SPACE CHARGE EFFECT IN PROTON ~INAC

s. Inagaki
National Laboratory for High Energy Physics
lThe longitudinal oscillation of a particle near the synchronous phase ofa linear accelerator is usually expressed bytJ., [ l J d. ( ,, •L )~l~l. 1'; ~s ar - n :J.e. r;. T.
= 21(..
-;i" -.;::c:> ( I - S) ........_¥. · f"l/- ~)Vlasov gives the space charge term S by
.J ~ J\ M1I$ ¥ ;[. A-x 4..; !i. T ~ ¥, (. 1/-- Ys )..,""'
where=(I)(2..)

11 "- = a."" A;llzi""'77~:-;:;:~IL:::t~;=::::;=::::;::;:::;==:=::::;::;(.n(a. ...""+~N(e.:l+i.)C a.,·T"~)<az""+itJ
no = ~ 0 I Eo = 120 Tin, a (u = x,y,z) is the half axis of the ellipsoid, andI is the current. Eq. 1 Mescribes the averaged motion in a whole cell. The
decreased phase stable region gives the upper limit of the current.
If we use the number of rf cycles dl needed for the synchronous particle
to traverse the distance dz and assume the change of Ss in a cell to be small,Eq. 1 becomestl ( t1..
,)..2 rr.. e J\. Eo T.
~£ 1-.L (1/-¥,ij= -111-oC.""~ 3 (Js ( / - $) _,.....lf. ·{1/--lf,)
Instead of using a single equation for an averaged field in a cell, we can use
two separate equations as follows •.2.

.t·~

J.J:>

"= - ?t f.l) )C.

(sJ.'Yf..(.fJ-=tK.'"'( 1-s/:F
where?t;1 =-(6 J-SF{71If a linac consists of gaps with rf cycles !1.1 and of drift tubes with rfcycles !1.2, we can define the transform matrix M1 and M2. The trace of themultiplication of two matrices gives the stable region. Fig.l shows thestability diagram for !1.1 = 0.3 and 0.25. If a = a = a
5 X 10- 3 m(lL~)max = Tr/3, then we get the maximum curre~t ofY0.349 A for conven~ional
proton linac.
.
The APF structure part of PIGMI project has seven superperiod, each ofwhich consists of four gaps and four drift tubes in four rf cycles. The
synchronous phase of the first gap is damped by 2" in every superperiod. The
stability region for this FODODOFO structure can be solved in the same way(Fig.2). This shows the remarkable increase for upperS' compared to a simple
FODO stru.cture. In the first cell with F = 1.341, S' is 0.062 and the upper
current is 367 mA. In the last cell F = 1.192, S' is 0.039 and the current
limit is 247 mA.
In the design of transverse dynamics, it is common to use the integral
form in a cell.
(!)
4 X= J.,. N/,
A X"= - 7C. < v.. T ~ 1~ }(..
(9)""'" ' " 1P·rJ J\.

-231-

where Vn is the peak voltage in the n-th gap. However we can make a different
approach for transverse dynamics just as the logitudinal one.

discribes the accurate transverse equation. By neglecing the magnetic term at
low beta region, approximating for ?E /~z and p by the value on the axis,
and assuming a constant beta value, E~. 10 reads to.t.""r(;nTI•"' -11-;t(£) randnt(£)= (Gt - St') Ft(t:>.)where F t' Gt' and St' are difined by_ _e_. (>.'A •? Ji,

Ft: - .J.m..c> "'F ( 1t.
6rt = (?E.
?land(.H(ArJ(O)¥ ),V-) I ( ?E,c-1f)I?Zs::. ~I c 7i1£.

(t'f}~'If),{ts-}

The suffix 1 difines the values of the first gap. In this expression the field
gradient along z is replaced by the average value with appropriate effective
length at the exit and the entrance of the drift tubes. Fig.3 shows the phasestable region for a simple APF structure, in which a superperiod consists oftwo gaps in two rf length.
The above arguments and cited papers should be referenced to LASL reports(LA-8287-MS rv LA-8289-MS (1980)). The auther thanks Drs. R, Gluckstern and K.
Crartdall for the discussion.
-o.'3- - - - 0.25

Cf.l

First
superperiodt- - Last
superperiod

stable region

20

Fig. l12....-,. F 45

• ~F'I

150

Fig.2

Stability diagram for£1 = 0 • 3 or 0 • 25 •Stable region for
FODODOFO structure.

~~-,~~~~~--~-,~,-~·~~~~-,,

p.

= -30°

S'0.1

sl~'~

-~~~~~.~~--±oc-'--~';:0...>..~,0180

sl= -60°~LY

1~,!::--'-.........,~

-p

Fig.3

ptlttbt•

0.1~~.~o~~,~o~~-~,.-o~-L~~-~~~~~,.o~.........,,~,o~~-~

Ps2

s2

Stable phase region for-232-S~

0.1.