THE EFFECTS OF SPACE HARMONICS AND
ATTENUATION ON LONGITUDINAL
ELECTRON MOTION
IN A
LINEAR
ACCELERATOR
DONALD P. MOONEY
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1963
ACKNOWLEDGEMENTS
The author wishes to express his sincere appreciation to
his supervisory committee for their counsel and encouragement.
lie is especially indebted to Dr. A. Ii. Wing and Dr. V. W. Dryden
for their constant guidance and to Prof. W. F. Fagen for his
project supervision. He also wishes to convey his gratitude
to the personnel of the University Computing Center who have been
helpful in the completion of this work.
The author wishes to thank his wife Elizabeth for en
couraging him throughout his graduate studies. lie also wishes
to thank Dr. Henry S. Blank for his friendship and assistance.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS  ii
LIST OF FIGURES  v
PREFACE  vii
Chapter
I. THE LINEAR ELECTRON ACCELERATOR  1
Introduction 
The University of Florida Linear Accelerator  2
Accelerator Structures  6
II. EQUATIONS OF LONGITUDINAL MOTION  13
Solution of Field Equations  13
Conventional Equations of Motion  20
Mode of Operation  24
Solution of Field Equations Considering Space Harmonics 25
Attenuation  32
Equations of Longitudinal Motion  33
Summary  34
III. SOLUTION OF EQUATIONS BY DIGITAL COMPUTER  36
Introduction  36
Bessels Equation for Complex Argument  37
Space Harmonic Amplitude  43
IV. SOLUTIONS OF EQUATIONS OF MOTION . 49
Introduction  49
Traveling Wave Prebunchr  49
The Uniform Section  57
V. RESULTS AND CONCLUSIONS  76
Traveling Wave Prebuncher Calculations  76
Uniform Section Calculations  76
Conclusions from Traveling Wave Prebuncher Results 77
Conclusions from Uniform Section Results  78
Summary of the Most Significant Conclusions  78
iii
TABLE OF CONTENTS continued
Page
APPENDICES  80
A. IBM 709/7090 FORTRAN PROGRAMS  81
B. METHOD FOR CALCULATING SIGNIFICANT HARMONICS  92
LIST OF REFERENCES  94
BIOGRAPHICAL SKETCH  96
LIST OF FIGURES
Figure Page
1. DiskLoaded 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 DiskLoaded Iterative Circular
Waveguide  14
5. Phase Lag, A, of Electron with Respect to the Stable
Phase Position  23
6. Brillouin Diagram  26
7. Instantaneous Distribution of E( ,a) at t = T/4  29
8. Main Program and Subroutines  38
9. Flow Diagram of Subroutine Bessel  41
10. Subroutine Bessel Continued  42
11. Flow Diagram of Main Program  47
12. Main Program Continued  48
13. Electron Phase Angle A vs. Normalized Axial Distance 5
6w= 0.5, r = 0, Nonharmonic Analysis  51
14. Electron Phase Angle A vs. Normalized Axial Distance S
Sj= 0.5, r = a/4, Nonharmonic Analysis  52
15. Electron Phase Angle Avs. Normalized Axial Distance ?
/6uJ= 0.5, r = a/2, Nonharmonic Analysis  53
16. Electron Phase Angle Avs. Normalized Axial Distance
A/)=0.5, r = 0, Space Harmonic Analysis  54
17. Electron Phase Angle Avs. Normalized Axial Distance
16tu= 0.5, r = a/4, Space Harmonic Analysis  55
Page
18. Electron Phase Angle A vs. Normalized Axial Distance 5
6ow= 0.5, r = a/2, Space Harmonic Analysis 
19. Normalized Electron Energy vs. Normalized Axial Distance 5
6,=0.5, r = 0, Nonharmonic Analysis ...
20. Normalized Electron Energy d vs. Normalized Axial Distance
6o=s0.5, r = a/2, Nonharmonic Analysis
21. Normalized Electron Energy vs. Normalized Axial Distance
S=0.5, r = 0, Space Harmonic Analysis
22. Normalized Electrin Energy K vs. Normalized Axial Distance
6,,=0.5, r = a/2, Space Harmonic Analysis 
23. Normalized Electron Energy K vs. Normalized Axial Distance S
/, =l.0, r = 0, Nonharmonic Analysis 
24. Normalized Electron Energy vs. Normalized Axial Distance
3S,=1.0, r = a/4, Nonharmonic Analysis 
25. Normalized Electron Energy K vs. Normalized Axial Distance
61,l.0, r = a/2, Nonharmonic Analysis  .........
26. Normalized Electron Energy I vs. Normalized Axial Distance S
)6U1.0, r = 0, Space Harmonic Analysis 
27. Normalized Electron Energy vs. Normalized Axial Distance 5
p=1.0, r = a/4, Space Harmonic Analysis ...
28. Normalized Electron Energy Yvs. Normalized Axial Distance
/6,=1.0, r = a/2, Space Harmonic Analysis
29. Electron Phase Angle A vs. Normalized Axial Distance
d =1.0, r = 0, Nonharmonic Analysis 
30. Electron Phase Angle 6 vs. Normalized Axial Distance S
r1=1.0, r = a/4, Nonharmonic Analysis
31. Electron Phase Angle A vs. Normalized Axial Distance S
/4,=1.0, r = a/2, Nonharmonic Analysis
32. Electron Phase Angle \vs. Normalized Axial Distance S
/z=1.0, r = o, Space Harmonic Analysis ...
33. Electron Phase Angle A'vs. Normalized Axial Distance
/e$=1.0, r = a/4, Space Harmonic Analysis ....
34. Electron Phase Angle & vs. Normalized Axial Distance
/, ,1.0, r = a/2, Space Harmonic Analysis 
PREFACE
This paper presents a method of analyzing longitudinal elec
tron motion in a diskloaded 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 diskloaded circular waveguide structures used in
linear accelerator applications. The advantage of using computer
techniques to calculate electron phase angle and energy in accelera
tor structures is that detailed numerical results can be obtained
to any desired accuracy. When experimental methods are used the
techniques involved are extremely complicated and expensive. These
experimental results are frequently ambiguous and their accuracy
depends upon the ingenuity of the investigator. Data from such
experiments must generally be interpreted with considerable uncer
tainty due to practical limitations imposed by existing techniques.
Previously existing analyses of longitudinal electron motion
in a linear electron accelerator do not take into consideration
space harmonics of the axial field existing in the accelerator struc
ture, and the attenuation in the structure. Both these factors are
important in dealing with the problem of an electron beam with a finite
radius which is treated in this paper. Here the equations governing
longitudinal motion for a traveling wave prebuncher and a uniform
accelerator section are derived considering space harmonics, struc
ture attenuation, and offaxis trajectories. These equations are
then solved using the IBM 709 digital computer and the results are
compared to existing data. This extension of existing knowledge
can be used to explain the consequences of previous approximations,
and offers greater accuracy for future design.
In Chapter I the basic operation of a linear accelerator is
discussed. In addition, the various components which can be combined
to form an accelerator structure are described, particularly those
structures which are analyzed in later chapters.
In Chapter II a solution is developed for the axial field
component in a traveling wave structure and the conventional equa
tions governing longitudinalmmotion in a sinusoidal field distribu
tion are derived. The solution for the axial field component is
then modified to account for the existence of space harmonics and
structure attenuation in order that new equations governing longi
tudinal motion can be developed including these factors.
In Chapter III the computer programs used to solve the equa
tions of longitudinal motion are explained. In Chapter IV the re
sults of computer solutions considering space harmonics and structure
attenuation are presented along with computer solutions of the con
ventional equations for two accelerator structure configurations.
Chapter V contains a summary of significant differences ob
served between the results of the conventional analysis and the results
of the analysis considering space harmonics and structure attenuation.
viii
The results of the analysis developed in this paper illustrate
that in a prebuncher structure space harmonics have a significant ef
fect on the phase distribution of the electrons and electron beam ra
dius has a small effect on the electron energy distribution, compared
to results predicted by previous theory. Results also verify the as
sumption that in a uniform accelerator structure the presence of
space harmonics results in a reduction in energy gain with negligible
effect on the phase distribution of the electrons, compared to that
obtained by previous theory.
All phase angle and energy calculations conducted for both the
conventional analysis and the space harmonic analysis developed in
this paper considered the effects of attenuation. This represents
an improvement over the previous analysis in which the effects of
attenuation on offaxis trajectories are neglected. Computations
neglecting the effects of attenuation were not made for this paper
since a mathematical model neglecting attenuation does not represent
a physically realizable system and therefore the computer time would
not have been justifiable.
CHAPTER I
THE LINEAR ELECTRON ACCELERATOR
Introduction
Recent developments in the design and construction of multi
megawatt klystron tubes have made available sources of electromag
netic energy in a frequency range not previously utilized in linear
accelerator applications. The Nuclear Engineering Department of the
College of Engineering at the University of Florida sponsored the
construction of a 10 million electron volt (Mev) Linear Electron
Accelerator designed to operate at a frequency of 5760 megacycles
per second.
As the name implies, a linear accelerator is a device in which
charged particles are accelerated in a straight line. This fact
differentiates a linear accelerator from an orbital accelerator in
which angular acceleration is imparted to a particle moving in a
curved path. In the Linac (Linear Electron Accelerator) acceleration
is accomplished by energy exchange from a traveling electromagnetic
wave to an electron beam which coexists with the wave. This fact
distinguishes the traveling wave linear accelerator from the stand
ing wave or static (Van de Graff) types.
Since an electron has finite mass, its velocity can never
equal the velocity of light and the phase velocity of the traveling
wave must be reduced so that energy can be transferred to the elec
tron. Reduction in the phase velocity of an electromagnetic wave
can be achieved by many different structures. The diskloaded
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 11megawatt 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. DiskLoaded 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 SAC225
threecavity klystron furnished by the Electronic Tube Division of
SperryRand 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 highpower klystron has output pulses of 3 megawatts power.
This rf power is supplied to the accelerator structure where it gener
ates the fields used for electron acceleration.
The electron gun which injects electrons into the accelera
tor structure has a backbombarded tantalum cathode button as a
thermionic electron source. The rear surface of the cathode is bom
barded by electrons from a second cathode which is a directlyheated
tungsten filament. This tungsten filament and the tantalum cathode
are referred to as the "inner diode." The bombarding electron stream
supplies the energy required to heat the tantalum cathode. The "outer
diode" is formed by the tantalum cathode and an anode immediately in
front. This anode is pulsed to turn on the electron beam and the
amplitude of the voltage pulse supplied to this anode determines the
magnitude of the beam current. The entire electron gun is maintained
at a negative potential with respect to the accelerator structure
which is electrically grounded for convenience and safety. This
potential difference is adjustable from 0 to 100 kilovolts and controls
STABILIZED
CONTROL KLYSTRON STA
FREQUENCY SOURCE
MODULATOR
AND
CONTROL CIRCUITS
Figure 2. Simplified Block Diagram of Linear Electron Accelerator System
the velocity with which the electrons enter the accelerator struc
ture. The Gun Control system supplies the necessary filament, bombar
der, accelerating, and anode pulse voltages to the electron gun.
The Accelerator Structure is basically the type of slow wave
structure illustrated in Figure 1. There are several other types of
structures or combinations thereof which could be utilized in this
system, the complexity of which is dictated by the efficiency desired
and ease of construction.
Accelerator Structures
The Accelerator Structure in Figure 2 can consist of as many
as three different types of components, each named for the function
it performs. These components are a prebuncher, a buncher, and a
uniform section. A prebuncher alters the electron phase distribu
tion while the beam velocity remains approximately equal to the in
jection velocity. A buncher section imparts an increase in electron
velocity as well as rearranging the phase distribution. A uniform
section is one in which the phase velocity of the electromagnetic
wave equals the velocity of light (c). This section is necessary
to achieve high electron energies. The uniform section may be used
alone, as in the prototype University of Florida Linac, or in combina
tion with either a prebuncher or buncher section, or with both.
Due to the problems encountered in high voltage dc systems
the greatest electron injection velocity into the accelerator struc
ture that can be reasonably obtained is one half the velocity of
light (O.Sc). If electrons with this velocity are injected directly
into a uniform section only a fraction of the available electrons
will be accepted and accelerated, due to the difference in velocity
between the particle and the wave. Increasing the accelerating com
ponent of the electric field in a uniform section will increase the
fraction of electrons accepted or bound to the wave. The limitation
now is the problem of high field emission and electrical breakdown in
the accelerator structure. A system of this type has a maximum ef
ficiency of acceptance of 50 per cent since the fields in the ac
celerator structure alternate in polarity each half cycle. When the
accelerating component of electric field is large enough to accelerate
all the electrons injected during one half cycle, electrons entering
the structure during the alternate half cycle will be rejected. The
reasonable thing to do is rearrange (bunch) the electron beam before
it enters the uniform section so that electrons which would normally
arrive during the decelerating portion of a cycle instead arrive with
the electrons which enter during an accelerating portion.
An ideal bunching system would generate electron bunches in
which all electrons have the same energy and phase position. Analysis
of electron trajectories has been made [1] and such a system is shown
to be physically unrealizable. For Linacs with reasonable output
energies, prebunching in phase is more important than prebunching in
energy because if all particles enter the uniform section at the same
phase the subsequent energy gain will be the same for all particles.
1Numbers in brackets refer to references at the end of the text.
The spread in energy at the output of the entire machine will be
the same as the energy spread at the output of the bunching system,
and the consequent energy differences will be insignificant compared
to the total electron energy at the output.
Klystron type prebunching is one method of accomplishing
the desired phase grouping of an electron beam. In this method the
velocity of the electrons is perturbed in a device which is usually
a resonant cavity. The perturbations result in the formation of
packets or bunches of electrons at some distance down the beam from
where the velocity perturbations occur. Murphy [2] described such
a system where the perturbing device was a resonant reentrant cavity
using a sinusoidal gap voltage. Smars [3] described the use of a
series of sinusoidally excited gaps separated by fieldfree drift
spaces.
An alternate method of prebunching employs a travelingwave
structure described by DOme [4]. The construction of this type of
device is the same as illustrated in Figure 1, with the dimensions ad
justed so that the phase velocity of the rf traveling wave is reduced
to a value slightly greater than or equal to the velocity of the
electron beam but less than the velocity of light. In this type of
prebuncher the electric field strength is very low and therefore the
amount of rf power required is comparable to that for a single cavity
system. As electrons travel through the structure they oscillate about
a reference phase position on the wave (illustrated in Figure 5). This
reference position will be called the point of stable phase since
electrons moving away from this point experience a restoring force
directed toward this phase position. Bunching occurs when electrons
are concentrated about this point of stable phase. Optimum structure
length is determined by the phase velocity of the electromagnetic
wave, the magnitude of the electric field and the allowable spread
in phase or electron energy. It has been proposed [5] that for struc
tures of reasonable length, the travelingwave prebuncher is the best
available device for reducing the phase and energy distribution of
electrons. The travelingwave prebuncher is one of the structures
analyzed in later chapters.
A type of buncher applicable to the accelerator system is a
special diskloaded waveguide, as shown in Figure 3, tapered so that
the phase velocity and field strength of the electromagnetic wave vary
along the length of the structure. At the end of the structure into
which electrons are injected the phase velocity is generally made
equal to the electron injection velocity. The electrons do not re
main at a constant phase angle with respect to the wave [6] but oscil
late about a point of stable phase. In this type of structure the
velocity at which the point of stable phase travels is increased with
axial distance by increasing the length of each successive cavity.
The electrons grouped near the point of stable phase gain energy
since they are continually accelerated toward the stable phase posi
tion. A corresponding increase in the accelerating component of the
electric field is necessary to keep the electrons in synchronism with
the wave. This increase in the magnitude of the field is accomplished
Figure 3. Buncher Section
by reducing the size of the disk hole in each successive disk. The
results of simultaneously increasing the phase velocity and the
magnitude of the accelerating field are reduction in the amplitude
of oscillation about the point of stable phase and increased electron
energy.
The section of the accelerator structure in which the elec
tron beam acquires most of its energy is the uniform section. This
is a diskloaded 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 constantgradient 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 travelingwave pre
buncher as recommended by DOme, and operated as a conventional uni
form section. In each case the analysis will take into consideration
12
space harmonics, offaxis 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 diskloaded circular waveguide type
of structure as a traveling wave prebuncher and as a uniform sec
tion of accelerator structure was discussed in Chapter I. In both
cases the solutions to the field equations are the same in form with
specific differences arising from parameter variations. Figure 4 is
a schematic drawing of a section of diskloaded circular waveguide
illustrating the dimensional notation used. The rationalized MKS
system of units will be used exclusively in this paper. The follow
ing analysis is restricted to structures which have rotational sym
metry.
In cylindrical structures the wave components are most con
veniently expressed in terms of cylindrical coordinates as shown
in Figure 4A. The axis of symmetry of the diskloaded wave guide
in Figure 4B.is oriented so that electrons travel in the positive P
direction. The structure is excited in such a manner that an axial
component of electric field is available to accelerate electrons
injected along the axis (TM mode). It is important to have exact
information about the fields in the region in which the electron
travels. However, because of the complex shape of the guide walls,
solutions of the field equations are inevitably somewhat inexact
r
0
.e 4A. Coordinate System
Figure 4B. Schematic Drawing of DiskLoaded Iterative Circular
Waveguide
and complicated. A brief analysis will be conducted in the conven
tional manner so that in later sections the departure from this analysis
can be emphasized.
Several methods of analysis [7,8] have been applied to the
solution of the field equations in a diskloaded circular waveguide.
One common assumption is that the fields in region I (r a) are the
type that exist in a circular waveguide. Various techniques were
then applied to match these fields to those existing between the
disks at r = a.
From Maxwell's equations
2.1 7XT= L T)
and
2.2 7XVXE = V(VE) V2E,
where = electric field (volts/meter)
IT = magnetic field intensity (amps/meter)
P = permeability of vacuum.
Equations 2.1 and 2.2 can be combined, resulting in
2.3 V(V.() V2VXT)
By definition
2.4 VXiT = 7 + _ ((T),
)t
where ( = permitivity of vacuum.
As in other analyses the effects of space charge will be
neglected. This assumption is justified by the fact that even in
lowgradient 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 103 meters.
Electron Velocity = 1.498 x 108 meters/second (0.5c).
Fundamental Phase Velocity = 2.997 x 108 meters/second (c).
It has been shown [9] that for electrons injected with a velocity of
0.5c into a uniform section operating at a frequency of 5670 mega
cycles, the lowest peak value of the axial component of electric
field for which electrons will be accepted is 4.388 x 106 volts/meter.
Under these conditions the ratio of the radial electric field due
to charge contained in the electron beam to the peak value of the
radial component of electric field in the structure is approximately
2 x 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 VXT= i .
Substitution of equation 2.5 into equation 2.3 yields
2.
2.6 V(V.E) V2 =p .
at2
Since the presence of the electron beam is assumed to have no effect
on the electric field distribution, e = 0 and
2.7 V'*E = 0.
Therefore equation 2.6 becomes the wave equation
b2F
2.8 72E 8=2.
t2
In cylindrical geometry the 3 component of equation 2.8 is
2EL + 1 21, + 1 2L 2 1: [ 2121
2.9 +
2 ,2 r ar r2 a2 2 t
Assuming Ez of the forward traveling wave to be of the conventional
form R e[L 0(r)eJt" 3 and assuming operation in the lowest order TM
mode, equation 2.9 becomes
2.10 d2E,(r) + 1 dE,(r) =K 2E (r)
dr2 r dr c o
where K2 = K2 +
K = o/c,
r = I + jP, and
c is velocity of light (meters/second), I is attenuation con
stant (nepers/meter), and 8 is the phase constant (radians/meter).
Initial conditions are
Eo(r3 = Eo(0) magnitude of the field on axis at = 0,
fr=0
do(r) 0.
dr
r=0
From wave guide theory it is known [10] that the remaining nonzero
field components can be expressed in terms of Ez.
2.11 Er = EZ
c
2.12 11 = .
0 K2 br
At this point in the analysis it is generally assumed that
attenuation is negligible, as done by Chu and Hansen [7]. Therefore
2.13 r2 = 2
and
2.14 Kc2 = K2 2.
Since
2.15 = L,
VP
where v is the phase velocity of rf wave, therefore
2.16 K2 ()2
/.lO K i G I 
It is necessary that vp c in order to optimize the interaction be
tween the traveling wave and particles of finite mass. Therefore from
equation 2.16, Kc2 0 and one solution of equation 2.10 can be written
directly for the rotationally symmetric system considered here.
2.17 Eo(r) = ClI^(Kr) ,
where o(Kcr) is a modified Bessel function of the first kind, zero
order.
For a differential equation of second order there is, in general,
a second solution with its associated arbitrary constant. The other
solution must [11] have a singularity at r = 0 and therefore can be
ignored since there is no conductor on the axis of the structure and
E must be finite on the axis.
Evaluation of equation 2.17 on axis yields
2.18 E = ReEoIo(Kcr)eJ(l't4)
Substitution of equation 2.18 into equations 2.11 and 2.12 yields
2.19 Er = Re c EoIl(Kcr)e(JtI)3,
c
2.20 H10 = Re e 4 Eoli(Kcr)e(tc ).
Equations 2.18, 2.19, and 2.20 are similar to those derived by Chu
and Hansen [7].
Conventional Equations of Motion
Axial electron displacement .' is measured from a transverse
plane through the center of the first disk, where electrons enter
the structure. This differs from the axial dimension which is taken
as + d/2 for simplification of the space harmonic amplitude analysis.
These variables are defined in Figure 4B.
The conventional equations, describing electron phase angle
and energy in a linear accelerator, are based on the assumption that
electrons travel on the axis (r = 0) and that radial velocity and an
gular velocity are negligible. These assumptions are reasonable since
a solenoidal magnetic field is used to confine the electron beam until
particle energies are achieved where relativistic stiffening of the
beam occurs. Lemnov [12] has shown that variations in radial position
can be restricted to within a few per cent of the beam radius and states:
"Oscillations in the r8 plane in the cases normally encountered in
practice have little influence upon the basic motion along the axis."
Considering only the axial component of force
2.21 F = (mc
The mass m of the electron can be determined from the well known
equation
2.22 m= 
(1 e2)1/2
wherein mo is the mass of electron (kilograms) at rest and fe is the
axial electron velocity divided by the velocity of light.
Let a normalized mass be defined as
2.23 Y= m/mo;
therefore
2.24 = (1 2)1/2
e
Distance can be normalized with respect to the guide wavelength
of the fundamental component of the electric field by the ratio
2.25 = '/g,
where is measured from the transverse place at which electrons
are injected. The prime notation is to differentiate y' from
which is measured from the center of a cavity for subsequent space
harmonic amplitude determination.
21v
2.26 g= 1 .
Equation 2.21 can be rewritten using equations 2.23, 2.24, and
2.25.
2.27 F = (moc2
The force exerted on an electron by the axial component of the elec
tric field is described by defining the phase lag angle delta (a) as
the relative phase of the electron with respect to the point of stable
phase as shown in Figure 5. The point of stable phase travels down
the structure at the phase velocity of the fundamental space harmonic.
For each electron, A is measured from the nearest point of stable
phase at the instant of injection and varies in accordance with the
integrated difference between the electron velocity and the phase
velocity (equation 2.30). From Figure 5 the axial electric field
force acting on the electron can be described by
2.28 F = eE? sinA,
where e is electron charge coulombss). Substitution of equation
2.28 into 2.27 yields
2.29 d = sinA,
where o = e normalized energy gain per wave length.
m oc2
A second equation describing longitudinal motion of the elec
tron can be derived by considering the change in Delta (A) caused by
a difference in the velocity of the electron and the phase velocity
of the electric field.
2.30 dA = (ve Vp) 2dt,
where ve is the electron velocity and vp is the phase velocity of
the electric field.
Phase Lag, 6, of Electron with Respect to the Stable
Phase Position
Figure 5.
A normalized phase velocity can be defined as the ratio of
the phase velocity of the electric field to the velocity of light.
2.31 /o =Vp/c.
Applying equations 2.31 and the definition of/'g 2.30 results in
2.32 d = 2 1 _1 .
where A is defined such that when A = 0 at = 0, then r= d/2 and
t = 3T/8.
Equations 2.29 and 2.32 are the two simultaneous first order
differential equations which describe longitudinal electron motion,
Use of these equations in conjunction with equation 2.18 is the con
ventional method of solving for electron motion in a linear accelerator.
Mode of Operation
In the literature pertaining to the theory and operation of
linear electron accelerators the word "mode" has two connotations.
In one application the word mode is used to describe the types of
waves associated with the propagation of electromagnetic energy
such as transverse electromagnetic (TEM) waves. A second application
is the use of the word mode to describe the phase shift per section
in a periodic structure. For example a cavity length equal to Ag/4
results in ff/2 phase shift per cavity, referred to as operation in
the If/2 mode.
Figure 6 is a plot of frequency vs. the phase shift per sec
tion showing the lowest frequency band of the infinite number of
pass bands which exist for a diskloaded waveguide. The manner in
which this diagram is constructed is discussed by Ginzton [13] who de
scribes how the group velocity and phase velocity of a wave propa
gating in the structure are determined from the Brillouin diagram.
At any frequency within the pass band shown the group velocity is the
slope of the curve at that ordinate and the phase velocity equals
the slope of a line from the origin to the above intersection.
The ff/2 mode of operation is the one most generally used in
traveling wave linear accelerator structures. One reason for this
is apparent from the Brillouin diagram since at the frequency correspond
ing to the 7;/2 mode the group velocity is maximum. This implies that
the fill time is reduced and electrons can be accelerated during a
longer portion of the pulse length. This can be an important design
factor as in the Mark III Stanford Linear Accelerator, wherein the fill
time is half the duration of each pulse. A second reason for choosing
the 7f/2 mode of operation is the ease with which measurements can be
made in the structure [14].
Solution of Field Equations Considering Space Harmonics
In the diskloaded structure shown in Figure 1 the boundary
conditions cannot be satisfied when one assumes that each component
of the electromagnetic field is a single sinusoid. Due to field
distortions at the disks and requirements imposed by the mode of
operation the field components will be composed of a series of space
slope of tangent = g
B\ I I
II I o.
II
I I/ i I
I I I I\
I I/ I I
I II I i
I I
i !
Brillouin Diagram
Frequency
27/d
x_
rf/2 ff/2 z2r p1
o 2 rrd/,Ag
Figure 6.
harmonics. The combined wave, including the space fundamental and
space harmonics, must travel in the positive z direction. This
requires that each space harmonic must have a phase constant lying
within a range in which is positive in the structure. A few such
harmonics are illustrated by the solid curves in Figure 6.
Considering space harmonics, the axial component of electric
field of the forward traveling wave can be more accurately described
by an infinite series of terms of the form of equation 2.18. Although
attenuation will be considered later, it is neglected here in order
to simplify the computation of the relative magnitudes of the space
harmonic components. The resultant field distribution and coordinate
system is shown in Figure 7.
n=oo
2.33 E(y,r) = Re Enl(Kcnr)ej (
n=oo
where
2.34 /6n =o 2
From equation 2.14 Kc varies with /n, therefore it is also
subscripted in terms of n. Eon is the amplitude of the nth space
harmonic at r = 0.
Let time t = T/4 be defined as that instant at which the
fundamental (axial field) component of the forward wave has its
maximum positive value at = d, the disk spacing. The axial dis
tribution of E(Q,r) at r = a, t = T/4, is assumed to have the
configuration shown in Figure 7 for one wave length of axial distance.
In constructing this distribution it was assumed that no field fring
ing occurs at the disk hole and that attenuation is negligible. The
axial component of electric field E(Q,a), illustrated in Figure 7,
can be described in terms of equation 2.34.
n =0O
2.35 E(,,a) = J Enlo(Kcna) sin (Pn.
n=00
Experimental methods of determining the ratio of the amplitude
of each space harmonic to the amplitude of the fundamental component
are described in detail in the literature [15,16]. The ratios cal
culated from structure dimensions by the following method are at
least as accurate as thoseobtainable by experimental methods [17].
The axial component of electric field shown in Figure 7 can be
described by a Fourier Series:
m=oo
2mr1
2.36 E(k,a) = Dm sin 2i
m=l
where L equals 4d and
/2
2 2m_ .
2.37 Dm = r E(3,a) sin L d,.
/2
Equation 2.37 can be written in terms of structure parameters.
3d
2.38 Dm = E( ,a) sin m d .
d+T
2
E( Y,A)
I I
d+al
2.
3 d4 r d
2 2.
3d2
2
Instantaneous Distribution of E( ,a) at t = T/4
Figure 7.
Equation 2.38 can be integrated, yielding
2.39 Dm= 2A cos mff[(1 ( +cos [m1(3 3 ])
2Ao 4 4 ( d
Substitution of integers into equation 2.39 shows that Dmexists
only for m odd. Equating coefficients of equations 2.35 and 2.36 under
the stipulation that
mit
2.40 n =2
results in
2.41 En = Dm
Io(Kcna)
where m = 1 + 4n .
This can be shown from equation 2.40 and the fact that Dm exists for
m equal to positive odd integers.
The ratio of the amplitude of the nth space harmonic to the
amplitude of the fundamental component can be expressed using equations
2.41 and 2.39.
2.42 En I(Kcoa) cos [(lJ) l1+4nl]cos[ (3 I 1+4nI]
Eo l1+4n I o(Kcba) cos [t(1+Z.)]cos [(31)]
A treatment of the backward wave would be identical in form
other than for the sign on ,n in equation 2.34. Any linear combina
tion of the forward and backward wave will satisfy the boundary con
ditions for the structure within the approximations made in the above
analysis.
The problem now remains to determine the magnitude of the
axial component of electric field. Since the input power (rf) to
the structure is known, the peak electric field at ,= 0 can be com
puted [18].
2.43 Eo = (2IroPo)1/2 volts/meter
where Po is input rf power (watts) and ro is shunt impedance (ohms/meter)
which is defined by equation 2.2 of reference 14.
Eo2
2.44 ro = 
where Eo is the peak value of the electric field at = 0 and dP/dk
is the rate at which power is dissipated in the walls of the structure.
As shown by equation 2.43, the axial electric field strength
in an electron accelerator varies as the square root of the power flow
ing in the structure. The shunt impedance per unit length ro is the
parameter which indicates the effectiveness of a given structure in
generating an accelerating electric field for a given power flow.
Shunt impedance is an experimentally determined quantity and the tech
niques for performing its measurement are discussed in the literature.
Some typical values are 4.73 x 107 ohms/m for the Stanford University
Mark III Linac [19] and 5.6 x 107 ohms/m for the University of Florida
Linac.
Attenuation
In a diskloaded waveguide structure the magnitudes of the
fields vary with axial distance as e'I where I is the attenuation
coefficient, which can be determined [14] from the relationship
2.45 I =
where vg is the group velocity at the angular frequence 'J and Q
is the unloaded Q of the structure considered as a resonator. The
energy in electron volts imparted to an electron passing through
the structure on the accelerating peak of the traveling wave has
been shown [20] to be
2.46 V = (2PoroI)/2( IL ) electron volts,
where Po is the magnitude of the input rf power, L is the length
of the structure, and ro is the shunt impedance per unit length
defined by equation 2.44.
From equation 2.46 the effect of attenuation on total electron
energy is obviously important. Also significant is the effect of
finite attenuation on the solution for the axial component of elec
tric field. With attenuation (I) other than zero the solution of
equation 2.10 contains Bessel functions with complex arguments and
such functions are not generally available in tabulated form. In
this event equation 2.14 becomes
2.47 Kcn2 = K2 + 12 n2 + j218n.
It has been shown [21] that the two equations of longitudinal motion,
equations 2.28 and 2.31, are best solved by numerical integration tech
niques. Use of a digital computer makes it feasible to solve equations
2.10 and 2.47 in a similar manner.
Equations of Longitudinal Motion
A new set of equations describing longitudinal electron motion
can now be developed in terms of the axial component of electric field
derived in equations 2.33 and 2.42. Since each space harmonic of the
traveling wave illustrated in Figure 6 will have a different normalized
phase velocity, equation 2.31 can be written for each harmonic.
2.48 d~.= 21f1. 
where An represents the relative phase of the electron to the nth
space harmonic.
In order to calculate the effects of the fundamental and 2p
space harmonics, equation 2.47 must be written
2.49 Ken2 = K2 + 12 n2 + j2I1n n = p,..., l,0,+l,...+p.
Since each space harmonic of the axial component of electric field
will affect the total energy of the particle, equation 2.28 can be
written
2.50 Re Z n( sin Ln
7X n=q
where cn was defined in conjunction with equation 2.28. The electric
field used to compute the phase angle and energy is described by
equation 2.33 with f/n replaced by P to account for the effects of
attenuation. Since Kn and n are both complex the real part of the
right hand side of equation 2.50 is significant.
Equation 2.48 must be calculated for the fundamental and
each of the 2p space harmonics being considered.
2.51 = 2 1
~~ 7_7
n = p, ..., 1, 0, +1, ..., +p.
Summary
Longitudinal electron motion in a diskloaded circular wave
guide, operating in the #I/2 mode, can be described by simultaneous
solution of the following equations. For convenience in programming
R is substituted for Eo in equation 2.10.
d2 1 dR 2
2.52 d +  + Kcn R = 0.
dr r dr
2.53 K n2 = 2 + K2 _n2 + j21Bn
E,
2.54 E
E5
2.55 d =
df
lo(Kcoa) cos [r/4(1+T/d) I 1+4n]cos[r/4(3r/d) 1+4n ]
11+4n Io(Kcna) cos [/4(1+lf/d)]cos [(/4(3/d)]
rn=p n0
"e Zc'n sindn n = p, ..., 1, 0, +1, ..., p.
n=q
2.56 eA = E2 n E eI
"oc7 EO pe
where Ep is the magnitude of the fundamental at r = 0.
2.57 E(,r)n Eo pe Re
2.58 = 20 
dl ^
o (Kcnr)ej (Wtn k)
n = p, ..., 1, 0, +1, ..., +p.
Due to the complexity of these equations it was necessary to
solve them using the IBM 709 digital computer. Programs developed
for this purpose are discussed in the following chapter.
CHAPTER III
SOLUTION OF EQUATIONS BY DIGITAL COMPUTER
Introduction
As summarized in Chapter II there are seven basic equations,
(plus one additional equation for each space harmonic considered)
which must be solved simultaneously by numerical methods in order
to determine longitudinal electron motion in a linear accelerator.
This chapter contains a discussion of the computer programs developed
for the purpose of solving these equations. These programs are
written in IBM 709/7090 FORTRAN language [22,23], and comprehension
of the subsequent sections requires a knowledge of the basic pre
cepts of this language. The RungeKutta [24] method of numerical
integration for fourth order accuracy was used in solving the dif
ferential equations.
Several of the equations which must be solved are functions
of radial position and space harmonic number only. These equations
need only be solved once for each space harmonic being considered
since only trajectories of constant radius are considered. Equa
tions 2.52 and 2.53 are numerically integrated over the range from
(radius) r = 0 to the maximum possible beam radius r = a for each
space harmonic being considered. This is accomplished in SUBROUTINE
1 Expressions in FORTRAN language will be written in capital
letters.
BESSEL and the resultant values of Io (Kcnr) are stored for later
use. Equation 2.54 defines the ratio of the amplitude of each space
harmonic to the amplitude of the fundamental component of electric
(axial) field. In SUBROUTINE FIELD equation 2.54 is solved for each
space harmonic being considered, anC these data are also stored in
such a way that they are available for solution of the equations of
axial motion which are functions of axial distance. These equations
of motion are solved in the MAIN program. Figure 8 illustrates the
relationship between the MAIN program and the two subroutines.
Bessels Equation for Complex Argument
The numerical integration of equation 2.51 is performed in
SUBROUTINE BESSEL using values of Ken computed by use of equation
2.53. As suggested by Ramo and Whinnery [25] let Kcn2 equal g2 so
that equation 2.51 can be written
3.1 dR + g2R = 0
dr2 r dr
the solution of which has the desired form
3.2 R = C10o(gr).
In order to transform equation 3.1 into a form more compatible with
dR
the method for numerical solution let d = P. Solution of equation
dr
3.1 now reduces to the solution of three simultaneous first order
ordinary differential equations.
3.3 r 1,
dr
Start mai program
Read inut data
Call seR l a
Continue
Cnll Fild
Continue
Solve equations 2.55
and 2.56 for space
harmonic amplitude at
the trajectory radius
Integrate equations
2.54 and 2.57
Compute electron energy
and phase
F End
SUBROUTINE BESSEL
Integrate equation 2.51 using
equation 2.52 for all space
harmonics of interest
SUBROUTINE FIELD
Compute the space harmonic
amplitude ratio
Figure 8. Main Program and Subroutines
dR
3.4 d= P,
dr
dr r
For use in the numerical integration process in SUBROUTINE BESSEL
variables and their associated derivatives can now be defined. Let
V and D denote variables and derivatives respectively. Let
3.6 V1 = r
V2 = R
V3 = P
therefore,
3.7 U1 = dr
dR
D2 =dr
and
dP
I) dP
3 U7
For use in the program each derivative must be defined in terms of
one or more variables from equation 3.6. This is accomplished by
using equations 3.3 through 3.5 in conjunction with equation 3.6.
The resultant expressions in FORTRAN language are in the LIST of
SUBROUTINE BESSEL shown in Appendix A.
From equation 2.52, Kcn is shown to be complex when structure
attenuation is considered; therefore, SUBROUTINE BESSEL was written
in complex notation [26]. A detailed logic flow diagram of SUBROUTINE
BESSEL is shown in Figures 9 and 10. The majority of the operations
indicated in the flow diagram are selfexplanatory, and the follow
ing discussion is merely to clarify some steps which might be ambiguous.
The results of the numerical integration of equation 2.52 for
each space harmonic considered were stored as a function of both space
harmonic number and radial position. The solution to equation 2.52
was defined as a subscripted variable RNUM'I.. The first subscript
(NUM) refers to the space harmonic number for which the integration
was performed. The second subscript (I) refers to the radial posi
tion associated with each numerical value. In FORTRAN language
DIMENSION subscripts are required to be nonzero positive integers;
therefore, it was necessary to define both subscripts in such a way
that negative space harmonics subscripts and fractional values of
radius could be described.
In order to make the computer program versatile it was written
in such a way that as many as eighteen space harmonics could be con
sidered, by allowing for a range of space harmonic subscripts from
minus nine to plus nine. This was accomplished by defining NUM equal
to ten plus the space harmonic subscript (n).
In the numerical integration of equation 2.52 the independent
variable (radius) r starts at zero and is stepped in increments H for
NCOUNT increments. Both H and NCOUNT are read into the program as
Flow Diagram of Subroutine Bessel
W x
Figure 9.
w x
Figure 10. Subroutine Bessel Continued
parameters so that different values may be used at the discretion of
the investigator. The maximum allowable value which can be read in
for NCOUNT is one hundred, which was used here. This limit is governed
by a DIMENSION statement contained in the program. In order to relate
a nonzero positive integer to a fractional value of radius the second
subscript I was defined.
3.8 I = r/H + 1,
where H is the incremental change in r for each integration step.
In order to minimize error, H should have a small fractional value.
For the data exhibited in this paper the value of H was 0.001.
The number of space harmonics considered by the program is
controlled by the values read in for NUM and NN. Each time the
subroutine has performed the integration of equation 2.52 over the
specified range of r, NUM is incremented by one. The entire inte
gration process is performed repeatedly for different space harmonics
until NUM equals NN and at this time control is returned to the MAIN
program. The remaining logic operations shown in Figures 9 and 10
deal with line count, paging and data display. These are merely for
the purpose of obtaining the output data in a concise, wellorganized
form.
Space Harmonic Amplitude
Having computed and stored the necessary values of Io(Kcnr),
equation 2.54 can now be solved. Equation 2.54 contains complex
terms and, since it need only be computed once for each space harmonic,
it is advantageous to compute and store En in another subroutine
names FIELD. En is the coefficient which specifies the ratio of the
amplitude of the nth space harmonic to the amplitude of the fundamental
peak electric field existing in the structure. Eon is stored as a
subscripted variable whose subscript is related to the same associated
space harmonic subscript (NUM) as was used in SUBROUTINE BESSLE. A
FORTRAN LIST of SUBROUTINE FIELD in included in Appendix A.
Before integration of equations 2.55 and 2.58 can be executed
it is necessary that equations 2.56 and 2.57 be solved once for each
space harmonic being considered. Since this operation must be re
peated for each increment of integration in normalized axial distance
(f) it is accomplished by a DO LOOP (an instruction causing the
execution of a prescribed number of repetitive operations) in the
MAIN program where ] is available. A value ofa, for each space
harmonic considered is then stored as a subscripted variable. The
subscript is also related to the same space harmonic subscript (NUM)
as was used in SUBROUTINE BESSEL.
Differential Equations of Longitudinal Motion
Equations 2.55 and 2.58 must be integrated simultaneously
over the normalized axial distance of interest for each space har
monic considered. This is accomplished by defining a new set of
two firstorder differential equations for each space harmonic con
sidered and then numerically integrating the set of firstorder
differential equations simultaneously.
For convenience in programming define An in terms of do.
3.9 An = o + (n o)j
where n is the space harmonic number, An is the phase angle of the
electron with respect to the nth space harmonic and ,n is the phase
constant of the nth space harmonic.
Equation 3.9 can be written in terms of normalized axial
distance and normalized phase velocity of the fundamental /o0.
3.10 An O + 8n.
Let k be defined as the number of space harmonics being considered
including the fundamental.
3.11 V1
V2
Vj
VP
Vp
f
= sin Aj
= cos Aj
j = n + 13
=j + k + 1
p = + k + 1
Therefore,
3.12 D1 = d
1 d
D = d
2 
D. = d_
ds
D d (sin Aj)
dS
j = n + 13
. = j + k + 1
D = c (cosA) p = + k + 1
For use in the program each derivative of equations 3.12 must be
defined in terms of one or more variables from equations 3.11. This
is accomplished by using equations 2.55 and 2.58 in conjunction with
3.10 and 3.11. The resultant expressions in FORTRAN language are
shown in the LIST of the MAIN program in Appendix A. A detailed
logic flow diagram of this program is shown in Figures 11 and 12.
There are three major loops in the MAIN program. The inner
most loop causes f to increment from zero to some predetermined
normalized axial distance (TERM) in steps of UI1. The middle loop in
crements the initial value of Ao at = 0 in steps of DELINC from
DELTAZ to ANGLE. Each increment corresponds to the initial phase
of one electron trajectory. In the analysis of the prebuncher section
DELTAZ was 1f and DELINC was 11/3. In the analysis of the uniform sec
tion DELTAZ was f1/2 and DELINC was ff/6. In the event that an elec
tron is injected an at initial phase angle which results in the elec
tron velocity being reduced to zero the program will recycle to the
next value of Ao until the value ANGLE is reached. The outermost loop
is a DO LOOP instructing the program to cycle through the entire tra
jectory analysis three times. The first set of trajectory calculations
are made for r = o. In each successive set of calculations r is in
creased by a/4 where a is the radius of the disk hole in the structure.
Write title page
Set line count
Write page heading
Write column heading
LINE = LINE + 7
Y Z
Flow Diagram of Main Program
Figure 11.
V Y
Figure 12. Main Program Continued
CHAPTER IV
SOLUTIONS OF EQUATIONS OF MOTION
Introduction
The preceding chapters of this paper have presented a
more accurate analysis of a constant phase velocity diskloaded
waveguide used as a prebuncher or as a uniformsection accelera
tor, by including the effects of space harmonics, structure at
tenuation and offaxis position on longitudinal electron motion.
In this chapter solutions of these equations are presented for a
particular traveling wave prebuncher and a particular uniform
section.
Also presented for comparison are solutions for the same
structures by what has been heretofore termed in this paper the
conventional analysis. For purposes of identification in the
succeeding parts of this paper this method is given the name non
harmonic analysis. The analysis provided by this paper is given
the name space harmonic analysis.
Traveling Wave Prebuncher
Both analyses were applied to a traveling wave prebuncher
having the following parameters:
Sa= 0.5
S e 0.5 at input
o<( = 0.10
I = 0.6838 neper/meter
f = 5670 megacycles/second
a = 0.01 meter
For purposes of investigating offaxis trajectories three
radii were chosen as specified below. The phase velocity of the
fundamental space harmonic was selected equal to the electron in
jection velocity.
It was decided to make computations over an axial distance
of four wavelengths so that two successive regions of close phase
groupings could be investigated. The three trajectory radii were
selected to investigate the effect of trajectory radius on the axial
distance to the regions of minimum phase spread.
Figures 13 through 18 are plots of phase angle A versus
normalized axial distance Figures 13 through 15 are phase
plots for the three trajectory radii r = 0, r = a/4, and r = a/2,
calculated by the nonharmonic analysis. Figures 16 through 18 are
corresponding phase plots for the same three trajectory radii cal
culated by the space harmonic analysis, for the same input power.
Curves identified by the same number have the same initial
value of L. The initial values chosen are if, 2 0, 0, , 23_
3 3 3 3
and /, identified respectively as curves numbers 1 through 7.
The purpose of this structure is to group electrons within
a small phase spread. Both methods of analysis show that this phase
7, __ ,,,,_____________________________________________
/.O /.5 2.0 2 .5 .3.0 3.5 4.0
X;
Figure 13. Electron Phase Angle A vs. Normalized Axial Distance
/, = 0.5, r = 0, Nonharmonic Analysis
.5 1.0 I .5 2.0 2.5 3.0 3.5 4.0
Figure 14. Electron Phase Angle A vs. Normalized Axial Distance
/, = 0.5, r = a/4, Nonharmonic Analysis
 o
.5 1.0 1.5 Xi 2.0 2.5 3.0 3.5 4.0
Figure 15. Electron Phase Angle A vs. Normalized Axial Distance
F r = 0.5, r = a/2, Nonharmonic Analysis
7
1 / / X 2,o 2,5 3.o 33. 4,
Figure 16. Electron Phase Angle A vs. Normalized Axial Distance
/O = 0.5, r = 0, Space Harmonic Analysis
/' Xi 2.0 2.15 3.0 J, 
Figure 17.
Electron Phase
/,= 0.5, r
Angle A vs. Normalized Axial Distance
= a/4, Space Harmonic Analysis
2
\J
A3<
4.0
Figure 18.
5.~j
2.0
2.5
Electron Phase Angle A vs. Normalized Axial Distance
/ = 0.5, r = a/2, Space Harmonic Analysis
o0
w
Q
3.S
grouping occurs. This is illustrated in particular by the inter
section of curves 2 and 6 which indicates that electrons injected
at do= 2S and 40 = 2ifhave the same value of A at particular
3 3
values of .
For each trajectory radius the nonharmonic analysis (Figures
13 through 15) shows a greater axial distance to the first phase
grouping than is obtained from the space harmonic analysis (Figures
16 through 18).
Figures 19 through 22 are plots of normalized energy K versus
normalized axial distance f. Figures 19 and 20 are energy plots
for trajectory radii r = 0, and r = a/2, calculated by the nonharmonic
analysis. Figures 21 and 22 are corresponding energy plots for the
same two trajectory radii, calculated by the space harmonic analysis,
for the same input power. As expected, all differences are small
because the traveling wave prebuncher is a lowgradient structure.
Therefore only results for the extreme radii are presented.
The Uniform Section
Both analyses were applied to uniform section having the
following parameters selected for the University of Florida Linac:
/ao = 1.0
/c = 0.5 at input
do =.Z.0
I = 1.0 neper/meter
f 5670 megacycles/second
a = 0.005 meter.
.47
/.2
/o
/.. Ao 2.. 3.o . 
Figure 19. Normalized Electron Energy 2 vs. Normalized Axial Distance
/,= 0.5, r = 0, Nonharmonic Analysis
/4,77
/1o /.5 Xi 2.0 2, 3. 3.C 4. o
Figure 20. Normalized Electron Energy Y vs. Normalized Axial Distance
/1 = 0.5, r = a/2, Nonharmonic Analysis
4
",7
3
1.5, X, 76
I .1. ______________~,~
2. 
Figure 21. Normalized Electron Energy Yvs. Normalized Axial Distance
So= 0.5, r = 0, Space Harmonic Analysis
__ I
A0C
60
/02
Figure 22. Normalized Electron Energy vs. Normalized Axial Distance
,,= 0.5, r = a/2, Space Harmonic Analysis
A structure length of four wavelengths was selected because
all the electrons which are accepted will have achieved an energy
of at least two Mev in this distance.
Figures 23 through 28 are plots of normalized energy versus
normalized axial distance Figures 23 through 25 are energy plots
for the three trajectory radii r = 0, r = a/4, and r = a/2, calculated
by the nonharmonic analysis. Figures 26 through 28 are corresponding
energy plots for the same three trajectory radii, calculated by the
space harmonic analysis, for the same input power.
There is no significant difference, among themselves, between
the plots of energy versus distance for different trajectory radii
from the nonharmonic analysis. A similar statement holds true for
the space harmonic analysis. There is a significant different be
tween the results of the two analysis. This can be observed by
comparison of Figures 23 and 26, Figures 24 and 27, and Figures
25 and 28.
The variation of electron phase angle with axial distance is
of secondary importance in the use of a uniform section as an accelera
tor. Nevertheless, calculations of phase angle vs. normalized axial dis
tance were made for the same conditions as in Figures 23 through 28,
and the results are presented in Figures 29 through 34.
Curves identified by the same number have the same initial
value of A. The initial values chosen are 0, 4 4 and .T
236 6 3 2
Curves for these initial values are identified respectively by numbers
1 through 7. Note that these initial phase angles are different from
those chosen for the traveling wave prebuncher.
Figure 23.
Normalized Electron Energy ?(vs. Normalized Axial Distance
43 = 1.0, r = 0, Nonharmonic Analysis
4.0
4
S
,5 / o /.5 Xi 20 2. 3.0 3.5
Figure 24. Normalized Electron Energy X vs. Normalized Axial Distance
/ ,= 1.0, r = a/4, Nonharmonic Analysis
5s /o /5 X; 2.0 2. 3.0 3.5 4,
Figure 25. Normalized Electron Energy d vs. Normalized Axial Distance
/8,= 1.0, r = a/2, Nonharmonic Analysis
7
Z.s
3.0
Figure 26. Normalized Electron Energy 6 vs. Normalized Axial Distance
,8 = 1.0, r = 0, Space Harmonic Analysis
'3
a
I
3.1
4.0
O0
0\
X't 2,0
2 /0O Xi 2 .s 3,0 " 4.o
Figure 27. Normalized Electron Energy Yvs. Normalized Axial Distance
/ = 1.0, r = a/4, Space Harmonic Analysis
/ o
7
/5S, ; x; O
2.5
3. 0
Figure 28. Normalized Electron Energy Y vs. Normalized Axial Distance
,g = 1.0, r = a/2, Space Harmonic Analysis
4. c
s /o /5 i 2.0 2.5 3.0 3.5 4.0
Figure 29. Electron Phase Angle 6 vs. Normalized Axial Distance
A,= 1.0, r = 0, Nonharmonic Analysis
3.0
Figure 30. Electron Phase Angle A vs. Normalized Axial Distance
/4o= 1.0, r = a/4, Nonharmonic Analysis
3.
4.0
*^~~~~~~~~       
*) ^     
7
\
.^ ^___
tj  ^ s ^ _________________________________~ 
O
1
I
*5s /o /.5" S 2.O 2.s 3.0 3.,5 4,0
Figure 31. Electron Phase Angle A vs. Normalized Axial Distance S
,/$ = 1.0, r = a/2, Nonharmonic Analysis
2. 5
Figure 32. Electron Phase Angle A vs. Normalized Axial Distance
/,= 1.0, r = 0, Space Harmonic Analysis
7r
4,0
!7
/.5
2.0
2.  
S/G 5" 20 2., 3.0 3., 4.0
A;
Figure 33. Electron Phase Angle A vs. Normalized Axial Distance
/4= 1.0, r = a/4, Space Harmonic Analysis
1
0 /.5 ,X,/ 20 2,5 3.3,0 4.
Figure 34. Electron Phase Angle A vs. Normalized Axial Distance
/6= 1.0, r = a/2, Space Harmonic Analysis
75
The greatest differences in the results of the two methods
are for those electrons which are not accepted, as shown by curves
numbered 1 and 7.
CHAPTER V
RESULTS AND CONCLUSIONS
Traveling Wave Prebuncher Calculations
Both methods of calculating phase angle versus axial dis
tance show that there are two phase groupings in the first four
wavelengths. The first phase grouping occurs at an axial distance
of approximately one wavelength. The second phase grouping occurs
at an axial distance of approximately 3.75 wavelengths.
Both methods of calculation show that the first phase group
ing has less phase spread than the second phase grouping.
Both methods show that for offaxis 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 onaxis and offaxis electrons, space
harmonics reduce the distance to the first phase grouping.
Calculations of energy versus distance (Figures 19 through
22) indicate that in the particular structure analysed in this paper
there is no significant difference between the results of the two
methods and that trajectory radius has little effect.
Uniform Section Calculations
Calculations of energy versus distance by both methods show
that trajectory radius has little effect.
The most significant difference between the results of the
two methods is that the space harmonic analysis indicates less energy
gain for the same input power.
With regard to phase variations with distance there was very
little difference between the results of the two methods for the chosen
structure. Owing to the cost of computer time a detailed analysis of
the structure acceptance angle was not made.
Conclusions from Traveling Wave Prebuncher Results
From the results of analysis of longitudinal motion in the par
ticular traveling wave prebuncher analyzed in this paper one can con
clude that the best bunching occurs at the first point of minimum phase
distribution. However, in the analysis of a structure with different
parameters the possibility of utilizing the second phase grouping
should not be neglected.
We have shown that both trajectory radius and space harmonics
have an effect on the axial distance to the first phase grouping,
and that the further the trajectory is from the axis, the nearer
to the point of injection is the location of the first phase group
ing. Although no formula for this relationship has been developed,
the effects should be taken into account in designing any prebuncher.
The energy versus axial distance calculations show that in this
structure the effects of electron trajectory radius and space harmonics
on electron energy are small. However, this structure has a low gradient
and a phase velocity equal to the injection velocity of the electron.
These conditions reduce the effects of trajectory radius and space
harmonics. Therefore it is not to be concluded that these effects
can always be neglected.
Conclusions from Uniform Section Results
Calculations of energy versus axial distance by both methods
indicate that in a uniform section the effects of trajectory radius
on energy gain is slight. This may be taken as a general conclusion.
The effects of space harmonics on energy gain, however, is not
negligible. The existence of space harmonics reduces the energy gain.
The only way to avoid this reduction is to design a structure in which
there is little energy in the harmonics.
Summary of the Most Significant Conclusions
In a traveling wave prebuncher space harmonics and offaxis
displacement reduce the axial distance to the first phase grouping.
This implies that selection of structure length on the basis of on
axis trajectories will result in significant error.
The predominant factor to consider in the analysis of longi
tudinal motion in a uniform section of accelerator structure is the
reduction in energy gain resulting from the existence of space har
monics.
The only way to avoid the resultant energy reduction in a uni
form section is to design a structure in which negligible rf power is
contained in the space harmonics. In the structure analyzed in this
paper the harmonic energy content in the prebuncher was approximately
one per cent and in the uniform section was approximately ten per cent.
It has been shown elsewhere that in designing a uniform sec
tion the disk hole radius can be reduced to obtain higher gradients.
We have shown here that reduction in disk hole radius results in a
greater percentage of power being contained in the space harmonics.
This implies that designing a uniform section for maximum allowable
gradient does not result in maximum energy transfer to the electrons.
We believe that the computer program for the solution of the
more complicated equations of motion, both of which are original in
this paper, are a significant contribution to the art of accelerator
design.
APPENDICES
APPENDIX A
IBM 709/7090 FORTRAN PROGRAMS
C LISTING OF FEB. 27,1963
C AXIAL EQUATIONS OF MOTION FOR NN NUM SPACE HARMONICS
I DIMENSION R(20,101),P(20,101),CC(1),EZ(20),E1(20)
DIMENSION V(65),D(65),AA(65),SUMD(65),ALFA(20),EO(20)
C NUM 10 IS THE MOST NEGATIVE SPACE HARMONIC BEING CONSIDERED
C RADIUS IS THE TRAJECTORY RADIUS FOR WHICH THE CALCULATION IS MADE
C NN 10 IS THE MOST POSITIVE SPACE HARMONIC BEING CONSIDERED
C A IS THE RADIUS OF THE DISK HOLE IN THE STRUCTURE
C W IS THE ANGULAR FREQUENCY
C BWO IS THE NORMALIZED PHASE VELOCITY OF THE FUNDAMENTAL
C TAU IS THE STRUCTURE DISK THICKNESS
C AO IS THE PEAK VALUE OF FUNDAMENTAL (AXIAL)
C DELTAO IS THE PHASE ANGLE AT WHICH THE FIRST ELECTRON ENTERS
C DELINC IS THE INCREMENT IN DELTAZ BETWEEN ELECTRONS
C ANGLE IS THE LIMITING VALUE OF DELTAZ TO BE CONSIDERED o0
C TERM DEFINES THE AXIAL DISTANCE OVER WHICH COMPUTATIONS ARE MADE
C BETAE IS THE NORMALIZED ELECTRON INJECTION VELOCITY
C GM IS THE STRUCTURE ATTENUATION IN NEPERS PER METER
C CYL IS THE LENGTH OF ONE CAVITY
C NCOUNT TELLS HOW MANY POINTS WILL BE COMPUTED IN BESSEL
C H IS THE INCREMENT IN RADIUS IN SUBROUTINE BESSEL
NPAGE = 1
WRITE OUTPUT TAPE 6,1
1 FORMAT(1H1,49X,20HELECTRON ACCELERATOR,30X,6HPAGE 1,/51X,18HPHASE
ENERGY STUDY//,53X,13HDONALD MOONEY//)
READ INPUT TAPE 5,2,NUM,NN,RADIUS,A,W,BWO,TAU,AO,DELTAO,DELINC,
1ANGLE,TERM ,BETAE,GM,CYL,NCOUNT,H
2 FORMAT(2(15),4(E15.6) / 4(E15.6) / 4(E15.6) / E15.6,15,E15.6)
WRITE OUTPUT TAPE 6,3,NUM,NN,RADIUS,A,W,BWO,TAU,AO,DELTAO,DELINC,
1ANGLE,TERM,BETAE,GM, CYL,NCOUNT, H
3 FORMAT(/ 10X,10HINPUT DATA / 2(15),4(5X,E15.6) / 5(5X,E15.6) /
14(5X,E15.6),4X,I5 / 4X,E15.6)
ONE = 1.0
TWO = 2.0
FOUR = 4.0
C = 2.99793E+08
PI = 3.14159
Q = 1.6E19
HH = 0.01
AK = W/C
RBW = 1./BWO
NPAGE = 1
LINE = 11
C2MO = 8.176E14
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 ONAXIS 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)**21.)))
L = K + KK + 1
LL= K + L = 1
D(L) = V(LL)*D(KK)
D(LL) = V(L) D(KK)
8 CONTINUE
DO 21 IJ = 1,65
D(IJ) = HH D(IJ) o0
SUMD(IJ) = TT D(IJ) + SUMD(IJ) L
21 V(IJ) = T D(IJ) + AA(IJ)
IF(V(2) 1.0) 300,300,26
300 LINE = LINE + 1
WRITE OUTPUT TAPE 6,301,V(1),V(2),ALFA(10),D(2),AA(2)
301 FORMAT(7X,F6.3,5X,17HELECTRON REJECTED, 4(5X,E15.6))
MM = 0
GO TO 113
26 CONTINUE
IF (LLL 2) 22,23,24
22 TT = 2.0
GO TO 100
23 T = 1.0
GO TO 100
24 TT = 1.0
100 CONTINUE
DO 25 JJJ = 1,65
25 V(JJJ) = SUMD(JJJ)/6.0+ AA(JJJ)
302 IF(MM 10) 112,110,110
110 LINE = LINE + 1
IF(LINE 57) 500,102,102
102 NPAGE = NPAGE + 1
WRITE OUTPUT TAPE 6,103,NPAGE
103 FORMAT(1H1,100X,5HPAGE ,14)
LINE = 1
GO TO 200
500 WRITE OUTPUT TAPE 6,111,V(1),V(13),V(2)
111 FORMAT(7X,F6.2,2(4X,E13.6))
MM = 0
112 CONTINUE
IF (V(1) TERM) 15,15,113
113 CONTINUE
IF(DELTAZ ANGLE) 114,114,14
114 CONTINUE
C DATA CARD ORDER IS NUM,NN,RADIUS,A,W,BWO,TAU,AO,DELTAZ,DELINC,
C ANGLE,TERM,BETAE,GM,CYL,NCOUNT,H
C THE FOLLOWING FORMAT IS USED TO READ INPUT DATA
C FORMAT(2(I5),4(E15.6)/4(E15.6)/4(E15.6)/E15.6,I5,E15.6)
CALL EXIT
END
SUBROUTINE FIELD(R,EO,NUM,NN,TAU,D,AO,H,A)
C LISTING OF 8 FEB. 1963
I DIMENSION R(20,101),P(20,101)
DIMENSION EO(20)
ONE = 1.0
PI = 3.14159
ETA = TAU/D
FOUR = 4.0
AMP = 2./PI
ONETA = 1.0 + ETA
THETA = 3.0 ETA
WRITE OUTPUT TAPE 6,1,NUM,NN,TAU,D,AO,H,A
1 FORMAT(/,10HINPUT DATA,/,2(4X,I5),5(4X,E15.6))
I = ((1. /H) A) + 1.
DO 3 K = 1,20
EO(K) = 0.0
3 CONTINUE
DO 2 L = NUM,NN
AN = L 10
B = ABSF(ONE + FOUR AN)
BB = (PI/FOUR) B
C = BB ONETA
D = BB THETA
EO(L) = (AMP/B) (COSF(C)COSF(D))/R(L,I)
2 CONTINUE
DO 5 J = NUM,NN
N = J 10
EO(J) = (EO(J) / EO(10)) AO
WRITE OUTPUT TAPE 6,4,N,EO(J)
4 FORMAT(/,17HHARMONIC NUMBER =,15,5X,16HELECTRIC FIELD =,E20.8)
5 CONTINUE
RETURN
END
SUBROUTINE BESSEL(R,P,NJK,NN,GM,W,BWO,H,NCOUNT)
C LISTING OF 14 FEB. VARIABLE LENGTH, PRINTS EVERY TENTH BIT
I DIMENSION AKC(20),R(20,101),V(3),D(3),AA(3),SUMD(3),P(20,101),
I 1A(1),CKA(20),SRAKC(20),Z(1)
C SOLUTION OF BESSELS EQUATION WITH COMPLEX ARGUMENTS
C R IS MODIFIED BESSEL FUNCTION OF FIRST KIND,ZERO ORDER
C P IS MODIFIED BESSEL FUNCTION OF FIRST KIND,FIRST ORDER
C V(1) IS RADIUS
C V(2) IS R
C V(3) IS P
C D(1) IS ONE
C D(2) IS DR/D(RADIUS)
C D(3) IS DP/D(RADIUS)
C IN R DIMENSION THE FIRST NUMBER REFERS TO 10 PLUS THE HARMONIC
C NUMBER AND SECOND NUMBER REFERS TO 1/H TIMES THE RADIUS
C NUM MINUS TEN TELLS THE FIRST SPACE HARMONIC TO BE CONSIDERED
C THE PROGRAM RUNS FROM NUM TO NN
C NN TELLS THE HARMONIC NUMBER LIMIT
C GM IS ATTENUATION IN NEPERS PER METER
C W IS ANGULAR FREQUENCY
C BWO IS NORMALIZED PHASE VELOCITY OF FUNDAMENTAL
C H IS INCREMENT IN RADIUS
NPAGE = 1
NUM = NJK
WRITE OUTPUT TAPE 6,2
2 FORMAT(1H1,49X,20HELECTRON ACCELERATOR, 30X,6HPAGE 1,/48X,24HCOMPLE
1X BESSELS EQUATION//53X,13HDONALD MOONEY//)
PI = 3.14159
C = 2.99793 E+08
LINE = 5
AMINUS = 1.0
SIX = 6.0
HALF = 0.5
WRITE OUTPUT TAPE 6,4,NUM,NN,GM,W,BWO,H
4 FORMAT(/,1OHINPUT DATA,/,2(4X,I5),3(4X,E20.8),4X,F8.7,/)
LINE = LINE + 3
AK = W/C
DO 9 L = 1,20
DO 9 LL = 1,101
I P(L,LL) = (0.,0.)
I R(L,LL) = (0.,0.)
9 CONTINUE
NUM = NUM 1
10 NUM = NUM + 1
MM = 0
I V(1) = (0.,0.)
I V(2) = (1.0,0.)
I V(3) = (0.,0.)
AN = NUM 10
N = AN
A(1) = GM
A(2) = (AK/BWO)*(1. + 4.*AN)
I GAMA = A
I CKA(NUM) = (AK**2) + (GAMA**2)
I AKC(NUM) = (CKA(NUM)) AMINUS
IF(LINE 50) 6,5,5
5 NPAGE = NPAGE + 1
LINE = 2
WRITE OUTPUT TAPE 6,7,NPAGE
7 FORMAT(1H1,100X,5HPAGE ,14)
6 WRITE OUTPUT TAPE 6,8,N
8 FORMAT(//,7X,42HNUMBER OF SPACE HARMONICS BEING CONSIDERED,2X,15)
WRITE OUTPUT TAPE 6,27
27 FORMAT(7X,6HRADIUS, 7X,9HREAL PART,7X,9HIMAGINARY,7X,15HDERIVATIVE
1 REAL,7X,9HIMAGINARY,10X,8HKCN REAL,10X,9HIMAGINARY,///)
LINE = LINE + 7
IF (LINE 8) 11,25,11
11 CONTINUE
I = V(1) (1./H) + 1.
C I IS 10000 TIMES THE VALUE OF RADIUS PLUS ONE
I R(NUM,I) = V(2)
I P(NUM,I) = V(3)
DO 12 J = 1,3
I AA(J) = V(J)
I 12 SUMD(J) =(0.0,0.0)
TT = 1.0
T = 0.5
C
C START INTEGRATION LOOP
C
DO 17 K=1,4
I D(1) = (1.,0.)
I D(2) = V(3)
IF(V(1)) 30,31,32
30 WRITE OUTPUT TAPE 6,33
33 FORMAT(/,15HNEGATIVE RADIUS,/)
K = 0
K=O
GO TO 29 o
I 31 D(3) = HALF AKC(NUM)
TO TO 34
I 32 D(3) = V(2) AKC(NUM) (V(3)/V(1))
34 CONTINUE
DO 13 J = 1,3
I D(J) = H* D(J)
I SUMD(J) = TT* D(J)+ SUMD(J)
I 13 V(J) = T D(J) + AA(J)
IF ( K2 ) 14,15,16
14 TT = 2.0
TO TO 17
15 T = 1.0
GO TO 17
16 TT = 1.0
17 CONTINUE
DO 18 KK =1,3
I 18 V(KK) =(SUMD(KK)/SIX) + AA(KK)
w
MM = MM + 1
IF (MM 10) 37,38,38
38 CONTINUE
LINE = LINE + 1
IF(LINE 57) 25,21, 21
21 NPAGE = NPAGE + 1
WRITE OUTPUT TAPE 6,22,NPAGE
22 FORMAT(1H1,100X,5HPAGE ,14)
LINE = 1
GO TO 6
25 CONTINUE
MUM = 20 + NUM
I SRAKC(NUM) = SQRTF(AKC(NUM))
IF(SRAKC(NUM)) 35,36,36
35 SRAKC(NUM) = AMINUS SRAKC(NUM)
36 CONTINUE
I Z = V(3)/SRAKC(NUM)
MM = 0
WRITE OUTPUT TAPE 6,26,V(1),V(2),V(5),Z(1),Z(2),SRAKC(NUM),
ISRAKC(MUM)
26 FORMAT(7X,F6.4,4X,E12.6,4X,E12.6,2(7X,E12.6),7X,E13.6,5X,E12.6)
37 CONTINUE
IF (I NCOUNT) 11,28,28
28 IF(NUM NN ) 10,29,29
29 CONTINUE
RETURN
END
