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Beam Breakup Mitigation by Ion Mobility in Plasma
Acceleration
A. Burov, S. Nagaitsev and V. LebedevFermilab, PO Box 500,
Batavia, IL 60510-5011
(Dated: August 11, 2018)
Moderate ion mobility provides a source of BNS damping in the
plasma wakefield acceleration,which may serve as an effective
remedy against the transverse instability of the trailing bunch.The
ion-related BNS parameter κ, proportional to the beam brightness,
is introduced as a singleparameter of the partial
integro-differential equation of the bunch collective motion. This
equationis further reduced to an ordinary differential equation,
which solutions are shown versus its singletime-space argument and
the BNS parameter. It is demonstrated that conditions of the
instabilitysuppression and emittance preservation at energy
efficient plasma acceleration leave for the ion BNSparameter κ
about an order of magnitude of its possible variation along the
acceleration line.
Plasma wakefield acceleration (PWA) suggests ex-tremely high
acceleration fields, so it is no surprise thatthis area of research
attracts interest of groups workingon future colliders, giving rise
to many publications, tar-geted at resolution of multiple
interrelated problems inthis challenging area. A special subset of
these problemsis associated with stability of both driving
(accelerating)and trailing (accelerated) bunches. The latter
problemappears to be harder than the former, since mismatchesat
every change of the driving bunch between positions ofthe two
bunches produce initial kicks for the instabilitydevelopment along
the acceleration line for one and thesame trailing bunch. From a
very general point of view,the PWA trailing bunch instability
belongs to the fam-ily of similar effects in linacs. Due to
interactions withthe surroundings, dipole perturbations at the head
of thebunch leave electromagnetic wake fields behind, thus act-ing
on the bunch tail. The kick felt by the test particleat the unit
trajectory length by a unit dipole momentof the leading particle is
known as the wake function,W⊥(ξ), where ξ is the separation between
the particles,see e.g. Ref. [1]. As a result, the tail dipole
oscillationsmay grow more and more, leading to the emittance
degra-dation. This sort of unbounded convective instability [2]is
known as the beam breakup in linacs [1]. Here we areconsidering the
acceleration of a short electron bunch inthe blowout regime, a
regime in which the fields of thedriver (laser or an electron
bunch) are so intense thatthey expel all plasma electrons, creating
a cavity filledwith pure ion plasma [3]. The longitudinal and
trans-verse electric fields inside this cavity are used to
accel-erate and focus the trailing electron bunch. The
excitedtransverse wake fields are very sensitive to the
apertureradius, which is the plasma bubble radius at the
bunchlocation, rb, for the PWA case: for the short bunches ofthe
interest, the wake function is inversely proportionalto the fourth
power of this radius, W⊥(ξ) ' 8ξΘ(ξ)/r4b ,where Θ(ξ) is the
Heaviside theta-function. Many de-tails on that can be found in the
recent Refs. [4, 5]. Toget the desired high acceleration, the
plasma bubble hasto be small, typically rb ' 50 − 100µm, compared
with1− 2cm for conventional colliders; thus, with the fourthpower
of the aperture in the transverse wake, the trans-verse instability
is by necessity one of the main obsta-
cles for the PWA colliders. From this, one may correctlyconclude
that there must be a relation between energyefficiency and beam
stability for PWA: while the formerrequires smaller bubbles, the
latter is lost with them.Such efficiency-instability relation has
been recently for-mulated and proved in Ref. [6]; here we reproduce
thisstatement for the reader’s convenience.
Let ηP < 1 be the PWA energy efficiency, i.e. the ra-tio of
the power of the trailing bunch acceleration to thepower of the
driving bunch deceleration. Further, let thewake parameter ηt be
the ratio of the bunch-averaged de-focusing by the transverse wake
fields to the main focus-ing of the bunch electrons by the ions
inside the bubble.The efficiency-instability relation of Ref. [6]
states that
ηt ≈η2P
4(1− ηP ). (1)
Thus, an increase in the efficiency indeed brings thebunch
closer to the instability threshold. To push backthis limitation
for a given wake parameter ηt, BNS damp-ing [7] can be used. The
main idea of this method isbased on a compensation of the wake
deflecting force inevery position of the bunch by additional
focusing. If thetwo terms cancel each other for the bunch
deflection asa whole, the instability would be effectively
suppressed.Let us assume that the relative focusing strength
δω⊥/ω⊥varies along the bunch by one or another reason;
then,neglecting, for simplicity sake, the variation of the
bunchline density, the equation of motion for the bunch nor-malized
local offsets X(ξ, µ) can be presented as
∂2X
∂µ2+
(1 + 2
δω⊥ω⊥
)X =
2ηtL2t
∫ ξ0
X(ξ′)(ξ − ξ′)dξ′ . (2)
Here Lt is the full bunch length, µ is the normalized time,dµ =
kpds/
√2γ with ds as a differential length along the
bunch motion, γ as the relativistic factor, kp =√
4πn0reas the relativistic Debye length; n0 is the plasma
den-sity and re = e
2/(mc2) is the electron classical radius. Itfollows from Eq. 2
that the bunch deflection as a wholewould evolve as stable
oscillations if the BNS compensa-tion condition is fulfilled:
2δω⊥ω⊥
= ηtξ2
L2t, (3)
FERMILAB-PUB-18-388-AD-APC
This manuscript has been authored by Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department
of Energy, Office of Science, Office of High Energy Physics.
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2
where the longitudinal position ξ is measured from thebunch
head. In conventional linear accelerators, this con-dition can be
fulfilled by momentum modulation δp/palong the bunch [7], using
that 2δω⊥/ω⊥ = −δp/p. ForPWA though, there is an additional
mechanism of thefocusing variation, associated with the ion
mobility inthe Coulomb field of the trailing bunch. Indeed,
plasmaions move in this field, causing variation of the ion
3Ddensity δni/ni. Assuming this variation to be small, it
isestimated as
δnini
= 2πntriξ2 , (4)
where ri is the ion classical radius and nt = Nt/(πLtb2)
is 3D density of the trailing bunch. Here we assume asimple
model of a homogeneous 3D density of the bunchparticles, with the
full transverse radius b and the fulllength Lt.
The ion density perturbations (4) entail similar varia-tions of
the electron focusing:
2δω⊥ω⊥
=δnini
= 2πntriξ2 . (5)
Note that both the sign and position dependence of thisfocusing
variation are the same as BNS compensation re-quires (3). Thus, the
latter can be presented as position-independent:
κ ≡ 2NtriLtb2ηt
=µ2iηt
= 1 , (6)
where µi =√
2NtriLt/b2 � 1 is the phase advance ofion’s oscillations in the
field of the trailing bunch.
The equation of motion (2) can be further simplifiedwith the
slow amplitudes x = X exp (iµ), measuring thepositions inside the
bunch as fractions of its full length,ζ = ξ/Lt and using slow time
τ = µηt. Then, it is trans-formed to the following equation on the
slow amplitudesx(ζ, τ ), with the constant initial condition,
∂x
∂τ= −iκζ2x+ 2i
∫ ζ0
x(ζ − ζ ′)ζ ′dζ ′ ;
x(ζ, 0) = 1 ; 0 ≤ ζ ≤ 1 .(7)
It is straightforward to show that solution of this equa-tion
x(ζ, τ ) with the specified initial condition has a scal-ing
invariance: it depends on its space and time argu-ments ζ and τ as
x(ζ, τ ) = x(1, ζ2τ) ≡ f(ζ2τ). In otherwords, the complex amplitude
of the oscillations at po-sition ζ and time τ is the same as at the
bunch tail andearlier time ζ2τ . This means that the partial
integro-differential equation (7) with the specified initial
condi-tion is equivalent to an ordinary one. Apparently,
thisordinary integro-differential equation has the simplestform
with the space-time argument u ≡ ζ
√2τ . With
x(ζ, τ ) = g(u), Eq. (7) reduces then to the following form
d g
d u= −iκug + 2i
u
∫ u0
g(u− u′)u′du′ ; g(0) = 1 , (8)
At κ = 0, i.e. without any damping, the solution atlarge
argument, u � 1, asymptotically tends to g(u) 'exp(3 i1/3(u/2)2/3),
omitting the pre-exponential factor.
FIG. 1. Natural logarithm of the bunch oscillation amplitudef(y)
≡ x(1, y), versus the scaled time y ≡ ζ2τ and the BNSparameter
κ.
Since the problem is reduced now to finding a func-tion of just
two parameters, gκ(u), from a linear ordinaryintegro-differential
equation (8), it can be easily solvednumerically for all
interesting cases; Fig. 1 presents theamplitude modulus x(1, τ) as
a 3D plot for 0 ≤ κ ≤ 1.2and 0 ≤ τ ≤ 100. Patterns of oscillations
x(ζ) along thebunch for a case of slight BNS overshooting, κ =
1.2,and more overshooting, κ = 2.0, for τ = 100 are shownin Figs. 2
and 3. A couple of things are worth noting in
0.2 0.4 0.6 0.8 1.0ζ
-0.6-0.4-0.2
log(x)log(x)arg(x)
FIG. 2. Natural logarithm and argument of the perturbationat
time τ = 100 and slightly overshooting BNS, κ = 1.2.
relation to Figs. 1 – 3. First, the instability is dramat-ically
weakened even with a moderate BNS parameterκ ' 0.2 − 0.5. Secondly,
for overshooting κ > 1, thebunch is stable; its initial
perturbation decoheres, cer-tainly contributing to the emittance
growth.
During acceleration, the ion-driven BNS parameter in-
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3
0.2 0.4 0.6 0.8 1.0ζ
-3-2-1123log(x)
log(x)arg(x)
FIG. 3. Same as Fig. 2, but for more BNS overshooting,κ =
2.0.
creases, κ ∝ √γ, so it cannot be the same at the begin-ning and
the end of the acceleration. Assuming the initialand final energies
differ by the factor of 100, it meansthat the ion-BNS parameter κ =
µ2i /ηt increases by afactor of 10. With acceleration efficiency ηP
= 50%, thewake parameter is given by Eq. (1), resulting in ηt = 0.1
.Thus, full BNS damping would occur if the ion’s phaseadvance is
not too small, µ2i ≥ 0.1. On the other hand,if this phase advance
is not at all small, µ2i ≥ 1, the ionscollapse inside the bunch,
and that leads to a dramatic
emittance growth. Thus, for the specified parameters,there is an
order of magnitude of possible variation of theion-BNS parameter κ
compatible with both stabilizationand emittance preservation. This,
in turn, supports thepossibility for the acceleration by 100 times,
say, from10 GeV to 1 TeV. Keeping in mind the significant helpof an
incomplete BNS compensation, demonstrated byFig 1, as well as
approximations of this model, one mayhope that this acceleration
range may be significantlylarger.
Transverse stabilization with the ion mobility takeninto account
was observed in simulations of Ref. [8] forNt = 4 · 109 cm−3, rms
bunch length σz = 6.4µm, rmstransverse size σ⊥ = 0.5µm, and the
energy efficiencywas ηP = 0.5. According to the
efficiency-instability rela-tion (1), it corresponds to the wake
parameter ηt = 0.13.For these parameters, the proton phase advance
is com-puted as µi = 0.56 rad, and the ion BNS parameterκ = 2.6,
assuming Lt ≈ 4σz and b ≈ 2σ⊥. Thus, theobserved stabilization is
in agreement with the suggestedion-driven BNS theory.
Fermilab is operated by Fermi Research Alliance, LLCunder
Contract No. DE-AC02-07CH11359 with theUnited States Department of
Energy.
[1] A. W. Chao, Physics of collective beam instabilities in
highenergy accelerators (Wiley, 1993).
[2] A. Burov, (2018), arXiv:1807.04887 [physics.acc-ph].[3] J.
B. Rosenzweig, B. Breizman, T. Katsouleas, and J. J.
Su, Phys. Rev. A 44, R6189 (1991).[4] G. Stupakov, Phys. Rev.
Accel. Beams 21, 041301 (2018).[5] P. Baxevanis and G. Stupakov,
Phys. Rev. Accel. Beams
21, 071301 (2018).
[6] V. Lebedev, A. Burov, and S. Nagaitsev, Phys. Rev. Ac-cel.
Beams 20, 121301 (2017).
[7] V. E. Balakin, A. V. Novokhatsky, and V. P.
Smirnov,Proceedings, 12th International Conference on High-Energy
Accelerators, HEACC 1983: Fermilab, Batavia,August 11-16, 1983,
Conf. Proc. C830811, 119 (1983).
[8] W. An, in FACET-II Science Workshop, SLAC, 2017(2017).
http://arxiv.org/abs/1807.04887http://dx.doi.org/10.1103/PhysRevA.44.R6189http://dx.doi.org/10.1103/PhysRevAccelBeams.21.041301http://dx.doi.org/10.1103/PhysRevAccelBeams.21.071301http://dx.doi.org/10.1103/PhysRevAccelBeams.21.071301http://dx.doi.org/10.1103/PhysRevAccelBeams.20.121301http://dx.doi.org/10.1103/PhysRevAccelBeams.20.121301https://conf.slac.stanford.edu/facet-2-2017/
Beam Breakup Mitigation by Ion Mobility in Plasma
AccelerationAbstractReferences