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Luminosity for Electron-Proton Collision in Tristan (Coasting Pr0ton Beam)

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Luminosity for Electron-Proton Collision in Tristan (Coasting Pr0ton Beam) KEK-76-9 52.1.1? J LUMINOSITY FOR ELECTRON-PROTON COLLISION IN TRISTAN (COASTING PR0TON BEAM) Toshio SUZUKI OCTOBER 1976 NATIONAL LABORATORY FOR HIGH ENERGY PHYSICS • OHO-MACHI, TSUKUBA-SUN IBARAKI, JAPAN KEK'Reports are available from Library National Laboratory for High Energy Physics Ohb-machi, Tsukiiba-gun Iharaki-ken, 300-32 JAPAN Phone: 02986^4-1171 Telex: 3652-534 (Domestic) (0)3652-534 (International) Cable: KEK0H0 LUMINOSITY FOR ELECTRON-PROTON COLLISION IN TRISTAN (COASTING PROTON BEAM) Toshio S0ZOKI National Laboratory for High Energy Physics Oho-machi, Tsukuba-gun, Ibaraki-ken, 300-32, Japan Abstract Luminosity for electron-proton collision in the intersecting storage accelerator TRISTAN has been studied. A bunched electron beam and a coasting proton beam are assumed. Parameters of the interaction region have been optimized with respect to linear tune shifts. With an electron current of 200 mA (8.4 x 10 electrons) and a proton cur­ rent of 14 A (6 x IO1* protons), luminosity of 2.8 x 1031 cm~2s-1 and 31 —2 —1 4 x 10JX cm s can be achieved in a straight interaction region in a conventional ring of 70 GeV protons and in a superconducting ring oi" ISO GeV protons, respectively. The electron energy is chosen to be 15 GeV. In a curved interaction region where collision occurs within a bending 31 magnet, long-range interactions are reduced and luminosity of 6 x 10 cnT^s-! and 8.5 x 103* cnT^s-^- can be achieved in a conventional ring and in a superconducting ring, respectively. §1. Introduction Luminosity for electron-proton;collision in a storage ring.has ; 1 2) 15 been studied by several authors ' '. Nishikawa ' derived approximate formulae for luminosity and linear tune shifts neglecting long-range forces and determined an optimized set of parameters for the interaction region. 2) Piwinski used the expressions for tune shifts taking the long-range forces into account, but assumed a round proton beam as in Keil et al3.} Recently, a tune shift formula for a beam having a Gaussian distrib­ ution and general elliptical cross-section was derived by Montague?' A corresponding formula of luminosity for collision between a bunched electron beam and a coasting proton beam was derived by Reggiero ' and the author ', so that consistent discussion can be made on the luminosity for electron-proton collision in a general case. The aim of this note is to determine an optimized set of parameters for the interaction region in TRISTAN and to estimate achievable luminosity. §2. Formulae of Luminosity and Tune Shifts Luminosity^ for collision between a bunched electron beam and a coasting proton beam is given by * ' 2 2 . 2 2(0-' + ) . J + a xe a"xp V Mx e xp V^ y e yp (1) where f is the revolution frequency of electrons, H is the total number of electrons, A_ is the line (number) density of prptbM,^do'xe, a_, a and o are horizontal (x) and vertical (y) rms beam sizes for ye jrtr electrons (e) and protons (p). x and x are the deviations of the electron and proton central trajectories from tha reference z-axis. In a straight interaction region, x = (j>z, e Y x = -<j>z, (2) P Y -1- where 2<t> is the crossing angle. Luminosity depends on the total number of electrons and is independent of the number of bunches. ,,in formula (1), horizontal crossing is assumed and the beams are assumed to have Gaussian distributions. For vertical crossing, the suffixes x and y are inter­ changed. If we neglect the variation of the beam sizes along the z-axis and if we assume a straight interaction region with a crossing angle 2$, we can integrate eq.(1) to obtain JL't e P 1 . (3) V ye yp Thus, it is seen that luminosity is sensitive to the crossing angle and the vertical beam sizes. If we assume a head-on collision and separate the beams at z = i£~,./2 , / VE %TT = f (4) yj xe xp y ye yp where the variation of beam sizes is also neglected. Small crossing angle 2iji or large length of the interaction region £, are preferable to HJT increase luminosity, but their values are limited by linear tune shifts. The linear tune shifts Av^ and Avye which protons impose on electrons are*' -r 32 xp (5) f r P (z) X. 7p where n TTu~^ + -^ ft fT Tfil vx>K) - •• ^ _ 1 A2 -1 -2- X2 A 2 r, e" ^ /rf X Fy(X, K) = -^ [1 - ^— + j== * (X, K)], (6) IT" — 1 ,/ — 1 vK X 1KX g *(X; K) * e" w(»\i. - —)/ -— w»v( _. ), - <s - v ^ axp a w(z) is the complex error function defined by z2 Z t2dt w(z)=e- (l + 2if e ). (7) x 15 r is the classical electron radius (2.8179 10" m) and Yg is the relativistic energy factor of electrons. The dispersion function is assumed to be zero in the interaction region. The limiting expressions of eq.(6) for K=l are 2 FX(X) - _| [<i + 2X )e~^ - 1], (8) 1 -X2 Fy(X) - -4 [1 - e * ] . For a quasl-collinear crossing, X = 0 and '*THT , x 1 a X o— re_ B„(zxe ) X_P Tnr 0 + e dz , - *THT V xp V 2 (9) £INT re «W» ». Avye = f i • dz . Trye yp xp yp 2 -3- If we assume that the variation of the proton beam sizes is negligible, but that fi and (3 vary as xe ye * ,2 ee(z) - Be + TF . (10) 6„ -e we obtain tie* . , a2 AM e p*xe . ,, ,_1 INT. xe a iry <j (a + a ) *IHTU 12 _*2 '» v 'e xp xp yp' Pxe (11) ye irr o (a + a ) ^IHT11 12 „*2 ; * •ep JV xp yp ye where fi and (3 denote the values at the interaction point. Long-range forces are neglected in these formulae. To minimize the linear tune shifts, the following relation should be satisfied * 9 TNT y 2J3 where the vertical tune shift Av is assumed to be larger than Avxe. The relation (12) was obtained by Nishikawa1' from simpler considerations. If we include long-range forces, an optimum value of 3 is larger than ye that given by eq. (12). This point is verified by numerical calculation and will be discussed later. We now consider the nature of long-range forces. For large &., F 3--i • x x2 ' F ~-i •' y "x *2 ' and the specific tune shifts will become r B X ^se* a e xe P dz HY 2 e Gfe- v (13) d(Av ) E X ye eV P 'e (s_ - x ) -4- For a straight interaction region 2 2 2 (xg - xp) = 4(}i z ;iv n and the dividend behaves as z . The numarator also bat!aves;,as z?_ffrom . • eq.CIO) and the contribution at large distances is constant. The maximum value of |3 is limited by physical consideraxons, however, and e -2 the specific tune shifts decrease faster than z and the integral con­ verges. For an accurate estimation of long-range forces, the exact knowledge of the behavior of the ^-function is required, Without the exact knowledge of the behavior of the fJ-funstion at large distances, the mavimiTm g-value is set to be 500 m. If the g-value calculated from the type of eq.(10) exceeds 500 m, it is set. to be 500 m. The integral is further cut off at z = ±30 m. Actually at these distances, the two beams will be in separate beam pipes and in separate magnets so that the electromagnetic interaction between ths two beams will be suppressed. The contribution from z < ±2 m is taken rather arbitrarily to be due to short-range interaction. Protons pass, in general, several electron bunches in one inter­ action region. The tune shits Av_ and Av_ which electorns impose xp yp on protons are expressed as''' N r g 2irB ((I + ff ) ^ Yp °te te ye D Jo <^^f > t^Cu) - In4lC«» , .ye te 2 l •wr P Vp yp ir -u x . 'p ye ye te Jo^ + «<^f>nin4<«). ye te 2 with , zd> .2 te a2 = o£ + a2 tan2(f te xe ze -5- Here, B is the number of electron bunches, r is the classical proton —18 D radius (1,535 x 10 m), .y_ is the proton energy in units of rest mass energy, and 0ze is the rms longitudinal electron beam size. I\l(u) is a modified Bessel function of order n + 1/2. For z =* 0 (short-range interaction only), N r / Av . e P *» ^ 2TTYB ffte(o-te + 0 ) » (15) N r B AV™ : eP yp 3T 2TIYP B ory e(o ry e + a,te. J' where * denotes the value at the interaction point. For z#) and an approximately round electron beam, the tune shifts due to long-range forces are expanded as A 2 % - 2„rY&ye + ate) 2u-[(4u+l)e- °-l + ^^ <u ~ I" ^2U^ + 5 £>} 0(^>)2}] . ye te + + ye te K r 6 Av = eP yP i ri - e"2u tf 2a " yp 2TTYPB aye(aye + o"te> + 3 1 2 2 0 ^ tl - i + (1 + l)e- «} + 0t(^>) ), . ye te ye te (Hi) •' §3. Collision Geometries Four possible geometries of an electron-proton interaction regian are shown in Fig.l. Case a) is a simple straight interaction region and the length &TNT where the two beams overlap is determined by the crossing angel 2$. Case b) is a curved interaction region originally proposed by Montague'1' and extensively studied in the electron-proton collision facility for LSK.9^' at CEEH.
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