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SPACE-CHARGE EFFECTS IN -ELECTRON AND -ELECTRON COLLIDING OR CROSSING BEAM RINGS F. AMMAN and D. RITSON* I.N.F.N.—Laboratori Nazionali di Frascati, Rome, Italy

I. INTRODUCTION that the effective charge-density achievable under the condition that the two beams still The interaction between two proton beams in interact is given by: storage rings has been studied, but since Liouville's theorem does not allow the achieve­ NB ~ k Q∆Q γ ment of beam densities higher than the transfer wh P 2Fre ( 'R ) (1) densities, it need not be considered as a strongly limiting factor. In electron-electron or elec­ NB_ k Q∆Q ~ γ ) (2) tron-positron storage rings, the strong damping wls p 2Fr ( R due to radiation losses causes the beam to e shrink to a cross section which can be as small for crossing beams, where: -4 2 as 10 cm . Accordingly, current densities up NB is the number of charges in the more to 103 or 104 amp/cm2 can be achieved by con­ intense beam, B; ventional means. k is the number of bunches per turn; Our results, which apply to the case of p is the number of interaction regions per electron-electron or electron-positron rings, turn; show that if two beams collide, the less intense w, h and l are the radial, vertical and beam A will be stable in the presence of beam B azimuthal dimensions of the beam; up to some limiting effective density of beam B δ is the angle between the direction of and will then break up into a diffuse halo the bunch axis of one of the beams around beam B. Therefore, the calculations and the median plane of the ring (see can be separated into an investigation of the Fig. 1); if 5 is different for the two point at which beam A becomes unstable and beams, in Eq. (2) the higher value of the breaks up, and an investigation of the point at two δ's must be used. which the diffuse halo becomes stable. Q is the number of betatron wavelengths The two calculations agree, and therefore around the ring, for the vertical give confidence that our description is correct. betatron mode; The calculations for the diffuse halo are made in ∆Q is the distance to the closest proper the approximation that the due to a ribbon resonance; of charge is constant, except close to the ribbon γ is the relativistic factor E/(mc2) plane, and changes sign as this plane is crossed. F =βQ/R is the "beat factor" in the inter­ We have also calculated whether arranging action region (β is the amplitude for the two beams to intersect, at an angle in factor); the vertical plane, can change the allowed R is the mean radius of the machine; 2 2 effective space-charge density. The results re = e /(mc ) is the classical electron radius. give substantially the same limits as those ob­ Since γ/R is proportional to the mean tained for the head-on collision of two beams. magnetic field along the ring, this result shows While the calculations and effects are compli­ that the effective charge density is almost a cated and tedious, the simple result is obtained constant irrespective of machine size or design, 12 *On leave of absence from Massachusetts Institute of Technology. when limited to values of the order of 10

471 Appendix particles/cm2. This is compared to 1014 to if ∆μ/p is small and (μ + ∆μ)/p is not close to a 1016 particles/cm2 which could be quite easily multiple of Π, we can rewrite (7) : achieved in the absence of the space-charge ∆μ/p Aβ/2 (8) limitation. Our calculations for the space-charge-limited If the next integral or half-integral resonance density agree with the calcula­ is AQ removed, the allowed change in phase tions made for the Stanford electron-electron shift is: storage ring.1 ∆Μ < 2Π ∆ Q (9) We have used the notation of Green and 2 or Courant and the method of solution follows P(Aβ)/2<2Π ∆ Q (10) their treatment. and, with (5), remembering that β = FR/Q, N 2k Q ∆ Q γ ) (11) II. HEAD-ON COLLISION; INSTABILITY LIMIT wh p 2Fre ( R The - or + sign in Eq. (11) means that, in the We consider the motion of an electron or a electron-electron case, the closest lower reson­ positron in beam A interacting head-on with a ance must be taken into account (∆Q must be more intense electron beam B. Beam B has negative), while in the positron-electron case, it k bunches, a total number of N , its B is the closest higher resonance which must be length is t, width w, height h, and the distribu­ considered. tion of charges is assumed to be uniform. If (Μ + ∆Μ)/P is very close to a multiple of π The impulse I given to an electron (or (which is, for instance, the case when p = l), positron) passing through one of the p inter­ the approximation in (8) is wrong by a factor action regions with vertical displacement z is: of 2; we can take it into account by redefining 4πN e2 p as a parameter which depends on the number I = ± B z for |z | ≤ h/2. (3) whkc of collision regions per turn, which is always bigger than 2, and is approximately equal to As a result, the region behaves like a thin the number of collision regions only when they lens with a transfer matrix Ti: are more than 2 per turn. 1 0 T = (4) i ±A 1 III. HEAD-ON COLLISION; STABLE DIFFUSE with: ORBITS 4πN r2 A = B (5) Consider the motion of an electron (or ywhk positron) in beam A executing an orbit which The unperturbed transfer matrix from one does not intersect beam B. We shall approxi­ collision region to the next, under the condi­ mate the impulse per unit length, I, given to tion that the number of periods of the magnetic the electron (or positron) in traversing the structure to be a multiple of the number of interaction region as: collision regions p, is: 4πN e2 I = ± B z for |z| ≥ h/2 (12) | | wklc α β |z| μ μ_ The resulting equation for the vertical Tm = 1 cos + sin (6) p -γ -α p motion is: where μ. is the phase shift of the betatron d2z 4πN r2 z + k(s)z = ± B L(s) (13) oscillations around the ring. ds2 γkwl |z| The resulting transfer matrix is the product where s is the distance along the equilibrium of (4) and (6); the phase shift between two orbit, k(s) represents the focusing properties collision regions becomes : of the machine, L(s) is equal to 1 within the p Μ + Aμ μ A/3 μ regions of interaction and to zero outside, and cos = cos ± sin (7) the length of interaction regions is l/2. V V 2 p 472 Appendix

The equation may be transformed, in the where L(Φ) is equal to 1 within the p interaction standard manner, in terms of the variables : regions and to zero outside. Because of the = -½ nonlinearity, Eq. (15) may have periodic solu­ η β z (14) tions (closed orbits) with periods 2πq in Φ, where q is an integer. into: 2 Let us expand, in Fourier series, the pertur­ d r, 4πre η + Q2η = ± Q2β3/2 L(Φ) (15) bation term over q turns : dΦ2 γkwl |η|

Q2 β3/2 4πreNB η Φ γkwhl |η| L( q ) ½ p s (16) Qβ N r . ηi• l η Φ+Φ .• 2 B e + i cos r i γkwq ∑ 2 r=∑l i=1 [ |ηi| |ηi| q ] where Φi are the coordinates of the p interaction in Eq. (19), for the/th interaction region, gives regions; ηi,-/|ηi| gives the sign of the perturba­ βNBrep tion in the interaction regions. zj = γw∆ Qk (24) The expansion (16) is approximate, and is valid under the assumption that The condition both for the validity of the (rl)/(4Rq)«l (17) potential used and for all the orbits of beam A to lie outside beam B is: Accordingly, it is limited to a certain harmonic term of order S; it can be seen a posteriori that zj≥h (25) this approximation is usually quite good. which finally gives the density limitation: The behaviour of the periodic solutions of Eq. (15) is dominated by the term of (16) for NB 2k Q ∆ Q γ + ) (26) which wh p 2Fre ( R r/qQ (18) The — sign is for electron-electron (∆Q must Taking only this term we have : be negative for the solution to be consistent) ½ p 2Qβ NBre cos(r/q)(Φ-Φi) ηi and the + sign is for the positron-electron η = ± ∑ 2 2 (19) interactions. γkwq i=1 Q - (r/q) |ηi| ∆Q should be the distance from the closest The solution, when (η) = 0, is consistent resonance of every order; in fact it can be for: easily seen that for q4 the treatment given Q>r/q for electron-electron interaction. here begins to fail; ∆Q, for higher values of q, (20) decreases, but at the same time the value of the Q

2 2 2πN e 2x-1 4πN e ZA z I = ± B + B for - x < A < 1 - x (27) { wkc l wkc lδB } ΔB and 2 2πNBe ZA ZA z I = ± for < - x or A > 1 - x (28) wkc δ Δ |zA| B B where x is the distance of the particle from the the electron-electron case, for a weak-focusing leading edge of the bunch, and δB = α+θB is ring, we have: the angle of the axis of beam B with respect to 1 the median plane of the ring. R NB re θA = -4F - } (30) If the density of beam A is very small, then γ wl k { 2 1 - θB0 and θBα, since the tilt angle θB is due As an example, with Q0.8, the lens_ effect to the effect of beam A. The Q-value of beam of the interaction region tends to drive close A will change (decreasing in the electron-electron to 0.5, which is the closest lower half-integral case, increasing in the positron-electron resonance. If we combine Eq. (29) with Eq. case), and the direction of the bunch axis will (30), at the space-charge limit ( = 0.5), we be tilted by an angle θA with respect to the obtain : direction, because of the interaction with beam B and as a result of its effective charge density. δB θA - 2 (31) The sign of the tilt angle depends on the closest integral resonance: in the electron- In this case, then, when the density of beam B is high enough to drive beam A into the half- electron case θA will be negative (see Fig. 1 for the definition of the sign of the angles) if the integral resonance, the tilt angle is negative and closest integral resonance is lower than the , is given by Eq. (31). and positive if the closest integral resonance is Because the tilt angle depends on , it is higher than the ; the opposite is true for the not easy to write down a general formula. positron-electron case. is the number of However, it can be calculated by introducing betatron wavelengths per turn due both to the the impulse given by Eq. (27) into the equations main-field focusing and to the lens effect of the of motion and solving them for each particular interaction region. case. If the crossing regions are more than one per turn, they can be arranged to make θ The density of beam B, which gives rise to the small. instability limit for crossing beams, comes out The tilt angle θA changes the length L of the to be the same as the one calculated for colliding interaction region along the direction , from beams, provided we substitute, for the height the initial value L0 (for θA = 0). of the beam, the equivalent height lδB : h L = (32) NB 2k Q∆Q γ 0 2α. ≤ to lδBw p 2Fre ( R ) (29) h L' = + lQA (33) All the considerations made for Eq. (11) are 2α 2α valid for Eq. (29). If we drop the assumption NA«NB, and With a method similar to the one used in consider beams of roughly equal density, then, Section III, we can find the tilt angle θA for if the effective densities are close to the limit the case of one interaction region per turn. In given by Eq. (29), the equilibrium will be un-

474 Appendix

tween the two beams, it seems more correct to take as a space-charge-limited density, a value at least a factor of two lower. Then, we obtain the values given in Section I. The interaction rates per interaction region, n, for colliding beams, are given by : = N NB f σ events/sec (34) A wh k and, for crossing beams, by n = N NB f σ events/sec (35) A wlδ k where f is the revolution frequency, and σ is the cross section in cm2. Fig. 1 Interaction region in a crossing beam ring. If we introduce the space-charge-limited densities for beam B with the numerical values: stable. It seems, then, that the crossing angle Q ∆ Q0.1 (weak focusing) 2α must be controlled to keep it large enough γ/R4: cm-1 to avoid these unstable situations. At the beginning of the present section we we find: made the assumption that α remains constant 4 × 1030 when Q changes of the order of one unit. If the h IA σ events/sec (36) crossing angle is obtained with perturbations p of low harmonic order, close to Q, the treat­ where IA is the current, in amperes, of the ment must be modified, since α will also depend weaker beam A and p the number of the inter­ on . action regions per turn. A final remark must be made about the case This result seems to set an effective upper of crossing beams. Every beam-interaction limit for the cross sections that can be inves­ effect has been calculated with the approxima­ tigated with colliding or crossing-beam rings. tion that the impulse given by beam B to a In the case of colliding beams, where the particle of beam A, outside of beam B, does not cross section of the beams is determined by depend on the distance from the axis of beam B. the radiation effects, one should provide some This approximation is correct when the dis­ means to increase this cross section if he wants tance, in a vertical plane, from the axis is to use, for IA, a value bigger than the one which smaller than the' radial width of the beam; is given by the space-charge-limited density this is not always the case for a crossing-beam times the unperturbed cross section of the beam ring. Nevertheless, we have used this approxi­ (which is of the order of 1 ma for a 750-Mev mation, which probably gives a pessimistic weak focusing ring). value for the space-charge-limited density; No problems arise in crossing-beam rings however, it permits us more readily to evaluate because, in this case, the effective beam cross the effects. section can be varied by changing the crossing angle. V. RESULTS REFERENCES The space-charge limit sets in at a charge density given by Eqs. (11), (26), and (29). 1. W. C. BARBER et al.: An Experiment on the Limits of —Internal Report of Stan­ With the densities given by these relations, ford University, HEPL 170, (June 1959). there should not be any collision between beams 2. G. K. GREEN and E. D. COURANT: The Proton A and B. If we allow for the interaction be­ Synchrotron—Handbuch der Physik, 44, 218-340.

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