Effect of the Space Charge Fields on Landau Damping of Dipole Transverse
Coherent Oscillations
D.V. Pestrikov, Budker INP, 630090 Novosibirsk, Russian Federation2Abstract
Without external non-linear fields the space charge fields ofan intense beam do not affect its dipole transverse coherent oscillations. However, an interference of e.g. a cubic
nonlinearity of the lattice focusing and of nonlinear spacecharge fields of such a beam changes the betatron tune
distributions in the beam and hence, changes the Landau
damping of its dipole coherent oscillations. For conventional coherent instabilities, which occur outside the latticeresonance stopbands, this interference changes the shape ofthe stability diagram of dipole coherent oscillations. Thiseffect can be used to recalculate the Laslett tine shift ofintense beam measuring the stability diagram of its dipole
oscillations, or the response functions of these oscillations.

1

INTRODUCTION

Important limitations on the operation performance of ion
beams occur due to their space charge fields. Typically, thestrengths of these limitations are described using the socalled Laslett tune shift of the beam, which is defined as
the value of the tune shift of betatron oscillations of small
amplitudes due to space charge fields. For a coasting beamof a round cross section consisting of non-relativistic ionsthe Laslett tune shift is estimated using∆νL =N e2.
4πM v 2 (1)Here, N e is the charge of the beam, M is the ion mass,v is its velocity,  is the transverse beam emittance. Dueto repulsion of particles, the space charge fields decrease
the tunes of the betatron oscillations. For small amplitudes of betatron oscillations relevant tune shift is equalto (−∆νL ). Outside the beam the space charge fields decay. Therefore, for the particles oscillating with the amplitudes substantially exceeding transverse beam sizes thesetune shifts tend to zero. Correspondingly, the space charge
fields increase the tune spreads in the beams and frequency
ranges where coherent oscillations are resonant with incoherent oscillations of particles which is required for Landau
damping. On its turn, this changes the shapes of the stability diagrams for these oscillations as well as the frequency
distributions in the transverse Schottky noise spectra in thespace charge dominated beams. These effects can be used
for direct measurements of the beam Laslett tune shift measuring the spectra of dipole coherent oscillations.
We report main results of studies on this subject. Moredetail can be found in [1]. To simplify calculations we assume nonrelativistic ion energies and a coasting ion beam.

QUASI-RIGID BEAM MODEL

In this Section we obtain dispersion equations for thedipole coherent oscillations assuming that the (vertical)
space charge force of the beam is described using a function F (y − d(t)), where d(t) is the vertical dipole momentum of the beam. We call it as the quasi-rigid beammodel (QRBM). Assuming also one-dimensional coherent oscillations, monochromatic beamand smoothed focusing approximation, we write y = 2I/(M ωy ) cos ψ,py = − 2IM ωy sin ψ, θ = ω0 t + φ, φ̇ = 0. Here, thesuffix 0 marks the values of the reference particle in thebeam, θ = s/R0 is the particle azimuth, Π = 2πR 0 is theperimeter of the closed orbit, M is the ion mass. Coherent oscillations of the beam are described using the Vlasovequation:py ∂f∂f∂f+ ω0+= M ωy2 y − b(θ)y 3 −∂t∂θM ∂yF (y − d, θ) − F2 (t, θ)](2)∂f.
∂pyHere, f = f (y, py , θ, t) is the beam distribution function, F2 (t, θ) is the force due to the beam wake-fields, theterm b(θ)y 3 describes the cubic nonlinearity of the guiding magnetic field. Linearizing Eq.(2) near a stationarystate f = f0 (I)δ(∆p) + δf , |δf |  f0 , F (y − d) F (y) − d(∂F/∂y),δf = e−iωteinθ
fmn (I)eimψ ,nm=±1dmn (ω) = dΓyfmn , (dΓ = dId∆p), and assuming thatthe working point of the ring is far from the lattice resonances, we obtain (Im(ω) > 0)
m(df0 /dI) 2IM ωy (F2 )mn− Ω(I)dmn ,fmn =∆ωmn − m(aI − Ω[I]) 2M ωy(3)where ∆ωmn = ω − nω0 − mωy , a = −3b/(4M 2ωy2 ) and1Ω(I) = 2IM ωy 2π0dψF (y) cos ψ2π(4)is the value of the frequency shift due to the space charge
force of the beam. If the beam wakes are described interms of the transverse coupling impedance, we substitutein Eq.(3) m(F2 )mn = −2M ωy Ωmn dmn and arrive at thedispersion equation of the problem:∆ωmn = Ωmn + ma

Q2 (∆ωmn , I).
Q1 (∆ωmn , I)(5)

According to Eq.(5) without external (cubic) nonlinearity
the frequency spread due to space charge fields produces
no Landau damping and no additional frequency shift ofthe dipole coherent oscillations (∆ω m = Ωmn , if a = 0).
Same feature of coherent spectra holds for stationary distributions providing no Landau damping.

1,21,0Im(Ωmn)/∆ω3Here, Ωmn is the coherent frequency shift calculated neglecting the frequency spreads in the beam, and (q = 1, 2)
 ∞I q df0 /dI. (6)dIQq (∆ωmn , I) =[∆ωmn − maI + mΩ(I)]00,80,60,40,20,0-123 STABILITY DIAGRAMS

Inspecting stability diagrams for different values of a, wefind out (Fig.1) that in the regions where a > 0, the spacecharge fields mainly increase the width of the stability diagrams substantially reducing limitations on the accessible values of coherent frequency shifts in this region. The
height of the stability diagram, i.e. the accessible increments of coherent modes are mainly determined by the frequency spread due to the cubic nonlinearity of the lattice
focusing (Fig.1). The left border and the center of the stability diagram shift in the left direction with an increase in∆νL . In this region, equations ∆ω(I) = aI − Ω(I) =const have unique roots in I. On the contrary, in theregion, where a < 0, an increase in the value of theLaslett tune shift is accompanied (Fig.2) by a decrease in
the height of the stability diagram, in the strong deformation of its shape and for large ∆ν L (from numerical calculations we have found that for ∆ν L > 6.4∆ν3 , where ∆ν3is the tune spread due to the cubic nonlinearity) in appearance of the region with Landau anti-damping. In this region, equations ∆ω(I) = const may have double roots inI, so that the oscillations can degrade ∆ω(I 1 ) = ∆ω(I2 ).
Solutions with anti-damping occur in the region, where thewidth of the degradation of incoherent oscillations is large
as compared to the beam r.m.s. radius (see in Fig.3).

-40

48Re(Ωmn)/∆ω312Figure 1: Stability diagram for dipole coherent oscillations.
Oscillations are stable below the curve; a > 0, from left toright ∆νL /∆ν3 is subsequently equal to 8, 4, 2 and 0.

1,2Im(Ωmn)/∆ω3

1,00,80,60,40,20,0-0,2-0,4-10-8-6-4-20Re(Ωmn)/∆ω32Figure 2: Stability diagram for dipole coherent oscillations.
Oscillations are stable below the curve; a < 0, from rightto left ∆νL /∆ν3 is subsequently equal to 0, 1, 2, 4, 6 and8.

-0,85-0,90∆ν(J)/∆νL

For a given distribution function f 0 , Eq.(5) can be used toplot the stability diagram of coherent oscillations. This is
the curve in the plane of the complex variable Ω mn separating the regions containing the roots corresponding to stableand unstable oscillations. It is obtained, if we vary in Eq.(5)the variable ∆ω mn slightly above the real axes. The frequency distribution and the frequency spread in the beam
depend on the sign of a. Correspondingly, the values of theLandau damping decrements and widths of the oscillation
stability diagrams also depend on the sign a. For numerical calculations below we assumed a Gaussian transverse
distribution of ions in the beam with equal horizontal andvertical emittances ( x = y = ) as well as equal average
β-functions of the lattice and hence, the round cross section of the beam. In this case, (I 0 (x) is the Bessel functionof the zero order of imaginary argument)
J4∆νL Ω(Jy , 0)
J=1 − exp −I0. (7)
ω0J4
4

-8-0,95-1,00-1,05-1,100,0

0,5

1,0

1,5(J/ε)

2,02,53,0

3,5

1/2

Figure 3: Dependence of the tune shift on the beam radiusJ/, a < 0 and ∆νL /∆ν3 = 6.4.

4SOFT BEAMS

5The calculations using the QRBM are not exactly selfconsistent. Recently, this fact was also studied in [3]for the beam-beam instability. For simplicity we takethe stationary distribution of a coasting beam in the form
f0 (I)δ(Ix )δ(∆p). Then, direct calculations of the spacecharge fields
result in the following integral equation√(fmn = IXdf0 /dI and Q = ∆ωmn − maI + mΩ(I))
 ∞

mN e2
df0X+
dJ
ΠI0
 I2  ∞mN e
df0
df0X+X,
dJJ
dJ
×dJ
Π
dJ0I

−QX = ΩmndJJ(8)I
where Ω(I) = [N e2 /(ΠI)] 0 dJf0 (J). The rhs in Eq.(8)differs from that in Eq.(3) by a defect function
 ∞

df0
mN e2
X(J) −dJΠI0 I2  ∞mN edf0
df0×X−X,dJJdJdJ
ΠdJ0I

D = mΩ(I)dJJ(9)which vanishes only in the case where X = const. That is
an exact solution to Eq.(8), if a = 0. Beyond the QRBM
Eq.(8) should be solved numerically. Comparing Figs.1
and 4, we find that for higher ∆ν L the stability diagrams
have smaller widths compared to that predicted by QRBM.
The shape of these diagrams becomes more symmetricaldue to their extension in the region Re(Ω mn )< 0, while its
maximum is shifted towards the origin.

1,2Im(Ωmn)/∆ω3

1,00,80,60,40,20,0-8-6-4-202468Re(Ωmn)/∆ω3Figure 4: Stability diagrams for dipole coherent oscillations using Eq.(8). Open circles: χ = ∆ν L /∆ν3 = 1, x:
χ = 2, full circles: χ = 8, full triangle: χ = 0 and solving
Eq.(8) solid line: χ = 0, direct formula (see, e.g in [2]),f0 = e−I/I0 /I0 .
If a = 0, or without Landau damping (e.g. if f 0 is a stepfunction) the eigenfrequencies of Eq.(8) do not depend on
the space charge fields similar to prediction using QRBM.

CONCLUSION

We studied stability conditions of dipole coherent oscillations in the coasting beam with strong space charge. Dispersion equations for such problems differ from that obtained neglecting the space charge forces. In particular, it
predicts no effect of the space charge fields on dipole coherent oscillations in the lattices with linear optics, or for thebeam distributions suppressing Landau damping. Althoughnonlinear dependencies of the space charge fields on theparticle offsets inside the beam do not result directly in theLandau damping of dipole coherent oscillations, its interference with the lattice focusing nonlinearity (for example,the cubic) can substantially modify the Landau damping
decrements and stability diagrams. These variations areespecially strong, if the lattice nonlinearity decreases thebetatron tunes. In the last case, the space charge fields ofthe beam can result in the Landau anti-damping of dipole
oscillations, if the Laslett tune shift is large enough.
For bunched beam, from general theory of coherent oscillations (see e.g. in [2]) and within the framework of
QRBM we can write the dispersion equation for dipole coherent oscillations of a bunched ion beam. If Ω m is thecomplex coherent frequency shift of the monochromaticbunch and provided that all particles in the bunch haveequal amplitudes of synchrotron oscillations, such dispersion equations for both betatron and synchro-betatronmodes is similar to Eq.(5) (∆ω m = ω − mωy − ms ωs ):∆ωm = Ωm + ma

Q2 (∆ωm , I)
,Q1 (∆ωm , I)

(10)where ωs is the frequency of synchrotron oscillations, mis the mode number of betatron oscillations and m s is themode number of synchrotron oscillations. For soft beams
the widths of stability diagrams are narrower than predicted
by QRBM.
The dependence of the width of the stability diagram on
the value of the Laslett tune shift of the beam can be used
to measure this important beam parameter using the spectraof dipole coherent oscillations.
We expect that increases in the widths of stability diagrams due to space charge fields will substantially increase
the thresholds of the mode-coupling instabilities in ion storage rings.
I thank V.V. Parkhomchuk for useful discussions. Thiswork was supported by the GSI-Budker-INP Research andDevelopment Contract #3.

6REFERENCES[1] Angert N., Blasche K., Boine-Frankenheim O., Dikansky N.,
et al., Study for a High-Intensity Heavy Ion Synchrotron withStrong Bunch Compression (Part I), Budker INP 2000-93,
Novosibirsk, 2000.
[2] Dikansky N.S., Pestrikov D.V., Physics of Intense Beams andStorage Rings, AIP Press, New York, 1994.
[3] K. Yokoya, Phys. Rev. ST Accel. Beams, 3, 124401, 2000.