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Landau anti-damping of the coherent beam–beam instability
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 588 (2008) 336–346 www.elsevier.com/locate/nima
Landau anti-damping of the coherent beam–beam instability D.V. Pestrikov Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russian Federation Received 20 December 2007; received in revised form 5 February 2008; accepted 8 February 2008 Available online 19 February 2008
Abstract Within the framework of a simplifying model and within the first approximation of the perturbation theory we discuss effects of the Landau damping on the stability of the coherent oscillations of short identical colliding bunches. Near the resonances n=ð2mÞ where n and m are integers these oscillations are unstable. Comparing the stopbands calculated for monochromatic bunches with those that are calculated taking into account the beam–beam tunespreads of the bunches, we have found that the latter increases the widths of the stopbands of coherent modes, thus resulting in the Landau anti-damping of the coherent beam–beam oscillations. r 2008 Elsevier B.V. All rights reserved. PACS: 29.20.Dh; 29.27.Bd Keywords: Beam–beam instability; Coherent oscillations; Colliders
1. Introduction The space charge forces of the counter-moving colliding bunches at the interaction point (IP) of a collider perturb the motions of particles. In circular colliders these beam–beam perturbations repeat periodically, resulting in numerous resonances. Besides, nonlinear dependencies of the space charge forces on particle coordinates in the bunches produce the tuneshifts and the tunespreads of the particle oscillations. For these reasons, the beam–beam perturbations result in the resonant instabilities of incoherent oscillations of particles as well as of the coherent oscillations of the colliding bunches. Both classes of these instabilities can limit the luminosity of a collider. Descriptions of such limitations are usually complicated by the fact that these manifold phenomena are generically selfconsistent. So, numerous simplifying assumptions are used to predict particular limitations. Theoretical studies of the limitations due to instabilities of the coherent beam–beam modes were started long time ago (see, e.g. in Refs. [1–8]). Using various approaches these studies have figured out that coherent oscillations of Tel.: +7 383 339 4412.
E-mail address: [email protected] 0168-9002/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.02.007
the colliding bunches may become unstable when the tunes of incoherent oscillations of particles approach nonlinear resonant values. Only the sum-type resonances produce the instabilities of the collective modes of the colliding bunches [5]. The values of the coherent tuneshifts of unstable beam–beam modes are equal to zero (e.g. in Ref. [5]). Therefore, some Landau damping of these unstable modes could be expected. However, a qualitative study made in Ref. [7] has shown that the beam–beam tunespread is not sufficient to suppress the beam–beam coherent instability and that near the resonant tunes, unstable coherent beam–beam modes can still exist. Using a different approach, this result was confirmed numerically for the beam–beam p-modes in Ref. [8]. We use the simplifying model described in Refs. [5,7,8] or in Ref. [9] to study the stability of the self-consistent coherent beam–beam oscillations of short colliding bunches. In this paper, we focus on quantitative predictions of the model and on the studying of various effects of the Landau damping on the stability of the coherent beam–beam oscillations. Using the technique developed in Ref. [10], we calculate the stopbands for coherent oscillations of colliding bunches taking into account and/or ignoring the beam–beam tunespreads. For the last case, we use to call colliding bunches as the monochromatic ones.
ARTICLE IN PRESS D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 588 (2008) 336–346
Comparing the results of such calculations, we find out that Landau damping due to the nonlinearity of the beam– beam forces, generally, increases the widths of the stopbands of coherent beam–beam modes. It also can change positions of the stopbands of unstable modes relative to the values of the resonant tunes. This means that together with the damping of the unstable modes of the monochromatic colliding bunches the beam–beam tunespreads result in the instabilities for the regions of the tunes where the modes of the monochromatic bunches were stable—i.e. in the Landau anti-damping. Such a Landau anti-damping is specific for the resonant instabilities of coherent oscillations near the sum-type coupling resonances [11]. More precisely, the calculations in Ref. [11] (or, in e.g. Ref. [9]), in particular, predict that two modes coupled near the sum-type resonance are unstable in the case, if only one of the coupled modes is damped. We remind the reader that Landau damping of collective modes occurs due to the particles, which are in the resonance with a beam mode. On its turn, coherent beam–beam oscillations become unstable near the tune nb ¼ n=ð2mÞ (where n and m are e.g. positive integers) due to the resonant coupling of the modes ðm; nÞ and ðm; nÞ. If the Landau damping resonance condition holds for a given value of the coherent tuneshift for the mode ðm; nÞ, it results in the damping of the mode. However, simultaneously this resonance condition is violated for the mode ðm; nÞ resulting in a zero damping decrement of the mode ðm; nÞ. The sum-type coupling of the damped and non-damped modes results in the mentioned Landau antidamping of the beam–beam modes [11]. 2. The model We consider collisions of two short counter-moving relativistic bunches, which move in separate storage rings and interact head-on at a single IP. In our calculations we assume a zero dispersion function at the IP. Then, unperturbed incoherent vertical (y) and horizontal (x) betatron oscillations of particles at the center of IP are described using the following expressions: sffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi Jy py0 y ¼ J y by cos cy ; py ¼ ¼ p sin cy R0 by sffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px0 Jx x ¼ J x bx ðsÞ cos cx ; px ¼ ¼ p sin cx R0 bx I x;y
pJ x;y ; ¼ 2
ðcx;y Þ0 ¼ nx;y .
(1)
Here, the couples ðI; cÞx;y stand for the action-phase variables of the unperturbed incoherent oscillations, P ¼ 2pR0 is the perimeter of the closed orbit, s ¼ R0 y is the path along the closed orbit, primes mean d=dy, p ¼ gMc is the value of the momentum of the reference particle, the values bx;y stand for b-functions of the horizontal and vertical oscillations at the center of IP. The colliding
337
bunches are assumed to be short. This means that the bunch lengths ss are supposed to be substantially smaller than the values of b-functions at the center of IP. For this reason we shall ignore in our calculations the synchrotron oscillations of the particles. In the action-phase variables incoherent betatron oscillations of the bunch particles are described using the Hamiltonian H 0 ¼ nx I x þ ny I y .
(2)
Passing the interaction region, the particles in the bunches are perturbed by the space charge forces of the countermoving bunch. Marking the colliding bunches by the symbols ð1Þ and ð2Þ, we describe such perturbations using the Lagrangians L1;2 and L2;1 , respectively. For example, the oscillations of the particles in the bunch (1) are described using the Hamiltonian H 1 ¼ H 0 L1;2 .
(3)
2.1. Coherent oscillations of bunches Without coherent oscillations the bunches are described using the distribution functions, which do not depend on the phase variables cx and cy . We write f ð1;2Þ ¼ f ð1;2Þ 0 ðIÞ;
I ¼ fI x ; I y g.
We assume that the distribution functions normalized using Z d2 If ð1;2Þ 0 ðIÞ ¼ 1.
(4) f ð1;2Þ 0 ðIÞ
are
(5)
We also assume that the bunch ð1Þ passes IP in the positive direction while the bunch ð2Þ—in the negative direction. Correspondingly, we shall write the longitudinal coordinates in the bunches ð1Þ and ð2Þ using z1;2 ¼ s R0 y, where y ¼ o0 t and o0 is the revolution frequency of the reference particle. Coherent oscillations of the bunches are described by small additions to f 0 , which are functions of the time t and are the periodic functions of the phase variables: ð1;2Þ f ð1;2Þ ¼ f ð1;2Þ 0 ðIÞ þ df
where df ð1;2Þ ¼
X
(6)
f ð1;2Þ m ðI; tÞ expðimx cx þ imy cy Þ,
m
m ¼ fmx ; my g.
(7)
Provided that the perturbations of coherent oscillations due to beam–beam kicks result in small variations of action variables of the particle oscillations I during the revolution period in the ring (2p=o0 ), the amplitudes f ð1;2Þ can be m calculated using the Vlasov equations which are linearized in df ð1;2Þ m . Using azimuth y as an independent variable, assuming the horizontal coherent oscillations (m ¼ fmx ; 0g) and neglecting the contributions of the fast oscillating terms in the linearized Vlasov equation, we find that the
ARTICLE IN PRESS D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 588 (2008) 336–346
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amplitudes f ð1;2Þ obey the following equations: m qf ð1Þ qf ð1Þ mx ð1Þ 0 þ imx nð1Þ ¼0 x f mx þ imx ðdL1;2 Þmx qy qI x qf ð2Þ qf ð2Þ mx ð2Þ 0 þ imx nð2Þ f þ im ðdL Þ ¼ 0. x 2;1 mx x mx qy qI x
Within the framework of our model Eqs. (8) can be rewritten in the following form:
(8)
qf ð1Þ ð1Þ m þ imnð1Þ x fm qy Z 1 1 qf ð1Þ N 2 e2 X ¼ imdT ðyÞ 0 imm3 dI 1 K m;m3 ðI; I 1 Þ qI c m ¼1 0 3
Together with Eq. (1), these equations imply that we also neglect the effects of the nonlinear beam–beam resonances on the incoherent oscillations of the particles and by possible effects of these resonances on the stability of the coherent beam–beam modes. For relativistic particles (gb1) we can calculate the space charge fields of the colliding bunches in the local field approximation, which results in e.g. Z 2N 2 e1 e2 dkx dky L1;2 ðI; c; yÞ ¼ expðikrÞ po0 k2x þ k2y Z 2 d I 1 d2 c1 dðs s2 Þ ð2pÞ3 expðikr1 Þf ð2Þ ðI 1 ; c1 ; yÞ
(9)
where kr ¼ kx xðI; c; yÞ þ ky yðI; c; yÞ and N 2 e2 or N 1 e1 are the charges of the bunches (2) or (1), respectively. For short bunches (bx;y bss ), the beam–beam instability occurs due to a sequence of short periodic kicks, which for e1 e2 ¼ e2 are described by the following expressions: Z dkx dky N 2 e2 dT ðyÞ expðikrÞ L1;2 ðI; c; yÞ ¼ pc k2x þ k2y Z 2 d I 1 d2 c1 expðikr1 Þf ð2Þ ðI 1 ; c1 ; yÞ. (10) ð2pÞ3 Here, dT ðyÞ is a periodic d-function: dT ðyÞ ¼
1 X
1 einy . 2p n¼1
qf ð2Þ ð2Þ m þ imnð2Þ x fm qy Z 1 1 qf ð2Þ N 1 e2 X ¼ imdT ðyÞ 0 imm3 dI 1 K m;m3 ðI; I 1 Þ qI c m ¼1 0 3
ðf ð1Þ m3 ðI 1 ; yÞ
þ
f ð1Þ m3 ðI 1 ; yÞÞ.
(11)
(13)
Here, we use 1 X
expðiz cos fÞ ¼
ik J k ðzÞeif
(14)
k¼1
where J k ðzÞ is the Bessel function, J k ðzÞ ¼ ð1Þk J k ðzÞ and J k ðzÞ ¼ ð1Þk J k ðzÞ, and we define qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi Z 1 dk ð1;2Þ J m k Ibð1Þ J m3 k I 1 bð2Þ . K m;m3 ðI; I 1 Þ ¼ k 1 Below, we assume that for the bare and for the dynamic b-functions at the IP holds the condition bð1Þ ¼ bð2Þ . Then, we find ð2;1Þ K ð1;2Þ m;m3 ðI; I 1 Þ ¼ K m;m3 ðI; I 1 Þ ¼ K m;m3 ðI; I 1 Þ
and
Z
K m;m3 ðI; I 1 Þ ¼ 2 0
1
pffiffiffiffiffi pffiffiffi dk J m ðk I ÞJ m3 ðk I 1 Þ k
m þ m3 ¼ 2l
(15)
K m;m3 ðI; I 1 Þ ¼ 0; m þ m3 ¼ 2l þ 1
Eqs. (8) are too complicated even for solving numerically. We simplify our calculations considering a special model, where the bunches have very flat distributions f ð1;2Þ ! dðI y Þf ð1;2Þ 0 0 ðI x Þ
ð2Þ ðf ð2Þ m3 ðI 1 ; yÞ þ f m3 ðI 1 ; yÞÞ
(16)
where l is an integer. Generally Eqs. (8) and (13) are simplified, when two identical bunches collide while passing through the rings with identical lattice focusing. For such bunches, we write ð2Þ f ð1Þ 0 ¼ f 0 ¼ f 0;
ð2Þ nð1Þ x ¼ nx ¼ nx
(17)
and find that the combinations (12)
and where the bunches execute coherent oscillations in the horizontal plane only (m ¼ fmx ; 0; 0g). In this model, the collective beam–beam modes are unstable near the sumð2Þ type resonances [5] m1 nð1Þ x þ m2 nx ¼ n, where n, m1 and m2 are integers (m1 m2 40). Qualitative predictions of such a model concerning the stability of coherent beam–beam oscillations have been already studied in our paper [7] (see also in the book Ref. [9]). Here we use the technique developed in Ref. [10] to focus on quantitative predictions of the model and on the studying of particular effects of Landau damping on the stability of the coherent beam– beam oscillations.
ð1Þ ð2Þ f ðÞ m ¼ fm fm
(18)
obey independent equations. This means that the functions describe the normal modes of identical colliding f ðÞ m bunches [6]. Within the framework of the discussed model the equations for f ðÞ m read qf ðÞ m þ imnx f ðÞ m qy 1 qf N 2 e2 X ¼ imdT ðyÞ 0 imm3 qI c m ¼1 3 Z 1 ðÞ dI 1 K m;m3 ðI; I 1 Þðf ðÞ m3 ðI 1 ; yÞ þ f m3 ðI 1 ; yÞÞ. 0
(19)
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2.2. Incoherent oscillations According to the general rules of the theory of coherent oscillations (see e.g. in Ref. [9]), the tunes nxð1;2Þ in Eqs. (8) and (13) should be calculated taking into account incoherent beam–beam kicks at IP. Within the framework of our model, the horizontal oscillations of particles in the bunch (1) are perturbed at IP by the force qLð0Þ N 2 e2 p 1;2 ¼ dT ðctÞ x b2 qx 2 Z 1 du ð2Þ px pffiffiffiffiffiffiffiffiffiffiffi f 0 u 2b2 1u 0
F x ¼ o0
(20)
where b1;2 are the values of the b-functions of the horizontal betatron oscillations at IP for the bunches (1) and (2). If ð1;2Þ are the horizontal emittances of the colliding bunches (f 0ð1;2Þ ð0Þ ¼ 1=ðpð1;2Þ Þ), then simple calculations (see also e.g. in Ref. [4]) result in the following expressions for betatron tunes nð1;2Þ and for b-functions of x the horizontal betatron oscillations at IP for the colliding bunches: cos mð1Þ ¼ cos m0
N 2 e2 b0 sin m0 pcð2Þ b2
b1 sin mð1Þ ¼ b0 sin m0
(21) 2
cos mð2Þ ¼ cos m0
N 1 e b0 sin m0 pcð1Þ b1
b2 sin mð2Þ ¼ b0 sin m0 .
(22)
Here, m ¼ 2pnx , m0 ¼ 2pnx0 . The values nx0 and b0 define the betatron tune and b-function of the horizontal oscillations calculated without collisions of the bunches— the bare tune and b-function. Eqs. (21) and (22) describe self-consistent variations of the betatron tunes and of the b-functions at IP due to the beam–beam interactions. Due to their self-consistency these equations have solutions and, therefore, predict stable betatron oscillations of particles for all values of m0 and b0 . For identical colliding bunches (N 1 ¼ N 2 ¼ N and ð1Þ ¼ ð2Þ ¼ ) Eqs. (21) and (22) read cos mð1Þ ¼ cos m0 B
b0 sin m0 b2
b1 sin mð1Þ ¼ b0 sin m0 cos mð2Þ ¼ cos m0 B
(23) b0 sin m0 b1
b2 sin mð2Þ ¼ b0 sin m0
(24)
where B ¼ 2px and x is the beam–beam parameter: x¼
Ne2 . 2ppc
(25)
In the realistic region of parameters Bo1, Eqs. (23) and (24) predict equal variations of the betatron tunes and b-functions for all values of nx0 . Besides, in the narrow region: ! 2 B 1 B2 1 arccos pm0 p arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Bo1 (26) 1 þ B2 1 þ B2 Eqs. (23) and (24) predict a flip-flop splitting of the betatron tunes and of the b-functions where e.g. the values b1 and b2 can differ substantially. For small values of x the width of such a flip-flop region is also small (Dm0 ’ 0:3B). In this paper we ignore the possibility of such splitting and, hence, assume equal self-consistent variations of the betatron tunes and of the b-functions for identical colliding bunches. Then, the solutions to Eqs. (23) and (24) yield qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos m0 B B2 þ sin2 m0 cos mx ¼ 1 þ B2 kppm0 pkp þ p (27) where k is an integer and cos m0 ¼ cos mx þ B sin mx .
(28)
For small values of x the tune shift due to the beam–beam interactions (nx nx0 ) is equal to x. However, with an increase in x the tune shift deviates from x especially in the regions, where nx0 approaches parametric resonances (Fig. 1). We remind the reader, that according to Eq. (27) the variations of the bare tune nx0 within the region k=2pnx0 pðk þ 1Þ=2 (where k is an integer) result in the variations of nx within the region k kþ1 þ Dx pnx p 2 2
(29)
2.0
1.5 (νx-ν0)/ξ
Since, generally, the functions f ð1;2Þ are linear combinations m of the modes f ðÞ , the coherent oscillations of identical m colliding bunches are stable only in the regions, where both ðÞ modes f ðþÞ m and f m are stable.
339
1
2
1.0
0.5
0.0 0.0
0.1
0.2
ν0
0.3
0.4
0.5
Fig. 1. Dependence of the beam–beam tune shift on the bare tune (nx0 ); line 1 x ¼ 0:05, line 2 x ¼ 0:005.
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1.0
3. Dispersion equations and modes We simplify the calculations of the Landau damping of the coherent oscillations of identical colliding bunches (N 1 ¼ N 2 ¼ N) assuming that the distribution function f 0 ðIÞ in Eq. (19) is an exponential:
0.8
νx
0.6
f0 ¼
0.4
1 x e ; p
x¼
I p
where is the horizontal bunch emittance. Substituting this expression in Eq. (33), we obtain
0.2
nx ðIÞ ¼ nx0 þ Dnx ð0Þ 0.0 0.0
0.2
0.4
νx0
0.6
0.8
1.0
Fig. 2. Dependence of the tune nx on the bare tune (nx0 ); horizontal betatron oscillations, x ¼ 0:05, only the square 0pnx0 p1 and 0pnx p1 is shown.
where
(36)
Now, substituting f 0 from Eq. (34) in Eq. (19) 1 X
iny f ðÞ m;n e
(37)
n¼1
(30)
This means that for a given value of x the spectra of betatron oscillations of the colliding bunches in nx have a slot just above a parametric resonance with the width Dx (Fig. 2). Since positions of the upper borders in Eq. (29) do not depend on x, the oscillations of particles in the colliding bunches cannot be perturbed by nonlinear resonances nx ¼ n=ð2mÞ, where n and m are natural numbers obeying the condition: k n k p p þ Dx . 2 2m 2
(35)
1 ex nx ðIÞ ¼ nx Dnx ð0Þ 1 . x
iny f ðÞ m ¼ e
1 1B arccos . 2p 1 þ B2
1 ex x
or
2
Dx ¼
(34)
f ðÞ m;n ¼
In wider regions of x and for the discussed model, the dependence of the tune shift on the amplitudes of the horizontal betatron oscillations of particles can be found using one-dimensional simulations, where the beam–beam kicks are described by the force, given in Eq. (20). For identical colliding bunches the result can be written in the form similar to Eq. (32): Z p I nx ðIÞ ¼ nx0 þ Dnx ð0Þ dI 1 f 0 ðI 1 Þ (33) I 0 where Dnx ð0Þ is the beam–beam tune shift of the linear betatron oscillations calculated using e.g. Eq. (27).
f ð1Þ m;n ¼
X ðÞ m ðIÞ n þ n þ mnx ðIÞ
(38)
and defining x¼
Ne2 2ppc
(39)
ðÞ we find X ðÞ m ¼ X m and Z 1 1 X ðÞ x dx1 imm3 K m;m3 ðx; x1 Þ X m ¼ mxe 0
X ðÞ m3 ðx1 ÞðS m3
(31)
Nonlinear dependencies of the beam–beam forces on transverse coordinates of particles result in the tune spreads in the colliding bunches. For small x, simple calculations in the first approximation of the perturbation theory and within the framework of the model result in Z Ne2 I ð1;2Þ nð1;2Þ ðIÞ ¼ n þ dI 1 f ð1;2Þ (32) x x0 0 ðI 1 Þ. 2pcI 0
X ðÞ m ðIÞ ; n þ n mnx ðIÞ
m3 ¼1
S m3 Þ.
(40)
Here, Sm ¼
1 X
1 . n þ n mnx ðxÞ n¼1
(41)
For small values of the beam–beam parameter x (e.g. B ¼ 2pxo1) and according to the general rules of the theory of the linear coherent oscillations (see, e.g. in Ref. [9]), the value of the eigentune n in Eq. (40) for the mode with the multipole number m is expected to be close to mnx . On the other hand, since the beam–beam interactions occur via conservative forces, the oscillation modes can be unstable only in the cases when the tunes nx approach the resonant values nx ¼ n=ð2mÞ, where n and m are integers. For simplicity, we assume that for small x the beam–beam resonances of the modes with different values of m do not interfere and, correspondingly, that we may omit on the right-hand side of Eq. (40) the off-diagonal terms m3 am. The resulting equation reads Z 1 x ðxÞ ¼ e dx1 V ðx1 ÞK m ðx; x1 ÞX ðÞ (42) X ðÞ m m ðx1 Þ 0
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where (m40; [12]) pffiffiffi Z 1 pffiffiffiffiffi J m ðk xÞJ m ðk x1 Þ dk K m ðx; x1 Þ ¼ 2 k 0 8 m=2 1 < ðx=x1 Þ ; xox1 ¼ m : ðx1 =xÞm=2 ; x4x1 : V ðxÞ ¼
2dðxÞ ; 2 z1 d2 ðxÞ
Im z1 40
Namely, using Eq. (48), we find Z x dpðÞ 1 ðÞ m ðxÞ ¼ mþ1 dx1 ex1 xm 1 V ðx1 Þpm ðx1 Þ dx x 0 (43)
(44)
1 n n ; mx 2
dðxÞ ¼
1 n nx ðxÞ x 2m
(45)
where n is the azimuthal number of the resonance harmonic. Substituting Eq. (43) in Eq. (42) and taking e.g. x m=2 ðÞ pm ðxÞ X ðÞ m ¼ e x
(46)
we obtain the following integral equation for a new unknown function pðÞ m ðxÞ: Z x 1 xm pðÞ dx1 1m ex1 V ðx1 ÞpðÞ m ðxÞ ¼ m ðx1 Þ m 0 x Z 1 1 þ dx1 ex1 V ðx1 ÞpðÞ m ðx1 Þ m x or pðÞ m ðxÞ
1 ¼ m
Z
1
dx1 ex1 V ðx1 ÞpðÞ m ðx1 Þ Z x 1 ðÞ þ dx1 ex1 xm 1 V ðx1 Þpm ðx1 Þ mxm 0 Z 1 x dx1 ex1 V ðx1 ÞpðÞ m ðx1 Þ. m 0 0
(47)
Since pðÞ m ð0Þ ¼
1 m
Z 0
wðxÞ x we obtain (w0 ¼ dw=dx)
dx1 ex1 V ðx1 ÞpðÞ m ðx1 Þ
we normalize the eigenfunctions using the condition pðÞ m ð0Þ ¼ 1. Then, Eq. (47) is replaced by the following nonhomogeneous integral equation for the function Z x 1 ðÞ ðxÞ ¼ 1 þ dx1 ex1 xm pðÞ m 1 V ðx1 Þpm ðx1 Þ mxm 0 Z 1 x dx1 ex1 V ðx1 ÞpðÞ (48) m ðx1 Þ m 0 and by the dispersion equation of the problem: Z 1 1 dxex V ðxÞpðÞ 1¼ m ðxÞ. m 0
(49)
The function pðÞ m ðxÞ in these equations depends also on the eigentune (z1 ) like on a parameter. For this reason, ðÞ sometimes we shall write pðÞ m ðz1 ; xÞ instead of pm ðxÞ. Following the paper [10], we transform the integral equation (48) in the differential equation for the function pðÞ m ðxÞ.
(51)
(52)
ðm 1Þ 0 ðm 1Þ ex w V ðxÞwðxÞ. (53) w¼ 2 x x x In order to provide pðÞ m ð0Þ ¼ 1, this equation should be solved with the boundary conditions: w00 þ
wð0Þ ¼ 0;
w0 ð0Þ ¼ 1.
(54)
The calculation of the integral in the dispersion equations (49) can be avoided using the transformations similar to those found in Ref. [10]. We note that according to Eq. (51), the following equations hold: Z 1 1 1¼ dxex V ðxÞpðÞ m ðxÞ m 0 Z 1 ðÞ 1 dx d mþ1 dpm ðxÞ x ¼ m 0 xm dx dx Z 1 ðÞ 1 dp ðxÞ dpðÞ ðxÞ x m ¼ dx m m dx dx 0 x¼1 1 dpðÞ ðxÞ x m ¼ pðÞ m ð1Þ þ 1. m dx x¼1 Since in the asymptotic region (x ! 1) solutions to Eq. (51) read dpðÞ C m ðxÞ ¼ mþ1 ; dx x
1
(50)
Substituting here1 pðÞ m ðxÞ ¼
and z1 ¼
and ðÞ d mþ1 dpm ðxÞ x ¼ ex xm V ðxÞpðÞ m ðxÞ. dx dx
341
pðÞ m ðxÞ ¼ C 1 þ
C ðm þ 1Þxm
where C and C 1 are arbitrary functions of z1 , we find dpðÞ m ðxÞ lim x ¼0 (55) x!1 dx and the dispersion equation in the following form: pðÞ m ðz1 ; 1Þ ¼ 0.
(56)
This fact simplifies substantially the calculations of the eigenvalues and of the stability conditions of coherent oscillations of the colliding bunches within the framework of the used model. Similar calculations for the mode ðþÞ result in the dispersion equation pðþÞ m ðz1 ; 1Þ ¼ 0 1
(57)
Corresponding equation in Ref. [7] is obtained, if we substitute ¼ xðmþ1Þ=2 wðxÞ in Eq. (51), which yields
pðÞ m ðxÞ
x e V ðxÞ m2 1 w00 ¼ w. 2 x 4x
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ðþÞ where pðþÞ ðxÞ=x and the function wðþÞ ðxÞ obeys m ðxÞ ¼ w Eqs. (53), (54) with the substitution V ðxÞ ! V ðxÞ:
lim pðÞ m ðz1 ; 1Þ ¼ 1.
0.4 pfin(-)
d2 wðÞ ðm 1Þ dwðÞ ðm 1Þ ðÞ þ w x x2 dx2 dx ex V ðxÞwðÞ ðxÞ. ¼ (58) x According to Eq. (48) or Eqs. (58) and (54) the functions pðÞ m ðz1 ; xÞ obey the asymptotic condition
0.6
0.2
0.0
(59)
jz1 j!1
This means that the roots of the dispersion equations (56) and (57) can be found only within a limited region near the origin in the plane of the complex variable z1 . We have made the calculations assuming the eigenvalue problem and the studying of the mode stability conditions. The oscillations spectra of the colliding bunches are found solving the initial value problem. Such calculations are similar to those described in e.g. Ref. [10].
-0.2 0.1 Im(Δνm)/(mξ)
0.01
1
Fig. 3. Dependence of e.g. pðÞ ðz1 ; 1Þ on z1 . Solid line—m ¼ 1, dashed line—m ¼ 2; x ¼ 0:05, for both curves nx corresponds to the point of the maximum increment.
4. Stopbands of nonmonochromatic bunches
1/2
0.6
2dðxÞ ; þ d2 ðxÞ
ImðnÞ . mx
0.4
0.2
Corresponding eigensolutions wðÞ ðr; xÞ are real functions while the dispersion equations (56) or (57) have real roots r. For this paper, the values of the increments of unstable modes as well as the widths of relevant stopbands and their positions in nx are found solving Eqs. (56)–(58) numerically. As far as the coefficients in Eq. (58) diverge at the point x ¼ 0, solutions to Eq. (58) near the origin were calculated (xpx0 ¼ 1010 ) using their series expansions in powers of xpx0 and the boundary conditions in Eq. (54):
0.1
r2
wðÞ ðxÞ ¼ x
2d x2 ðr2 þ d2 Þ ð1 þ mÞ 2
þ
r¼
2
2
2
2
4d ð1 þ mÞðbðr d Þ þ 2dðr þ d ÞÞ 3 x (61) 2ðr2 þ d2 Þ2 ðm þ 2Þð1 þ mÞ
where b ¼ Dnx ð0Þ=x. Solutions to Eq. (58) for wider region xXx0 were calculated using wðÞ ðx0 Þ and ðdwðÞ dxÞx¼x0 from Eq. (61) as the boundary values. Trying several different large values of x, we found that the asymptote x ! 1 occurs, if we take e.g. x ¼ 80. From two examples depicted, e.g., dependencies of pðÞ m ðz1 ; 1Þ on z1 (Fig. 3), we see that within the stopband and for the given x and nx the dispersion equations (56), (57) may have one or several roots z1 . The code solving the dispersion equations (56) and (57) calculated only the maximum increment of the collective modes.
1/6 2/10 1/4 3/10 2/6 3/8 4/10
1/10
0.3
(60)
V ðxÞ ¼
1/8
0.5
Im(Δν)/ξ
As it was found qualitatively in Ref. [7], near the resonances nx ’ n=ð2mÞ Eq. (42) always has solutions with eigenvalues Reðz1 Þ ¼ 0 and Imðz1 Þa0. Since pðÞ m ðz1 ; 1Þ are functions of z21 , these solutions describe the unstable modes. Substituting z1 ¼ ir in Eq. (44), we find that in such cases V ðxÞ is a real function
0.0 0.0
0.1
0.2
νx
0.3
0.4
0.5
Fig. 4. Dependencies of the increments of the coherent beam–beam modes on the tune of the horizontal incoherent betatron oscillations (nx ). Modes 1pmp5, arrows show positions of relevant stopbands near particular resonances, wider curves to the right from the resonance nx ¼ n=ð2mÞ depict the increments of the modes ðþÞ, x ¼ 0:05.
Results of such calculations for the modes ðÞ with 1pmp5 and 1pnp4 are depicted in Fig. 4. In this picture the increment of an oscillation mode starts from a zero value at some particular value nx;in , passes the maximum value and then decreases to zero again. Those curves depict the stopbands of coherent oscillations. Although only the segment 0pnx p12 is shown in Fig. 4, the stopbands for higher or lower values of nx appear periodically with a period in nx of 12. For dipole oscillations m ¼ 1 we have found no roots of the dispersion equation (57) for 0pnx p12. This result means that only ðÞ dipole mode has the stopband below the resonance nx ¼ 12. The stopband for the mode ð; 2Þ starts slightly below the resonance nx ¼ 14. The lower ends of all other found stopbands are found to be close to the corresponding resonant tunes nx ¼ n=ð2mÞ.
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Numerical values of the maximum increments and of the widths of the stopbands for modes ðÞ and 1pmp3, which are shown in Fig. 4, are in general agreement with similar results reported in Ref. [8] and obtained using a different approach. The deviations from the numbers found in Ref. [8] occur due to thedifferences in the dispersion properties of incoherent betatron oscillations of the particles taken into account in this paper and in Ref. [8]. In particular, due to this difference the values of the maximum increments for the modes mX3 are a bit smaller in the region nx o14 than that above nx ¼ 14. The maximum increments of the modes near the resonances nx ¼ 1=ð2mÞ tend to zero, when nx approaches the border value Dx from Eq. (30). This fact is in general agreement with the experimental observations of the beam–beam instabilities in the electron–positron colliders. Although it is not shown here, a decrease in x results in the decreases in the values of the mode increments (Im n) and in the narrowing of the widths of the stopbands in nx . However, the ratios Im n=x and of these widths to x remain the same. 5. Stopbands of monochromatic bunches Effects of Landau damping on the stability of coherent oscillations of the colliding bunches were studied comparing the stopbands, described in the previous subsection, with those, calculated using the distribution f 0 from Eq. (34) and neglecting the tunespread of incoherent betatron oscillations due to the beam–beam interactions. The modes of such monochromatic bunches are described using Eq. (42) Z 1 ðÞ x X m ðxÞ ¼ e dx1 V ðx1 ÞK m ðx; x1 ÞX ðÞ (62) m ðx1 Þ 0
where we put V ðxÞ ¼
1 1 . z1 d z1 þ d
z1 is defined in Eq. (45), but 1 n nx d¼ x 2m
the eigennumbers L of Eq. (67) are real positive numbers for modes ðÞ. For the modes ðþÞ the eigennumbers are negative numbers L. Therefore, for the monochromatic colliding bunches the eigentunes of the collective modes due to their beam–beam interactions are determined using the dispersion equation z21 ¼ d2 2Ld ¼ ðd LÞ2 L2
(64)
(65)
as a new eigenvalue and defining x X ðÞ ðxÞ ¼ exp PðÞ ðxÞ (66) m 2 m we transform Eq. (62) into the integral equation with a symmetrical kernel Z 1 x=2 LPðÞ ðxÞ ¼ e dx1 ex1 =2 PðÞ m m ðx1 Þ 0 pffiffiffi Z 1 pffiffiffiffiffi J m ðk xÞJ m ðk x1 Þ dk. (67) 2 k 0
(68)
where the sign þ is taken for the mode ðÞ, L is an eigenvalue of the integral equation Z 1 ðþÞ x=2 LPm ðxÞ ¼ e dx1 ex1 =2 PðþÞ m ðx1 Þ 0 pffiffiffi Z 1 pffiffiffiffiffi J m ðk xÞJ m ðk x1 Þ dk (69) 2 k 0 x=2 or (PðþÞ PðxÞ) m ðxÞ ¼ e Z x 1 m=2 LPðxÞ ¼ dx1 x1 ex1 Pðx1 Þ mxm=2 0 Z xm=2 1 ex1 Pðx1 Þ þ dx1 . m=2 m x x
(70)
1
Substituting PðxÞ ¼ xm=2 pðL; xÞ in Eq. (70), we find Z x 1 x1 LpðL; xÞ ¼ dx1 xm pðL; x1 Þ 1e mxm 0 Z 1 1 þ dx1 ex1 pðL; x1 Þ m x
(71)
and that the eigennumbers L are calculated solving the dispersion equation (72)
where pðL; xÞ ¼ wðxÞ=x and m1 0 m1 w w x x2 ex ¼ wðxÞ; wð0Þ ¼ 0; w0 ð0Þ ¼ 1. (73) Lx Note also, that according to Eq. (70), for a given value of m the sum of all L is equal to 1=m. The eigenvalues L were calculated solving Eqs. (73) and (72) numerically. The results of these calculations clearly indicated an existence of the well-defined ground state mode with the maximum value of L (see e.g. in Fig. 5). On the other hand, Eqs. (69) or (70) are the homogeneous integral equations with symmetrical and positively defined kernels. The maximum eigenvalues to Eq. (70) can be evaluated using the extrema properties of its eigenvalues (e.g. in Ref. [13]). Taking as a probe ground state solution to Eq. (69) the function rffiffiffiffiffiffi a ðþÞ Pm ðxÞ ¼ (74) ðaxÞm=2 exa=2 m! w00 þ
does not depend on x. The kernel K m ðx; x1 Þ is defined in Eq. (43). Using 1 1 1 ¼ L z1 d z1 þ d
Since 2 Z 1 Z 1 pffiffiffi ðÞ dk x=2 dxe J m ðk xÞPm ðxÞ 40 2 k 0 0
pðL; 1Þ ¼ 0 (63)
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344
For this reason, the stopbands and maximum increments of unstable modes of the monochromatic colliding bunches were calculated by substituting Lmax from Eq. (76) in Eq. (68). The oscillation increments r ¼ Im n=x reach their maximum values rmax ¼ mLmax when d ¼ Lmax for the modes ðÞ and when d ¼ Lmax for the modes ðþÞ. The stopbands are placed below the resonance 2Lmax pdp0 for the modes ðÞ and above the resonance 0pdp2Lmax for the modes ðþÞ. According to Eqs. (68), (76) and (30) the dipole mode ðþÞ is stable provided that the value of x is in a realistic region (actually, if xo0:2872).
0.3
pfin(Λ)
0.2 0.1 0.0 -0.1 -0.2
6. Discussion 0.1 Λ
1
Fig. 5. Dependence of pðL; 1Þ on L for m ¼ 1, the first root of pðL; 1Þ defines Lmax .
0.6 0.5
Λmax
0.4 0.3 0.2 0.1 0.0
1
2
3 m
4
5
Fig. 6. Dependence of Lmax on the multipole number m; open circles— using numerical solutions of Eqs. (73) and (72), solid line—using Eq. (76).
calculating the probe eigenvalue Z 2amþ1 1 dxxm=2 exðaþ1Þ=2 LðaÞ ¼ m! 0 pffiffiffi Z 1 Z 1 pffiffiffiffiffi J m ðk xÞJ m ðk x1 Þ m=2 dk dx1 x1 ex1 ðaþ1Þ=2 k 0 0 4 amþ1 ¼ (75) m ða þ 1Þmþ2 and defining the value of a using the equation dLðaÞ=da ¼ 0, we find Lmax ¼
4 mðm þ 2Þ 1þ
1 1 mþ1
mþ1 .
(76)
Note that for large multipole numbers m (mb1) the value Lmax varies proportionally to 1=m2 . The calculations of the maximum eigenvalues Lmax using Eq. (76) and numerical solutions of Eqs. (73) and (72) agree well (Fig. 6).
Comparing the curves in Fig. 7, we find that for the dipole coherent mode ðÞ of the colliding bunches Landau damping results only in minor changes of the stopband of the monochromatic bunches. The maximum values of the increments almost coincide, Landau damping suppress the oscillations within a narrow band A0 A and decreases the oscillation increments within the segment AB. Within the segment BC the beam–beam tune spread slightly increases the increments of unstable modes hence, resulting in some Landau anti-damping. Stronger Landau anti-damping indicates the stopbands of the modes with higher betatron multipole numbers (mX2, e.g. in Figs. 8 and 9). For such modes, the beam–beam tunespread although decreases the values of the maximum increments, moves the lower border of the stopbands of the modes ðÞ towards the resonant tune n=ð2mÞ and substantially increases the widths of the stopbands. Except for the case m ¼ 2, the stopbands of the multipole modes ðÞ are placed above the resonant tunes almost entirely. Hence, the beam–beam tunespread suppressing the modes of the monochromatic colliding bunches opens new wide regions of the tunes nx where the oscillations become unstable. The described Landau 0.6 B
0.5
Im(Δν)/ξ
0.01
0.4 0.3 0.2 0.1 0.0
A' 0.44
C
A 0.45
0.46
0.47 νx
0.48
0.49
0.50
Fig. 7. Dependence of the increment of the dipole mode ðÞ on the tune of the horizontal incoherent oscillations nx . Solid line—Landau damped coherent oscillations, dashed line—monochromatic bunches.
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anti-damping of the coherent oscillations of the colliding bunches is a generic phenomenon for the coherent beam–beam interactions. These instabilities occur due to the coupling of the modes m and m near the sum-type ð2Þ resonance mðnð1Þ x þ nx Þ ¼ 2mnx ¼ n. According to general properties of such instabilities [11] any damping can stabilize such coherent modes only in the case, when both coupled modes are damped sufficiently strongly. Otherwise, the oscillations become unstable [11].
unstable modes and of the positions of the stopbands of these modes in betatron tunes. Obtained numerical results are in a good agreement with the qualitative predictions previously obtained in Ref. [7]. These results do not contradict experimental observations of the limitations due to the beam–beam instabilities in the electron–positron colliders. Also the results of the calculations of the maximum mode increments and of the widths of the stopbands in the tunes for the modes ðÞ which have been obtained in this paper are in good numerical agreement with that obtained in Ref. [8] using a different technique. Comparing the spectra of coherent oscillations which are calculated taking into account and/or ignoring the beam–beam tunespread we have found out the Landau anti-damping of coherent oscillations of colliding bunches. Namely, together with the damping of the modes of the monochromatic colliding bunches the beam–beam tunespread results in the instability of coherent oscillations for the regions of betatron tunes nx where coherent oscillations of monochromatic bunches were stable. Effects of this antidamping increase with an increase in the value of the betatron multipole number m. It is almost negligible for the dipole modes, but for the modes with mX2 the calculations ignoring the beam–beam tunespread result in strong underestimation of the widths of the stopbands of coherent oscillations of the colliding bunches as well as in wrong positions of these stopbands relative the resonant values of the tunes nx . Generally, effects of the beam–beam tunespread decrease the maximum values of the oscillation increments as compared to those calculated for monochromatic bunches. We made the calculations for the eigenvalue problem focusing on the studying of the mode stability conditions. The oscillations spectra of the colliding bunches can be calculated solving an initial value problem. Relevant calculations for the problems with the space charge forces are described in e.g. Ref. [10]. We simplified our half-analytic calculations ignoring possible effects of incoherent beam–beam resonances on the stability collective beam–beam modes assuming that only small amount of the bunch particles are captured in the resonance buckets. If the incoherent resonances are strong and/or are wide enough in nx , the described calculations may predict the results which are not reliable (see, e.g. in Ref. [14]). In such cases, the stability collective beam–beam modes should be studied using numerical simulations.
7. Conclusions
References
(-)
(+)
0.4 (-)
(+)
Im(Δν)/ξ
0.3
0.2
0.1
0.0
0.23
0.24
0.25
0.26 νx
0.27
0.28
0.29
0.30
Fig. 8. The stopbands of coherent oscillations near the resonance 14. Solid line—Landau damped coherent oscillations, dashed line—monochromatic bunches, the symbols ðÞ mark the curves for modes ðÞ.
0.35
(-)
(+)
0.30
Im(Δν)/ξ
0.25 0.20 0.15
(-)
(+)
0.10 0.05 0.00 0.15
0.16
0.17
0.18 νx
0.19
0.20
0.21
Fig. 9. The same as in Fig. 8, but near the resonance 16.
Within the framework of a simplifying model we have studied the influence of the beam–beam tunespread on the stability of coherent oscillations of short identical colliding bunches. We found that the technique similar to that developed in Ref. [10] enables the calculation of the dispersion equations describing the spectra of coherent beam–beam modes, the calculations of the increments of
[1] V.I. Balbekov, A.A. Kolomensky, Sov. J. Tech. Phys. 37 (1967) 2021. [2] A. Piwinski, in: Proceedings of the 8th International Conference on High Energy Accelerators, CERN, Geneva, 1971, p. 357. [3] Ya.S. Derbenev, in: Proceedings of the 3rd All-union Conference on Accelerators, Moscow, 1972, p. 382. [4] E.A. Simonov, in: Proceedings of the 3-D Advanced ICFA Beam Dynamics Workshop on Beam–Beam Effects in Circular Colliders, Novosibirsk, 1989, p. 106.
ARTICLE IN PRESS 346 [5] [6] [7] [8] [9]
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N. Dikansky, D. Pestrikov, Part. Acc. 12 (1982) 27. A.W. Chao, R.D. Ruth, Part. Acc. 16 (1985) 201. D. Pestrikov, SLAC-PUB-5510, SLAC, 1991. P. Zenkevich, K. Yokoya, Part. Acc. 40 (1993) 229. N.S. Dikansky, D.V. Pestrikov, Physics of Intense Beams and Storage Rings, AIP Press, New York, 1994. [10] D.V. Pestrikov, Nucl. Instr. and Meth. A 578/1 (2007) 65.
[11] Ya.S. Derbenev, N.S. Dikansky, in: Proceedings of All Union Particle Accelerator Conference, Moscow, vol. 2, 1970, p. 391. [12] I.S. Gradshteyn, I.M. Ryzhik, in: A. Jeffrey (Ed.), Table of Integrals, Series, and Products, fifth ed. Academic Press, New York, 1994. [13] V.I. Smirnov, Course of Higher Mathematic, vol. 4, Moscow, Fizmatgiz, 1958. [14] L. Jin, J. Shi, G.H. Hoffstaetter, Phys. Rev. E 71 (2005) 036501.
Nuclear Instruments and Methods in Physics Research A 588 (2008) 336–346 www.elsevier.com/locate/nima
Landau anti-damping of the coherent beam–beam instability D.V. Pestrikov Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russian Federation Received 20 December 2007; received in revised form 5 February 2008; accepted 8 February 2008 Available online 19 February 2008
Abstract Within the framework of a simplifying model and within the first approximation of the perturbation theory we discuss effects of the Landau damping on the stability of the coherent oscillations of short identical colliding bunches. Near the resonances n=ð2mÞ where n and m are integers these oscillations are unstable. Comparing the stopbands calculated for monochromatic bunches with those that are calculated taking into account the beam–beam tunespreads of the bunches, we have found that the latter increases the widths of the stopbands of coherent modes, thus resulting in the Landau anti-damping of the coherent beam–beam oscillations. r 2008 Elsevier B.V. All rights reserved. PACS: 29.20.Dh; 29.27.Bd Keywords: Beam–beam instability; Coherent oscillations; Colliders
1. Introduction The space charge forces of the counter-moving colliding bunches at the interaction point (IP) of a collider perturb the motions of particles. In circular colliders these beam–beam perturbations repeat periodically, resulting in numerous resonances. Besides, nonlinear dependencies of the space charge forces on particle coordinates in the bunches produce the tuneshifts and the tunespreads of the particle oscillations. For these reasons, the beam–beam perturbations result in the resonant instabilities of incoherent oscillations of particles as well as of the coherent oscillations of the colliding bunches. Both classes of these instabilities can limit the luminosity of a collider. Descriptions of such limitations are usually complicated by the fact that these manifold phenomena are generically selfconsistent. So, numerous simplifying assumptions are used to predict particular limitations. Theoretical studies of the limitations due to instabilities of the coherent beam–beam modes were started long time ago (see, e.g. in Refs. [1–8]). Using various approaches these studies have figured out that coherent oscillations of Tel.: +7 383 339 4412.
E-mail address: [email protected] 0168-9002/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.02.007
the colliding bunches may become unstable when the tunes of incoherent oscillations of particles approach nonlinear resonant values. Only the sum-type resonances produce the instabilities of the collective modes of the colliding bunches [5]. The values of the coherent tuneshifts of unstable beam–beam modes are equal to zero (e.g. in Ref. [5]). Therefore, some Landau damping of these unstable modes could be expected. However, a qualitative study made in Ref. [7] has shown that the beam–beam tunespread is not sufficient to suppress the beam–beam coherent instability and that near the resonant tunes, unstable coherent beam–beam modes can still exist. Using a different approach, this result was confirmed numerically for the beam–beam p-modes in Ref. [8]. We use the simplifying model described in Refs. [5,7,8] or in Ref. [9] to study the stability of the self-consistent coherent beam–beam oscillations of short colliding bunches. In this paper, we focus on quantitative predictions of the model and on the studying of various effects of the Landau damping on the stability of the coherent beam–beam oscillations. Using the technique developed in Ref. [10], we calculate the stopbands for coherent oscillations of colliding bunches taking into account and/or ignoring the beam–beam tunespreads. For the last case, we use to call colliding bunches as the monochromatic ones.
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Comparing the results of such calculations, we find out that Landau damping due to the nonlinearity of the beam– beam forces, generally, increases the widths of the stopbands of coherent beam–beam modes. It also can change positions of the stopbands of unstable modes relative to the values of the resonant tunes. This means that together with the damping of the unstable modes of the monochromatic colliding bunches the beam–beam tunespreads result in the instabilities for the regions of the tunes where the modes of the monochromatic bunches were stable—i.e. in the Landau anti-damping. Such a Landau anti-damping is specific for the resonant instabilities of coherent oscillations near the sum-type coupling resonances [11]. More precisely, the calculations in Ref. [11] (or, in e.g. Ref. [9]), in particular, predict that two modes coupled near the sum-type resonance are unstable in the case, if only one of the coupled modes is damped. We remind the reader that Landau damping of collective modes occurs due to the particles, which are in the resonance with a beam mode. On its turn, coherent beam–beam oscillations become unstable near the tune nb ¼ n=ð2mÞ (where n and m are e.g. positive integers) due to the resonant coupling of the modes ðm; nÞ and ðm; nÞ. If the Landau damping resonance condition holds for a given value of the coherent tuneshift for the mode ðm; nÞ, it results in the damping of the mode. However, simultaneously this resonance condition is violated for the mode ðm; nÞ resulting in a zero damping decrement of the mode ðm; nÞ. The sum-type coupling of the damped and non-damped modes results in the mentioned Landau antidamping of the beam–beam modes [11]. 2. The model We consider collisions of two short counter-moving relativistic bunches, which move in separate storage rings and interact head-on at a single IP. In our calculations we assume a zero dispersion function at the IP. Then, unperturbed incoherent vertical (y) and horizontal (x) betatron oscillations of particles at the center of IP are described using the following expressions: sffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi Jy py0 y ¼ J y by cos cy ; py ¼ ¼ p sin cy R0 by sffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px0 Jx x ¼ J x bx ðsÞ cos cx ; px ¼ ¼ p sin cx R0 bx I x;y
pJ x;y ; ¼ 2
ðcx;y Þ0 ¼ nx;y .
(1)
Here, the couples ðI; cÞx;y stand for the action-phase variables of the unperturbed incoherent oscillations, P ¼ 2pR0 is the perimeter of the closed orbit, s ¼ R0 y is the path along the closed orbit, primes mean d=dy, p ¼ gMc is the value of the momentum of the reference particle, the values bx;y stand for b-functions of the horizontal and vertical oscillations at the center of IP. The colliding
337
bunches are assumed to be short. This means that the bunch lengths ss are supposed to be substantially smaller than the values of b-functions at the center of IP. For this reason we shall ignore in our calculations the synchrotron oscillations of the particles. In the action-phase variables incoherent betatron oscillations of the bunch particles are described using the Hamiltonian H 0 ¼ nx I x þ ny I y .
(2)
Passing the interaction region, the particles in the bunches are perturbed by the space charge forces of the countermoving bunch. Marking the colliding bunches by the symbols ð1Þ and ð2Þ, we describe such perturbations using the Lagrangians L1;2 and L2;1 , respectively. For example, the oscillations of the particles in the bunch (1) are described using the Hamiltonian H 1 ¼ H 0 L1;2 .
(3)
2.1. Coherent oscillations of bunches Without coherent oscillations the bunches are described using the distribution functions, which do not depend on the phase variables cx and cy . We write f ð1;2Þ ¼ f ð1;2Þ 0 ðIÞ;
I ¼ fI x ; I y g.
We assume that the distribution functions normalized using Z d2 If ð1;2Þ 0 ðIÞ ¼ 1.
(4) f ð1;2Þ 0 ðIÞ
are
(5)
We also assume that the bunch ð1Þ passes IP in the positive direction while the bunch ð2Þ—in the negative direction. Correspondingly, we shall write the longitudinal coordinates in the bunches ð1Þ and ð2Þ using z1;2 ¼ s R0 y, where y ¼ o0 t and o0 is the revolution frequency of the reference particle. Coherent oscillations of the bunches are described by small additions to f 0 , which are functions of the time t and are the periodic functions of the phase variables: ð1;2Þ f ð1;2Þ ¼ f ð1;2Þ 0 ðIÞ þ df
where df ð1;2Þ ¼
X
(6)
f ð1;2Þ m ðI; tÞ expðimx cx þ imy cy Þ,
m
m ¼ fmx ; my g.
(7)
Provided that the perturbations of coherent oscillations due to beam–beam kicks result in small variations of action variables of the particle oscillations I during the revolution period in the ring (2p=o0 ), the amplitudes f ð1;2Þ can be m calculated using the Vlasov equations which are linearized in df ð1;2Þ m . Using azimuth y as an independent variable, assuming the horizontal coherent oscillations (m ¼ fmx ; 0g) and neglecting the contributions of the fast oscillating terms in the linearized Vlasov equation, we find that the
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amplitudes f ð1;2Þ obey the following equations: m qf ð1Þ qf ð1Þ mx ð1Þ 0 þ imx nð1Þ ¼0 x f mx þ imx ðdL1;2 Þmx qy qI x qf ð2Þ qf ð2Þ mx ð2Þ 0 þ imx nð2Þ f þ im ðdL Þ ¼ 0. x 2;1 mx x mx qy qI x
Within the framework of our model Eqs. (8) can be rewritten in the following form:
(8)
qf ð1Þ ð1Þ m þ imnð1Þ x fm qy Z 1 1 qf ð1Þ N 2 e2 X ¼ imdT ðyÞ 0 imm3 dI 1 K m;m3 ðI; I 1 Þ qI c m ¼1 0 3
Together with Eq. (1), these equations imply that we also neglect the effects of the nonlinear beam–beam resonances on the incoherent oscillations of the particles and by possible effects of these resonances on the stability of the coherent beam–beam modes. For relativistic particles (gb1) we can calculate the space charge fields of the colliding bunches in the local field approximation, which results in e.g. Z 2N 2 e1 e2 dkx dky L1;2 ðI; c; yÞ ¼ expðikrÞ po0 k2x þ k2y Z 2 d I 1 d2 c1 dðs s2 Þ ð2pÞ3 expðikr1 Þf ð2Þ ðI 1 ; c1 ; yÞ
(9)
where kr ¼ kx xðI; c; yÞ þ ky yðI; c; yÞ and N 2 e2 or N 1 e1 are the charges of the bunches (2) or (1), respectively. For short bunches (bx;y bss ), the beam–beam instability occurs due to a sequence of short periodic kicks, which for e1 e2 ¼ e2 are described by the following expressions: Z dkx dky N 2 e2 dT ðyÞ expðikrÞ L1;2 ðI; c; yÞ ¼ pc k2x þ k2y Z 2 d I 1 d2 c1 expðikr1 Þf ð2Þ ðI 1 ; c1 ; yÞ. (10) ð2pÞ3 Here, dT ðyÞ is a periodic d-function: dT ðyÞ ¼
1 X
1 einy . 2p n¼1
qf ð2Þ ð2Þ m þ imnð2Þ x fm qy Z 1 1 qf ð2Þ N 1 e2 X ¼ imdT ðyÞ 0 imm3 dI 1 K m;m3 ðI; I 1 Þ qI c m ¼1 0 3
ðf ð1Þ m3 ðI 1 ; yÞ
þ
f ð1Þ m3 ðI 1 ; yÞÞ.
(11)
(13)
Here, we use 1 X
expðiz cos fÞ ¼
ik J k ðzÞeif
(14)
k¼1
where J k ðzÞ is the Bessel function, J k ðzÞ ¼ ð1Þk J k ðzÞ and J k ðzÞ ¼ ð1Þk J k ðzÞ, and we define qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi Z 1 dk ð1;2Þ J m k Ibð1Þ J m3 k I 1 bð2Þ . K m;m3 ðI; I 1 Þ ¼ k 1 Below, we assume that for the bare and for the dynamic b-functions at the IP holds the condition bð1Þ ¼ bð2Þ . Then, we find ð2;1Þ K ð1;2Þ m;m3 ðI; I 1 Þ ¼ K m;m3 ðI; I 1 Þ ¼ K m;m3 ðI; I 1 Þ
and
Z
K m;m3 ðI; I 1 Þ ¼ 2 0
1
pffiffiffiffiffi pffiffiffi dk J m ðk I ÞJ m3 ðk I 1 Þ k
m þ m3 ¼ 2l
(15)
K m;m3 ðI; I 1 Þ ¼ 0; m þ m3 ¼ 2l þ 1
Eqs. (8) are too complicated even for solving numerically. We simplify our calculations considering a special model, where the bunches have very flat distributions f ð1;2Þ ! dðI y Þf ð1;2Þ 0 0 ðI x Þ
ð2Þ ðf ð2Þ m3 ðI 1 ; yÞ þ f m3 ðI 1 ; yÞÞ
(16)
where l is an integer. Generally Eqs. (8) and (13) are simplified, when two identical bunches collide while passing through the rings with identical lattice focusing. For such bunches, we write ð2Þ f ð1Þ 0 ¼ f 0 ¼ f 0;
ð2Þ nð1Þ x ¼ nx ¼ nx
(17)
and find that the combinations (12)
and where the bunches execute coherent oscillations in the horizontal plane only (m ¼ fmx ; 0; 0g). In this model, the collective beam–beam modes are unstable near the sumð2Þ type resonances [5] m1 nð1Þ x þ m2 nx ¼ n, where n, m1 and m2 are integers (m1 m2 40). Qualitative predictions of such a model concerning the stability of coherent beam–beam oscillations have been already studied in our paper [7] (see also in the book Ref. [9]). Here we use the technique developed in Ref. [10] to focus on quantitative predictions of the model and on the studying of particular effects of Landau damping on the stability of the coherent beam– beam oscillations.
ð1Þ ð2Þ f ðÞ m ¼ fm fm
(18)
obey independent equations. This means that the functions describe the normal modes of identical colliding f ðÞ m bunches [6]. Within the framework of the discussed model the equations for f ðÞ m read qf ðÞ m þ imnx f ðÞ m qy 1 qf N 2 e2 X ¼ imdT ðyÞ 0 imm3 qI c m ¼1 3 Z 1 ðÞ dI 1 K m;m3 ðI; I 1 Þðf ðÞ m3 ðI 1 ; yÞ þ f m3 ðI 1 ; yÞÞ. 0
(19)
ARTICLE IN PRESS D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 588 (2008) 336–346
2.2. Incoherent oscillations According to the general rules of the theory of coherent oscillations (see e.g. in Ref. [9]), the tunes nxð1;2Þ in Eqs. (8) and (13) should be calculated taking into account incoherent beam–beam kicks at IP. Within the framework of our model, the horizontal oscillations of particles in the bunch (1) are perturbed at IP by the force qLð0Þ N 2 e2 p 1;2 ¼ dT ðctÞ x b2 qx 2 Z 1 du ð2Þ px pffiffiffiffiffiffiffiffiffiffiffi f 0 u 2b2 1u 0
F x ¼ o0
(20)
where b1;2 are the values of the b-functions of the horizontal betatron oscillations at IP for the bunches (1) and (2). If ð1;2Þ are the horizontal emittances of the colliding bunches (f 0ð1;2Þ ð0Þ ¼ 1=ðpð1;2Þ Þ), then simple calculations (see also e.g. in Ref. [4]) result in the following expressions for betatron tunes nð1;2Þ and for b-functions of x the horizontal betatron oscillations at IP for the colliding bunches: cos mð1Þ ¼ cos m0
N 2 e2 b0 sin m0 pcð2Þ b2
b1 sin mð1Þ ¼ b0 sin m0
(21) 2
cos mð2Þ ¼ cos m0
N 1 e b0 sin m0 pcð1Þ b1
b2 sin mð2Þ ¼ b0 sin m0 .
(22)
Here, m ¼ 2pnx , m0 ¼ 2pnx0 . The values nx0 and b0 define the betatron tune and b-function of the horizontal oscillations calculated without collisions of the bunches— the bare tune and b-function. Eqs. (21) and (22) describe self-consistent variations of the betatron tunes and of the b-functions at IP due to the beam–beam interactions. Due to their self-consistency these equations have solutions and, therefore, predict stable betatron oscillations of particles for all values of m0 and b0 . For identical colliding bunches (N 1 ¼ N 2 ¼ N and ð1Þ ¼ ð2Þ ¼ ) Eqs. (21) and (22) read cos mð1Þ ¼ cos m0 B
b0 sin m0 b2
b1 sin mð1Þ ¼ b0 sin m0 cos mð2Þ ¼ cos m0 B
(23) b0 sin m0 b1
b2 sin mð2Þ ¼ b0 sin m0
(24)
where B ¼ 2px and x is the beam–beam parameter: x¼
Ne2 . 2ppc
(25)
In the realistic region of parameters Bo1, Eqs. (23) and (24) predict equal variations of the betatron tunes and b-functions for all values of nx0 . Besides, in the narrow region: ! 2 B 1 B2 1 arccos pm0 p arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Bo1 (26) 1 þ B2 1 þ B2 Eqs. (23) and (24) predict a flip-flop splitting of the betatron tunes and of the b-functions where e.g. the values b1 and b2 can differ substantially. For small values of x the width of such a flip-flop region is also small (Dm0 ’ 0:3B). In this paper we ignore the possibility of such splitting and, hence, assume equal self-consistent variations of the betatron tunes and of the b-functions for identical colliding bunches. Then, the solutions to Eqs. (23) and (24) yield qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos m0 B B2 þ sin2 m0 cos mx ¼ 1 þ B2 kppm0 pkp þ p (27) where k is an integer and cos m0 ¼ cos mx þ B sin mx .
(28)
For small values of x the tune shift due to the beam–beam interactions (nx nx0 ) is equal to x. However, with an increase in x the tune shift deviates from x especially in the regions, where nx0 approaches parametric resonances (Fig. 1). We remind the reader, that according to Eq. (27) the variations of the bare tune nx0 within the region k=2pnx0 pðk þ 1Þ=2 (where k is an integer) result in the variations of nx within the region k kþ1 þ Dx pnx p 2 2
(29)
2.0
1.5 (νx-ν0)/ξ
Since, generally, the functions f ð1;2Þ are linear combinations m of the modes f ðÞ , the coherent oscillations of identical m colliding bunches are stable only in the regions, where both ðÞ modes f ðþÞ m and f m are stable.
339
1
2
1.0
0.5
0.0 0.0
0.1
0.2
ν0
0.3
0.4
0.5
Fig. 1. Dependence of the beam–beam tune shift on the bare tune (nx0 ); line 1 x ¼ 0:05, line 2 x ¼ 0:005.
ARTICLE IN PRESS D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 588 (2008) 336–346
340
1.0
3. Dispersion equations and modes We simplify the calculations of the Landau damping of the coherent oscillations of identical colliding bunches (N 1 ¼ N 2 ¼ N) assuming that the distribution function f 0 ðIÞ in Eq. (19) is an exponential:
0.8
νx
0.6
f0 ¼
0.4
1 x e ; p
x¼
I p
where is the horizontal bunch emittance. Substituting this expression in Eq. (33), we obtain
0.2
nx ðIÞ ¼ nx0 þ Dnx ð0Þ 0.0 0.0
0.2
0.4
νx0
0.6
0.8
1.0
Fig. 2. Dependence of the tune nx on the bare tune (nx0 ); horizontal betatron oscillations, x ¼ 0:05, only the square 0pnx0 p1 and 0pnx p1 is shown.
where
(36)
Now, substituting f 0 from Eq. (34) in Eq. (19) 1 X
iny f ðÞ m;n e
(37)
n¼1
(30)
This means that for a given value of x the spectra of betatron oscillations of the colliding bunches in nx have a slot just above a parametric resonance with the width Dx (Fig. 2). Since positions of the upper borders in Eq. (29) do not depend on x, the oscillations of particles in the colliding bunches cannot be perturbed by nonlinear resonances nx ¼ n=ð2mÞ, where n and m are natural numbers obeying the condition: k n k p p þ Dx . 2 2m 2
(35)
1 ex nx ðIÞ ¼ nx Dnx ð0Þ 1 . x
iny f ðÞ m ¼ e
1 1B arccos . 2p 1 þ B2
1 ex x
or
2
Dx ¼
(34)
f ðÞ m;n ¼
In wider regions of x and for the discussed model, the dependence of the tune shift on the amplitudes of the horizontal betatron oscillations of particles can be found using one-dimensional simulations, where the beam–beam kicks are described by the force, given in Eq. (20). For identical colliding bunches the result can be written in the form similar to Eq. (32): Z p I nx ðIÞ ¼ nx0 þ Dnx ð0Þ dI 1 f 0 ðI 1 Þ (33) I 0 where Dnx ð0Þ is the beam–beam tune shift of the linear betatron oscillations calculated using e.g. Eq. (27).
f ð1Þ m;n ¼
X ðÞ m ðIÞ n þ n þ mnx ðIÞ
(38)
and defining x¼
Ne2 2ppc
(39)
ðÞ we find X ðÞ m ¼ X m and Z 1 1 X ðÞ x dx1 imm3 K m;m3 ðx; x1 Þ X m ¼ mxe 0
X ðÞ m3 ðx1 ÞðS m3
(31)
Nonlinear dependencies of the beam–beam forces on transverse coordinates of particles result in the tune spreads in the colliding bunches. For small x, simple calculations in the first approximation of the perturbation theory and within the framework of the model result in Z Ne2 I ð1;2Þ nð1;2Þ ðIÞ ¼ n þ dI 1 f ð1;2Þ (32) x x0 0 ðI 1 Þ. 2pcI 0
X ðÞ m ðIÞ ; n þ n mnx ðIÞ
m3 ¼1
S m3 Þ.
(40)
Here, Sm ¼
1 X
1 . n þ n mnx ðxÞ n¼1
(41)
For small values of the beam–beam parameter x (e.g. B ¼ 2pxo1) and according to the general rules of the theory of the linear coherent oscillations (see, e.g. in Ref. [9]), the value of the eigentune n in Eq. (40) for the mode with the multipole number m is expected to be close to mnx . On the other hand, since the beam–beam interactions occur via conservative forces, the oscillation modes can be unstable only in the cases when the tunes nx approach the resonant values nx ¼ n=ð2mÞ, where n and m are integers. For simplicity, we assume that for small x the beam–beam resonances of the modes with different values of m do not interfere and, correspondingly, that we may omit on the right-hand side of Eq. (40) the off-diagonal terms m3 am. The resulting equation reads Z 1 x ðxÞ ¼ e dx1 V ðx1 ÞK m ðx; x1 ÞX ðÞ (42) X ðÞ m m ðx1 Þ 0
ARTICLE IN PRESS D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 588 (2008) 336–346
where (m40; [12]) pffiffiffi Z 1 pffiffiffiffiffi J m ðk xÞJ m ðk x1 Þ dk K m ðx; x1 Þ ¼ 2 k 0 8 m=2 1 < ðx=x1 Þ ; xox1 ¼ m : ðx1 =xÞm=2 ; x4x1 : V ðxÞ ¼
2dðxÞ ; 2 z1 d2 ðxÞ
Im z1 40
Namely, using Eq. (48), we find Z x dpðÞ 1 ðÞ m ðxÞ ¼ mþ1 dx1 ex1 xm 1 V ðx1 Þpm ðx1 Þ dx x 0 (43)
(44)
1 n n ; mx 2
dðxÞ ¼
1 n nx ðxÞ x 2m
(45)
where n is the azimuthal number of the resonance harmonic. Substituting Eq. (43) in Eq. (42) and taking e.g. x m=2 ðÞ pm ðxÞ X ðÞ m ¼ e x
(46)
we obtain the following integral equation for a new unknown function pðÞ m ðxÞ: Z x 1 xm pðÞ dx1 1m ex1 V ðx1 ÞpðÞ m ðxÞ ¼ m ðx1 Þ m 0 x Z 1 1 þ dx1 ex1 V ðx1 ÞpðÞ m ðx1 Þ m x or pðÞ m ðxÞ
1 ¼ m
Z
1
dx1 ex1 V ðx1 ÞpðÞ m ðx1 Þ Z x 1 ðÞ þ dx1 ex1 xm 1 V ðx1 Þpm ðx1 Þ mxm 0 Z 1 x dx1 ex1 V ðx1 ÞpðÞ m ðx1 Þ. m 0 0
(47)
Since pðÞ m ð0Þ ¼
1 m
Z 0
wðxÞ x we obtain (w0 ¼ dw=dx)
dx1 ex1 V ðx1 ÞpðÞ m ðx1 Þ
we normalize the eigenfunctions using the condition pðÞ m ð0Þ ¼ 1. Then, Eq. (47) is replaced by the following nonhomogeneous integral equation for the function Z x 1 ðÞ ðxÞ ¼ 1 þ dx1 ex1 xm pðÞ m 1 V ðx1 Þpm ðx1 Þ mxm 0 Z 1 x dx1 ex1 V ðx1 ÞpðÞ (48) m ðx1 Þ m 0 and by the dispersion equation of the problem: Z 1 1 dxex V ðxÞpðÞ 1¼ m ðxÞ. m 0
(49)
The function pðÞ m ðxÞ in these equations depends also on the eigentune (z1 ) like on a parameter. For this reason, ðÞ sometimes we shall write pðÞ m ðz1 ; xÞ instead of pm ðxÞ. Following the paper [10], we transform the integral equation (48) in the differential equation for the function pðÞ m ðxÞ.
(51)
(52)
ðm 1Þ 0 ðm 1Þ ex w V ðxÞwðxÞ. (53) w¼ 2 x x x In order to provide pðÞ m ð0Þ ¼ 1, this equation should be solved with the boundary conditions: w00 þ
wð0Þ ¼ 0;
w0 ð0Þ ¼ 1.
(54)
The calculation of the integral in the dispersion equations (49) can be avoided using the transformations similar to those found in Ref. [10]. We note that according to Eq. (51), the following equations hold: Z 1 1 1¼ dxex V ðxÞpðÞ m ðxÞ m 0 Z 1 ðÞ 1 dx d mþ1 dpm ðxÞ x ¼ m 0 xm dx dx Z 1 ðÞ 1 dp ðxÞ dpðÞ ðxÞ x m ¼ dx m m dx dx 0 x¼1 1 dpðÞ ðxÞ x m ¼ pðÞ m ð1Þ þ 1. m dx x¼1 Since in the asymptotic region (x ! 1) solutions to Eq. (51) read dpðÞ C m ðxÞ ¼ mþ1 ; dx x
1
(50)
Substituting here1 pðÞ m ðxÞ ¼
and z1 ¼
and ðÞ d mþ1 dpm ðxÞ x ¼ ex xm V ðxÞpðÞ m ðxÞ. dx dx
341
pðÞ m ðxÞ ¼ C 1 þ
C ðm þ 1Þxm
where C and C 1 are arbitrary functions of z1 , we find dpðÞ m ðxÞ lim x ¼0 (55) x!1 dx and the dispersion equation in the following form: pðÞ m ðz1 ; 1Þ ¼ 0.
(56)
This fact simplifies substantially the calculations of the eigenvalues and of the stability conditions of coherent oscillations of the colliding bunches within the framework of the used model. Similar calculations for the mode ðþÞ result in the dispersion equation pðþÞ m ðz1 ; 1Þ ¼ 0 1
(57)
Corresponding equation in Ref. [7] is obtained, if we substitute ¼ xðmþ1Þ=2 wðxÞ in Eq. (51), which yields
pðÞ m ðxÞ
x e V ðxÞ m2 1 w00 ¼ w. 2 x 4x
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342
ðþÞ where pðþÞ ðxÞ=x and the function wðþÞ ðxÞ obeys m ðxÞ ¼ w Eqs. (53), (54) with the substitution V ðxÞ ! V ðxÞ:
lim pðÞ m ðz1 ; 1Þ ¼ 1.
0.4 pfin(-)
d2 wðÞ ðm 1Þ dwðÞ ðm 1Þ ðÞ þ w x x2 dx2 dx ex V ðxÞwðÞ ðxÞ. ¼ (58) x According to Eq. (48) or Eqs. (58) and (54) the functions pðÞ m ðz1 ; xÞ obey the asymptotic condition
0.6
0.2
0.0
(59)
jz1 j!1
This means that the roots of the dispersion equations (56) and (57) can be found only within a limited region near the origin in the plane of the complex variable z1 . We have made the calculations assuming the eigenvalue problem and the studying of the mode stability conditions. The oscillations spectra of the colliding bunches are found solving the initial value problem. Such calculations are similar to those described in e.g. Ref. [10].
-0.2 0.1 Im(Δνm)/(mξ)
0.01
1
Fig. 3. Dependence of e.g. pðÞ ðz1 ; 1Þ on z1 . Solid line—m ¼ 1, dashed line—m ¼ 2; x ¼ 0:05, for both curves nx corresponds to the point of the maximum increment.
4. Stopbands of nonmonochromatic bunches
1/2
0.6
2dðxÞ ; þ d2 ðxÞ
ImðnÞ . mx
0.4
0.2
Corresponding eigensolutions wðÞ ðr; xÞ are real functions while the dispersion equations (56) or (57) have real roots r. For this paper, the values of the increments of unstable modes as well as the widths of relevant stopbands and their positions in nx are found solving Eqs. (56)–(58) numerically. As far as the coefficients in Eq. (58) diverge at the point x ¼ 0, solutions to Eq. (58) near the origin were calculated (xpx0 ¼ 1010 ) using their series expansions in powers of xpx0 and the boundary conditions in Eq. (54):
0.1
r2
wðÞ ðxÞ ¼ x
2d x2 ðr2 þ d2 Þ ð1 þ mÞ 2
þ
r¼
2
2
2
2
4d ð1 þ mÞðbðr d Þ þ 2dðr þ d ÞÞ 3 x (61) 2ðr2 þ d2 Þ2 ðm þ 2Þð1 þ mÞ
where b ¼ Dnx ð0Þ=x. Solutions to Eq. (58) for wider region xXx0 were calculated using wðÞ ðx0 Þ and ðdwðÞ dxÞx¼x0 from Eq. (61) as the boundary values. Trying several different large values of x, we found that the asymptote x ! 1 occurs, if we take e.g. x ¼ 80. From two examples depicted, e.g., dependencies of pðÞ m ðz1 ; 1Þ on z1 (Fig. 3), we see that within the stopband and for the given x and nx the dispersion equations (56), (57) may have one or several roots z1 . The code solving the dispersion equations (56) and (57) calculated only the maximum increment of the collective modes.
1/6 2/10 1/4 3/10 2/6 3/8 4/10
1/10
0.3
(60)
V ðxÞ ¼
1/8
0.5
Im(Δν)/ξ
As it was found qualitatively in Ref. [7], near the resonances nx ’ n=ð2mÞ Eq. (42) always has solutions with eigenvalues Reðz1 Þ ¼ 0 and Imðz1 Þa0. Since pðÞ m ðz1 ; 1Þ are functions of z21 , these solutions describe the unstable modes. Substituting z1 ¼ ir in Eq. (44), we find that in such cases V ðxÞ is a real function
0.0 0.0
0.1
0.2
νx
0.3
0.4
0.5
Fig. 4. Dependencies of the increments of the coherent beam–beam modes on the tune of the horizontal incoherent betatron oscillations (nx ). Modes 1pmp5, arrows show positions of relevant stopbands near particular resonances, wider curves to the right from the resonance nx ¼ n=ð2mÞ depict the increments of the modes ðþÞ, x ¼ 0:05.
Results of such calculations for the modes ðÞ with 1pmp5 and 1pnp4 are depicted in Fig. 4. In this picture the increment of an oscillation mode starts from a zero value at some particular value nx;in , passes the maximum value and then decreases to zero again. Those curves depict the stopbands of coherent oscillations. Although only the segment 0pnx p12 is shown in Fig. 4, the stopbands for higher or lower values of nx appear periodically with a period in nx of 12. For dipole oscillations m ¼ 1 we have found no roots of the dispersion equation (57) for 0pnx p12. This result means that only ðÞ dipole mode has the stopband below the resonance nx ¼ 12. The stopband for the mode ð; 2Þ starts slightly below the resonance nx ¼ 14. The lower ends of all other found stopbands are found to be close to the corresponding resonant tunes nx ¼ n=ð2mÞ.
ARTICLE IN PRESS D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 588 (2008) 336–346
Numerical values of the maximum increments and of the widths of the stopbands for modes ðÞ and 1pmp3, which are shown in Fig. 4, are in general agreement with similar results reported in Ref. [8] and obtained using a different approach. The deviations from the numbers found in Ref. [8] occur due to thedifferences in the dispersion properties of incoherent betatron oscillations of the particles taken into account in this paper and in Ref. [8]. In particular, due to this difference the values of the maximum increments for the modes mX3 are a bit smaller in the region nx o14 than that above nx ¼ 14. The maximum increments of the modes near the resonances nx ¼ 1=ð2mÞ tend to zero, when nx approaches the border value Dx from Eq. (30). This fact is in general agreement with the experimental observations of the beam–beam instabilities in the electron–positron colliders. Although it is not shown here, a decrease in x results in the decreases in the values of the mode increments (Im n) and in the narrowing of the widths of the stopbands in nx . However, the ratios Im n=x and of these widths to x remain the same. 5. Stopbands of monochromatic bunches Effects of Landau damping on the stability of coherent oscillations of the colliding bunches were studied comparing the stopbands, described in the previous subsection, with those, calculated using the distribution f 0 from Eq. (34) and neglecting the tunespread of incoherent betatron oscillations due to the beam–beam interactions. The modes of such monochromatic bunches are described using Eq. (42) Z 1 ðÞ x X m ðxÞ ¼ e dx1 V ðx1 ÞK m ðx; x1 ÞX ðÞ (62) m ðx1 Þ 0
where we put V ðxÞ ¼
1 1 . z1 d z1 þ d
z1 is defined in Eq. (45), but 1 n nx d¼ x 2m
the eigennumbers L of Eq. (67) are real positive numbers for modes ðÞ. For the modes ðþÞ the eigennumbers are negative numbers L. Therefore, for the monochromatic colliding bunches the eigentunes of the collective modes due to their beam–beam interactions are determined using the dispersion equation z21 ¼ d2 2Ld ¼ ðd LÞ2 L2
(64)
(65)
as a new eigenvalue and defining x X ðÞ ðxÞ ¼ exp PðÞ ðxÞ (66) m 2 m we transform Eq. (62) into the integral equation with a symmetrical kernel Z 1 x=2 LPðÞ ðxÞ ¼ e dx1 ex1 =2 PðÞ m m ðx1 Þ 0 pffiffiffi Z 1 pffiffiffiffiffi J m ðk xÞJ m ðk x1 Þ dk. (67) 2 k 0
(68)
where the sign þ is taken for the mode ðÞ, L is an eigenvalue of the integral equation Z 1 ðþÞ x=2 LPm ðxÞ ¼ e dx1 ex1 =2 PðþÞ m ðx1 Þ 0 pffiffiffi Z 1 pffiffiffiffiffi J m ðk xÞJ m ðk x1 Þ dk (69) 2 k 0 x=2 or (PðþÞ PðxÞ) m ðxÞ ¼ e Z x 1 m=2 LPðxÞ ¼ dx1 x1 ex1 Pðx1 Þ mxm=2 0 Z xm=2 1 ex1 Pðx1 Þ þ dx1 . m=2 m x x
(70)
1
Substituting PðxÞ ¼ xm=2 pðL; xÞ in Eq. (70), we find Z x 1 x1 LpðL; xÞ ¼ dx1 xm pðL; x1 Þ 1e mxm 0 Z 1 1 þ dx1 ex1 pðL; x1 Þ m x
(71)
and that the eigennumbers L are calculated solving the dispersion equation (72)
where pðL; xÞ ¼ wðxÞ=x and m1 0 m1 w w x x2 ex ¼ wðxÞ; wð0Þ ¼ 0; w0 ð0Þ ¼ 1. (73) Lx Note also, that according to Eq. (70), for a given value of m the sum of all L is equal to 1=m. The eigenvalues L were calculated solving Eqs. (73) and (72) numerically. The results of these calculations clearly indicated an existence of the well-defined ground state mode with the maximum value of L (see e.g. in Fig. 5). On the other hand, Eqs. (69) or (70) are the homogeneous integral equations with symmetrical and positively defined kernels. The maximum eigenvalues to Eq. (70) can be evaluated using the extrema properties of its eigenvalues (e.g. in Ref. [13]). Taking as a probe ground state solution to Eq. (69) the function rffiffiffiffiffiffi a ðþÞ Pm ðxÞ ¼ (74) ðaxÞm=2 exa=2 m! w00 þ
does not depend on x. The kernel K m ðx; x1 Þ is defined in Eq. (43). Using 1 1 1 ¼ L z1 d z1 þ d
Since 2 Z 1 Z 1 pffiffiffi ðÞ dk x=2 dxe J m ðk xÞPm ðxÞ 40 2 k 0 0
pðL; 1Þ ¼ 0 (63)
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For this reason, the stopbands and maximum increments of unstable modes of the monochromatic colliding bunches were calculated by substituting Lmax from Eq. (76) in Eq. (68). The oscillation increments r ¼ Im n=x reach their maximum values rmax ¼ mLmax when d ¼ Lmax for the modes ðÞ and when d ¼ Lmax for the modes ðþÞ. The stopbands are placed below the resonance 2Lmax pdp0 for the modes ðÞ and above the resonance 0pdp2Lmax for the modes ðþÞ. According to Eqs. (68), (76) and (30) the dipole mode ðþÞ is stable provided that the value of x is in a realistic region (actually, if xo0:2872).
0.3
pfin(Λ)
0.2 0.1 0.0 -0.1 -0.2
6. Discussion 0.1 Λ
1
Fig. 5. Dependence of pðL; 1Þ on L for m ¼ 1, the first root of pðL; 1Þ defines Lmax .
0.6 0.5
Λmax
0.4 0.3 0.2 0.1 0.0
1
2
3 m
4
5
Fig. 6. Dependence of Lmax on the multipole number m; open circles— using numerical solutions of Eqs. (73) and (72), solid line—using Eq. (76).
calculating the probe eigenvalue Z 2amþ1 1 dxxm=2 exðaþ1Þ=2 LðaÞ ¼ m! 0 pffiffiffi Z 1 Z 1 pffiffiffiffiffi J m ðk xÞJ m ðk x1 Þ m=2 dk dx1 x1 ex1 ðaþ1Þ=2 k 0 0 4 amþ1 ¼ (75) m ða þ 1Þmþ2 and defining the value of a using the equation dLðaÞ=da ¼ 0, we find Lmax ¼
4 mðm þ 2Þ 1þ
1 1 mþ1
mþ1 .
(76)
Note that for large multipole numbers m (mb1) the value Lmax varies proportionally to 1=m2 . The calculations of the maximum eigenvalues Lmax using Eq. (76) and numerical solutions of Eqs. (73) and (72) agree well (Fig. 6).
Comparing the curves in Fig. 7, we find that for the dipole coherent mode ðÞ of the colliding bunches Landau damping results only in minor changes of the stopband of the monochromatic bunches. The maximum values of the increments almost coincide, Landau damping suppress the oscillations within a narrow band A0 A and decreases the oscillation increments within the segment AB. Within the segment BC the beam–beam tune spread slightly increases the increments of unstable modes hence, resulting in some Landau anti-damping. Stronger Landau anti-damping indicates the stopbands of the modes with higher betatron multipole numbers (mX2, e.g. in Figs. 8 and 9). For such modes, the beam–beam tunespread although decreases the values of the maximum increments, moves the lower border of the stopbands of the modes ðÞ towards the resonant tune n=ð2mÞ and substantially increases the widths of the stopbands. Except for the case m ¼ 2, the stopbands of the multipole modes ðÞ are placed above the resonant tunes almost entirely. Hence, the beam–beam tunespread suppressing the modes of the monochromatic colliding bunches opens new wide regions of the tunes nx where the oscillations become unstable. The described Landau 0.6 B
0.5
Im(Δν)/ξ
0.01
0.4 0.3 0.2 0.1 0.0
A' 0.44
C
A 0.45
0.46
0.47 νx
0.48
0.49
0.50
Fig. 7. Dependence of the increment of the dipole mode ðÞ on the tune of the horizontal incoherent oscillations nx . Solid line—Landau damped coherent oscillations, dashed line—monochromatic bunches.
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anti-damping of the coherent oscillations of the colliding bunches is a generic phenomenon for the coherent beam–beam interactions. These instabilities occur due to the coupling of the modes m and m near the sum-type ð2Þ resonance mðnð1Þ x þ nx Þ ¼ 2mnx ¼ n. According to general properties of such instabilities [11] any damping can stabilize such coherent modes only in the case, when both coupled modes are damped sufficiently strongly. Otherwise, the oscillations become unstable [11].
unstable modes and of the positions of the stopbands of these modes in betatron tunes. Obtained numerical results are in a good agreement with the qualitative predictions previously obtained in Ref. [7]. These results do not contradict experimental observations of the limitations due to the beam–beam instabilities in the electron–positron colliders. Also the results of the calculations of the maximum mode increments and of the widths of the stopbands in the tunes for the modes ðÞ which have been obtained in this paper are in good numerical agreement with that obtained in Ref. [8] using a different technique. Comparing the spectra of coherent oscillations which are calculated taking into account and/or ignoring the beam–beam tunespread we have found out the Landau anti-damping of coherent oscillations of colliding bunches. Namely, together with the damping of the modes of the monochromatic colliding bunches the beam–beam tunespread results in the instability of coherent oscillations for the regions of betatron tunes nx where coherent oscillations of monochromatic bunches were stable. Effects of this antidamping increase with an increase in the value of the betatron multipole number m. It is almost negligible for the dipole modes, but for the modes with mX2 the calculations ignoring the beam–beam tunespread result in strong underestimation of the widths of the stopbands of coherent oscillations of the colliding bunches as well as in wrong positions of these stopbands relative the resonant values of the tunes nx . Generally, effects of the beam–beam tunespread decrease the maximum values of the oscillation increments as compared to those calculated for monochromatic bunches. We made the calculations for the eigenvalue problem focusing on the studying of the mode stability conditions. The oscillations spectra of the colliding bunches can be calculated solving an initial value problem. Relevant calculations for the problems with the space charge forces are described in e.g. Ref. [10]. We simplified our half-analytic calculations ignoring possible effects of incoherent beam–beam resonances on the stability collective beam–beam modes assuming that only small amount of the bunch particles are captured in the resonance buckets. If the incoherent resonances are strong and/or are wide enough in nx , the described calculations may predict the results which are not reliable (see, e.g. in Ref. [14]). In such cases, the stability collective beam–beam modes should be studied using numerical simulations.
7. Conclusions
References
(-)
(+)
0.4 (-)
(+)
Im(Δν)/ξ
0.3
0.2
0.1
0.0
0.23
0.24
0.25
0.26 νx
0.27
0.28
0.29
0.30
Fig. 8. The stopbands of coherent oscillations near the resonance 14. Solid line—Landau damped coherent oscillations, dashed line—monochromatic bunches, the symbols ðÞ mark the curves for modes ðÞ.
0.35
(-)
(+)
0.30
Im(Δν)/ξ
0.25 0.20 0.15
(-)
(+)
0.10 0.05 0.00 0.15
0.16
0.17
0.18 νx
0.19
0.20
0.21
Fig. 9. The same as in Fig. 8, but near the resonance 16.
Within the framework of a simplifying model we have studied the influence of the beam–beam tunespread on the stability of coherent oscillations of short identical colliding bunches. We found that the technique similar to that developed in Ref. [10] enables the calculation of the dispersion equations describing the spectra of coherent beam–beam modes, the calculations of the increments of
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