advertisement

© 1971 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. T’&lS ENVELOPE EQUATIONS WITH SPACE CHARGE' Frank J. Sacherer CERN, Geneva, Switzerland Summary Envelope equations for a continuous beam with uniform charge density and elliptical cross-section were first derived by Kapchinsky and Vladimirsky*(K-V). In the K-V equations 3re not restricted to uniformly fact, charged beams, but are equally valid for any charge distribution with elliptical symmetry, provided the beam boundary and emittnnce are defined by rms (root-mean‘This results because (i) the second square) values. moments of any particle distribution depend only on the linear part of the force (determined by least squares while (ii) this linear part of the force in method), turn depends only on the second moments of the distribuThis is also true in practice for three-dimention. sional bunched beams with ellipsoidal symmetry, and allows the formulation of envelope equations that include the effect of space charge on bunch length and energy spread. The utility of this rms approach was first demonstrated by L3postolle3 for stationary distributions. Subsequently, Gluckstern4 proved that the rms version equ3tions remain valid for all continuous of the K-V In this report these rebeams with axial symmetry. sults are extended to continuous beams with elliptical symmetry as well as to bunched beams with ellipsoidal. form, and also to one-dimensional motion. Yoment Consider single-particle an ensemble equntions moment equations, namely tile equation for each moment involves the higher moments in an endless hierarchy. However, if the self-force is derived from the froespace Poisson equation, xF, depends mainly on the second moments and very little, if at all., on the lligher moments. This will be demonstrated in the following sections. The remaining term “Fs is associated with cmittance growth; we will avoid considering it by assuming that the rms emittnnce is either constant, or that its time dependence 2 pr i or-i . Then 2 is given in terms of x2, z, two equations of (4) form by (5), and the first set. They can be combined to give the K-V type (6) where X is particles the D = that obey rms value, Z = ;/ x2 . The space-charge term in tllis equation 1~3s 3n inIf wo define the linear p3rt teresting interpretation. where c(t) is determined of the force Fs(x,t) as e(t)x, by minimizing the dif fercnce equations of (5) is known and E(t) a closed equation: for the 3 fixed [e(t)x / t, - Fs(x,t))’ where n(x,t) n(x,t) = ,/ f(x,p,t) dx dp, (7) tllen (8) ;:zp (1) i, = F(x,t) where F(x,t) self-force, trary particle includes both F = Fe + F,. distribution the rms envelope In other words, an the linear part of the forces, sceuarcs method. , the external force and the Averaging (1) over an arbif(x,p,t), we obtain ;=; (2) p=F=F,, It is convenient The assumption form. equivalent to setting has the form equation depends only determined by lenst to put equation (4) into matrix of constant rns emittancc is jZs = e(t)Y$. Then equation (4) h = Fo + aFT where the last equation follows because Fs = 0 by Newton’s third law. (We neglect the small magnetic selfIf Fe(x,t) is non-linear forces due to internal motion.) in x, the second equation of (2) involves the higher moments 7=c x of the distribution. for linear exHowever, ternal forces, Fe f -K(t)x, equations (2) involve only the first moments x and 5, and therefore the centre-of mass motion depends only on the external force, .. x + K(t); = 0 , (3) and not on the detailed form of the distribution. the remainder of this paper we consider only linear ternal forces. The second 7 - moments of f(x,p,t) satisfy the 7 equations -y xp = xp + xp = p 7 where higher = 2 2 - K(t)? + -XF, = -2li(t)xp + 2 pF, ) the terms3s and 3, moments xn and 5. 5 is are usually functions of This is 3 general feature (4) the of the covariance matrix t xp (10) .-I xp 7 1 and F is 1 ! 0 F= In ex- =22==2 7 where (9) (11) -K(t) Equation M is the I + F(t) + c(t) 0, (9) is equivalent to u(t + dt) = Mn(t)MT where infinitesimal transfer matrix X(t + dt, t) = dt. This procedure is easily extended to two For three dimensions, dimensions. the 6 x 6 matrix includes cross-correlation terms such the 6 x 6 force matrix F may 5’ , . . . . while linear coupling terms from both space-charge forces. The three-dimensional equivalent of 1105 and three correlation as 5, include and external (9) has been gate effects forces tion zero tions incorporated into program TRANSPORT’ to invcstiboth longitudinal and transverse space-charge in transfer lines”. In many cases the external will not involve coup1 ing and the cross-correlaterms between the different directions will be or close to zero. In this case the envelope equareduce to the R-V form (h) for each direction. One-dimensional -~ _~ envelope --~ where g = 1 + 2 In (pipe radius/beam corresponding envelope equation is (17) where N is the number of particles X2 = $ [ jk(z, ‘The envelope equation = 4ien(x,t) hl is long in the only the and this is . (12) 1 ; + K(t);: - AC - $ 5 = 0 ) x x3 per as unit (13) area dimensionless parameter u> 2 1 d(x) dx 7 h(x’) dx’ 0 -Ii --~ x i =in 1 x2h(x) dx % [ iu I)) uniform, 11(x) Farabolic h(x) specifies the =o for for 0 C) d) ~a”SSlZl, h(x) I,($ :u~llow, the values Envelope of \1 arc = ; for In this given Table dz (18) for continuous beams -- the solution c3 to Poisson’s equation + s) % is (23) ’ where x -< 1 x j 1 T=L+.L. a2 a similar (21) expression cxz b2 + s + s for t m 1. ” which -u suggests r I cos e^C With the performed the change of r sin 0 = -A--c--G' new variables, giving the integration dr N is the @ can be dr’ . (24) r can be evaluated with the help .x n(x,y) dx dy = ab number of -w where Then (23) over n(r”)2nr’ co /i 0 = ____ cu The remaining integrals of the definition 9 second type of one-dimensional envelope equation arises in the study of longitudinal oscillations of a bunched beam inside a conducting pipe’. The longitudinal self-field is determined by (22) variables 0 N= there- -w -XL: =- 4?iea3b2 n(r’)2xr x a+b I for the range 01 distributions likely to be encountered in practice, the variation in hi is negligible and the rms envelope motion will be accurately described by Cq. (14) with constant 41, for example Ai 3 l/6. is x dy (b2 + ,+ co rllus, The term z Y’ m 3A 1 ’ equations , I [YiqAl h’(z) . ..a C‘ x with fore in ‘Table case x -< 1 x :a 1 -x?,/2 E = L F2‘,-x2JQ . 1 2 “i (10 distribution. for - XT) dz ] (15) = + = ;(I and In the absence of cross-correlations and coupling terms, the envelope equations have the form (13) where the space-charge terms involve the average mx and KY. These averages will depend only on the second moments 2 and q and not on the higher moments Drovided the 1 a) bunch with values of hz listed in Table 1. For this case of a shielded electric field, the envelope equation does depend on the type of distribution. However, if the form of the distribution varies only slightly during its evolution, for example remains within the range uniformparabolic-Gaussian, then the envelope equation (17) can be used with confidence. in Ay the and Acre h(x) = (l/N)n(x) For the four distributions per -tx> is where M is tlw nwbcr of particles r; z . Thi.s equation can be written where and the equations For a beam in free space that is very z-direction and very r;ide in the y-direction, x-component of the self-force is important, obtained from the Poisson equation ai. ax radius), particle i 0 n(r’)Znr per unit dr , (25) length. K t(z,t) = -eg an(z,t) ___ a2 , (16) Q(r) 1106 = ab i 0 n(r’2)2m-’ dr’ (26) number is the becomes of within particles radius r, and with (24) Eq. the normalization u) 10 EL h(ri)r2 2ea = x a+b 2 - Qh-'11 [N dr’ dr = 1 . (37) (27) > / 0 which is easily XE + rN% =-z-.eN2a x this and the Using envelope equations I. ; The parameter X3 depends only weakly on the type of distribution as shown in Table 1. Thus for practical distrithe dependence of the envelope equations on the butions, type of distribution can be neglected. The same statement also applies if cross-correlations or linear external coupling forces are present; in this case the more general matrix form (9) of the rms equations can be used. integrated, a + b expression Kx(t)x - (7-8) ,+, for E 2 -.K. z e?N _ - 23 Y) the we obtain Conclusion 1 ~ ” = 0 A rather surprising and useful result has been found for beams in free space, namely that the linear part of the self-field depends mainly on the rms size of the distribution and only very weakly on its exact form. Using this result, envelope equations for the rms beam size have been derived that are exact for continuous beams of elliptical synmwtry, and in practice also valid for bunched beams of ellipsoidal form. The main restriction in applying these equations is that the time dependence of the rms emittnnce must be known 2 pricri . G+“y (29) $- + KY(t)7 - 5 - 1‘2N-i” Y =” . E+$ These equations are identical to the K-V equations if 7, Ey are replaced by the physithe ms values X, E,, cal boundary for a uniform distribution, namely they are not restricted to the x = 2 j;, . . . . However, K-V distribution but are valid for any distribution with the elliptical symmetry (1.9). Envelope for equations The procedure bunched beams in with for bunched two-dimensions the ellipsoidal Possible uses of the equations include the specification of stationary or matched states in the presence For example, tile periodic solution c>i of space charge. Eq. (35) for alternating-gradient structures, including radio frequency cavities, specifies the matched beam size (botlr longitudinal and transverse) as a function The largest matched of rms emittances and intensity. size attainable without exceeding aperture limits or bucket size determines a space-charge limit. For a envelope oscillations about beam matched in this way, the periodic solution are suppressed, although higher modes of oscillations (sextupole, octupole, etc.) nay Suppression of the higher modes will require OCC”tZ. as vet undetermined, on the higher moments constraints. of the distribution. Anotiler use is the design of lowenergy beam transfer lines. beams can be repeated symmetry (30) The electric field is” w & x n(T) = 2neabcx (a2 + &(b’ ds , + s) ‘4 + s)%(cz (31) 0 where x2 T=--+ ai -and with XC can x analogous be reduced N is b* expressions to the form (32) +L, + s C2 for + E - eNLA3 xc where YZ + s x = -- the number ad x of Y and c = . The term per bunclr and _-.-g,=; i h[~T+d~)%[i; + sj s ’ (j4) 3 (1+ s) The integral in tic integrals of evaluation with easier and also velope equation 1. F. Sacherer, charge, 2. I.M. Kapchinsky and V.V. Vladimirsky, Conf. on High-Energy Accelerators tation, CERN 1959, p. 274-288. 3. P. Lapostolle, Quelques proprietcs essentielles des effets de la charge d’espace dans des faisceaux continus, CERN internal report, CERN/ISR/DI/70-36. 4. R. Gluckstern, Batavia. 5. K.K. Brown 6. F.3. Sacherer 7. A. Sdrenssen, The effect of strong space-charge forces at transition, report, MPS/int. MUlFF!67-2. 8. The electric field for a uniform1.y charged ellipseid or elliptical cylinder is given, for example, by Foundations of Potential Theory (Dover Kellog, Publications, New York, 1953), p. 192. His derivation is easily generalized to include any or ellipsoidal charge distribution, as elliptical was pointed out by B. Houssais, Rennes University (private communication). (33) kcx particles References s (34) can be expressed in terms of ellipthe second kind, but direct numerical the Gaussian integration method is quick and accurate. The complete enfor X is (35) where A3 =-L 3 Jli u) w, h(r2)r4 dr h(r’)r’ dr OF r dp (36) 1107 RMS envelope CERN internal equations with space repurt, SI/DL/70-12. Discussion and S.K. and at Howry, T.R. 1970 SIAC-91 Sherwood, Proc. Int. and Instrumen- Linac Conference, (1970). Tliis \:onferencr. longitudinal CERN internal