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The effects of space harmonics and attenuation on longitudinal electron motion in a linear accelerator.

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The effects of space harmonics and attenuation on longitudinal electron motion in a linear accelerator.
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The effects of space harmonics and attenuation on longitudinal electron motion in a linear accelerator.
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Mooney, Donald Francis
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University of Florida
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THE EFFECTS OF SPACE HARMONICS AND

ATTENUATION ON LONGITUDINAL


ELECTRON MOTION


IN A


LINEAR


ACCELERATOR


DONALD F. 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. Ile is especially indebted to Dr. A. If. Wing and Dr. V. W. Dryden for their constant guidance and to Prof. W. F. Fagen for his project supervision. Ile 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 encouraging him throughout his graduate studies. Ile 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 --------------------------------------------- I
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
Vessels Equation for Complex Argument -------------------- 37
Space Harmonic Amplitude --------------------------------- 43

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

Introduction --------------------------------------------- 49
Traveling Wave Prebunchcr -------------------------------- 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

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

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

7. Instantaneous Distribution of E(Ja) 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 L vs. Normalized Axial Distance 5
,6w= 0.5, r = 0, Nonharmonic Analysis ---------------------51

14. Electron Phase Angle .A vs. Normalized Axial Distance S
18j=0. 5, r = a/4, Nonharmonic Analysis -------------------52

15. Electron Phase Angle & vs. Normalized Axial Distance
/3 uI= 0.5,, r = a/2, Nonh'arionic Analysis -----------------53

16. Electron Phase Angle Avs. Normalized Axial Distance
A,)=.5,r = 0, Space Harmonic Analysis -------------------54

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











Page


18. Electron Phase Angle A vs. Normalized Axial Distance 5
,&w= 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 vs. Normalized Axial Distance
,6w=0.5, r = a/2, Nonharmonic Analysis

21. Normalized Electron Energy ' vs. Normalized Axial Distance
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
/8 =I.0, r = 0, Nonharmonic Analysis

24. Normalized Electron Energy ' vs. Normalized Axial Distance �
IS =1.0, r = a/4, Nonharmonic Analysis

25. Normalized Electron Energy K vs. Normalized Axial Distance
, l.0, r = a/2, Nonharmonic Analysis ---------------------26. Normalized Electron Energy I vs. Normalized Axial Distance S
,v 1.0, r = 0, Space Harmonic Analysis

27. Normalized Electron Energy ' vs. Normalized Axial Distance 5
p=.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
,dI.0, r = 0, Nonharmonic Analysis

30. Electron Phase Angle A vs. Normalized Axial Distance S
r = a/4, Nonharmonic Analysis---------------------31. Electron Phase Angle A vs. Normalized Axial Distance S
/4,=I.0, r = a/2, Nonharmonic Analysis---------------------32. Electron Phase Angle Lvs. Normalized Axial Distance S
A z1.0, r = o, Space Harmonic Analysis.

33. Electron Phase Angle A'vs. Normalized Axial Distance
=1.0, r = a/4, Space Harmonic Analysis ------------------34. Electron Phase Angle & vs. Normalized Axial Distance
/5,z1.0, r = a/2, Space Harmonic Analysis
















PREFACE


This paper presents a method of analyzing longitudinal electron motion in a disk-loaded circular waveguide structure, taking into account the effects of space harmonics and structure attenuation. 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 accelerator 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 uncertainty 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 structure, 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, structure 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 equations governing longitudinalmmotion in a sinusoidal field distribution 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 longitudinal motion can be developed including these factors.

In Chapter III the computer programs used to solve the equations of longitudinal motion are explained. In Chapter IV the results of computer solutions considering space harmonics and structure attenuation are presented along with computer solutions of the conventional equations for two accelerator structure configurations.

Chapter V contains a summary of significant differences observed 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 effect on the phase distribution of the electrons and electron beam radius has a small effect on the electron energy distribution, compared to results predicted by previous theory. Results also verify the assumption 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 multimegawatt klystron tubes have made available sources of electromagnetic 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 standing 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 electron. 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 generates the fields used for electron acceleration.

The electron gun which injects electrons into the accelerator structure has a back-bombarded tantalum cathode button as a thermionics electron source. The rear surface of the cathode is bombarded 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































CONTROL KLYSTRON STABILIZED
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 structure. The Gun Control system supplies the necessary flmn, bombarder, 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 distribution while the beam velocity remains approximately equal to the injection 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 combination 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 structure that can be reasonably obtained is one half the velocity of












light (0.5c). 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 component 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 efficiency of acceptance of 50 per cent since the fields in the accelerator 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 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. lNumbers 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 D~rne [4]. The construction of this type of device is the same as illustrated in Figure 1, with the dimensions adjusted 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 structures 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 butcher 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 remain at a constant phase angle with respect to the wave [6] but oscillate 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 position. 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 electron beam acquires most of its energy is the uniform section. This is a disk-loaded cylindrical waveguide, as shown in Figure 1, dimensioned so that the phase velocity of the electromagnetic wave equals the velocity of light. There are two types of uniform sections referred 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 structure. 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 velocity 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 longitudinal equations of motion will be investigated in later chapters for this second type of structure operated as a traveling-wave prebuncher as recommended by DOme, and operated as a conventional uniform 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 section 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 following analysis is restricted to structures which have rotational symmetry,

In cylindrical structures the wave components are most conveniently 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 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 (TH 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 conventional 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 VX= - 2L


and


2.2 7XVXE a V(V.E) -V2-F, 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(7v. - E) - (VX_.


By definition


2.4 vxiT = 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 structure. 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 O.5c into a uniform section operating at a frequency of 5670 megacycles, 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 bean 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 vr i














Substitution of equation 2.5 into equation 2.3 yields


2.6 V(V.T) -V" =pE _.
at2


Since the presence of the electron beam is assumed to have no effect on the electric field distribution, = 0 and


2.7 V' .= 0.


Therefore equation 2.6 becomes the wave equation


2.8 72'E = 2-
t2

In cylindrical geometry the 3 component of equation 2.8 is

~2L, + 1 2L, + 1 ?2L3 2:, 212
2.9 :r2 r pr r2 2


Assuming Ez of the forward traveling wave to be of the conventional form e[L0(r)eJetl'r)3 and assuming operation in the lowest order TM mode, equation 2.9 becomes


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

where - K2 = K2 + r 2


K = w/cs

f = I + jP, and

c is velocity of light (meters/second), I is attenuation constant (nepers/meter), and 8 is the phase constant (radians/meter).













Initial conditions are


Eo(r3 = Eo(O) magnitude of the field on axis at = 0,
|Ir=O


dEo(r) -.
(Ir
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 ) E " Z-'


c
2.12 tt1 = - 7( F .
0 K 2 br


At this point in the analysis it is generally assumed that

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


2.13 r2 = P2 and


2.14 -Kc2 K2 - p2. Since


2.15
Vp

where Vp is the phase velocity of rf wave, therefore


2.16 -Kc _) (7,)2.













It is necessary that Vp c in order to optimize the interaction between 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) = ClII (Kcr)3


where Io(Kc r) 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(wt4) Substitution of equation 2.18 into equations 2.11 and 2.12 yields


2.19 Er = Re- 0 EoIl(Kcr)e(Jot- )3,
c

2.20 110 = l~e - 4 EOIl(Kcr)e~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 -1 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 angular 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 rG plane in the cases normally encountered in practice have little influence upon the basic motion along the axis."

Considering only the axial component o f force


2.21 F d (mc 2


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


2.22 m m 0
p e 2)1/2













wherein mo is the mass of electron (kilograms) at rest and be 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 _,6 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


where is measured from the transverse place at which electrons

are injected. The prime notation is to differentiate ' from which is measured from the center of a cavity for subsequent space harmonic amplitude determination.

21,
2.26 = :: .


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

2.25.


2.27 F .d- (MOWC2\
d
g

The force exerted on an electron by the axial component of the electric 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, L6 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? - eF- sinA~


where e is electron charge (coulombs). Substitution of equation

2.28 into 2.27 yields


2.29 d sn6


where d - e normalized energy gain per wave length.
rn OC2

A second equation describing longitudinal motion of the electron 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 d4 = (ye - vp) 21L"dt,



where ye is the electron velocity and vp is the phase velocity of the electric field.
































































Figure S.


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













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 AL vpIc.


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


2.32 dA = 2 1 /- _L


where L is defined such that when A = 0 at "=0, thnrd/2 and t = 3T/8.


Equations 2.29 and 2.32 are the two simultaneous first order differential equations which describe longitudinal electron motionii Use of these equations in conjunction with equation 2.18 is the conventional 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 tAg/4 results in ff/2 phase shift per cavity, referred to as operation in the I/2 mode.












Figure 6 is a plot of frequency vs. the phase shift per section 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 describes how the group velocity and phase velocity of a wave propagating 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 corresponding to the /;/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 pulsc 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 i imposed by the mode of operation the field components will be composed of a series of space














slope of tangent = Vg


\I s I

SI /I i -.

II I

I I I I

I I I I
I I II
I I I I
II I I i


-f/2 ff/2 7 2 /7'
115o 2 7rd/Ag


Brillouin Diagram


Frequency 27/d
x _


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 dwis 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.


2.33 E(y,r) =Re n~I En0(Kr)e j (.tin ), n= -co

where


2.34 '6n + -do * n~


From equation 2.14 Kc varies with i 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 distribution of EQ,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 fringing occurs at the disk hole and that attenuation is negligible. The axial component of electric field E(,a), illustrated in Figure 7, can be described in terms of equation 2.34.

n =O
2.35 E(,,a) = Enlo(Kcna) sin (Pn-.
n=-c

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 calculated from structure dimensions by the following method are at least as accurate as those obtainable by experimental methods [17].

The axial component of electric field shown in Figure 7 can be described by a Fourier Series: 2.36 E(ka) Z Dm sin
m=1

where L equals 4d and
V/2

2.37 Dm r E(3,a) sinl L d .


-L2

Equation 2.37 can be written in terms of structure parameters.



2.38 Dm = .2 E(.,a) sin m d?.
d
2T















E( YCA)


I I


I I
I I


d+l P.


-3d4"r -d-


3d
2


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


Figure 7.












Equation 2.38 can be integrated, yielding


2.39 Dm =2ALcoS 1mff (I o [mLf3)f


Substitution of integers into equation 2.39 shows that Drnexists only for m odd. Equating coefficients of equations 2.35 and 2.36 under the stipulation that

mit
2.40 On 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.


= (ca cos [q(1 1+4n !]'eos[ [f(3-J)11 +4n I]
2.42 En I�(K1 a(
Eo 11+4n 1 1o (Kc-na) Io S[ (1+Z) I-cos [9(3-]1) 4 d 4 d

A treatment of the backward wave would be identical in form other than for the sign on 6n in equation 2.34. Any linear combination of the forward and backward wave will satisfy the boundary conditions 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 computed [18].


2.43 E0 = (21ropo) 1/2 volts/meter


where P0 is input rf power (watts) and ro is shunt impedance (ohms/meter) which is defined by equation 2.2 of reference 14.


2.44 ro E



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 flowing 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 techniques for performing its measurement are discussed in the literature. Some typical values are 4.73 x 107 ohms/rn for the Stanford University Mark III Linac [19] and 5.6 x 107 ohms/rn for the University of Florida Linac.













Attenuation

In a disk-loaded waveguide structure the magnitudes of the

fields vary with axial distance as e-Ik where I is the attenuation coefficient, which can be determined [141 from the relationship


2.45 I '0_where vg9 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 = (2P~r0I) 2 ( eI ) electron volts, where P0 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 electric 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 Kcn 2 ,K2 + J2 - n 2 + j21)8n-













It has been shown [21) that the two equations of longitudinal motion, equations 2.28 and 2.31, are best solved by numerical integration techniques. 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,&n. 2 7Y ~ .*


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 Kcn 2 =K2 + J2 -Pn2 + j21Pn n =p.,-l#Q,+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' (, sin An3


whereG n 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 fdn replaced by Pn to account for the effects of attenuation. Since Ken 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
dS n e


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.

d2R 1dR 2
2.52 d + ---- + Kcn R = 0.
dr r dr



2.53 Kcn2 = 12 + K2 _8 n2 + j21n


En
2.54 - =
Eo


2.55 dl =
di


Io (Kcoa) cos [r/4(1+T/d) I +4n]-cos [/4 (3-3/d) l+4n ] I1+4n io(Kcna) cos [/4(1+ l/d)]-cos[f/4(3-T/d)] "Re Zcn sinAn n = -p, ., -1, 0, +1, ., p.
n=q


2.56 eA2 = ~ E e-If
mP


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














2.57 E(,r)n (Eo) pC Re



2.58 2-0 = 2 -


Io (Kcnr) e (t -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 consi dered) 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 precepts of this language. The Runge-Kutta [24] method of numerical integration for fourth order accuracy was used in solving the differential 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. Equations 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 accompli shed in SUBROUTINE



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













BESSEL and the resultant values of 1 0 (K cn r) 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, ane 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 Kcn computed by use of equation

2.53. As suggested by Ramo and Whinnery [25] let Kcn 2 equal -g2 so that equation 2.51 can be written


3.1 d2R + I -g2ij = 0
dr2 r dr

the solution of which has the desired form


3.2 R =C110(gr).


In order to transform equation 3.1 into a form more compatible with the method for numerical solution let dT-P.Sltoofeuin dr
3.1 now reduces to the solution of three simultaneous first order ordinary differential equations.


3.3 dr 1,
dr














Start maiI program [ Read input data I Call Bessel


I Continue

I1
CV11 ril-ld


SUBROUTINE BESSEL

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



SUBROUTINE FIELD Compute the space harmonic amplitude ratio


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


End


Figure 8. Main Program and Subroutines














dR P

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 D1 = dr


dR



and


I) dP


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, Kc 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 following 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 RNUMf "* The first subscript (NUN) refers to the space harmonic number for which the integration was performed. The second subscript (I) refers to the radial position 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 considered, 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 11 for NCOUNT increments. Both ff and NCOUNT are read into the program as






























































Flow Diagram of Subroutine Bessel


IV x


Figure 9.



























































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, 1 = r/11 + 1,


where 11 is the incremental change in r for each integration step. In order to minimize error, 11 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 integration 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 conciseJ, 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 pcak 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 repeated for each increment of integration in normalized axial distance

(f) it is accomplished d by a DO LOOP (an instruction causing the execution of a prescribed number of repetitive operations) in the MAIN program where fis available. A value of.,,,, 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 harmonic considered. This is accomplished by defining a new set of two first-order differential equations for each space harmonic considered and then numerically integrating the set of first-order differential equations simultaneously.












For convenience in programming define ,n in terms of do.


3.9 An =8 + (fn where n is the space harmonic number, An is the phase angle of the electron with respect to the nth space harmonic and Ign 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 0o"


3.10 An =0 + 8n1.


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


3.11 V1

V2 Vj



Vp


3



= sinA A = cos Aj


j=n + 13 j=j +k+ 1 p =R + k + 1


Therefore,


312 D= d5
=dl

D2 dD. --dti
df

D = d (sin Aj)
d3


j = n + 13


.9= j + k + 1















D = d Cos Aj) 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 innermost loop causes to increment from zero to some predetermined normalized axial distance (TERM) in steps of 1111. The middle loop increments the initial value of at at 0 in steps of DI3LINC from DELTAZ to ANGLE. Each increment corresponds to the initial phase of one electron trajectory. In the analysis of the prebuncher section DELTAZ was 17' and DELINC was 11/3. In the analysis of the uniform section DELTAZ was ff12 and DELING was ff1t6. In the event that an electron is injected an at initial phase angle which results in the electron 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 trajectory analysis three times. The first set of trajectory calculations are made for r = o. In each successive set of calculations r is increased 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


Figure 11.


Flow Diagram of Main Program











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 accelerator, by including the effects of space harmonics, structure attenuation 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 nonharmonic 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:

0. 5

Pe 0.5 at input













CX0 =Q0.10

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 injection 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 4 versus normalized axial distance .Figures 13 through 15 are phase plots for the three trajectory radii r = 0, r = af4, and r =a/2, calculated by the nonharmonic analysis. Figures 16 through 18 are corresponding phase plots for the same three trajectory radii. calculated 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 vf, 2 Z4 f 00 -24 33' 3 3
and -)Y, 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



























4











7


01.0 iO 2,5 .3.0 3.5- 4.0


Figure 13. Electron Phase Angle a vs. Normalized Axial Distance
16, = 0.5, r = 0, Nonharmonic Analysis








































7


1.0 I X 2.0 2.S- 3.0 3.5 4.0

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



























44 W











-T7


.5 .o0 1.5 Xi 2.0 2. .5 3.0 3.5 4.0
Figure 15. Electron Phase Angle 6 vs. Normalized Axial Distance
FA = 0.5, r = a/2, Nonharmonic Analysis


























L,
4 6











7


S/10 / X 2,o , 3.o 3. 4,e

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














































4.0


/'- Xi 2.0 2.5 3.0 .3 -


Figure 17.


Electron Phase
, = 0.5, r


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




























o


5


2.0


Figure 18.


2.5


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


3.5-













grouping occurs. This is illustrated in particular by the intersection of curves 2 and 6 which indicates that electrons injected at 60=2 and 40= -2lrfhave the same value of 6 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 d versus normalized axial distance 5.Figures 19 and 20 are energy plots for trajectory radii r - 0, and r = a/2, calculated by the nonharinonic. 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:


19 = 1. 0

6e= 0.5 at input

O(0 . 0

1 1.0 neper/meter

f 5670 megacycles/second

a =0.005 meter.


















/.4











/.2
1/,2


Ia,7








/o
0 /.Y A 2.- 5. o J.Figure 19. Normalized Electron Energy & vs. Normalized Axial Distance f/,= 0.5, r = 0, Nonharmonic Analysis































/,4, 7 "/ 4









,o /5 X; 2.o 2, J. a. r 4o

Figure 20. Normalized Electron Energy Yvs. Normalized Axial Distance J
/0, = 0.5, r = a/2, Nonharmonic Analysis




































1,7


'.5, X, .


2. T'


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


2


-A C


























60




4









2,0 7

Figure 22. Normalized Electron Energy vs. Normalized Axial Distance
,,C = 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 between 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 accelerator. Nevertheless, calculations of phase angle vs. normalized axial distance 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 ff 00 -4 and
2 6 3 2
Curves for these initial values are identified respectively by numbers I 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
,40= 1.0, r = 0, Nonharmonic Analysis


4.0





















S6






3



2




o5 / /. X; 2,0 ;."3,0 ,"

Figure 24. Normalized Electron Energy X vs. Normalized Axial Distance
i5',= 1.0, r = a/4, Nonharmonic Analysis



















5







.3



2




,5 0 /5 0 2.) 2.5 3. o 3.5 4,,o

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




























4 '3




a




I


Z.5S


36)


Figure 26. Normalized Electron Energy dvs. Normalized Axial Distance
,c = 1.0, r = 0, Space Harmonic Analysis


4
S








----------G

2.


3. -


4,0


0'
C%


I." X 1 ,0





















5"


4




-3



2




/ 25 5.0 2-1 4o

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


























G
z












7


/5~ x;


2.5


3.0


4,c


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


/0






































. /o /5 i 2.0 2.5 3.0 3.5 4.0

Figure 29. Electron Phase Angle 6 vs. Normalized Axial Distance
c= 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


77
N\


3.-


4.0























0




[.=,.














-7
/" 20 2.s- 3.0 4.

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










































-77


/.5-


2.O0


2. Y


Figure 32. Electron Phase Angle 2 vs. Normalized Axial Distance
,d= 1.0, r = 0, Space Harmonic Analysis


4,0


4
5




7
































2.A











/0 Y." 2,0 2.5 3.0 3.,S 4.0
A';
Figure 33. Electron Phase Angle A vs. Normalized Axial Distance S
A4,= 1.0, r = a/4, Space Harmonic Analysis








































1-5 X0, 210 3,0 .

Figure 34. Electron Phase Angle L\ vs. Normalized Axial DistanceS
/6w - 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 I and 7.
















CHAPTER V

RESULTS AND CONCLUSIONS


Traveling Wave Prebuncher Calculations

Both methods of calculating phase angle versus axial distance 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 grouping 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 particular traveling wave prebuncher analyzed in this paper one can conclude 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 grouping. 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 onaxis trajectories will result in significant error.

The predominant factor to consider in the analysis of longitudinal motion in a uniform section of accelerator structure is the reduction in energy gain resulting from the existence of space harmonics.

The only way to avoid the resultant energy reduction in a uniform 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 section 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.

Ile 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 00
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) o
SUND(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 ,I4) 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













UmH

MflUMJA03 9 (8"OZH'= l"ISI DIJI3,UB'IBH91'Xg'9'= UHMIRN DIHIDOflVHHLT'/).VKIHOa I
(f)oa'mC'v'9 dVI- Jaino aIITm ov * ((o-loa / (r)oa) = (r)o o01 - P=r xx' ifi = r o(I

(I 'I)1/((I)asoD-(D)Iaso() * (/dl) = ('I)O H VJ.ML * U B = (I VJ2O * B = 3 e � (InOA/Id) = me (MV * urm + HND)asr = 9 01 -rl = K w 'wIn = g O(I

0o'o0 = (x)o Og*I = ou(
T + (V * (H/ T)) = I ((9g.T''Xk)g'(gI'XV)Z'/'V.LVG J ldIHOI'/)JVNWHOA I V'H'OV'a'flVL'& 'flM'T'9 BdVL JJlJLUO ULIlIM VIH - 0 E = VJU3}IH VJZ + '0I = VJSMO Id/'z = dNV 0 = UflOa (I/fvIL = Viz 69TVI'C = Id 0*I = NO
(0oz)oa oIs Iw
(IOT'O)d'(IOtI'O)g NOISMWII I
C96 '~IA 8 AD D IJSII 3 (v'H'ov''nV.L-' 'Rwn'oat'u).naIa aIJ fOnuoIB











/' L'8A' X80'x' )C'(SI'Xz)'/' 'Jad[NI HO I-/)VIWOa V H'0h1I' M'D' N)N'RI I'V'9 HdVl JLdJiO IIUM 9'= A IVHt
0"9 = XIS
0" I- = SIMIRV
= amiaw
80+H C6L66Z = 3
6G9'I' = Id
(//A1OW SN0oIVNW(HIT'XC//0OILVflb2S SrIXSSS Xi WId0OHPV 'XS*/' I SOVdH9 'X0C 'UOVUS'133OV O?0IllHIHOZ'X6V' IHI)tVIMIOA z Z'9 SdVL JIldJ�iO aSIUM

I = aDVdN
SIIIVU 9I JA MKIII SI H 3 rIvJmaINVUAm dO AIIOO'IHA SVHtd UIZIIVMO SI OSI 3 AD3Nflb UA UIflNDV SI I 3 HUMJ~ Ud SUdN NI NO)LLVnSUV SI Ai) 3 J II'I HuaOwn DINDHVH ill SaI MMi 3 S01 oWfmN WOda sNflu vuDOoud il 3 mSUSGISNO r9 01 3INDtWVIH H3VdS SUII sll STI7 MIL SfLINI WIN 3
SnIIuv s ill S57(1 H/T O SZITa US9WflN UDO3S ([MV HHBEflN 3
DINOMVH MlL Snfl'Id O O SUH HU HHUfEN ISHIIl HIL HDISNHlWIa H 9I 3 (sniaim)a/da sI ()a 3
(sfIuvu)au/a si (m)a 3 HMO SI ()a 3
d SI ()A 3
U SI (7)A 3
snIUaV SI ()A 3 UUciuO LSmIA'a(II SuIA do NOIjz)NflA ISSu s UIII(aIOw SI d 3
UHGuO OUHZ'NIX SHIAJ lO 1D3IJMlNAd ISSg (1I1ECOR SI U 3
SJixa WIDuV X'1dWD 3D ILI mOIVflb SISSr ta O NDIfl1OS 3
(T)Z'(OZ)DXH'SI'(O)v'()VT I
'(IOI'O)d'(C)CIMS'(C)W'(g)'(C)A'(TO')U'(O)OXV OISNRIUI I
1I1 HINSI AU las SiMIud 'I mi.t nri sVIUVA "5A VI IO DMILSII 3 (INflO3N' H'OMS A I~ && M' uY' IdssssB &IlfOUSls











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 ,I4)
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= 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(1H,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),
1SRAKC(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




Full Text

PAGE 1

THE EFFECTS OF SP ACE HARMONICS AND ATTENUATION ON LONGITUDINAL ELECTRON MOTION IN A LINEAR ACCELERATOR By DONALD F. MOONEY A DISSERTATION PRESENTED TO THE GRADUATE COUNOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1963

PAGE 2

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.H. 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. ii

PAGE 3

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS------------------------------------------------ii LIST OF FIGURES-------------------------------------------------v PREFACE---------------------------------------------------------vii Chapter I. TIIE LINEAR ELECTRON ACCELERATOR-------------------------1 Introduction--------------------------------------------1 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 Hannonics--25 Attenuation---------------------------------------------32 Equations of Longitudinal Motion------------------------33 Summary-------------------------------------------------34 III. SOLUTION OF EQUATIONS BY DIGITAL cmtPUTER ---------------36 Introduction--------------------------------------------36 Bessels Equation for Complex Argument-------------------37 Space Harmonic Amplitude--------------------------------43 IV. SOLUTIONS OF EQUATIONS OF MOTION-•----------------------49 Introduction--------------------------------------------49 Traveling Wave Prebunchcr -------------------------------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

PAGE 4

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 iv

PAGE 5

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 l1avcguide --------------------------------------------14 5. Phase Lag, A, of Electron with Respect to the Stable Phase Position-------------------------------------------23 6. Brillouin Diagram----------------------------------------26 7. Instantaneous Distribution of E(J,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) 8w= 0.5, r = O, Nonharmonic Analysis-------------------51 14. Electron Phase Angle ..A vs. Normalized Axial Distance ) f3w= 0.5, r = a/4, Nonharmonic Analysis-----------------52 15. Electron Phase Angle A vs. Normalized Axial Distance f j3u.J= 0.5, r = a/2, Nonharmonic Analysis --------------53 16. Electron Phase Angle D.vs. Normalized Axial Distance 5 ft,0=0.S, r = O, Space Harmonic Analysis ------------------54 17. Electron Phase Angle A vs. Normalized Axial Distance 5 /3 tu= 0.5, r = a/4, Space Harmonic Analysis --------------55 V

PAGE 6

Page 18. Electron Phase Angle /J. vs. Normalized Axial Distance J P w = 0. 5, r = a/2, Space Harmonic Analysis ----------------56 19. Normalized Electron Energy~ vs. Normalized Axial Distance 5 Sw=O.S, r = O, Nonharmonic Analysis -----------------------58 20. Normalized Electron Energy~ vs. Norr:ialized Axial Distance 3 BwcO.S, r = n/2, Nonharmon1c Analysis---------------------59 21. Normalized Electron Energy t vs. Nonnalized Axial Distance 5 .Bw"" 0 .5, r = O, Space Harmonic Analysis --------------------60 22. Normalized Electrin Energy t vs. Normalized Axial Distance f ,.ct)::. 0. 5, r = a/2, Space Harmonic Analysis ------------------61 23. Nomalized Electron Energy o vs. Normalized Axial Distance 5 ,Bw=l.O, r = O, Nonharmonic Analysis-----------------------63 24. Normalized Electron Energy t vs. Normalized Axial Distance~ tSw~l.O, r = a/4, Nonharmonic Analysis---------------------64 25. Normalized Electron Energy t vs. Normalized Axial Distance) ;S'w~l.O, r = a/2, Nonharmonic Analysis---------------------65 26. Normalized Electron Energy 'I vs. Normalized Axial Distance fiw 1.0, r = 0, Space Harmonic Analysis --------------------66 27. NRrmalized Electron Energy t vs. Normalized Axial Distance J P'w=l.O, r = a/4, Space Harmonic Analysis ------------------67 28. Normalized Electron Energy X vs. Normalized Axial Distance~, /Jw-= 1.0, r = a/2, Space Harmonic Analysis ------------------68 29. Electron Phase Angle /J. vs. Nonnalized Axial Distance 5 ,8w=1.0, r = O, Nonharmonic Analysis-----------------------69 30. Electron Phase Angle b. vs. Normalized Axial Distance 5 ,6'~=1.0, r = a/4, Nonharmonic Analysis---------------------70 31. Electron Phase Angle b. vs. Normalized Axial Distance~ ,6'Uj:: 1.0, r = a/2, Nonharmonic Analysis ---------------------71 32. Electron Phase Angle b. vs. Normalized Axial Distance 5 A.o= 1.0, r = o, Space Harmonic Analysis --------------------72 33. Electron Phase Angle D, vs. Normalized Axial Distance 5 fJIA.) = 1.0, r = a/4, Space Harmonic Analysis ------------------73 34. Electron Phase Angle t.. vs. Normalized Axial Distance 5 p',,.,=-1.0, r = a/2, Space Harmonic Analysis ------------------74 vi

PAGE 7

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 arc 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 vii

PAGE 8

accelerator section are derived considering space harmonics, struc ture attenuation, and off-axis trajectories. These equations are then solved using the IB~I 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 longitudinalrnmotion in a sinusoidal field distribu tion are clerived. The solution for the axial field component is then modifiecl 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

PAGE 9

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. . ix

PAGE 10

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
PAGE 11

2 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

PAGE 12

Figure 1. Disk-Loaded Circular Waveguide

PAGE 13

4 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 arc 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 accelerator structure has a back-bombarded tantalum cathode button as a therr.1ionic 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 Oto 100 kilovolts and controls

PAGE 14

....... ACCELERATOR ELECTRON GUN ,/ STRUCTURE I\ I ' l/ STABILIZED GUN CONTROL KLYSTRON --...... FREQUENCY SOURCE I \ I\ /\ f MODULATOR AND CONTROL CIRCUITS Figure 2. Simplified Block Diagram of Linear Electron Accelerator System

PAGE 15

6 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 Tl1e 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 de systems the greatest electron injection velocity into the accelerator struc ture that can be reasonably obtained is one half the velocity of

PAGE 16

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

PAGE 17

8 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 D0me [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

PAGE 18

9 electrons moving away from this point experience a restoring force directed toward this phase position. Bunching occurs when electrons are concentrate
PAGE 19

Figure 3. Buncher Section 0

PAGE 20

11 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 D8me, and operated as a conventional uni form section. In each case the analysis will take into consideration

PAGE 21

12 space harmonics, off-axis trajectories, and structure attenuation. The results of this analysis will be compared with the results of previous theory.

PAGE 22

CHAPTER II EQUATIONS OF LONGITUDINAL ~IOTION Solution of Field Equations The application of the disk-loaded circular waveguide type of structure as a traveling wave prebuncher and as a unifonn 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 arc 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; 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 (TI! 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 13

PAGE 23

14 r .c 4A. Coordinate System +--d t----f--.. ------. T I-}-+ 2a l.. Figure 4B. Schematic Drawing of Disk-Loaded Iterative Circular Waveguide

PAGE 24

15 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 VXE = -fr (pIT) , and 2.2 VXVXt = V(V•E) -V2ir, where E = electric field (volts/meter) IT= magnetic field intensity (amps/meter) = permeability of vacuum. Equations 2.1 and 2.2 can be combined, resulting in By definition 2.4 Vx -11 = -J + ...R.. (l-I') ~t ; ' where ( = permitivity of vacuum.

PAGE 25

16 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 (O.Sc). Fundamental Phase Velocity= 2.997 x 108 meters/second (c). It has been shown [9] that for electrons injected with a velocity of o.sc 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 10 6 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 103 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

PAGE 26

17 Substitution of equation 2.5 into equation 2.3 yields Since the presence of the electron beam is assumed to have no effect on the electric field distribution, e = 0 and 2.7 V'IT = o. Therefore equation 2.6 becomes the wave equation In cylindrical geometry the) component of equation 2.8 is 2.9 Assuming E 2 of the forward traveling wave to be of the conventional form Re [E 0 (r) cjwtr)J and assuming operation in the lowest order Tr.I mode, equation 2.9 becomes 2.10 d2E 9 (r) + .!_ dE 0 (r) = dr2 r dr -K 2 E (r) C O ' K = w/c, r = I + j (3, and c is velocity of light (meters/second), I is attenuation con stant (nepers/meter), and /j is the phase constant (radians/meter).

PAGE 27

Initial conditions are dEo(r) dr r=O = o. 18 magnitude of the field on axis at j' = O, From wave guide theory it is known (10] that the remaining nonzero field components can be expressed in terms of Ez• 2.11 2.12 a 11Ez K 2 or C At this point in the analysis it is generally assumed that attenuation is negligible, as done by Chu and Hansen [7]. Therefore 2.13 r 2 = fi 2 , and Since where vp is the phase velocity of rf wave, therefore 2.16 -K/ = ( f )2 (~Y.

PAGE 28

19 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. where I 0 (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 J must be finite on the axis. Evaluation of equation 2.17 on axis yields Substitution of equation 2.18 into equations 2.11 and 2.12 yields Equations 2.18, 2.19, and 2.20 are similar to those derived by Chu and Hansen (7]

PAGE 29

20 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 1 , + 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 = O) 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 rQ 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 The mass m of the electron can be determined from the well known equation mo 2.22 m = -------,

PAGE 30

21 wherein m 0 is the mass of electron (kilograms) at rest and /3e is the axial electron velocity divided by the velocity of light. Let a normalized mass be defined as therefore Distance}' can be normalized with respect to the guide wavelength of the fundamental component of the electric field by the ratio where )' is measured from the transverse place at which electrons are injected. The prime notation is to differentiate)' from which is measured from the center of a cavity for subsequent space harmonic amplitude determination. Equation 2.21 can be rewritten using equations 2.23, 2.24, and 2.25. 2.27 F = .d_ (mo'6c2) } dr ~g The force exerted on an electron by the axial component of the elec tric field is described by defining the phase lag angle delta(~) as the relative phase of the electron with respect to the point of stable

PAGE 31

22 phase as shown in Figures. The point of stable phase travels down the structure at the phase velocity of the fundamental space harmonic. For each electron, D. 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 where e is electron charge (coulombs). Substitution of equation 2.28 into 2.27 yields 2.29 where d t . A TT = o1 sin~, j eE , ). = .:..::.i:.:g_, normalized m 0 c2 energy gain per wave length. 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. where Ve is the electron velocity and Vp is the phase velocity of the electric field.

PAGE 32

23 -~ 11 Figure S. Phase Lag, A, of Electron with Respect to the Stable Phase Position

PAGE 33

24 A normalized phase velocity can be defined as the ratio of the phase velocity of the electric field to the velocity of light. Applying equations 2.31 and the definition of/~ 2.30 results in 2.32 where 1:. is defined such that when 6. = 0 at J' = O, then}= 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 t\g/4 results in rf/2 phase shift per cavity, referred to as operation in the ff/2 mode.

PAGE 34

25 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 /f/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 71'/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 71/2 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 l 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

PAGE 35

' ' \ \ \ ' I \ \ -rf/2 ' ' Frequency 27fd To slope of tangent= vg -. I I I slope= vp I I I I rf/2 /3c I 11 ' ' ' Figure 6. Brillouin Diagram ' ' 2/'/ ' ' \ N a

PAGE 36

27 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 tis 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 tl1e 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. 2.33 E (~, r) where 2 34 f;) a + 2n rr. f-'n = ~o d From equation 2.14 Kc varies with /3n, therefore it is also subscripted in terms of n. E 0 n is the amplitude of the nth space harmonic at r = o. 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(g,r) at r = a, t = T/4, is assumed to have the

PAGE 37

28 configuration shown in Figure 7 for one wave length of axial distance. In constructing this distribution it was asswned that no field fring ing occurs at the disk hole and that attenuation is negligible. The axial component of electric field E(J,a), illustrated in Figure 7, can be described in terms of equation 2.34. n =ex> 2.35 E(;,a) = L EnI 0 (Kcna) sin Cfn-'i n=-oo 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 tho~ obtainable by experimental methods [17). The axiai component of electric field shown in Figure 7 can be described by a Fourier Series: m= a:> 2.36 E(y,a) = E Dm sin 2 ~ 71 l' mz:l where L equals 4d and 1/2 Dm f f E(},a) -1/2 2.37 . 2mf{ d SlJl L, t Equation 2.37 can be written in terms of structure parameters. 3d-1' --r 2 .38 0,, = ;/( E (;,,•l sin Jdj• /4+1 -r

PAGE 38

2. Figure 7. Instantaneous Distribution of E(}•a) at t = T/4 N \0

PAGE 39

30 Equation 2.38 can be integrated, yielding 2.39 2A ,. Dm = _)cos [!!!.!( (1 + -I.)] m'lf l 4 d cos Substitution of integers into equation 2.39 shows that Dmexists only form odd. Equating coefficients of equations 2.35 and 2.36 under the stipulation that 2.40 results in 2.41 where m = j 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 !:n_ = __ 1_ 0 _(_K_co_a_) __ E 0 ll+4n I 1 0 (Kc.,,a) cos[ ) 1+4n ]-cos["(3)ll+4n] cos[ (l+!.)]-cos[!!(3-~) 4 d 4 d A treatment of the backward wave would be identical in form other than for the sign on/Jn 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.

PAGE 40

31 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 y= 0 can be com puted [18]. 2.43 1/2 Eo = (2IroPo) volts/meter where P 0 is input rf power (watts) and r 0 is shunt impedance (ohms/meter) which is defined by equation 2.2 of reference 14. 2.44 E 2 0 ro = -, dP -cir where Eo is the peak value of the electric field at j = 0 and dP/d~ 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 r 0 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 10 7 ohms/m for the Stanford University Mark III Linne [19] and 5.6 x 10 7 ohms/m for the University of Florida Linac.

PAGE 41

32 Attenuation In a disk-loaded waveguide structure the magnitudes of the fields vary with axial distance} as e1 , where I is the attenuation coefficient, which can be determined [14] from the relationship 2.45 where v g is the group velocity at the angular frequence w 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 where P 0 is the magnitude of the input rf power, Lis the length of the structure, and r 0 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 avnilable in tabulated form. In this event equation 2.14 becomes

PAGE 42

33 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 wherehn 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 n = -p, , -1,0,+1, +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 where 'n was defined in conjunction with equation 2.28. The electric field used to compute the phase angle and energy is described by

PAGE 43

34 equation 2.33 with,6'n replaced by rn to account for the effects of attenuation. Since Ken and rn 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 2r,/..J_ ..L) \Ai~ /Jc n = -p, , -1, O, +l, , +p. Summary Longitudinal electron motion in a disk-loaded circular wave guide, operating in the ///2 mode, can be described by simultaneous solution of the following equations. For convenience in programming R is substituted for E 0 in equation 2.10. 2.52 2.53 2.54 Io(Kcoa) lcos[11'/4(l+T/
PAGE 44

35 2.58 n = -p, , -1, O, +l, , +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.

PAGE 45

CHAPTER III SOLUTION OF EQUATIONS BY DIGITAL cmtPUTER Introduction As summarized in Chapter II there are seven basic equations, (plus one additional equation for each space hannonic 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 1 (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 arc 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 languvge will be written in capital letters. 36

PAGE 46

37 BESSEL and the resultant values of I 0 (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 d2R + .!. -g2!' = 0 dr2 r dr ' ' the solution of which has the desired form In order to transform equation 3.1 into a fonn more compatible with the method for numerical solution let i = P. Solution of equation 3.1 now reduces to the solution of three simultaneous first order ordinary differential equations. 3.3 dr = l dr ,

PAGE 47

Continue 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 hase 38 SUBROlITINE BESSEL Integrate equation 2.51 using equation 2.52 for all space harmonics of interest SUBROlITINE FIELD Compute the space harmonic amplitude ratio Figure 8. Main Program and Subroutines

PAGE 48

3.4 3.5 dR = P, dr dP dr p + r 39 For use in the numerical integration process in SUBROUTINE BESSEL variables and their associated derivatives can now be defined. Let V and O denote variables and derivatives respectively. Let 3.6 V1 = r Vz = R V3 = p therefore, 3.7 01 dr = dr 02 dR cir and 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.

PAGE 49

40 From equation 2.52, Ken 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 following 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. Doth Hand NCOUNT are read into the program as

PAGE 50

w X < 41 SUBROUTINE BESS L .. Write title page Set line count Nill! = N M + 1 Write new page number LINE = 2 Write page heading Write column heading LINE= LINE+ 7 y Figure 9. Flow Diagram of Subroutine Bessel z

PAGE 51

w X 42 y LINE = LINE + 1 > Write new page number LINE= 1 Go to 6 > Return <. < Figure 10. Subroutine Bessel Continued z

PAGE 52

43 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 + I, where His 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 ~IAIN 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 I 0 (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,

PAGE 53

44 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. E 0 n is stored as a subscripted variable whose subscript is related to the same associated space harmonic subscript (NU~!) 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 Cf) it is accomplished by a 00 LOOP (an instruction causing the execution of a prescribed number of repetitive operations) in the MAIN program where J is available. A value of 'l'I for each space harmonic considered is then stored as a subscripted variable. The subscript is also related to the same space harmonic subscript (Nm!) 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.

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45 For convenience in programming define ~n in terms of ~ 0 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 5, and normalized phase velocity of the fundamental f1wo• Let k be defined as the number of space harmonics being considered including the fundamental. 3.11 V1 = f Vz = l( V J = f;,_. J VJ. = sin f1 j VP = cos Aj Therefore, 3.12 D2 = d~ rlf D. = dl>.j J df D1l = (sin ilj) j = n + 13 Q=j+k+l p=l2+k+l j=n+l3 .) = j + k + 1

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46 p=..Q+k+l 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 HAIN 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 S to increment from zero to some predetennined normalized axial distance (TERM) in steps of Ill!. The middle loop in crements the initial value of ..1 0 at 5 = 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 rt and DELINC was rf/3. In the analysis of the uniform sec tion DELTAZ was Tf/2 and DELINC was rf/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 A 0 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.

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V w < = 47 Write title page Set line count Call BESSEL Call FIELD Write new page number LINE= 2 Write page heading Write column heading LINE= LINE+ 7 Co inue X Figure 11. Flow Diagram of Main Program y z

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w 48 X LINE= LINE+ 1 Write new page number LINE = 1 Go to 6 Figure 12. Main Program Continued V Y Z

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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: /3 wo = o.s fle = o.s at input 49

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so 0(0 = 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 b. versus normalized axial distance J. 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 6.. The initial values chosen are If, 2 1f, ll., 0, -lf, -3.lf, 3 3 3 3 and -ff, 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

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+11~------'-----!...---+--------i------+-------l------+------1--------.1 ~o~-----4--=--~-_.,...~----.,..--+------+-------4--,~--'-~-----r-+-----~ ...J :.u .::, tri--------11---~7'----+-------1------1-------1--------+------+------I.O z.o 2.5" .3. 0 Xi Figure 13. Electron Phase Angle A vs. Normalized Axial Distance 5 ;8'.A.1 = O.S, r = O, Nonharmonic Analysis 3.S 4.0

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rt~-----+==:==========l===========i========:::!=======t::::=====t======l I <( so1-------+--=-~r-~~---~-+-----+------+-..,..,c.._--C',::---lf-----,~--+----..,.__----1 Ul 0 .5 1.0 J.S Xi 2.0 3.0 Figure 14. Electron Phase Angle 6vs. Normalized Axial Distance /3w = 0.5, r = a/4, Nonharmonic Analysis 3.S 4.0 0, N

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nL=J===='~====L======r==c==i __ J -nl----.,J.:~=='======'=====il=====!=:===::::::J[__ __ -J__ ___, .5 Figure 15. I oO 1.s Xi 2.0 2.5" Electron Phase Angle 6 vs. Normalized Axial Distance J J5'w = O.S, r = a/2, Nonharmonic Analysis 3.5" 4.0 V, t.,:I

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I
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I ~o~----~~-~-~~~--~~-----~----~~~-~~~---~-~--~-~ 0 7 -ff,--7---,---,---,---,----r---==f===:::=:=1 /o Xi 2.0 Figure 17. Electron Phase Angle.::l vs. Nonnalized Axial Distance S ,.dc.o= 0.5, r = a/4, Space Harmonic Analysis .J.S 4.0 (J1 u,

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TTt----+-_!,_--+------t-------j----====::t:::::::----t-----t----, S 01-----~L-~-4.-~:+----.,.&...---1-------+----_,.~~----,4------'r+----+-~ Ul 0 -rr,--,__:_7 __ -t---===t====t====:::=t====--,--,--, /. () /S Xi 2.0 2.5 3.o Figure 18. Electron Phase Angle .L\vs. Normalized Axial Distance f ,,6'-'-' = o.s. r = a/2, Space Hannonic Analysis 3.S 40 V, a,

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57 grouping occurs. This is illustrated in particular by the inter section of curves 2 and 6 which indicates that electrons injected at 6 = 21l and c. O 3 . o values of". = -Whave the same value of~ at particular 3 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 't versus normalized axial distance J• Figures 19 and 20 are energy plots for trajectory radii r = O, and r = a/2, calculated by the nonhannonic 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: /1wo = l.O /ic = o.s at input I= 1.0 neper/meter fa 5670 megacycles/second a= 0.005 meter.

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-< :;;: "' /-4t------jf------+-----t------+-------1------~----t-------1 /.3 /. 2 /.O '-------1.::=------:-1-:-----:-----L------1-----,.1_____ L__ ___ _j'------' ,5' / () /.S2.0 3.S4-.o 3.o Figure 19. Normalized Electron Energy tvs. Normalized Axial Distance 5 ,,6'<-..>= 0.5, r = O, Nonharmonic Analysis u, co

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j (.:, /4'------'-----+-----+-----+------+----+-----4------l /.3 12 /QL____ J_ ____ J_ ____ J_ ____ J_ ____ -'------'------.J-----~ /,o IS Xi 2,0 2.s s.o 3,S 4.o ,.5 Figure 20. Normalized Electron Energy vs. Normalized Axial Distance J /Sw= 0.5, r = a/2, Nonharmonic Analysis V,

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/,4,-....----+-----+------+------1-----~-----+------1-----1 /3t-------+-----+-----t------+-----+-----+-------4-----1 J.0'------.....;._ ____ _,_ _____ ,__ ____ _._ ____ __._ _____ ...,__ ____ -1____ __, ,,.. . ...; /0 .l. 0 2.s 3.0 Figure 21. Normalized Electron Energy ';/vs. Normalized Axial Distance J ,6'c.o = o.s, r = O, Space Harmonic Analysis ' 0

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-< c.:, /41-------!..-----L------i-----+-----+-----+------i-----t /3 /Z /01-------'------'------J-------'-------'------------_.__-----' I v /, 5" 2, o Z .5' Xi Figure 22. Normalized Electron Energy ;(vs. Normalized Axial Distance 5 A:_,= 0. S, r = a/2, Space Harmonic Analysis 4-.0 Q\ I-'

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62 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 arc plots of normalized energy 't versus normalized axial distance ) Figures 23 through 25 are energy plots for the three trajectory radii r = o, 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 /1. The initial values chosen are 1I, Ji, JI. o, -JJ. -,/J; and -ff 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.

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I 1.o 15 Xi 2.o 2.5 3. r) Figure 23. Normalized Electron Energy (5' vs. Normalized Axial Distance ;:L = 1. 0, r = O, Nonharmonic Analysis 4.o

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I ,s /. 0 /.S Xi 2.0 3-0 Figure 24. Normalized Electron Energy~ vs. Nonnalized Axial Distance J ,6w= 1.0, r = a/4, Nonharmonic Analysis 3,5 4.0

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,~z_ __ _J_ ___ _L ___ ___JL_ __ __;:~:::::=:::::_ _ _.,l_ ___ ---L, ____ ,L_ __ ----J /_ 0 IS Xi 2,0 2.S 3. D 4-,o Figure 25. Normalized Electron Energy t vs. Normalized Axial Distance ) ;6'u.>= 1.o. r = a/2t Nonharmonic Analysis

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/.~ Xi 2,o 2.S 3.0 Figure 26. Nonnalized Electron Energy t vs. Normalized Axial Distance ) #cv = 1.0, r = O, Space Harmonic Analysis 4.P

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G /.o Xi 2.o 2 . .s 3.u Figure 27. Normalized Electron Energy vs. Normalized Axial Distance /S'w = l.o. r = a/4. Space Harmonic Analysis 4.o

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3 ,~"""'----.1..-----.l...-----.l..---~-==:..L.---__,;;===-,-----~-----'--""-=a..-----' .s/o /S Xi 2.0 2.s .3. o 3.s 4.c Figure 28. Normalized Electron Energy 't vs. Normalized Axial Distance .) P<.v = 1.0, r = a/2, Space Harmonic Analysis ' 00

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7L------+----+-----+----t------t----r----r----"--1 .s Figure 29. /o /..5 Xi 2.0 Electron Phase Angle .1 vs. Normalized Axial Distance) ~c.c,;= 1.0, r = o, Nonharmonic Analysis 4-0

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1 1--------+-----+-------t-----+-----~-----+----+------1 0 _.,, Xi 2.0 2.S .;.o Figure 30. Electron Phase Angle /J. vs. Normalized Axial Distance 5 !'fw = 1.0, r = a/4, Nonharmonic Analysis 4.0 .....i 0

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'o/z 1-----+-----+------+-----+------+-----+------+------1 0 Xi 2,0 2.S" 3,0 Figure 31. Electron Phase Angle fl vs. Normalized Axial Distance S ,,6c.v = 1.0, r = a/2, Nonharmonic Analysis 3.S' 4,0 -...J .,_

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/T/2L----~----+----+-----+----+----+-------ir-----i /0 Xi Figure 32. Electron Phase Angle Li vs. Normalized Axial Distance S ftw= 1.0, r = O., Space Harmonic Analysis -...J N

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fl/21-------1------4------+-----+------+----i------+--------I Figure 33. /0 I.S Xi 2.0 2. , :., 3,0 Electron Phase Angle~ vs. Normalized Axial Distance J ,6w= 1.0. r = a/4, Space Hannonic Analysis 3.S -..J

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Figure 34. Electron Phase Angle .6 vs. Normalized Axial Distance~ ,,6w• 1.0, r = a/2, Space Harmonic Analysis --..J ""'

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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.

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CHAPTER V RESULTS AND CONCLUSIONS Traveling Nave 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. 76

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77 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 Prcbuncher 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.

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78 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.

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79 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 hannonics. This implies that designing a uniform section for maximum allowable gradient does not result in maximum energy transfer to the electrons. \Je 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.

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APPENDICES

PAGE 90

APPENDIX A IBM 709/7090 FORTRAN PROGRAHS

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C LISTING OF FEB. 27,1963 C AXL\L EQUATIONS OF MOTION FOR NN NUM SPACE HARMONICS I DIMENSION R(20,101),P(20,101),CC(l),EZ(20),El(20) DIMENSION V(65),D(65),AA(65),SUMD(65),ALFA(20),E0(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 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 C\VITY C NCOUNT TELLS HOW MANY POINTS WILL BE COMPUTED IN BESSEL C HIS THE INCREMENT IN RADIUS IN SUBROUTINE BESSEL NPAGE = l WRITE OUTPUl' TAPE 6,1 1 FORMAT(1Hl,49X,20HELECTRON ACCELERATOR,30X,6HPAGE l,/51X,18HPHASE !ENERGY STUDY//,53X,13HDONALD MOONEY//) READ INPUT TAPE 5,2,NUJl,NN,RADIUS,A,W,BWO,TAU,AO,DELTAO,DELINC, lANGLE,TERM,BETAE,GM,CYL,NCOUNT,H 2 FORMAT(2(I5),4(El5.6) / 4(El5.6) / 4(El5.6) / El5.6,I5,El5.6) WRITE OUTPUl' TAPE 6,3,NUM,NN,RADIUS,A,W,BWO,TAU,AO,DELTAO,DELINC, lANGLE,TERM,BETAE,GM,CYL,NCOUNT,H 3 FORMAT(/ lOX,lOHINPUl' DATA/ 2(15),4(5X,El5.6) / 5(5X,El5.6) / 14(5X,El5.6),4X,I5 / 4X,El5.6) ONE= 1.0 0:, N

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C C C C 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 = l 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) THE FOLimING DO STATEMENT MAY BE REMOVED TO LIMIT TRAJECTORY CALCUIATIONS TO ON-AXIS ELECTRONS DO 114 KEND = 1,3 RADIUS= RADIUS+ (A/4.0) DELTAZ = DELTAO + DELINC GAMAZ = 1.0/SQRTF(l. BETAE**2) K = NN NUM 14 DELTAZ = DELTAZ DELINC V(l) = 0.0 V(2) = GAMAZ DO 5 J = NUM,NN AN = J 10 L = J + 3 V(L) = DELTAZ M=K+L+l V(M) = SINF(V(L)) N=K+M+l V(N) = COSF(V(L))

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5 CONTINUE IF (LINE 50) 200,600,600 600 NPAGE = NPAGE + 1 WRITE OUTPUT TAPE 6,601,NPAGE 601 FORMAT(lHl,l00X,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 =,El3.6,/ 1, 7X, 10HPARAMETERS,5X, l0HDELTA ZERO,8Xl0H GAMA ZERO,9X, l0HWAVE BETA 2 /22X,El3.6,2(5X,El3.6),//7X,6HLA1dDA ,7X,10H DELTA ,7X,10H GA 3MA ,//) LINE= LINE+ 7 IF (LINE 8) 18,500,18 18 MM= 0 15 CONTINUE MM=IIM+l I= RADIUS/ H + 1. C ATI' =-GM*V(l).-i'WO*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 START INTEGRATION LOOP C DO 100 LLL = 1,4 D(l) = 1.0 DD= 0.0 DO 7 JJ = NUM,NN L = JJ + K + 4 M=L+K+l 00 .;:.

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CC(l) = V(L) CC(2) = V(M) I El(JJ) = R(JJ,I) CC D(2) = El(JJ) * ALFA(JJ) + DD DD= D(2) 7 CONTINUE DO 8 J = NUM,NN KK = J + 3 AN= J 10 D(KK) = TWO*Pl*(RBW*(ONE +FOUR*AN) -(V(2)/SQRTF(V(2)**2-l.))) L = K + KK + l LL = K + L = 1 D(L) = V(LL)*D(KK) D(LL) = -V(L) * D(KK) 8 CONTINUE 00 21 IJ = 1,65 D(IJ) = HH * D(IJ) SUMD(IJ) =Tr* D(IJ) + SUMD(IJ) 21 V(IJ) = T * D(IJ) + AA(IJ) IF(V(2) 1.0) 300,300,26 300 LINE= LINE+ 1 WRITE Ol.1l'Plrr TAPE 6,301,V(l),V(2),ALFA(l0),D(2),AA(2) 301 FORMAT(7X,F6.3,5X,17HELECTRON REJECTED, 4(5X,El5.6)) Ml4 = 0 GO TO 113 26 CONTINUE IF (LLL 2) 22,23,24 22 Tr= 2.0 GO TO 100 23 T = 1.0 GO TO 100 24 Tr= 1.0 100 CONTINUE 00 25 JJJ = 1,65 25 V(JJJ) = SUMD(JJJ)/6.0+ M(JJJ) co (,/1

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302 IF(MM 10) 112,110,110 110 LINE= LINE+ 1 IF(LINE 57) 500,102,102 102 NPAGE = NPAGE + 1 WRITE OtrrPtrr TAPE 6,103,NPAGE 103 FORMAT(1Hl,100X,5HPAGE ,14) LINE= 1 GO TO 200 500 WRITE OtrrPtrr TAPE 6,lll,V(l),V(l3),V(2) 111 FORMAT(7X,F6.2,2(4X,El3.6)) MM = 0 112 CONTINUE IF (V(l) TERM) 15,15,113 113 CONTINUE IF(DELTAZ ANGLE) 114,114,14 114 CONTINUE C DATA C\RD ORDER IS NUM,NN,RADIUS,A,W,BWO,TAU,AO,DELTAZ,DELINC, C ANGLE,TERll,BETAE,GM,CYL,NCOUNT,H C THE FOLLOWING FORMAT IS USED TO READ INPUT DATA C FORMAT(2(I5),4(El5.6)/4(El5.6)/4(El5.6)/El5,6,I5,El5.6) C\LL EXIT END 00 '

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SUBROUTINE FIEID(R,EO,NUM,NN,TAU,D,AO,H,A) C LISTING OF 8 FEB. 1963 I DIMENSION R(20,101),P(20,101) DIMENSION E0(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(/,lOHINPUT DA.TA,/,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(lO)) * AO WRITE OUTPUT TAPE 6,4,N,EO(J) 4 FORMAT(/,17HHARMONIC NUMBER =,15,5X,16HELECTRIC FIELD =,E20.8) 5 CONTINUE RETURN END co -..J

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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(l),CXA(20),SRAKC(20),Z(l) C SOLUTION OF BESSE~ EQUATION WITH COMPLEX ARGUMENTS C R IS MODIFIED BESSEL FUNCTION OF FIRST KIND,ZERO ORDER C PIS MODIFIED BESSEL FUNCTION OF FIRST KIND,FIRST ORDER C V(l) IS RADIUS C V(2) IS R C V(3) ISP C D(l) 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 C C C C C THE PROGRAJ.t RUNS FROM NUM TONN NN TELLS THE HARMONIC NUMBER LIMIT GM IS ATTENUATION IN NEPERS PER METER W IS ANGULAR FREQUENCY BWO IS NORMALIZED PHASE VELOCITY OF FUNDAMENTAL H IS INCREMENT IN RADIUS NPAGE == 1 Nml == NJK WRITE OUTPUT TAPE 6,2 2 FORMAT(1Hl,49X,20HELECTRON ACCELERATOR,30X,6HPAGE l,/48X,24HCOMPLE lX 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(/,lOHINPtrr DATA,/,2(4X,15),3(4X,E20.8),4X,F8.7,/) 00 00

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LINE = LINE + 3 AK= W/C DO 9 L = 1,20 00 9 LL = 1,101 I P{L,LL) = (0.,0.) I R{L,LL) = {0.,0.) 9 CONTINUE NUM = NUM 1 10 NtJM = NUM + 1 MM = 0 I V{l) = (0. ,0.) I V(2) = (1.0,0.) I V(3) = {0.,0.) AN = NUM 10 N = AN A(l) = GM A{2) = (AK/BWO)*(l. + 4.•AN) I GAMA=A I CKA{NUM) = (AK**2) + (GAMA**2) I AKC{NUM) = {CKA(Nt.JM)) * AMINUS IF(LINE 50) 6,5,5 5 NPAGE = NPAGE + 1 LINE = 2 WRITE Otn'PUT TAPE 6,7,NPAGE 7 FORMAT{1Hl,100X,5HPAGE ,14) 6 WRITE Otrl'PUT TAPE 6,8,N 8 FORMAT(//,7X,42HNUMBER OF SPACE HARMONICS BEING CONSIDERED,2X,15) WRITE Otrl'PUT 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(l) * (1./H) + 1. C I IS 10000 TD!ES THE VALUE OF RADIUS PLUS ONE 00 t.O

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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 INTEGRA.TION LOOP C DO 17 K=l,4 I D(l} = (1.,0.) I D(2) = V(3) IF(V(l)) 30,31,32 30 WRITE OUTPUT TAPE 6,33 33 FORMAT(/,15HNEGATIVE RADIUS,/} K = 0 GO TO 29 I 31 D(3} =HALF* AKC(NUM) TO TO 34 I 32 D(3) = V(2) * AKC(NUM) (V(3)/V(l}) 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 00 18 KK =1,3 I 18 V(KK) =(SUMD(KK)/SIX) + AA(KK) '-0 0

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MM = MM + l IF (MM 10) 37,38,38 38 CONTINUE LINE= LINE+ 1 IF(LINE 57) 25,21, 21 21 NPAGE = NPAGE + 1 WRITE OUTPtrr TAPE 6,22,NPAGE 22 FORMAT(1Hl,100X,5HPAGE ,I4) 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 OUTPtrr TAPE 6,26,V(l),V(2),V(5),Z(l),Z(2),SRAKC(NUM), lSRAKC(UUM) 26 FORMAT(7X,F6.4,4X,El2.6,4X,El2.6,2(7X,El2.6),7X,El3.6,5X,El2.6) 37 CONTINUE IF (I NCOUNT) 11,28,28 28 IF(NUM NN) 10,29,29 29 CONTINUE RETURN END '-0 ....

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APPENDIX B HARMONIC CONTENT In order to determine the number of space harmonic components which should be considered, it is necessary to determine the ampli tude of each cor.1ponent. The total rf power Pr can be described as the sum of the power contained in the fundamental and 2p space harmonic cor.iponents. Assuming that the shunt impedance is the same for all components the power contained in each harmonic can be described in terms of the square of the amplitude of electric field. B.2 2 + Ep, where E 0 is the magnitude of the axial component of electric field. In most structures to which this paper is applicable four space harmonics plus the fundamental are sufficient to describe more than 99.9 por cent of the total rf power. This is easily verified for any particular structure using the output from sub routine FIELD. B.3 % = 92

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93 The program is written in terms of the peak value of the fundamental (AO) which is read in as data. For convenience in pro gramming AO is substituted for E 0 in the text. When considering space harmonics the magnitude of the fundamental (AO) must be re duced by a correction factor (Cr) which accounts for the power con tained in the neglected harmonic components. B.4 E Cr = -9.. Er Therefore B.S AO= (Cr)(AO). This correction factor must be calculated by the investigator.

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LIST OF REFERENCES 1. Ponce de Leon, J.M., "Design of the Bunching Section of the Stanford Mark IV Linear Accelerator." M.L. Report No. 205, Stnaford University, Palo Alto, California, June, 1955. 2. Murphy, J. A., "A Single Cavity Prebuncher for a Linear Elec tron Accelerator," M.S. Thesis, University of Florida, Decem ber, 1962. 3. Srnars, E. A., "Prebunching by Velocity Modulation in Linear Accelerators," M.L. Report No. 303, Stanford University, Palo Alto, California, April, 1956. 4. D6me, G., "Electron Bunching by Uniform Sections of Disk Loaded Waveguide, Part A; General Study," M. L. Report No. 780-A, Stanford University, Palo Alto, California, December, 1960. 5. Dome, G., p. 70. 6. Chodorow, et al., "Stanford High-Energy Linear Electron Accelera tor," M:t'.Report No. 258, Stanford University, Palo Alto, California, February, 1955, p. 185. 7. Chu, E. L., and Hansen, N. w., ''The Theory of Disk-Loaded Waveguides," J. Appl. Phys. 18: 996-1008 (1947) 8. Chu, E. L., "The Theory of Linear Electron Accelerators," ~I. L. Report No. 140, Stanford University, Palo Alto, California, May, 1951. 9. Chodorow, !.!_., p. 180. 10. Hutter, R. G. E., "Beam and Wave Electronics in Microwave Tubes," D. Van Nostrand Company, Inc., Princeton, New Jersey, p. 32. 11. Ramo, s. and Whinnery, J. R., "Fields and Waves in Modern Radio," John Wiley and Sons, Inc., New Yo.rk, Second Edition, p. 154. 12. Lemnov, S.P. and Tyagunov, G. A., "Radial Oscillations of Particles in a Linear Electron Accelerator," Translation, Accelerators, Collection of Articles, No. FTD-TT-61-242, Foreign Technology Division, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio. 94

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95 LIST OF REFERENCES continued 13. Ginzton, E. L., "Microwave Measurements," McGraw-Hill Book Com pany, New York, 1957, p. 453. 14. ~lallory, K. B., "A Comparison of the Predicted and Observed Per formance of a Billion-Volt Electron Accelerator," High Energy Physics Laboratory Report No. 46, Standord University, Palo Alto, California. 15. Ginzton, E. L.' p. 452. 16. Mallory, K. B., p. 29. 17. Mallory, K. B. P• 30. 18. Chodorow, .:! !!1:, p. 138 19. Chodorow, .:! !.!_., p. 141 20. Neal, R. B •' "Design of Linear Electron Accelerators with Beam Loading," M. L. Report No. 379, Standord University, Palo Alto, California, March, 1957. 21. Lemnov, s. P. and Tyagunov, G. A. 22. I.B.~I. Reference Manual 709/7090 FORTRAN Programming System. 23. McCracken, Daniel D., "A Guide to FORTRAN Programming," John !Viley and Sons, Inc., New York. 24. Liebowitz, B. H., "Numerical Solutions to Differential Equations," Electro-Technology, Volume 29, No. 4, April, 1962. 25. Ramo, s. and Whinnery, J. R., p. 154. 26. I.13.M. Bulletin J 28-6114, "32K Complex Algebra."

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BIOGRAPllICAL SKETCH Donald F. Mooney was born in Ft. Wayne, Indiana, on March 7, 1932. In January, 1953, he graduated from Florida State University with the degree of Bachelor of Science in Arts and Sciences. Having completed Air Force R.O.T.C. he received a commission and served a two-year tour of duty os Radar Maintenance Officer. After separa tion from the service in April, 1955, he was employed by RCA at Cape Canaveral, Florida, until his return to school in September, 1955, at the University of Florida. He graduated with the degree of Bachelor of Electrical Engineering from the University of Florida in June, 1957. During the course of his undergraduate studies he was employed as a student assistant by the Engineering Experiment Station at the University of Florida. Immediately after his gradua tion, he enrolled in the graduate division of the University of Florida, and received the degrees of Master of Science in Engineer ing in January, 1959, and Doctor of Philosophy in August, 1963. While pursuing his graduate studies, he held a graduate research assistantship in the Department of Nuclear Engineering. 96

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been ap proved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the de gree of Doctor of Philosophy in Engineering. June, 1963 Supervisory Committee: A-aat& 7 Dean, College of Engineering Dean, Graduate School