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 = -
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( _. ), -
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) - _| [
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 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 + ^^ } 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. This geometry is suitable for increasing luminoisty by quasi-collinear collision and decreasing the long-range interaction -6- by separation with bending magnets. The length Jl™, is determined by the magnetic field. The maximum field is, however, limited to about 3 kG because the background due to synchrotron radiation; ,wjllbeconfi: enormous with a higher field for electrons. Case c) is simply a variant of case b) and case d) is an intermediate geometry between^ a) and c). We study the cases a) and b) in detail in this note. If we neglect a variation of beam widths, the length &m^ is given by w + w ftlHT- % P . (I?) for a straight interaction region of case a), where we and w_ denote the horizontal half widths of the electron and proton beams, respectively. For a curved section of case b) £riT = /8(we + wp)p , (18) where p denotes the radius of curvature of the magnet. We assume that the proton beam is rigid and w_>w. - - - For Gaussian beams,fcr™. canno t be uniquely determined because a tail effect affects luminosity. In this note, we define *•_-_, as the length within which 90 % of the events occur. §4. Straight Interaction Region The energy of the electron beam is set at 15 GeV and that of the pro ton beam is set to be either 70 GeV (conventional magnet) or 180 GeV (super conducting magnet). The electron current is assumed to be 200 m& (8.4 x 10l2 electrons). This current is required to: achieve lumitiosii^ cjf 10 cnf's-! for electron-positron collision!*^ The proton current is assumed to be 14 A (6 x 10^-* protons). The values of emittances assumed in this note are summarized in the Appendix. The maximum allowable"1 linear tune shifts are taken to be 0.06 for electrons and 0.005 for protons. (1) Conventional ring (15 GeVe x 70 GeVp) The rms eoittance of the proton beam is 2.91 x'lO"7 irm-rad horizontally, TCP = 5.80 x 10"8 mn-rad vertically. -7- The coefficient of coupling K for electrons is, unless otherwise stated, set at K = 0.05 so that the rms emittance of the electron beam is —S 7.1 * 10 -nm'rad horizontally, ire, • = { -9 e 3.6 * 10 irm-rad vertically. * * a - "* We optimize the parameters 0 , (3 » fi . , fL and 4? ^ yP ^e* i "-"'?•'••.-••:: With the parameters we first optimize Bye. The result is shown in Fig.2. For the short-range tune shift, an optimum value of Bye is about 0.5 mi which rougly agrees with eq. (12). If long-range forces are taken into account;, an optimum value of By is aoout lm. Luminosity changes very little with fJye because the electron vertical emittance is much smaller than the proton emittance and luminosity is governed by the proton beam size as shown by eq.(3). We set 8ye = lm. We then study the dependence on B . The result is shown in Fig.3. Luminosity and the stoi-t^range tune shift decrease roughly proportionally, but the total tune shift: decreases more slowly. This is consistent with eqs.(3), (11) and (13) in that luminosity land the short-range tune shift show almost the same dependence on the proton beam blight whereas the long-range tune shift is independent of the beam size. A smaller value it * • of Q seems preferable and we set {5 = lm, which will be the practical yp yp minimum from the requirement on the lattice structure. £ A i A = With the parameters B = 1.5 m, Bye = 1m, Bxp 5m, B = 1m, we study the dependence on <(•• The result is shown in Fig.4. The tune shift and luminosity decrease with increasing We further study the dependence on B^* Bxe and K. When B_ is changed, luminosity and time shifts change only a little, but &XNT can be made smaller with smalle;: g!|L. With $ = lm, JIJJJJ = 0..6 m. When Bx« is changed, o£and Av (tot) change very little. The value of Avx(tot) decreases with increasing 6* , but even with 0xe = lm, Avs(tot) = -0.031 and is acceptable. Luminosity gradually decreases -with increasing K. -8- With K = 0.05, «£ = 2.8 x 1031 cm~2s-1 and with K = 1, /_= 2.1 x 1031 cm" s . With the parameters thus determined (K = 0*05), the proton tune shifts and the minimum number of electron bunches are calculated according to eq.(15), taking into account only short-range forces. The limit set by the vertical tune shift is more serious and it gives Bj,^ = 237- Maximum allowable value of B is 34 m. With 400 MHz EF frequency for the electron ring, the harmonic number is about 2700. Thus, the number of bunches is much smaller than the number of RF buckets. The long-range forces will be small, but if these become serious, we can increase the vertical electron beam height by increasing K. When K = 1, the electron beam is roughly round and the expression (16) can be used. With B = 2700, the distances between bunches is about 75 cm and the long-range forces are evaluated at z^xn (n = ±1, ±2, ±3, -••) up to z.= ±30 m. The result is Av (tot) = 3 x 10"^ and is quite acceptable. (2) Superconducting ring (15 GeVe x 180 GeVp) With the normalized emittances of 22 innm mrad horizontally and 4.4 irrnm mrad vertically, the same procedure is applied to the superconducting ring. Optimum parameters are 6*e are rather insensitive to luminosity and the tune shifts, but chosen to be 5 m and 1.5 m, respectively. The corresponding luminosity is 4.0 x io31 cm~2s-1 with K = 0.05 and the total vertical tune shift is 0.06. Since the protons are more rigid (large y), the proton tune shift problem is less severe, than the case of 70 GeV conventional ring. The minimum number of electron bunches is B . = 96 with K = 0.05 if only short-range interactions are taken into account. In a superconducting ring, it is preferable to exchange the hori zontal and vertical emittances in order to make the aperture as round as possible. Inrth'i s case, the optimum luminosity is 2.4 x 10JJ- cm-is~x. Since the horizontal emittance is smaller, vertical crossing of the interaction geometry is preferable to increase luminosity. In this case, the luminosity of about 4 x 10— csT^s-! will be obtained. -9- §5. Curved Interaction Region Maximum lr-i. .. _, Limited by linear tune shifts. In the straight interaction region studied in the previous section, long-range forces play an important role. In order to reduce the contribution from long- range forces and to increase luminosity by head-on collision, a curved interaction region, in which interaction takes place in a bending magaet, has been proposed?) This possibility will be studied in this section. A parameter, which defines Hj^ or the length within which the two beams overlap appreciably, is the field strength of the bending magnet. The maximum field strength is limited to about 3 kG because the background due to synchrotron radiation will become enormous with a higher field. The length of the magnet is chosen rather arbitrarily to be 6 m. (1) Conventional ring (15 GeVe * 70 GeVp) With a bending field B = 2.5 kG, B is first optimized. In a curved interaction region, the long-range forces play little role and the tune shifts are determined mainly by the short-range contribution. An optimum value of g* is,thus, about 0.6 m. With B = 2.25 kG, B* = 5m, B* = lm, _2 -1 B*e = 1.5m and B* = 0.6m, luminosity achieved is 6.0 x 1Q31 cm s , Avye = 0.057 and £INT = 1.8 m. (2) Superconducting ring (15 GeVe x 180 GeVp) Optimum values are searched for B , $ and B. With B—, = 5m, Byp = 1.5m, B|e = 1.5m, B*e = 0.5 m and B = 2.75 kG, luminosity is 8.5 x 2 _1 1031 cm~ s , Avy = 0.059 and J!™, = 1.3m. The value of B* is chosen to be somewhat larger to reduce the electron' tune shift and thus to reduce the magnetic field. §6. Conclusion Sensitive parameters which determine luminosity and tune shifts for electron-proton collision in horizontal crossing are B , B, $ and B. Smaller {£_, Bxe and smaller horizontal enittance are preferable t-o decrease £,_,—,- but they are insensitive to luminosity and time shifts. Optimum luminosities are 2.8 x 10 -10- 31 -2 -1 11 —9 -1 luminosities are 6 x ioJ cm s and 8.5 x 10 cm s for 15 GeVe 70 GeVp collision and 15 GeVe X" 180 GeVp collision, respeci^e^i "The. corresponding strengths of the magnetic field are 2.25 kG and 2.75 %G, respectively. If these fields are too high due to the background of synchrotron radiation11', additional measures such as introducing a taper in a magnetic field, introducing a crossing angle, increasing the proton beam height etc. will be necessary. In these cases, ma-rimim achievable luminosity will become somewhat lower. Acknowledgement The author wishes to thank Professor T. Nishikawa for his interest in this work and helpful discussions. He also wishes to thank Mr. S. Kamada for his help in numerical calculation. Discussions with Dr. M.J. Lee were also helpful. References 1). X. Nishikawa: "KEK Future Project", Proc. U.S.-Japan Seminar on High Energy Accelerator Science, Tokyo and Tsukuba, p.209 (1973). T. Nishikawa: "A Preliminary Design of Tri-Hing Instersecting Storage Accelerators in Nippon, TRISTAN", Proc. IX-th Intern. Conf. on High Energy Accelerators, SLAG, p.584 (1974). 2) A. Piwinski: "Betatron Frequency Shifts for PETRA", DESY HL-73/2 (1973). 3) E. Keil, C. Pellegrini and A.M. Sessler: "Tune Shifts for Particle Beams Crossing at Small Angles in the Low-g Section of a Storage Ring", Nuel. Instrum. & Methods, 118, 165 (1974). 4) B.W. Montague: "Calculation of Luminosity and Beamr-Beam Detuning in Coasting-Beam Interaction Regions", CERN/ISR-GS/75-36 (1975). 5) A.6. Ruggiero: "The Luminosity from the Collisions of Two Unequal and Not-Round Beams", FNAL-FN-271 (1975). 6) T. Suzuki: "General Formulae of Luminosity for Various Types of Colliding Beam Machines", KEK-76-3 (1976). 7) M. Abramowitz and I.A. Stegun: "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", National Bureau of Standards Applied Mathematics Series - 55 (1964). -11- 8) B.W. Montague: "Curvilinear Interaction Regions fot Storage Rings", CERN-ISR.TDI/72-44, (1972). V;'?J:V :':y-;"// --:V' 9) e-p Working Group: "Some Remarks about e-p Interaction"iS#gions"j CERN-ISR-GS/75-33 (1975)." iO) S. Kamada and T. Suzuki: "Luminosity for Electron-Positron Collision in TRISTAN", KEK-76-6 (1976). 11) H.F. Hoffmann: "Interactions of the Synchrotron Radiation with the Proton Beam in the LSR e-p Interaction Regions", CERN-ISR-LTD/75-53, (1975). 12) T. Suzuki: "Orbit Analysis of the KEK Synchrotron", KEK-74-4 (1974). 13) J.D. McCarthey and Y. Kimura: "Design of Fast Extraction System for the KEK Proton Synchrotron", KEK-74-16 (1975). 14) J. Tanaka, H. Baba, I. Sato, S. Inagaki, S. Anami, T. Kakuyama, T. Takenaka, T. Terayama and H. Matsumoto: "Operation of the KEK 20 MeV Injector Linac", submitted to 1976 Linear Accelerator Conference, Chalk River (1976). 15) K. Sato: Accelerator Study Note (in Japanese) ASN-18, KEK (1976). 16) M. Koabyashi: "A Feasibility Study fdr the Charge Exchange Injec tion into the KEK Booster Proton Synchrotron", KEK-76-2 (1976). 17) M. Sands: "The Physics of Electron Storage Rings. An Introduction" in "Physics with Intersecting Storage Rings , Proceedings of the International School of Physics, Enrico Fermi, Course XLVT" ed. by B. Touschek (Academic Press, 1971). 18) A.S. King, M.J. Lee and W.W. Lee: "MAGIC, A Computer Code for Design Studies of Insertion and Storage Rings', SLAC-183, RL-75-110 and EPIC/MC/86 (1975). A.S. King and M.J. Lee: "Some up-to-date Additions to MAGIC fdr PEP Desing Studies", PEP-134 (1975). 19) S. Kamada, M.J. Lee and T. Suzuki: "Finding an Electron-Ring Lattice for TRISTAN with MAGIC:, KEK-76-1 (1976). -12- Appendix Emittance Expected in TRISTAN At the present time, the beam characteristics of the present 12 GeV proton synchrotron are not veil known. The design of the electron ring for TRISTAN is also not yet completed. Thus, only crude estimation for the emittance can be made at present. We describe the assumptions which led to the values of emittance used in this note. (I) Proton Ring Emittance in proton rings is determined by the property of the injector (ion source and linac), by the injection scheme and by the emittance growth during the accelerating periods in the booster, 12 GeV proton synchrotron and TRISTAN. Without the emittance growth, the quantity TTE-norm called the normalized emittance is conserved. It is related to the emittance ire by the relation ^norm = m'** ' - where fj and y are usual relativistic Lorentz factors. In the design of the present 12 GeV synchrotron, ' the normalized emittance of the linac is assumed to be 10 irmm-mrad. Horizontal multi- turn injection scheme is performed for the booster and the normalized acceptance of the booster is chosen to be 50 irmm-mrad horizontally and 10 irmm-mrad vertically. Emittance growth of factor two is assumed in the booster and the normalized acceptance of the main ring is chosen to be 100 mnm-mrad horizontally and 20 irmm.mrad vertically. The emittance growth of factor two is also assumed in the main ring and the emittance of the main ring is 200 Tnaii"B..*ad horizontally and 40 Trmm-mrad verticallyi* We assume no further emittance growth in TRISTAN. If we assume bunch-by-bunch beam transfer to TRISTAN using a ticker magnet, the emittance is conserved during beam transfer. I£rire use a shaving extraction system designed for the bubble chamber channel, the'... horizontal emittance is reduced to about 35 % of the circulating beam and the normalized: horizontal emittance is about 70 Tnan-mrad. The measured emittance of the linac is consistent with the design if we assume that the design emittance gives three standard deviations of the Gaussian beam size!*) The emittanee measured in the transport -13-r line"' from the booster to the main ring is 68 •nn8B.jnraa' horizontally and 28 lima»mrad vertically if we assume that the emittances correspond ts/ three standard deviations of the Gaussian beam size. The horizontal emittance is smaller than the design, but the vertical emittance is larger. This may be due to the fact that the vertical tune crosses the third-integral resonance during acceleration in the booster. Further, there may be some horizontal-vertical coupling. The emittance of the main ring has not been measured. We thus assume the design values for the present 12 GeV proton syn chrotron, assume that there is no further emittance growth is TRISTAN and assume that the design emittance corresponds to three standard deviations of the Gaussian beam size. Then, the rms (normalized) emit tance iTe_- ~„.^, is rms,norm 22 TTmm-mrad horizontally, enas,norm " 4.4 ttmm.mrad vertically, if we assume bunch-by-bunch transfer, and the rms horizontal (normalized) emittance is 7.7 itmm>mrad if we assume a shaving extraction. In the superconducting ring, it may be preferable to exchange the horizontal and vertical emittances by skew quadrupole magnets in the beam transport line to make the aperture as round as possible. In this case, the values of horizontal and vertical eaittances are interchanged. Possibilities of reducing the emittanee of the present synchrotron are also under discussion. One is to increase the intensity of the ion source and the linac and to perform single-turn injection into to the booster instead of multi-turn Injection. Another is to usemulti-turn injection of negatively charged hydrogen atoms. ' (II) Electron Ring The emittance of an electron beam in a storage ring is determined by the balance between radiation damping and quantum fluctuation of syn chrotron radiation (natural emittance). If there is no vertical bend, the natural emittance in the vertical phase space is quite small and is mainly determined by horizontal-vertical coupling. The horizontal emittance e_ under no coupling is given by17) -14- C Y2 Jx - 1 - D * 1 ^mag " 5? jmag I{ ^ + «*' * W2>- « f • p is the radius of curvature, y is ths relatiyistic energy factor, {3 is the betatron amplitude function of Courant and Snyder,n is the dispersion function and the prime denotes differentiation with respect to orbit length z. The integral is done over bending magnets, a is the momentum compaction factor, R is the average radius of the curved section and Vx is the horizontal tune of the curved section. With the approximation a * -K , (A.3) £Q is expressed as o -6 E eT)R EQ * 1.47 x io where E is the electron energy in GeV. Putting the coefficient of coupling K as e K «-*• (0 < K < 1), (A.5) where E^ and Ey denote horizontal and vertical emittances,ex and Ey are expressed as17' 1 (A.6) K E = ee y l + K o • An exact calculation of Ej. can be done by a computer program MAGIC7 but it has been shown1**) that a simple analytic calculation according to -15- eq.(A.A) agrees fairly well with the computer calculation. With a pre 19 7 liminary electron ring lattice ^ with Vx = 12, £Q = 2,64 x io~ m-rad, at 15 GeV. A lattice with vx = 20 is preferable from beam transfer con siderations on the proton ring. Luminosity of electron-positron collision is much influenced by the tune shift limit and large emittance is pre ferable! ' In the case of electron-proton collision, however,- smaller electron emittance is preferable from lumiposity considerations and the ^une shift problem can be eased by increasing the number of electron bunches. Thus, the lattice with V_. = 20 is assumed in the note. -8 Then, with R = 200m, p = 110.8m, vx = 20 and E = 15 GeV, eQ = 7.5 * 10 m-rad. -16- a) b) c) d) FIG.1 POSSIBLE COLLISION GEOMETRIES -17- 4fc It?*?? 1 **-#.* - 3 • »Ji *.* • AM ftf *•) Att 4j/j«CShorf) ar l.o RG.2 LUMINOSITY AND TUNE SHIFT vs p,e Z '•of cm2? faMl€ - m 0.62 • as\ •&ol FIG.3 LUMINOSITY AND TUNE SHIFT vs pj -18- 1*» ,\ - 2 • f A/* \\\ /•*I-T 3 •*•/ \\ xy^ i • I /x 1 **/ —_, 1 i FIG. 4 LUMINOSITY AND TUNE SHJFT vs -19-