1Y/, CERN-PS/KJ25

STABIL~TY DIAGRAM FOR LINACS WITH ALTERNATING

GRADIENT FOCUSil~G LENSESo

by Kjell Johnsen.

L Introduction

ln'i. this report a stability diagram will be given for an AG focused linaco

To obtain this diagram the following assumptions have been made:

i) The focusing system is periodlc with period Lo

ii) The defocusing .force dua to the accelerating wave changes negligibly over

one period of the free oscillations. Th:j)a meania amoag other things that we assume

the period of the phase oscillatiors to ba long compared with the period of the

free oscillations, ali.d that th~, en~$'W gain per period of the free oscilla:tions ' is smallo

iii} All lenses are identical, but every second one is turned 90° with respect

to the other ones.

The first two assumptions are made in order to enable ua to apply the

ordinary stabil~ty condition for a Hill equation. It is believed that this approach

gives a reasonably good accuracy, and can be used for the first steps in designing

a linac with AG focusing. However, a final design must be computed tiu'ough numeri­

cally. It looks as if the results obtained here are conservative, as all changes

with energy in the linac parameters tend to improve the stability. This, however,

is not true when the coupling between phase oscillations and free oscillations is

CQ~aidercdc This may cause resonance effects and ther~~o~e a build-up of oscillation

amplitudes. This may not be dangerous for as short an accelerator as the one we

are planning, but a careful examination of thie problem will be needed. It will,

however, not be treated in this report.

The stability diagram will here be given for two extreme cases: the very

simple case of point lenses, and the case of no field-free sections. It will then

be seen that if the proper parameters are chosen to represent the linac and the

lena syst~m the two diagrams are so similar that the difference between them oan

be neglected. As actual cases will lie between these two extremes, it can be con-

" '

eluded that the dilig;:·a:t• W8 he:vf.! :found can be u.socJ. for· any prac\:ica.1 Jens

arrangeman t ,

2 ·' Point !@!~

By a point lens is in thia ~or1nf)C'ti on me~nt an infinitely thin four .... pole

lens that changes the deriv-ati.<re of a particle orbit by an amount ! 6x~ where

x is the displacement of the orbit .fr.om the axis., The upper sign is for a de­

focusing lena (in {;he X·,-direc~tion) 9 and the lower sign is for a focusiri.g lensQ

In a linear leris 6 is independent of x ,-.

The .relation between a point: just in front of a lena end just after it ia

Between the lenses the equa.Uon of the particle orbit is

d.2x -7' ~ kz " 0 ( 2) dz~

whe1-e k l.s considered to be constant (if the assumpti'on ii ) is satisfied)" '!'his

eon.-. .. .,..._ on the Unac 14".Jlletere,1 such as accelerating fie.l.d 9 phaM ftlo­

oltf, ,._. tlllCl•~ ~etc,.~ !t can be shown that

k Jt!::_ -"-· ,.'11

( '·' ,, ~ ·' ~lll If· " I

"" m,:; 'O

where E J.S the ampli tudP of the. acce1era.ting f:ieldp >. the free=space wa'l?-elengt.J:, 0

~ the phase angle, .mea.st~red froill i.;.ha pe11..lt of the wave, ani.t f1 ie the particle valv,!1 t.Y

over the ve.loci ty cf 1ii;h t ._

The .eolution of (2) and tte derivative of the solution CM. be written a~'!

)

r cosh'fk

1

z ']' . ·f1? \ . ( x )-

-~..... in n. .. 'I z

) I "'' ....

Ykf 0

) \ {i2 sinh\f"k1 z

l (4) f r-1 \ XV \ ll' \

(:oshVk z \ 0 I

I

where x and x' are the ini tia.l values,, 0 0

The transfer ma. tx·ix o-o'sr one period 1 is consequently

T { c .. s/{k') (l vk's c o

0

0) (5)

l ' where

C :; cosh(k' 1/2 S /!:: sinhfk'' L/2 (6)

Half of the tTace of T is

cos µ ~ c2. ~ s2 = l. (61/2)2 (- s _ )2

2 fj2I,/2 (7)

and the stabi .l ity condl ti o~ :l B;, as WIS lar0-;1 ~ tha. t

=1 <cosµ< l (a)

However~ we get a more convenient expression if we rear.range (7) in the

following way :

Let the defocueing forces in (7) tend to zeroo We then get cosµ for

what can conveniently be called the "emptt' system11 Leo the same system with

the RF switched offg but still. assuming the particles to have the proper mQc. '

mentwn in each point" The car.responding µ we give the index O~ and get 2

COBµ :. l "' 1 ( 61/2) ( 9) 0 2

This is then a parameter characterizi:lg the le118 system orily0 and containing no

linac parameter apart from tbe particle m~ntum~

· · -· · · · -proru . thi s-·_equat1 on we then aubsti tu:te in ( 7) !Or- {6L/2l-and·- obtain

1'\ r

.. c~ ~~ ,sl!'.

'fhe tvo limits of the stability region. are given b:ir the hlo c·1n·ves mw obta:Jns ~· I ) by putting !'..'OS p .C. ·- 1 in tlO ..

'fhe lirni ting curve cos p ;:; 1 ( µ;:.{)) is gi Yen

·1-coe ;io ,~ 2k(L/2)2 (11)

and the oth<):r lim .. i ting cu.rve .. cos µ. -:;: ~J.. ( ~n) J.a g.iven by

l -Nl6 ~; ., ~( · k )(t/? -., 2 ,-ot.2(y:;: [,/?' 0

by

lr. i1g l thi1;tk ;..,IJ i.L.ii tlllf!: '"'Ur'le~ f\UH' ~ 4l'aft U ootW a.rt•• ~ ..

region between the cu."98 ia tbe atabh "'1•· 911 p11•1tll (~ "•)' . characterhine-: the lene system ituelf ~. is 810?18' the horizontal axt.e'.' si'ld k(.1.~h:)':

is &long the vert:lcal 1m.s,.

Jafore wa din.cuss this die.gram in more '~etaj_1 w~ .sh&.11 co1:-n~ide:c

another eX!:1.mp.lt1.

't'he 0 ther extl"'lme caoo is a f ocusj_ng aya-te:ro. Iii th no f i~ ld f'ree

sections 'l'bi:o can be 1mr.lu~d through :in the same wuy a.s abo"\Te. bu·t the t~a.1cu1a:tl.om1.

will not be gh,.on here,. The result .. i:'Jowever. can be x.i1·esented t:n exactly the

same WlJ.y as ~.::. UH~ Gase of point la:o.ses, .. and is shown by -the fully dral!n curv·ea

marked ~· . ··· C v.H" F : It :i .. n li'ig , ~ .

A~' u.ot'tced, the difference betw·een the l'~iaul ts in th5 two ex tr.em~

'!.'he d:i.e.,gr{:l.i'.T, cau theirf'.<fo1·e be used for· a;,~:y J.en::1 :;:.r1'!.1.rlf~~l<rl.';;\.l:. )f · U!•::·

parameters c:~w.J-.~H.:tl'lrizlng the lene ayotem only cos µ Br.1 t::irs the diacu81:;11:iu of th~

How tc ob i;e.tn th::.i cos '' f'',., ·v

CERN~-PS/KJ25

4o A Dlscuaeion of a Posaible ~ or.ldng Region Inaid..e tb.e Stability Diagram~

The defocusing problems are moat serious near the input end of the

linac, and we shall therefore only consider that part of the accelerator'-' That

lll8anB that there are particles performing phase oscillations to the very limits of the

phase stable regiono That 888in means that if k in eqo (2) on Figol for the

ayncbronoua particle is k , there are particles in tha bunch experienci!J8 a radial . 8 .

force corresponding to k:: -k8

(which to~ these particles acts focusing) .. We

now atate that also these particles must be inside the stability diagram 11 1.e~

tlllir ..td.ltc point ............ tbl ...... u. . .. ,, ....... -- *' .. WGl'll1as poiat ot tM ...,_ ,.nt.oi. .n "' ..... -- - ... '" "7 tarniDC theµ=~ aar'9 ................. , .... W).

The muimum k a particle in a bunch oan 9zperi._,. -i.•

k = ~sin fv/ain 1J1 ll8X 8

0 _J 0 where cp'~ 21J>8

for cp8

( 45 and 1p 0 = JY2 for cp;> 45 o

'l'heae particles must have their working point below the µ = 0 line. This giV'es

a new limiting curve for the working point of the synchronous particleo Thia

curn i• obtained by multiplying the ordinates for the µ = 0 cune b7 sin rp8

/s'in. cp'.

Two auch cn.irvea ~ve been drawn into the diagram in Flgol (the curves marked

" ~ 60° and ~ = 30°)., 8 B .

It is thus noticed that the region for t.he working point of the

aynchronoua particle ia much smaller than the main stability region, and consists

only ot the triangular shaped ar~OAB.

At low energies, Su.ch as ve have near tbe injection end of the 11.nac,

the aooelerating field will be limited by the max1mum "f8.lue one can choose for k 8

from the atabilit;y diagram. That mean.a that we want to vorkJnear the. inJection

emJin the top corner ot t.he stability triangle (point A).. If' this point i• chosen

a.a tbe working point there are particles in the .bunch with working pointe ever;yvhere

on the vertical line through this top point, quite out to the stability limits

µ = 0 and µ = 110 Due to the phase oscillaticma a particle 91i working point mo'Tes

verticully along thir> line in the stability diagram .

.M;. th8 energy of the pDfiieles incres.s~s the 1'f01'ki ng ])Oin t of

the synchronous particles moves" and then in general d ownwarcls and. to the left

o~ the diagrai11 1 so that 1~e get fw·ther arUlJ' f:rnnl the illHilt~bility l:Lr:d.ts;, j_f we

do not at the sa.rne tilne increase e:i ther t"f1e accelerat:l:l{~ fiGld or !\/ or both,

to make up for this motion.~ !!:specially neu.r ·the inj~~c~;1on it is de>sirtmble to

do the.t in ordo1· to reacll the maximum field ;:~.s quir~kl.;r as :pos~:l.bJ.o . 'l1h.is cannot

be doue contir1uously,. but only a-t the begirmi:og of eQ.<::h accelerating aectton •

.As an flXBmple we shall consider the conditions at the i11.J~c-·t..ioh v•~

of the firet tiection of a helix accelara·tor for O./i .. 50 MeV t focused. by magnetlc

four-pole lern3et5

\'ie choose the following basic p$.l'&".iet~rz ~

Pree apace wayelengtb

Synchronous phase (measurad

from tbe peak of the wav-e)

Period of focusing system

Q1 \1 - 60 'S

0

1, :::; o~s r,l

The choice .of wavelength j,s ms,inly go11er.ned b;r ooli.Ji.:.:~di&..'II'ete1~ cor.1aic2erations. /:,

large phase angle is chosen in order to get a. idd.e t:rapp:i.ng I'e)gion. In choosing

the period. of the focusj.i~· system I assumed i;ht'l.t 'th('1 1snseB ougl1t to us at least

twice as 1ung a:s che d.tameter of the system .J.t if; go:i.u~~ t.l ~~l1rround, 'l""hi» d.i.o..1Uete:c

is 8-10 cm, and con.sequently i have chosen for this e:w.rnple L ~-= <L? l!l ail a reasor!able

per~.00. ..

J:>z·om the oi;auili t'J iJJ.agram w~ find fo"1.~ poin ~ A, which lfe., a.ooordin{~

to the consid~ratton.s made in aeotiou 4; choose au the 1r1orldng point

O!'

We eubsti tute for k (L/2) 2 in eqo (3) and find E == 308 kV/mf. and the corresponding 8

energy gain per unit length is E cos 'P ::: 154 ke V/m " 8

As noticed we get a rather low accelerating field, but as the energy

increases~ one can, as already mentioned~ increase the field coneiderablyo For

instancev if the first section accelerates only up to l MeV 1• we can at the input

end of the next section employ almost three times as strong fields as in the first

section a

To find an approximate value for the field gradient needed in the

tour-pol• lenaea t.o get th9 requi.r.d 009 i1 o~ about -0.29, ,,. 096leot poMlibl• 0

t1•14-tree aectiona am fiDl tbrtn tt.t. a tt•lAl CNU•t ot O,/a I. 4'10 O 1 ••/•

1• ••ded, which ahould. be qllite..., to .-Sa.e.

60 Concluding Remarkso

In the discussion of which region of the stability diagram can be

used for the working point of the synchronous particle 9 it was assumed that no

particle should at any time of its acceleration be alloved·to get outside the + main stability limits cos µ = ~ 1.

This may~ for several reasons., ~ a stricter requirement than

actually neededo Firstlyi as the particle is accelerated~ the stability conditions

improve because its wlocity increases, and a particle that at injection is just

outside the stability regionp may after a while, and before its radial amplitude

has increased appreciably, find itself inside the stable regiono Secondlyw the

accelerator is not infinitely 10118~ but 9 on the contraey ~ rather short .. and even

if some particles are part of the time a little outside the stability limits~ the

amplitude may not have time to grow to a dangerous magnitude o

. This, however, is parUy counteracted by the fact that the stability

conditions are based on the assumption that the amplitude should not grow to infinity,

whereas ve cannot permit the amplitude to grow outside the size cf the vacuum chamber~

If that bad been considered when stating the stability conditions 9 a narrower

stability region would habe been foundo

CERN-PS/KJ25

Th\9ee considerations again show that for detailed analysj.e of a

proton li.nac numeri~al computations are need.ado But for the first estimates

the method desc.ribed here should give satisfactory reaultso

From the example given~ it can also be concluded that it now

88elD8 practically possible to fOCUB a helix type of linac by means of the alter­

nating gradient methodo This means that we sh.all have to consider the helix in

compariaon Yi~ the type using drift tubes for our injection machine, A report

containing aome very tentatiie tables comparing these two types will be iaeued

sb.ortlyo

01nhu, .. ,. .. .,1, ...

41

.,

-9-

... (). I

- {), 2.

-t>.J

-P. '/-

--as

-{). 6

- ().1

-1),g

W' ...---+~~-~~-1-~-4-----¥-

.-..-1-~-+-·~+---l-~-+-~~·ll-/ ~+---+~-+--vi -~JI

-l.1>

Fig .. 1.. Stability diagram for linacs with

alternatlne;-gradient focusing.

CERN-PS/KJ2 5/