W. W. HANSEN LABORATORXES OF PHYSICS STANFORD UNIVERSITY STANFORD, CALIFORNIA

1

M - 201

BEAM DYNAMICS OF THE

PROJECT M ACCELERATOR

BY R. H. Helm

W. K. H., Panofsky

Internal Memorandum

Project M Report No. 201 November 1960

W. W. Hansen Laborat.ories of Physics Stanford University Stanford, California.

I

TABU OF CONTENTS

Page

1 . . . . . . . . . . . . . . . . . . . . . . . 1 . Introduction

3 2 . 1 Beam Qual i ty ; Phase Space . . . . . . . . . . . . . . 3 2.2 Formulation of t he Problem . . . . . . . . . . . . . 5 2.3 Misalignment Ef fec ts . . . . . . . . . . . . . . . . 6 2.4 External Magnetic Fields . . . . . . . . . . . . . .

2.4.1 Steer ing . . . . . . . . . . . . . . . . . . . 2.4.2 Degaussing . . . . . . . . . . . . . . . . . . 9 2.4.3 Focusing . . . . . . . . . . . . . . . . . . . 10 2.4.4 Aberrations . . . . . . . . . . . . . . . . . 12

2.5 Asymmetries . . . . . . . . . . . . . . . . . . . . . 13 2 .5 .1 Accelerator Asymmetries . . . . . . . . . . . 13 2.5.2 Coupler Asymmetries . . . . . . . . . . . . . 13

Accelerating F ie ld . . . . . . . . . . . . . . . . . 19 2.6.1 Effec ts Linear i n the Energy Gradient . . . . 19

2 . Transverse Beam Dynamics . . . . . . . . . . . . . . . . .

7 7

2.6 Effect of Axial Variations i n Phase and

2.6.2 Effec ts Quadratic i n the Energy Gradient . . . 20 2.7 Sca t te r ing by Residual G a s . . . . . . . . . . . . . 24

2.7.1 Multiple Sca t te r ing . . . . . . . . . . . . . 24

2.7.2 Single Coulomb Sca t te r ing . . . . . . . . . . 25

3 . Transverse Force Equations and Numerical Results . . . . . 27

3.1 Summary of t he Transverse Force Equations . . . . . . 27

3.2 Numerical E s t i m a t e s of Transverse Forces . . . . . . 28

3.2.1 Misalignment . . . . . . . . . . . . . . . . . 28

3.2.2 Coupler asymmetry . . . . . . . . . . . . . . 29 3.2.3 Axial va r i a t ions i n acce lera t ing f i e l d . . . . 31 3.2.4 "Noise" forces . . . . . . . . . . . . . . . . 31 3.2.5 Focusing f i e l d s . . . . . . . . . . . . . . . 32

3.3 Special Solutions . . . . . . . . . . . . . . . . . . 32

3.4 Approximate Treatment of the Noise Forces . . . . . . 38

4 . Summary . . . . . . . . . . . . . . . . . . . . . . . . . 41

..... .. ~- -I

T 7 - T

% 1. INTRODUCTION

We shall not present here a comprehensive consideration of the beam dynamics of the electron linear accelerator; for details, the

reader is referred to the publications of Chu (ML-140), Chu and Hansen.. . r~1i-66, ML-771, Chodorow et al. [Rev. sci. Instr. - 26, 134 (1955) 1, and Smars (ML-252). ing relevant facts emerge:

From these and other papers the follow-

1) tic) sections of a linear accelerator, the motion is longitudinally

neutral, i.e., neither phase stable nor phase unstable; the relative

phase between particle and fields depends on the accuracy of control of

the phase and phase velocity within the accelerator structure and the

other radio-frequency components.

With the exception of the first few (not completely relativis-

2) In the nonrelativistic section, phase stability and bunching action are attainable at the expense of radial defocusing. This de-

focusing can be counteracted by external weak or strong focusing lenses.

3) No complete analysis has been made of the theoretical phase

acceptance of the nonrelativistic entrance section of an electron

linear accelerator with external lens focusing which takes into account

the coupling between longitudinal and radial dynamics. Analyses of 0 particular cases have shown that a phase acceptance in excess of 120

is easily possible; final bunching angles depend on the feasible length

of the nonrelativistic section.

(4) The current capacity of an electron accelerator (particularly at short wavelength) is limited by electron beam loading. In an accel- erator specifically designed for highest current operation, very heav-

ily loaded operation is possible; in this Proposal a loading of the

order of 10% is adopted.

(5) An ideal continuous linear accelerator guiding electrons approaching the velocity of light has very simple properties regarding

the radial behavior of the beam. The radial electric force is balanced by a magnetic force which is 2

-Be 'e times the electric force, where

* This section is quoted from the original "Proposal for a Two-

Mile Linear Electron Accelerator," Stanford University, Stanford, California, April, 1957.

- I -

i s the p a r t i c l e ve loc i ty i n u n i t s of the ve loc i ty of l i g h t .

sult, t h e radial momentum i s very near ly a constant of the motion of

the p a r t i c l e s ; s ince the longi tudina l momentum increases l i n e a r l y with

t h e axial d is tance , the angle of the beam w i t h the a x i s va r i e s as

7 = m, and the beam radius increases logar i thmica l ly w i t h

the axial d is tance ,

A s the r e -

( 6 ) The logarithmic radial divergence does not l ead t o a ser ious

beam loss, Also, res idual gas s c a t t e r i n g e f f e c t s a r e s m a l l . I n addi-

t i on , very small strong-focusing l enses are sufficient t o reduce these

r e s idua l l o s ses t o zero; such lenses lead , however, t.0 somewhat IC-

creased f i n a l angular divergence ( f o r d e t a i l s see M L 2 0 2 ) .

(End of quotation from 1957 Proposal )

- 2 -

2 (I TRANSVERSE BEAM DYNAMICS

The present discussion will concern the main accelerator. The - injector region, which may be composed of the first 10 to 100 feet of

the accelerator, will not be considered here except as a source of beam current for the main accelerator.

From the point of view of a beam-dynamics study the requirements

to be met are:

injected beam;

ity at the far end of the machine.

(1) transmission of a large fraction (near 100%) of the

(2) attainment of a beam having sufyiciently good qual-

2.1 BEAM QUALITY; PHASE SPACE

The "quality" of the beam may be defined in terms of the phase

space, proportional to

PP r jj.. dxdydzdp dp dpz X Y

(2.1-1)

where the integration is over a phase-space volume which includes the

trajectories of all particles; (x, y, z) and (px, py, p,) are respect-

ively the coordinates and their conjugate momenta. According to

Liouville's theorem this quantity will be conserved in any system for

which the equations of motion can be expressed in the Hamiltonian form.

An appropriate form of the phase-space integral for accelerator beam- optics is

nr n

or for highly relativistic particles

(2J-2)

where 7 is a measure of the kinetic energy, x' = dx/dz, y' = dy/dz,

and 8 is the phase angle spread of electron bunches. In beam-dynamics

problems the paraxial formulation, corresponding to first-order optics, *

* See, for example, P. A. Sturrock, Static and Dynamic Electron

Optics, Cambridge University Press (1955)

- 3 -

T I

usually will be sufficient. The quantities x, y, 8 , yx', 7ys, and

67/7 are assumed to be always small.

always a linear transformation relating the coordinates

(yx'), (7y'), 673 to their initial values. A n alternate statement of

Liouville's theorem in this case is that the matrix of this transforma-

tion will always have a unit determinant, since this is the condition

for a unit Jacobian determinant in a linear transformation.

In this approximation there is

[x, y, 8 ,

The most useful quality in defining the optical quality of a beam

or projection of the phase-space volume is the transverse phase space

on the transverse coordinates: - -'

Evidently a beam of small size and small. angular divergence must have a

small transverse phase space.

be considering systems in which, with proper choice of the x and y

axes, the x and y motions are decoupled, so that

In the present discussion we w i l l usually

Thus, the x and y projections of the phase space are meaningful

individually.

The transverse phase space generally will be considered for a

given energy 7, but in systems involving transverse deflections, for

example, there may be first-order coupling between energy and the trans-

verse coordinates (e.@;. , dispersion terms), so that for a finite spec- trum the transverse phase space need not be conserved. In processes

involving deflection or focusing by rf fields, there may a l s o be cou-

pling between phase angle 8 and transverse coordinates, leading again

to nonconservation of the transverse phase space.

--

Other processes leading effectively to phase-space nonconservation

might include scattering, collimation, and randomly varying forces

(because such effects involve degrees of freedom which would not, be in-

cluded in formulating the Hamiltonian of the system). In a region in which there are no accelerating fields, 7 is

constant for a given particle, and the transverse phase space is

T

- 4 -

proportional to

(2.1-6)

(assuming x and y motions to be decoupled) with both ss dxdx' and

IJ' dydy' conserved except for dispersion effects. We shall spea.k rather loosely of the "phase space" as &.AxD or @yC,y' where Ax,

ny define the maximum beam size at a given point, and &!, the

maximum angular divergence.

quirements in the beam-switching and experimental areas are expected to

impose a tolerance of

Beam optical and energy resolution re-

0 An energy spread of Ay = f 0.5% - y and phase spread of At3 = 10 are

presumed

As will be seen, these requirements place certain tolerances on machine alignment, symmetry (particularly coupler symmetry), and the

phase space and spectrum of the injected beam.

2.2 FORMULATION OF TKF: PROBLEM *

Some of the transverse forces, notably phase focusing in an ideal

uniform accelerator, and space-charge forces, vary approximately as

(1 - Be) a y

istic energies are reached. Such forces may be extremely important

in the initial portion of the injection region but will be assumed to

be negligible for the present discussion.

2 -2 and thus rapidly become unimportant as highly relativ-

The following list summarizes effects which may be important in

the highly relativistic region: 1. Misalignment.

2 External magnetic fields, including steering (to compen- sate misalignment and stray fields, for example), degaussing (to com-

pensate the earth's field), focusing fields, and aberrations in the

above fields

* See, however, Section 2.6.

- 5 -

I

3. Deviations from axial symmetry in the accelerator struc-

ture, particularly in the couplers which feed rf power.

4. "Modulation" effects related to local variations in phase

and amplitude of the rf fields relative to the electrons.

5. Electron scattering by residual gas in the accelerator,,

6. Possible unknown effects.

Suitable equations of motion for the highly relativistic electrons

are given by

d A

d A ( 2 0 243)

dY heEZ(z) cos 8

dl; mc = a COS e (2.2-lc) 2 - =

where 5 = z/h, 6 = x/X, 7) = y/A; z is measured along the accelera-

tor axis; h is the rf wavelength; the prime denotes the derivatfve 2 with respect to 5 ; y 2 electron energy in mc units; fl, f2 are the

transverse forces Fx, Fy in dimensionless units; 8 ( 5 ) is the phase

angle of the crest of the rf accelerating wave relative to the electron;

EZ is the amplitude of the rf accelerating wave; and a ( c ) is EZ

in dimensionless units.

It has been assumed that the transverse motions are small and that

the velocity in the z-direction is nearly equal to the velocity of

light. Terms of order higher than the first in ', 7 ', and l/y have been dropped in going from time derivatives to c-derivatives.

We now consider these effects in greater detail.

2.3 MISALIGNME3VT EFFECTS

Misalignment can arise in two ways: mechanical errors in joining

the sections together, and bends arising from displacements in the sup-

porting structure. The first effect can amount to something like

f .001 in. per 10-ft section, and the second effect, perhaps f 1/8 in.,

- 6 -

per 320 ft sector. step-wise discontinuity in either the displacement or angle of the

accelerator axis.

be characterized by a curvature vector

Evidently neither effect can contribute any large

Hence, the misalignment of the axis at any point can

where

to a straight line. (Ea, 7,) are the coordinates of the accelerator axis referred

If we express the equation of motion in terms of a transverse

coordinate system whose origin is the accelerator axis, the curvatuye

introduces a centrifugal force term:

11 N

fal = -75,

1 0 N

fa2 = -7Va

(2.3-la)

( 2 3 -1b)

p s 2 Terms of order (5,) , (v:)~ have been dropped.

2.4.1 EXTERNAL MAGNETIC FIELDS: STEERING

The steering field will be applied to correct for misalignments, stray magnetic fields, and possibly for accelerator asymmetry (see

Sec. 2.5). The force exerted by the net magnetic field is given by

XeB

me Y 2 = b 2

fml - 2

heBx

2 me -b 1 fm2---- E

(2 a &-la)

(2.4-lb)

-f where terms of order v x B’, are valid either if the field is transverse - or if transverse velocities

have been dropped, i,eo, the expressions

- 7 -

T ’ 1

are small.

these will be treated as focusing fields and aberrations. For the present we ignore transverse variations in b; +-

In practice the steering field might consist of sets of Helmholz

c o i l s spaced along the machine at intervals of perhaps 300 ft.

be a sufficiently good approximation for the present to treat this field as an axially continuous, slowly varying field, the criterion

being

It will

I

Az << accelerator radius (a) ' 'max 9

is the maximum bend in a given steering period, and aZ E 'ma, where

is the length of a steering period; this condition is fairly well satis- field if 6'' - 3 x lo-', Az - 10 4 cm, and a 1 cm,

maX In this continuous approximation the average field applied to

counteract misalignment is given by equating the magnetic force

[Eq. (2.4-1)l to the centrifugal force [Eq. (2.3-l)]:

where (7,) is the mean energy. Hence the net force on a correctly

steered beam is

(2 e 4-3a)

where Ay/r is the relative deviation from mean beam energy. Correct

steering in the y-direction gives

(2.4-3b)

We can now estimate the beam spread due to misalignment. From

Eqs. (2.2-1) and (2.4-3), neglecting other forces, we have

- 8 -

Considering the ''worst" case of

possible value, t he equation in t eg ra t e s t o 6; everywhere equal t o i t s maximum

(2 ,4-4)

where 7 ' = d7/d(

co r rec t ly s teered i n i t i a l l y [ s ( O ) = k'(O> = 01. Using the est imate

above, E;(max) - 3 x rad/(1000 wavelengths), and taking

~ 7 / 7 = we obtain

i s assumed constant and the p a r t i c l e i s assumed t o be

a t the end of the machine, which i s bare ly to l e rab le . However, i f

we wished t o t ranspor t a low-energy and high-energy beam simultaneously,

s o t h a t

x (max) <<acce le ra to r radius , unless s t rong focusing i s t o be used,

i n which case the requirement mi.ght be relaxed somewhat,

AY/7 s 1, it i s evident t h a t we would require

a

2 - 4 . 2 EXTERNAL MAGNETIC FIELDS: DEGAUSSING

The magnetic f i e l d of the ea r th has a maximum component ( i n the

v e r t i c a l d i rec t ion)of - 0.43 gauss i n t h e Stanford a rea , I f uncom-

pensated, t h i s would, by in tegra t ion of Eq. (2*4-1) , give a de f l ec t ion

of

B ( 2 4-6) Y 8 X Z - z

EO

f o r an e l ec t ron in jec ted along the ax i s of a s t r a i g h t machine, where

the quan t i t i e s a r e as defined f o r Eq. ( 2 * 2 - 1 ) -

f i e l d of

or 4 meters i n 10,000 f t .

efPect by a f a c t o r of N L x lo3 6x 2 0,1 em.

A t y p i c a l acce lera t ing

- 300 esu/cm would thus give 8x Z 0.4 cm per 10 f t sec t ion ,

Shielding o r compensating t o reduce the

i s thus necessary i f we require

It i s poss ib le t h a t degaussing c o i l s w i l l be used r a t h e r than

magnetic shielding, s ince s t ee r ing t o compensate misalignment and s t r a y

f i e l d s i s necessary i n any event. With independent sec tor control of

1 I

degaussing fields, it is entirely possible that steering and degaussing

could be applied by the same set of coils. In this connection we note

that the field necessary to compensate a misalignment is given by

V Bo2 = - 300 AEL (2.4-7)

where V = beam energy is in ev.

ment is - 3 x radian, and aZ = 300 ft, we find that B 0.5

gauss at 45 Bev, so that the steering field is of the same order of magnitude as the degaussing field.

Thus if the maximum angular misalign-

Design tolerance on the degaussing field coils will be set by

allowable field non-uniformity across the accelerator hole.

fects will be discussed under

Such ef- "Field Aberrations, '* Section 2.4.4.

2.4.3 EXTERNAL MAGNETIC FIELDS: FOCUSING

If focusing f ie lds are used, we assume that they will be of the

ordinary strong-focusing quadrupole type, for which the forces are

given by

Ae aBy f = - - x = Q ( 5 k a mc2 ax

(2.4-8a)

(2.4-8%)

where

The condition

ax ay

results from our assumption that beam-current (space-charge) Porces

are negligible.

If thin quadrupole lenses are used, the effect of a lens may be

stated as

- 10 -

where f i s t h e foca l length of t he t h i n l e n s i n u n i t s of 1. I n

matrix notat ion,

( 2 4-lob)

The matrix f o r a sec t ion of acce le ra to r having no t ransverse forces and

constant energy gradient 7 ' i s given i n e i t h e r t he ( 5 , E ) or ( ( , 7 )

plane by

where

71

7 ' 70

1 a =-log- . 21

(2.4-11)

- 11 -

L

"Continuous" approximation. If one uses closely spaced quadru-

poles of alternating sign and slowly varying strength and spacing, the

net effect in averaging over a focusing period is given approximately

by an equivalent force

(2 4-12a)

( 2 4-l2b)

where 45) tance between quadrupoles of opposite sign), and

value of the magnitude of Q ( 5 ) over a half period:

is the half period of quadrupole spacing (i-e., the dis-

q is the average

Equation (2.4-12) should be useful approximation if 4 5 ) and q(5)

are slowly varying, and if 45) << f, where the focal length f of a

quadrupole element is given by

2.4.4 EXTEXNAL MAGNETIC FIE&DS: ABERRATIONS

In Section 2.4.1 it was pointed out that the earth's field must be reduced by a factor of

- gauss of average residual field. This means that the steering

and degaussing fields should have no systematic variation greater than

- gauss across the accelerator hole. A dimensional tolerance in the order of 1% on the steering and degaussing coils seems to be

- 4 X lo3 on the average, which results in

- 12 -

ind ica ted . T h i s question and t h e question of aber ra t ions i n poss ib le

strong-focusing lenses w i l l be considered i n a fu tu re memo on. beam

dynamics .

2.5.1 ACCELERATOR ASYMMETRIES

The dimensional to le rances i n the acce le ra to r s t ruc tu re a r e such

t h a t one would not expect any s ign i f i can t devia t ion from a x i a l symmetry,

e spec ia l ly i n averaging over many disc-loading per iods. However, it i s

en t iye ly poss ib le t h a t asymmetry or at least nonuniPormity may e x i s t

a t t he rf couplers.

2 - 5 . 2 COUPLER ASYMMETRIES

Assume the coupler introduces a f i e l d per turba t ion f o r which the

axial e l e c t r i c component i s given by

e =_ - [e3(x, y, z ) I cos (ut + $) Z

(2.5-1)

where i s the phase of t he per turba t ion r e l a t i v e t o t h e accelera , t ing

wave,

we expect t h a t

If t h e ( z , x ) plane i s the plane of symmetry of the coupler,

The corresponding vect.or p o t e n t i a l i s

where

the coupler region i s , by the well-iinown de f l ec t ion theorem:

k = w/c =: 2 ~ r / X . The t ransverse impulse on an e l ec t ron t r ave r s ing *

* W. K. H . Panofsky and W . A . Wenzel, Rev, Se i . I n s t r . - 27, 967,

- 13 -

dz e ?! (2.5-3) z z X

C V C Z

coupler

N - _ - - e s i n ( e + pI) 1 2 d z kc

(2.5-3)

where 8 i s t h e mean phase of t h e e l ec t ron r e l a t i v e t o t h e acce le ra t ing

f i e l d i n t h e region of t h e asymmetry. Trans i t time i s neglected, and

we assume v = const. , v >> v and t h a t t h e in t eg ra t ion i s over a

region where t h e per turb ing f i e l d vanishes a t and outside t h e boundaries. Z Z Y'

I n dimensionless u n i t s ,

s i n ( 0 + pI) d 7 s 9 = -

2n

2 where 5 = x/X, f l = z/X, ( ) =: d/dc, a3 = ee X/(mc ), and

7 = 11 - (vz/c2)] -2 3

e lec t ron energy i n mc2 u n i t s . L J

Equally or ien ted couplers.

i s given by

d(76 ' 1 N A(7S ' 1 - -

a -e

The average impulse per unit,

E s i n ( e + $)

where & = L/X = number of wavelengths between couplers, and

(2.5-4)

Jc oupl e r

- 14 -

2 Assuming dy/dC = Q: cos 8 constant , where Q: = eEoh/mc ) (Eo = acce l -

e r a t i n g f i e l d amplitude averaged over L ) , then Eq. (2.5-5) i n t eg ra t e s t o

(Ykl)o y E s i n ( e + p) k = E , + l og - +

a cos 8 cos e Y O

Thus t h e net e f f e c t of t he coupler asymmetry i s

- (I; - 5,) E s i n ( e + @) a cos e sg (2.5-7d)

I f we assume t h a t t he per turbing f i e l d i s confined t o a region

Az = one disc-spacing, then from the def in ing equations

~z x dez a - E = - - - N

2 r r ~ E~ ax

SO t h a t

( 2.5-8)

Alternat ing coupler o r i en ta t ion . I f the couplers are a l t e r n a t e l y

or ien ted i n opposite d i r ec t ions , we average t h e t ransverse impulse over

two consecutive sec t ions . From Eq. (2.5-4)

- 15 -

J

2 {E! s i n ( 0 + $) (2.5-10)

3 where E ! 2 &/as. Terms of order E ~ , s t , &, 5 , e t c . , have been

neglected [E i s as defined by Eq. (2.5-6) 1. It i s evident t h a t i n t h i s

case the e f f e c t i s s m a l l .

"Vertical" de f l ec t ion , It has been assumed t h a t ae /dy = 0 ( 3 i - 0

e3 because the coupler i s symmetrical about the ( z , x ) plane. Since

may have a quadrat ic term i n y the re may be de f l ec t ion i n the y

d i r ec t ion which by analogy with Eqs. (205-3 ) , (2.5-5) would be given by

where

Jc ouple r

and 7 = y/A.

and can be cancelled by a weak t ransverse gradient of magnetic f i e l d .

Coupler asymmetry i n the presence of a s t ee r ing f i e l d . Consider

Again the lowest order term i s l i n e a r i n the def1ect.ion

the case of equal ly or iented couplers. Assume t h a t a s t ee r ing f i e l d

- 16 -

B = B ( z ) i s present (aB/ay = aB/ax = 0 f o r the moment). Then

Eq. (2.5-5) becomes

(2.5-12)

where

Note t h a t B i s e s s e n t i a l l y a component of t he degaussing f i e l d

and t h a t the coupler asymmetry i s equivalent t o an addi t iona l v e r t i c a l

component of magnetic f i e l d ( i f $ # 0 ) .

Y

Correct s t ee r ing w i l l r e s u l t when the t ransverse force i s essen-

t i a l l y cancelled;

N b = - s i n $ €0

AZ A de = - - - 3 ,

2 r t ~ ax N

0

where a r e f e r s t o the average acce lera t ion . 0

Equation (2.5-13) may be r e s t a t ed :

B ' Z - - A' (A -- "'1> Eo s i n $ 2rtL Eo ax Y

(2 e 5-13)

(2.5-14)

Subs t i tu t ing (2.7-13) i n t o (2 .5- l2) , and c a l l i n g ( a - ao)/ao = Aa/a,

or, f o r s m a l l phase angle 0 ,

(2.5-16)

where

"i keeping only terms dependent on E, ( e . g . , neglect ing o ther t ransverse

forces and i n i t i a l angular spread) we have f o r t he small-8 case

= "rea l" o r in-phase component = E cos $; and

- "imaginary" o r out-of-phase component = E s i n In t eg ra t ing and

I f w e assume t h a t 8 and a r e e s s e n t i a l l y random, uncorrelated

funct ions we may est imate t h a t

where now <Sgp and (66) a r e in t e rp re t ed as beam divergence and

s i z e respect ively, (e) i s the rms phase spread, and <(&/a,) i s

the rms deviat ion of the acce lera t ion parameter

= - = beam-spectrum width . (:) $.

- 18 -

2.6 EFFECT OF AXIAL VARIATIONS I N PHASE AND ACCEL;ERATING FIELD

2.6 .1 EFFECTS LINEAR I N THE ENERGY GRADIENT

Let us use z and r for the a x i a l var iab les and r a d i a l var iables ,

and adopt t he notion:

5 = 2 / 1 9

T = ct/h,

P = r/A, fi = phase ve loc i ty of f i e l d s i n guide,

2 a ( z ) = XeE/(mc ),

E ( z ) = c r e s t value of e l e c t r i c f i e l d ,

2

8 = phase angle of p a r t i c l e r e l a t i v e t o c r e s t

( e > 0 f o r p a r t i c l e ahead of c r e s t ) ;

me = r e s t energy of p a r t i c l e ,

l e t a dot over a l e t t e r denote

space charge and cor rec t t o l i n e a r terms i n

pendent of f i e l d shape (assumed only t o have a x i a l symmetry), i s

[see Eq. (6.59b), ML-1401:

a/&. The radial equation, neglect ing

p, but otherwise inde-

d flolp 1 p aa -(yb) = - (1 - p t ) s i n e - ’;--- cos 8 dC N rJ 2 a(

(2.6-1)

W e s h a l l not assume t h a t

tend t o uni ty; instead, we s h a l l assume (as corresponds t o the real

s i t u a t i o n ) t h a t

s t ruc ture . However, fi and the phase angle a r e r e l a t e d by

fi, t he phase ve loc i ty i n the guide, w i l l

B w i l l be governed by the imperfections i n t,he guide

(2.6-2)

S i m p l i a i n g and ignoring the r a d i a l va r i a t ion of a (which would only

a f f e c t nonlinear terms i n

i n the l i m i t of -+ 1,

p , which a r e already neglected) , we obtain

- 19 -

d I d 1 % YTpa

dS 2 2 dl: Y - ( y p ' ) = - - - (ap COS e ) + - - (a COS e ) .+ s i n 8 (2.6-3)

where the prime ( ' ) denotes

g iv ing a r a d i a l focusing or defocusing term depending l i n e a r l y on t.he d/d(. The last, term i s the usual term

.- 2 r a d i a l var iab le but varying as

e l e c t r i c and magnetic f i e l d s . We w i l l not consider t h i s term f u r t h e r

s ince i t s e f f e c t has been discussed. The remaining terms become:

y due t o the near cance l la t ion of

*

(2.6-4)

f i n a l

a cos e - dp d( (2.6-5) dS (YP 'Ifinal = (YP9init ial *

i n i t i a l

s ince a: cos 0 i s the dimensionless energy gain per wavelength. Hence

the e f f e c t of f i e l d nonuniformities and phase va r i a t ions i s equivalent

t o a f i n a l beam divergence corresponding t o a foca l length given by -

1/F = 1/2L (2.6-7)

where L i s the e n t i r e length of t he machine. This e f f e c t i s thus

negl ig ib le . W e r e t a i n the conclusion, previously demonstrated under

more l i m i t i n g conditions, t h a t the r a d i a l momentum proport ional t o y p '

i s a constant of t h e motion.

2.6.2 EFFECTS QUADRArIC I N THE ENERGY GRADIENT

Several spec ia l e f f e c t s a r i s i n g from axial va r i a t ions may be

considered:

* For references, see See. 1, p. 1, t h i s repor t .

- 20 -

(1) Effec t of per iodic feeds. Equation (2.6-4) may be wr i t t en

a l t e r n a t e l y as

where

Because of the per iodic nature of t he power leve

can wr i te

(2.6-9

between couplers, we

a i s the value of a cos 8 at the start of a sec t ion , and YO where

a = X I

per iodic nature of t he f i e l d :

i s the vol tage a t tenuat ion coe f f i c i en t . The sketch shows the

I I I I r;

Coupler Po sit ions

- 21 -

In t eg ra t ion of Eq. (2.6-8) across one of t he d i s c o n t i n u i t i e s gives

1 ' I (7P ' )2 = ( 7 P ' ) l - 2 ( 7 2 - 71) P 1

1 ' (2.6-9)

and

where A = a& = IA&. I n matrix nota t ion (Sec. 2.4.3),

o r

The d i scon t inu i ty a c t s as a th inconverg ing lens of foca l l eng th

r,/F = 7; (1 - e-A) .

It i s evident that the r e s t of t h e sec t ion (from poin t 2 t o poin t 3 i n the f igu re ) w i l l a c t as a t h i c k l e n s with a negative foca l length .

As a f i rs t approximation it i s r ead i ly shown t h a t t he sec t ion

represented by t h e matrix product

2+3 i s

- 22 -

where d12 & 1 &/2, and y/F Z ('1/2) 7; (1 - e-*); i . e , , t h e e f f e c t i s e s s e n t i a l l y t h a t of a t h i n diverging l e n s a t t.he center of

the sec t ion , The approximation assumes &yl << y ar_d A < 1.

The ne t e f f e c t of t h i s a l te rna t ing-gradien t s i t u a t i o n ma,y be found

wi th the help of t he above matrices.

per iod gives Averaging the o r b i t s over a reed

which def ines a force f A given by

- A where 7 ' = y; [ ( l - e ) / A ] = average value of a cos 0 .

a similar focusing force but t h e e f f e c t w i l l be much smaller than t h e

per iodic coupler e f f e c t .

The per iodic f i e l d va r i a t ion assoc ia ted with disc- loading w i l l g ive

(2) Effec t OP an unpowered region. If power i s turned off i n an

< 5 5 el + A, Eq. (2-6-8) can be in tegra ted (1 - acce le ra to r sec t ion ,

across t h e region i n a s t ra ight-forward manner.

transforms [ : q ) from el t o el + i s given by

The matrix which

( 2.6 -12 )

- 23 -

T

which is equivalent to a thick lens of focal length

Here y t is of course the accelerating field in the regions ( < cl and 5 > cl i- A. If 5 , is small or A is large, the effect may be

large enough to affect the beam focusing considerably. For instance,

if c1 + Y o / Y = 100 ft and A = 100 ft, then F = 400 ft,

(3) Random axial field variations. Local variations in the

acceleration parameter y ' , arising from rf phase and amplitude mcdu-

lation and construction errors in the accelerator, will contribute a

"noise" term of the form indicated by Eq. (2.6-8), where y ' " now repre-

sents the random modulation seen by the electron.

this effect is to replace

order of

One way to estimate

by its rms value, which might be of the y"

0.ly' 3 10-3 y t (2.6-13) (13 -T-=

for a 1O-ft section (see Sec. 3.4).

2.7 SCATTERING BY RESIDUAL GAS

2.7.1 MULTIPLE SCATTERING *

Neal gives an expression for the rms scattering by residual gas

in the accelerator, which may be written

* R. B. Neal, "A High Energy Linear Electron Accelerator," ML Report

No. 185 (February 1953). multiple scattering which substantially overestimates the effect.

Nealgs result is based on a model of the

- 24 -

where

V = 21 Mev

Vf = final energy at the end of the machine

z = length of the machine

S

f X = radiation length of residual gas at atmospheric pressure

4 0 Z 3 x 10 em (assuming air)

p/po = residual gas pressure relative to atmospheric pressure.

5 -8 -5 FOP Yf = 15 Bev, z = 3 x 10 em, and p/po = 10 (ioeQ, - 10 mm Hg, which is about the maximum allowable pressure for breakdown

reasons), the expression gives

f

Hence, this effect can always be ignored.

2.7.2 SINGLE COULOMB SCATTERING

Neglecting nuclear form factors and recoil terms (a valid assump-

tion for the small momentum transfers involved), we can write the

Coulomb-scattering cross-section per unit solid angle as

do 4r0 2 2 Z

where Z is the atomic number of the residual gas, and r = 2.81 x em is the classical electron radius. The total cross section down to a minimum angle 'Din is then 0

d 'min

The minimum angle can be either:

screening radius according to the Fermi-Thomas model, (a) the angle corresponding tlr, the

- 25 -

I

n 1 1 1 1 1

y mc 0.6 Z1I3 a. y 137 = 157 y , (2.7-4) - = - - - - B

min 0

2 for Z = 7 and a = */(me ); or (b) an angle comparable to the angle at which the normal orbits traverse the accelerator. We can take this number to be approximately

0

1 e - - ,

10 Y min (2.7-4' )

0 corresponding to an injection angle of about 5 at 100 kev injection

energy (or to a final angular divergence of - f 0.5 x

10 Bev).

radian at

The cross-section based on assumption (a) is

2 2 -18 2 (5 = 431 r2 0 Z - 157 = 1.2 x 10 cm , 0 (2.7-5)

(assuming

the usual orbit angles given by (2.7-4') is Z = 7), while the number based on angles corresponding to

= 5 x cm 2 . (2.7-5 )

2 17 The number of atoms/cm at a pressure of mm Hg is 2.2 x 10

(assuming a diatomic gas). Hence the fraction f undergoing any

scattering is

while only a fraction f', given by

f' = 0.01 , (2.7-6 ' )

undergoes scattering through an angle comparable to the normal radial angles.

- 26 -

where

3 . TRANSVERSE FORCE EQUATIONS AND NUMERICALI RESULTS

3.1 SUMMARY OF THE TRANSVERSE FORCE EQUATIONS

We can now write down the "complete" equations of motion:

Al ( y y ) ' = fal + fd + fa + fcl + f

+ f (noise) 1

+ f (noise) 2

y ' = a cos e

fal = - YSa

fa2 = - YTa

Mi sal i gnme nt I External Mag. Field

XeB

me - Y =

fml- 2

XeBx 2 mc

Mag. Quadrupoles I l e aB f = - -

2 a me ax = Ea(()

Xe dBx y = - VQ(5) f = - - - .

2 Q2 me ay

(2.2-la)

(2.2-lb)

(2 2-14

(2.3-la)

(2.3-ib)

( 2.4-la)

(2.4-lb)

( 2.4-8a)

(2.4-8b)

- 27 -

( 2 - 5 - 5 )

Coupler Per turbat ion (equal ly or iented couplers)

(2*5-11)

s i n (e + pI)

2Tr.e

- f c l - -

q s i n ( e + $)

2Tr.e fc2 = -

16 Y Per iodic Feeds

fa - - -

(2.6-11)

The noise forces f (no i se ) , f2(noise) represent t he random p a r t of

t he energy gradient va r i a t ion , combined with other randomly varying

forces such as l o c a l l y uncompensated s t r a y f i e l d s .

1

We assume that the beam i s co r rec t ly s teered so t h a t t he net

e f f e c t of misalignment and s t ee r ing i s given by

fal + b2 - - - Ay ( re; ) Y

(2.4-3a)

(2.4-3b)

+ Properly these terms should be included i n

t o r curvature (6 f (noise) s ince the accelera-

11 11

is e s s e n t i a l l y a random var iab le . a’ 7,)

3.2 NUMERICAL ESTIMATES OF TRANSVERSE FORCES

(1) Misalignment. The maximum curvature t h a t the acce lera tor can 4

have i s on the order of f 10-

sec t ion ( t r 3 0 ) , or

radian increment per acce lera tor

- 28 -

(see See. 2.3). Hence, if &,/7 '2 l$, we have at 45 Bev (Tor 'y - 10 5 )

However, if we wish to transmit high- and low-energy beams simult,aneous-

ly, Ay/y - 1, so that in this case steering is useless and we have ( 3.2-ib)

Treating the misalignment term as a random variable, we might estimate the rms value of the curvature as

10-5 per 10 ft section a /

or

(3.2-ic j

See Sec. 3.4 for an approximate treatment of the effect of the noise forces on final beam size.

(2) Coupler asymmetry. Measurements of coupler asymmetry by *

W. J. Gallagher indicate that, for existing couplers,

A de

Eo ax 3 - - - = 0.6

(assuming $ = 0), from which, by Eqs . (2 .5 -5 ) and (2*5-8),

LIZ x de N ---(->I y'tane

2n-e ax fcl

z - 1.1 x 10-3 yt tan e ( 3 e 2-2)

where L/Az 90 is appropriate to Project M design. Integration of

* Private communication.

- 29 -

(3.2-2) (neglect ing other forces , and taking

gives

6; = 0 and 7 >> y o )

or , taking 8 0.2 radian maximum phase spread and the t o t a l l ength

of t he machine as A t f 3 x 10 cm, 5

Thus the t ransverse phase volume i s

evident t h a t asymmetries of t h i s magnitude cannot be to le ra ted ; t h e

coupler asymmetry would have t o be reduced by a f ac to r

above example.

Use of t he a l t e r n a t i n g coupler o r i en ta t ion i s one method of re -

I S E ' l x lSxl N 1.3 x It i s

N 30 from the *

moving the constant term of t he coupler force.

i s replaced by Eq. (2.5-lO), and

I n t h i s case Eq. (2.5-5)

s i n ( 8 + $) N _ - -

fcl 2rra

N

(again assuming $ = 0) . Gallagher s measurements ind ica te

( 3.2-5)

* Recent couplers have been developed by G. A. Leow and 0.

Altenmueller (Linear Electron Accelerator Studies , S t a tus Report, M. L., No. 741, Microwave Laboratory, Stanford University) having (A/E~) (he3/&) < 0.05.

- 30 -

from which

E 1.1 x kyg tan e fcl - -

Equation (3.2-6) probably represents a fairly conservative upper limit for the coupler perturbation force if suitably symmetrized couplel-s

are used. See Sec. 3.3, Case (3), for the effect of such a force on the final angular divergence of the beam.

(3) Axial variations in accelerating field. The attenuation per

section, A = IL, is expected to be - 0.6 for Project M. Hence,

from Eq. (2.6-11) we get

(4) "Noiset1 forces. In addition to the misalignment term, we

have the random forces associated with locally uncompensated stray

magnetic fields and random axial rf modulations. The stray magnetic

fields might have an rms magnitude of, say,

field, averaged over a steering period of - 320 feet. This would be

equivalent to

o f the earth's

4 (Xe/mc2) (B) = 10- rms per 320 ft sector

or

- 31 -

.. 1

The rf modulation t e r m , from the est imate of Sec, 2.6 (3), might

have an r m s magnitude of

( 5 ) Focusing f i e l d s . Some externa l focusing w i l l almost c e r t a i n l y

be necessary t o compensate for t h e forces which have been discussed so

f a r , as w e l l as f o r i n i t i a l beam divergence. Idea l ly we would l i k e t o

apply a focusing force of the type defined i n Eq. (2.4-12), which would

j u s t cancel t he t ransverse forces . However, t h i s i s not possible be-

cause the t ransverse rorces do not a l l have the same funct ional form as

t h a t given by Eq. (2.4-12), i. e . , t he re a r e force terms depending on

phase angle and energy deviat ion as wel l as random, time-dependent

forces . Hence, i f we impose the condi t ion of - 106 transmission,

t he focusing funct ion should meet the requirements t h a t

t ransverse forces a r e e s s e n t i a l l y cancelled (which w i l l over-focus a

l a rge f r a c t i o n of t he e l ec t rons ) ;

s t i l l contained within the l i m i t p

focusing funct ions w i l l be discussed i n the next s ec t ion ) ; and

f i n a l bean "qual i ty" as defined i n Sec. 2 .1 should be a t an acceptable

l e v e l

( a ) the maximum

(b ) the o r b i t s of a l l e lec t rons a r e

2 a /h 2 0 .1 ( t h e choice of l-llaX

( c ) t h e

3.3 SPECIAL SOLUTIONS

We may now wr i te down the force equations i n a form more s u i t a b l e

f o r ana lys i s :

N N N

where fO1, fO2, gl, g2, fl, f2, 6fl, 6f2 a r e funct ions of 5 ; -r"

represents all "noise" forces and 6f represents terms which a r e

nonlinear i n ( s , ~ ) . From the discussion of Sec. 3.1, we have

( t = 0, 7 = 0 ) = coupler asymmetry force ( 3 0 3 - 2 4 N - f O 1 - f c l

- 32 -

'coupler

N with analogous expressions f o r P 02, g2' To the extent that f is a

truly random variable we can characterize it by an rms value:

- 1

T h a t is, the sum of the stray magnetic field effect,s, random accelerating

Pield modulation effects, and misalignment effects,

large fluctuations in these terms, e.g., shut-down accelerator sectiops

or large misalignments, needs to be treated separately.

The question o r

The numerical estimates of section 3.2 give

fO1 N = - 10- 3 7 1 tan e (asymmetric coupler) (3.3-3)

Let us consider solutions to the problem in several special cases:

Case (\ 1) . Assumptions :

(a) (b) y ' = a cos 0 is assumed constant, i.e., y = y 9 ( *

All transverse forces are neglected.

This problem has the familiar solution

y ' ~ = yO5A = c0nstan.t

(3.3-41

I€ we maximize the product and 5 , fixed, we

obtain

- 33 -

3 4 em (100 Yt), yp = 3 x 10

= 300 (150 Mev), t,he requirement on the beam Trom

Taking a = AS f (15 Bev), and the injector is

1 cm, z 0 = hCo = 3 x 10

Thus to get 100% transmission even in an ideal machine the tolerance on injector beam quality is quite small. It appears that some focusing is unavoidable at least in the initial part of the main machine.

*

Case (2). Assumptions:

(a) Transverse forces are neglected.

(b) y ' is assumed constant

(e) Focusing forces are introduced in the continuous approxi-

mation of Eq. (2.4-12). **

The orbit equations now have the form of (2.4-12) :

The solutions depend on the form of A( e ) , q( 5 ) . Case (2a)

(3.3-6) become Assume A( 5 ) q( 5 ) = constant. The solutions of Eq.

* For the Mark I11 Accelerator, experience indicates that

-4 xo6A 5 10 cm is a more realistic value.

** This approximation is introduced to simplify the discussion. The

exact treatment of alternating-gradient focusing is given in "External Magnetic Focusing Devices for the Mark I11 Accelerator," J. A. McIntyre, R. L, K y h l , and W. K. H. Panofslqr, ML Report No. 202 (July 1953).

- 34 '-

where cos CD = (o/gmax, and K = - A q f 2 y I . Maximizing the i n i t i a l phase

space (Eo!;) i n t h i s case gives 0 = f r(/4 and

As a numerical example, suppo

= 10 cm, which i s a r a the r

and

-3

= 0.1, we have Emax

4 7 ~

*YO 4-Y

e we wish t o allow

modest requirement. Then f o r y = 300 (xo(;),, = h ( ( ~ ~ ~ ) ~ ~ ,

0

max - 5,G I - 12.0 t 2

A q =

’max

or

n q = 360 gauss

( 3 ” 3 - 9 4

(3.3-9b)

Quadrupoles 1 JY, long would need only a gradient 01 aDout ~ r i gauss/crn.

Obviously very moderate s ized quadrupoles a r e s u f f i c i e n t .

The configuration nq = constant has one worthwhile advantage:

i f i d e n t i c a l t h i n lenses a r e used, then nq = A( go, where AC i s

the length of the t h i n quadrupoles, and qo i s the I’ield gradient

within t h e lens ; t h e ac tua l quadrupole spacing fl i s completely

a r b i t r a r y , subject t o the condition that

A be slowly varying.

A<< IPI = y/ (qAc) and t h a t

This design would a l s o be advantageous i P we wished t o i n j e c t a

low-energy beam at some a r b i t r a r y point along the acce lera tor .

- 35 -

2 2 2 Case (2b) . Assume nq = (As), ( y / y o ) . Then the solut ion of

Eq. (3.3-6) i s

where

I f k i s f a i r l y l a rge ,

where I) i s an a r b i t r a r y parameter. Maximizing the i n i t i a l phase space

i n t h i s case gives

(Ad 0

4 7 ~

2 Emax ( E E ' )

O O m a x ( 3 " 3-12]

i . e . , the i n i t i a l value of hq The same r e s u l t would be obtained i n any case i n which the most diver-

gent in jec ted e lec t rons go through the maximum amplitude of t ransverse

o s c i l l a t i o n i n a dis tance i n which the r e l a t i v e energy increment i s

i s e s s e n t i a l l y the same as i n Case ( 2 a ) .

s m a l l .

Case ( 3 ) . Assumptions:

(a) Transverse forces of t he form g 1 5 , g2q [Eq- (30 3-1) I are present .

"Noise" forces and aberrat ions a r e ignored, and asym-

metry forces a r e assumed negl ig ib le . (b)

( c ) y ' = a cos 8 i s assumed constant

- 36 -

(d) Focusing forces are introduced i n the continuous approxi-

mation Eq. (2.4-12).

The o r b i t equation i n 5 now becomes ( ( E 1 ) 2 (g l /y l ) where

from Eq. (3.3-2) and (2.4-12)

or, using the numerical es t imate [Eq. (3-3-3) 1,

Neglecting the second term (per iodic feed e f f e c t ) , and assuming the

focusing funct ion t o be given by example (2b) above, t he o r b i t s a r e

again given by Eqs. (3.3-11), where now

The increase of t he f i n a l angular divergence i s given, with t h e help

of Eq. (3.3-11), as

= all, and h q 1 2 5 Taking y 1 = 1.5, yo = 300, yf = 10 L a x

[from Eq. (3.3-9) 3 , we f ind

Hence f o r a reasonable range of phase angle

t ransverse forces i s negl ig ib le i n the presence of a focusing 3 f i e l d

l a rge enough t o contain in j ec t ed e lec t rons having

( 0 = +_ 0.1) t he e f f e c t of

= 10- . - 37 -

3.4 APPROXIMATE TWATMENT OF THE NOISE FORCES

As a f i r s t approximation, assume t h a t t h e t ransverse coordinates

can be represented by

5 = s , + i

where

ignored, and << 5 Then Eq. (3.3-la) can be wr i t t en (0 i s a so lu t ion of t he equation i n which noise forces a r e

1'

Neglecting the energy v a r i a t i o n over a s ing le acce le ra to r s ec t ion and

in t eg ra t ing over the i - t h sec t ion ,

o r

2 Using the form of Eq. (3.3-2c) f o r ?, and summing (sei) over a l l

sec t ions , one obtains

where

If = yf/7 & = t o t a l number of sec t ions , and

of o r b i t amplitude.

Io = 7,/r'& = number of 10 f t s ec t ions - i n i n j e c t o r , 2 6 = mean square value

- 38 -

Consider the var ious terms separately:

(1) St ray magnetic f i e l d s :

= 0.28 (B > cm/gauss Y

Io = 10, kf, = 300 em, and E = 50 kv/cm = 167 esu, For

have an r m s value of - 0.3 gauss over a 10 f t s ec t ion beyond the 100 C t

po in t .

down t o "< 5 x

0 where

(6x) < 0.1 em, the random component of s t r a y magnetic f ' ield could

This condi t ion ce r t a in ly w i l l be m e t i n keeping t h e n e t f i e l d

gauss, averaged over t h e whole machine.

( 2 ) Random rf modulations:

In t e rp re t ing (r's) / y as the r e l a t i v e f luc tua t ion i n acce le ra t ion

per sec t ion (perhaps 5% r m s ) , it i s evident t h a t no ser ious defocusing

can r e s u l t from t h i s eWect .

(3) Misalignment e f f e c t .

-2 where it has been assumed t h a t

acce le ra to r sec t ions = 1000; ( $ 5 ; ) i s in t e rp re t ed as t he angular

misalignment per 10 f t sec t ion . The allowable rms angular de f l ec t ion

pe r 10 f t sec t ion could be as l a rge as - 2 x radians; however,

t h i s would requi re an excessively sho r t s t ee r ing per iod .

Ay/y = 10 and I. f = t o t a l number or"

Evidently,

- 39 -

the p r a c t i c a l requirement of s t ee r ing per iods of the order of a 320 f t

s ec to r imposes a much s t ronger condition on the alignment, namely

- 0.2 cm max def l ec t ion i n 320 f t .

- 40 -

4. SUMMARY

The main conclusions of the present study may be stated as follows:

(1) There appears to be no known reason why a beam of reasonably

good quality cannot be transmitted rrom the 100 ft point to the end of the machine with essentially no l o s s in current,

(2) The quantities (xpx)),2x) (fly) are conserved in the IlBX

presence of all known transverse forces except those arising rrom

coupler asymmetry, misalignment, and stray magnetic field. The toler-

ances imposed are estimated as

(a) Coupler asymmetry: ( h/E2) (bE2/bx) 2 .02 (a value of 0.05 has been obtained in an existing coupler).

(b) Magnetic field: uncompensated field - 0.04 gauss rms value per 320 ft sector: - 5 x gauss average value over entire

machine.

(e) Misalignment: - 0.2 cm maximum per steering period;

(3) A requirement for the beam quality of ( r dr/dz) 2 lom5 em - 10 cm maximum €or the whole machine €or a 1% spectrum.

m a X at the end of the machine implies (r dr/dz)mx cm at the 100 ft

point by virtue of the above approximate conservation law. (4) If we allow (r dr/dz)max to be as large as - cm at the

100 ft point, then quadrupole focusing is necessary but the quadrupoles

could be quite weak:

€or a typical focusing period near the 100 ft point.

1 (aB /ax) dz of the order of a Yew hundred gauss Y

( 5 ) Further analysis is needed in the following problems:

(a) Design of an actual focusing system (with discrete

quadrupoles rather than the continuous approximation used here).

(b) Effects of large, nonrandom bends and other perturbations.

(c) The conditions under which two beams having different

energies can be transmitted simultaneously.

(d) Extension of the whole problem of beam dynamics through

the injection region.

(e) The specific possibility of using fairly strong quadru- Such a system offers the possibility of containing

* pole focusing.

* The authors are indebted to K. L. Brown f o r summarizing and

emphasizing the advantages of this possibility.

- 41 -

T

beams of d i f f e r e n t energies i n the presence of f i n i t e misalignments and

s t r a y f i e l d s (without r e so r t ing t o pulsed s t ee r ing ) , and i n f a c t might

e l iminate the necess i ty f o r s t ee r ing and g r e a t l y increase the alignment

to le rances . c

- 42 -