• TABLE OF CONTENTS
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 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 Preface
 The linear electron accelerato...
 Equations of longitudinal...
 Solution of equations by digital...
 Solutions of equations of...
 Results and conclusions
 Appendix
 Reference
 Biographical sketch
 Copyright














Title: effects of space harmonics and attenuation on longitudinal electron motion in a linear accelerator.
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Title: effects of space harmonics and attenuation on longitudinal electron motion in a linear accelerator.
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Figures
        Page v
        Page vi
    Preface
        Page vii
        Page viii
        Page ix
    The linear electron accelerator
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
    Equations of longitudinal motion
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
    Solution of equations by digital computer
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
    Solutions of equations of motion
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
    Results and conclusions
        Page 76
        Page 77
        Page 78
        Page 79
    Appendix
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
    Reference
        Page 94
        Page 95
    Biographical sketch
        Page 96
        Page 97
    Copyright
        Copyright
Full Text










THE EFFECTS OF SPACE HARMONICS AND

ATTENUATION ON LONGITUDINAL


ELECTRON MOTION


IN A


LINEAR


ACCELERATOR


DONALD P. MOONEY







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








UNIVERSITY OF FLORIDA
August, 1963
















ACKNOWLEDGEMENTS


The author wishes to express his sincere appreciation to

his supervisory committee for their counsel and encouragement.

lie is especially indebted to Dr. A. Ii. Wing and Dr. V. W. Dryden

for their constant guidance and to Prof. W. F. Fagen for his

project supervision. He also wishes to convey his gratitude

to the personnel of the University Computing Center who have been

helpful in the completion of this work.

The author wishes to thank his wife Elizabeth for en-

couraging him throughout his graduate studies. lie also wishes

to thank Dr. Henry S. Blank for his friendship and assistance.

















TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ------------------------------------------------- ii

LIST OF FIGURES -------------------------------------------------- v

PREFACE --------------------------------------------------------- vii

Chapter

I. THE LINEAR ELECTRON ACCELERATOR -------------------------- 1

Introduction --------------------------------------------
The University of Florida Linear Accelerator ------------ 2
Accelerator Structures ---------------------------------- 6

II. EQUATIONS OF LONGITUDINAL MOTION ------------------------- 13

Solution of Field Equations ------------------------------ 13
Conventional Equations of Motion ------------------------- 20
Mode of Operation --------------------------------------- 24
Solution of Field Equations Considering Space Harmonics--- 25
Attenuation ---------------------------------------------- 32
Equations of Longitudinal Motion ------------------------ 33
Summary -------------------------------------------------- 34

III. SOLUTION OF EQUATIONS BY DIGITAL COMPUTER --------------- 36

Introduction --------------------------------------------- 36
Bessels Equation for Complex Argument -------------------- 37
Space Harmonic Amplitude --------------------------------- 43

IV. SOLUTIONS OF EQUATIONS OF MOTION ---.-------------------- 49

Introduction --------------------------------------------- 49
Traveling Wave Prebunchr -------------------------------- 49
The Uniform Section -------------------------------------- 57

V. RESULTS AND CONCLUSIONS --------------------------------- 76

Traveling Wave Prebuncher Calculations ------------------- 76
Uniform Section Calculations ----------------------------- 76
Conclusions from Traveling Wave Prebuncher Results-------- 77
Conclusions from Uniform Section Results ----------------- 78
Summary of the Most Significant Conclusions -------------- 78



iii













TABLE OF CONTENTS continued
Page

APPENDICES --------------------------------------------------------- 80

A. IBM 709/7090 FORTRAN PROGRAMS ------------------------------ 81

B. METHOD FOR CALCULATING SIGNIFICANT HARMONICS --------------- 92

LIST OF REFERENCES ------------------------------------------------- 94

BIOGRAPHICAL SKETCH ------------------------------------------------ 96
















LIST OF FIGURES


Figure Page

1. Disk-Loaded Circular Waveguide --------------------------- 3

2. Simplified Block Diagram of Linear Electron Accelerator
System ---------------------------------------------------- 5

3. Buncher Section ------------------------------------------- 10

4. A. Coordinate System ------------------------------------- 14
B. Schematic Drawing of Disk-Loaded Iterative Circular
Waveguide --------------------------------------------- 14

5. Phase Lag, A, of Electron with Respect to the Stable
Phase Position ------------------------------------------ 23

6. Brillouin Diagram ---------------------------------------- 26

7. Instantaneous Distribution of E( ,a) at t = T/4 ----------- 29

8. Main Program and Subroutines ------------------------------ 38

9. Flow Diagram of Subroutine Bessel ------------------------- 41

10. Subroutine Bessel Continued ------------------------------- 42

11. Flow Diagram of Main Program ------------------------------ 47

12. Main Program Continued ------------------------------------ 48

13. Electron Phase Angle A vs. Normalized Axial Distance 5
6w= 0.5, r = 0, Nonharmonic Analysis -------------------- 51

14. Electron Phase Angle A vs. Normalized Axial Distance S
Sj= 0.5, r = a/4, Nonharmonic Analysis --------------- 52

15. Electron Phase Angle Avs. Normalized Axial Distance ?
/6uJ= 0.5, r = a/2, Nonharmonic Analysis --------------- 53

16. Electron Phase Angle Avs. Normalized Axial Distance
A/)=0.5, r = 0, Space Harmonic Analysis ------------------- 54

17. Electron Phase Angle Avs. Normalized Axial Distance
16tu= 0.5, r = a/4, Space Harmonic Analysis ------------- 55











Page


18. Electron Phase Angle A vs. Normalized Axial Distance 5
6ow= 0.5, r = a/2, Space Harmonic Analysis ----------------

19. Normalized Electron Energy vs. Normalized Axial Distance 5
6,=0.5, r = 0, Nonharmonic Analysis --------------------...

20. Normalized Electron Energy d vs. Normalized Axial Distance
6o=s0.5, r = a/2, Nonharmonic Analysis----------------

21. Normalized Electron Energy vs. Normalized Axial Distance
S=0.5, r = 0, Space Harmonic Analysis-----------------

22. Normalized Electrin Energy K vs. Normalized Axial Distance
6,,=0.5, r = a/2, Space Harmonic Analysis ------------------

23. Normalized Electron Energy K vs. Normalized Axial Distance S
/, =l.0, r = 0, Nonharmonic Analysis ------------

24. Normalized Electron Energy vs. Normalized Axial Distance
3S,=1.0, r = a/4, Nonharmonic Analysis -------

25. Normalized Electron Energy K vs. Normalized Axial Distance
61,l.0, r = a/2, Nonharmonic Analysis ------- --.........

26. Normalized Electron Energy I vs. Normalized Axial Distance S
)6U1.0, r = 0, Space Harmonic Analysis ----------

27. Normalized Electron Energy vs. Normalized Axial Distance 5
p-=1.0, r = a/4, Space Harmonic Analysis -------------..--.

28. Normalized Electron Energy Yvs. Normalized Axial Distance
/6,=1.0, r = a/2, Space Harmonic Analysis-------------------

29. Electron Phase Angle A vs. Normalized Axial Distance
d =1.0, r = 0, Nonharmonic Analysis -----------

30. Electron Phase Angle 6 vs. Normalized Axial Distance S
r1=1.0, r = a/4, Nonharmonic Analysis---------------------

31. Electron Phase Angle A vs. Normalized Axial Distance S
/4,=1.0, r = a/2, Nonharmonic Analysis----------------------

32. Electron Phase Angle \vs. Normalized Axial Distance S
/z=1.0, r = o, Space Harmonic Analysis ----------------...

33. Electron Phase Angle A'vs. Normalized Axial Distance
/e$=1.0, r = a/4, Space Harmonic Analysis ----------....

34. Electron Phase Angle & vs. Normalized Axial Distance
/, ,1.0, r = a/2, Space Harmonic Analysis --------------
















PREFACE


This paper presents a method of analyzing longitudinal elec-

tron motion in a disk-loaded circular waveguide structure, taking

into account the effects of space harmonics and structure attenua-

tion. A generalized computer program is developed which is capable

of calculating electron phase angle and energy versus distance in

the type of disk-loaded circular waveguide structures used in

linear accelerator applications. The advantage of using computer

techniques to calculate electron phase angle and energy in accelera-

tor structures is that detailed numerical results can be obtained

to any desired accuracy. When experimental methods are used the

techniques involved are extremely complicated and expensive. These

experimental results are frequently ambiguous and their accuracy

depends upon the ingenuity of the investigator. Data from such

experiments must generally be interpreted with considerable uncer-

tainty due to practical limitations imposed by existing techniques.

Previously existing analyses of longitudinal electron motion

in a linear electron accelerator do not take into consideration

space harmonics of the axial field existing in the accelerator struc-

ture, and the attenuation in the structure. Both these factors are

important in dealing with the problem of an electron beam with a finite

radius which is treated in this paper. Here the equations governing

longitudinal motion for a traveling wave prebuncher and a uniform













accelerator section are derived considering space harmonics, struc-

ture attenuation, and off-axis trajectories. These equations are

then solved using the IBM 709 digital computer and the results are

compared to existing data. This extension of existing knowledge

can be used to explain the consequences of previous approximations,

and offers greater accuracy for future design.

In Chapter I the basic operation of a linear accelerator is

discussed. In addition, the various components which can be combined

to form an accelerator structure are described, particularly those

structures which are analyzed in later chapters.

In Chapter II a solution is developed for the axial field

component in a traveling wave structure and the conventional equa-

tions governing longitudinalmmotion in a sinusoidal field distribu-

tion are derived. The solution for the axial field component is

then modified to account for the existence of space harmonics and

structure attenuation in order that new equations governing longi-

tudinal motion can be developed including these factors.

In Chapter III the computer programs used to solve the equa-

tions of longitudinal motion are explained. In Chapter IV the re-

sults of computer solutions considering space harmonics and structure

attenuation are presented along with computer solutions of the con-

ventional equations for two accelerator structure configurations.

Chapter V contains a summary of significant differences ob-

served between the results of the conventional analysis and the results

of the analysis considering space harmonics and structure attenuation.


viii













The results of the analysis developed in this paper illustrate

that in a prebuncher structure space harmonics have a significant ef-

fect on the phase distribution of the electrons and electron beam ra-

dius has a small effect on the electron energy distribution, compared

to results predicted by previous theory. Results also verify the as-

sumption that in a uniform accelerator structure the presence of

space harmonics results in a reduction in energy gain with negligible

effect on the phase distribution of the electrons, compared to that

obtained by previous theory.

All phase angle and energy calculations conducted for both the

conventional analysis and the space harmonic analysis developed in

this paper considered the effects of attenuation. This represents

an improvement over the previous analysis in which the effects of

attenuation on off-axis trajectories are neglected. Computations

neglecting the effects of attenuation were not made for this paper

since a mathematical model neglecting attenuation does not represent

a physically realizable system and therefore the computer time would

not have been justifiable.
















CHAPTER I

THE LINEAR ELECTRON ACCELERATOR


Introduction

Recent developments in the design and construction of multi-

megawatt klystron tubes have made available sources of electromag-

netic energy in a frequency range not previously utilized in linear

accelerator applications. The Nuclear Engineering Department of the

College of Engineering at the University of Florida sponsored the

construction of a 10 million electron volt (Mev) Linear Electron

Accelerator designed to operate at a frequency of 5760 megacycles

per second.

As the name implies, a linear accelerator is a device in which

charged particles are accelerated in a straight line. This fact

differentiates a linear accelerator from an orbital accelerator in

which angular acceleration is imparted to a particle moving in a

curved path. In the Linac (Linear Electron Accelerator) acceleration

is accomplished by energy exchange from a traveling electromagnetic

wave to an electron beam which coexists with the wave. This fact

distinguishes the traveling wave linear accelerator from the stand-

ing wave or static (Van de Graff) types.

Since an electron has finite mass, its velocity can never

equal the velocity of light and the phase velocity of the traveling

wave must be reduced so that energy can be transferred to the elec-

tron. Reduction in the phase velocity of an electromagnetic wave













can be achieved by many different structures. The disk-loaded

circular waveguide shown in Figure 1 is the simplest configuration

capable of transmitting enough microwave power to achieve reasonable

electron accelerator. This is the type of structure which will be

investigated in later chapters.


The University of Florida Linear Accelerator

Figure 2 is a simplified block diagram of a typical linear

accelerator system. In the University of Florida "Linac" the high

voltage power supply modulator and associated control circuits furnish

three outputs. One output is an 11-megawatt high voltage pulse five

microseconds long at a repetition rate of 60 pulses per second to

the klystron rf source. The second output is a timing pulse to the

stabilized frequency source. The third output is a pulse to the

electron gun control delayed in time with respect to the other two

outputs in order that the accelerator structure may fill with rf

energy from the klystron before the electron beam is turned on. This

"fill time" is required because in a slow wave structure the group

velocity is generally a small fraction of the phase velocity.

The frequency of the source must be stabilized to within 1

part in 6000 since the phase velocity of the traveling wave in the

accelerator structure is very sensitive to the frequency of the

power source. The rf signal is initially generated by a reflex

klystron operated from a stabilized power supply. This signal is then

amplified by two stages of pulsed klystron amplifiers to obtain the











































Figure 1. Disk-Loaded Circular Waveguide













pulse power of 10 kilowatts, necessary to drive the main klystron

amplifier (pulse power times pulse duration is the energy of the

pulse).

The klystron amplifier indicated in Figure 2 is a SAC-225

three-cavity klystron furnished by the Electronic Tube Division of

Sperry-Rand Corporation. Pulses with a power of 11 megawatts are

supplied to this klystron by the modulator. A pulsed rf input signal

of 10 kilowatts power is furnished by the stabilized frequency source.

This high-power klystron has output pulses of 3 megawatts power.

This rf power is supplied to the accelerator structure where it gener-

ates the fields used for electron acceleration.

The electron gun which injects electrons into the accelera-

tor structure has a back-bombarded tantalum cathode button as a

thermionic electron source. The rear surface of the cathode is bom-

barded by electrons from a second cathode which is a directly-heated

tungsten filament. This tungsten filament and the tantalum cathode

are referred to as the "inner diode." The bombarding electron stream

supplies the energy required to heat the tantalum cathode. The "outer

diode" is formed by the tantalum cathode and an anode immediately in

front. This anode is pulsed to turn on the electron beam and the

amplitude of the voltage pulse supplied to this anode determines the

magnitude of the beam current. The entire electron gun is maintained

at a negative potential with respect to the accelerator structure

which is electrically grounded for convenience and safety. This

potential difference is adjustable from 0 to 100 kilovolts and controls






























STABILIZED
CONTROL KLYSTRON STA
FREQUENCY SOURCE







MODULATOR
AND
CONTROL CIRCUITS





Figure 2. Simplified Block Diagram of Linear Electron Accelerator System
















the velocity with which the electrons enter the accelerator struc-

ture. The Gun Control system supplies the necessary filament, bombar-

der, accelerating, and anode pulse voltages to the electron gun.

The Accelerator Structure is basically the type of slow wave

structure illustrated in Figure 1. There are several other types of

structures or combinations thereof which could be utilized in this

system, the complexity of which is dictated by the efficiency desired

and ease of construction.


Accelerator Structures

The Accelerator Structure in Figure 2 can consist of as many

as three different types of components, each named for the function

it performs. These components are a prebuncher, a buncher, and a

uniform section. A prebuncher alters the electron phase distribu-

tion while the beam velocity remains approximately equal to the in-

jection velocity. A buncher section imparts an increase in electron

velocity as well as rearranging the phase distribution. A uniform

section is one in which the phase velocity of the electromagnetic

wave equals the velocity of light (c). This section is necessary

to achieve high electron energies. The uniform section may be used

alone, as in the prototype University of Florida Linac, or in combina-

tion with either a prebuncher or buncher section, or with both.

Due to the problems encountered in high voltage dc systems

the greatest electron injection velocity into the accelerator struc-

ture that can be reasonably obtained is one half the velocity of












light (O.Sc). If electrons with this velocity are injected directly

into a uniform section only a fraction of the available electrons

will be accepted and accelerated, due to the difference in velocity

between the particle and the wave. Increasing the accelerating com-

ponent of the electric field in a uniform section will increase the

fraction of electrons accepted or bound to the wave. The limitation

now is the problem of high field emission and electrical breakdown in

the accelerator structure. A system of this type has a maximum ef-

ficiency of acceptance of 50 per cent since the fields in the ac-

celerator structure alternate in polarity each half cycle. When the

accelerating component of electric field is large enough to accelerate

all the electrons injected during one half cycle, electrons entering

the structure during the alternate half cycle will be rejected. The

reasonable thing to do is rearrange (bunch) the electron beam before

it enters the uniform section so that electrons which would normally

arrive during the decelerating portion of a cycle instead arrive with

the electrons which enter during an accelerating portion.

An ideal bunching system would generate electron bunches in

which all electrons have the same energy and phase position. Analysis

of electron trajectories has been made [1] and such a system is shown

to be physically unrealizable. For Linacs with reasonable output

energies, prebunching in phase is more important than prebunching in

energy because if all particles enter the uniform section at the same

phase the subsequent energy gain will be the same for all particles.


1Numbers in brackets refer to references at the end of the text.













The spread in energy at the output of the entire machine will be

the same as the energy spread at the output of the bunching system,

and the consequent energy differences will be insignificant compared

to the total electron energy at the output.

Klystron type prebunching is one method of accomplishing

the desired phase grouping of an electron beam. In this method the

velocity of the electrons is perturbed in a device which is usually

a resonant cavity. The perturbations result in the formation of

packets or bunches of electrons at some distance down the beam from

where the velocity perturbations occur. Murphy [2] described such

a system where the perturbing device was a resonant re-entrant cavity

using a sinusoidal gap voltage. Smars [3] described the use of a

series of sinusoidally excited gaps separated by field-free drift

spaces.

An alternate method of prebunching employs a traveling-wave

structure described by DOme [4]. The construction of this type of

device is the same as illustrated in Figure 1, with the dimensions ad-

justed so that the phase velocity of the rf traveling wave is reduced

to a value slightly greater than or equal to the velocity of the

electron beam but less than the velocity of light. In this type of

prebuncher the electric field strength is very low and therefore the

amount of rf power required is comparable to that for a single cavity

system. As electrons travel through the structure they oscillate about

a reference phase position on the wave (illustrated in Figure 5). This

reference position will be called the point of stable phase since













electrons moving away from this point experience a restoring force

directed toward this phase position. Bunching occurs when electrons

are concentrated about this point of stable phase. Optimum structure

length is determined by the phase velocity of the electromagnetic

wave, the magnitude of the electric field and the allowable spread

in phase or electron energy. It has been proposed [5] that for struc-

tures of reasonable length, the traveling-wave prebuncher is the best

available device for reducing the phase and energy distribution of

electrons. The traveling-wave prebuncher is one of the structures

analyzed in later chapters.

A type of buncher applicable to the accelerator system is a

special disk-loaded waveguide, as shown in Figure 3, tapered so that

the phase velocity and field strength of the electromagnetic wave vary

along the length of the structure. At the end of the structure into

which electrons are injected the phase velocity is generally made

equal to the electron injection velocity. The electrons do not re-

main at a constant phase angle with respect to the wave [6] but oscil-

late about a point of stable phase. In this type of structure the

velocity at which the point of stable phase travels is increased with

axial distance by increasing the length of each successive cavity.

The electrons grouped near the point of stable phase gain energy

since they are continually accelerated toward the stable phase posi-

tion. A corresponding increase in the accelerating component of the

electric field is necessary to keep the electrons in synchronism with

the wave. This increase in the magnitude of the field is accomplished












































Figure 3. Buncher Section













by reducing the size of the disk hole in each successive disk. The

results of simultaneously increasing the phase velocity and the

magnitude of the accelerating field are reduction in the amplitude

of oscillation about the point of stable phase and increased electron

energy.

The section of the accelerator structure in which the elec-

tron beam acquires most of its energy is the uniform section. This

is a disk-loaded cylindrical waveguide, as shown in Figure 1, dimen-

sioned so that the phase velocity of the electromagnetic wave equals

the velocity of light. There are two types of uniform sections re-

ferred to in the literature. A constant-gradient uniform section is

one in which the phase velocity remains equal to the velocity of light

and the field strength remains constant down the length of the struc-

ture. This is accomplished by reducing the iris diameter slightly on

successive disks to compensate for reduced power flow as a result of

power dissipated in the structure walls. The second type of uniform

section is one in which the phase velocity remains equal to the velo-

city of light and all structure dimensions remain unchanged with

length. Therefore the field strength is reduced with distance due

to losses in the structure. The advantage of the latter structure is

the simplicity of construction and associated testing. The longi-

tudinal equations of motion will be investigated in later chapters

for this second type of structure operated as a traveling-wave pre-

buncher as recommended by DOme, and operated as a conventional uni-

form section. In each case the analysis will take into consideration








12




space harmonics, off-axis trajectories, and structure attenuation.

The results of this analysis will be compared with the results of

previous theory.
















CHAPTER II

EQUATIONS OF LONGITUDINAL MOTION


Solution of Field Equations

The application of the disk-loaded circular waveguide type

of structure as a traveling wave prebuncher and as a uniform sec-

tion of accelerator structure was discussed in Chapter I. In both

cases the solutions to the field equations are the same in form with

specific differences arising from parameter variations. Figure 4 is

a schematic drawing of a section of disk-loaded circular waveguide

illustrating the dimensional notation used. The rationalized MKS

system of units will be used exclusively in this paper. The follow-

ing analysis is restricted to structures which have rotational sym-

metry.

In cylindrical structures the wave components are most con-

veniently expressed in terms of cylindrical coordinates as shown

in Figure 4A. The axis of symmetry of the disk-loaded wave guide

in Figure 4B.is oriented so that electrons travel in the positive P

direction. The structure is excited in such a manner that an axial

component of electric field is available to accelerate electrons

injected along the axis (TM mode). It is important to have exact

information about the fields in the region in which the electron

travels. However, because of the complex shape of the guide walls,

solutions of the field equations are inevitably somewhat inexact

















r
0


.e 4A. Coordinate System


Figure 4B. Schematic Drawing of Disk-Loaded Iterative Circular
Waveguide












and complicated. A brief analysis will be conducted in the conven-

tional manner so that in later sections the departure from this analysis

can be emphasized.

Several methods of analysis [7,8] have been applied to the

solution of the field equations in a disk-loaded circular waveguide.

One common assumption is that the fields in region I (r a) are the

type that exist in a circular waveguide. Various techniques were

then applied to match these fields to those existing between the

disks at r = a.

From Maxwell's equations


2.1 7XT= L T)-


and


2.2 7XVXE = V(V-E) -V2E,


where = electric field (volts/meter)

IT = magnetic field intensity (amps/meter)

P = permeability of vacuum.


Equations 2.1 and 2.2 can be combined, resulting in


2.3 V(V.() V2VXT)


By definition


2.4 VXiT = 7 + _- ((T),
)t


where ( = permitivity of vacuum.












As in other analyses the effects of space charge will be

neglected. This assumption is justified by the fact that even in

low-gradient structures the coulombic repulsion force is very small

compared to the forces exerted by the fields existing in the struc-

ture. As a typical example, consider the specifications at the

input to the University of Florida Linac.


Beam Current = 0.1 ampere.

Beam Diameter = 2.54 x 10-3 meters.

Electron Velocity = 1.498 x 108 meters/second (0.5c).

Fundamental Phase Velocity = 2.997 x 108 meters/second (c).


It has been shown [9] that for electrons injected with a velocity of

0.5c into a uniform section operating at a frequency of 5670 mega-

cycles, the lowest peak value of the axial component of electric

field for which electrons will be accepted is 4.388 x 106 volts/meter.

Under these conditions the ratio of the radial electric field due

to charge contained in the electron beam to the peak value of the

radial component of electric field in the structure is approximately

2 x 10-3. Therefore the presence of the electron beam is assumed to

have no effect on the fields in the structure and trajectory analyses

are conducted from a ballistic approach.

In a homogeneous nonconducting medium


2.5 VXT= i .













Substitution of equation 2.5 into equation 2.3 yields

2-.
2.6 V(V.E) -V2 =p .
at2

Since the presence of the electron beam is assumed to have no effect

on the electric field distribution, e = 0 and


2.7 V'*E = 0.


Therefore equation 2.6 becomes the wave equation

b2F
2.8 72E 8=2-.
t2

In cylindrical geometry the 3 component of equation 2.8 is

2EL + 1 21, + 1 2L 2 1: [ 2121
2.9 +
2 ,2 r ar r2 a2 2 t


Assuming Ez of the forward traveling wave to be of the conventional

form R e[L 0(r)eJt" 3 and assuming operation in the lowest order TM

mode, equation 2.9 becomes

2.10 d2E,(r) + 1 dE,(r) =-K 2E (r)
dr2 r dr c o

where K2 = K2 +

K = o/c,

r = I + jP, and

c is velocity of light (meters/second), I is attenuation con-

stant (nepers/meter), and 8 is the phase constant (radians/meter).













Initial conditions are


Eo(r3 = Eo(0) magnitude of the field on axis at = 0,
fr=0


do(r) 0.
dr
r=0


From wave guide theory it is known [10] that the remaining nonzero

field components can be expressed in terms of Ez.


2.11 Er =- EZ
c


2.12 11 = .
0 K2 br


At this point in the analysis it is generally assumed that

attenuation is negligible, as done by Chu and Hansen [7]. Therefore


2.13 r2 = 2


and


2.14 -Kc2 = K2 2.


Since


2.15 = L,
VP

where v is the phase velocity of rf wave, therefore


2.16 -K2 ()2
/.lO -K i -G I -












It is necessary that vp c in order to optimize the interaction be-

tween the traveling wave and particles of finite mass. Therefore from

equation 2.16, Kc2 0 and one solution of equation 2.10 can be written

directly for the rotationally symmetric system considered here.

2.17 Eo(r) = ClI^(Kr) ,

where o(Kcr) is a modified Bessel function of the first kind, zero

order.

For a differential equation of second order there is, in general,

a second solution with its associated arbitrary constant. The other

solution must [11] have a singularity at r = 0 and therefore can be

ignored since there is no conductor on the axis of the structure and

E must be finite on the axis.

Evaluation of equation 2.17 on axis yields

2.18 E = ReEoIo(Kcr)eJ(l't4)

Substitution of equation 2.18 into equations 2.11 and 2.12 yields

2.19 Er = Re c EoIl(Kcr)e(JtI-)3,
c

2.20 H10 = Re e -4 Eoli(Kcr)e(t-c ).

Equations 2.18, 2.19, and 2.20 are similar to those derived by Chu

and Hansen [7].













Conventional Equations of Motion

Axial electron displacement .' is measured from a transverse

plane through the center of the first disk, where electrons enter

the structure. This differs from the axial dimension which is taken

as + d/2 for simplification of the space harmonic amplitude analysis.

These variables are defined in Figure 4B.

The conventional equations, describing electron phase angle

and energy in a linear accelerator, are based on the assumption that

electrons travel on the axis (r = 0) and that radial velocity and an-

gular velocity are negligible. These assumptions are reasonable since

a solenoidal magnetic field is used to confine the electron beam until

particle energies are achieved where relativistic stiffening of the

beam occurs. Lemnov [12] has shown that variations in radial position

can be restricted to within a few per cent of the beam radius and states:

"Oscillations in the r8 plane in the cases normally encountered in

practice have little influence upon the basic motion along the axis."

Considering only the axial component of force


2.21 F = (mc


The mass m of the electron can be determined from the well known

equation


2.22 m= -
(1 e2)1/2













wherein mo is the mass of electron (kilograms) at rest and fe is the

axial electron velocity divided by the velocity of light.

Let a normalized mass be defined as


2.23 Y= m/mo;


therefore


2.24 = (1 2)-1/2
e

Distance can be normalized with respect to the guide wavelength

of the fundamental component of the electric field by the ratio


2.25 = '/g,


where is measured from the transverse place at which electrons

are injected. The prime notation is to differentiate y' from

which is measured from the center of a cavity for subsequent space

harmonic amplitude determination.

21v
2.26 g= -1 .


Equation 2.21 can be rewritten using equations 2.23, 2.24, and

2.25.


2.27 F = (moc2


The force exerted on an electron by the axial component of the elec-

tric field is described by defining the phase lag angle delta (a) as

the relative phase of the electron with respect to the point of stable













phase as shown in Figure 5. The point of stable phase travels down

the structure at the phase velocity of the fundamental space harmonic.

For each electron, A is measured from the nearest point of stable

phase at the instant of injection and varies in accordance with the

integrated difference between the electron velocity and the phase

velocity (equation 2.30). From Figure 5 the axial electric field

force acting on the electron can be described by


2.28 F = eE? sinA,


where e is electron charge coulombss). Substitution of equation

2.28 into 2.27 yields


2.29 d -= -sinA,


where o = e normalized energy gain per wave length.
m oc2

A second equation describing longitudinal motion of the elec-

tron can be derived by considering the change in Delta (A) caused by

a difference in the velocity of the electron and the phase velocity

of the electric field.


2.30 dA = (ve Vp) 2-dt,



where ve is the electron velocity and vp is the phase velocity of

the electric field.
































































Phase Lag, 6, of Electron with Respect to the Stable
Phase Position


Figure 5.












A normalized phase velocity can be defined as the ratio of

the phase velocity of the electric field to the velocity of light.


2.31 /o =Vp/c.


Applying equations 2.31 and the definition of/'g 2.30 results in


2.32 d = 2 1 _1 .


where A is defined such that when A = 0 at = 0, then r= d/2 and

t = 3T/8.


Equations 2.29 and 2.32 are the two simultaneous first order

differential equations which describe longitudinal electron motion,

Use of these equations in conjunction with equation 2.18 is the con-

ventional method of solving for electron motion in a linear accelerator.


Mode of Operation

In the literature pertaining to the theory and operation of

linear electron accelerators the word "mode" has two connotations.

In one application the word mode is used to describe the types of

waves associated with the propagation of electromagnetic energy

such as transverse electromagnetic (TEM) waves. A second application

is the use of the word mode to describe the phase shift per section

in a periodic structure. For example a cavity length equal to Ag/4

results in ff/2 phase shift per cavity, referred to as operation in

the If/2 mode.












Figure 6 is a plot of frequency vs. the phase shift per sec-

tion showing the lowest frequency band of the infinite number of

pass bands which exist for a disk-loaded waveguide. The manner in

which this diagram is constructed is discussed by Ginzton [13] who de-

scribes how the group velocity and phase velocity of a wave propa-

gating in the structure are determined from the Brillouin diagram.

At any frequency within the pass band shown the group velocity is the

slope of the curve at that ordinate and the phase velocity equals

the slope of a line from the origin to the above intersection.

The ff/2 mode of operation is the one most generally used in

traveling wave linear accelerator structures. One reason for this

is apparent from the Brillouin diagram since at the frequency correspond-

ing to the 7;/2 mode the group velocity is maximum. This implies that

the fill time is reduced and electrons can be accelerated during a

longer portion of the pulse length. This can be an important design

factor as in the Mark III Stanford Linear Accelerator, wherein the fill

time is half the duration of each pulse. A second reason for choosing

the 7f/2 mode of operation is the ease with which measurements can be

made in the structure [14].


Solution of Field Equations Considering Space Harmonics

In the disk-loaded structure shown in Figure 1 the boundary

conditions cannot be satisfied when one assumes that each component

of the electromagnetic field is a single sinusoid. Due to field

distortions at the disks and requirements imposed by the mode of

operation the field components will be composed of a series of space














slope of tangent = g


B\ I I

II I o.



II

I I/ i I
I I I I\

I I/ I I
I II I i
I I
i !


Brillouin Diagram


Frequency
27/d
x_


-rf/2 ff/2 z2r p1
o 2 rrd/,Ag


Figure 6.













harmonics. The combined wave, including the space fundamental and

space harmonics, must travel in the positive z direction. This

requires that each space harmonic must have a phase constant lying

within a range in which is positive in the structure. A few such

harmonics are illustrated by the solid curves in Figure 6.

Considering space harmonics, the axial component of electric

field of the forward traveling wave can be more accurately described

by an infinite series of terms of the form of equation 2.18. Although

attenuation will be considered later, it is neglected here in order

to simplify the computation of the relative magnitudes of the space

harmonic components. The resultant field distribution and coordinate

system is shown in Figure 7.

n=oo
2.33 E(y,r) = Re Enl(Kcnr)ej ( n=-oo

where


2.34 /6n =o 2


From equation 2.14 Kc varies with /n, therefore it is also

subscripted in terms of n. Eon is the amplitude of the nth space

harmonic at r = 0.

Let time t = T/4 be defined as that instant at which the

fundamental (axial field) component of the forward wave has its

maximum positive value at = d, the disk spacing. The axial dis-

tribution of E(Q,r) at r = a, t = T/4, is assumed to have the












configuration shown in Figure 7 for one wave length of axial distance.

In constructing this distribution it was assumed that no field fring-

ing occurs at the disk hole and that attenuation is negligible. The

axial component of electric field E(Q,a), illustrated in Figure 7,

can be described in terms of equation 2.34.

n =0O
2.35 E(,,a) = J Enlo(Kcna) sin (Pn-.
n=-00

Experimental methods of determining the ratio of the amplitude

of each space harmonic to the amplitude of the fundamental component

are described in detail in the literature [15,16]. The ratios cal-

culated from structure dimensions by the following method are at

least as accurate as thoseobtainable by experimental methods [17].

The axial component of electric field shown in Figure 7 can be

described by a Fourier Series:

m=oo
2mr1
2.36 E(k,a) = Dm sin 2i
m=l

where L equals 4d and

/2

2 2m_ .
2.37 Dm = r E(3,a) sin L d,.

-/2

Equation 2.37 can be written in terms of structure parameters.
3d-

2.38 Dm = E( ,a) sin m d .

d+T
2















E( Y,A)


I I


d+al
2.


-3 d4 r -d-
2 2.


3d-2
2


Instantaneous Distribution of E( ,a) at t = T/4


Figure 7.












Equation 2.38 can be integrated, yielding


2.39 Dm= 2A cos mff[(1 ( -+cos [m1(3- 3 ])
2Ao 4 4 ( d

Substitution of integers into equation 2.39 shows that Dmexists

only for m odd. Equating coefficients of equations 2.35 and 2.36 under

the stipulation that

mit
2.40 n -=2


results in


2.41 En = Dm
Io(Kcna)

where m = 1 + 4n .


This can be shown from equation 2.40 and the fact that Dm exists for

m equal to positive odd integers.

The ratio of the amplitude of the nth space harmonic to the

amplitude of the fundamental component can be expressed using equations

2.41 and 2.39.



2.42 En I(Kcoa) cos [(lJ) l1+4nl]-cos[ (3- I 1+4nI]
Eo l1+4n I o(Kcba) cos [t(1+Z.)]-cos [(3-1)]


A treatment of the backward wave would be identical in form

other than for the sign on ,n in equation 2.34. Any linear combina-

tion of the forward and backward wave will satisfy the boundary con-

ditions for the structure within the approximations made in the above

analysis.












The problem now remains to determine the magnitude of the

axial component of electric field. Since the input power (rf) to

the structure is known, the peak electric field at ,= 0 can be com-

puted [18].


2.43 Eo = (2IroPo)1/2 volts/meter


where Po is input rf power (watts) and ro is shunt impedance (ohms/meter)

which is defined by equation 2.2 of reference 14.

Eo2
2.44 ro = -



where Eo is the peak value of the electric field at = 0 and dP/dk

is the rate at which power is dissipated in the walls of the structure.

As shown by equation 2.43, the axial electric field strength

in an electron accelerator varies as the square root of the power flow-

ing in the structure. The shunt impedance per unit length ro is the

parameter which indicates the effectiveness of a given structure in

generating an accelerating electric field for a given power flow.

Shunt impedance is an experimentally determined quantity and the tech-

niques for performing its measurement are discussed in the literature.

Some typical values are 4.73 x 107 ohms/m for the Stanford University

Mark III Linac [19] and 5.6 x 107 ohms/m for the University of Florida

Linac.













Attenuation

In a disk-loaded waveguide structure the magnitudes of the

fields vary with axial distance as e'-I where I is the attenuation

coefficient, which can be determined [14] from the relationship


2.45 I =


where vg is the group velocity at the angular frequence 'J and Q

is the unloaded Q of the structure considered as a resonator. The

energy in electron volts imparted to an electron passing through

the structure on the accelerating peak of the traveling wave has

been shown [20] to be


2.46 V = (2PoroI)/2( IL ) electron volts,


where Po is the magnitude of the input rf power, L is the length

of the structure, and ro is the shunt impedance per unit length

defined by equation 2.44.

From equation 2.46 the effect of attenuation on total electron

energy is obviously important. Also significant is the effect of

finite attenuation on the solution for the axial component of elec-

tric field. With attenuation (I) other than zero the solution of

equation 2.10 contains Bessel functions with complex arguments and

such functions are not generally available in tabulated form. In

this event equation 2.14 becomes


2.47 Kcn2 = K2 + 12 n2 + j218n.













It has been shown [21] that the two equations of longitudinal motion,

equations 2.28 and 2.31, are best solved by numerical integration tech-

niques. Use of a digital computer makes it feasible to solve equations

2.10 and 2.47 in a similar manner.


Equations of Longitudinal Motion

A new set of equations describing longitudinal electron motion

can now be developed in terms of the axial component of electric field

derived in equations 2.33 and 2.42. Since each space harmonic of the

traveling wave illustrated in Figure 6 will have a different normalized

phase velocity, equation 2.31 can be written for each harmonic.


2.48 d~.= 21f1.- -


where An represents the relative phase of the electron to the nth

space harmonic.

In order to calculate the effects of the fundamental and 2p

space harmonics, equation 2.47 must be written


2.49 Ken2 = K2 + 12 n2 + j2I1n n = -p,..., -l,0,+l,...+p.


Since each space harmonic of the axial component of electric field

will affect the total energy of the particle, equation 2.28 can be

written


2.50 -Re Z n( sin Ln
7X n=q

where cn was defined in conjunction with equation 2.28. The electric

field used to compute the phase angle and energy is described by











equation 2.33 with f/n replaced by P to account for the effects of

attenuation. Since Kn and n are both complex the real part of the

right hand side of equation 2.50 is significant.

Equation 2.48 must be calculated for the fundamental and

each of the 2p space harmonics being considered.


2.51 = 2 1
~~ 7_7


n = -p, ..., -1, 0, +1, ..., +p.


Summary
Longitudinal electron motion in a disk-loaded circular wave

guide, operating in the #I/2 mode, can be described by simultaneous

solution of the following equations. For convenience in programming

R is substituted for Eo in equation 2.10.

d2 1 dR 2
2.52 d- + --- + Kcn R = 0.
dr r dr


2.53 K n2 = 2 + K2 _n2 + j21Bn


E,
2.54 -E
E5

2.55 d =
df


lo(Kcoa) cos [r/4(1+T/d) I 1+4n]-cos[r/4(3-r/d) 1+4n ]
11+4n Io(Kcna) cos [/4(1+lf/d)]-cos [(/4(3-/d)]
rn=p n0
"-e Zc'n sindn n = -p, ..., -1, 0, +1, ..., p.
n=q


2.56 eA = E2 n E e-I
"oc7 EO pe


where Ep is the magnitude of the fundamental at r = 0.












2.57 E(,r)n Eo pe Re


2.58 = 2-0 -
dl ^


o (Kcnr)ej (Wt--n k)


n = -p, ..., -1, 0, +1, ..., +p.


Due to the complexity of these equations it was necessary to

solve them using the IBM 709 digital computer. Programs developed
for this purpose are discussed in the following chapter.















CHAPTER III

SOLUTION OF EQUATIONS BY DIGITAL COMPUTER


Introduction

As summarized in Chapter II there are seven basic equations,

(plus one additional equation for each space harmonic considered)

which must be solved simultaneously by numerical methods in order

to determine longitudinal electron motion in a linear accelerator.

This chapter contains a discussion of the computer programs developed

for the purpose of solving these equations. These programs are

written in IBM 709/7090 FORTRAN language [22,23], and comprehension

of the subsequent sections requires a knowledge of the basic pre-

cepts of this language. The Runge-Kutta [24] method of numerical

integration for fourth order accuracy was used in solving the dif-

ferential equations.

Several of the equations which must be solved are functions

of radial position and space harmonic number only. These equations

need only be solved once for each space harmonic being considered

since only trajectories of constant radius are considered. Equa-

tions 2.52 and 2.53 are numerically integrated over the range from

(radius) r = 0 to the maximum possible beam radius r = a for each

space harmonic being considered. This is accomplished in SUBROUTINE



1 Expressions in FORTRAN language will be written in capital
letters.












BESSEL and the resultant values of Io (Kcnr) are stored for later

use. Equation 2.54 defines the ratio of the amplitude of each space

harmonic to the amplitude of the fundamental component of electric

(axial) field. In SUBROUTINE FIELD equation 2.54 is solved for each

space harmonic being considered, anC these data are also stored in

such a way that they are available for solution of the equations of

axial motion which are functions of axial distance. These equations

of motion are solved in the MAIN program. Figure 8 illustrates the

relationship between the MAIN program and the two subroutines.


Bessels Equation for Complex Argument

The numerical integration of equation 2.51 is performed in

SUBROUTINE BESSEL using values of Ken computed by use of equation

2.53. As suggested by Ramo and Whinnery [25] let Kcn2 equal -g2 so

that equation 2.51 can be written


3.1 dR + g2R = 0
dr2 r dr

the solution of which has the desired form


3.2 R = C10o(gr).


In order to transform equation 3.1 into a form more compatible with
dR
the method for numerical solution let -d = P. Solution of equation
dr
3.1 now reduces to the solution of three simultaneous first order

ordinary differential equations.


3.3 r 1,
dr













Start mai program


Read inut data


Call seR l a


Continue


Cnll Fild


Continue



Solve equations 2.55
and 2.56 for space
harmonic amplitude at
the trajectory radius



Integrate equations
2.54 and 2.57



Compute electron energy
and phase


F End


SUBROUTINE BESSEL

Integrate equation 2.51 using
equation 2.52 for all space
harmonics of interest


SUBROUTINE FIELD
Compute the space harmonic
amplitude ratio


Figure 8. Main Program and Subroutines














dR
3.4 d= P,
dr


dr r

For use in the numerical integration process in SUBROUTINE BESSEL

variables and their associated derivatives can now be defined. Let

V and D denote variables and derivatives respectively. Let


3.6 V1 = r

V2 = R

V3 = P


therefore,


3.7 U1 = dr

dR
D2 =-dr


and


dP
I) dP
3 U-7

For use in the program each derivative must be defined in terms of

one or more variables from equation 3.6. This is accomplished by

using equations 3.3 through 3.5 in conjunction with equation 3.6.

The resultant expressions in FORTRAN language are in the LIST of

SUBROUTINE BESSEL shown in Appendix A.













From equation 2.52, Kcn is shown to be complex when structure

attenuation is considered; therefore, SUBROUTINE BESSEL was written

in complex notation [26]. A detailed logic flow diagram of SUBROUTINE

BESSEL is shown in Figures 9 and 10. The majority of the operations

indicated in the flow diagram are self-explanatory, and the follow-

ing discussion is merely to clarify some steps which might be ambiguous.

The results of the numerical integration of equation 2.52 for

each space harmonic considered were stored as a function of both space

harmonic number and radial position. The solution to equation 2.52

was defined as a subscripted variable RNUM'I.. The first subscript

(NUM) refers to the space harmonic number for which the integration

was performed. The second subscript (I) refers to the radial posi-

tion associated with each numerical value. In FORTRAN language

DIMENSION subscripts are required to be nonzero positive integers;

therefore, it was necessary to define both subscripts in such a way

that negative space harmonics subscripts and fractional values of

radius could be described.

In order to make the computer program versatile it was written

in such a way that as many as eighteen space harmonics could be con-

sidered, by allowing for a range of space harmonic subscripts from

minus nine to plus nine. This was accomplished by defining NUM equal

to ten plus the space harmonic subscript (n).

In the numerical integration of equation 2.52 the independent

variable (radius) r starts at zero and is stepped in increments H for

NCOUNT increments. Both H and NCOUNT are read into the program as































































Flow Diagram of Subroutine Bessel


W x


Figure 9.












w x


Figure 10. Subroutine Bessel Continued












parameters so that different values may be used at the discretion of

the investigator. The maximum allowable value which can be read in

for NCOUNT is one hundred, which was used here. This limit is governed

by a DIMENSION statement contained in the program. In order to relate

a nonzero positive integer to a fractional value of radius the second

subscript I was defined.


3.8 I = r/H + 1,


where H is the incremental change in r for each integration step.

In order to minimize error, H should have a small fractional value.

For the data exhibited in this paper the value of H was 0.001.

The number of space harmonics considered by the program is

controlled by the values read in for NUM and NN. Each time the

subroutine has performed the integration of equation 2.52 over the

specified range of r, NUM is incremented by one. The entire inte-

gration process is performed repeatedly for different space harmonics

until NUM equals NN and at this time control is returned to the MAIN

program. The remaining logic operations shown in Figures 9 and 10

deal with line count, paging and data display. These are merely for

the purpose of obtaining the output data in a concise, well-organized

form.


Space Harmonic Amplitude

Having computed and stored the necessary values of Io(Kcnr),

equation 2.54 can now be solved. Equation 2.54 contains complex

terms and, since it need only be computed once for each space harmonic,












it is advantageous to compute and store En in another subroutine

names FIELD. En is the coefficient which specifies the ratio of the

amplitude of the nth space harmonic to the amplitude of the fundamental

peak electric field existing in the structure. Eon is stored as a

subscripted variable whose subscript is related to the same associated

space harmonic subscript (NUM) as was used in SUBROUTINE BESSLE. A

FORTRAN LIST of SUBROUTINE FIELD in included in Appendix A.

Before integration of equations 2.55 and 2.58 can be executed

it is necessary that equations 2.56 and 2.57 be solved once for each

space harmonic being considered. Since this operation must be re-

peated for each increment of integration in normalized axial distance

(f) it is accomplished by a DO LOOP (an instruction causing the

execution of a prescribed number of repetitive operations) in the

MAIN program where ] is available. A value ofa, for each space

harmonic considered is then stored as a subscripted variable. The

subscript is also related to the same space harmonic subscript (NUM)

as was used in SUBROUTINE BESSEL.


Differential Equations of Longitudinal Motion

Equations 2.55 and 2.58 must be integrated simultaneously

over the normalized axial distance of interest for each space har-

monic considered. This is accomplished by defining a new set of

two first-order differential equations for each space harmonic con-

sidered and then numerically integrating the set of first-order

differential equations simultaneously.












For convenience in programming define An in terms of do.


3.9 An = o + (n o)j

where n is the space harmonic number, An is the phase angle of the

electron with respect to the nth space harmonic and ,n is the phase

constant of the nth space harmonic.

Equation 3.9 can be written in terms of normalized axial

distance and normalized phase velocity of the fundamental /o0.


3.10 An O + 8n.


Let k be defined as the number of space harmonics being considered

including the fundamental.


3.11 V1

V2
Vj

VP

Vp


f




= sin Aj

= cos Aj


j = n + 13

=j + k + 1

p = + k + 1


Therefore,


3.12 D1 = d
1 d

D = d
2 -

D. =- d_
ds


D d (sin Aj)
dS


j = n + 13


. = j + k + 1














D = c (cosA) p = + k + 1


For use in the program each derivative of equations 3.12 must be

defined in terms of one or more variables from equations 3.11. This

is accomplished by using equations 2.55 and 2.58 in conjunction with

3.10 and 3.11. The resultant expressions in FORTRAN language are

shown in the LIST of the MAIN program in Appendix A. A detailed

logic flow diagram of this program is shown in Figures 11 and 12.

There are three major loops in the MAIN program. The inner-

most loop causes f to increment from zero to some predetermined

normalized axial distance (TERM) in steps of UI1. The middle loop in-

crements the initial value of Ao at = 0 in steps of DELINC from

DELTAZ to ANGLE. Each increment corresponds to the initial phase

of one electron trajectory. In the analysis of the prebuncher section

DELTAZ was 1f and DELINC was 11/3. In the analysis of the uniform sec-

tion DELTAZ was f1/2 and DELINC was ff/6. In the event that an elec-

tron is injected an at initial phase angle which results in the elec-

tron velocity being reduced to zero the program will recycle to the

next value of Ao until the value ANGLE is reached. The outermost loop

is a DO LOOP instructing the program to cycle through the entire tra-

jectory analysis three times. The first set of trajectory calculations

are made for r = o. In each successive set of calculations r is in-

creased by a/4 where a is the radius of the disk hole in the structure.













Write title page
Set line count


Write page heading
Write column heading
LINE = LINE + 7


Y Z


Flow Diagram of Main Program


Figure 11.











V Y


Figure 12. Main Program Continued


















CHAPTER IV

SOLUTIONS OF EQUATIONS OF MOTION


Introduction

The preceding chapters of this paper have presented a

more accurate analysis of a constant phase velocity disk-loaded

waveguide used as a prebuncher or as a uniform-section accelera-

tor, by including the effects of space harmonics, structure at-

tenuation and off-axis position on longitudinal electron motion.

In this chapter solutions of these equations are presented for a

particular traveling wave prebuncher and a particular uniform

section.

Also presented for comparison are solutions for the same

structures by what has been heretofore termed in this paper the

conventional analysis. For purposes of identification in the

succeeding parts of this paper this method is given the name non-

harmonic analysis. The analysis provided by this paper is given

the name space harmonic analysis.


Traveling Wave Prebuncher

Both analyses were applied to a traveling wave prebuncher

having the following parameters:

Sa= 0.5

S e 0.5 at input













o<( = 0.10

I = 0.6838 neper/meter

f = 5670 megacycles/second

a = 0.01 meter


For purposes of investigating off-axis trajectories three

radii were chosen as specified below. The phase velocity of the

fundamental space harmonic was selected equal to the electron in-

jection velocity.

It was decided to make computations over an axial distance

of four wavelengths so that two successive regions of close phase

groupings could be investigated. The three trajectory radii were

selected to investigate the effect of trajectory radius on the axial

distance to the regions of minimum phase spread.

Figures 13 through 18 are plots of phase angle A versus

normalized axial distance Figures 13 through 15 are phase

plots for the three trajectory radii r = 0, r = a/4, and r = a/2,

calculated by the nonharmonic analysis. Figures 16 through 18 are

corresponding phase plots for the same three trajectory radii cal-

culated by the space harmonic analysis, for the same input power.

Curves identified by the same number have the same initial

value of L. The initial values chosen are if, 2 0, 0, -, -23_
3 3 3 3
and -/, identified respectively as curves numbers 1 through 7.

The purpose of this structure is to group electrons within

a small phase spread. Both methods of analysis show that this phase









































7, __ ,,,,_____________________________________________


/.O /.5 2.0 2 .5 .3.0 3.5 4.0
X;

Figure 13. Electron Phase Angle A vs. Normalized Axial Distance
/, = 0.5, r = 0, Nonharmonic Analysis










































.5 1.0 I .5 2.0 2.5 3.0 3.5 4.0

Figure 14. Electron Phase Angle A vs. Normalized Axial Distance
/, = 0.5, r = a/4, Nonharmonic Analysis
































- o








.5 1.0 1.5 Xi 2.0 2.5 3.0 3.5 4.0
Figure 15. Electron Phase Angle A vs. Normalized Axial Distance
F r = 0.5, r = a/2, Nonharmonic Analysis







































7


1 / / X 2,o 2,5 3.o 33.- 4,

Figure 16. Electron Phase Angle A vs. Normalized Axial Distance
/O = 0.5, r = 0, Space Harmonic Analysis









































/'- Xi 2.0 2-.15 3.0 J, -


Figure 17.


Electron Phase
/,= 0.5, r


Angle A vs. Normalized Axial Distance
= a/4, Space Harmonic Analysis


2

\J
A3<


4.0







































Figure 18.


5.~j


2.0


2.5-


Electron Phase Angle A vs. Normalized Axial Distance
/ = 0.5, r = a/2, Space Harmonic Analysis


o0
w
Q


3.-S-













grouping occurs. This is illustrated in particular by the inter-

section of curves 2 and 6 which indicates that electrons injected

at do= 2S and 40 = -2ifhave the same value of A at particular
3 3
values of .

For each trajectory radius the nonharmonic analysis (Figures

13 through 15) shows a greater axial distance to the first phase

grouping than is obtained from the space harmonic analysis (Figures

16 through 18).

Figures 19 through 22 are plots of normalized energy K versus

normalized axial distance f. Figures 19 and 20 are energy plots

for trajectory radii r = 0, and r = a/2, calculated by the nonharmonic

analysis. Figures 21 and 22 are corresponding energy plots for the

same two trajectory radii, calculated by the space harmonic analysis,

for the same input power. As expected, all differences are small

because the traveling wave prebuncher is a low-gradient structure.

Therefore only results for the extreme radii are presented.


The Uniform Section

Both analyses were applied to uniform section having the

following parameters selected for the University of Florida Linac:


/ao = 1.0

/c = 0.5 at input

do =.Z.0

I = 1.0 neper/meter

f 5670 megacycles/second

a = 0.005 meter.

























.47




/.2










/o
/.. Ao 2..- 3.o .- -

Figure 19. Normalized Electron Energy 2 vs. Normalized Axial Distance
/,= 0.5, r = 0, Nonharmonic Analysis

































/4,77










/1o /.5 Xi 2.0 2, 3. 3.C 4. o

Figure 20. Normalized Electron Energy Y vs. Normalized Axial Distance
/1 = 0.5, r = a/2, Nonharmonic Analysis





































4
",7
3


1.5, X, 76


I .1. ______________~,~


2. -


Figure 21. Normalized Electron Energy Yvs. Normalized Axial Distance
So= 0.5, r = 0, Space Harmonic Analysis


__ I


-A0C


























60








/02








Figure 22. Normalized Electron Energy vs. Normalized Axial Distance
,,= 0.5, r = a/2, Space Harmonic Analysis












A structure length of four wavelengths was selected because

all the electrons which are accepted will have achieved an energy

of at least two Mev in this distance.

Figures 23 through 28 are plots of normalized energy versus

normalized axial distance Figures 23 through 25 are energy plots

for the three trajectory radii r = 0, r = a/4, and r = a/2, calculated

by the nonharmonic analysis. Figures 26 through 28 are corresponding

energy plots for the same three trajectory radii, calculated by the

space harmonic analysis, for the same input power.

There is no significant difference, among themselves, between

the plots of energy versus distance for different trajectory radii

from the nonharmonic analysis. A similar statement holds true for

the space harmonic analysis. There is a significant different be-

tween the results of the two analysis. This can be observed by

comparison of Figures 23 and 26, Figures 24 and 27, and Figures

25 and 28.

The variation of electron phase angle with axial distance is

of secondary importance in the use of a uniform section as an accelera-

tor. Nevertheless, calculations of phase angle vs. normalized axial dis-

tance were made for the same conditions as in Figures 23 through 28,

and the results are presented in Figures 29 through 34.

Curves identified by the same number have the same initial

value of A. The initial values chosen are 0, 4 -4 and -.T
236 6 3 2
Curves for these initial values are identified respectively by numbers

1 through 7. Note that these initial phase angles are different from

those chosen for the traveling wave prebuncher.





























Figure 23.


Normalized Electron Energy ?(vs. Normalized Axial Distance
43 = 1.0, r = 0, Nonharmonic Analysis


4.0


4























S-



















,5 / o /.5 Xi 20 2. 3.0 3.5

Figure 24. Normalized Electron Energy X vs. Normalized Axial Distance
/ ,= 1.0, r = a/4, Nonharmonic Analysis









































5s /o /5 X; 2.0 2-. 3.0 3.5 4-,

Figure 25. Normalized Electron Energy d vs. Normalized Axial Distance
/8,= 1.0, r = a/2, Nonharmonic Analysis





































7


Z.s


3.0


Figure 26. Normalized Electron Energy 6 vs. Normalized Axial Distance
,8 = 1.0, r = 0, Space Harmonic Analysis


'3




a




I


3.-1


4.0


O0
0\


X't 2,0










































2 /0O Xi 2 .s 3,0 -" 4.o

Figure 27. Normalized Electron Energy Yvs. Normalized Axial Distance
/ = 1.0, r = a/4, Space Harmonic Analysis














































/ o


7


/5S, ; x; O


2.5


3. 0


Figure 28. Normalized Electron Energy Y vs. Normalized Axial Distance
,g = 1.0, r = a/2, Space Harmonic Analysis


4. c







































s /o /5 i 2.0 2.5 3.0 3.5 4.0

Figure 29. Electron Phase Angle 6 vs. Normalized Axial Distance
A,= 1.0, r = 0, Nonharmonic Analysis














































3.0


Figure 30. Electron Phase Angle A vs. Normalized Axial Distance
/4o= 1.0, r = a/4, Nonharmonic Analysis


3.-


4.0


*^~~~~~~~~ ---- --- ------ --- --- ------- --------





*) ^ - - - ----------------------------- -
7






\



.^ ^___





tj- ---- ^ s ^ _________________________________~ --


























O






1--------





-I-








*5s /o /.5" S 2.O 2.s- 3.0 3.,5 4,0

Figure 31. Electron Phase Angle A vs. Normalized Axial Distance S
,/$ = 1.0, r = a/2, Nonharmonic Analysis



























2. 5


Figure 32. Electron Phase Angle A vs. Normalized Axial Distance
/,= 1.0, r = 0, Space Harmonic Analysis


-7r


4,0


!7


/.-5


2.0






























2. ------ ----






S/G 5" 20 2., 3.0 3., 4.0
A;
Figure 33. Electron Phase Angle A vs. Normalized Axial Distance
/4= 1.0, r = a/4, Space Harmonic Analysis



























-1

















0 /.5 ,X,/ 20 2,5 3.3,0 4.

Figure 34. Electron Phase Angle A vs. Normalized Axial Distance
/6=- 1.0, r = a/2, Space Harmonic Analysis





75






The greatest differences in the results of the two methods

are for those electrons which are not accepted, as shown by curves

numbered 1 and 7.
















CHAPTER V

RESULTS AND CONCLUSIONS


Traveling Wave Prebuncher Calculations

Both methods of calculating phase angle versus axial dis-

tance show that there are two phase groupings in the first four

wavelengths. The first phase grouping occurs at an axial distance

of approximately one wavelength. The second phase grouping occurs

at an axial distance of approximately 3.75 wavelengths.

Both methods of calculation show that the first phase group-

ing has less phase spread than the second phase grouping.

Both methods show that for off-axis electrons, the greater

the radial distance from the axis, the smaller the axial distance

to the first phase grouping.

The most significant difference between the results of the

two analyses is that for both on-axis and off-axis electrons, space

harmonics reduce the distance to the first phase grouping.

Calculations of energy versus distance (Figures 19 through

22) indicate that in the particular structure analysed in this paper

there is no significant difference between the results of the two

methods and that trajectory radius has little effect.


Uniform Section Calculations

Calculations of energy versus distance by both methods show

that trajectory radius has little effect.












The most significant difference between the results of the

two methods is that the space harmonic analysis indicates less energy

gain for the same input power.

With regard to phase variations with distance there was very

little difference between the results of the two methods for the chosen

structure. Owing to the cost of computer time a detailed analysis of

the structure acceptance angle was not made.


Conclusions from Traveling Wave Prebuncher Results

From the results of analysis of longitudinal motion in the par-

ticular traveling wave prebuncher analyzed in this paper one can con-

clude that the best bunching occurs at the first point of minimum phase

distribution. However, in the analysis of a structure with different

parameters the possibility of utilizing the second phase grouping

should not be neglected.

We have shown that both trajectory radius and space harmonics

have an effect on the axial distance to the first phase grouping,

and that the further the trajectory is from the axis, the nearer

to the point of injection is the location of the first phase group-

ing. Although no formula for this relationship has been developed,

the effects should be taken into account in designing any prebuncher.

The energy versus axial distance calculations show that in this

structure the effects of electron trajectory radius and space harmonics

on electron energy are small. However, this structure has a low gradient

and a phase velocity equal to the injection velocity of the electron.












These conditions reduce the effects of trajectory radius and space

harmonics. Therefore it is not to be concluded that these effects

can always be neglected.


Conclusions from Uniform Section Results

Calculations of energy versus axial distance by both methods

indicate that in a uniform section the effects of trajectory radius

on energy gain is slight. This may be taken as a general conclusion.

The effects of space harmonics on energy gain, however, is not

negligible. The existence of space harmonics reduces the energy gain.

The only way to avoid this reduction is to design a structure in which

there is little energy in the harmonics.


Summary of the Most Significant Conclusions

In a traveling wave prebuncher space harmonics and off-axis

displacement reduce the axial distance to the first phase grouping.

This implies that selection of structure length on the basis of on-

axis trajectories will result in significant error.

The predominant factor to consider in the analysis of longi-

tudinal motion in a uniform section of accelerator structure is the

reduction in energy gain resulting from the existence of space har-

monics.

The only way to avoid the resultant energy reduction in a uni-

form section is to design a structure in which negligible rf power is

contained in the space harmonics. In the structure analyzed in this

paper the harmonic energy content in the prebuncher was approximately

one per cent and in the uniform section was approximately ten per cent.












It has been shown elsewhere that in designing a uniform sec-

tion the disk hole radius can be reduced to obtain higher gradients.

We have shown here that reduction in disk hole radius results in a

greater percentage of power being contained in the space harmonics.

This implies that designing a uniform section for maximum allowable

gradient does not result in maximum energy transfer to the electrons.

We believe that the computer program for the solution of the

more complicated equations of motion, both of which are original in

this paper, are a significant contribution to the art of accelerator

design.




























APPENDICES


























APPENDIX A

IBM 709/7090 FORTRAN PROGRAMS













C LISTING OF FEB. 27,1963
C AXIAL EQUATIONS OF MOTION FOR NN NUM SPACE HARMONICS
I DIMENSION R(20,101),P(20,101),CC(1),EZ(20),E1(20)
DIMENSION V(65),D(65),AA(65),SUMD(65),ALFA(20),EO(20)
C NUM 10 IS THE MOST NEGATIVE SPACE HARMONIC BEING CONSIDERED
C RADIUS IS THE TRAJECTORY RADIUS FOR WHICH THE CALCULATION IS MADE
C NN 10 IS THE MOST POSITIVE SPACE HARMONIC BEING CONSIDERED
C A IS THE RADIUS OF THE DISK HOLE IN THE STRUCTURE
C W IS THE ANGULAR FREQUENCY
C BWO IS THE NORMALIZED PHASE VELOCITY OF THE FUNDAMENTAL
C TAU IS THE STRUCTURE DISK THICKNESS
C AO IS THE PEAK VALUE OF FUNDAMENTAL (AXIAL)
C DELTAO IS THE PHASE ANGLE AT WHICH THE FIRST ELECTRON ENTERS
C DELINC IS THE INCREMENT IN DELTAZ BETWEEN ELECTRONS
C ANGLE IS THE LIMITING VALUE OF DELTAZ TO BE CONSIDERED o0
C TERM DEFINES THE AXIAL DISTANCE OVER WHICH COMPUTATIONS ARE MADE
C BETAE IS THE NORMALIZED ELECTRON INJECTION VELOCITY
C GM IS THE STRUCTURE ATTENUATION IN NEPERS PER METER
C CYL IS THE LENGTH OF ONE CAVITY
C NCOUNT TELLS HOW MANY POINTS WILL BE COMPUTED IN BESSEL
C H IS THE INCREMENT IN RADIUS IN SUBROUTINE BESSEL
NPAGE = 1
WRITE OUTPUT TAPE 6,1
1 FORMAT(1H1,49X,20HELECTRON ACCELERATOR,30X,6HPAGE 1,/51X,18HPHASE
ENERGY STUDY//,53X,13HDONALD MOONEY//)
READ INPUT TAPE 5,2,NUM,NN,RADIUS,A,W,BWO,TAU,AO,DELTAO,DELINC,
1ANGLE,TERM ,BETAE,GM,CYL,NCOUNT,H
2 FORMAT(2(15),4(E15.6) / 4(E15.6) / 4(E15.6) / E15.6,15,E15.6)
WRITE OUTPUT TAPE 6,3,NUM,NN,RADIUS,A,W,BWO,TAU,AO,DELTAO,DELINC,
1ANGLE,TERM,BETAE,GM, CYL,NCOUNT, H
3 FORMAT(/ 10X,10HINPUT DATA / 2(15),4(5X,E15.6) / 5(5X,E15.6) /
14(5X,E15.6),4X,I5 / 4X,E15.6)
ONE = 1.0











TWO = 2.0
FOUR = 4.0
C = 2.99793E+08
PI = 3.14159
Q = 1.6E-19
HH = 0.01
AK = W/C
RBW = 1./BWO
NPAGE = 1
LINE = 11
C2MO = 8.176E-14
CALL BESSEL(R,P,NUM,NN,GM,W,BWO,H,NCOUNT)
CALL FIELD(R,EO,NUM,NN,TAU,CYL,AO,H,A)
LINE = LINE + 10
RADIUS = RADIUS (A/4.0)
C
C THE FOLLOWING DO STATEMENT MAY BE REMOVED TO LIMIT TRAJECTORY oo
C CALCULATIONS TO ON-AXIS ELECTRONS
C
DO 114 KEND = 1,3
RADIUS = RADIUS + (A/4.0)
DELTAZ = DELTAO + DELINC
GAMAZ = 1.0/SQRTF(1. BETAE**2)
K = NN NUM
14 DELTAZ = DELTAZ DELINC
V(1) = 0.0
V(2) = GAMAZ
DO 5 J = NUM,NN
AN = J 10
L=J + 3
V(L) = DELTAZ
M = K + L + 1
V(M) = SINF(V(L))
N = K + M+ 1
V(N) = COSF(V(L))










5 CONTINUE
IF (LINE 50) 200,600,600
600 NPAGE = NPAGE + 1
WRITE OUTPUT TAPE 6,601,NPAGE
601 FORMAT(1H1,100X,5HPAGE ,14)
LINE = 2
200 WRITE OUTPUT TAPE 6,201,AN,RADIUS,DELTAZ,GAMAZ,BWO
201 FORMAT(/7X,25HNUMBER OF SPACE HARMONIC ,F6.0,5X,8HRADIUS =,E13.6,/
1,7X,10HPARAMETERS,5X,10HDELTA ZERO,8X1OH GAMA ZERO,9X,10HWAVE BETA
2 /22X,E13.6,2(5X,E13.6),//7X,6HLAMDA ,7X,10H DELTA ,7X,10H GA
3MA ,//)
LINE = LINE + 7
IF (LINE 8) 18,500,18
18 MM = 0
15 CONTINUE
MM = MM + 1
I = RADIUS / H + 1.
ATT =-GM*V(1)*TWO*PI*C/W
DO 6 JJ = NUM,NN
ALFA(JJ) = (36.90812E+02 / W) EO(JJ) EXPF(ATT)
6 CONTINUE
DO 17 J = 1,65
AA(J) = V(J)
17 SUMD(J) = 0.0
TT = 1.0
T = 0.5
C
C START INTEGRATION LOOP
C
DO 100 LLL = 1,4
D(1) = 1.0
DD = 0.0
DO 7 JJ = NUM,NN
L =JJ + K + 4
M=L + K + 1










CC(1) = V(L)
CC(2) = V(M)
I E1(JJ) = R(JJ,I) CC
D(2) = E1(JJ) ALFA(JJ) + DD
DD = D(2)
7 CONTINUE
DO 8 J = NUM,NN
KK = J + 3
AN = J 10
D(KK) = TWO*PI*(RBW*(ONE +FOUR*AN) -(V(2)/SQRTF(V(2)**2-1.)))
L = K + KK + 1
LL= K + L = 1
D(L) = V(LL)*D(KK)
D(LL) = -V(L) D(KK)
8 CONTINUE
DO 21 IJ = 1,65
D(IJ) = HH D(IJ) o0
SUMD(IJ) = TT D(IJ) + SUMD(IJ) L
21 V(IJ) = T D(IJ) + AA(IJ)
IF(V(2) 1.0) 300,300,26
300 LINE = LINE + 1
WRITE OUTPUT TAPE 6,301,V(1),V(2),ALFA(10),D(2),AA(2)
301 FORMAT(7X,F6.3,5X,17HELECTRON REJECTED, 4(5X,E15.6))
MM = 0
GO TO 113
26 CONTINUE
IF (LLL 2) 22,23,24
22 TT = 2.0
GO TO 100
23 T = 1.0
GO TO 100
24 TT = 1.0
100 CONTINUE
DO 25 JJJ = 1,65
25 V(JJJ) = SUMD(JJJ)/6.0+ AA(JJJ)










302 IF(MM 10) 112,110,110
110 LINE = LINE + 1
IF(LINE 57) 500,102,102
102 NPAGE = NPAGE + 1
WRITE OUTPUT TAPE 6,103,NPAGE
103 FORMAT(1H1,100X,5HPAGE ,14)
LINE = 1
GO TO 200
500 WRITE OUTPUT TAPE 6,111,V(1),V(13),V(2)
111 FORMAT(7X,F6.2,2(4X,E13.6))
MM = 0
112 CONTINUE
IF (V(1) TERM) 15,15,113
113 CONTINUE
IF(DELTAZ ANGLE) 114,114,14
114 CONTINUE
C DATA CARD ORDER IS NUM,NN,RADIUS,A,W,BWO,TAU,AO,DELTAZ,DELINC,
C ANGLE,TERM,BETAE,GM,CYL,NCOUNT,H
C THE FOLLOWING FORMAT IS USED TO READ INPUT DATA
C FORMAT(2(I5),4(E15.6)/4(E15.6)/4(E15.6)/E15.6,I5,E15.6)
CALL EXIT
END











SUBROUTINE FIELD(R,EO,NUM,NN,TAU,D,AO,H,A)
C LISTING OF 8 FEB. 1963
I DIMENSION R(20,101),P(20,101)
DIMENSION EO(20)
ONE = 1.0
PI = 3.14159
ETA = TAU/D
FOUR = 4.0
AMP = 2./PI
ONETA = 1.0 + ETA
THETA = 3.0 ETA
WRITE OUTPUT TAPE 6,1,NUM,NN,TAU,D,AO,H,A
1 FORMAT(/,10HINPUT DATA,/,2(4X,I5),5(4X,E15.6))
I = ((1. /H) A) + 1.
DO 3 K = 1,20
EO(K) = 0.0
3 CONTINUE
DO 2 L = NUM,NN
AN = L 10
B = ABSF(ONE + FOUR AN)
BB = (PI/FOUR) B
C = BB ONETA
D = BB THETA
EO(L) = (AMP/B) (COSF(C)-COSF(D))/R(L,I)
2 CONTINUE
DO 5 J = NUM,NN
N = J 10
EO(J) = (EO(J) / EO(10)) AO
WRITE OUTPUT TAPE 6,4,N,EO(J)
4 FORMAT(/,17HHARMONIC NUMBER =,15,5X,16HELECTRIC FIELD =,E20.8)
5 CONTINUE
RETURN
END











SUBROUTINE BESSEL(R,P,NJK,NN,GM,W,BWO,H,NCOUNT)
C LISTING OF 14 FEB. VARIABLE LENGTH, PRINTS EVERY TENTH BIT
I DIMENSION AKC(20),R(20,101),V(3),D(3),AA(3),SUMD(3),P(20,101),
I 1A(1),CKA(20),SRAKC(20),Z(1)
C SOLUTION OF BESSELS EQUATION WITH COMPLEX ARGUMENTS
C R IS MODIFIED BESSEL FUNCTION OF FIRST KIND,ZERO ORDER
C P IS MODIFIED BESSEL FUNCTION OF FIRST KIND,FIRST ORDER
C V(1) IS RADIUS
C V(2) IS R
C V(3) IS P
C D(1) IS ONE
C D(2) IS DR/D(RADIUS)
C D(3) IS DP/D(RADIUS)
C IN R DIMENSION THE FIRST NUMBER REFERS TO 10 PLUS THE HARMONIC
C NUMBER AND SECOND NUMBER REFERS TO 1/H TIMES THE RADIUS
C NUM MINUS TEN TELLS THE FIRST SPACE HARMONIC TO BE CONSIDERED
C THE PROGRAM RUNS FROM NUM TO NN
C NN TELLS THE HARMONIC NUMBER LIMIT
C GM IS ATTENUATION IN NEPERS PER METER
C W IS ANGULAR FREQUENCY
C BWO IS NORMALIZED PHASE VELOCITY OF FUNDAMENTAL
C H IS INCREMENT IN RADIUS
NPAGE = 1
NUM = NJK
WRITE OUTPUT TAPE 6,2
2 FORMAT(1H1,49X,20HELECTRON ACCELERATOR, 30X,6HPAGE 1,/48X,24HCOMPLE
1X BESSELS EQUATION//53X,13HDONALD MOONEY//)
PI = 3.14159
C = 2.99793 E+08
LINE = 5
AMINUS = -1.0
SIX = 6.0
HALF = 0.5
WRITE OUTPUT TAPE 6,4,NUM,NN,GM,W,BWO,H
4 FORMAT(/,1OHINPUT DATA,/,2(4X,I5),3(4X,E20.8),4X,F8.7,/)











LINE = LINE + 3
AK = W/C
DO 9 L = 1,20
DO 9 LL = 1,101
I P(L,LL) = (0.,0.)
I R(L,LL) = (0.,0.)
9 CONTINUE
NUM = NUM 1
10 NUM = NUM + 1
MM = 0
I V(1) = (0.,0.)
I V(2) = (1.0,0.)
I V(3) = (0.,0.)
AN = NUM 10
N = AN
A(1) = GM
A(2) = (AK/BWO)*(1. + 4.*AN)
I GAMA = A
I CKA(NUM) = (AK**2) + (GAMA**2)
I AKC(NUM) = (CKA(NUM)) AMINUS
IF(LINE 50) 6,5,5
5 NPAGE = NPAGE + 1
LINE = 2
WRITE OUTPUT TAPE 6,7,NPAGE
7 FORMAT(1H1,100X,5HPAGE ,14)
6 WRITE OUTPUT TAPE 6,8,N
8 FORMAT(//,7X,42HNUMBER OF SPACE HARMONICS BEING CONSIDERED,2X,15)
WRITE OUTPUT TAPE 6,27
27 FORMAT(7X,6HRADIUS, 7X,9HREAL PART,7X,9HIMAGINARY,7X,15HDERIVATIVE
1 REAL,7X,9HIMAGINARY,10X,8HKCN REAL,10X,9HIMAGINARY,///)
LINE = LINE + 7
IF (LINE 8) 11,25,11
11 CONTINUE
I = V(1) (1./H) + 1.
C I IS 10000 TIMES THE VALUE OF RADIUS PLUS ONE










I R(NUM,I) = V(2)
I P(NUM,I) = V(3)
DO 12 J = 1,3
I AA(J) = V(J)
I 12 SUMD(J) =(0.0,0.0)
TT = 1.0
T = 0.5
C
C START INTEGRATION LOOP
C
DO 17 K=1,4
I D(1) = (1.,0.)
I D(2) = V(3)
IF(V(1)) 30,31,32
30 WRITE OUTPUT TAPE 6,33
33 FORMAT(/,15HNEGATIVE RADIUS,/)
K = 0
K=O
GO TO 29 o
I 31 D(3) = HALF AKC(NUM)
TO TO 34
I 32 D(3) = V(2) AKC(NUM) (V(3)/V(1))
34 CONTINUE
DO 13 J = 1,3
I D(J) = H* D(J)
I SUMD(J) = TT* D(J)+ SUMD(J)
I 13 V(J) = T D(J) + AA(J)
IF ( K-2 ) 14,15,16
14 TT = 2.0
TO TO 17
15 T = 1.0
GO TO 17
16 TT = 1.0
17 CONTINUE
DO 18 KK =1,3
I 18 V(KK) =(SUMD(KK)/SIX) + AA(KK)




-w


MM = MM + 1
IF (MM 10) 37,38,38
38 CONTINUE
LINE = LINE + 1
IF(LINE 57) 25,21, 21
21 NPAGE = NPAGE + 1
WRITE OUTPUT TAPE 6,22,NPAGE
22 FORMAT(1H1,100X,5HPAGE ,14)
LINE = 1
GO TO 6
25 CONTINUE
MUM = 20 + NUM
I SRAKC(NUM) = SQRTF(AKC(NUM))
IF(SRAKC(NUM)) 35,36,36
35 SRAKC(NUM) = AMINUS SRAKC(NUM)
36 CONTINUE
I Z = V(3)/SRAKC(NUM)
MM = 0
WRITE OUTPUT TAPE 6,26,V(1),V(2),V(5),Z(1),Z(2),SRAKC(NUM),
ISRAKC(MUM)
26 FORMAT(7X,F6.4,4X,E12.6,4X,E12.6,2(7X,E12.6),7X,E13.6,5X,E12.6)
37 CONTINUE
IF (I NCOUNT) 11,28,28
28 IF(NUM NN ) 10,29,29
29 CONTINUE
RETURN
END




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