LIMITING
CURRENT
DENSITY
IN A L I N E A R
ION ACCELERATOR
A. V. Z o t o v and V. A. T e p l y a k o v
UDC 621.384.62
The differential equation of the separatrix is found for the initial section of a linear ion accelerator in which the diameter of the group of particles is much greater than its length; the solution of the equation is obtained for three forms of stationary charge-density distribution function in phase space. The solution enables the limiting or saturation current density in the accelerator to be determined. A knowledge of the limitations imposed on the c u r r e n t p a s s e d by an a c c e l e r a t o r either in the longitudinal or t r a n s v e r s e direction is not only valuable in selecting the acceleration conditions c o r r e c t l y but also in deciding the type of focusing to be used. In this paper we shall estimate the limiting c u r r e n t density associated with the stability of longitudinal motion. The limiting c u r r e n t density determined by the capabilities of the focusing channel was analyzed in [1, 2]. The p a r t i c l e s of the a c c e l e r a t e d group are acted upon by the phasing f o r c e of the hf field, equal to sin ~p~~ 2~Az vU ~-~ 9 ~, where N~ok -- E , , , is the amplitude of the a c c e l e r a t i n g field, e , ; U is the m a x i m u m energy
evU
increment of the particle in the gap, N~o~ is the length of the a c c e l e r a t i n g period, ~c is the synchronous phase, and Az is the distance between any given particle and the synchronous particle. The Coulomb force of longitudinal repulsion between the p a r t i c l e s in the group is given by the expression V E = I'~ ~ - . ~ / ~ a n d ; equals ~, _el / / / ,lTTL,~~_z
where j is the c u r r e n t density of particles at a given
point, = ~20a, E is the Coulomb field of the group, and kZ is the f o r m factor of the group. F o r the f o r m factor of a u n i f o r m l y charged ellipsoid of revolution we m a y write the simple f o r m u l a R
kz ~
I
-- e
--0,4 - ~zm,
which gives almost exact values over the whole range of ellipsoid-semiaxis ratios R / A Z m . (The constant 0.4 is derived from the condition kz= 1/3 at R / A Z m-~ i). For a stationary charge distribution, the cur ~rent density in the group does not exceed
9
2nx'U
sin ~0c
2nEro sin qL
(i)
At j = Jer the phasing and repulsive f o r c e s are equal. This estimate, however, is too c o a r s e . A m o r e rigorous estimate of the limiting c u r r e n t is obtained by analyzing the equations of motion on the a s s u m p tion that the group is inthe form o f a u n i f o r m l y - c h a r g e d ellipsoid of revolution [3, 4]. It is shown in [5] that this approximation is exact: it is possible to have stationary distribution functions (in phase space) such that uniform charge density is maintained over the whole of an ellipsoidal group. Strictly speaking, however, this proof is only valid to a linear approximation. The limitation on the c u r r e n t (with r e s p e c t to longitudinal motion) is mainly due to nonlinear effects. A m o r e rigorous " s e l f - c o n s i s t e n t " solution is given in [6]. The assumption made by the authors of [6] that the group is in the form of a cylinder, n e v e r theless deprives this of its generality. It m a y be shown that the solution obtained in [4] on the assumption of a group in the f o r m of a uniformly charged ellipsoid is also as s t r i c t in the nonlinear approximation as is the solution of [6]. The difference is that these solutions are obtained for different distribution functions. It should be noted that only methods of s o l u t i o n - not n u m e r i c a l calculations - are considered in [3, 4, 6]. T r a n s l a t e d f r o m Atomnaya t~nergiya, Vol. 21, No. 5, pp.356-360, November, 1966. Original article submitted May 23, 1966.
1034
i
/
0,08
I .
../
i-I
.
.
.
]
ji -I
0,0~
0.02
0.1
0
0.2
0.3
0.4
0.5
o.g
O.,8
0.7
~e
Fig. I. Limiting current density as a function of the spacecharge parameter for three types of distribution functions: A; ....... B; ......... C.
//" 0.14
/
0.f2
/
"~
I :~ ~ " ~
"~'~
a.lg
OO8
/y /
0.O5 O.O4 0.O2 -&8
/ -a.7
/
51 . . . .
f
-o.5
iS," -0.5 -0.'~
-a3
-0,2
J
-0.1
\
0
,
#]
,11 a1
0.2
0.3 ~-~o
F i g . 2. S e p a r a t r i c e s f o r v a r i o u s v a l u e s of s p a c e - c h a r g e p a r a m e t e r ~0 and t h r e e f o r m s of d i s t r i b u t i o n s A; . . . . . . . B; . . . . . . . . .
Co
W e w e r e i n t e r e s t e d in an a c c e l e r a t o r in w h i c h the v o l t a g e a p p l i e d to the g a p s w a s k e p t c o n s t a n t (not the a v e r a g e field). In a d d i t i o n to t h i s , w e m a d e the f o l l o w i n g a s s u m p t i o n s : 1) the d i a m e t e r of the g r o u p w a s m u c h g r e a t e r than i t s length; 2) the v e l o c i t y of the i o n s w a s s m a l l (tic << 1), i t a l s o v a r i e d l i t t l e o v e r the gap, i n w h i c h i t w a s i n d e p e n d e n t of r a d i u s , and o v e r the p e r i o d of p h a s e o s c i l l a t i o n ; 3) a l l the p a r t i c l e s w e r e a c c o m m o d a t e d i n t h e g r o u p ; 4) t h e s c r e e n i n g e f f e c t of t h e d r i f t w a s not t a k e n into a c c o u n t . Except for the first, these are the conventionalassumptions. The f i r s t a s s u m p t i o n e n a b l e s u s to m a k e a o n e - d i m e n s i o n a l a p p r o x i m a t i o n (with r e s p e c t to z): A l l the p a r t i c l e s l i e in the p l a n e of the l a y e r , and the a c c e l e r a t o r a x i s (z) i s p e r p e n d i c u l a r to t h i s . The Hamittonian
&e (,, w.
n)--~" ~-e ~ N _+_ vU
[1112cos(po- ~i. (,~.~+ , ) 7j - -T-'vf~~ ~ E (,, ,,) d , ,J
(2)
1035
d e s c r i b e s a l m o s t c o n s e r v a t i v e m o t i o n . H e r e n = t / T i s the n u m b e r of t h e g a p , ~ = m c 2 / e = 9 4 0 m V f o r a p r o t o n , 2~ ~ EY Az ts the p h a s e d i s t a n c e of the p a r t i c l e f r o m the s y n c h r o n o u s p a r t i c l e , r = ~ - - me; tic i s the v e l o c i t y of the s y n c h r o n o u s p a r t i c l e , and ~ i s the p h a s e of the s y n c h r o n o u s p a r t i c l e w i t h o u t t a k i n g a c c o u n t of s p a c e c h a r g e . If the particle distribution is described by the function/=: i ( ~ g r - - ~), satisfying the kinetic equation df/ch] = 0, then it is stationary, since the Hamiltonian may be considered as an integral of the motion (oY/Ygr -- const). T h e c h a r g e d e n s i t y and c u r r e n t d e n s i t y in the g r o u p r e s p e c t i v e l y e q u a l :
p (*)i
i (~a
I 4 ! ( d @ - ~ ) dw ~ 4op (%
- ~)u,<,; i (,) =
The b o u n d a r i e s of the g r o u p a r e d e t e r m i n e d by the p r o j e c t i o n of the b o u n d a r y p h a s e c u r v e
~3)
~ (% w).= dd gr on the r a x i s .
The s t a t e s of a l l the p a r t i c l e s in the g r o u p l i e i n s i d e c u r v e (3).
In o r d e r to find the d i f f e r e n t i a l e q u a t i o n of the l i m i t i n g p h a s e t r a i e c t o r y , l e t us t a k e the s e c o n d 0.
d e r i v a t i v e with r e s p e c t to r f o r both s i d e s of e x p r e s s i o n (3), t a k i n g ~ A s a r e s u l t of t h i s we o b t a i n s/" (ID ! s~n (%-; , ) sirt ~c
H e r e g - v~;~
i (y) _ o. /cr
(4)
sinm ' , the c u r r e n t d e n s i t y i s c o n s i d e r e d a function of y, s i n c e we a r e only c o n s i d e r i n g
p o i n t s on c u r v e (3), w h i c h d e f i n e s a r e l a t i o n s h i p b e t w e e n y a n d r k z = 1. The p a r t i c l e w i t h r
J c r i s g i v e n by e x p r e s s i o n (1), w h e r e
0 w i l l be s y n c h r o n o u s and the bounding p h a s e t r a j e c t o r y w i l l be the s e p a r a t r i x i f
v' (o) =o,
.q"(tO< o; ]
The point ~ c in the phase plane
(5)
~;"(~<,) > o. ,/
y (,~,) = .,' (q,~) = o,
~, w is an isolated singularity of the "saddle"
type.
Let us find the solution of (4) satisfying condition (5) for three types of distribution function a
A. f (,, ~) = V
~
;
~ ~ fo~
B. [(%w)=bSt'(aC/dgr--~162 { Ofor
d7~gr dd; Q%'dgr , ~a,~;
c. f(,, w ) - - c l / ~ - ~ ' , w h e r e a, b, and c a r e n o r m a l i z i n g c o n s t a n t s . w,
Case* e = ]/~
f
'
0
gr
i . e . , the c h a r g e d e n s i t y in the g r o u p i s c o n s t a n t ,
o oV
L e t u s w r i t e the e q u a t i o n of t h e bounding t r a j e c t o r y
y,,(~)q sin(~c+~,) sin (Pc
~ 0 : 0 , S0= i(0) / cr '
w h e r e j(0) i s t h e c u r r e n t d e n s i t y in the c e n t e r of the g r o u p .
1036
The s o l u t i o n s a t i s f y i n g b o u n d a r y c o n d i t i o n s (5) has the f o r m g(~))= sin@r sin (pc
{-~o-~--r
bC.
where sin (Tc-~ ~e) sin %
C = ~ ctg % We find the v a l u e ofr
~o ~
.
f r o m the e q u a t i o n -- ~osin % $r = cos (% q- $~) -- cos %.
A v e r a g i n g o v e r the a c c e l e r a t i n g p e r i o d , the c u r r e n t d e n s i t y i s
i
9 av
=~
i
~h ~
3 .
(6)
.I ](qo)&p ~ - ~ - ] c r t g % ~ o ( l - - ~ o )
~c F o r j(0) = 0.5 J c r t h e m e a n c u r r e n t d e n s i t y i s m a x i m u m : .max 3 /av ~ ] c r
.
3 Em sin2q9c tg(pc=-~-. ~cos(pe
(7)
F o r a u n i f o r m l y c h a r g e d e l l i p s o i d a l group, A . D . V l a s o v o b t a i n e d a f o r m u l a [4, Eq. (2.53)] f o r the m e a n beam current ~.3R~Em sin qPcq% rS~, (I--SM) I
y~,kz
which in the case of an "infinitely" flattened group
O. i 8
(k z ~ i) gives the limiting current density in the form
/ max s -av
sin ~c ~,
T h i s e x p r e s s i o n a g r e e s s a t i s f a c t o r i l y with f o r m u l a (7). C a s e B.
The c h a r g e i s d i s t r i b u t e d u n i f o r m l y i n s i d e the s e p a r a t r i x , a s in [6,7].
The c h a r g e d e n s i t y
I~gr P=2b
~ dw=2bl/-~gr.
The e q u a t i o n of the b o u n d i n g t r a j e c t o r y is
y. (~) [ sin (~c4-a~)
If g~
sin(pc
To i/ ~ :
0
"
The b r a n c h of the s o l u t i o n r e q u i r e d f o r o u r p u r p o s e s is g i v e n by the s e r i e s oo
y ( , ) = Y, a,, n! ' n~O
where
ao =g(O); a f = O ; % = ~ o - - 1 ; a a = - - c t g % ;
a4 -= t -- 2--7~ (1-- ~o);'%
--2-7~-)ctg%;etc.
In o r d e r to find the s o l u t i o n , l e t u s e x p a n d sin(gOc+~) i n s e r i e s and r i d o u r s e l v e s of the r a d i c a l . The c o n s t a n t y(0) is g i v e n by the c o n d i t i o n r = ~bc -- a d o u b l e root. If i n the e x p a n s i o n f o r y(r we c o n f i n e o u r s e l v e s to the t e r m of the t h i r d d e g r e e , we a g a i n o b t a i n e x p r e s s i o n s (6) and (7) for the c u r r e n t d e n s i t y . A m o r e a c c u r a t e s o l u t i o n of the e q u a t i o n for C a s e B w a s o b t a i n e d by i n t e g r a t i n g the e q u a t i o n on a c o m p u t e r , the s e r i e s of the r e s u l t a n t s o l u t i o n c o n v e r g i n g v e r y s l o w l y f o r ~0 > 0.5. C a s e C.
_2 c_] / ~
T
t/ W'agr--W~ d w =
[5~
~-V~
W~gr'"
1037
The equation of the bounding trajectory, ,,(,)~ ~n (~r162 sin(pc
b e c o m e s linear.
~ ,s-Y~-
The solution is obtained f r o m the e x p r e s s i o n V (*) = C, Sh ' " - ' /
~o
,
y (o)
~,.I ~-(6 ~ i ~ ) -~ ~.C+-~-(o)
w h e r e C 1 and X a r e constants d e t e r m i n e d by condition (5). cendental equation.
sin ((p~ + , )
sJn~
In o r d e r to find y(0) we have to solve a t r a n s -
F i g u r e 1 shows the m e a n c u r r e n t density J a v / Jcr as a function of the c u r r e n t density in the c e n t e r of the g r o u p j(0)/jc r f o r all t h r e e c a s e s c o n s i d e r e d and t h r e e values of s y n c h r o n o u s p h a s e ~Pc, equal to 20, 30, and 40 ~. F i g u r e 2 shows the s e p a r a t r i c e s f o r ec = 20~ In o r d e r to follow the v a r i a t i o n in the s y n c h r o n o u s p h a s e r e s u l t i n g f r o m Coulomb f o r c e s , the s e p a r a t r i c e s a r e plotted as functions of ~-~0, w h e r e ~00 is the s y n c h r o n o u s p h a s e at z e r o c u r r e n t . The equation of the s e p a r a t r i x is found f r o m solution (4) and c o n d i tion (5). F o r p ( y ) = 0 ~1= V'gsinq~=~
r~U
" 2t~~
-- V sinq)o-~ocosq)o +sinq~-qJcos~o.
F i g u r e 2 p r e s e n t s t h r e e s e r i e s of c u r v e s : I) the s e p a r a t r i x for z e r o c u r r e n t ; II) the s e p a r a t r i c e s f o r t0 = j ( 0 ) / J c r = 0.2; Ill) t h e s e p a r a t r i c e s
m a x
c o r r e s p o n d i n g to m a x i m u m a v e r a g e c u r r e n t d e n s i t y j a v 9m a x
.
m a x - .
f o r the t h r e e d i s t r i b u t i o n s : A) t0 = 0.5, J a y / J c r = 0.042; B) ~0 = 0.7, j a v 9m a x
.
/J c r
/ J c r = 0.041; C) ~0 = 0.8,
,.
3 av / j c r = 0.039. Thus the a d m i s s i b l e a v e r a g e c u r r e n t density is a l m o s t independent of the f o r m of the d i s t r i b u t i o n function used. This indicates that the f o r m u l a f o r the a v e r a g e c u r r e n t density in an a c c e l e r a t o r with s h o r t g r o u p s of l a r g e d i a m e t e r is valid f o r p r a c t i c a l l y all d i s t r i b u t i o n functions. It should be noted that the m a x i m u m a v e r a g e c u r r e n t density is a c h i e v e d f o r a c e r t a i n o p t i m u m peak c u r r e n t density, depending on the f o r m of the d i s t r i b u t i o n function. LITERATURE I.
2. 3. 4. 5. 6. 7.
1038
CITED
I. K a p c h i n s k y and V. V l a d i m i r s k i j , P r o c . Internat. Conf. H. E n e r g y A c c e l e r a t o r s and I n s t r u m e n t a tion, CERN, p. 274 (1959). I. M. Kapchinskii, A t o m n a y a t~nergiya, 13~ 235 (1962). A . I . A k h i e z e r et al., T h e o r y and C a l c u l a t i o n s of L i n e a r A c c e l e r a t o r s [in R u s s i a n ] , Moscow, A t o m i z d a t (1965). A . D . Vlasov, T h e o r y of L i n e a r A c c e l e r a t o r s [in R u s s i a n ] , Moscow, A t o m i z d a t (1965). B.I. B o n d a r e v and A . D . Vlasov, A t o m n a y a l~nergiya, 1_1_1,423 (1965). I.M. Kapchinskii and A. S. K r o n r o d , in the Book: T r a n s a c t i o n s of the I n t e r n a t i o n a l C o n f e r e n c e on A c c e l e r a t o r s , Dubna, (1963)[in R u s s i a n ] , Moscow, Atomizdat(1964). C. Nilsen and A. S e s s l e r , Rev. Scient. I n s t r u m . , 3_O0,80 (1960).