Effect of the frequency spread on the transverse beam break-up instability of a bunch in storage rings

Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Effect of the frequency spread on the transverse beam break-up instability of a bunch in storage rings D.V. Pestrikov a,b, a b

Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russian Federation Novosibirsk State University, 630090 Novosibirsk, Russian Federation

a r t i c l e in fo

abstract

Article history: Received 23 September 2010 Accepted 23 September 2010 Available online 1 October 2010

We study modifications of the beam break-up instability of transverse coherent oscillations of a single bunch which occur in storage rings due to the frequency spread of incoherent oscillations of the bunch particles. We assume that the frequency spread is produced by the octupole nonlinearity of the lattice focusing. Provided that the bunch wakefields decay longer than the revolution period of particles, the frequency spread results in the Landau damping of the self-consisting modes of the bunch and in the cumulative decoherence of the beam break-up part of the coherent oscillations. We study resulting features of coherent signals in the time-domain. & 2010 Elsevier B.V. All rights reserved.

Keywords: Storage rings Coherent oscillations Singleturn and multiturn wakes Landau damping Beam break-up instability

1. Introduction In this paper we continue the studies of the effects of the longterm memory of the bunch wakes on the beam break-up instability of the transverse coherent oscillations of a single bunch in storage rings [1]. These instabilities are assumed to be so fast that the amplitudes of coherent oscillations vary substantially faster than the periods of small synchrotron oscillations of the bunch particles 2p=os . Generally, this problem can simulate the asymptotic at very large bunch intensities of the instability of the transverse coherent oscillations due to the coupling of the synchro-betatron modes of the bunch. Besides, such instabilities were observed in the operating storage rings (e.g. in Ref. [2]). In the paper [1] we have studied the long-term memory effects on the beam break-up instability of a monochromatic single bunch in storage rings. Due to effects of the wake memory the coherent signal of a bunch, executing fast transverse betatron oscillations, consists of two parts. The first part describes the singleturn (singlebunch) effect. It is namely the beam break-up instability of the bunch. The amplitudes p of ffiffiffiffiffiffiffi this part rise in time in a non-exponential law (typically pexpð t=tÞ, t is the instability rise time). The second part of the signal gives the multiturn effect which results in the self-consistent modes of coherent oscillations of the bunch. This part of the coherent signal is presented by the sum of the collective modes, which exponentially rise, or decay in

 Correspondence address: Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russian Federation. E-mail address: [email protected]

0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.09.127

time depending on the stability of a mode. It turned out, that in the case of fast coherent oscillations of the monochromatic bunch the total sum of the contributions of all collective modes also result in a non-exponential growth of this part of the coherent signal. Compared to the beam break-up part the contribution of self-consisting modes contained in Ref. [1] a suppressing factor, describing the wake decay during the particle revolution period. Due to non-exponential increases of the amplitudes such instability could be damped (at least, asymptotically) by any effect, resulting in additional exponential decay of coherent amplitudes. Those could be due to effects of the wideband feedback damping systems, or the beam cooling. However, in operating storage rings the frequencies of the betatron oscillations of the bunch particles are distributed within some frequency spread. So, the bunch is not monochromatic. That may occur due to the ring lattice nonlinearity, or due to the lattice chromaticity. The frequency spread of incoherent betatron oscillations results in the cumulative decoherence of the beam break-up part and in the Landau damping of the collective modes. These effects change the features of the fast coherent oscillations of the monochromatic and of non-monochromatic bunches. In particular, the transverse instabilities of a non-monochromatic bunch may have thresholds, which were absent for coherent oscillations of the monochromatic bunch [1]. We simplify the calculations assuming that the frequency spread of incoherent oscillations is produced by the octupole nonlinearity of the ring lattice. Besides, we assume that the bunch wakes are described using the localized transverse impedance of a special kind. In this paper, we still assume that the longitudinal mobility of the bunch particles is suppressed for some reason.

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D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

2. Models and definitions Following Ref. [1] we assume that the amplitudes of the transverse coherent oscillations of the bunch vary in time substantially faster than the frequency of synchrotron oscillations of the particles os . As was shown in Ref. [3] in such a region of parameters coherent oscillations of the bunch can be described using the quasi-coasting bunch approximation where the synchrotron incoherent oscillations of particles are ignored. In this case and assuming the smoothed focusing approximation, incoherent oscillations of particles near the closed orbit are described using sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi pny I R0 I y¼ sinc cosc, py ¼  R0 pny

the Fourier transform in time Z 1 @fm ¼ fmð0Þ ðI, fÞiofm ðoÞ, dt eiot @t 0

Im o 4 0

ð7Þ

where fmð0Þ ðI, fÞ ¼ fm ðI, f,0Þ is the initial value of the harmonic fm ðI, f,tÞ. After this Fourier transform Eq. (6) reads sffiffiffiffiffiffiffiffiffiffi ð0Þ if ðI, fÞ dG=dI R0 I þ mr0 ðfÞ F ðoÞ: ð8Þ fm ðI, f, oÞ ¼ m omoy omoy 2pny y Assuming the bunch wake is described using the localized transverse impedance and neglecting in Eq. (6) the fast oscillating terms, we write Z 1 1 Ne2 o0 X dfu Fy ðoÞ ¼  einf Z? ðmo0 ny þ no0 Þ Dm ðfu, oÞeinfu P n ¼ 1 1 2p ð9Þ

dc ¼ oy , dt

y ¼ o0 t þ f,

o0 ap df ¼ Dp: dt p

ð1Þ

Here, y ¼ s=R0 is the particle azimuthal position along the closed orbit, 2pR0 is the perimeter of the closed orbit, oy is the frequency of the vertical betatron oscillations, p is the momentum of the synchronous particle, p þ Dp is the particle momentum, ap ¼ ð1=g2 Þa is the slip factor of the ring, o0 is the revolution frequency of the synchronous particle of the bunch. Below, we neglect the effects of the momentum spread in the bunch and assume that the frequencies of the betatron oscillations of the bunch particles differ due to octupole fields in the ring lattice.1 For simplicity, we shall write

oy ¼ o0 ny þ o0 AI:

ð2Þ

Eq. (1) generate the canonical transformation from the variables (y,py) to the action phase variables ðI, cÞ of the unperturbed vertical betatron oscillations. If the value of the tune ny is not close to the lattice resonances, then without coherent oscillations the beam is described by the distribution function f0 ¼

1 dðDpÞGðIÞr0 ðfÞ: 2p

ð3Þ

We assume that the function f0 is normalized using the following conditions: Z 1 Z 1 dIGðIÞ ¼ 1, df r0 ðfÞ ¼ 1: ð4Þ 0

1

Following Ref. [1] we assume that the bunch length is substantially shorter than the closed orbit perimeter. So, the integration limits over f can be extended from the segment ½0,2p to the region jfj o 1. The vertical dipole coherent oscillations of the bunch are described by small addition to the distribution function f0. If the distribution function of the beam with coherent oscillations is f, we write f ¼ f0 ðI, Dp, fÞ þ

dðDpÞ X 2p

fm ðI, f,tÞeimc :

ð5Þ

where sffiffiffiffiffiffiffiffiffiffi Z pffiffiffi R0 dIu Iufm ðIu, f, oÞ Dm ðf, oÞ ¼ 2pny

ð10Þ

is the Fourier amplitude of the bunch dipole momentum corresponding to fm ðI, f, oÞ. Following Ref. [1] we simplify our calculations using the model (see, e.g. in Ref. [4]), where Z? ðoÞ ¼

icZ 0 b2? ðo þiGo0 Þ

ð11Þ

b? is the pipe radius. Relevant particle wake exponentially decay in time: Z? ðtÞ ¼ 

cZ 0 o0 Gt e , b2?

t Z0:

ð12Þ

For this reason, Eq. (11) describes the wake with a memory. After the revolution period in the ring 2p=o0 the bunch meets the wake, which is left from the previous turn. According to Eq. (12), the amplitude of this residual wake is proportional to a factor expð2pGÞ and, therefore, it is small, if the wake memory parameter 2pG is large. Substituting Eq. (11) in Eq. (9) and Eq. (9) in Eq. (8), we find sffiffiffiffiffiffiffiffiffiffi ð0Þ if m ðI, fÞ r ðfÞð@G=@IÞ R0 I mNe2 cZ 0 fm ¼ þi 0 Dom dom Dom dom 2pny Pb2? Z 1 1 inf X e dfu  Dm ðfu, oÞeinfu , mny þ n þ iG 1 2p n ¼ 1 or, using Eq. (10), Dm ðf, oÞ ¼ Dð0Þ m ðf, oÞiW m r0 ðfÞ

Z

1

dfu SðffuÞDm ðfu, oÞeinfu :

1

ð13Þ Here, S¼

1 1 X einf 2p n ¼ 1 n þ mny þ iG

¼ iexpððimny þiGÞfÞ½HðfÞ þ Lm 

ð14Þ

m ¼ 71

For small and fast coherent oscillations the amplitudes fm are calculated using the linearized Vlasov equation:   @fm @y dG ¼0 ð6Þ þimoy fm þ Fðy,tÞ r0 ðfÞ @c m dI @t where Fðy,tÞ is the average value of the force due to the bunch wake perturbing the particle oscillations. Eq. (6) is solved using

HðfÞ is the Heaviside step function,

Lm ¼

expð2piðmny þ iGÞÞ 1expð2piðmny þiGÞÞ

ð15Þ

sffiffiffiffiffiffiffiffiffiffi Z pffiffi ð0Þ 1 R0 Ifm ðI, fÞ oÞ ¼ i dI 2pny 0 Dom dom

ð16Þ

ð0Þ Dm ðf,

Dom ¼ omo0 ny , dom ¼ mo0 AI, 1

Those can be either real octupole fields, or strong sextupole fields of the lattice.

Wm ¼ Om

Z

1

dI 0

Ið@G=@IÞ

Dom dom ðIÞ

,

Im Dom 4 0

ð17Þ

D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

and 2

Om ¼

mNe cZ 0 : 4ppny b2?

ð18Þ

We remind the reader that the value Om =2 yields the singleturn part of the coherent frequency shift of the monochromatic bunch (dom ¼ 0) of zero length (in Ref. [1] for more detail). The integral equation which is similar to Eq. (13) was solved in Ref. [1]. Repeating these calculations (see also in Appendix A), we obtain Z 1 ð0Þ Dm ðf, oÞ ¼ Dð0Þ dfu Dm ðfu, oÞeG2 ðf, fuÞ m ðf, oÞWm r0 ðfÞ f Z Wm r0 ðfÞB 1 ð0Þ  dfu Dm ðfu, oÞeG1 ðf, fuÞ ð19Þ Wm ð1Be Þ 1 where B¼

Lm 1þ Lm

¼ expð2piðmny þiGÞÞ

ð20Þ

G1 ðf, fuÞ ¼ iðmny þiGÞðfufÞWm F G2 ðf, fuÞ ¼ iðmny þiGÞðfufÞWm

Z

ð21Þ

where zm,k ¼ Om,k =dy , dy ¼ o0 jAjI0 is the betatron frequency spread, gm ¼ mA=jAj, and the parameter zm ¼ Dom =dy varies along the real axes. This stability diagram is shown in Fig. 1 (solid line). On the same graph full circles and full squares show the mode maps (zm,k ) for zm ¼ Om =ðmdy Þ ¼ 20 and for zm ¼ 10. In this example the bunch wake decays rather quickly (2pG ¼ 10). For this reason, the maximum values of the increments of unstable modes (Om =ð4pGÞ) are not large. As is seen in Fig. 1 the modes can only approach to the stability border for zm ¼ 10. Similar map for the bunch with zm ¼ 20 indicate more unstable modes. For negative values of A the points on the stability diagram are reflected relative to the axes Im z. In this case, the modes which were unstable for A 4 0 may occur inside the stability diagram (Fig. 2). More unstable modes can be found, if the wake memory parameter 2pG decreases. According to Eq. (25) or, (28), the monochromatic bunch approximation holds in the region, where jzm,k j b1. In the region, where jzm,k j5 1, collective modes decay due to Landau damping with the decrements, which are proportional to the bunch frequency spread. Since the values jzm,k j decrease with an increase in the mode number k, for bunches with a finite frequency spread

fu f

df00 r0 ðf00 Þ

ð22Þ

and 1 00

00

df r 0 ðf Þ þ

f

Z

1.0

fu 00

1

00

df r0 ðf Þ:

ð23Þ

For the bunches with smooth and non-singular frequency distribution functions the first two terms in Eq. (19) have no pole singularities in the whole plane of the complex variable Dom . These terms describe the bunch decoherence after initial kick and the effect of the beam break-up instability on such a decoherence. The third term describes the collective modes of the bunch due to a finite long-term memory of the wake. As a function of the complex variable Dom this term has simple poles at the values of Dom which are the roots of the dispersion equation 1BeWm ðDom Þ ¼ 0

0.5 Im (ζ)

F¼

Z

0.0

-0.5

-1.0

ð24Þ

-8

or Z

Om 2piðmny þ kÞ2pG

1

0

dI

IðdG=dIÞ

Dom,k dom ðIÞ

,

Im Dom,k 4 0: ð25Þ

Here, integers k¼0,71, 72,y give the mode numbers of the collective modes. For stable modes (Im Dom,k o0) the right-hand sides in Eqs. (17) and (25) should be analytically continued in the lower half-plane of Dom . We remind the reader that collective modes of a monochromatic bunch (dom ðIÞ ¼ 0) are stable provided that mðmny þ kÞ 4 0 [1]. According to Eq. (25) for the bunches with smooth distribution functions in the action variables of incoherent betatron oscillations Landau damping can stabilize the coherent betatron oscillations. Namely, the modes with the mode numbers k are stable, if their complex frequency shifts

Om,k ¼

Om

are found inside the stability diagram (e.g. in Ref. [5]). For example, for a bunch with an exponential distribution function   1 I GðIÞ ¼ exp  ð27Þ I0 I0 the equation for the stability diagram reads 1 ¼ zm,k

0

1

x

xe dx, zm gm x þ i0

-4

-2

0 Re (ζ)

2

4

6

8

1.0

0.5

0.0

ð26Þ

2piðmny þ kÞ2pG

Z

-6

Fig. 1. Stability diagram for an exponential distribution in actions of the vertical incoherent betatron oscillations (solid line) and the collective mode map (full circles and squares). Collective modes are stable below the border curve, m ¼1, 2pG ¼ 10, Om =dy ¼ 20 (circles) and 10 (squares), A 4 0.

Im (ζ)

1¼

569

m ¼ 71

-0.5

-1.0 -8

ð28Þ

-6

-4

-2

0 Re (ζ)

2

Fig. 2. Same as in Fig. 1, but A o 0.

4

6

8

570

D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

the stable modes can be found even in the regions, where the values of the ratios zm ¼ Om =dy are large. That reduces an amount of unstable collective modes in the coherent signal of a nonmonochromatic bunch compared to the monochromatic one.

where ð0Þ Xm ðf,tÞ ¼

¼

Z

1

dDom iDom t e 1 2pi 1

Z

1 0

yey

Dom gm dy y

dy ð38Þ

ð1 þ ig m dy tÞ2

3. Time-domain solutions An inverse Fourier transform Z dDom iDom t e Dm ðf, Dom Þ, eimo0 ny t Dm ðf,tÞ ¼ 2p C

Em ðf,tÞ ¼ Im Dom 40

where Rm is a factor, which is proportional to the kick strength. Substituting this expression in Eq. (16), we find Z 1 yey Wm ð0Þ dy ¼ ir0 ðfÞDð0Þ Dm ðf, oÞ ¼ ir0 ðfÞDð0Þ ð32Þ m m Dom gm dy y Om 0 where

Dm ðf, oÞ r0 ðfÞDð0Þ m

ð34Þ

or, using Eq. (19) Z 1 2 dfu r0 ðfuÞeG2 ðf, fuÞ dm ¼ Wm Wm f



2 Wm B Wm

ð1Be

Z

1

f

dfu r0 ðfuÞeG2 ðf, fuÞ ð39Þ

Pm ðf,tÞ ¼

B

Z

Om

C

2 dDom eiDom t Wm ðDom Þ 2pi ð1BeWm Þ

Z

1

1

dfu r0 ðfuÞeG1 ðf, fuÞ : ð40Þ

ð0Þ Xm ðf,tÞ

In these equations the term describes the decoherence of the low-intensity bunch after its initial transverse kick, the term Eðf,tÞ describes the effect of the beam break-up instability on this decoherence and the term Pðf,tÞ describes effect of collective modes on the dipole coherent oscillations of the bunch. As an example, we simplify further calculations assuming ( 1, jfjr D 1 r0 ðfÞ ¼ ð41Þ 2D 0, jfj4 D where 2D is the bunch length. Assuming also jmjny D 5 1, GD 5 1 and calculating in Eq. (35) the integrals over fu, we obtain      f Wm dm ðf, Dom Þ ¼ Wm Wm 1exp  1 D 2     BeWm f Wm  Wm exp 1þ Wm D 2 ð1Be Þ     f Wm exp  1 ð42Þ D 2

Em ðf,tÞ ¼

1

Z

Om

C

     dDom iDom t f Wm e Wm 1exp  1 2pi D 2 ð43Þ

and Pm ðf,tÞ ¼

Z

dDom eiDom tWm Wm ð1BeWm Þ C 2pi         f Wm f Wm  exp 1þ exp  1 : D 2 D 2 B

Om

ð44Þ

Þ

1 1

dfu r0 ðfuÞeG1 ðf, fuÞ :

ð35Þ

Substituting here Eq. (35), we write eimo0 ny t Dm ðf,tÞ ð0Þ r0 ðfÞDm

These expressions show that for head-on particles of the bunch (f ¼ D) the contribution to the coherent signal (Xm) of the singleturn beam break-up oscillations vanishes (Em ðf ¼ D,tÞ ¼ 0). For the tail-on particles (f ¼ D) Z 1 dDom iDom t e Em ðf ¼ D,tÞ ¼ Wm ð1eWm Þ ð45Þ Om C 2pi Pm ðf ¼ D,tÞ ¼

With these definitions we rewrite Eq. (29) in the following form: ð0Þ Z r ðfÞDm dDom iDom t e dm ðDom Þ: ð36Þ eimo0 ny t Dm ðf,tÞ ¼ 0 Om C 2pi

Xm ðf,tÞ ¼

C

Z

ð33Þ

Now, we define dm ðDom Þ ¼ iOm

Om

dDom iDom t 2 e Wm ðDom Þ 2pi

and

For monochromatic bunches (dy ¼ 0) this function has a pole singularity at the point Dom ¼ 0. For the bunches with a finite frequency spread (dy a0) the function Wm ðDom Þ is an analytic function of Dom in the plane of the complex variable Dom with the cut along the real semi-axis Dom Z0 (gm 4 0), or with the cut along the real semi-axis Dom r0, if gm o 0. We also assume that initial coherent oscillations of the bunch are produced by a weak transverse kick in the vertical direction of equal strength for all bunch particles. Therefore, we shall write pffiffi I=I Ie 0 fmð0Þ ðI, fÞ ¼ Rm r0 ðfÞ ð31Þ I0

sffiffiffiffiffiffiffiffiffiffi R0 I0 : 2pny

Z

ð29Þ

yields the solution to the problem in the time-domain. In this expression the integration contour C comes from Dom ¼ 1 to Dom ¼ 1 parallel to the axis Re Dom and above the highest increment of unstable collective modes. For the bunches with smooth (and non-singular) frequency distributions the integrand in Eq. (29) is more complicated function of Dom than we met in Ref. [1]. For this reason, most results below will be found calculating the integral in Eq. (29) numerically. We simplify these calculations assuming that G(I) is defined by Eq. (27). Then, for example, the function Wm ðDom Þ reads Z 1 yey dy, Im zm 4 0, m ¼ 71: ð30Þ Wm ¼ zm z m gm y 0

ð0Þ ¼ Rm Dm

1

ð0Þ ¼ Xm þ Em ðf,tÞ þ Pm ðf,tÞ

ð37Þ

1

Om

Z C

dDom iDom t BeWm e Wm ð1eWm Þ 2pi ð1BeWm Þ

ð46Þ

the contribution of the beam break-up oscillations to Xm can dominate, if the wake memory is short enough (jBj5 1). In the last case, the amplitudes of coherent oscillations vary rapidly along the bunch. In these equations the integration contour in the plane of the complex variable Dom should go above the increment of the maximal unstable mode. For particular values of zm positions of such contours were found varying Dom parallel to the axes ReðDom Þ, plotting the hodographs of the function 1BeWm and searching a hodograph curve which does not encircle the origin.

D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

5

571

105

4 103

2 1

Amplitude

Im (1-Bexp (-Wm))

3

0 -1

101

10-1

-2 10-3

-3 -4 -1

0

1 2 3 Re (1-Bexp (-Wm))

4

5

0

20

40

60

80

100

|Ωm|t

Fig. 3. Hodographs of 1BeWm for Im Dom ¼ 0:25  jOm j (dashed line) and for Im Dom ¼ 0:35  jOm j (solid line), zm ¼ 20, 2pG ¼ 10, gm ¼1.

For example, for zm ¼ 20 and 2pG ¼ 10 the dashed curve in Fig. 3 (Im Dom ¼ 0:25  jOm j) encircles the origin once. It means that one of the roots of the dispersion equation 1 ¼ BeWm has its imaginary part exceeding Im Dom ¼ 0:25  jOm j and hence, the integration contour in Eq. (40) should go higher than Im Dom ¼ 0:25  jOm j. The solid line in Fig. 3 shows the hodograph line which was calculated taking Im Dom ¼ 0:35  jOm j. This line does not encircle the origin and hence, we conclude that for zm ¼ 20 and 2pG ¼ 10 the increments of the collective modes do not exceed the value of Im Dom ¼ 0:35  jOm j. So, for this case we can take the integration contour in Eq. (40) above the line Im Dom ¼ 0:35  jOm j. On the other hand, to prevent too long computing times one should avoid unreasonable large values of Im Dom .

Fig. 4. Dependence of the amplitude of coherent oscillations jXm j (upper curve) and of the amplitude of the contributions of collective modes jPðf,tÞj (lower curve) on time. Open circles show the fitting curves, zm ¼ 20, 2pG ¼ 10, f ¼ D.

107 106 105 104 103 102 101 100

3.1. Short-range memory wakes For simplicity, we calculate the integrals over Dom in Eqs. (39) and (40) using the same integration contour. Besides, in this and in the next subsections we make the calculations assuming for simplicity that my ¼1 and A 4 0. The results of numerical calculations of the amplitudes of coherent oscillations jXm ðf,tÞj taken for e.g. zm ¼ 20 and 2pG ¼ 10 show (solid lines in Fig. 4) the beam break-up type dependence on time for both jXm ðf,tÞj and for the contribution of the collective modes jPðf,tÞj. Open circles in this plot show that at large times these functions can be fitted reasonable well using the following simple formulae: pffiffiffiffiffiffiffiffiffiffiffiffi! jOm jt jXm ðD,tÞj ¼ 0:34  exp ð47Þ 0:86 and jPm ðD,tÞj ¼ expð2pG þ ðjOm jtÞ0:61 Þ:

10-5

6

|Xm|

-2

ð48Þ

These results as well as the fact that the beam break-up contributions to the amplitudes of coherent oscillations dominates for short memory wakes qualitatively agree with predictions obtained in Ref. [1] for a monochromatic bunch. Similar dependencies of the amplitudes on time were found for the same G and zm ¼ 10 and for the time interval jOm jt r 100. However, for larger time intervals (jOm jt r 250) and provided that the value of zm r 10 the amplitudes of coherent oscillations jXm ðD,tÞj can reach a maximum and then decay (Fig. 5). Such dependencies of the amplitudes of coherent oscillations on time

10-1

0

50

100

150

200

250

|Ωm|t Fig. 5. Dependence of the amplitude of coherent oscillations jXm j on time. From top to bottom zm ¼ 15, 12, 10, 9 and 7; 2pG ¼ 10, f ¼ D.

are specific for the beam break-up phenomenon. Closer inspection (e.g. in Figs. 6 and 7) shows that for smaller values of zm the amplitudes jXm j can deviate from its beam break-up part jEm j, which can start be damped by the frequency spread of the bunch earlier than its self-consistent part jPm j. This fact can reflect a differences in the damping mechanisms for these two contributions. The function jEm j decays since the betatron frequency spread of the bunch results in the cumulative decoherence of its oscillations. The self-consistent part jPm j decays due to Landau damping of the collective modes of the bunch and due to the interference of the bunch self-consistent modes. Starting from zm ¼ 4 the function jXm ðD,tÞj reaches a maximum in the region jOm jt 4 25 (Fig. 8). In the region jOm jt Z 50 the decay of the amplitude jXm j can be fitted well using the formula:   jOm jt jXm j ¼ 25  exp  : ð49Þ 3:5  2pG In all these cases the contributions of beam break-up instability to the amplitudes of coherent oscillations exceed the contributions

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D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

of the self-consistent modes of the bunch. That is a consequence of a short memory of the bunch wake.

1.0x104

Amplitude

8.0x103

3.2. Long-range memory wakes

6.0x103

An increase in the duration of the wake memory (a decrease in

G) enlarges the contributions of the self-consistent modes to the 4.0x103 2.0x103 0.0 0

50

100

150

200

250

|Ωm|t Fig. 6. Dependence of the amplitude of coherent oscillations on time. Solid line—jXm j, dashed line—jPm j, open circles—jEm j; zm ¼ 10, 2pG ¼ 10, f ¼ D.

amplitudes of the bunch oscillations. Simultaneously, a faster decrease of increments of unstable modes of monochromatic bunches with an increase in the mode number may result in an exponential (or, in an almost exponential) asymptotic rises of the oscillation amplitudes in time (Fig. 9). Since the maximum increments of the monochromatic modes vary in proportion to Om =ð4pGÞ, a decrease in G, generally, increases the rates of the oscillation amplitudes (compare e.g. Figs. 5 and 9). For example, comparing relative contributions of the functions Pm ðD,tÞ and Em ðD,tÞ to jXm j (Fig. 10) we find that after initial transient period (jOm jt r 50) the contributions of the bunch modes (jPm j) to jXm j dominate and that in the asymptotic region the amplitude of coherent oscillations exponentially increases in time. In particular, in the asymptotic region (jOm jt Z70) the function jXm j can be

107 200

106 105 104 |Xm|

Amplitude

150

100

103 102

50

101 100

0 0

50

100

150

200

10-1

250

|Ωm|t

0

20

40

60

80

100

|Ωm|t

Fig. 7. Same as in Fig. 6, but zm ¼ 7.

Fig. 9. Dependence of the amplitude of coherent oscillations jXm j on time. From top to bottom zm ¼ 10, 8, 6, 5, 4.5 and 4; 2pG ¼ 1, f ¼ D.

10

106 105

8

104 |Xm|

Amplitude

6

4

103 102 101

2

100 0

10-1 0

50

100

150

200

250

|Ωm|t Fig. 8. Dependence of the amplitude of coherent oscillations jXm j (solid line) on time. Open circles—fitting curve from Eq. (49); zm ¼ 4, 2pG ¼ 10, f ¼ D.

0

20

60

40

80

100

|Ωm|t Fig. 10. Dependence of the amplitude of coherent oscillations on time. Solid line—jXm j, dashed line—jPm j, dots—jEm j; zm ¼ 8, 2pG ¼ 1, f ¼ D.

D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

fitted well using the formula   jOm jt , 2pG ¼ 1: jXm j ¼ 0:76  exp 7:87

ð50Þ

Inspecting the graph in Fig. 9, we conclude that for the wake with the memory parameter 2pG ¼ 1 the threshold of the instability is inside the region 4:5 Z zm Z4. Near the threshold coherent oscillations of the bunch decay using two stages (Fig. 11). After initial fast decay, which can be fitted well using the formula   jOm jt , 2pG ¼ 1, jOm jt r20, jXm j ¼ 1:85  exp  ð51Þ 25 the amplitudes of oscillations begin to decrease slower. During this stage a decrease in jXm j with time can be fitted using the formula   jOm jt , 2pG ¼ 1, jOm jt Z 20: ð52Þ jXm j ¼ 0:9  exp  155 This feature disappears, if the bunch and the wake parameters are well below the threshold of the instability (compare open circles

1.5

573

and squares in Fig. 12). For example, for zm ¼ 2 a decrease in the amplitude of coherent oscillations can be fitted well using one exponential function (Fig. 12)   jOm jt , jXm j ¼ 1:2  exp  5:5

2pG ¼ 1:

ð53Þ

Below the threshold of instability the character of the dependence of the amplitudes of coherent oscillations on time depends both on zm and on value of the memory parameter 2pG (e.g. Figs. 13 and 14). Just below the threshold (zm ¼ 4) and provided that the wake memory is not very short (e.g. 2pG r 4) the amplitudes decay in two stages where their initial rather fast decreases are followed by substantially slower ones. Closer inspection shows that such a behavior occurs due to misphasing of the functions Pm(t) and Em(t). For example, the data depicted in Fig. 15 show that the functions jPm ðtÞj and jEm ðtÞj can exceed the value jXm ðtÞj. That occurs in the regions, where the functions Pm(t) and Em(t) compensate each other. For a twice smaller bunch intensity (zm ¼ 2) the amplitudes of coherent oscillations decrease without changing the slope (e.g. in Fig. 14). The fact that a two stage oscillation amplitude decay occurs due to the contributions of the

8 1.0

|Xm|

6 |Xm|

0.5

0.0

4

2 0

20

40

60

80

100

|Ωm|t

0

Fig. 11. Dependence of the amplitude of coherent oscillations jXm j on time (open circles). Solid line—result of the fit using Eq. (51), dashed line—using Eq. (52); zm ¼ 4, 2pG ¼ 1, f ¼ D.

0

40

20

60

80

100

|Ωm|t Fig. 13. Dependence of the amplitude of coherent oscillations jXm j on time. From top to bottom 2pG ¼ 10, 4, 2, 1, zm ¼ 4, f ¼ D.

1.2 1.4

1.0

1.2 0.8

|Xm|

|Xm|

1.0 0.6 0.4

0.8 0.6

0.2

0.4

0.0

0.2 0

20

40

60

80

100

|Ωm|t Fig. 12. Dependence of the amplitude of coherent oscillations jXm j on time. Open circles—zm ¼ 2, solid line—result of the fit using Eq. (53), open squares—zm ¼ 4; 2pG ¼ 1, f ¼ D.

0.0 0

20

40

60 |Ωm|t

Fig. 14. Same as in Fig. 13, but zm ¼ 2.

80

100

574

D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

9 1.0

8 7

0.5

5

Im (ζ)

Amplitude

6

4

0.0

3 2

-0.5

1 0 -1

-1.0 0

20

40

60

80

-6

100

|Ωm|t Fig. 15. Dependencies of the amplitudes of coherent oscillations on time. Solid line—jXm j, dashed line—jPm j, dots—jEm j; 2pG ¼ 4, zm ¼ 4, f ¼ D.

-4

-2

0

2

4

6

8

Re (ζ) Fig. 17. Stability diagram (solid line) and the collective mode map (the first  50 modes) of a monochromatic bunch (full circles). Collective modes are stable below the border curve, my ¼ 1, 2pG ¼ 2, zm ¼ 6:873, A o 0.

3.0 1.0

2.5

0.5

1.5 Im (ζ)

Im(Δωm) / δy

2.0

1.0 0.5 0.0

0.0

-0.5

-0.5 -1.0

-1.0 0

2

4

6

8 ζm

10

12

14

16

-6

-4

-2

0 Re (ζ)

2

4

6

Fig. 16. Dependencies of the oscillation increments (ImðDom Þ) on the number of particles in the bunch (zm ). Open circles—2pG ¼ 1, full circles—2pG ¼ 2, full squares—2pG ¼ 4, the lines show results of the linear fits; f ¼ D.

Fig. 18. Same as in Fig. 17, but zm ¼ 4:406 and A 4 0.

bunch modes can be verified inspecting the upper graph in Fig. 13, where the mode contributions are negligible small (2pG ¼ 10). Such calculations were used to study the dependencies of the oscillation increments (Im Dom ) on zm for several values of the memory parameter 2pG (Fig. 16). Above the threshold these data show a linear increase in the increment on the number of particles in the bunch (zm ). The slopes of the fitting lines slowly decrease with an increase in the value of the memory parameter. Below the threshold of the instability the oscillations decay due to their Landau damping with the decrements (Im Dom ), which weakly depend on the value of zm .

lower bunch intensities they asymptotically decay in time. Such a behavior of the amplitudes of unstable coherent oscillations corresponds to the instability threshold. According to general expectations, the instability threshold occurs when the most unstable mode of the bunch oscillations reaches the stability diagram (e.g. in Figs. 17 and 18). In this paper, instead of calculations of the eigenfrequencies of the threshold modes we find the thresholds inspecting the dependencies of the amplitudes of coherent oscillations on time. In the simplest case we define that a threshold occur in the region of parameters, where the amplitudes of coherent oscillations do not depend on time asymptotically, or tend to an asymptotically constant value. The data depicted in Figs. 19 and 20 show that such a requirement can be very sensitive to the value of zm . So, the threshold value zm ¼ zth can be found with a good accuracy. Using such a recipe we calculate the dependence of zth on e.g. the value of the wake memory parameter 2pG. As is seen from Fig. 21, the threshold values of zm ¼ zth ðGÞ depend on the sign of the parameter A. Namely, the lattices with A o0 provide higher threshold bunch currents than that with A 4 0. This phenomenon occurs due to the differences in the Landau damping of the collective

3.3. Instability thresholds According to data depicted in Figs. 5 and 9 we can expect that for certain values of the bunch intensity (jOm j) and of the frequency spread of the bunch (dy ) an initial increase in the amplitudes of coherent oscillations will saturate (or, will saturate on average) in time. For higher bunch intensities the amplitudes of coherent oscillations asymptotically increase, while for the

D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

16

2.4

14

2.2

12 |Pm (-Δ,|Ωm|t)|

2.0

|Xm|

1.8 1.6 1.4

8 6

2

1.0

0

0.8

0

50

100

150

200

250

0

|Ωmt| Fig. 19. Dependencies of the oscillation amplitudes on time. From top to bottom zm ¼ 5:657, 5.6544 and 5.650; 2pG ¼ 2, f ¼ D, A o 0.

100

200 |Ωmt|

300

400

Fig. 22. Dependencies of the contributions of the bunch collective modes jPm j on time. Solid line—A 4 0, zm ¼ 4:406, dashed line—A o 0, zm ¼ 6:873; 2pG ¼ 3.

modes in the lattices providing a positive, or negative sign of the parameter A. According to Eq. (26) the frequency shift of coherent oscillations of a monochromatic bunch in our model is negative, if my 40 (it is positive, if my o 0). It means that for the lattices where A o0 the frequencies of the collective modes of a monochromatic bunch appear inside the spectrum of incoherent oscillations of the bunch. So, the collective modes can more effectively exchange their energies with the bunch particles, which results in their stronger Landau damping and, correspondingly, in higher instability thresholds. This effect can be seen in Figs. 19 and 20 as well as in Fig. 22, where the modulations of the coherent signal (jXm j) and of the contribution of the collective modes (jPm j) decay longer in the case, when A 4 0. Such dependence of decays of transverse coherent oscillations on the sign of the octupole lattice nonlinearity qualitatively agree with observations reported in e.g. Ref. [6].

2.5

|Xm|

2.0

1.5

1.0 50

0

100

150

200

4. Conclusions

250

|Ωm|t Fig. 20. Dependencies of the oscillation amplitudes on time. From top to bottom zm ¼ 4:15, 4.1445 and 4.14; 2pG ¼ 2, f ¼ D, A 4 0.

18 16 14 12 ζth

10

4

1.2

10 8 6 4 2

575

0

1

2

3

4

5

6

7

8

9

2πΓ Fig. 21. Dependencies of the threshold value of zm on 2pG. Full circles—A 4 0, open circles—A o 0, f ¼ D.

Using several simplifying models we have studied the effects of the frequency spread on the stability of the fast transverse coherent oscillations of a single bunch in a storage ring. In our calculations we neglected the possibility of the bunch cooling. It implies that the assumed frequency spread of incoherent oscillations of particles substantially exceeds the values of the cooling decrements. For simplicity, we have limited our studies by the case, when the frequency spread is produced by the octupole nonlinearity of the ring lattice. Besides, we assumed that the transverse bunch sizes and the lattice parameters enable the calculations in the one-dimensional frequency dispersion approximation, where the betatron frequencies of particles are determined by Eq. (2). We also have simplified our calculations assuming that the bunch linear density is a step function of the distance inside the bunch, which is defined in Eq. (41). The most of results concerning the time-domain evolutions of the amplitudes of coherent oscillations have been obtained using numerical integration in Eqs. (43) and (44). For this reason, we could avoid necessities of solving dispersion equations in Eq. (25) with subsequent calculations of the contributions of particular collective modes to the bunch coherent signal. The required integration contour in the plane of the complex variable Dom was set above the increment of the most unstable collective mode inspecting the hodographs for the function in the left-hand side of the dispersion Eq. (24). Initial conditions for such calculations

576

D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

were set assuming that initial coherent oscillations of the bunch are produced by a weak transverse kick in the vertical direction of equal strength for all bunch particles Eq. (31). For the short-range wakes (2pG Z10) and small frequency shifts (e.g. jzm j Z 20) results of numerical calculations are in general agreement with those, reported in Ref. [1] (e.g. in Fig. 4). Due to short memory the amplitudes of coherent oscillations are mainly determined by the contribution of the beam break-up phenomenon. The amplitude of the coherent signal varies along the bunch rapidly. An increase in the frequency spread, or a decrease in the bunch intensity below, say, jzm j C 9 results in a decay of the oscillation amplitudes at large times (e.g. Fig. 5). Qualitatively, the damping of coherent signal coincides with a decay of its beam break-up component (e.g. in Fig. 7). For the long-range wakes (e.g. 2pG o 10) coherent oscillations indicate stronger contributions to their features of the collective modes. If the wake memory parameter is not very large (e.g. 2pG ¼ 128) the contributions of the collective modes to the coherent signal of the bunch can overcome that due to the beam break-up effect at least at large times. In the intermediate asymptotic region in time the amplitudes of the coherent signals of bunches with large intensity can rise exponentially (e.g. in Fig. 10). With a decrease in the bunch intensity coherent oscillations indicate an instability threshold. In the rings with A 40 approaching this threshold is accompanied by the modulations of the amplitudes of coherent signals. Close to the threshold coherent signals decay via two-stages, when an initial fast decrease of the amplitudes in time is followed by a substantially slower one (e.g. in Fig. 11). Above the thresholds the oscillation increments linearly increase with the bunch intensity (e.g. Fig. 16). The thresholds of the instability occur when the increment of the most unstable collective mode reaches the border of the stability diagram of coherent oscillations. The value of the threshold current (zth ) depends on the sign of the parameter A. For the model wake, which is used in this paper, the lattices providing A o0 enable higher instability thresholds than that providing A 40. The higher thresholds reflect stronger Landau damping of the modes for the cases, where the collective mode frequencies are found inside the frequency spectrum of incoherent oscillations of particles. Qualitatively, this fact is illustrated by the data depicted in Fig. 22. Since the eigenmodes in the problem appear due to the wake memory the threshold bunch intensities (zth ), generally, depend on the value of the tune of incoherent betatron oscillations. For considered cases, the amplitudes of coherent oscillations should enable an application of the theory of the linear coherent oscillations. That reduces possible values of initial amplitudes of coherent oscillations of the bunch.

iLm

Z

1 1

dfuexpðiðmny þiGÞðfufÞÞDð0Þ m ðfu, oÞ:

ðA:3Þ

Defining here pm ðfÞ ¼ wðf, oÞexpðiðmny þ iGÞfÞ, ð0Þ pð0Þ m ðfÞ ¼ wm ðfÞexpðiðmny þiGÞfÞ

ðA:4Þ

we transform Eq. (A.2) to the following form: Z 1 dfur0 ðfuÞpm ðfuÞWm Lm Cm pm ðfÞ ¼ pð0Þ m ðfÞWm

ðA:5Þ

where Z Cm ¼

ðA:6Þ

f

1 1

dfu r0 ðfuÞpm ðfuÞ:

Repeating the calculations given in Appendix A of the paper [1], we obtain Z 1 Wm B ð0Þ ðfÞ dfu r0 ðfuÞpð0Þ pm ¼ pm m ðfuÞexpðWm FÞ ð1BeWm Þ 1 ! Z Z fu

1

ð0Þ dfu r0 ðfuÞpm ðfuÞexp Wm

Wm f

f

df00 r0 ðf00 Þ

ðA:7Þ

or, using Eq. (A.4),

wðf, oÞ ¼ wð0Þ m ðfÞ

Z

Wm B ð1Be

Wm

Þ

1

ð0Þ dfu r0 ðfuÞwm ðfuÞeG1 ðf, fuÞ þKðfÞ

1

ðA:8Þ where KðfÞ ¼ Wm

Z

1

f

G2 ðf, fuÞ dfu r0 ðfuÞwð0Þ m ðfuÞe

ðA:9Þ

and G1 ðf, fuÞ ¼ iðmny þ iGÞðfufÞWm F Z

G2 ðf, fuÞ ¼ iðmny þ iGÞðfufÞWm

ðA:10Þ fu

f

df00 r0 ðf00 Þ:

ðA:11Þ

Now, we note that Z 1 ð0Þ K¼ dfuwm ðfuÞexpðiðmny þ iGÞðfufÞÞ f " !# Z fu d 00 00 exp Wm  df r0 ðf Þ dfu f Z 1 dwð0Þ ðfuÞ G2 ðf, fuÞ G2 ðf, fuÞ e Þwð0Þ dfu m ¼ lim ðwð0Þ m ðfuÞe m ðfÞ dfu fu-1 f Z 1 ð0Þ iðmny þ iGÞ dfuwm ðfuÞeG2 ðf, fuÞ : f

Appendix A. Solution to Eq. (13) Substituting here The integral Eq. (13) can be solved defining Dm ðf, o

Þ ¼ Dð0Þ m ðf,

oÞiW m r0 ðfÞwm ðfÞ:

ðA:1Þ

Substituting this expression in Eq. (13), we find that a new unknown function wm ðfÞ obeys the integral equation Z 1 wm ðfÞ ¼ wð0Þ dfuexpðiðmny þ iGÞðfufÞÞr0 ðfuÞwm ðfuÞ m ðfÞWm f Z 1 Wm Lm dfuexpðiðmny þ iGÞðfufÞÞr0 ðfuÞwm ðfuÞ 1

dwð0Þ ð0Þ m ð0Þ ¼ iDm ðf, oÞiðmny þ iGÞwm ðfÞ df and lim ðexpðiðmny þiGÞfuÞwð0Þ m ðfuÞÞ Z 1 ð0Þ dfu Dm ðfu, oÞ  expðiðmny þ iGÞfuÞ ¼ iLm

fu-1

wð0Þ m ðfÞ ¼ i

we find ð0Þ m ðfÞiLm exp

K ¼ w Z

Z

1 00

f

f

expðiðmny þ iGÞðfufÞÞi

00

!Z

df r0 ðf Þ

Wm

1

dfuexpðiðmny þ iGÞðfufÞÞDð0Þ m ðfu, oÞ

ðA:13Þ

1

ðA:2Þ where

ðA:12Þ

Z

1

f

1

1

ð0Þ dfu Dm ðfu, oÞ

G2 ðf, fuÞ dfu Dð0Þ m ðfu, oÞe

D.V. Pestrikov / Nuclear Instruments and Methods in Physics Research A 624 (2010) 567–577

Z 1 ð0Þ þ iðmny þiGÞ dfu wm ðfuÞeG2 ðf, fuÞ iðmny f Z 1 G2 ðf, fuÞ þ iGÞ dfu wð0Þ m ðfuÞe

iðmny þ iGÞ

Z

K1 ¼ ið1 þ Lm Þð1BeWm Þexp Wm

!

1

df00 r0 ðf00 Þ

f

Z

ð0Þ dfu Dm ðfu, oÞexpðiðmny þ iGÞðfufÞÞ 1 Z 1 BK 1 ðfÞ ð0Þ i þ dfu Dm ðfu, oÞeG2 ðf, fuÞ : Wm ð1Be Þ f

K1 ðfÞ ¼ Wm

1

1

ð0Þ G1 ðf, fuÞ : m ðfuÞe

dfu r0 ðfuÞw

dfuDð0Þ m ðfu,

Substituting this expression in Eq. (A.14), we obtain ! Z 1 00 00 wðf, oÞ ¼ iLm exp Wm df r0 ðf Þ Z

ðA:15Þ

þ

1

Using Eq. (A.13) and !

1

df00 r0 ðf00 ÞWm

f

ð0Þ lim expðiðmny þ iGÞfuÞwm ðfuÞ Z 1 dfu Dð0Þ ¼ ið1 þ Lm Þ m ðfu, oÞ  expðiðmny þ iGÞfuÞ 1

lim expðWm FÞ ¼ exp Wm

Z

!

1

df00 r0 ðf00 Þ

iB

Z

1

Wm

ðA:16Þ

f

Recalling Eq. (A.1), we arrive at Eq. (19).

!

1

f

df00 r0 ðf00 Þ

we rewrite K1 in the following form: ! Z 1 K1 ¼ ið1 þ Lm Þð1BeWm Þexp Wm df00 r0 ðf00 Þ f

1

1

Z

ð0Þ dfu Dm ðfu, oÞeG1 ðf, fuÞ ð1Be Þ 1 Z 1 ð0Þ i dfu Dm ðfu, oÞeG2 ðf, fuÞ :

fu-1



ð1BeWm Þexp Wm

or

wðf, oÞ ¼

as well as

Z1 1

iBð1 þ Lm Þ

f

Z

fu-1



ð0Þ dfu Dm ðfu, oÞexpðiðmny þiGÞðfufÞÞ

ð1BeWm Þ f Z 1 ð0Þ  dfuexpðiðmny þ iGÞðfufÞÞDm ðfu, oÞ 1 Z 1 iB ð0Þ dfu Dm ðfu, oÞeG1 ðf, fuÞ  Wm ð1Be Þ 1 Z 1 G2 ðf, fuÞ i dfuDð0Þ m ðfu, oÞe

G1 ðf, fuÞ G1 ðf, fuÞ K1 ¼ lim ðwð0Þ Þ lim ðwð0Þ Þ m ðfuÞe m ðfuÞe fu-1 fu-1 Z 1 d ð0Þ dfu eG1 ðf, fuÞ w ðfuÞ  dfu m 1 Z 1 G1 ðf, fuÞ dfu wð0Þ : iðmny þ iGÞ m ðfuÞe

Z

f

1



Now, we write

fu-1

oÞ  expðiðmny þ iGÞðfufÞÞ

1

1

lim expðWm FÞ ¼ exp Wm

df00 r0 ðf00 Þ

G1 ðf, fuÞ dfuDð0Þ : m ðfu, oÞe

i

Here,

!

1

f

Z1 1 ðA:14Þ

Z

1



1



Z

G1 ðf, fuÞ dfuwð0Þ m ðfuÞe

or

or Z

1 1

f

wðf, oÞ ¼ iLm exp Wm

Z

577

ð0Þ dfuexpðiðmny þ iGÞðfufÞÞDm ðfu, oÞ ð0Þ G1 ðf, fuÞ dfuðiDð0Þ m ðfu, oÞiðmny þ iGÞwm ðfuÞÞe

References [1] D.V. Pestrikov, Nucl. Instr. and Meth. A 603 (2009) 214. [2] P. Kernel, R. Nagaoka, J.-L. Revol, G. Besnier, in: Proceedings of EPAC 2000, 2000, p. 1133. [3] D.V. Pestrikov, Part. Accel. 41 (1) (1993) 13. [4] D.V. Pestrikov, Nucl. Instr. and Meth. A 562 (2006) 12. [5] N.S. Dikansky, D.V. Pestrikov, Physics of Intense Beams and Storage Rings, AIP PRESS, New York, 1994. [6] Y. Kobayashi, K. Ohmi, in: Proceedings of EPAC98, 1998, p. 1288.