Transverse modes and instabilities of a bunched beam with space charge and resistive wall impedance

0.26 MB·English

by  V. Balbekov

FERMILAB-PUB-11-598-APC
Transverse modes and instabilities of a bunched beam
with space charge and resistive wall impedance
V. Balbekov
Fermi National Accelerator Laboratory
P.O. Box 500, Batavia, Illinois 60510∗
(Dated: January 25, 2012)
Transverse instability of a bunched beam is investigated with synchrotron oscillations, space
charge,andresistivewallwakefieldtakenintoaccount. Boxcarmodelisusedforall-roundanalysis,
and Gaussian distribution is invoked for details. The beam spectrum, instability growth rate and
2 effectsofchromaticityarestudiedinawiderangeofparameters,bothwithhead-tailandcollective
1 bunchinteractionsincluded. Effectsoftheinternalbunchoscillationsontheofcollectiveinstabilities
0 is investigated thoroughly. Landau damping caused by the space charge tune spread is discussed,
2 and theinstability thresholds of different modes of Gaussian bunch are estimated.

n
a PACSnumbers: 29.27.Bd
J

4
I. INTRODUCTION well,andithasbeenshownthatthismorerigorouseffect
2
can be caused only by positive wake (for comparison,

] Transverse instability of a bunch in a ring accelera- resistive wall creates a negative wake).
h tor has been considered independently by Pellegrini [1] Asinglebunchwasinvestigatedinthe mentionedarti-
p
- and Sands [2] with synchrotronoscillationsand some in- cles because only short-range forces have been accepted
c ternal degrees of freedom of the bunch taken into ac- in all the cases. More general theory is developed in
c count (“head-tail instability”). Later Sacherer have in- this paper where a bunched beam with arbitrary num-
a
. vestigated the effect in depth including dependence of ber of the bunches is examined taking into account the
s the bunch eigenmodes on amplitude of synchrotron os- space charge, intra-bunch and bunch-to-bunch interac-
c
i cillations (“radial modes” [3]). tions. Both kinds of the interactions affect the beam
s eigenmodesandtheinstabilitygrowthrate,althoughone
y Space charge field of the bunch was not taken into ac-
ofthemcandominateisspecificsituation. Justfromthis
h countinthesepioneerworks. Itsrolewasstudiedfirstin
p Ref. [4] where it has been shown that the space charge standpointthecommonusedtermslike“collectivemodes
[ tunespreadresultsinLandaudampingwhichsuppresses instability” or “head-tail instability” are treated in the

1 many of the head-tail modes, much like other sources of paper.
Thepresentationisfocusedfirstontheresistivewallin-
v the incoherent tune spread.
0 However,the lastresulthas beenobtainedwith anas- stability[7]–[9]buttheresultscanberathereasyadapted
1 sumptionthatthe space chargetune shift issignificantly for other known wakefields. The boxcar model is inten-
1 sively used to get a general outlook of the problem in a
less than the synchrotron tune. A closer examination of
5 widerangeofparameters. MorerealisticGaussianbunch
the problem in Ref. [5] has led to the conclusion that
.
1 almost all head-tail modes are prone to Landau damp- is closely examined in a limiting case of low synchrotron
0 frequency,althoughtheoppositecaseofhighfrequencyis
ing till then the space charge tune shift is about twice
2 also invoked to estimate thresholds of Landau damping.
less then the synchrotron tune. The damping vanishes
1 Both betatron and synchrotron bare oscillations are
when the shift becomes more, lower eigenmodes being
:
v free from the decay first. The lowest (rigid) mode is the implied to be linear in this paper. The nonlinear effects
i unquestionably requirea specialinvestigation,especially
X only universal exception from the rule being potentially
as an additional factors of Landau damping.
unstable with any space charge. Sometimes 1–2 next
r
a modes can demonstrate similar behavior, in dependence
on the bunch shape.
However, no wakefield was included in the analysis of II. BUNCHED BEAM GENERAL EQUATIONS
Ref. [5] in fact, so the results might be interpreted only
as conditions which permit the bunch instability but do With space charge and wakefield taken into account,
not determine its characteristics completely. Effects of equation of coherent betatron oscillations of a bunched
short wakes were actually studied in Ref [6] where the beam in the rest frame can be written in the form [6]:
instabilitygrowthratehasbeenfoundatdifferentbunch
parameters like its length, chromaticity, etc. Transverse ω ∂Y
Q Y +iQ +∆Q(Y Y¯)
modes couplinginstability wasconsideredinthe workas Ω − 0 s∂φ −
(cid:18) 0 (cid:19)
∞
ω
′
= 2 exp i(θ θ) Q ζ
0
− Ω − −
Zθ (cid:20) (cid:18) 0 (cid:19)(cid:21)
∗Electronicaddress: [email protected] q(θ′ θ)Y¯(θ′)ρ(θ′)dθ′ (1)
× −
2
where θ is longitudinal coordinate (azimuth), ω is fre- where k = 1,2,...,M. There are the head-tail modes
quency of the coherent oscillations, Q and Ω are cen- inside the groups, each satisfying the equation:
0 0
tralbetatrontune andrevolutionfrequency(inthelabo-
∂Y
ratoryframe),∆Q(θ)isthe spacechargetune shiftaver- νY +iQ +∆Q(Y Y¯)
s
∂φ −
agedovertransversedirections,Q andφaresynchrotron
s
tune and phase, ζ = −ξ/η is normalized chromaticity, = 2 θ0exp[ iλ(θ′ θ)]q(θ′ θ)Y¯(θ′)ρ(θ′)dθ′
and q(θ) is normalizedwakefieldwhich specific form will − − −
Zθ
be defined later. The bunch shape in the longitudinal ∞ θ0
phase space (θ,u) is described by a distribution function +2 exp( 2πimκ) exp[ iλ(θ′ θ)]
F(θ,u) with corresponding linear density: m=1 − Z−θ0 − −
X
∞ q 2πm +θ′ θ Y¯(θ)ρ(θ′)dθ′ (7)
ρ(θ)= F(θ,u)du (2) × M −
Z−∞ where 2(cid:16)θ isthebunch(cid:17)length,and κ=(k ω/Ω )/M
0 0
(normalization of the functions will be specified later). (k Q )/M. Note that the head-tail mod−es are not au≃-
0
The function Y(θ,u) is an average transverse displace- ton−omous formations but each of them more or less de-
ment of particles located in the point (θ,u) of the longi- pends on the collective mode which it falls.
tudinal space, multiplied by the factor exp[i(Q0+ζ)θ]. The variable τ = θ/θ0 will be used further as a new
Additional averaging of this function over momentum is longitudinal coordinate having a range [–1,1], and the
denoted as Y¯(θ) being defined by the expression: normalization condition will be imposed
∞
1
ρ(θ)Y¯(t,θ)= F(θ,u)Y(t,θ,u)du (3) ρ(τ)dτ =1. (8)
Z−∞ Z−1
It is more convenient to represent the right hand part
of Eq. (1) as a sum over all the bunches preceding the
III. RESISTIVE WAKE
examined one (including all preceding turns):
νY +iQ ∂Yn +∆Q(Y Y¯ ) ResistivewallinstabilitywaspredictedfirstitRef.[7]–
n s n n
∂φ − [8]. Corresponding wakefield function q(θ) is negative,
end and can be presented in the convenient form:
=2 exp[ iλ(θ′ θ)]q(θ′ θ)Y¯ (θ′)ρ (θ′)dθ′
n n
Z∞θ − − 2πm −′ q(θ)=−q0r2θπ, q0 = 2απrγ0βNQbR0b23yr2Ωπ0σ (9)
+2 exp iλ +θ θ
− M −
m=n+1 Z (cid:20) (cid:18) (cid:19)(cid:21) where Nb is number of particles per bunch, r0 =
X (m) e2/mc2 is classic radius of the particle, R is the machine
2πm
q +θ′ θ Y¯ (θ′)ρ (θ′)dθ′ (4) radius, σ is specific conductivity of the beam pipe, by is
m m
× M − its semi-height, and α is the pipe form-factor (α= 1 for
(cid:18) (cid:19)
aroundpipe). ThenonecanrewriteEq.(7)intheform:
where ν = ω/Ω Q , λ = ζ ν ζ, M is num-
0 0
− −′ ≃ ′
ber of bun′ches in the beam, ′ρm(θ ) = ρ(2πm/M +θ ), νY +iQ ∂Y +∆Q(Y Y¯)= 2q 2π exp(iλθ τ)
and Ym(θ ) = Y(2πm/M +θ ). Any integral in this ex- s∂φ − − 0 θ 0
pression is taken over one of the bunches which are not r 0
presumedyettobeidentical(inparticular,someofthem 1 y(τ′)dτ′ ∞ 1 y(τ′)dτ′
+ exp( 2πimκ)
msaatiysfibeedeimnpatnyy).cHasoew:ever,periodicityconditionsmustbe ×"Zτ √τ′−τ mX=1 − Z−1 √Tm+τ′−τ#
(10)
ρ (θ)=ρ (θ),
m+M m where T = 2π/Mθ is the bunch spacing in terms of
0
Y¯m+M(θ)=Y¯m(θ)exp[ 2πi(Q0+ζ)] (5) the variable τ, and the notation is used for a shortness:
−
y(τ) = Y¯(τ)ρ(τ)exp( iλθ τ). Square root in the last
0
− ′
integral not much depends on the addition (τ τ), so
A. Symmetric beam the expansioninto Taylorseries can be applied−resulting
in:
In subsequent sections we will consider symmet-
∂Y
ric beams composed of identical equidistant bunches: νY +iQs +∆Q(Y Y¯) q0exp(iλθ0τ)
∂φ − ≃
ρ (θ) = ρ(θ). It would be checked than that all the
sonlutions of Eq. (4) fall into M groups which are known 2 2π 1 y(τ′)dτ′ +√M V (κ) 1 y(τ′)dτ′
as collective modes: ×(cid:20)− rθ0 Zτ √τ′−τ (cid:16) 1 Z−1
Y¯(k)(θ)=Y¯(θ)exp 2πin(Q +ζ k) (6) V (κ)B 1 y(τ′)(τ′ τ)dτ′ (11)
n (cid:20) M 0 − (cid:21) − 2 Z−1 − (cid:17)(cid:21)
3
1.0 6
n=0 m=2
n=1
4 n=2
0.5
n=3 m=1
2 n=4
0.0
) Q
κV( ∆/m 0 m=0
−0.5 νn,
Re V |κ|1/2 −2
Im V1|κ|1/2 m=−1
1
−1.0 Re V −4
2
Im V
2
m=−2
−1.5 −6
−0.50 −0.25 0.00 0.25 0.50 0.0 0.5 1.0 1.5 2.0 2.5 3.0
κ µ = Q /∆Q
(cid:24)(cid:19)s
FIG.1: Functionsp|κ|V1(κ)andV2(κ) (resistivewallwake). FIG. 2: Eigentunes of theboxcar bunch
where B =Mθ0/π is bunch factor, and Yn,m(τ,u) with eigentunes νn,m where m = n,n −
2,...,2 n, n. They satisfy the orthogonality condi-
∞ exp( 2πimκ) tions: − −
V (κ)= 2 − , (12)
1
− √m
∗
m=1 FY Y dτdu δ δ (15)
X∞ n1,m1 n2,m2 ∝ n1,n2 m1,m2
1 exp( 2πimκ) ZZ
V (κ)= − (13)
2 −2 m√m All the eigentunes are real numbers. Some of them are
m=1
X plotted in Fig. 2 where the space charge tune shift is
Boththefunctionsareperiodical,oneperiodofV2(κ)be- used for a scaling. At µ ≡ Qs/∆Q = 0, the eigen-
ing plotted in Fig. 1. Because the function V1(κ) has tunes νn,n take start from the point ν = 0 that is
thesingularities,itsplotismultipliedbyaperiodicfactor ω/Ω0 = Q0 (the bare tune). Other eigentunes νn,m<n
κ for a convenience. start from the point ν = ∆Q which corresponds the
|It|goeswithoutsayingthatformoftheseplotsdepends incoherenttunewiththesp−acechargeincluded: ω/Ω0 =
pon the wakefield. However, in most cases it does not Q0 ∆Q. At µ 1, all eigentunes acquire the ad-
change a general structure of Eq. (11) as well as many ditio−ns δν ∝ Q2s/∆≪Q. In this limiting case, the eigen-
subsequent inferences. functions have about linear polarization: Yn,n depends
mostly on longitudinal coordinate τ at any n, and all
other functions Y depend mostly on momentum
n,m<n
IV. BOXCAR MODEL u (many examples are given in Ref. [5]). However,
these solutions merge together at µ 1, forming multi-
≫
poles Y (A)exp(imφ) with different dependence from
The boxcar model is described by the expressions: n,m
synchrotronamplitude. These radial modes Y (A) are
n,m
1 1 bornbyLegendrepolynomialsofpowersn= m, m +2,
F = , ρ(τ)= at τ <1 (14) | | | |
etc. NotethatthefunctionsY (A)arethelowest(min-
2π√1 τ2 u2 2 | | n,n
− − imallyoscillating)radialmodeswithgivenn=m. Their
where u is normalized transverse momentum conjugated eigentunes are n(n + 1)/2 Q2/∆Q at small µ and
× s
with the longitudinal coordinate τ: u2 = A2 τ2, A = nQs at large one. The functions Yn,m<n are treated
− ∼
τ . A virtue of this model is that all solutions of usually as higher radial modes.
max
Eq. (11) are exactly known at q = 0 with any param-
0
eters Q and ∆Q, as it was shown first in Ref. [3] and
s
developed in detail in Ref. [5]. This circumstance gives A. Low wake
a greatpossibility to overlookthe wakefieldproducedef-
fects. Required information is shortly reminded below. The study of the boxcar model is continued in this
All the solutions of Eq. (11) with q = 0 are derivable subsection with an assumption that the wake is small
fromLegendrepolynomials Y¯(τ)=P (τ), n=0,1,2,... enoughtoapply the perturbationmethods(applicability
n
At any n, there are n + 1 different eigenfunctions of this approximation will be discussed in Sec. V). Then
4
1.0 0.0
(0,0)
n=0
n=1
0.8 −0.2
n=2
(1,1)
n=3
0.6 −0.4
Q (2,2) n
ν∆/n,m 0.4 (3,3) (1,−1) Re f −0.6 nn==01
n=2
(2,−2)
(3,−3) n=3
0.2 (2,0) −0.8 n=4
(3,1)&(3,−1) n=5
0.0 −1.0
0 1 2 3 4 5 0 2 4 6 8 10
µ = Q /∆Q θζ
(cid:24)(cid:19)s 0
FIG. 3: Functions Λn,m(µ) of the boxcar bunch. Numbers FIG.4: FunctionsRefn(θ0ζ)oftheboxcarbunch. Theright-
(n,m) are given near each line. hand parts of these even functions are shown.
λ = ζ in Eq. (11), and the additions to the eigentunes
can be presented in the form:
n=0
∆ν =q Λ (µ) n=1
n,m 0 n,m 0.4
× n=2
2 2πf (θ ζ)+√M g (θ ζ)V (κ)+ih (θ ζ)V (κ)B n=3
n 0 n 0 1 n 0 2
(cid:20) rθ0 (cid:16) (cid:17)(cid:21) 0.3 nn==45
(16)
n
with the coefficients: m f 0.2
I
−1
0.1
Λ (µ)= F Y 2dτdu (17)
n,m n,m
| |
(cid:20)Z Z (cid:21)
1 1 y(τ′)dτ′ 0.0
∗
f (θ ζ)= y (τ)dτ (18)
n 0 −Z−1 n Zτ √τ′−τ
1 2 −0.1 0 2 4 6 8 10
g (θ ζ)= y (τ)dτ (19)
n 0 n θζ
(cid:12)Z−1 (cid:12) 0
1 (cid:12) 1 (cid:12)
h (θ ζ)=2Im y∗(τ)τdτ(cid:12) y (τ′)dτ(cid:12)′ (20)
n 0 n n
Z−1 Z−1 FIG.5: FunctionsImfn(θ0ζ)oftheboxcarbunch. Theright-
hand parts of these odd functions are shown.
Generally, these relations are applicable to a bunch of
anyformwithlowwakefield. However,theboxcarmodel
is the only known case at present which allows to inves-
modes Y . It is very understandable result because
tigate the problem in depth because its eigenfunctions n,m<n
higherradialmodesarepolarizedmostlyinu-directionat
Y (τ,u) andy (τ)=P (τ)ρ(τ)exp( iθ ζτ)arereally
m,n n n 0
− small µ, so that the global bunch displacement at given
knownwithanyµ. Theresultsarepresentedgraphically
azimuth and, correspondingly, excited wakefield should
in Figs. 3–7 and commented below.
be relatively small in this limiting case.
The coefficients Λ depend only on the ratio µ =
n,m
Q /∆Q, and does not depend on the bunch length or In contrast with this, the part of Eq. (16) in square
s
chromaticity (Fig. 3). They describe a general effect of brackets describes effects of the bunch length and chro-
synchrotron oscillations and space charge on transverse maticity on different n–modes. There are three terms
coherentmotionofthebunches,includingpossibleinsta- here which are concerned with different physical effects.
bility growth rate. It is seen that the space charge tune Interaction of particles inside the bunch is described
shiftenhancesinfluenceofthewakefieldonthelowestra- by the coefficients f (θ ζ) some of which are plotted in
n 0
dialmodesY butdepressesitsinfluenceonthehigher Figs.4–5. With non-zerochromaticity,this partis capa-
n,n
5
1.0
n=0
0.4
0.8 n=1
n=2
n=3 0.2
0.6 n=4
n=5
n n 0.0
g h
0.4
n=0
−0.2
n=1
n=2
0.2 −0.4 n=3
n=4
n=5
0.0 −0.6
0 2 4 6 8 10 0 2 4 6 8 10
θζ θζ
0 0
FIG. 6: Functions gn(θ0ζ) of the boxcar bunch. The right- FIG. 7: Functions hn(θ0ζ) of the boxcar bunch. The right-
hand parts of these even functions are shown. hand parts of these even functions are shown.
ble to cause the head-tail instability of different modes upon the different character of the chromatic effects in
which basic properties were predicted in earliest works coasting and bunched beams.
[1]–[2]. It is a single-bunch effect which is proportional Infirstcase,chromaticityleads toa spreadofincoher-
tothebunchpopulationanddoesnotdependonnumber ent betatron frequencies which phenomenon can cause
of the bunches. Landaudamping resulting intotalsuppressions(preven-
Second term in Eq. (16) describes the main effect of tion) of instability.
the collective interaction of the bunches (factor V (κ) in
1 In contrast with this, averagemomenta of all the par-
Fig. 1) including its dependence on chromaticity (factor
ticlesareequalizedinthe bunchthroughsynchrotronos-
g (θ ζ)). As a rule, the term gives a maximal contribu-
n 0 cillations. Insuchconditions,chromaticitydoesnotpro-
tion to the tune shift, especially if the beam consists of
vide a systematic tune spread and cannot bring about
many bunches. Indeed, with M 1 one can get κ =
Landau damping at once. There is no questions that
≫
(k Q )/M 1 that is V (κ) (1+κ/κ)/ κ
0 1 additional slip of betatron phases of particles with re-
− ≪ ≃ | | | |
for the most unstable modes (k > Q ). Then the total
0 spect to the coherent field phase affects the interaction
p p
contribution of this term to the tune is:
andcan change the instability growthrate. As it follows
q MΛ g (θ ζ) k Q from Fig. 6, switch of the instability peaks to the higher
0 n,m n 0 0
∆νn,m 1+i − (21) internal modes is the most descriptive result of this.
≃ |k−Q0| (cid:18) |k−Q0|(cid:19) However,itisnecessarytotakeintoaccountalsoother
With Eq. (9) forpq used, it gives the expression: then chromaticity factors which can result in Landau
0
damping. In particular, space charge tune spread itself
∆ω αr NRδ(ω) ω canproducethiseffectinbunchedbeams. Asitisshown
nm 0
= 1+i Λ g (θ ζ) (22)
Ω 2πγQ b3 ω n,m n 0 in Ref. [4]–[5], the higher internal modes are more sensi-
0 0 y (cid:18) | |(cid:19) tive to this kind of Landaudamping. Therefore,one can
where N = MN is the total beam intensity, ω = expect that the above mentioned shift of the instability
b
Ω (k Q ) is the coherent frequency in the laboratory peaksiscapabletosuppressallinternalbunchmodesex-
0 0
−
frame, and δ(ω) is corresponding skin depth. First part cept the rigid one which is not prone to this kind of the
of the formula coincides with well known expression for damping [5]. However, this problem cannot be solved in
resistive wall instability of a coasting beam [7]–[8]. The frames of the boxcar model which ignores this part of
factors Λ describe an impact of the bunching upon the tune spread at all. Therefore, we postpone its de-
n,m
different head-tail modes, including their dependence on tailed analysis to Sec. VI where more realistic Gaussian
synchrotronfrequencyandspacechargetune shift. Note distribution will be invoked.
that the coefficient of the most important rigid mode A peculiarity of a single bunch beam is that its insta-
Λ = 1 independently on the mentioned parameters, bility is possible in a restricted region of parameters. As
0,0
that is the bunching does not affect this mode. it follows from Fig. 1 at M = 1, imaginary part of the
Thelasttermg (θ ζ)inEq.(22)describesaninfluence coefficients V and V is positive at 0<k Q <0.5. It
n 0 1 2 0
−
of chromaticity. In this regard, it is pertinent to dwell meansthat,withoutchromaticity,theinstabilityisfeasi-
6
ble only if betatron tune is located between half-integer As it is shown in Ref. [5] and [10], the eigentunes of
and next integer (e.g. Q =0.75 but not 0.25). This re- Eq.(23) are n(n+1)Q2/∆Q in orderof value. There-
0 ∼ s
sultwasfirstobtainedinRef.[9]forashortbunchwhere fore,withrathersmallsynchrotronfrequencyandfornot
the chromaticity was ignored by the model. Actually, it very high modes, it can be simplified using the approxi-
is apparentfrom Eq. (16) that the chromaticity is an es- mation ν ∆Q which results in:
| |≪
sentialfactor,mainlybecauseittriggersthehead-tailin-
teraction. For example, without chromaticity the bunch U2(τ)Y¯′′(τ)=R(τ) (25)
is quite stable at k Q= 0.25, B =0.5. However,the
− − with
rigid mode becomes unstable if θ ζ = 1 obtaining the
0
growth rate Im∆ν (0.35 0.26)q =0.09q (first U2ρ′ ν∆Q
0,0 ≃ − 0 0 R(τ)= τ + Y¯′ Y¯
term in this formula is the head-tail contribution, and ρ − Q2
(cid:18) (cid:19) s
second one – turn by turn interaction). Of course, chro-
q ∆Qexp(iλθ τ) 2π 1 y(τ′)dτ′
maticity of opposite sigh could prevent such a situation + 0 0 2
but higher modes instability would be enforced by this, Q2s (cid:20)− rθ0 Zτ √τ′−τ
as it follows from Fig. 6. 1 1
+√M V y(τ′)dτ′ V B y(τ′)(τ′ τ)dτ′ (26)
intTerhaectliaosntwtehrimchidnesEcqri.b(e1s6t)heisfiaeldpavratrioaftitohneocfopllreeccteivde- (cid:18) 1Z−1 − 2 Z−1 − (cid:19)(cid:21)
ing bunches inside the considered one. Actually, only Boundary conditions of this equation are evident from
the nearest bunch gives a noticeable contribution to this the relation U2( 1) = 0 which follows from definition
±
part so that the result does not depend on number of (24)andwillbereinforcedbyexamplesinthesubsequent
bunches, in practice. Influence of this addition looks sections. Therefore,any appropriatesolution ofEq. (25)
much like to the head-tail interaction which statement should satisfy the relations:
canbecheckedbycomparisonofFigs.5and7. However,
R( 1)=0. (27)
theeffectstronglydependsoncollectivemodeasitisde-
±
scribed by the coefficient V (κ). For the example above,
2 BecauseEq.(25)islinearanduniform, initialconditions
its contribution to the rigid mode instability is about Y¯(1) = 1, R(1) = 0 can be used in practice to cal-
q h (θ ζ)/16 which is less then the “usual” head-tail
− 0 0 0 culate the function R(τ) everywhere with any trial ν,
effect in order of magnitude. Probably, this part of the
and to separate thereafter the eigentunes ν assuring
n
resistive wake is negligible at any conditions.
the condition R( 1) = 0 [5]-[6]. The method is espe-
−
cially effective at q = 0 to determine the basic modes.
0
In particular, it confirms that Legendre polynomials are
V. LOW SYNCHROTRON FREQUENCY solutions of the boxcar bunch. Generally, it is easy to
show that the basic solutions of any bunch are regular
The boxcar model gives a broad outlook of the prob- functions satisfying the orthogonality conditions:
lem but maybe it omits some important details being
1
notsufficientlyrealisticitself. Thereforeanotherpointof Y¯ (τ)Y¯ (τ)ρ(τ)dτ =δ (28)
n1 n2 n1,n2
view is developed in this section based on the approach Z−1
Q ∆Q which is rather characteristic of many proton
msac≪hines. As it is shown in previous section and illus- Therefore, with enough small q0, additions to the eigen-
tunes can be found by standard perturbation methods
trated by Fig. 3, space charge suppresses all the modes
which way results in expression like Eq. (16):
Y (τ,u)inthislimit. Thereforethefollowingresults
n,m<n
areactuallyconcernedonlywiththemodesY (τ)which
n,n 2π
willbedenotedlatersimplyasYn(τ,u). FollowingRef.[6] ∆νn =q0 2 θ fn+√M gnV1+ihnV2B (29)
with resistive wakefield added, one can show that the (cid:20) r 0 (cid:16) (cid:17)(cid:21)
space part of this function satisfies the equation: Eqs.(18)–(20)canbeusedaswellwithappropriateeigen-
functions y (τ) = Y¯ (τ)ρ(τ)exp( iθ ζτ) to calculate
U2(τ)Y¯′′(τ) τ + U2∆Qρ′(τ) Y¯′(τ)+ ν(∆Q+ν)Y¯ the factors fnn, gn, hnn All the co−effic0ients Λn,n = 1 in
− (∆Q+ν)ρ Q2 this case due to normalization condition (28).
s
h i
q ∆Qexp(iλθ τ) 2π 1 y(τ′)dτ′
0 0
= 2
Q2s (cid:20)− rθ0 Zτ √τ′−τ A. Gaussian bunch with low wake
1 1
+√M V y(τ′)dτ′ V B y(τ′)(τ′ τ)dτ′ (23)
1 2
(cid:18) Z−1 − Z−1 − (cid:19)(cid:21) TruncatedGaussianbunchisconsideredinthissubsec-
tion for a comprehensive investigation of the instability.
where Its distribution function is:
1 C 1 A2
U2(τ)= F(τ,u)u2du (24) F = exp − 1 at A 1 (30)
ρ(τ) 2√2σ 2σ2 − ≤
Z (cid:18) (cid:19)
7
wherethenormalizingcoefficientC dependsonσ. Other
involved functions are: 0.0
√π 2CT3 2T2 −0.2
ρ(τ)=C exp(T2)erf(T) T 1+
2 − ≃ 3 5
(cid:18) (cid:19) (cid:18) (cid:19) −0.4
(31)
−0.6
and e fn n=0
R −0.8 n=1
2CT3 2σ2T2 n=2
U2 =σ2 1 (32)
− 3ρ ≃ 5 −1.0 n=3
(cid:18) (cid:19)
n=4
−1.2 n=5
whereT2 =(1 τ2)/(2σ2),andapproximateexpressions
−
at τ 1 are added for references.
| |≃ −1.4
The case σ = 1/3 (3σ truncation, C = 0.016) is ac- 0 2 4 6 8 10 12 14 16 18 20
tually consideredbelow. Six basic normalizedeigenfunc- θζ
0
tions of the bunch are shown in Fig. 8. Their eigen-
tunes haveaform ν =α Q2/∆Q with the coefficients
n n s c
αn whicharealsopresentedinFig.8inthe brackets(for FIG. 9: Functions Refn(θ0ζ) of truncated Gaussian bunch.
comparison: α =n(n+1)/2fortheboxcarmodel). Here The right-handparts of theseeven functions are shown.
n
and further, the subindex c marks the bunch center.
The coefficients f , g and g are plotted in Figs. 9–
n n n
12 which look much like Figs. 4–7 of the boxcar model.
Of course, it is necessary to take into account that the
0.5
Gaussian bunch has less rms length in comparison with
the boxcar one (1/3 instead 1/√3), so the Gaussian 0.4
plots should be proportionally wider. The absence of
secondary oscillations is well explicable because of more 0.3
smooth bunch shapes. With these reservations, one can
n
assert that the boxcar model provides an adequate de- m f 0.2
I n=0
scriptionofthebunchcoherentinstability,atleastwithin
0.1 n=1
the limit of low synchrotronfrequency.
n=2
0.0
n=3
−0.1 n=4
n=5
3.0 −0.2
0 2 4 6 8 10 12 14 16 18 20
θζ
2.0 0
1.0 FIG. 10: Functions Imfn(θ0ζ) of truncated Gaussian bunch.
The right-handparts of theseodd functions are shown.
n 0.0
Y
B. Expanded low wake approach
−1.0
n=0 (0)
n=1 (1.31) Formally, relation (29) is applicable if the spectrum
n=2 (4.20)
−2.0 n=3 (8.61) shifts ∆νn are small in comparison with distances be-
n=4 (14.5) tween the basic spectrum lines which are
n=5 (22.0)
−3.0
−1.0 −0.5 0.0 0.5 1.0 ν ν =(α α ) Q2s (n+1) Q2s
τ n+1− n n+1− n ∆Q ≃ ∆Q
c c
Therefore,conditionofapplicabilityofEq.(29)isforthe
lowest mode:
FIG. 8: Normalized eigenfunctions of truncated Gaussian
bunch (0th–5th modes). Corresponding eigennumbers are Q2
given in theparentheses. ∆ν0 s (33)
| |≪ ∆Q
c
8
1.0 0.3
n=0
n=0
n=1
0.8 n=1 0.2 n=2
n=2 n=3
n=3 n=4
n=5
0.6 n=4 0.1
n=5
n n
g h
0.4 0.0
0.2 −0.1
0.0 −0.2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
θζ θζ
0 0
FIG.11: Functionsgn(θ0ζ)oftruncatedGaussianbunch. The FIG. 12: Functions hn(θ0ζ) of truncated Gaussian bunch.
right-hand parts of these even functions are shown. The right-handparts of theseodd functions are shown.
Theleft-handpartofthisexpressionisclosetotheinsta- Υ (τ)arenotorthogonal,incontrastwithY¯ . Therefore
n n
bilitygrowthratewhichisessentiallylessof1probablyin we turn back to the approximation Υ = Y¯ which is cer-
allpracticalcases(e.g. ∆ν0 <0.1). However,theright- tainlyacceptableatlowintra-bunchinteractionanddoes
| |
handpartcanbestillless,forexample0.052/0.25=0.01. not violate the overallstructure of Eq.(36). It results in
The example demonstrates that a violation of condition the dispersion equation
(33)is quite possible occasion,especially whencollective
∞
modes instabilities of a multi-bunchbeamare examined. q V √Mg (θ ζ)
0 1 n 0
1= (37)
Therefore,we considerthis importantcaseingreaterde-
ν ν q 8π/θ f +ih V B√M
tail, without the assumption that the multi-bunch con- nX=0 − n− 0 0 n n 2
tribution is small. (cid:16)p (cid:17)
As a first step, we need to solve Eq. (25) at V = withthesamecoefficientsfn, gn, hnasbefore(Figs.9–12
1
0 to know the basic modes of this case. Let us denote for Gaussian bunch). With an additional condition
corresponding eigenfunctions and eigentunes as Υ and
n Q2
υn. Thensolutionofthe totalequationcanbe presented q0 V1 √M s (38)
in the form: | | ≪ ∆Qc
1 the low wakefield approximation is totally satisfied, and
Y¯(τ)=q V (κ)√M Y¯(τ′)ρ(τ′)exp( iζθ τ′)dτ′
0 1 ∞Z−1 − 0 theAenqoutahteironeagsiyvecsasteheissazemreorcehsruolmtaastitchiteyewarhleienrsEuqm.((2397)).
e Υ (τ)
n n holds the only term n = 0 because g (0) = δ . It
(34) n n,0
× ν υn means that solely the rigid internal mode can be excited
n=0 −
X without chromaticity. Because it can appear inside any
where en are coefficients of the expansion: collective mode, the resulting tune shift is:
∞
∆ω 2π k Q
exp(iλθ0τ)= enΥn(τ) (35) =q 2 f (0)+√MV − 0 (39)
0 0 1
Ω θ M
nX=0 0 (cid:20) r 0 (cid:18) (cid:19)(cid:21)
It immediately results in the dispersion equation: This expression formally coincides with Eq. (29) at ζ =
∞ e 1 0, but can be applied at any value of the coefficient
1=q0V1√M n Υn(τ)ρ(τ)exp( iζθ0τ)dτ V1(κ).
n=0ν−υn Z−1 − Generally, series (37) contains restricted number of
X
(36) summands which conclusion follows from Figs. (6) and
In principle, new eigenfunctions Υ (τ) and eigentunes (11). Inparticular,onecanseethatthetermsn=0and
n
υ could be found by the same method which was used 1 give major contributions at θ ζ < 3. Eq. (37) has
n 0
for Y¯ (τ). However, it would be a more difficult prob- twoactualsolutionsinthis case|,at|lea∼stoneofthembe-
n
lem to calculate the coefficients e because the functions ingunstable. IntheextremecasewheninverseofEq.(38)
n
9
inequality is fulfilled, one of the eigentunes is real, and Anadditionalimportantpropertyofbunchedbeamsis
another tune is: that, with a coherent frequency ω, the particles undergo
an influence of harmonics of frequencies ω+jΩ where
∆ω s
Ω ≃q0V1(κ)√M g0(θ0ζ)+g1(θ0ζ) Ωs is synchrotron frequency, and j is integer. An in-
0 tenseenergytransferispossibleifanyofthesefrequencies
αr NRδ(ω)(cid:2) ω (cid:3)
0 1+i g (θ ζ)+g (θ ζ) (40) falls in the incoherent region. Fast look in Fig. 2 reveals
≃ 2πγQ0b3y (cid:18) |ω|(cid:19) 0 0 1 0 that harmonics j =m have the most chances to do this.
(cid:2) (cid:3) Therefore,more informativegraphcanbe obtainedfrom
Ratherweakdependenceofthisexpressiononchromatic-
Fig. 2 by a transformation of each curve ν (µ) to the
ity engages an attention. In relative units, the addition n,m
ν mµ. The results are presented in Fig. 13 by the
is 1 at θ0ζ = 0 and about 0.8 at θ0ζ = 3. However, sonl,imd−lines of the same color as in Fig. 2.
it should be mentioned again that the low synchrotron
Averaged over synchrotronphases incoherent tunes of
frequency limit is considered here. Role of this factor is
the truncated Gaussian bunch lie in the region 1 <
discussed in the next section. −
ν/∆Q < 0.274. Drawingcorrespondingboundaryline
c
−
inFig.13andassumingthatthecoherenttunesofGaus-
sian bunch have about the same behavior as in the box-
VI. THE INSTABILITY THRESHOLD
car model, one can make the conclusions: (i) the lowest
(rigid)mode (0,0)is unstable withany µ; (ii) the higher
It could be concluded from previous analysis that the
modes (n, n) canbe unstable atµ< 0.5;(iii) the more
boxcar is a quite adequate model for investigationof the ∼
is n the less is corresponding threshold of µ; (iv) all
bunchedbeaminstability,andonlyminorandalmostob-
higher radial modes like (n, m < n) are stable in any
viouschangesareneededformorerealisticdistributions.
case.
However, it would be a premature conclusion because
These conclusions are in a reasonable agreement with
the space charge tune spread is ignored at all in the box-
results of Ref. [5], according which only the rigid mode
car model. Meanwhile, at certain situations the spread
(0,0) of Gaussian bunch is unstable at µ 1. One
can cause Landau damping and suppress many unstable ≫
can anticipate from Fig. (13) that thresholds of all other
modes of a real bunch, as it has been shown in Ref. [4]–
modes are located at µ < 0.5. It is a region where the
[5]. Therigidintra-bunchmodeistheonlyoccasionwhen
lowµ approximationcould be fitted to refine the thresh-
this mechanismdoes notworkandcannotpreventinsta-
olds of Gaussianbunch. To accomplishthis, we consider
bility at any combination of parameters.
Eq.(23)with q =0 but without the additionalsimplifi-
Unfortunately, at present the problem is adequately 0
cation ν ∆Q . Correspondingly, boundary conditions
covered only in the limiting cases µ 1 and µ 1. In ≪ c
≫ ≪ (27) should be changed by the following one:
first case, this kind of Landau damping really works and
suppresses almost all internal modes [5]. However, the
ν2
′
mechanism is turned off in opposite limiting case which Y ( 1)=
± Q2
was just the subject of previous section. The only con- s
clusion can be done from these facts: this stabilization
mechanism has a threshold character and actually begin
toworkatµ> 1. Thegoalofthissectionistogetmore
∼ 0.0
exact estimation of the thresholds. Because the space
n=0
chargewill be treated here as a promotingand dominat-
n=1
ing factor, the wakefield is excluded from the analysis.
−0.2 n=2
ItshouldberemindedpreliminarythatLandaudamp- Incoherent boundary
n=3
ing arises when coherent frequency penetrates rather of Gaussian bunch
deeply in a region of incoherent tunes of the system. (0) −0.4 n=4
Q
Then the particles which own tunes are located below ∆
or above the coherent tune could be exited in contra- µ)/
m
phases by the coherent field, transforming its energy to − −0.6
m
the incoherent form (beam heating). Such a mechanism νn,
affects the beam decoherence and creates the instability (
−0.8
threshold.
Well known practical recommendation follows from
this statement for coasting beams: incoherent tune
−1.0
spreadshouldexceedthespacechargetuneshifttoavoid 0.0 0.5 1.0 1.5 2.0 2.5 3.0
the instability. In principle, a wake field (e.g. the resis- µ = Q/∆Q(0)
s
tive wall contribution) affects this criterion, however,its
influence is small in practice if the space charge domi-
nates in the impedance budget. The last is just the case FIG.13: Transformedeigentunes: solidlines–boxcarmodel;
of our study. dashed lines - Gaussian bunchwith 3σ truncation.
10
VII. SUMMARY AND CONCLUSION
TABLE I: Instability thresholds of Gaussian bunch with
3σ truncation (lower radial modes (n,n)).
Transverse instability of a bunched beam is studied
n 0 1 2 3 4
with synchrotron oscillations, space charge, and wake-
Qs/∆Qc < ∞ 0.63 0.21 0.13 0.095 field taken into account. Resistive wall wakefield is con-
sidered as the most universal and practically important
case. However, many results have a common sense and,
Obtained eigentunes are plotted in Fig. (13) by dashed withsmallchanges,canbeadaptedtootherwakes. Box-
lines above the Gaussian incoherent boundary. The car model of the bunch is extensively used in the paper
crossing points which are the expected thresholds of the to get a general outline of the problem in wide range of
head-tail modes are presented in Table 1. parameters. A realistic Gaussian distribution is invoked
Distributions with more abrupt bunch boundaries in some cases for comparison and more detailed investi-
demonstratesimilarbehaviorbuthavehigherthresholds. gations of important problems like Landau damping.
For example, three modes of parabolic bunch retain the Eigenfunctionsandeigentunesofthebeamareinvesti-
stability at µ [5]. The boxcar bunch is an extreme gatedwithbothintra-bunchandinter-bunchinteractions
→∞
case which is unaffected by this sort of damping at all. taken into account. Contributions of the interactions to
An important conclusion follows from this or similar the instability growthrate are studied overa wide range
table,concerninganinfluenceofchromaticityinrealcon- of the parameters, including effects of the bunch length
ditions. As might be expected from Figs. 6 and 11, an and chromaticity. It is shown that known head-tail and
increase of chromaticity should not affect drastically the collective modes instabilities are the extreme cases when
instability growth rate because its main effect is a sim- one type of the interaction certainly dominates. How-
ple switchofthe bunch oscillationsfroma lowerinternal ever, an essential influence of the intra-bunch degrees of
modetoahigherone. Forexample,itfollowsfromFig.11 freedomonthecollectiveinstabilitiesisespeciallymarked
thatthemostunstableinternalmodesofGaussianbunch andinvestigatedindetail. Inparticular,itis shownthat
are: n =1 at θ0ζ =4, and n=2 at θ0ζ =6, in both a variability of the internal bunch modes explains why
| | | |
casestheinstabilitygrowthratebeingabout0.45inused theinstabilitygrowthratedepends onthebunchparam-
relative units (the head-tail contribution is neglected in etersincludingspacechargetuneshift,synchrotrontune,
the estimationsbecauseitisreasonablysmallinamulti- bunch length, and chromaticity.
bunch beam). However, now we must take into account It is emphasized that the space charge tune spread
thattheseresultswereobtainedatµ=0. Letusconsider can cause Landau damping and suppress the instability
thecaseµ=0.4asananotherexample. Thenthemodes (other than the space charge sources of the tune spread
n 2cannotappearbeing suppressedbyLandaudamp- are not considered in the paper). The phenomenon ap-
≥
ing. Generally,only 0th and1stinternalmodes couldbe pears at rather large ratio of synchrotron frequency to
unstable in this case, and chromaticity θ0ζ 10 is suf- the spacechargetune shift, lowerinternalmodes obtain-
| |≃
ficient to suppress both of them that is to reach a total ing the stability at larger the ratio. Several modes of
beam stability. Gaussian bunch are considered in the paper, and their
Of course, Table 1 is only an estimation of the thresh- thresholds are estimated by comparison of the limiting
olds because approximateEq. (23) lies in its foundation. cases.
It would be a good idea to solve more general Eq. (7)
with arbitrary Q and ∆Q , and to use the results for
s c
exact instability thresholds of realistic bunches. Note
VIII. ACKNOWLEDGMENTS
that all eigennumbers of this equation are real. There-
fore, the lack of regular solutions with real ν at some
combination of synchrotron frequency and space charge FNAL is operated by Fermi Research Alliance, LLC
tune shiftwouldbe asignofLandaudamping. However, undercontractNo.DE-AC02-07CH11395withtheUnited
the boxcar model is the only solved case at present. States Department of Energy.
[1] C. Pellegrini, NuovoCimento A 64, 447 (1969). [7] L. Lasslett, V. Neil, and A. Sessler, Rev. Sci. Instr. 36
[2] M. Sands, SLACTN-69-8 (1969). 436 (1965).
[3] F. Sacherer, CERN-SI-BR-72-5(1972). [8] V.BalbekovandA.Kolomensky,AtomnayaEnergiya19,
[4] V.Balbekov, Zh.Tekh. Fiz. 46, 1470 (1976) [Sov.Phys. 126 (1965). Plasma Physics 8, 323 (1966)
Tech.Phys., Vol.21, No.7, 837 (1976)]. [9] N. Dikansky and A. Skrinsky, Atomnaya Energiya 21,
[5] V. Balbekov, Phys. Rev. ST Accel. Beams 12, 124402 176 (1966).
(2009). [10] A. Burov, Phys. Rev. ST Accel. Beams 12, 044202 and
[6] V. Balbekov, Phys. Rev. ST Accel. Beams 14, 094401 109901 (2009).
(2011).