KEK-76-8

A BUNCHED PROTON BEAM IN TRISTAN

Koji TAKIKAWA

OCTOBER 1976

NATIONAL LABORATORY FOR HIGH ENERGY PHYSICS

OHO-MACHI, TSUKUBA-GUN IBARAKI, JAPAN

KEK Reports are available from Library National Laboratory for High Energy Physics Oho-machi, Tsukuba-gun Ibaraki-ken, 300-32 JAPAN

Phone: 02986-4-1171 Telex: 3652-534 (Domestic)

(0)3652-534 ( In te rna t iona l ) Cable: KEKOHO

A BUNCHED PROTON BEAM IN TRISTAN

Koji TAKIKAWA National Laboratory for High Energy Physics Oho-machi, Tsukuba-gun, Ibaraki-ken, 300-32, Japan

Abstract

A bunched proton beam scheme In TRISTAN phase I is discussed, where one of the TRISTAN rings is exclusively used for protons and the other ring for electrons (positrons). A 70 GeV proton beam of 5 x 10 protons in a bunch will give the luminosity of about 3 x 1 0 ^ cm-2.sec""l by a head-on collision against a 17 GeV electron beam of 200 mA. A feasibility consideration on the bunched proton beam scheme shows that a tight tolerance is necessary for a manipulation of proton bunch and that a longitudinal instability of proton bunch may be a stumbling block at the beam intensity of 5 x 10 protons .in a bunch, forcing us to operate at a lower intensity.

1. Introduction

During the 1976 Hakone Summer Study on e-p collision physics in TRISTAN phase I, a considerable interest was shown to a bunched proton beam operation. The length if proton bunch should be about 1 nr or less if one wants to define the source points of collision unambiguously with detectors of moderate sice. Now the bunch length of 12 GeV protons from Main Ring is about 4.2 m. So, if it is possible to injeet the Ma-tsi Ring bunches successively into one of the TRISTAN rings (hereafter called as Ring 1) and reduce the bunch length to 1 m at 70 GeV, Ring 1 can be design­ed exclusively for protons and the other ring, Sing 2, for electrons and/or positrons. This is an attractive feature in contrast with the case of coasting proton beam* where Ring 1 is used for a dual purpose; proton acceleration and electron (positron) acceleration and storage.

The TRISTAN has an average radius of R = 324 a = 6 times the Main Ring radius and the Main Ring pulse consists of 9 bunches. A natural choice of number of bunches in TRISTAN is 54, with bunch separation of 37.7 m. Then, electrons and protons can be collided head-on. In this report we shall give som* preliminary considerations on such a bunched proton beam operation; luminosity, scheme to make a short proton bunch, tolerances and stability of bunch.

Besides the above outlined scheme there are other alternatives for bunched proton beam operation. For example, one is to bunch the 70 GeV coasting beam in Ring 2 which is formed by acceleration in Sing 1 and subsequent transfer ;ind rf stacking in Ring 2. In this case Ring 1 is » used for electrons -ind protons. This scheme is a natural (-intension of the coasting beam operation. The other is to extract the Main Ring pulses by "shaving", inject them into Ring 1 by the rf-stacking method and accelerate to 70 GeV. Ring 1 is used exclusively for protons and Ring 2 for electrons. In the two cases mentioned above, the 70 GeV proton beam consists of numerous number of bunches with bunch separation of the same order of bunch length, thus resembling the coasting beam in a sense. In this report we shall not consider these cases, leaving them to discussions at a later time when the parameters of the coasting beam operation are well settled.

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2. Luminosity

Neglecting the effects of beam size variation in the collision region, the luminosity for head-on collision is given by

f BH H , JLJL _ J ^ B = = ^ ( 1 ) 2ir / *2 , *2 Fi /a + a la *2 *2

ye yp

where N is the number of particles in a bunch, f the revolution frequency, B the number of bunches (with fB denoting the number of collisions per unit-time) and a* the rms beam size at the center of collision region. The suffix x(y) refers to the horizontal (vertical) plane and the suffix e(p) the electron (proton).

Again neglecting the beam size variation in the collision region, 2)

the beam-beam tune-shift i s given by * r B N

A v = «* xe £ { 2 )

xe 2irv * . * * . w xe 2irye a (o + a ) xp xp yp

r B _ N Av„. = - | J f i - x g_

y e 2 ^ e a* (a* + a * ) yp xp yp'

Aw. r 6 H

P xp ?_ xp 2irv * , * * » r ' p a (a + a ) xe xe ye

r B - H AvT O=-i-?P- e

yp 2nY * , * * » z T P .a (a + 5 ) ye xe ye'

where r (r ) i s the c lass ica l electron (proton) radius, y the usual r e l a t i v i s t i c factor and B* the 8-function at the center of co l l i s i on region.

. For a dispersion-free low-B configuration of proton ring with typical values of 8 m 1.5 a and B = 1.0 m, we have a _ = 0.67 mm and 0__ = Hxp yp ^P yp 0.32 mm. Here the 2-a normalized emittances of proton beam are assumed to be horizontally 90 x 10 ir m-rad and vertically 30 x 10 n m-rad.

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Then the electron tune-shift at 17 GeV is given by

A v x e = 0.020 x I O " 1 2 N p g ^ (3) A v y e - 0.043 x 10~ 1 2 N p Py e

If we take the electron tune-shift limit to be 0. 06?> the above expression gives the upper limits of electron 0 -values. For a typical value of N = 5 x 1 0 1 1 , we get

p£ e < 6.0 m (4) gy e < 2.9 m

The 500 MeV Booster has already achieved the intensity of H„ » 5 * 1 0 1 1

per bunch. We believe that by the time the TRISTAH project starts the 12 GeV Main Ring can deliver at least 5 x 10-"- protons per bunch.

The lower limits of electron P -values are given from a consideration of the proton tune-shift. Too small values of (5 are, of coarse, discarded from the lattice considerations. Following H. Sands*), the electron beam size is related to the ^-function by

a2

e x i + K 80 ' <5>

e„ " i + K go

W h e r e C RY 2 fi

= -9 =• = 0.133 x io"° 80 J * P v x

-13 Here C- = 3.84 x 10 m, R = 324 m, p = 130 m, J x = horizontal damping partition number = 1, v x = 20 and K = coefficient of horizontal-vertical coupling in betatron oscillations. In a dispersion-free insertion we have o" = o xg = beam width due to betatron oscillation. Usually the electron beam height is smaller than the beam width and sc the proton vertical tune-shift is larger than the horizontal one. Then, from the condition Av™, < Av„ = tolerable proton tune-shift, we get

pxe pye - 2 Y p g0 % ^

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* it it where the term a + a in Eq.(2) is replaced hy a . The tolerable

xe ye ™ tune-shift Av p for bunched proton beam is not known. If we put Av_ " 0.005 somewhat arbitrarily, we get

•J? B* > 143 (7) xe ye •

where the following parameter values are assumed; y = 75.6 (70 GeV), 12 K = 0.1 and BN = 8.5 x 10 corresponding to circulating current of

200 tnA. In order to avoid a proton bunch colliding an electron bunch aore than once in a collision region, the bunch separation should be larger than, say, 30 m for the case of head-on collision. By injecting successive 6 Main Ring pulses into proton ring, we have B = 6 x 9 » 54, with a bunch separation of 37.7 m satisfying the above criterion. For B = 54, Eq. (7) gives

I 6* 3* > 2.65 (8) xe ye -

* Eqs. (4) and (8) give the constraints that the electron 6 -values

should obey. Let us take B* e =3.0 mand f5ye = 2.0 m. Then we have O = 0.60 mm and o" =0.16 mm. For the above described parameter values, the luminosity and the beam-beam tune-shift are given by

L = 3.1 x l o 3 1 cm~ 2-sec _ 1 (9)

Av = 0.030 Av = 0.043 xe ye

Av^ = 0.0017 AVyp = 0.0042

3. Proton short bunch scheme

We assume the following parameter values for the 12 GeV proton bunch from the Main Ring.

Ap/p = ±1 x 10~ 3 or Ap/moc = ±13.75 x 10 3

A = bunch area = irAij) Ap/nigc = 1.51 x 10" £ B = full length of bunch = (A<j>/ir) (2irR/h) = 4.19 m

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The mean radius of the Main Ring is R = 54 in .rod the harmonic number is h = 9.

The TRISTAN has R = 6 * 54 = 324 m. Sis successive Main Ring pulses are Injected into TRISTAN in about IS sec and captured by a rf system operating in h = 5 x 54 = 270 with rf frequency f r f = 39.6 MHz. The number of proton bunches is 54. We choose a high harmonic number h = 270 instead of h = 54 because of its larger spread in synchrotron oscil­lation frequency which is effective to Landau-damp the longitudinal in­stabilities (see §5). The rf voltage required to match the proton bunch having Ad) = ±5 x 20° = ±100°, Ap/mgC = ±13.75 x 1 0 - 3 and A = 7.55 x 1 0 - 2

is found from the relationship"

-^ = 7he7 T» B, *!> I =L= (10)

where

E = 12.94 GeV Y = 13.8 corresponding to 12 GeV

|n| = |Y~ 2 - Y~ 2| = 2.76 x io~ 3 for yfc = 20

Y(<J>S, $j) = normalized half height of bunch = 1.08 for d)s = 0° and ^ = 100° .

From Eq.(10), we get V = 25.8 kV for the rf voltage necessary to capture the bunch.

Acceleration from 12 GeV to 70 GeV will be made in- about 10 sec. The necessary acceleration voltage is given by V sin $ s = 39.4 kV. By choosing d> = 30° we get V = 78.8 kV. The adiabatic damping of phase

6) amplitude is given by the relationship

1 1 A4> « [eV cos'$_] 5 Ihl/Yl 4 (U>

At the end of acceleration the TRISTAN is flat-topped. By putting 6 = 0 ° and V = 78.8 kV at the end of acceleration, the bunch parameters

s are

A(J> = ±47.4° X,B - 2.0 m Ap/n^c = A/irA<j> = ±29.0 x 10~ 3 Ap/p = ±3.8 x 10~ 4

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The adiabatic damping due to acceleration is not enough to make &g = 1.0 m. Further reduction of bunch length is achieved by an increase of rf voltage at the flat top of 70 GeV. For this, we transfer the proton bunch to another rf system operating in h » 2 x 270 = 540 with f - = 79.4 MHz. The rf voltage necessary to match the bunch having ^ » * 2 x 47.4° = ±94.8°, Ap/m^c = +29.0 x io~ 3 and A = 15.1 x lo" 2 is found from Eq.(10) to be V = 38.1 kV, where Y(<fis, * x ) = 1-04 for <j>s = 0" and $x = 94.8°. The rf voltage necessary to make A<j> = 47.7* corresponding to £ B = 1.0 m is found from Eq.(ll) to be V = 594 kV.

Fig.l shots an illustration of the above scheme.

V

"V

V

8 / V

8 / V

' / S W A \ J

inject* on Acceleration storage

Fie.i

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4. Tolerances

In the short bunch scheme described in §3, the proton bunch Is transferred to a separate rf system twice, at the injection stage and at the storage stage. Bunch shape was matched to the rf bucket by choosing rf voltage appropriately. In this section we shall derive tolerances for synchroniza­tion and momentum matching.

At the injection stage the bunch has A$ = ±100a in 39.6 MBz rf wave and Ap/p = ±1 x 10 , while the rf bucket has A(j> = ±180° and Ap/p « ±1.31 x 10~ . Thus, the momentum and phase tolerances during transfer should be, say

&- = +1.5 x i o - 4 6<j> = +15° (13)

At the storage stage the bunch has A(j) =.±94.8° in 79.4 MHz rf wave and Ap/p = ±3.8 x 10 , while the rf bucket has A$ = ±180° and Ap/p =

-4 ±5.2 x 10 . Thus, the momentum and phase tolerances during transfer should be, say

P̂- = ±5.7 x io~ 5 5$ = ±14° (14) P

The above phase tolerances correspond to timing tolerances St = ±1.0 ns for the injection stage and 6t = ±0.5 ns for the storage stage. These will require a considerable effort.

The momentum of the beam in accelerators Is fixed by defining any two of the three quantities; magnetic field B, rf frequency f, and beam radial position x. The accuracy in defining parameters required for a given tolerances in momentum is, however, different for a different set of parameters 7)

l i l B and f

P • —. 1

Tl *3 SB B

(15)

If each term on the right-hand side has to be accurate to (l/V^) (6p/p), we get, for the transfer at injection

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(if>MR = *1"8 X 10"6 <lfW " ±8"2 * 10"5 <16> where the suffix MR refers to Main Ring at 12 GeV. For the transfer at the storage stage, the same criterion gives

^ T R = ± 9- 4 X 1 0" 8 C^|) T R,= ± 3 . 8 X 1 0 - 5 (17)

where the suffix TR refers to TRISTAN at 70 GeV.

(2) f and x

p Y f + 2 x ( 1 8 )

T t P

For the transfer at injection, the criterion gives

(%TR = ± 5' 6 X 1 0~ 7 { 1 9 >

^ T R = ± 2' 5 X 1 0~ 5 ° r ( 6 x ) T R = ± 0' 0 5 i m a t X p = 2 m •

For the transfer at the storage stage, the criterion gives

^fv= ± 7 , 1 x io~9 (20)

(xr }TR = ± 2 ' 8 x 1 0~ 6 o r < 6 X > T R " ± 0- 0 0 6 no at x p = 2 B .

(3) B and x

«£ = .§* + «§ (21)

P Xp B

For the transfer at injection, the criterion gives

^ V = ±1-1 - 10"A (22)

(—)„„ = ±1.1 x 10~ 4 or (6x)__ = ±0.2'mm at x - 2 m . X-j MR, HK P -8-

For the transfer at the storage stage, the criterion gives

^ T R •' ** * 1 0 ~ 5 ^ 2 3 )

(•—>TP = ± 4 x 1 0 ~ 5 o r 6 x * ° - 0 8 m a t i = 2 i . X IK. p

Among the three sets of parameters, the case (3) is the best. Even then, the accuracy requirements are almost on the boundary of today's technical feasibility. Especially, the accuracy requirements for the transfer at the storage stage may be difficult to meet. Errors in phase and momentum lead to a coherent dipole oscillation. It will probably be necessary to correct the errors by sensing the dipole oscillation and by working some adequate feedback system. A feedback system to stabilize only one bunch is not difficult, but the problem is to stabilize 54 bunches at a time.

5. Longitudinal stability

A bunched beam can be unstable against longitudinal coherent oscil-lations. A stability criterion is given by F. Sacherer. A convenient measure for the strength of instability is the coherent frequency shift Aci)_ = u) - mil , where m is the mode number (m=l for dipole modes, m=2 for quadrupole modes, etc), (n the frequency of the coherent oscillation and oi the synchrotron oscillation frequency. The growth rate of oscillation is given by 1/T = Imfco in the absence of frequency spreads, m

If within-bunch frequency spreads are taken into account, the stability criterion is

S > _± \ M I (24) r ' m1

•m

where S is the spread in u s between center and edge of the bunch due to the non-linearity of the rf waveform. A convenient approximation for S is

s

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where A$ is the bunch half length in rf rad., the value of which was estimated for each stage of the short bunch scheme in §3. It is now clear why we have chosen high harmonic rf systems.

The contributions to the coherent frequency shift Am due to per­fectly conducting walls, resistive walls and resonators are given in Ref.8.

Perfectly conducting walls

For a bunch with parabolic line -?=.««••« v*

Aui = Vm Aid (26) m sc where

Ad) 0) 'sc l Zsc / k | X M 2 S C = 0.152 S C M

s V c o s $ s M 3

Z„ sc 2 g Y ^

g c =• 1 + 2 Jin b/a

I is the total current in M bunches; B is the bunching factor (bunch length/bunch separation); Z s c is the longitudinal coupling impedance for mode k; Zn = 377 ohm; a is the beam radius; b is the vacuum chamber radius.

The space charge frequency shift given by Eq. (26) is real and no growth results, since the image force induced on the beam is nonresistive if the walls are perfectly conducting. However, it can bring the beam onto the edge of stability region or on the outside of it, thus making ineffective the Landau damping due to the spread S. Then the beam would be unstable against any small resistive components. Note the strong dependence of Aw on bunch length.

The space charge frequency shift Au^ = Au) s c for dipole nodes is shown in Table 1. The vacuum chamber radius is assumed to be b = 25 ma. With 1 x 10 protons in a bunch the beam is outside the stability region in each stage and with 1 x 10 protons the beam is well inside the stability region.

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Resistive walls

The effect of a smooth round vacuum chamber on the dipole mode-is

s Ts

where

6(Q) = J , e2* i s<Vs 5/ 2

and 6/b is the ratio of skin depth at the revolution frequency (On to the vacuum chamber radius fa; Q is the number of synchrotron oscillations per revolution, Q s = to /(oQ; n is the index specifying the mode of coupled motion of bunches, n = 0, 1, . . ., M-l.

Since |ReG| < 1 and |lmG| < 1, the effect of the resistive walls may be estimated by specifying |Awj|, the values of which are shown as IAWJJRW in Table 1. The effect of resistive walls is small and may be neglected.

Resonators

We assume a cavity or resonant element chracterized by a shunt resistance R_, resonant frequency f r e s and quality factor Q. Then

— = 0.159 = S-r- # D F m (28) b) V cos <b Bh IB s T s

F is the form factor that specifies the efficiency with which the resonator can drive a given mode m. In general, mode m is most efficiently driven when the resonator frequency is f r e s = m f c ri t, where

fcrit = 1 f 0 < 2 9>

f„ is the revolution frequency, f Q = (00/2ir. For these frequencies, the maximum value of F m is approximately 1/A. The factor D depends on the attenuation of the induced signal between bunches, and on the ratio k

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k = fres^ fO (30)

This factor D is in general a complex number. Since |ReI>| < 1 and |lmD| < 1, the effect of resonator may be estimated by

s /m

Typical values of the frequency shift for dipole modes is shown as j&djj^, , ̂ in Table 1, for a typical value of R s = 5 kSJ. In the table also shown are the resonant frequency f^gg ** ̂ crit ^^-ch is most effective for driving dipole modes and the threshold impedance necessary to drive the dipole mode R , which is determined by

|Aa>s I + |AJU1 due to resonator | = T S (32)

R^/k denotes the impedance divided by the coasting beam mode number k given by Eq.(30). An impedance R m/k of ten ohms or so will be achievable by careful design of the vacuum chamber and all components inside. The relevant parts are chamber cross-section variation, bellows, cavity gap, pick-up electrodes, kickers etc and, last but not least, the collision regions.

From Table 1 we see that N„ = 2 x 1 0 1 1 protons in a bunch is a 11 safe number that we can guarantee. With Ng = 5 * 10 protons in a bunch

the beam is stable in the storage stage if the longitudinal impedance R^/k can be kept below 3.8 Q (a big if). In the injection stage and final stage of acceleration, the beam is unstable with N_ = 5 The dipole and quadrupole modes can be stabilized by a feedback system acting on the phase and amplitude of rf voltage.' The problem is, however, to stabilize 54 bunches at a time and another nuissance is the higher mode oscillations which is difficult to stabilize with such a feedback system. A cavity operating at a higher harmonic of the rf frequency can increase the Landau damping and provide a longitudinal stability for all modes?) However, it is questionable whether this method can be applied to our case, where the bunch length is of the same order of the higher harmonic rf waveform.

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6. Concluding Remarks

A scheme to make a i m proton bunch in TRISTAN Ring 1 is described. In this scheme, Ring 1 is used exclusively for protons and Ring 2 for electrons and/or positrons. It is shown that a tight tolerance is required for a successful operation of the scheme. With a 17 GeV electron beam of 200 mA circulating current, a 70 GeV proton beam of 5 x 10 pro-

31 —2 —3. tons in a bunch will give the luminosity of about 3 * 10 cm 'see . By the time the TRISTAN project starts, the 12 GeV Main Ring will deliver at least 5 x 10 protons per bunch. However, the problem lies in longitu­dinal stability of the bunch. Further studies on bunch stability, both theoretical and experimental, are needed to start the TRISTAN phase I with the bunched proton beam. In this context, it should be noted that accel­erator studies on the Booster and Main Ring can provide some useful Infor­mations on beam transfer and bunch stability.

Acknowledgements

The author would like to thank Prof. T. Nishikawa and Dr. T. Suzuki for helpful discussions.

References

1) T. Suzuki, KEK-76-3 (1976). 2) A. Piwinski, Nucl. Instrum. Methods, 81, 199 (1970). 3) SPEAR Group, IEEE Trans., NS, 20, No.3, 838 (1973). 4) M. Sands, SIAC-121 (1970). 5) I. Gumowski, CERN MPS/Int. RF 67-1 (1967). 6) I. Gumowski, CERN MPS/Int. RF 67-6 (1967). 7) J. Peterson and L.C. Teng, "1975 ISABELLE SUMMER STUDY", BNL 20550,

Vol.11, p.470. 8) F. Sacherer, IEEE Trans., NS-20, No.3, 825 (1973). 9) P. Bramham et al, Proc. of 9th Int. Conf. on High Energy Accelerators,

p.359, SLAC (1974).

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Table 1. Computed parameters for dipole aodea of instability

N B = 1 x 10 12 N B - 5 x i o u N B = 1 x 1 0 U

Storage stage ( ms = 1200, f r e 8 = 150 MHz )

s/H 0.011 0.011 0.011

K C / U B 0.017 0.0085 0.0017

I^IIRWK 2.9 x 10" •7 1.5 x 10-7 0.29 x io~7

l^ l l cv .Skf l^s 0.0064 0.0032 0.0006 R

00 - 3.9 kfl 73 kfl

Rjk - 3.8 a 72 £2

Final stage of acceleration ( u s = 3 1 ° . fres = 75 MHz )

S / 4 U g 0.011 0.011 0.011 Au>8c/u)s 0.032 0.016 0.0032

NIIRWK 4.4 x 10" -6 2.2 x 10-6 0.44 x 10-6

lAuilcv,5kn/us 0.049 0.025 0.0049 R

00 - - 8 kfi

R /k 00'

- - 16 Q

Injection stage ( U)s = 450, f r e s = 36 MHz )

S/4u>s 0.048 0.048 0.0-S A i i ) s c / t o s

0.22 0.11 0.022

lA<»l|R W/u)B 1.4 x 10 -5 0.7 x 10"5 0.14 x 10-5

l^l lcV.Skf l^s 0.070 0.035 0.007 18 left

*J* ' - - 74 ft

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