Third harmonic simulated buncher

NUCLEAR

INSTRUMENTS

AND

METHODS

THIRD

61 (1968) 221-225;

HARMONIC

0

SIMULATED

C. GOLDSTEIN

NORTH-HOLLAND

PUBLISHING

co.

BUNCHER

and A. LAISNE

Institut de Physique Nu&aire

d’Orsay, Orsay, France

Received 28 December 1967 Without any bunching, the theoretical efficiency of the linac phase acceptance q=3jvsj/2n is 16.6% (P)~= -20”). With a classical buncher this efficiency increases up to 65%. We propose a high efficiency buncher using the fundamental frequency and its second

harmonic to simulate the third one. The theoretical efficiency of this system is 88 %. Practically, it is possible to achieve a gain of 5 with a third harmonic simulated buncher instead of 3 with a classical buncher.

1. General

buncher would require a saw toothed be applied at the linac’s frequency.

The study of a high efficiency buncher was required for the injector of the CEVILlm3) because it was designed to accelerate multicharged heavy ions for which the efficiency of the sources are quite poor. Moreover, the use of grid focusing introduces a loss of intensity by reducing the phase acceptance and by capture. The ions are continuously extracted from the source to the static with a velocity u0 which corresponds potential V,. Between the source and the linac the available length is AB = L, where B is the middle of the first accelerating gap. Between A et B, we can put one or more gaps where the electric fields are supposed to be uniform by using grids at the input and the output of the gaps. These gaps constitute a buncher of which purpose consists in putting the greatest number of particules into the phase space accepted by the linac. This space limited by the separatrix, is given by the following relation4) : (A W/W’+(4/~)(4m)

The velocity at the output 0,{1

+(v,/VO)(dn))3

of the buncher =

%tl

with x1 = T/;/V,,. Let us assume that Z, = fv,T by the synchronous ion, during AB = L = K,Z, = fKlv,T.

is the path length, run the half rf period and

= 0,

where cp is the phase of an ion with a relative energy deviation A W/ W. For our accelerator this relation is plotted (fig. 1). The parameters are such as: e/m

=

0.1 x LO*;

E, = 1.75 MV/m ;

u,, = 1.675 x lo6 m/s;

f=

25 MC;

T = 0.85; cpS= -20”.

2. Ideal buncher The ideal buncher would increase the velocity of the ions at the output of the source, proportionally to their phase cp, the reference being the synchronous ion. By definition, this one will not have any velocity modulation in the buncher and will arrive in the linac with the synchronous phase cpS= -20”. This property of the

Fig. 1. Phase space of the linac.

221

would be:

+($xldn)>~

%U(M)l~

.(sincp-cpcosq,-qo,coscp,+sincp,)

V(cp) to

--nscpzz.

V(cp) = l/l(Pk

uA =

voltage

222

C.GOLDSTElN

AND

A.LAlSNE

At B, the phase shift of a given ion is: A% = cP+(Wa,)-W/a,)

- cp(l -$K,x,).

If in this relation +Krxr = 1; we get dqo, E 0 independant of cp. The phase bunching is complete. The relative energy error of an ion is: A W/ W = (vi -u;)/u;

= xl((p/n).

The longitudinal phase diagram A W/ W =~(Acp,) is reduced to a vertical straight line, which must be contained inside of the separatrix. This condition gives K, 2 20. On the other hand K, has an upper limit given by the stability of the injection voltage because we have: cp+A(Acp) = wL/(u,+Ao,)-(wL/o,)(l-Av,/o,) and A(Acp) = -~7~K~Al/~/l’~. As far as we can have AV,,/ V, = + 5 x 10d4 and A(Aq) = k 2” then K, 5 44. This ideal buncher would work with 20 5 K, 2 44. Such a buncher cannot be really considered in the present state of the technology, because it is extremely difficult to generate a high enough saw-tooth voltage at 25 MC. However, we must take into account that a practical buncher must give a relative energy error in respect with phase, which is as close as possible from the saw-tooth.

Fig. 3. Phase space diagram for the first harmonic buncher.

4. Harmonic buncher 4.1. THESECONDHARMONIC

BUNCHEROFTHEARGONNE NATIONAL LABORATORY"~)

3. First harmonic buncher system The roughest approximation of the saw-tooth is given by a sinusoidal voltage. The easiest is to use an electrode on a length 2, = &,T, with an applied voltage V= V, sin of, in respect with two grounded electrodes (fig. 2). This gives a whole system of two

g&T! Fig. 2. First harmonic buncher.

gaps A and A’. The bunching is maximum at B such as A’B = Z’ = +K,v,T. It is possible to show, using a first order calculation that:

AW/W((p,)

= x,[sincp,+sin(cp,+$rx,

sincp,)],

where x1 = V, / V, and p = frtx,(2K, + 1). In this case, the maximum efficiency increases from q = 3((p,)/2~ = 16,6:/i, to 659; for x1 = 0.012 and K, = 44 (fig. 3).

In order to improve the approximation of the tooth, we can imagine a buncher made of several as close as possible with applied voltages which respond to the Fourier analysis of the saw-tooth. is to say: V = C ((-l)“+r/n}sin(ncp).

sawgaps corThat

On this basis, a study has been done at Argonne. The results for the theoretical efficiencies, with cps= -26”, are as it follows: q = 65o/0 fundamental only; q = Sly/, fundamental and second harmonic; q = 86% fundamental, second and third harmonics; q = 89% fundamental, second, third and fourth harmonics. Fig. 4 shows the curves A W/ W =f(q) for these four cases. The practical development has been limited to 2 frequencies and is now being done. 4.2. THIRD HARMONIC SIMULATED BUNCHER') A sinusoidal

voltage

V, sin (wt) is applied in a gap A.

THIRD

HARMONIC

SIMULATED

223

BUNCHER

-‘4 fondonlentol hormonique

+

fondamental + normonique 2,3.4

2et 3

(d)

(Cl

Fig. 4. .I W/ W=f(g~) for 1, 12, 123, 1234 harmonics.

At the output of this gap the phase diagram is given by the curve “a” of fig. 5. After a given drift length AB, the phase diagram is changed and gives the curve “b”. At this point B, if we put an other gap on which we apply V,sin(2wt), curve “c”, the resulting curve “d” presents two maxima; in other words, we obtain a combination of the fundamental, second and third harmonics. The first accelerating gap of the linac is at point C. We can write: x2

=

BC = $K,u,T,

V,/~o,

After simplifying,

the equations

c( = K,nx,. are the following:

ug = uO[l ++x,sincp,++x,sin(2cp,-cxsincp,)], AW/W

(AW/W)/.u,

anal-

= Y,

we have to resolve the following

system :

X = qo,-2asinqo,, I; = sincp,+flsin(2X). * The theoretical conversion efficiency of a klystron in which the bunching is given by two cavities at the same frequency separated by a drift tube is Sl.5o/O. We can notice that this two cavities simulate the second harmonic. By replacing the second cavity by a second harmonic cavity which will simulate the third harmonic, the optimum efficiency is 85.7%.

= ~,sin~,,-x,sin(2~~-c~sin~~).

We can also write the last equation AW/W

for the three first terms of the saw-tooth Fourier ysis. Thus we have simulated a third harmonic*. Now, if we put Aq, = X,

on the form:

= x1 sincp,+x,[sin(2q,)cos(ccsincp,)-cos(2~,)sin(c~sin~,J].

The development of cos (U sin (P,,,)and sin (a sin (Pi) in Bessel functions and the substitution of sin (ncp*) cos (pep,) by sums of sinus gives: AWjW

= [xl +x,{J,(cc)-J,(x)}]sinq,+ +

~2CJ0(tl)-Jq(‘~)lsin(2~,)-

-xZ[J,(cc)+J,(cl)Jsin(3q,)+.... Then we can say that, with c( N 1.25 and /I = xz/xl 1: 0.55, the coefficients are approximately the same as

Fig. 5. iJ W/ W=f(p)

for the third harmonic simulated buncher.

224

C. GOLDSTEIN

The problem consists in finding CIand p so that the function Y =f(X) lays as long as possible between two parallel lines, with a slope IZ, their distance at X is dX nearly equal to 3 I(p4I . This is the concept used by Tchebychev in function analysis. We could use the Tchebychev analysis, but it seemed more convenient to remark that the slope at the origin of the function Y =f(X), must be approximately the same as the slope of the curve obtained with the three first terms of the Fourier analysis of the sawtooth. This slope is equal to 1. So we obtain a relation between c1and /I. Effectively we have the slope: m = (dY/dX),,,O

= [{cos~,J(l

-2cccos40A)}+

+2@os(2X)],,=, This gives as possible

= {l/(1-2c()}+2/I.

ranges for m and /I:

Practically, at a distance K,Z, from the first harmonic buncher, we put a second drift length +Z, which determines two gaps B and B’. The bunching takes place at the point C with B’C = )K2v,,T (fig. 6). We put x2 = V,/ V, and obtain the following equations: a’ = v,,/v,,

a = v,iv,, (PA’= (PB’ =

VA 4%

+

4’B

day

&’

+ +/b,

a’ = (a* -x1 sin qA,)+,

=

40A*+K1dar, 2%,‘,

4% vpc

5. Experimental

results

We have built a prototype of such a buncher. tested on a 10 drift tubes linac. The characteristics were as it follows: e/m = i’;rx lO’(N+), V, = 85 kV,

f=

E0 = 1.2 MV/m, 21 MC,

=

%,‘+

=

‘ps = - 20”.

With these values, the acceptance diagram is the same as the linac’s one. Then we have similar conditions of work with a reduction of scale Z, = 2.5 cm instead of 3.35 cm. The operating method was the following: Tn a first step, we adjust the fundamental phase of the first harmonic buncher in connection with the linac rf with K, = 44, x1 = 0.012 (V, N 900 V). Between the bunched beam and the unbunched beam, we find a gain of 3 after this adjustment. In the second step, with x1 = 0.02 (V, N 1500 V) and x* = -0,014 (V, N 1050 V), we optimize the phase of the second harmonic. We find a gain of 5, which is in good agreement, with the theory. Nevertheless, we have to point out the following restrictions : L!!@ w lo-*

h-%>

K&b’,

b = (a”+x2sin~,)~~,

b’ = (b*-xX2sin+,,)* and Acp = ‘pc -‘pcs with ‘pcs = R( 1.5 + K, + K,). The relative energy error is given by: AW/W(cp,)

= xI(sin~~-sin~A?)+x,(sin&-sin&).

We calculate A WI Wand Aq with x1, x2, K, and K2 as parameters on the UNIVAC 1007 computer by a systematic exploration in the previously defined range. The upper limit given by the injection voltage stability, (K, + K2 + 1.5 s 44) fixes the choice of x1. In our case x1 = 0.02, the highest efficiency is given by the following values: x1 = 0.02, x2 = -0.014, K, = 11, Kz = 3 1S. We obtain in this case a theoretical efficiency higher than 88% (fig. 7).

!fl ?

_?I? aY1 ‘7

C I

I m:V,:;h:

Fig. 6. Third harmonic simulated buncher.

It was

T= 0.85,

b’ = U&I,,

b = dv,,

=

AND A. LAISNE

Fig. 7. Phase space diagram for the THS buncher.

THIRD

HARMONIC

a. The beam intensity was low, several tens of PA; b. We do not know exactly the synchronous phase of the 10 drift tubes linac. We think it is possible to obtain a gain between 4 and 5 for the 1 MeV/nucleon linac*. These values will be verified within six months, since the buncher should be in operation at this time. * This buncher

can be used for the axial injection in a cyclotron if the injection velocity is large in respect of the natural dispersion of the ion source. It is certainly possible to take into account the effect of debunching between the mirror and the first accelerating gap of the cyclotron, by adjusting properly the amplitude of the three harmonics.

SIMULATED

BUNCHER

225

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IP/N