A METHOD TO CONTROL !:;p/p OF THE DEBUNCHED BEAM AT KEK-PS MAIN RING
Shigeshi NINOMIYA and Hidetoshi NAKAGAWA
National Laboratory for High Energy Physics
Oho-machi, Tsukuba-gun, Ibaraki-ekn, 305, Japanq2

INTRODUCTION

3= cos~o(q- z- tan~o- ~The momentum spread of the debunched beam isvaried by this method which is used in the normal
operation of slow extraction mode at KEK-PS main -ring.
The forced oscillation of quadrupole mode is excited
before debunching, and RF voltage is switched off at aThe momentum
phase of the quadrupole oscillation.
spread of the debunched beam is determined by thisphase.

+ ...

Since the terms in Eqs. (l) containing q 2 and q 3
can be neglected for small amplitude oscillation, Eqs.
(l) describe the motion whose trajectory is given bya circle. Therefore Eqs. (1) are already normalized.
In order to separate the oscillation of the centerof mass (coherent part) and the relative motion (incoherent part), we divide the parameters into twoparts;

A PICTURE OF QUADRUPOLE OSCILLATION
Consider the distribution of protons in l:;p/pThe quadrupole oscillation will not~ f-phase plane.
b~ observed when the bunch is matched with the trajectory of a synchrotron oscillation. If the axis of thephase plane are normalized such that the synchrotron
motion is represented by a circle, the matched bunch
occupies the area enclosed by a circle.
When the occupation of protons in the bunch is aellipse in the normalized phase plane, the ellipse
rotates around the center of the bunch with the frequency of synchrotron oscillation as shown in Fig. l.
Since we can observe the projection of the distribution
onto the ~ f-axis, the oscillation of bunch height (orwidth) is SEserved on the bunch monitor. The frequencyof the observed oscillation is the twice of that of theThis is a picture of thesynchrotron oscillation.
quadrupole oscillation.

q

q + 6qpp + 6p

Then Eqs. (1) can be divided
tions. For coherent part;

into two setsof equa-(2)q- ap+ w1

and for incoherent part;(3)

where terms higher than 6q 2 are neglected. Note thatEqs. (2) are just the same as the phase equations (l),while Eqs. (3) describe the motion whose frequency isslightly different from that of synchrotron oscillation.
In order to describe the quadrupole oscillation,
we use the following transformations.

ffi·m~f $.,, \fPrf
Fig. 1 The rotation of the bunch for one half
period of the synchrotron oscillation.

(4)

We get the following equations taking the differentiations of Eqs. ( 4) with respect to time, andusing Eqs. (3),EXCITATION OF QUADRUPOLE OSCILLATION
In order to excite the_ quadrupole oscillation,"phase-shake method", which was previously used for
artificial blowup of the emittance during acceleration, is employed.
The method given by H.G. Hereward, which gives
only qualitative behaviour of the oscillati~n, is used
for explaining the mechanism of this method 1 •The phase equations expanded to 3rd order in q = ~- ~ 0 are

z = (2b 2 q + 3b 3q 2 )yx(2a + 2bzq + 3b3q 2 )yy- (2a + 2bzq + 3b3q 2 )x

Taking the average for particles, and assuming z0p(l)X- ap + w1p

where pt;E ( - 2~nh )1/2
eVEcos~oS

2~Eanda- 20;(~ = A= canst.)

(6)

2ay

where higher order terms in frequency are neglected
since they are much smaller than 2a. Eqs. (6) describe
the motion with oscillation frequency 2a, and they contain the terms of the external forces. As it is easily
understood from the transformation (4), the quantity~
has a large value when the bunch is high (or narrow inwidth), and vice versa.
The phase-shake is now described by Eqs. (6). In
this case q is given by,- nheVcos~ 0 ) 1 / 2 (synchrotron frequency),bz(5)+ (2bzq + 3b3q 2 )ztan~o1;;-aThe other notations are the same as that of Refs. 1)
and 2). The synchrotron frequency a is positive below
transition, negative above. In Eqs. (1), the following
expansion is used.

q

K sinw tq

where K is a constant depending upon the amplitude ofphase-shake and wq the frequency of the shake.

280

In case of no acceleration where ~0 = 0 or 1r,
therefore b2 = 0, Eqs. (6) can be written as a 2ndorder differential equation;

We assume that the RF voltage is turned off at
the time t 1· The transfer matrix from t1 to t2 (> t 1)
is given by
01 'T =(1V(t2-t1)/p1)(7)

The solution describing the forced oscillation is givenby
3b 3AK 2x Ct){a (cos2w t - cos2at)
4(a 2 - w2 )
qq

where

V = /!,w R = Rlnlw (t,p/p),

R is the machine radi-ous, w the r~volution frequency, and n = 1/y~ - 1/y 2 .

The ellipse

parameters at the time t2

are givenby+ 2 (a 2 - w~)sin2at}

(8)

Y1

The resonance occurs at w
a. If we put w = a + ow
and assume 2owt << 1' Eq. ~8) can be rewritteil as- 43 b3AK 2 t sin 2atx(t)(9)+ (1 - af){V(t 2During acceleration, q and q2 terms can excite
the quadrupole mode. Since the effect from ~ 2 term has
already been shown, the equation containing q term must
be solved;

A half width of the bunch at t2 is given by
q2(ES2/1r)

1/2

A= q1t1 - 2a1VCt2 - t1)/(S1P1)+ (1 + af){V(t2- t1)/(S)P1)J2) 1/2
x + 4a 2 x = 4ab2AK sinw t

Assume that q2 = 2q1, in other words,
t2 is one half of the height at t1.

q

The solution describing the forced oscillation isV(t2- t1)/p1 = a1S1/(1+af) +w
4ab 2AK(sinw t - ~ sin2at)(2a) 2 - w2q2a/{a1S1/(1+af)} 2 + 3Sf/(1+af)q

It also resonates at w

2a, and is written asqx(t)bunch height at

Substituting V,= b2AKt{cos2at - sin2at/(2a)}These results explain why the phase-shake during
acceleration is effective at the frequencies of aand/or 2a.
DESCRIPTION OF THE BUNCH ROTATION
In order to calculate the momentum spread, we use
the ellipse parameters a , S , andy for describing thebunch in phase space. The bunch ellipse is given by

From Eq. (12), we can calculate the value of t,p/p. ,..If
we take the value of full width of the bunch for q1,Eq. (12) gives the full width of t,p/p.
THE EXPERIMENT AND RESULTS

where E is the area of the bunch in phase space, p andq are the same quantities used in Eqs. (1).
If the bunch height is a maximum at t = 0, theellipse is given by
E/1r = (qt/So + Sopfi)(with 0 < So < 1)

The integration of Eqs. (1), with b2 = b3
0 and w1
0, gives the transfer matrix of the particle from t = D
to t 1•R =(cos at 1 ,- sin at 1 )sin at1 ,cos at1

The transfer matrix of the eylipse parameters can becalculated from this matrix5 .
And we can get theellipse parameters at the time t1

The setup of the equipments is shown in Fig. 2.
The RF voltage of the cavity is switched off at 20 msec
after P3 (the end of acceleration) pulse. We can adjust the momentum spread of the coasting beam by changing the delay time of the pulse which triggers thesine-wave generator. The examples of the forced oscillation of the quadrupole mode are shown in Figs. 3 ~ 5.
The variation in momentum spread is plotted in Fig. 6.
The shaking amplitude in RF phase is nearly 20° ofangle. In calculating the momentum spread, we used themeasured values of q1 and tdb at each delay time, andused Eq. (12) with So = 0.5.
The quantity ~1 is thevalue of half 'width measured at nearly 10 % of thebunch height.
CONCLUSION
Though the variation of the momentum spread isonly from ± 0. 4 % to ± 0. 2 %, this is_ a useful method
to regulate the momentum spread of the extracted beam.
When we adjust t,p/p = ± 0;4 %, the RF structures in thespill of the slow extraction are not observed.

(So - 1/So)sin at1 cos at1(Socos 2at1 + sin 2at1/So)

ACKNOWLEDGEMENTS
The equation of S1 givest 1;

a half widthof the bunch at
The authoers wish to express their appreciation
to Assist. Professor S. Hiramatsu for his discussions.
They also express their appreciation to Professors T.
Kamei and M. Kondo for their interest on this work.

(10)
where qo (= (ESo/7r) 112 ) is a half width of the bunch att = 0.

281

REFERENCES
1)
2)
3)
4)
5)

Variable
Delay

H.G. Hereward, Proc. Int. Conf. on High Energy
Accelerators, 236, (1961).
E.D. Courant and H.S. Snyder, Ann. Phys. 3, 1,(1958).
D. Boussard, CERN MPS/SR/Note 70-18.
E.C. Raka, IEEE NS-16, No.3, 182, (1969).
C. Bovet et al., CERN/MPS-SI/Int (1970).

RFOscillator

Sine -WoveGenerator

1 - - - RF Power Amp.

Fig. 2 The setup of the equipments.
Fig. 3 The forced oscillation of the quadrupolemode. The RF voltage is switched off at thephase of a maximum Ap/p.( Delay time= 16msec.)
1st trace; Fast intensity monitor.
2nd trace; Phase difference between thebunch and the RF voltage.
3rd trace; RF voltage.

Fig. 4 Delay time

Fig. 5 Delay time

15 msec.

14 msec.

16P/PlHw [%]0.4

Fig. 6 The variation in momentum
spread vs. delay time.

Ill0.3

•

000
Ill00

•

0.2

•

O.l~-1~6--------1~5--------1·4------~13
Delay Time[ m sec]282