1Y/, CERN-PS/KJ25 DIAGRAM FOR LINACS WITH ALTERNATING GRADIENT 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 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 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 This may cause resonance effects and 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 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-
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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~