated from the fitted recoil

energies according to the formula

 

.. ‘/ £5- -
EO - 0.261 + 537 0.511 (A 18)
(with R in eV and E0 in MeV)

The fit for this set is fairly good, X2 for the standards
is 1.1, and the largest deviation is A%. The results of
applying the equation R1 = axi + b to the remaining peaks
in the spectrum are given in Table IV-3: the errors in
the recoil energy are all in the neighborhood of 15%,

the fit to the decay energies closer to 10%. The worst
fit is for the 29P case, which is not well separated from
the tail of the 3°S.

The mass-28 peak actually gives a somewhat better fit
than was expected, considering the decay scheme of 28A1,
shown in Figure A-2. The B decay of 28Al is to the 2+
excited state of 28Si which decays via a y ray to the 0+
state. Also the decay of the 28Al is by negatron (B-)
decay whereas the rest of the decays we observe are by

positron decays. Since the collector plate is lO-mil

Table IV-3.

9O

Fit to KCl Spectra Recoil Energies (no F-GT

 

 

 

 

 

 

 

Correction).

Calibrated

R yfit EO Efit
A (eV) (eV) % Error (MeV) (MeV) % Error
26 380. A97. A.5 3.81 3.90 2.A
27 362. 355. 1.8 3.79 3.7A 1.3
30 A3A. A2A. 2.A A.AA A.38 1.3

= 1.

Predicted

R yflt E0 Efit
A (eV) (eV) % Error (MeV) (MeV) % Error
28 273. 302. 11. 2.86 3.08 7.7
29 359. A28. 19. 3.92 A.32 10.
31 A10. A73. 15. A.37 A.7A 8.5
39 A9A. A23. 1A. 5.5 5.05 8.2

 

91

2.3! min

   
   

ZBAI

 

 

Op = 4.635
7
W
ZBSI

Figure A-2. Decay of 28Al.

92

stainless steel, it will stop 8's of an energy of less
than about 1 MeV, which means that the lower energy events
are discriminated against; however, the plastic (NE-102)
has roughly a 30% efficiency for detection of y rays of
energy 1.8 MeV, so that if the y ray is emitted in the
"right" direction, it has some probability for serving as
a valid start signal and can result in a detectable TOF
event. The estimation of the y detection efficiency was
accomplished by using a nomogram developed by Roulstan

and Naqui (R057). In the case of the positron-emitters,
even if the low-energy B's are stopped by the collector
plate, the accompanying annihilation radiation is suitable
for serving as a start event. Using the same procedure
mentioned earlier, the gamma efficiency for 511-keV annihila-
tion radiation is about 50%.

The second case we attempted to fit with our recoil
energy parameterization was the TOF spectrum resulting from
70-MeV 3He on a pure aluminum target. Unfortunately there
are only four peaks with reasonable statistics and shape.
The 28A1 peak is very broad and shaped distinctly dif-
ferently from the common TOF peaks, and the 29P peak has
very poor statistics. The peaks chosen for the "known"
energies are the three corresponding to masses 23, 25,
and 27, respectively; the results of these fits are given
in Table IV-A. The x2 for this fit was 7.A, and,-as shown,

the fit errors are in the neighborhood of a few percent.

93

 

 

Table IV-A. Fit for Al Spectrum -- No F-GT Correction.
R fit EO Efit

A (eV) (eV) % Error (MeV) (MeV) % Error

23 287. 286. 0.3 3.0 3.03 1.0

25 299. 331. 11.0 3.26 3.A5 6.0

27 362. 325. 10.0 3.79 3.56 6.0

 

9A

Fermi and Gamow-Teller Distributions

One assumption that is implicit in the parametric treat-
ment of the recoil energy is that all allowable energies
are equally probable and all the various decays have the
same initial energy distributions. This is not actually
true, for, as is known from B-decay theory, the distribu-
tion function for the recoil energy is dependent upon the
"type" of decay the parent nucleus undergoes. I will not
endeavor to present a full treatment of B-decay theory
here, there being a large number of well-known and more
appropriate texts for such a task. However, I will give
a somewhat pedestrian and brief discussion of general
results and terminology from what is called allowed B-decay
theory.

In the B-decay of a nucleus, an electron and an anti-
neutrino are emitted, and each of these particles has a
spin l/2. If the total spin of the decaying nucleus is
unchanged, then, in order to conserve total angular momentum
the electron and antineutrino must have antiparallel spins.
This type of decay is often called a "Fermi" decay. If
the total spin of the parent changes by l (in units of h),
then the electron and antineutrino must carry off one unit
of angular momentum, ;;g;, they have parallel spins. This
type of decay is known as "Gamow-Teller" decay. Common

terminology in B-decay theory attributes Fermi decay solely

95

to a "vector" interaction and Gamow Teller to an "axial-
vector interaction." Because these two interactions are
"effectively" different, the probability that a recoiling
nucleus has a given energy is dependent upon the type of I
interaction that gives rise to the decay.

In Figure A-3 are shown the distribution functions for
the recoil energies corresponding to the two types of decay
considered here. The distribution functions for both cases
are skewed towards higher energies, but the one correspond-
ing to the so-called Fermi decay is obviously more sharply
peaked, favoring higher-energy recoils. When one takes into
consideration these two distributions, it becomes reason-
able to ask whether the effects of the initial energy
distribution are in some way contained in the observed peak
shapes. If that is the case, then, even if there is only
a gross relationship, it is very useful and highly intriguing.
Assuming the initial energy distribution effects are some-
how preserved, we need to quantify the influence of the
distribution upon the TOF peak shapes. The majority of
decays observed are actually mixed, that is, they result
from a combination of the Fermi and Gamow-Teller decay
modes. In an attempt to take this account, we have included
as a multiplicative fixed parameter the quantity, D
5 (fF + «fGT) where fF and fGT are the fractions of transi-
tion probability due to the Fermi and Gamow-Teller modes,

respectively. The factor a is a strength parameter that

96

 

PROBABILITY

 

 

 

 

Figure A-3.

 

Fermi-Gamow-Teller recoil energy distribu-
tions.

 

97

is a measure of the relative contribution of the Gamow-
Teller mode to the FWHM of a composite energy distribution.
(It is estimated by taking the ratio of FWHM's for the two
distributions shown in Figure A-3. A rough estimate in
most cases gives a = 0.5.) The quantities fF and fGT can
be found if the ft value of the decay is known.

If we define IMFI and IMGTI as the matrix elements
for the vector and axial vector interaction, and Cv and CA
as dimensionless coupling constants for the vector and

axial vector interactions, then

_ 61A3 _
ft 5 IM I2 + (CVI2 IN I2 (u 19)
F '6; GT

(with CA CV = 1.25)

Now, fF and fGT are defined as

MF 2
(u-20)

 

c 2
2 2
IMF! + (51) IM

A

cr'

So in order to obtain fF and fGT’ we need the ft value for

the decay and a value for either IMFI2 or IMGTI2. In

98

most cases of interest to us, the decays will be simple
mirror transitions or at most decays that are classified

as allowed, in which instances there are simple rules to

 

Calculate IMF . The ft values can be calculated, (see
(Wu66)) or obtained from the literature. The combination
ft and IMF] allow one to calculate fF and fGT’ which repre-
sent the fraction of decay strength due to the Fermi and
Gamow-Teller decay modes, respectively.

In Table IV-5 are given the ft values and calculated
values of IMF], fF, fGT, and D for all the known decays

observed as described earlier in this chapter is followed,
CW
3
CP3
where DA is totally determined by the particular decay.

except now the independent variable is DA - A -

The results of this new parameterization applied to the
KCl spectrum are given in Table IV-6. X2 for the linear
least squares fit to the "known" cases was 1.8, not sig-
nificantly different from the simpler initial parameteriza-
tion, perhaps a little worse. The same seems to hold true
for the fit to the "unknowns." However, referring to
Table IV-7 where the results of the new parametric fit for
the A1 spectrum are given, a very obvious improvement in
the fit to the "known" peaks is evidenced. In fact, x2
for this fit is 0.0A, a very large and significant reduc-
tion from the x2 of 7.A for the simpler parametric form.
But the fit to the 27Si "unknown" is very poor in this

instance.

99

Table IV-5. Fermi-Gamow-Teller Parameters for Peaks in KCl

 

 

Spectrum.

A ft IMFI2 fF fGT D x'

26 3162 2 1.0 0.0 1.0 0.97
27 3981 l 0.65 0.35 0.85 0.61
28 79A3 0 0.0 1.0 0.58 0.26
29 5012 1 0.82 0.18 0.92 1.05
30 3162 2 1.0 0.0 1.0 1.12
31 5012 l 0.82 0.18 0.92 1.28
39 3981 l 0.65 0.35 0.85 0.9A

 

100

Table IV—6. Recoil Fit for KCl Spectrum (With F-GT Cor-

 

 

 

 

 

rection).

Calibrated

R Rfit EO E:f‘it
A (eV) (eV) % Error) (MeV) (MeV) % Error
26 381. A01. 5. 3.81 3.92 3.0
27 362. 356. 1.6 3.79 3.75 1.0
30 A3A. A20. 3.A A.AA A.36 1.8
Predicted
28 273. 308. 13. 2.87 3.53 23.
29 360. A11. 1A.O 3.92 A.23 8.0
31 A09. AAO. 8.0 A.37 A.55 A.0
39 A9A. 397. 20.0 5.5 A.88 11.0

 

Table IV-7. Fit to Al Spectrum Recoil Energies (With F-GT
Correction).

 

 

R Rfit EO Efit
A (eV) (eV) % Error (MeV) (MeV) % Error
23 287. 28A. ' .7 3.0 3.01 .A8
25 299. 297. .8 3.26 3.2A .61

27 362. 362. .1 3.79 3.788 .05

 

CHAPTER V

MODELING THE TOF SYSTEM

I. Introduction

The results of the empirical fitting to the observed
recoil energies in the previous section were encouraging
enough to elicit further investigation. As a means of
theoretically justifying our assumption about the recoil
energy contribution to the peak broadening, we decided to
mathematically model our TOF system and the resulting
particle trajectories.

The first step in modeling our TOF system was to
obtain a reasonable set of electric fields. The system
is made up of elements of finite lengths and edges whose
effects should not be totally ignored. As with most
practical problems, analytic solutions exist only for
idealized cases. For real geometries we often have to
settle for approximate solutions. To this end numerical
methods are especially well—suited. To obtain the electro-
static potentials for our system we have to solve Laplace's
equation subject to the appropriate boundary conditions.

The iterative technique of relaxation is a standard method

101

102

for obtaining approximate solutions to Laplace's equation.
Once we have a reasonable set of fields, we can calculate
particle trajectories. This means that we have to inte-
grate equations of motion for charged particles. To ac-
complish this we used a combination of Runge-Kutta and
Predictor-Corrector algorithms. Since these numerical
methods are an integral part of our modeling study, we

will present a discussion of them in some depth. This is
especially true for the relaxation methods we employed
because of what we feel is their growing importance in this

age of digital computers.

II. Relaxation Techniques
A. Preliminary

The solution of Laplace's equation for a system of
conductors is one of the most common problems in electro-
statics. Students are often taught the usual analytic
methods of solution: images, Green's functions, and ex-
pansions in orthogonal functions. The powerful yet in-
elegant numerical techniques are rarely presented as alter-
nate approaches. For the more complicated numerical tech-
niques, this exclusion is understandable. However, the
relaxation technique, which we shall discuss here, is so
simple and well—suited to boundary-value problems, that

once learned, it provides a valuable tool for constructing

103

solutions for many common and specific problems. When
used in conjunction with digital computers available today,
relaxation techniques are able to handle problems that are
beyond the scope of practical analytic solution.

In this section we present a straightforward discussion
and derivation of relaxation and the related over-relaxation
techniques. We then apply these techniques, both to simple
problems such as would arise in a classroom and also to
slightly more complicated problems such as have arisen
from our own experimental work.

Occasionally, in discussions of Laplace's equation,
students are told that it implies that the potential at a
given point is the average of the potential on a surface
enclosing the point. Then, to demonstrate this, the poten-
tial at the center of a charged, hollow, spherical conductor
is shown to be the same as the average on the surface. Such
an example serves the purpose well enough for the specific
case of spherical symmetry, but the implications of Laplace's
equation need not be so restricted. We feel that it is
reasonable to demonstrate to students that the general

equation

var = 0, (5-1)

can be interpreted as a differential statement of the fact

that the solution at a specific point is just the average

10A

of the solutions over a surface of any shape surrounding
thegmdxu;of interest. If such a demonstration can be easily
given, it will greatly enhance and supplement specific
examples. (As will be shown, application of relaxation
techniques to boundary-value problems in electrostatics

is intimately and obviously dependent upon the differential
validity of the averaged nature of solutions to Laplace's
equation.)

The presentation of relaxation methods in electricity
and magnetism texts is nearly nonexistent. A notable ex-
ception is the introductory text, Volume II of the Berkeley
Physics Course. Yet even here the student must delve into
the recesses of an appendix on advanced problems. There
a recipe, applicable only to a two dimensional Cartesian
problem, is given. The basis for the prescription and its
extension to other orthogonal systems is not discussed.
With a few elementary preliminaries we will attempt to
correct such exclusions and present relaxation techniques
in a manner that students of electrostatics can appreciate

and employ, having access to only modest sized computer.

B. Mathematical Preliminaries

To those of us who worked our way through differential
calculus with mild chagrin, the definition of a derivative
is perhaps familiar yet seldom used. However, for numerical

solutions the finite divided-differences that define a

105

derivative as a limit are essential to replacing a dif-
ferential equation by an analog algebraic expression.

Recalling the definition of the derivative at a point

 

x0, we have
f(X +Ax) - f(X )
df(x) = 11m 0 0 (5-2)
dx Ax

= Ax+o
x xO

This expression gives an approximation to the deriva-

tive when Ax is small; that is,

df x)l f(xO+Ax) - f(xo)
dX Ax ° (5-3)

 

Of course, this is an approximation and the associated
error is on the order of Ax. As shown in the Appendix IV,

a better approximation is given by

df;x) f(xO+Ax) - f(xO—Ax). (5 u)
2Ax ’ -

Q.
H
II

and for the second derivative,

106

d2f(x) f(xO+Ax) - 2f(xo) + f(xO-Ax) ( )
= 5'5
x=x (AX)2

 

These last two expressions have associated errors on the
order of (Ax)2, which shows the advantages of choosing Ax
small. The extension of these expressions to partial deriva-
tives is straightforward and given in Appendix IV.

Now let us consider Laplace's equation for a two-

dimensional Cartesian system, at the point (xo,yo):

32¢(x ) 82¢(X.y) _
____E£Z_ + -——_7T-_' - 0. (5-6)
3x 3y -
x=xo X‘XO
y=yo y=yo

 

 

The analogous expression in terms of finite differences is

¢(xO+Ax,yO) - 29(xo,yo) + 9(xO-Axo.yo)

(Ay)2

+

 

¢(xo.yo+Ay) - 29(x03y0) + ¢(xo.yO-Ay) = 0. (5_7)
(A102 '

 

Rearranging this last expression, we have

107

 

9(xo,yo) = J; 2 {©(xO+Ax,yO) + 9(x0-Ax,yo)
2<l+(E§) )
+ (93)2 (©(X y +Ay) + 9(x y -Av))}. (5-8)
Ay 0’ 0 0"0 ' °

If we choose Ax = Ay E h we obtain the simple expression,

¢<xo,yo> = % {(xo+h,yo)+¢(XO-h,yo)+¢(xo,yo+h)+¢(xo,yO-h)}
<5—9)

this makes it clear that the potential at x ) is equal

o’yo
to the average of the potentials at points around (xo,yo).

Furthermore, it is simple to obtain equivalent expres-
sions when we need to deal with different coordinate systems.

Consider the Laplace equation for an axisymmetric system

in cylindrical coordinates, O = ¢(o,z):

2
v20 = 3 No.2) +

30

I\.)

(0.2) + a 3 (0,2) = 0. (5-10)

0 dz

0'

3

 

DIH
”I

At the point (90,20) the equivalent finite-difference

expression is

108

(b - _
.(oO+A0,zo)+¢(oO Ao.zo) 2¢(OO,ZO)
(Ap)2

9(00+Ao,zo)=¢(oo=Ao,zo)
2A0

 

 

+._1_
pO

¢(p ,z +Az)+¢(p ,z -Az)-2¢(o ,z )
+ O 0 0 3 ° ° = 0 . (5-11)
(AZ)

 

Rearrangement yields

 

9(00.20) = J; 2 {9(00+Ao,zo)(1+Ao/200)+¢(CO-Ao.zo)(l-Ao/200)
2(1+(E%) )
+ <%%)2 (¢(oO,zO+Az)+9(oo,zO-AZ))}. (5—12)

In this last equation it appears that we have a singular-

ity for 00 = 0, but for axial symmetry,

0)
'6-
II
0

(5-13)

0)
0

Therefore, by L'H8pita1's rule,

109

 

 

“"122 ..ail
o +0 o 80 302
o o po 00
At 0 = 0, LaPlace's equation is
32¢ 32¢ _
27+ 2-0,
30 Bz

¢(O+Ap,zo) = ¢(0-Ap,zo)

(5 15)

(5-16)

(5-17)

Rearrangement of the finite-difference equation equivalent

to Equation 5-15 gives, for p0 = 0.

1

 

¢(O,ZO) = {A¢(Ap,zo)+(%%)2(¢(O,zO+Az)

2(2+(%%)2)

+ 9(0,zO-Az))} ,

(5-18)

110

which yields the starting point for applying relaxation tech-

niques.

C. Techniques for Calculation

If one employs the appropriate expressions for the
potential given in the preceding section, boundary-value
problems become very amenable to iterative techniques of
solution such as relaxation and its extension, overrelaxa-
tion. To employ either of these, our first step is to
replace the continuous region of interest with a grid net-
work as shown in Figure 5-1. For convenience we set up the
grid so that conductor and boundary surfaces lie along grid
lines. Next we make guesses for the value of the potential
at every point that is pg§_fixed by the problem. Such points
will be called "free" points. A convenient initial guess
for the free points would be all zeroes; although this will
work, it is very inefficient. Intuitively, it seems that
an initialization of the free points based on the analytic
solution to a similar but less difficult problem is a
reasonable first guess.

Relaxation consists of replacing the value of the
potential at a particular free point by the average of its
neighboring points according to Equation 5-8, or 5-12 and
5-18, whichever is appropriate to the coordinate-system.

This procedure is applied to each free point in the network,

111

 

 

 

 

 

 

 

 

 

3 6.3 7.3 8.2
I
j 2 5.2 6.2 7.2
I 4.0 4.0 4.0
0
o I 2. 3
| -—D
Figure 5-1. Two-dimensional x-y grid with 9

points after one iterations.

ég's at free

grid spacing

is the same in the x and y directions.
x = 1A, y = jA, where i, j are integers.

112

constituting one iteration. Iterations are continued until
a reasonable convergence criterion is met. We have monitored
the maximum percent change effected through one iteration.

For the Laplace problem it has been found that a modi-
fication of relaxation, called successive over-relaxation
(SOR), may significantly accelerate convergence (Am69),
(F060). Here the change in the distribution of values ob-
tained from each iteration of relaxation is similar to a
diffusion process; i.e., SOR acts to enhance the diffusing
corrections by "amplifying" the averaging process.

To utilize SOR one must choose a relaxation factor,
1 < m < 2. For the case of equal 0 and z mesh steps, an
estimate of the optimum m can be obtained from the expres-

sion

~ 2

wept " W , (5'19)

where A is the mesh step size.
Denoting ¢ij as the value at (i,j) obtained from the

8th iteration of ordinary relaxation, we get an "over-

SOR

relaxed" value of 013 by applying

SOR =

1-1 A
ij '

c (l-u)<I>lJ + 0013 (5-20)

113
Then we replace ¢ij by ¢§3R° Choosing w = 1, reduces over-
relaxation to relaxation.

Perhaps an example will more fully demonstrate. Ref-
ererring to Figure 5-1, suppose the values shown at each
point were obtained after five iterations. Consider the
point, (i = 2, j = 2); 922 = 6.2, and using Equation 5-8,
we obtain

Relaxation Step:

032 = 1/A(A.07+7.3+5.2+7.2) = 5.925 . (5-21)

If we are using only relaxation, we would move to the next
point and compute the average of its neighbors. However,
if we are using SOR with w = 1.5, our next step is

Over-Relaxation Step:

SOR

6
°22

5
(l-l.5)o22 + 1.5 922

-.5-6.2 + 1.5-5.925

5.7875 (5-22)

Next we take 932 to be 5.7875 and move on to the

next point to repeat the procedure.

11A

D. Applying Relaxation and Successive Over-Relaxation

A typical problem in electrostatics that is easily
solved analytically is the case of two coaxial infinitely
long cylinders held at different potentials. Referring to
Figure 5-2, the closed form of the solution is found to

be

9(0) = (AV/£n(a/b))°£n(o/b) . (5-23)

This case provides an instructive application of the tech-
niques and a simple means of comparing the convergence
rates of relaxation and SOR.

We mock-up the infinite length by fixing the potential,
at both ends of a finite coaxial length, to be the analytic
solution. An equal 0 and z mesh size of 0.25 was employed.
One hundred iterations were performed with a Xerox 2-7
computer, first using ordinary relaxation and then SOR
with w = 1.75. In both cases an initialization of 9(0) =
50/(l+p) was used. The results for each, with their
respective errors, at representative p points are given in
Table V-l, after 10, 50, and 100 iterations. In the right-
hand column the exact values from an analytic solution are
given for comparison. Inspection of the table makes it
apparent that the SOR method converges much more rapidly
than the relaxation technique (w = 1.0). The time saved

in computing can be significant.

115

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116

Table V-l. Comparison of Relaxation and SOR Results for
Infinite Coaxial Conductors.

 

 

 

 

 

 

 

 

 

Numerical % Numerical %
Value Error Value Error
Exact
Solution
0 = 1.0 w = 1.75
After 10 Iterations

1.5 53.67 3A.9 72.92 11.5 82.33A
5.5 7.7A 70.2 13.30 A8.8 25.96A
9.5 2.36 6.1 l.Ao 36.9 2.228

After 50 Iterations
1.5 69.90 15.1 81.31 1.3 82.33A
5.5 8.92 65.6 23.92 7.9 25.96A
9.5 1.2 A6.2 2.05 7.9 2.228

After 100 Iterations
1.5 75.09 8.9 82.33 0.069 82.33A
5.5 l2.A0 52.3 25.80 0.62 25.96A
9.5 1.08 51.6 2.21 0.6A 2.228

 

. 117

As an example of the usefulness of the SOR technique
the example of a finite cylindrical conductor coaxial with
a larger radius and longer closed cylinder held at a dif-
ferent potential has been solved. This geometry is shown
in Figure 5—3. Such a deceptively simple looking problem
is beyond closed form solution, but by employing SOR methods
(Equation 5-12 and 5-18) the problem can be solved in short
order. The matrix map of solutions results in the equi-
potentials plotted in Figure S-U.

In Figure 5-5 we show a second cylindrically symmetric
system that is intractable by analytic methods, yet presents
no real difficulties for an SOR treatment. (This particular
case is actually a prototype of an electrostatic particle
focusing system that we employ in a recoil—mass time-of-
flight spectrometer.) The potentials obtained from an
SOR treatment of the appropriately scaled system will be
used in calculating trajectories in such a region. The
equipotentials that result from an SOR solution of this
electrostatic problem are shown in Figure 5—6.

After gaining some experience in field calculations and
the estimation of the best relaxation factor to use, we
began the calculation of the potentials that correspond
to our TOF system. The calculations were accomplished in
two parts: (1) the acceleration zone and (2) the drift zone.
This was possible because the grounded mesh separating the

acceleration zone provides a natural boundary surface for

118

 

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122

both regions. This allows us to use the mesh plane as a

boundary surface for both portions of the calculation.

Acceleration Zone

The acceleration region consists of a circular stain-
less steel plate held at 6 kV, a fine wire mesh disc at
ground and a boron nitride (BN3) insulating support connect—
ing the two plates. Basically it is a cylindrical capacitor
with a dielectric "ring" connecting the two plates. The
radii of the charges plate and the mesh disc were both 1.83
cm. The length of the dielectric and therefore, the separa-
tion of theplates was 1.67 cm. The thickness of the insulat-
ing support was o.u7 cm. The inclusion of the eielectric
in our model complicates the calculation somewhat since we
no longer have pure Dirichlet boundary conditions on all
surfaces. At the surface of the dielectric we have
Neumann boundary conditions. Dirichlet boundary condi-
tions mean that the value of the function is specified on
a particular surface while Neumann boundary conditions
mean that a condition on the derivatives of the function
are specified. In electrostatics the relevant Neumann
boundary condition is that the normal component of the
electric displacement vector, Dn must be continuous. As
a result the averaging equation for the potential presented

in the previous section is not valid at the surface of the

123

dielectric. However the continuity of the normal component
of the electric displacement can be translated into an
equivalent expression using the finite divided differences
of the potentials. Using this method one can then obtain
a new expression for the potential at the surface of the

dielectric. The relevant equation is

¢ = (Q +

1,3 i-l,J €D ' ¢i+1,3)/(1 + 8D) (5-2u)

where so and ED are the dielectric constants for vacuum

(60 = 1) and BN3 (5D = “.08), respectively. This expres-
sion was used at the inner and outer surfaces of the di-
electric. A further complication in this treatment was
that the Neumann condition on the outer surface leaves the
potential unfixed on any surface in the rho direction which
is an unstable situation. There are a number of ways to
deal with this problem (see (Ac70)), the easiest and most
straightforward is to "enclose" the entire acceleration
zone in a large cylindrical box at a fixed potential.
Physically this is very reasonable since in actuality the
acceleration zone is encased in a vacuum chamber that is
held at ground. This was the approach we chose to utilize.
Our mesh step was .0235 cm enabling us to place 20 nodes
in the insulator along the c direction. Two hundred
iterations were performed until the maximum fractional

change in the mesh was less than a part in 10“. Shown in

129

Figure 5-7 is the equipotentials resulting from the over-
relaxation calculations for the acceleration zone. The
first nine lines represent drops in potential of 0.6 kV

as one moves from left to right —- the direction of motion
for a positively charged species. The spacing of these
equipotentials is fairly regular which indicates the
electric field strength is uniform. This is especially
true along the axis of symmetry (O = 0) as can be seen from
Figure 5-8. The top curve is a plot of the Z component of
the electric field (at o = 0) as a function of distance
from the charged plate. This component is relatively
constant throughout the acceleration zone, the total drop
in field strength is about 5% of the maximum. From these
considerations one would expect a positively charged ion

to undergo a uniform acceleration in the Z direction. Now
if we follow one of the equipotentials in the first half of
this zone it begins to bow slightly as we move away from

o = 0. This indicates the presence of a p component of the
electric field. The lower curve in Figure 5-8 shows the

0 component of E as a function of the distance from the
symmetry axis. This particular curve was calculated at

Z = .5 cm. As required by the Neumann boundary conditions
the electric displacement normal to the surface is con-
tinuous at the interface of facuum and the dielectric.
However, the electric field component normal to the surface

has a discontinuity at this interface. This implies the

125

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127

presence of a surface charge. There is indeed a surface
charge, it-is the induced polarization charge resulting from
the external applied E field.

The 0 component of the field is much weaker than the
Z component, its maximum value is about .08 times the Z
part. Nonetheless it is a defocusing mechanism through
much of the acceleration zone. This effect is most pro—
nounced in the first three quarters of the region. Also
the field in the 9 direction increases as the distance from
the center increases. Therefore particles starting out
far from the center of the collection plate should be most
strongly affected. This is one reason for focusing the

spray on center as best as possible.

Drift Zone

Once the recoil ions have passed through the wire mesh
disc it enters the drift zone region. The region is made
up of a small radius wire (0.0025 cm) concentric with a
much larger radius pipe. Ideally this was meant to provide
a logarithmic potential to focus the recoils onto the CEMA
detectors at the end of the flight path. It was hoped that
only an electric field in the p direction would result from
this configuration. Unfortunately the wire can't be in—
finitely long so it has to start and end somewhere. This
means that near the end of the wire the field are not purely

functions of 0, they have a dependence on Z that is very

128

difficult, if not impossible, to determine analytically.
Another nonideal aspect to the real geometry is a sudden
expansion in the radius of the outer conductor. This ex-
pansion, which occurs about 1/3 m from the acceleration
region, results in a change in radius from 1.83 cm to 5.2
cm. This discontinuity preserves the symmetry in the

0 direction but definitely destroys the Z symmetry in the
area of the expansion. Hereafter I will refer to the area
of the radius change as the expansion region. Obviously

we must again employ numerical methods to solve Laplace's
equation for the potentials in the flight tube. In setting
up the problem there are immediate differences from the
calculation on the acceleration zone. The first difference
is an advantage. In the flight tube region there are only
conducting boundaries to deal with, so only Dirichlet condi—
tions are relevant. This means the relaxation codes are a
bit simpler. The second difference is a practical dis-
advantage. As we pointed out earlier the step A should be
scaled to characteristic linear dimension of the problem
geometry. However in the flight tube region the most
obvious scale is the radius of the wire -— 0.0025 em. But
if we used a A = 0.0025 cm for our .5 cm x 100 cm system

we would require an array of nearly 80 million elements --
totally unrealistic for our computer. Increasing the step
size alleviates the problem of array size only at the ex-

pense of potentially large errors creeping into the

129

calculation since the error goes as the square of the step
size. In our studies on the relaxation technique applied

to the geometry of Figure 5-3 we found that the potentials
approached the idealized infinite wire case as we moved
away from the wire end. At a distance on the order of the
separation of the wire from the ground plane the potentials
calculated are well approximated by a logarithmic form. In
the flight tube the separation of the wire and the ground
disc is about 7 cm so after roughly 10 em down the wire we
thought that the use of a logarithm in 0 would sufficiently
describe the potential field distribution. This effectively
reduced the number of array elements required for the Z
direction. We still had to contend with the problem of

the small radius of the wire for the remaining regions.
Following a more intuitive than rigorous argument we decided
to use the actual wire voltage as the boundary value for
the appropriate p = 0 points (i.e., points in the wire)

and to use a boundary at one small step size from the p = 0
points. The fixed value of the potential on this surface
was obtained from the logarithmic form discussed previously.
Basically we replaced the "real" wire with a "pseudo"-wire
of a more manageable radius. This final approximation
allows us to reduce our array size to a realistic number
for computation. The final mesh step size was 0.0235 cm.

In the expansion region we again used the method of employ-

ing the logarithmic form for the potential at a reasonable

130

distance from the radius expansion. The results of the
calculations are shown in Figure 5-9 as a number of equi-
potentials. The effect of the wire ends are clearly dis-
played at both ends. At the end nearest the acceleration
zone the equipotentials resemble cylindrical waves, the cor-
responding electric fields will have both 0 and 2 components.
So in this region the ESPG is not focusing only in the radial
direction. In fact, near the axis of the wire the effect

of the particle guide is to accelerate predominately in the
z direction. At the other end the field acts to focus the
particles on the CEMA detectors and also to deaccelerate
them in the z direction. At the expansion region we can
easily see the field "response" to the sudden radius change,
the equipotentials seem to "expand" to fill the suddenly
enlarged region of space available. The effect of this

part of the fields will be a perturbation on particle tra-
Jectories that had been established as stable by the time
they reach this part of the system. In a sense we can view
the expansion as resulting in an easing of the attractive
force that a charge particle "feels" as it passes through
this region. The net effect is a defocusing of particles
that had made it down a third of the flight path length.

In the remaining regions the equipotentials are parallel to
the wire with no real 2 dependence as we should expect.
These regions, away from the discontinuities, make up about
80% of the system length and act to focus in the radial

direction only.

131

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132

Calculation of Particle Trajectories

 

The relaxation calculations Just described allow us to
obtain the electrostatic forces acting on a charged particle
passing through our system. We are now in a position to
calculate particle trajectories. This means we will inte-
grate an appropriate set of equations of motion. For our
case this set will be the Lagrange equations for a charged
particle in an external electric field. In cylindrical co-

ordinates we have

m -—§ = D¢ ‘ q -— (5-25)

m -—§-<p ¢) = -q 39 (5—26)

In-——§ = -q-g; <5—27)

For our case the electrostatic potential, ¢ is independent
of the angle ¢ so the quantity mp2$ is a constant of the
motion. Actually this quantity represents the angular

momentum of the particle about the z axis, so we have

L = m02$ <5-28)

133

This constant reduces the number of equations to be solved
to two and allows us to eliminate the variable 6 from the
remaining set. The resulting equations, after some simple

algebraics are:

 

2 2
9.2 = L _ 9 22 (5-29)
dt2 m2p3 m ap
232.- _ 9 23 (5-30)
dt2 - m Bz

The electrostatic potentials and their derivatives are
obtained from our calculated potentials by interpolation.
We have used a Lagrange double 3-point form which is dis-
cussed in more detail in Appendix III.

Now we must choose a method for integrating the equations
of motion. The most common numerical algorithms are the
Runge-Kutta and the Predictor-Corrector methods. Each of
these are prototypes of a general class of integrators.

Even though these approaches are well documented and
developed, the treatments are all in numerical analysis
texts. For the sake of clarity I will give a brief descrip-
tion and comparison of the Runge Kutta and Predictor-Cor-
rector algorithms.

For this discussion we will restrict ourselves to first

l3u

order equations since the extension to higher order equa-
tions is straightforward. By definition a first order equa-

tion has the form

53(- = f(x,.v) (5-31)

We want to obtain a solution y(x) that satisfies (5-31)
and a specified initial condition. Since we can't solve
for y(x) analytically, we approximate the true solution
at discrete subintervals of the independent variable x.

Considering equal sized subintervals the step size, A is

A = _H_ (5-32)

where n is the number of subintervals and [a,b] is the range
of the integration. With this defintion of the step size,
our problem is to obtain approximations, yi to y(x) at a

set of base points, xi where

Xi = x0 + iA (5-33)

135

and

Y1 = Y(Xi) (5-3“)

The various procedures to accomplish this goal can be
divided into two groups, one step and multistep methods.
Runge-Kutta methods are a common example of one step
methods. These allow one to calculate y1+1 using only the
differential equation and information at the point x1,
(i.e., yi). Multistep methods require values of yi and
f(xi, yi) for several different points, earlier and possibly
later points. The predictor-corrector methods are a class
of multistep methods.

All Runge-Kutta methods have the form

Y1+1 = v, + A ° s<xi, yi, A) (5-35)

Here g is simply an approximation to f(x,y) on the sub-
interval xi 3 x g x1+l. The usual procedure is to express
g as a linear combination of derivative evaluations on the
integration interval. Each evaluation is used to predict

a better estimate of yi+l which will then be used to obtain
an improved derivative estimate. For our calculations

we started with a fourth order method due to S. Gill (Ca69).

The algorithm is

y1+1 ‘ Vi

with

+

136

A
3 (K1 + ai o K2 + a2 - K3 + Ku)

2(1 - 1//§)

2(1 + 1//§)

f(xi, Y1)

f(xi + A,y1 + AoKl/2)
f(xi+A/2,yi+(- % + 1//2)AK1)

f(xi+A,y1-A°K2//2+(l+l//2)A-K3)

(5-36)

(5-37)

(5-38)

(5-39)

(S-UO)

(5-41)

(5-42)

The K's are evaluated successively in ascending order.

Although the formulas appear a bit long when translated

into a computer code the algorithm is very simple and com-

pact.

The predictor-corrector algorithms are more complex

than the corresponding Runge-Kutta routines.

of brevity, we will present a cursory description.

fuller and more rigorous discussion see (Ha62).

For the sake

For a

137

first step in the predictor-corrector method is to fit a
polynomial to the three most recent points of the deriva-
tive then extrapolating this polynomial to the next step
and integrating the fitted polynomial to obtain an increment
that gives yp. This yp is the predicted value. This value
is then used in the original differential equation to ob-
tain a better estimate of the derivative at the point we
previously extrapolated to, i.e., Xi+l' Now we again fit
a polynomial but this time to our new derivative estimate
and the derivative at earlier points. We integrate under
this new polynomial to obtain a second estimate of y at
the next x step. This second estimate is called the cor-
rected value, yo. Next we check the difference between

yp and ye. If it is too large we can repeat the procedure
with a smaller step size. If too small we can proceed
with a larger step size. The most important point to note
is that at each step forward we have available an internal
measure of the accuracy of the integration. The measure
is the difference between the predicted value and the cor-
rected value. One disadvantage of the multistep method

is that one must use another method to start the integra-
tion. This is because in the very first part of the
algorithm a polynomial is fitted to earlier points that
are not available at the outset. For our computations

we started with the Gill form of the-Runge-Kutta routine.

After advancing three steps with the Runge-Kutta routine

138

we then switched over to Hamming's Predictor-Corrector

method (Ca69). The form of this algorithm is

 

Predictor
p = + “A (f - f + 2f ) (5-u3)
1+1 y1-3 ‘3‘ 1. 1-1 1-2
Modifier
_ 112
m1+1 ‘ p1+1 + I‘I'(Ci ‘ p1) (5'““)
Corrector

 

C1+1 = 1/8 (9yi ‘ y1-2 + 3A<f<x1+1,m1+1) + 2f1 ‘ f1-1))

 

(5-45)
Error
e = -—9-— (c - > (5446)
1+1 121 1+1 p1+1
Final Value
y1+1 = c1+1 ‘ p1+1 (5'u7)

The term ei+l’ the truncation error, is checked at each step
to determine whether the step size should be changed.
If we compare the two algorithms just presented it can

be seen that the Runge-Kutta routine requires four separate

139

evaluations of the derivative term while the Hamming pre-
dictor-corrector only needs two evaluations of f(x,y).
Generally this difference will mean that the predictor-
corrector algorithm should be faster than the Runge-Kutta.
In an attempt to see how significant the difference in speed
of the two algorithms, we ran our computer routines for each
algorithm on a number of different analytical functions.

In Table V-2 are the results of the comparison, in
every case the Runge-Kutta algorithm required at least “0%
more time than the predictor-corrector routine. This is a
significant difference. When we begin calculating tra-
Jectories in our calculated fields the derivative evalua-
tions involve a number of interpolations in the field map.
These interpolations can be very time—consuming so we want
to keep these derivative evaluations to a minimum. This is
another advantage of the predictor-corrector method.

For the actual trajectory calculations we used the Hamming

method with a variable step size. Whenever the truncation

5

error term was greater than 1 x 10' the integration step

size was cut in half andtflueprocedure continued. If the
error term was less than 1/50 of the upper limit on the
error then the integration step size was doubled. This

was very useful for our problem since the particle traverses
long regions of slowly varying potentials on much of its
flight path so the step size can be very large. In the

more rapidly changing regions the step size reduces in

1H0

Table V-2. Comparison of Speeds for Runge-Kutta vs.
Predictor-Corrector Routines.

 

Time of Calculation
for 5000 Steps (in seconds)

 

 

Function Predictor-Corrector Runge-Kutta RK/PC
tanh(x) - X 15 23 1.5
eXp(x) - x - 1. 15 21 l.A
sech(x) - 1. 16 22 1.“
cosech(x) + l 19 31 1.6
cosh(x) - 1. ' 15 21 l.“
sinh(x) - x . 15 21 1.”
log(x) - x + l. 16 22 l.“
x + l/X 16 22 l.“
cos(x) 15 21 1.“
sin(x) 15 20 1.3

 

1Ul

response. In addition to the error monitor we checked the
constancy of the particle's total energy. Since we have an
electrostatic problem the sum of the kinetic and potential
energy should be a constant. This was true for our calcula-
tions to one part in 105. Special care had to be taken in
crossing from the acceleration zone into the drift zone. If
one does not insure that one of the integration steps
"lands" on the grounded disc position then it appears that
the electric field in the acceleration region continues to
act on the particle a short distance into the weak field
zone. The result of this is strong nonconservation of
energy and very incorrect results. After integrating down
the length of the flight tube the particle's 0 position was
checked to determine whether it was within the radius of

the CEMA detectors (“CEMA = 1.25 cm). If it was, then the
time, p position and velocities are recorded and the event

is considered a successful hit.

CHAPTER VI

THEORETICAL MODELING RESULTS

In the previous chapter we presented the tools that
will be required to model a specific time-of-flight peak
that has been experimentally observed. In the following
section we shall present the formal aspects of the problem.
This will include clearer definitions of 1) our approach,
2) the particular isotope chosen for study, and 3) the
most effective use of the mathematical tools discussed

previously.

1. Approach

Basically our problem is that we have, as a physical
observable, a distribution of events as a function of their
associated times-of-flight through the SIEGFRIED system.

Our goal is to relate the observed distribution to a reason-
able set of initial conditions that will be related to the
particular isotope of interest. Since we are interested

in correlating the particles' initial recoil energy to the
observed peak broadening, the kinetic energy is an important
initial parameter that must be specified. Also, the source

of the recoils is a finite size spot, so we should include

192

1H3

this fact in the initial distribution. Fortunately the
source spot is approximately circular and well-centered on
the collection plate. After passage through the SIEGFRIED
TOF system the initial particle distribution in position
and energy is transformed into the observed distribution in
time-of-flight. If we denote the initial distribution as
NO(E,DO) and the final distribution Pf(T), then the process

of measurement can be represented as shown below:

NO(E,oO) ———- SIEGFRIED —-—~ Pf.(T)

Mathematically we will consider the action of SIEGFRIED
on the initial distribution as expressed by an integral

transform of the form:

Pf(T) = fffS(Egpo:Tspf)NO(EapO)dedEdof (6‘1)

Here S(E,pO,T,pf) represents the transformation of a
particular kinetic energy E and position p0 distribution
in the energy and position interval dEde. (The effect
due to the finite size of the detector is taken into account
by integrating over final positions with of less than or
equal to Rdet’ the radius of the CEMA detectors.)

Ideally S(E,pO,T,pf) and NO(E,pO) would be expressible
in a closed form. Then we could merely integrate and solve

the problem easily. Even more appropriate would be to

14M

invert our approach: suppose we express our integral trans—

form, Equation 5-1, as

Pf(T) a§ENO(E,oO) (6-2)

Here §Z7represents the integral transform operator. Now
if an inverse oqug, exists, then we can invert the problem.

Denoting the inverse bycgy'l we get from Equation 6-2
_ -l
NO(E,OO) -Q me (6-3)

This last form says that, by using 3‘1 on the observed
final distribution Pf(T), we can reconstruct the initial
distribution function. That is precisely what we would
like to do -- if we could.

The problem is that the kernel of the transform, 8
is not expressible in a simple form. We do know that
solving the equations of motion for a set of initial condi-
tions that result from a particular distribution will allow
us to follow the time evolution of the distribution. This
is the functional effect of S on the initial distribution,
but we cannot "find" an inverse, so we are basically solv-
ing the problem in an unavoidably roundabout manner. We
will assume a given initial distribution, predict a test
distribution F(t) and compare this with Pf(T), the ob-

served distribution.

145

2. Choice for the Study

The particular isotope I have chosen for this modeling
study is 23Mg. It is a positron emitter with a half-life
of 12.1 sec. The decay energy is typical of isotopes seen
by SIEGFRIED (QEC = b.056 MeV). 91% of the decay proceeds
from the 3/2+ ground state of 23Mg to the 3/2+ ground
state of 23Na. The decay has 70% Fermi fraction and 30%
Gamow-Teller fraction (as defined in Chapter IV).

Experimentally 23Mg is seen very strongly in the TOF
spectrum resulting from 27A1 + 70-MeV 3He (See Figure 3-1).
Its TOF peak is the largest feature in the spectrum and
is very well separated from other peaks in the spectrum.
The shape is typical of TOF peaks we have observed, and
the high number of counts in this peak allows us to fit it
very well with SAMPO. This gives us good estimates of

peak parameters such as the width and the tailing.

Finite Source

To take account of the finite source size we considered
the source to be made up of 8 concentric rings. Each has

a mean radius of n°(.075 cm) with n = 1,2,...8.

1M6

Initial Conditions

An important part of our modeling will be the integra-
tion of a large number of particle trajectories. While
the integration codes are fast (roughly six complete
integrations per minute) we want to be as efficient as pos-
sible. A very significant increase in efficiency can be
obtained by eliminating initial conditions that result in
trajectories that strike the boundaries of the system or
are otherwise unreasonable. This is an important considera-
tion and will be the focus of this section.

The maximum initial recoil energy, R is fixed by the
decay energy as shown in Appendix II. If we denote the
maximum kinetic energy of the electron by T0 (in MeV)
and the mass of the recoil by A (in amu) then the maximum
kinetic energy of the recoil R (in eV) can be obtained

by the following equation from Appendix II

R — iilT

_ A T + 1.022) (6-4)

0(0

This is a bound on the square of the velocities. For

23Mg the maximum kinetic energy of the electron is 3.03
-MeV. (For positron emitters T0 = QEC - 2mec2). The result-
ing maximum recoil kinetic energy is 290 eV.

By using a gross model of the TOF system and some

1“?

simple energy conservation arguments we were able to ob-
tain rough limits on the transverse velocity and the
angular momentum. To demonstrate this we use the geometry
of Figure 6-1. In region I the electric field has only a
Z component, at Z = d is a ground plane and region II has
a potential that is purely logarithmic in p. This means
that a particle of mass m and charge q starting from Z = 0
experiences only accelerations in the Z direction in region
I and only 0 accelerations in region 11. The system is
cylindrically symmetric and so the angular momentum about
the Z axis (L) is constant.

Consider a particle of mass m and charge q starting
from Z = 0, O = O with an initial kinetic energy R. In

0
cylindrical coordinates

.2 .
R = 1/2m(p + L2/(m p )2 + 22) (6-5)
0 o 0
Since it starts from the plate held at 6 kV its potential
energy is 6 keV. Denoting the potential energy by U, we
have
U = 6 keV

The total energy, E of the particle is conserved so

E = R + UO = constant (6-7)

1118

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>x®
URN OuN

.__".

 

O
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v .1 E J.

 

 

149
In crossing the region I only the velocity in the Z direc-
tion is changed. When the particle reaches the ground
plane it has a total Z velocity, Z(d)

2(a) = z: + /"2U'O/—m (6—8)

The time necessary to cross region I is td, which is ob-

tained in Appendix I as

 

. . 2U
= £11751- [-z +122 + ——9 ] (6-9)
0

The p velocity is not constant if the angular momentum is

nonzero, at Z = d and t = td we have

 

O m

2
2 L 1 1
- _- _ - +—<—§-—§> <6—1o>
0(td) - ad -‘/0 2 po 0

There is drift across the region in the direction given by

Equation AI-19, which for t = td gives

 

[(6§ + (L/moo)2)t + 600012 + (L/m)2
pd = 0(td) = p0

' 2 2
(popo) + (L/m) (6-11)

150

Now we consider the particle to be infinitesimally far
from the ground disc at Z = d but now in region II where
it fields a force due to the logarithmic potential in 0.
We assume the p position is given by Equation 6-11 and the
O velocity by Equation 6-10. Again the total energy is
constant but in addition the velocity in the Z direction-

is also constant so defining T as follows:
T = 1/2m (52 + L2/m292) (6-12)
Then the quantity
T + W = constant = Ed

where W is the potential energy. In region II, W is

V
_ o
W - EHTE737 £n(p/b) (6-13)

V0 is the potential on the center wire, a is the radius
of the outer conductor. Thus, at the beginning of region

II

‘2
E mod mL2 V
d - 2

+ 3—2- 4' mm a0 b £n(Od/b) (6-1“)
O
d

151

Defining new quantities T¢ and To as

2
L
T (p) = L“— (6—15)
¢ 202
m52
To = T (6-16)
we have
Vo

If p = b, the particle has struck the outer boundary and

its p velocity 6 and potential energy W are both zero, so:

Ed = T¢(b) (At 0 = b) (6-18)

From Equation 6-17 we have

V
Ed = Tp(d) + T¢(d) + EaTé%BT-£n(pd/b) = T¢(b)
(6-19)

From Equations 6-10 and 6-16,

Tp<pd> = Tp<po> + T¢<oo><l-<oo/od>2> <6—2o>

152
T¢(b> = T¢(oo)'(oO/b)2 (6-21)

Rearranging Equation 6-19 and substituting, we have

-v

Tp(oo) + T¢(oo)(1-(oO/b)2) = aob £n(pd/b) (6-22)

n
which gives bounds on the initial 0 position and velocity
and angular momentum. Suppose initially the angular momentum

is zero, then

Tmax

-V
0 (p0) = E57526? 2n(pd/b) (for T¢(po) = 0 (6-23)

with

This fixes the bound on the maximum initial 0 velocity.
If L = 0, any initial velocity less than this maximum will
not hit the outer wall. If we want the corresponding
maximum angular momentum for 50 = 0, then Tp(po) = 0

For Tp(po) = 0
max<p

O) = -vO 2n<od/b)/{2n<a/b)-(1-(pO/b)2)} (6-25)

with

153

 

2
_ ,7 L.
pd - DO 1"" mp (6-26)
0
Equation 6-25 sets a limit on the maximum angular momentum
for a given initial 0 value, p0. Actually, Equations 6-23

and 6-25 are transcendental, since T and TD enter into the

¢
logarithmic term through pd. Nevertheless, they can be
solved graphically accurately enough for our purposes.

This was done and the results compared with limits on these
quantities obtained from the actual numerical integration

of the equations of motion as described in Chapter V.

The results are given in Table VI—l. Tpred are the limits
obtained from our simple argument just presented, and Tnum
are the corresponding bounds obtained from the numerical
integration. Considering the simplicity of our gross model,
the agreement is very good. The most severe deviations occur
for larger values of po, where our assumptions about the
field uniformity (pure Z field in region I, pure log(p)

in region II) are most strained.

Once a simple form such as Equation 6-23 or 6-25 is
available to estimate bounds, we can obtain numerical
bounds, in a much more efficient manner. This is how the
bounding curves shown in Figure 6—2 were obtained. To a
precision of less than 1 eV the limits were independent of
the Z velocity over the total range of UZ.

To reiterate the function of these limit curves: they

Table VI-l. Comparison of Bound Values.

Tgax values

159

 

 

 

 

 

p Tpred Tnum
O O 0
(cm) (eV) (eV)
.15 26 2A
.3 21 19
.us 18 in
.6 1A 10

Tglax values

pred num

p0 T¢ T
(cm) (eV) (eV)

.15 31 29

.3 27 28

.AS 23 26

.6 20 2A

 

30.

 

155

 

 

Figure 6-2.

To - T¢ bound curves for several initial

positions.

30

156

allow one to determine quickly if a set of initial veloci-
ties and position will lead to a trajectory that cannot
span the flight length. For instance, suppose one set of
initial conditions we might choose to integrate corresponds

= 30 eV. oReferring to

to 00 = 0.15, To 10 eV, and T

¢
Figure 6-2, we see that this set determines a point that
lies above the bound curve labeled p0 = 0.15. This tells
us that the resulting trajectory will end up striking the
wall of the outer conductor and thus eliminates considera-
tion of these initial conditions.

The usefulness of these limit curves cannot be over-
emphasized. They allowed us to eliminate quickly a very
large number of integrations and to focus in great depth
on potentially "successful" trajectories. By successful
I mean trajectories that end with the final 9 position
less than the radius of the CEMA detectors, which was
taken as 1.25 cm, and the Z position equal to 110 cm,
which is the length of SIEGFRIED.

When an event was successful, the time, initial and
final positions, and velocities and total energy were
filed. The results are best described graphically as
shown in Figure 6-3. The curves here represent successful
events for various initial p values as functions of the

quantities T and TD defined earlier. The interpretation

¢

of these curves is that, for a set of initial conditions

corresponding to a point below the curve (for the associated

157

 

 

 

l5
, =0.|5
11¢ P C33
(eV) . p.” Qo.45
IO .0 Pa .05
5__ 9
O I
C) 5 IC) l5
-Ep (9\/)

Figure 6-3. Tp - T¢ hit efficiency curves.

158

00 value), we will obtain a successful event. In a manner
of speaking, these plots represent detector hit efficiency
curves. Comparison of Figures 6-2 and 6-3 seems to imply
that both the limit curves and the hit efficiency curves
have similar structure or relationships between T¢ and

To' If one considers the area bounded by a particular
curve in either Figure 6-2 or 6-3, we have a measure of
the efficiency of the system. Comparing equivalent areas
on each graph, it appears that roughly half the particles
that make it down the flight tube without striking the
outer wall, will be detected.

Although our system contains a number of apparently
disruptive elements, the trajectories seem to have a smooth,
consistent behavior. This is borne out by Figures 6-u
and 6—5, which are plots of the final 0 position (of)
as a function of the quantity T¢ for various values of the
p velocity. There is apparently some type of harmonic
behavior exhibited by these curves. Note that the maximum
of each curve appears closer to the origin in Figures
6-H and 6-5 in a very regular pattern dependent on the
p velocity. Actually, it appears that all the curves
(except T0 = 0) are very similar but differ in some type
of phase factor dependent on To. It would be a useful
exercise to attempt a decomposition of these curves by
Fourier decomposition. The sharp takeoff in amplitude

shown by the curves in Figure 6-5 suggests some type of

Bessel functions might be reasonable for the analysis.

159

LS"—

(cm)

 

 

0.0 J |

 

Figure 6-8. Graphs of final p position as function of T¢
(for fixed initial 0 velocities).

160

 

 

2“
J
'/
LES"' I i,
/ /
I; .
I
1 I
: / 1
x / I
P I.O— .‘\ 4': I ,l
f \ \ .
(cm) x ‘y,’ I1 I,
4 °\. '5‘ ' III
\ i. \ I I
5— " V -
. \f \l I
1 lf‘LsJ‘( I
“¥"'TP’7 \
Tpla TPJ86.9)85
l
0.00 5 IO
T4, (eV)

Figure 6-5. Graphs of final p position as function of T¢
(for fixed initial 0 velocities).

161

Initial Energy Distributions

 

In our parametric fits to the observed peak widths
discussed in Chapter IV, we assumed various forms for
energy distributions for the recoil ions. These forms
included (1) a flat energy distribution, (2) a Fermi distribu-
tion, (3) a Gamow-Teller decay distribution, and (A) mixed
F-GT distributions. To include such distributions in our
modeling study means we will be making a number of different
choices for the term NO(E,OO) in Equation 6-1. The flat
distribution means that NO is constant. The Fermi and
Gamow-Teller distributions are shown in Figure 6-6. These
plots were obtained from formulas given in (Jo63). The
distribution function for the mixed decay corresponding
to 23Mg is shown in Figure 6-7. Even though we will be
using the various distribution functions just mentioned to
approximate the solution of Equation 6-1, this does not mean
that we will need to reintegrate the equations of motion
for each distribution. That would be slow, wasteful and
redundant. As stated earlier, our real difficulty lies in
obtaining a means of evaluating the effect of the kernel
function S(E,pO,T,pf). In plain terms, we need to know

what of and T result from an initial E and p this is

03
the function of S(E,QO,T,pf). Exactly how often a par—
ticular E and p0 must be considered is the function of
NO(E,pO). Integration of the equations of motion will

yield the necessary information about S(E,pO,T,pf).

162

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NO(E,DO) is ours to vary. In representing the distributions,
we divide the energy range bounded by Equation 5-M into a
set of ten subintervals of a set length. So, in practice,

our approximation to Equation 6-1 can be expressed as

10
Pf(T) = 3:1 aiS(Ej,Tj) NO(EJ)(AEJ) (6-27)
and
10 (
P (T) = X p T ) (6-28)
f j=l f J

We have suppressed the p arguments for convenience. The
index i refers to the ith bin in the energy interval and
AEi is the width of the ith bin. The term “i is a weight
factor. For the flat distribution NO(E1)AEi is constant,

so we have

aiSJ (6-29)

165
with
SJ 5 S(EJ,TJ) (6-30)

Thus, by evaluating the predicted Pglat for a flat distribu-

tion, we are able to obtain the elements SJ which are the
most important and elusive quantities.
The combination of Equation 6-27 and 6-28 allow us to

recast our problem in the form of matrix equations.
P = S N (6-31)

We construct the S matrix from Equation 6-29 and then
proceed to apply Equation 6-31 to the various initial

distribution functions of interest.

Results of Modeling

The determination of S was accomplished as just dis-
cussed and led to a number of reasonable conclusions.
First, the TOF of a particle is completely a function of
the Z velocity within a few nanoseconds of the 5 usec TOF.
While not surprising, it is something that could not be
stated definitively before the calculations because of
the field nonuniformities. Second, the effect of the kernel
function S will not transform the flat energy distribution

into the observed form. The result is basically still a

166

square wave as we input. In a way, the resolution of the
difficulty seems to be hinted at by the first conclusion.
This tells us that the TOF is dependent on the component
of the momentum in the Z direction. Instead of using a
flat energy distribution, it seemed reasonable to attempt
a flat momentum distribution. Actually we used a flat
distribution in the Z component of the momentum since the
results are independent of the other components. In addi-
tion to the independence of the observed distribution on
the transverse components, the initial distribution in
these components are severely restricted by the limit curves
of Figure 6-2 and reduce to a constant, multiplying the
distribution in the Z momentum.

The results of the calculation using the flat momentum
distribution gives a set of functional values of P(Ti)
at unequally spaced values of T1, the time-of-flight for
the ith bin. Treating these as functional values at un-
equally spaced base points (Ti's), I used a Newton Divided
Difference Interpolation (See (Ca69)) to generate inter-
polate values at equally spaced time intervals. These
values were then used to generate the plot shown in Figure
6-8. A similar procedure was followed for the other
distribution functions. For the Fermi and Gamow-Teller.
initial distributions, the distribution functions that are
the momentum equivalents of Figures 6-2 and 6-3, respec-

+
tively, were used as the NO(P,pO). The distribution

167

 

 

 

 

 

 

2000‘—

21mg
.J
UJ
Z
2:
<1
I
L)
\\
f2

- ~46

Z
:3
O
C)

75-:

' I35 ——-4

O _- - - I ., - -11-.. -.I nsec 1 _-..--.-..l mi]
4.5 4.7 4.9 5.|
TOF (psec)
Figure 6-8. Theoretical TOF peak predicted from flat

momentum distribution.

 

168

Table VI-2. Comparison of Peak Widths for Various Distribu-

 

 

tions.
Obs. flat Fermi GT mix
Peak No No No Nn
Base Width
(nsec) 1&5 135 13 135 135
Width at
1 Max 90 75 120 130 120
(nsec)
Width at
5 Max 20 16 70 105 100

(nsec)

 

169

corresponding to the correct admixture of F and GT decay
ror 23Mg was constructed by a linear combination of the F

and GT distribution. The form used was

Pmix = a PF + (l - oc)PGT (6-32)
here a = .7 for the case of 23Mg.

The predicted TOF distributions corresponding to the
Fermi and Gamow-Teller initial distributions are shown in
Figures 6-9 and 6-10, respectively. In Figure 6-11 is
shown the observed TOF spectrum resulting from the reaction
27A1 + 70-MeV 3He which was discussed in Chapter III.

The focus of interest is the large 23Mg peak that we are
attempting to simulate. On each of the 23Mg peaks in
Figures 6-8, 6-9, 6-10, and 6-11 are given the full widths
at the base, at .1 peak maximum and half maximum, all in
nanoseconds.

A visual comparison of the three predicted TOF dis-
tributions with the experimental 23Mg peak heavily favors
the peak shown in Figure 6-8 as the best reconstruction.
The peak resulting from the Gamow-Teller distribution
(Figure 6-10) has a tail on the short time side as the
experimental peak, but is clearly the wrong shape. In
addition, this peak is far too broad except at the base.
The Fermi distribution gives rise to a peak (Figure 6-9)

that is actually a rough mirror image of the experimental.

170

 

 

 

 

 

 

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FERMI DECAY
23
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Lu
2:
2:
<I
:I
L)
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U)
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2:
:3
<3
L)
o I .
45 47 49 SJ

TOF (nsec)

Figure 6-9. Theoretical TOF peak predicted from Fermi
distribution.

171

 

 

 

 

 

 

 

2000
GAM OW -TEL|_ER DECAY
23hflg
..I
LLI
Z
Z
<[
I
U
\
(I) HJ5-*
F_.
Z
3
O
U
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45 47 49 5J

TOF (used

Figure 6-10. Theoretical TOF peak predicted from Gamow—
Teller distribution.

172

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173

By this I mean that it has a sharp cutoff on the short

TOF side and tailing on the longer TOF side which is the
opposite of the experimental 23Mg peak. The peak in Figure
6-8 results from a flat momentum distribution and is an
excellent reproduction of the experimental peak. On the short
TOF side there is the long bowed-tailing that is so obvious
in Figure 6-11. Then the distribution comes to a sharp
spike with the sudden dropoff on the long TOF side in the
same manner as the experimental peak. The peak resulting
from the initial distribution from Equation 6-32 has a shape
similar to that of Figure 6-9, except that it is broader at
.1 maximum and half maximum (See Table VI-2).

Quantitatively the Nglat (flat initial momentum distribu-
tion) gives the best predictions also. Referring to Table
VI—2, the base width predicted is within 6.9% of the experi-
mental width, the width at 0.1 maximum is within 16% and
the full width at half-maximum (FWHM) is 25% off. This com-
pares very well with the various widths predicted from the
Fermi, GT and mixed distributions. These last three
distributions result in widths at .1 max and half—max that
are from 50% to 500% in error! The base widths for all
the distributions are all the same since they correspond
to the same definite limiting type events. That is, the
base widths are set by the time differences between a
particle starting out with 0.0 initial recoil energy and a

particle starting out with the maximum velocity in the Z

l7“

direction. Both cases can only occur in one way and so
are independent of the initial distribution assumed.

In many experimental instances, one uses the position
of the peak maximum as the best estimate of the position
of the centroid and then uses the associated time-of-flight
to calculate the mass from Equation 3-2 (in Chapter III).
The position of the peak maximum for the observed 23Mg is
M910 nsec. The "flat momentum" peak has a predicted posi—
tion of the maximum at 4915 nsec and the Gamow-Teller peak
at “908 nsec; both are in good agreement with the real peak.
The Fermi distribution peak of Figure 6-9 has its maximum
at #800 nsec, which is well off the correct position. In
fact, if we use Equation 3-2 with TOF = “.8 msec and HV
= 6 kV, we get A = 22.0 —- off by one complete mass unit.
This is not the case for the observed peak but does demon-
strate the care that should be taken in choosing positions
(on the peak of interest) to calculate the mass.

Although a little surprising, the simple flat distribu-
tion is the most successful in explaining the observed
23Mg peak. The success is fairly impressive; (1) the
peak position is correct to 5 nsec, which is a .1% error,
(2) the base width is correct to 6%. Thus, theoretical
broadening accounts for nearly 94% of the observed peak
broadening. The simple model demonstrates the importance
of the recoil energy on the observed peak broadening very

clearly.

CHAPTER VII

CONCLUSIONS

In this thesis I have tried to present experimental
results and theoretical analyses concerning the SEIGFRIED
Recoil Mass Identification system. Before closing, I will
briefly summarize the work presented and the conclusions
drawn throughout this work.

First, it was demonstrated that the system works well
in a number of different mass regions, although the chemi-
cal nature of the HeJRT system can be a real concern.
Second, the observed TOF peaks were shown to contain in-
formation about the recoil ion's initial kinetic energy,
which is a direct measure of the associated decay energy.
To demonstrate the relation between the recoil energy and
the peak shapes, I adopted a double-edged approach. First,
I showed that a parameterization of the observed peak
parameters yields reasonable predictions of the recoil
energy and the decay energy. Second, I developed a reason-
able theoretical model using numerical methods and used
this model to simulate a theoretical peak corresponding to
23Mg which I observed experimentally. The results of the

modeling study were very successful. Using a simple

175

176

distribution, the experimental peak shape is reproduced
very closely. The broadening of the peak was shown to be
due to the recoil energy, and the theoretical broadening
from this was 95% of the observed broadening. The overall
success of our study in the experimental and theoretical
studies proves the viability of using TOF system such as
SIEGFRIED to identify mass products (measure A) and measure
decay energies through the recoil energy. Combining these
two measurements in a simple system provides the nuclear
experimentalist with very useful and different tool for

the study of nuclei far from stability.

APPENDICES

APPENDIX I

As an approximate model of the time-of4flight system,
we will use the geometry shown in Figure AI-l. In region
I, the acceleration zone, there is a uniform electric field
E with only a Z component. In region II there is a field-
free drift zone of length 2. Thus, a particle of mass m
and charge q is uniformly accelerated across region I and
then drifts through region II.

Using cylindrical coordinates (p,¢,Z), the classical

Lagrange equations of motion are:

 

fi- (m6) = moi»2 - q 3—‘3- (AI-1)
§%-(m92$) = Oq %% (AI-2)
£5- <mé> = -q 3% (AI-3)

In region I
E = g 2 (AI-u)
¢ = V(l - Z/d) (AI-5)
ad}- (mz) = q g (AI-6)

177

178

.H-H< mpswfim

 

 

I-
*-- ---
N ’

 

 

 

 

179

Integrating twice with respect to time,

 

2
- - 9.12.

z - Z0 + Zot +md 2 (AI-7)
where Z0 is the velocity in the Z direction at t =-O. We
consider all particles starting from Z - 0 so ZO = 0.

Z = Z t + QY~§E (AI-8)
o md 22
Solving for t, we have
_I£1_° 42 2gVZ
t - qV IZO + Z0 + md (AI‘9)

When Z = d, the end of region I, the particle undergoes no
further acceleration. Defining T = t(z=d), the acceleration

time, we have from Equation AI+9

 

md ° 2 2 V
=— - + + -
r qV { ZO J20 -%— (AI 10)

Since 0 is independent of o and p in both regions I

and II, the remaining equations of motion are simple.

180

First, 3¢/B¢ = 0, so Equation AI-2 tells us that mo2$ is
a constant of motion. Actually, this is the angular momentum
about the Z axis. Thus, we have

L E mp2¢ = constant (AI-ll)

Using L to eliminate ¢ from Equation AI—l, we have for

the 9 equation

2 2
g_% = £5 3? (AI-l2)
dt m 0

Equation AI—l2, in spite of its simple form, is an example
of a second-order nonlinear differential equation. While
nonlinear differential equations of any order are notoriously
difficult in general, this particular equation can be solved.
Since appropriate references were not found we will solve
this in a step by step manner!
Defining a new independent variable x,
Lt

X = IT (AI-l3)

Equation AI-12 becomes

__5 = __ (AI-1A1

181

Defining u = g%, we have

 

 

 

 

du l
0
Integration gives
2 _ 2 _ 3; _ j;
u uo - 2 2 (AI-l6)
oo p
or
9.9.- 2 .1. ._1._
dx - uO + 2 2 (AI-l7)
O
0
Integrating from x = 0 to x, we have
x = 2{ .J 013 +013) 2 -1 - uopo} (AI-18)
1+(uO OOp )
Inverting to solve for p, we get
[(u2 + 1/p2) x+u p ]2 + l
p(x) = ___o 0 0 0 (AI-19)
u2 + l/ 2
o 0o
L, and Do

or in terms of t,

182

 

 

JE(5303+(L/M)2)t/p:+éoool2 + (L/m)2

p(t) = 00

(AI-2o)
(popo)2 + (L/m)2

In region II there are no external forces acting on
the particle, so the equations of motion are trivial. The
only one we need to look at is Equation AI-3. For a par-

ticle "starting" from Z = d we have

2
g_% = o (AI-21)
dt
Z = VZ a constant (AI-22)
and
Z = vzt + d (AI-23)

where t is measured from time of passing poind d. From
the discussion concerning region I, we know that when Z = d

its velocity Zd is

 

2 = J 22 + 3%?- (AI-2A)

Thus

183

We are interested in the time a particle takes to drift
through region II. Calling this time T from Equation AI—23,

we have

 

T = (l - d)/VZ (AI-26)
T = (z — d)/ z: + §%y_ (AI-27)

This will be called the drift time. The total time of
flight is the sum of the drift time and the acceleration

time:

 

 

 

TOF = T + T (AI-28)
(2 d) md 2 2 v
TOF - + av {-20 + ‘Jz + —9—- } (AI-29)

APPENDIX II

A 8 decay is actually a three-body event, involving
the decaying nucleus, the 8 particle, and a neutrino. If

we define the following momenta and energies:

F momentum vector of daughter nucleus of mass M
E electron momentum vector

3 neutrino momentum vector

Eo total decay energy

E total electron energy

Ev total neutrino energy

R

recoil energy

then the appropriate conservation laws take the forms:

E0 = E + Ev + R (AII-l)
q = Ev/C (All-2)
E2 = p2C2 + m2cu ' (AII-3)
R = r2/2M (AII-A)

18A

185
? = E + E (All-5)
2 2 + 2

q + 2pq cose (All-5)

since R << E + Ev

so = E + EV (AII-7)
2 E3 (EO-E)2
C C

According to relativistic energy-momentum relation for the

electron,

p2 = (E2 - m2o“)/c2 (All-9)

 

M
II

(E -E) 2 2 u
(E2-m2cu)/02 + (EO‘E)2/C2 + Co JE 'c'm C 0039

(AII-lO)

 

2 .
R = gfi" 1'2 (E2-m 0“) + (EC-E)2 + (EC—E) \1E2-m2ci cos

2M0
(AII—ll)

We want R in terms of the kinetic energy of the electron,

since for a“ decay the QB in the literature represents

 

186

the maximum kinetic energy of the electron. Defining

l as the electron kinetic energy, we have

 

 

R = 1'2 T(T+2mc2) + (TO-T)2 + (TO—T) VH<T+2mc2) cose
2Mc
(AII-l2)
R is a maximum for 6 = O and T - To, so
T (T 2)
o o + 2mc
R = (AII-l3)

 

Max 2Mc2

If we have To and mCZ in units of MeV, we can get RMax

in eV by the formula

- 536.8
R - A (TO(TO + 1.022)) (AII-lu)

where A is the mass number of the nucleus.

 

APPENDIX III

In any interpolation scheme, one starts with values
of a function f(x), that is known (or conveniently ob-
tained) at definite values of the independent variable x.
The actual interpolation is estimating the values of the
function for arguments between the base points x0, x1,
x2,..., xn at which the functional values of f(xo), f(xl),
f(x2), ..., f(xn) are known. The most common method used
to solve such problems is to assume that a polynomial
p(x) is a suitable approximation to the function f(x).
This polynomial is forced to agree with the known values
of f(x) and then used to predict an approximation to f(x)
at x values that do not coincide with the base points.

The algorithm used for the field interpolations in
this thesis was the Lagrangian Double 3-Point Interpolation
scheme. The basis for this algorithm is that a linear
combination of two quadratics is used for the interpolation
polynomial. Referring to Figure AIII-l, the two quadratic

polynomials are ¢l(x) and ¢2(x).

ax + bx + c (AIII—l)

¢1(X)

kx + 1x + m (AIII-2)

¢2(X)

187

 

188

f(x)

(2: )
«2: 4"

’43 "'_""‘~ —.
f0) 4’2”)

 

 

 

-3 -2 -l O+IL +2 *3 x

Figure AIII-l.

189

At the points x: =l,0,+1 we have for unit Spacing

¢1(-1) = f(-1) (AIII—3a)
¢2(o) = f(O) (AIII-3b)
¢l(1) = f(l) (AIII-3c)

and at the points x: 0,1,2 we require

¢2(o) = f(O) (AIII-Aa)
¢2(l) = f(l) (AIII-Ab)
¢2(2) = f(2) (AIII-Ac)

These sets of simultaneous equations can be solved to
obtain the coefficients a, b, c, k, 1, and m in terms of

the f(xi)'s. Now the interpolation polynomials become

¢1(x) = 2 x + 2 x + f0 (AIII-5)

 

 

(f +f —2r ) (Af —f -3f )
¢1(x) = O S l x2 + l 2; O x + f0 (AIII-6)

 

 

with fi 2 f(xi)o

190

For the Lagrangian Double 3-Point algorithm we use an

interpolating polynomial, F(x) which has the form

F(X) = (l-x)¢1(X) + X¢2(x) (AIII-7)

For our purposes we also need the derivative of F(x).

¢
Liécfl : -¢l(x) + (l-X)%?1(X)
d¢2 8
+ ¢2(x) + X 71-12-(X) (AIII' )

More convenient forms for use with computer routines

f(0)°{1 + 3x3/2 - 5x2/2}

F(x)

f(l)°{X/2 + 2x2 - 3x3/2}

+

+

f(2)-{x3/2 - x2/2}

f(-1)-{x2 - x/2 - x3/2} (AIII-9)

+

dF(x)

dx = f(O)-{9x2/2 = 5x}

+ f(l)-{l/2 + ux - 9x2/2}
+ f(2)-{3x2/2 - x}

+ f(-l)°{2x - 1/2 - 3x2/2} (AIII-lO)

1": “"“"'fi_

APPENDIX IV

To obtain approximate expressions for derivatives,
we made use of a Taylor expansion. Consider development

of the series for f(xO + Ax,yo) about the point (xo,yo):

 

 

 

2 2
3 f
f(xo+Ax,yO) = f(xo’yo) = f(Xo’yo) + AX 5% + (if? : 2
. x
3 3 n n
+(Ax) 8f +H.+(Ax) 3f +0“ (Anni)

where, |xo,yO means the derivative is evaluated at the

).

Rearranging and dividing by Ax, we have

point (xo,yO

3f - f(xO+Ax,yO) - f(xo,yo) Ax 2f
' IT

Q)

X AX
O’yo X x

Q)
N
u

o’yo (AIV—2)

plus higher terms in Ax, or,

191

192

f(x +Ax,y ) - f(x ,y )
.3; :: 0 0 O O + 0(Ax) . (AN-3)
x yO AX

 

0’

where 0(Ax) represents the order-of-magnitude error as-
sociated by the given approximation. Equation AIV-3 repre-
sents the "forward-difference" approximation to the deriva-
tive.

We could also obtain a "backward-difference" approxima-

 

tion by expansion of f(xO-Ax,yo):

 

 

 

 

2 2
3f Ax a f
'xofiyO xo’yo
(Ax)§_33f n (Ax)n ant
+ ... + (-1) + .,
3! 3x3 n! axn
xo’yo xo’yo (ATV-A)
which can be rearranged to yield
~ f(x y ) - f(x - x y )
3.1: ~ 0’ 0 0 ’0 + 0( x) . (AIV-S)
x x y Ax

o’ o

193

As indicated, both the forward- and backward-difference
approximations carry associated errors that are on the order
of Ax. To obtain a better approximation to the derivative,

one can subtract Equation AIV-A from AIV-l to obtain

 

 

_ 8f
f(xO+Ax,yO) - f(xO-Ax,yo) - 2Ax 5;
XO’yO
3 3
8x xo’yo
plus higher terms.
This can be rearranged to yield
f(x +Ax,yo) - f(x -Ax,y )
%§' : O 2Ax O O + 0((Ax)2) . (AIV—7)
xo’yo

This is called the "central-difference" approximation.
Comparing the cutoff errors associated with approximations
AIV-3, AIV-5, and AIV-7, it is evident that for Ax < 1,
expression AIV-7 carries the smallest error.

To obtain an approximate expression for the second
derivative, add Equation AIV-l and AIV-A and divide by
(Ax)2 to obtain

19L!

f(xO+Ax,yO) + f(xO—Ax,v ) 32

 

 

 

0.0 = f
2 2f<xo’yo) + 2
(Ax) 3x
Xo’yo
2 A
A
+ (1X2) 3 f; , (ATV-8)
8x
Xo’yo
plus higher terms,
or,
32f ~ f(xO+Ax,yO) + f(xO-Ax,yo) - 2f(x0,yo) + O(( 2)
5 ~ 2 Ax)
3x x y (AK)
0’ o (AIV-9)

As in the "central-difference" approximation to the first
derivative, this expression has an associated error on the

order of (Ax)2.

BIBLIOGRAPHY

(Ac70)

(Am69)

(B173)

(Ca69)

(Ca63)
(Ed76)

(F060)

(Gr65)

(Ha62)

(Jo63)

(Ju71)

(K073)

(LeB76)

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V

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