,,

Io Introduc'l;ion., -----

CE.RN~·PS/K3 30 lfov, . ·_. ·: · J.9~i6

INJECTION PHILOSOPHY FOR SYNCHROTRONS .R)

The injec:ton system is a very important part of any acoelers.tor~ and the beam

intensity one can expect to get out of the accelerator depends very much on having

the ic"ight deaigi~ for the injectoro By injector is in this respect meant everything up

to the synchrot~: ono In most large synchrotron projects om plans to use a linear

accelerator ae :injectoro Designs for such linear accelerators exiet, both for electrons

and for protons., However, such ma.chines are constructed for nucleE.r physics experiments9

for medical purposes or even for industrial applications, and the requirements put on

these machines for such purposes may be quite different from the requirements one would

like to impose on an injector.

The purpose of this lecture is to show the principles one sho'+ld follow in

analysing the injection problems to find the special features one would like an injection

ayetem to haveo I do not know the new German 6 GeV project well enough to be able to say

vhat would be the result of such an analysis ·carried out on that project, but I hope

that the method~ to be described will be useful, and we shall also during the lecture

be able to draw a few conclusions relevant to that projecto

IIo LioUTille 0s Theoremo

The equations of 1110tion o! the particles in an accelerator can~ lfi th the normal

approximations rmd assumptions, be derived from a Hamiltoniano This makes it possible to

apply some of th~ general theorems of Hamiltonian dynamics in anal.ysing the beam dynamics l

of particle accularatorso

Probably the most useful of these theorems, &nd the only one I shall use in this

lecture~ is the ao-oalled Liouville~s theoremo This states that if the motion of the

particles can ba derived from a Hamiltonian~ tm density of pointe in phase space -====~"'======;==-==-=======----="'!"""1"====================~=====================

~) Lecture given at the Seminar on PQrticle accelerators at the University of Bonn 3rd-6th Octobar, 19560

K !l~I~ ~ DE f-'A1:\f El"i f.: tff T 0~6190D21.DEC. 5 6

I

- 2 - C"ERN-PS/KJ 30

representing a group of particles does not change with time. Others prefer to state

the theorem in the following way: particles move in phase space like an incompressible

fluido

The motion of a particle is in general a three.dimensional problem in ordinary

space, and the phase space would therefore be a six-dimensional space, and it would

be very cumbersome if we had to consider the motion in this space in order to get the

information wanted. There are cases when that would be necessary, but in general one

can separate the motion in the various directions into independent one-dimensional

motions. For example, the longitudinal motion is usually independent of the transverse

motion, or the so-oalled betatron oscillations. Further, when the analysis is not too

detailed, one _hormally al.so assumes that there is negligible coupling between

horizontal and vertical transverse oscillationo

Although it should be kept in mind that these are approximations, they are

justifiable approximations for moat purposes, and we shall therefore introduce them

here., This simplifies. the problems. Instead of a six-dimensional phase space, we now

have to operate with only three independenttwo-dimensional phase spaces (phase planes).

Further, as the vertical and the horizontal betatron oscillations a.re of the same

nature, we only have to consider tbs two motions, the phase oscillation and s~ the

horizontal. betatron oscillations, and Liouville 0 s theorem now states that in each of

the corresponding phase planes tbs particle density round a given particle does not

change with time.

We could now continue to discuss in such general terms the problems involved in

determining optimum injection conditions that no reference would be needed to which

type of motion we are considering~ However, in practice, the problems are so different

that I prefer to split the problem al.ready here and first consider the betatron

oscillations and afterwards the longitudinal motiono This has also the advantage that

we have something more concrete to ~ncentrate on in the analysis.

IIIo Free Transverse Oscillations (Betatron Oscillations).

We ahall first consider the free transverse oscillations, the so-called betatron

oscillations, confining ourselves to the horizontal oscillations only. As already

mentioned, this means neglecting coupling between horizontal and vertical oscillations

and in that case the analysis applies equally to the vertical betatron oscillationso '

C:,i iN-PS/KJ 30

1) Defini1ion of acceptance.

·rbe optical system under consideration, in our case the linear

accelerator or the synchrotron, is characterizttd by a few quanti tiee of special

intere~t in this discuesion8 th& most important one of these being the

sok~calJ .ad acceptance o

!.et us assume that the half-width of the available horizontal arerture

is ac Thia is in general not the width of the vacuum chamber but the width .

a:f"ter E<pace for phase oscillations, misalignments» etc~ e h~ been subtracted,,

Le·t us further, for the momentp assume that the particle motion is sinuso.idalo

This iE1 nearly correct for a fixed gradient machine, and ir:J a ro\lgh approxima­

tion f,,r an alternating-gradient machine o We shall later consider the Ao Go

maclU.llEi· in more detail o

With this assumption a particle will in the phase-plane move in an

ellipSf1, as indicated in figure lo

-a

,,"' I I

' '-

--

-b Figo l

a

l!Urther9 all the ellipses described by the various partieles in the beain

'Will have the same shape, ioeo the axis ratio ~./! ic-. t~ same far all

.e:W.pse:Jo Here 1 and ~x are the amplitudes of the transverse displacement and

transvorse momentum of a particle reapectivelyo

The largest of these ellipses one can draw without goirig outside the

availa"ole ,aperture.± a is drawn in full on Figolo It is then readily noticed

tbat it we inject a particle in such a way that its point in thia phase plane

is out'.lide this ·fully drawn ellipse, the particle will sooner or later hit the

wall a::id get losto If, on the other hand, the particle is injected inside thif~

fully ..irawn ellipsep it will au.rviveo Consequently, tbs focusing system can

only a1Jcept tMt part of the phase plam that is inside t:ki! ellipeeo One

therefore cells the area of this limiting ellipse the accxrnta.nst~ of the system ..

If one ooneiders the horizontal betatron oscillations one talks about the

= 4 -

hc-rizorita:t acceptan:e, and oorrespondingly the vertical acceptance for the

'1erttci1t oacillatio:'l.Bo Generally cne triel3 to make the two acceptances equal

and talks ju.st· a.bou!; the acceptan.cs of t:OO synchrotron"

As pointed o-it a.bove 9 this limiting cu:rve is always an ellipse 9 and a.a it

ie~ aMkl1a.:rd always t ·:> keep a 'It in the expressions, it has become conventional

to d.t:ftne the acceptance as the araa divided by 1t ? or if N stands for

acceptence

N ~ ab (1)

A more convenient w.g,y, in some cases, to write the acceptance is

(2)

The ratio~ fJ/x is given by the focusing properties only, and for a given

focusing strength the acceptance :1.s seen to be proPQrtion~ to the square of

tbe aperture. For sinusoidal osoillations1 as we have dealt with so far, the

acceptance can be written in ternw of the wavelength ha of tha oscillations

(3)

Here Pt is the longitudinal momentum of' the particle. For convenience one

often 1U1asures the momenta in terms of m c 9 in which case N has the 0

d:7..mension of length. o

Qw.te often one also finda that tha acceptance is defined in terms of

ang1.tlar spread x 0 instead of p , If we call the acceptance defined in this ' ' J:

lll\Y A ve get

and in:::tead of (3) 11e then get

2 A~ 2~ a /'A')

(4) ,

(5)

- 5 - C~HN-PS/KJ 30

The relation between A and N is given by

(6)

It should be noted that ( x 0 ~) are not canonically conjugate

variables in a system vith acceleration, where PL changes, and that one there­

fore hae ,no area preservation in this planeo It is therefore preferable to

work in the ( x PP:z:) - plane and use N if one .c.o').eiders accelaratore. But as

the relation between N and A is very simple, it does not make much difference

which definition one uses, as long as one keeps in mind the fundamental

difference of the variables in the two syatemso

Let us now consider what happens in an AoG. synchrotron. Siguraeirsson

(1952) was the first J:erson to show that one could use ooneiderations like this

also with AoG• focusing, only would one get ellipses of different sha~s

according to what part of the structure one considerso Let us consider that

part of the structure that gets the largest transversal excursiono That is the

mid-point of a focusing eectiOno Observing in this point, we again find that

the particle describes en ellipse in phase space, only does it so by discrete

point plottingo Again we therefore get a limiting ellipse, which, for this

symmetry point, has the same orientation as indicated in Figolo A particle

being outside this ellipse at injection will sooner or later be lost and a

particle being inside will surviveo We therefore use the same definition of

acceptance as before : the area of this limiting ellipse divided by 1to •

The symmetry point of a focusing section has here been chosen as the

reference point for oonv<tnienoe, as it is easy to find the limiting ellipse

in this caseo In any other point there will be a limiting ellipse of' the same

area but of a different ahapeo If om ia inte~~ted in finding any of these

other limiting ellipses, it is easier to do this by a suitable matrix

tnnaformation or the ellipse in the symmatr.r point chosen, mald.D« use of the

transfer matrix from the symmetry pn.\llt to th~ point in queationo

The axis ratio and the area o'! the limiting c:'lllil>Se 1n the symnatry

point can be found from the transfer matrix over one period of the structure ,,

en 6 Kt• ( ~ '; '' ,,; t 'S/KJ 30 . L;.:.L·.

I shall not go thxough the calculation hero but only give thrt result:

where 'l'jk are the matrix elements defined by

(9)

vhsre t~ indices 1~2 indicate the symmetry point in question and the oi:JS

being one period away"

2) D&fini tion of &ii ttance., Simple beam charu(.Jlg eystemso

Lat us then consider what the beam looks like in this same planep how

it changes with time and how we can change 1 ts shape by simple devices.

Especially tbs ls.st problem. is of .importanoe in solving injection problemao

If we take the beam at a certain point, say just after t,ne linear

accelerator, the beam will occupy a certain area in the phase pla.nep tor

instance o.s indicated in .Fige2o

.Fi.go 2

"': 7 -

If we now follow the particlesw represented by this area~ as time goes

on~ it follows from Liouville 0 ei tbeore11. that the area is an invariant. ThiP

invariant ls an important property of the beam and is called "emi ttancs11" R

In order tei get confonnity between the definition of emittance 81ld the definition

of acceptance P one does not in practice use this definition of end ttance

direotly, but defines instead the emittanee as this area divided by 1to

I:f' the beam has gone through a rather loDg optical system, like for

instance a linear accelerator where it bas been aperture 11.mi ted, it will

look considerably more regular than indicated in Figo2a in fact it shOul.d

be clear from the previous section that it is marly ellipse ah.ai:ed, aa

indicated in Figo3o

/

/ //

/

/ ""

/ /

( ,,/'"

' ---=- - --

----- ....--....... ,.,- ' / / / /

/ /

/

/ /

Figo '

lC

Fig., 4

Altl'oough the emittanctt of the beam. is invariant, its shape can be

modified considerably by rather simple beam shaping deviceso Thi111 is indeed

important, as we Shall see later that it ia very desirable lo be able to makB

such changeeo First we shal.1 9 as examples, consider two of the very simplest

beam shaping devices, a d1"1.tt space and a point lenso

a) Drift .... space o

It is self evident what. is meant by a drift~spaceo It is a space where

there are no.forces present that can change pxo Such drift-spaces are

unavoidable, whether they are use:f'Ul. or not, but often they can be made usef'ulo

Mathematically the effect of a pui-e drift s~ce can be wr1 tten:

• The notation is not quite universally established in this field yet and should be taken with some resenation.

)

- 8 11911'1 OE.Hlki: '3/KJ }0

(10)

Here '( is directly proportions.al to the drift ... length» but depends also

on many other factors, such as how close one is to velocity of light and,

of coursep on the system of variables one has choseno

The dotted curve in Fi.go} shows ·what happens to the beam wh~n trans­

versing such a drift space .. Since the equation (10) gives a linear relation

between p and x at the two ends of the drift space, the shape of the ·beam

is still an ellipse 9 the area is unchanged and the maximum p is unchanged,

and 1 t is easy to construct the new ellipse o One only needs to know the d-ift

of one point, say the point (O~p)o

b) Po:ffit ~o

A point lens is by definition a system that changes only p . and mt x o

The effect of a linear point lens, which is the only one we shall consider9

can be written

C:) = (:: : ) (~) (11)

where 6 is the ,.strength" of the lens, closely related to its .focal length

in an ordinary optical lanso

.,, 9 -

The effect of such~ lens is shown in Figo4o.As a starting ellipse :tn

I have chosen tho result of the operation/Figc3~ and 6 ia so chosen that

the final ellipse agai::i ha.a its axes coinciding with tha co~ordinate axea"

What thia axample shove in particular is that by these two simple operations

we .have been able to transfar an ellipse with x~,p orientation into another

ellipse with the same orientation but with quite a diffe~nt axis ratio 11 the

area of all ellipses being t~ same. Thia is an important ,r·etru.lt that we are

going to make u5e of latero

J.t is further noticed that by o~ratione like this it is possible to

transfer any ellipse into an;y other ellipse of the same area but with

arbitr~1.:f a~ia ratios and orientationso

}) Beam Matchingo

With the two previous sections as a basis, it is now easy to pass over

to beam matching~ Let us first take the 'particular example of the emittanca

of the injected beam being equal to the acceptance of the synchrotronp that

the shape of the beam in the phase plane is an ellipse, but that the shape

and orientation of this emittance ellipse is different from the acceptance

ellipse, aS indicated .in Figo5o

emittance

aeoeptance

Figo 5

Ii ie then 1)bvious that we shave off a portion of the hoam equ&.l to

the }X:lrt of the eud:ttanca that is outside the accepte.nc.'"'·

fa \f.ewr, it :i.s now poesiblei, for inBtanc-e by uning eLnple bean; shaping

devfoe5" a.e indicated in Fi.go~ and 4~ to gt\te ths emittance r:}J.llpse the sama

aha.Jj(l 6tlc •Jrientation as the acceptance, and. thus one just a.ccepts all tlie

})3.rlic:le" into the aynchrotrono Thie is what we usually mean. by beam matchingo

One impo1 tant fact to be noticed is that i.t is no aim in itself to get tm

cross sectton (in configuration spac~) as small as possibJ.A., In Fig,, 5 for

insttm.ce" the ti'allSverse dimension of the beam is already too small o Al tholJ8'h

this :ls f. ·very obvious conclusion. from considerations like this~ it is often

diffJ.cul i to get the idea round to the people making for instance Van der Graaff

gene:rat1n·s or linear accelerators9 who seem to thi.nk that their machines are

better~ ths smaller spot size they can make .,

If the beein emittance is smaller than. the acceptance~ one still wants

a good nwtch if possible, i~e. the two ellipses having the same shape and

orientatjon at the injection point, although being diffGrent in sizeo This is

b&eauae or.e wants to occupy as little aa poasible of the vacuum chamber, as

one uu:.y in the end have less aperture than expected, or the barun emittance may

be larger than expected o

It' tha emi ttanoe i• larger than the acceptance, the complete matching

is not so important as long as one matches ao well that one gets full overlap

of tha al Upses. However, perfect matching is still advantageous~ as the density

in the phase plane is in £'Emeral highest mar th& centre and decreases outward.so

Ir. p:r.actice the beam emittance may not be quite an ellipse~ but 19rhaps

more as :indicated in Figo2o In that case one chooses the ellipse that fits

best, and matches that one o

Tl!e emi ttanco and shape of the beam injected into the synchrotron is

mainly determined by the optics of the injr:ictor (linear accelerator), and what

we hfave c;alled a beam matching is in fact as much a matching between two

machl.:neao Realizing this we also understand that the requirelllf~nts one wants to

put on the linac are very much determined by the synchrotron acceptance and by

the possibilities one has to put in matching devicesp si.ich as drift-spaces and

lenses~

.,, 11 -

I :n practics, the group in which I tmrk only dealt with problems like

these in proton w.chineso In principle$ the problems are the same for proton

and electron ma.chines~ but the practical. Sf>lutions may differ considerably.

I shall therefore not go into detail of th•:l solutions chosen by CERN o

Bowever~ a few nUJrerical figures for possible eleotron machines may be

of intersat. I then use as a basis some figures that are available for the

Cambridga Project in the UoS.;A. (Livingston.a, 1956)P as that project is similar

to ti/.e 110w Germain project in many respects •

.Jlss~:og an available aperture iltf .± 12 mm one gets

H ~ Oo45 x l0-3m for 20 MeV injection

~ Oo9 x 10-3m for 40 MeV injection

lt has been. difficult to find any figures f~r existing electron

linear e.ceelerators, but reasonable -ctalues seem to be a spot size of 10 mm

diameter and divergence of .± t x 10-~ radians at about i5 ... 20 Me Vo This

would give an emittance of

If one could increase the beam current from a linao by pemitting it to give a

larger emitt,:nce, one would do so, preferably by asking for a larger spot Size 9 L.

again an example that 011S should discourage the ~r of the injecticr to

produce the smallest possible spoto

Al though the beam shaping devices nentioned earlier a.re simple P one

should not forget that matching is needed in two planes, which complicates

the problem considerablyo One should thel"Elfore plan the whole injection system

such that a minimum of beam shaping is needed. This is not easy in proton

maohinsa, where the f oousing of the linear accelerator is a very tricky problem

in iteeJ.f ~ In electron machines, however, the focusing problem is considerably

easier, and one should therefore keep in mind the possibility of getting the . proper beam shape by suitable design of the lina.c foous~ng systemo

Pf" Plm;l' and Eno:· Q' Oscillations (.Synchrob:·on Oscillations '., "'-"'~·---o". ,..,,.____ ,..________________ ~

~ ·o Eihall n1 w- pass over ti) th~ lcmgi tu<.li.nal motion m1r: cons:>.der the

'oacHlai:'.ons nbou·: the phase s·httonar-j' poi.n:t:o These a.re th:: SO·=~alled phase

ose:.illa1:·.ons, or Gynchrotron oscillations" It should then firat of' all be

recalla( ~;hat in 1m electron synchrotron o:ne would always bJect above the

trausitHm energ:r:. and in general far abov'3Q Thia and the f~"ct that the

injecte~ olectronu are far up in th9 rolat ivi.stic regioni mtkes the phase

trappin£: ;?roblem fook different in an Ao~o synclrrotron for e>•lectrons as from

that of p:rotonsa

fiince injef,tion is ah~ve trans! tion~ the phase stationary point is on

the :fulling side of too radio-frequency wave, at !l distance qi8

from th.a

zero 1:~rossing" As a variable we use A ip ~· which is tm deviation in phase

C·f th1 particl1::i under consideration fro:n <p8

o The corraaponding can1Jm.cally

conj11ttate' variuble is 4p/m0

cf., where ~p is the devie.tion of the loDgi"~

tucli..J!Hl momenti:.1m of the particle from too longitudinal momentum of the

phal*· rJtationary particle. Since the pa:rticle velocity iE: close to that of 2

ligbf~, this conjugate variable can also be written AE/D' c ~ wherE• IE is 0

·the e:nergy dif:f erence between the particle in question ar:.d the phase

statjona.ry particieo

hJ before na car: in such a pha~~ pl'lnfJ draw the limiting curve for the

acceptance of J,articles. This looks somewhat like indicated in Figo6a.

1r,p/m c 0

----:----+-~·~

A'P

Figo 6b

.~ radia1 oscillation is also associated with the energy oscillation~

lf tl:·f;;ae radial oscillations are so lart~ that the trapping ia limited by

the r~i.dial aperture rather than by the -phase stability limits, a curve

f'urth~·r in than the one given in Figo6a is limiting the tra.pping efficiency~

for example the curve indicated in Figoobo In a proton synchrotron with

AaGa focusingt one has usually phase U~nitation, -whereas it looks as if, one

may easily get aperture limitation in a:o. electron synchrotrong

The curve in Figo6a is not an ellipse because the forces acting on the

phase oscillations are rather non-linearo Particle trajectories insi~e this

limiting curve will therefore not be ellipses either, but as these tra~

jectories depart from the limiting ourvH they approach more and more

ellipses, and for most purposes the linear approximation to. the phase

oscillations is sufficient" ~f we have :rather strong aperture limitation

the curve in Figo 6b will therefore be ~lose to an ellipse~

The beam from the linear ~ccelerator will come in bunches as indicated

AP/m c 0

cp

If. the frequency of the linear acceJ.erator is high comps.red with that

of the synchrotron, there will be many of these bunches within the trapping

range in Figo6, and we can treat the benm more or less like a continuous

beamo This will always be tbs case in proton ayn~hrotronso

In electron synchrotrons there is 9 however, the possibility of having

a harmonic relationship between the linnc and the synchrotron, say by

running tbs linac on the second harmonic} of the synchrotron, or Etven o:n the

fundamentalo The injection problems are somewhat differ~nt in these two

caaes, and we shall treat them sepa.rateJ.yo

. .

' ' 'I' · '. [ :

> 1 this c.~ :11 the beam ie conaiderec1 to be a contini.ic\1s beam as

' I

i.rp · ;!il c. !)

Fig .. 8b

'Ibo si tuatl.on after about a quarter of a synchrotron oscillation is tl! .

ehown ~.n FigoSb o The bending ove-.r/t'be baam is due to the non--linaarityo Th:Ll

l!Ollc-·] inearity ulso leads to an increase of the effective area of the beam in

i: he I l>ase planf'· ~ as the beam Winds round. and round in phase space effecti vel~

takir.6 all the ap&co inside the li.mi.ting c'ilrve. This may seem like a violation

c1f Lfouville 0a theor.emp but~ of course~ it is not. It is sormwhat analogous

t.o wil:,ding a tl1read into a ballo There will be plenty of air making the

'roJ.tuu: of the ball larger than the vol111oe of the thread itself, whereas the

three.a has not ~hanged its volumeo 1-lowe·rer, the analogy breaks down on one

imp01-t:an-t poini ~ The ball can usually b13 unwound. 9 also in practice. Tm beam

can .1n princip:ler- but not in practice. 'rhe process is in practice unro­

c·oinwD.bla, alth1ugh it may be ~versibl1l in principleo Hareward (1954}P who

:firn1. describe':' such a process in an ac~lerator, cal.lad it filamantatioiiv

F:LgoBa now indicates wha·t requiremei:lts to put on the phase trapping

region and the parameter.a dete:r.mining i !:~ a.'1.d on th1:1 baron its"'lf.,

Erstly 0 or;e wants the width in pha3e to be as large as poasibleo That

u·eanf; that one should work with a large RF. voltage compared with the

1.i.ecam:~:i.ry acco; 0:ration~ .or in other worisv the working poj.nt on the RF wa.w

abouJ d. be rathe:J' clcse to the zero oros3:Lng, In proton machines one hae to

1"trH.:i:1 a comp:r(:nLise ~ usually auoh that the trapping width is about 'It ( 50 o/o;

In electron mad1.i.nes, however, the acce laration needed later to compensate

i;ha n~diation · NJses ia so large that 0:1e can: easily get a trapping width SD!.Plitude

:naa:i:- 2n~ or .1 00 o/oo One shall in fact ha-ve to/nvdu.1at~ the RF in order

not 'i;e> come to.: cloae to tha zero croestng" 'r.:b:R.t is for other reasons 11 but

orw Ei'!;ill g9ta a fairly large tr~:p.iiittt;; i'rldth~ ~rtainly aomewhare between

50 o/n ~ JOO '' "'"

thE1 next requireraent ia on the width in energy of the ti·apping region,

or c.n i;h.E~ depth of the potential we.11~ as we r.Uao may call it~wbich shoulO. be

ama:11 enough not to gjve apei-turo limitation. · Here ie ot'lft of the

between electron maohi.:ngs and proton machinEHJo Xn AoG,

:9roton .:iynchrot!'Ona the depth of the pntr:>rrf;ial well is usuall:y rather smal 1

and the associated radial displacement ie a small fraction of tm vacuum

oharnbe:r,. To get such a situation is vei"'Y difficult with electron machims,

One liaa to work on a vecy high ham.onfop and ae ah"'eady mentionedv have

wnp~_j .. tude modulation of the RF to give a relatively small RF voltage at

injection compared with at ejectiono In the Dambridge project~ for instance~

it :i.s planned to work on the 350th harmonic and to amplitude l'JJOdulatep and

ati!l they plan to have a horizontal apextl.ire a little more than 3 times ·ctie

vertical aperture, which they certainly reed .. Personally I feel one might

cor1aider to go up to the 700th hamonic, i,,eo about 1000 Mc/S o

let us then consider the requirements om would like to impose on tm beaw.~. From fi~;n 8a it is seen that we want its energy spread to be an.all

com:r:ered with the depth of the pot.ential well, say Y4 of this deptho

'1'.'here isl' however, another effect and that is the betatron oscillation

tha"c is associated wi.th an energy spread,. A pa.rt:i.cle with an energy

deviation of &J~ from the correct sneJ.·gy will have its ~losed orbit moved

by fir~ given in the extreme relativistic approxi.mation by !Jr/rm= ex IJE/E~

whex"t:"l et is the momentum compaction fa~tor. A 2,5 o/o er.iergy spread gives

for example in -Che Cambridge machine e. fh: of about 3 cm L' If this parti1;:1.e

:.1.s ::.m.rnohed on i;o the same orbit as the particle with f.'orroct energy~ tbiB

diapJ.a.wment hall to be taken up by a betatron oscilliition Oif ~ cm amplit11d~h

Thi1:. is serious ; and if nothing is don.:1 about it~ it w:i.ll either increase

the requirement on the vacuum chamber width or lead to particle losso

Ona can roduce this betatron oscillation amplitude by using the prope:r

double bending cm the in:flector to pJ.a1:.-a each of the injected partial.ea on

its closed orbit. J or at least near to :.\.t" Thia is a complication, and a.

rather unnecessnry one for protonso For electron mac-.hines, however, it should

bo looked into :Jeriouslyo Probably ow UJB.Y get the energy spread of the

li:oa~ below the figl.U"a indicated above~ but perhaps not so very f'ar below~

~Jne should notice that one has to wake any wante:;d :L1.fl'ovement of the

{-m~::·,;y opread in the .linea:x- acceJ.erator :i.tself9 as thCJ:\3 is nothing one can

do to the energy spread between the linear accelerato1· and the synehrotrono

Thi 8 may seem .surprising if we r~member the close analogy there is be"b..i~en

the phase oscillations and the betatron oscillationsw and for the betatron

oscUlations lie discussed earlier very simple beam shaping deviceao

Such beam Bha:pi:o.g devices do indeed also exist for the phaoo oscil.lationsp

and· i;.he so-caUed debuncherr which has been described in a C.'ERN-~port

{Jobnsen~ 1955), wai'I invented just for the vecy purpose of reducing the

ene:cgy spread of the beam going into a synchrotrono The only draw-back~

however~ is that although it probably uorks beautifully for proton machines,

it wHl require completely unrealistic drift-spaces when the p;i.rticles are

as htgbly relativistic as electrons of 20 - 50 M3Vo

If we run the linear accelerator o:n the same frequency as the

synchrotron or on say its second ha.rmo~1ic, we get a qui-te different situation

from the one just described~ This ie e possibility that has been touched upon

sava1•al ti.Joos also for proton synchrot:r·ons, but seems to be unreal:i.stic for

such machinesc. One should thsn either use t..h.e fundamental or the· second

harmonic (which already throws away half of ·the particles)" Any higher

hanQanic: 9 al though doing no harm11 does not seem to give improvement over

the case of no harmon..'lc relationo

F'or simplfoi ty we here consider the case of dr.l.iri:ag the linac a.t the

fundamen.tal cf the frequency of the synchr.:itron$ althciugh the arguments

apply equally well t:o the case of the If:econd. harmonic"

I.n tact~ the case is now completely analogous to the casa of matching

the te"&atron oe;cillations (section lIL))o ~~e bunches have a certain

shape and a certa:i.n siz&, and one has to match"' if possV:ile* by giving the

trak!ping reg.ton (Figa6) the same shape e.s the buncheso The non,·linearities

make this not quite possible for very large buncheeo However~ in practice 9

one would then match by giving the elli.pses riear the origin the same shape o

The bunches wi.11 probably not be larger than the linear :region of t.he phase

trapp:ing area, If one runs on ths funde::nentaJ.~ one wocld in this way P in

,·.

...... \ ... ? "'

p:x1.nciple~ be a'ule w trap 100 o/o of the p<lrticles,. on the second. harmonic

50 o/oo A further advantage would be that the synchrotron oscillations would

bo smaller, as the area. near the edges of tm ·phase trapping region will

not be filla~L

I have not had time to investigate in detail if it is possible to do

this in practice~ but if it is not possible to get a complete match~ it

stilJ. looks as if much can be gained by this method even if one has to be

satisfied with an incomplete matcho

One difficulty one has with this matcn over th~ matchi.ng of the

betatron oscillations, is the difficuHy mentioned earHer when H ~1as shown

that a debuncher does not wom in the case of a'l electron synchrotron., One

cannot rely upon any bunch-shaping device between the linear accelerator

e.nd. the synchrotron. All matching of the S'.r'l!lchrotron oscillations must be

done by adjusting the parameters of the synchrotron and possibly the early

part of the linear accelerator (the bu.uching section)u

There does not seem to be any difficulty in making a linac run on the

second harmonic of an electron synchrotron if this is run at about 500 N.c/s,,

However~ if. one wants to run on the fundamental j one would probably prefer

not to run the linac much lower than at about 1000 Mc/a and one would trere~

fore have to increase the synchrotron frequency to about this value. This ia outside .

rather / what one is used to~ but there does oot seem to be ~ fundamental

reason why this should not be doneu In my opinion, it would have advantages

that makes it worth oonaideringo

4) )i'reguency tolerancesu

At last we shall briefly consider frequency tclnrariceso A frequency erro1·

shifts the phase stationary point outwards or inwards. This shift must

therefore be small compared with the radial width of the phase trapping

regiono Rather careful considerations of timing requirements and possible

frequency errors are therefore needed for proton synchrotrons (Hereward,

Johnsen, Lapostolle, 1956L whereas the problem is simpler for electron

machinea, and we shall consider those here.

CEfff: .. } ':~;/KJ 30

If LLe frequency ie sh;.fted by IA f 0 arid we c:alJ the corresponding radial

abifi: xf r wa have the relation (Courant~ Livingston and Seyderr, 1952)

/;,f/f:; (J.2)

and jf we insert for instance the Cembrtdge parameters~ we get

-4 -1 3 x 10 om

Since this is a fixed frequency machir.e, the difficulty in keeping

the :frequency error better than 10-4 is small~ and effects of frequency

exrors can therefore be considered negligible.

Vo Conclusionso

I have here mainly tried to give the philoephy one should follow in

tryirit: to a nalyss the problems one gets involved in when wanting to optimize

in,jection conditions for a eynchrotrono For proton synchrotrons the problems

ha·re been studied in consider.ably more detail than I have found it preferable

to ropoJ~t here., I can refer to CERN reports by Lapoatolle (1955), Hereward

(1954f l955) and myself' (Johnsen, 1952v 1953 and 1955L. and a summing up of the

main reaul ts will be found in the Proceedings from the Conference on Accelerators!

held J.a::1t June in Geru\ve (Hareward, Johnsen and Le.postolle~ 1956)0

The matching of the betatron oscillations are not dissimilar in the case

of electrons from that· of protons, but the problems involved should be kept in

mind l'l'hen the requirements for an injector is going to be specified~ The question

of space is important~ as beam shaping devices require more space the more

difference there is between the two systems to be matched (longer drift-spaces)o

(; ' 3/KJ 30

Fo~ the syr.(jhrotron oaciUat;.ons thE< condi tfons are no dlf:ta:r-ent that

a few r;~·11.ierical e:ir.am.p1ea may throw some e1.:t .:ra. JJ.ght: on the problemsp al th<>l1Bh

theee mrueric:Eil examples should ba taken \l.d. th ~ll.l z-eservatj.ons,,

~l?J~~l.o Assuurlng the main parameters RV "in 1;he C8lllbrid~ projeot

(Livi1~ston, 1956)P but assuming no amplitude tlOdvl.ation ,~j.' the RF., we fiud

that ·the radial half-width of the trappir.&£~ region 1.s 10-30 cm depending on

the DC bi.as chosen and on the injection einergy. This is i.Illpoasible, and RF

modllle.tion must be incorporated to get t.h:l.s &Dlnll enough~ although there·

are waya of pressing it down a littleo

~~-!!,_G_,,. Same ae above but with RF amplitude being smaller by a fao1;or of

about lC at injection, gives about .t 3 cm width., If one :further incNaaea the

h.arn.oru.c number to say 700 it should be ~ossible to come down to say .t 2 cmo

DotiJ. these figures are reascmableg but they put rather strict energy.,apread.

i-equiratrenta on the linear accelerator~ of the order of.± l o/op preferably

lesso

There is or,.e question I have not rai ead yotp that ie whether there may

b& al tematives t-0 the linear accelerator a.a an· injector. ~L'he only other

possibility seeme. to be the microtron~ With es~cially tha nice feature of

low ene:rgy spread. On the other hand, the low energy spread is obt.5linad :tn

the mic:rotron becP.luse of the aruall stable region,, 'l'he same csn perhaps also

be obtaiLed in a linear accelerator if oDa for instance is pl'i'tpa.Ted to reduce I

the pha.s~ trappir:,g region :ln the bunching section of the H.nac and consequently

lt.ei tra:pping .affi~iency.,

H1Jwewr~ I 'fael that in the analysts of cll the injeetiori probbms for

an ale¢tron synchrotron all possibla aolutiona should be included till they

can be dlsrEtga:rded on safe teclmical grounds~ fM both tha lineEU" accelerator

and the microtron a.re still in the pictureo

Ka Johnsen

/jdw

•

References:

OOURANT~ EoDo~ LJYINGSTON1, MoS01 and SNYDERr H~~io r 1952

HEBE\iARD9 H~ Go I• ~rQJINSEN 9 Kc 9 8Jld LAPOSTOLLE, Pnp J.9560

HEREWARD, Ho Go p J.9550

SIGURGEIRSsoN~ To~ 19520

Physo Reva .§§_p 11900

"Problems of Injection"., CERN Symposium on High Energy Acceleratorr, and Pion Physics~ June 1956~ Proceedings Volol. 0 ppo 179-1910

"Energy Sproad and Phaee-fodusing in Particle Accelerators"? CERN/PS/HGB/lo

"A note on injection timihg and tolerances for an AoGo Synchrotron"; CERN/PS/HGH/5o

0 Bla:ee Oscillations" 11 CERN/PS/i:J/lla

"Frequency and Momentum tolerances at Injection", CERN/PS/KJ/l8o

"'l'he "Debuncher'' : A device for reduoing the energy spread in a linac11

11 r:ERN/PS/U/29

"Etude graphique des trajectOiresa Rep~~q­tation da.ns 1 aespace des phases", CERN/PS/Pl. 0 2"

"Injection dans le synchrotron,, R6glage de l'acceptanc& du synchrotron."o CERN/PS/PLo3n

"The Cambri~ Electron Accelerator"o CERN Symposium on High Energy Accelerators and Pion Physics, June 19560 Proceedings~ Voloi9 ppn 439-4460

"Betatron Oscillations in the Strong Focusing Synchrotron"~ CERN/T/TS-2"