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Transverse Equilibria in Linear Collider Beam-Beam
CollisionsJ.B. Rosenzweig
UCLA Dept. of Physics, 405 Hilgard Ave., Los Angeles, CA
90024Pisin ChenStanford Linear Accelerator Center, Stanford, CA
94309
AbstractIt has been ohserved in simulations of the beam-beam
in-
teraction in linear colliders that a near equilibrium
pinchedstate of the colliding beams develops when the
disruptionparameter is large (D >> l)[l]. In this state the
beamtransverse density distributions are peaked at center, withlong
tails. WC present here an analytical model of theequilibrium
approached by the beams, that of a general-ized Bennett pinch[2]
which develops through collisionlessdamping due to the strong
nonlinearity of the beam-beaminteract,ion. In order to calculate
the equilibrium pinchedbeani size, an estimation of the rms
emittance growth ismade which takes into account the partial
adiabaticity ofthe collision. This pinched beam size is used to
derive theluminosity enhancement factors whose scaling is in
agree-nlent with the simulation results for both D and
thermalfactor A = (T,/,@* large, and explains the previously
notedcubic relationship between round and flat beam enhance-ment
factors.
IntroductionThe calculation of the luminosity enhancement of
linear
collider beam-beam collisions due to the mutual strongfocusing,
or disruption, of the beams has been tradi-tionally calculated[ l]
by use of particle-in-cell computercodes. These numerical
calculations solve for clectromag-netic fields and the motion of
the particles which generatethese fields self-consistently. The
emergence of near equi-librium pinch-confined transverse beam
profiles in the limitthat the disruption parameter D,,, =
2Nr*ear/~gZ,Y(gZ +~7~) >> 1 has been noted; in this regime
the beam particlesundergo multiple betatron oscillations during the
collision.It is proposed here that these near equilibrium states
areapproached through collisionless damping due to mixingand
filamentation in phase space, in analogy to a simi-lar phenomena
found in self-focusing beams in plasmas[3].The expected luminosity
enhancement obtained in thisstate is calculated in this paper.
Since the approach to thisequilibrium ent.ails examining very
nonlinear phase spacedynamics approximations are necessary,
especially with re-gards to the calculation of the emittance growth
induced by
filamentation. A model for this emittance growth, basedon 0.
Andersons theory of space charge induced emittancegrowth[4], is
employed, which then allows a calculation ofthe luminosity
enhancement which is in fairly good agree-ment with the values
obtained by simulation.
Maxwell-Vlasov EquilibriaThe equilibria we are proposing to
study are of the type
known as Maxwell-Vlasov equilibria, which are obtained bylooking
for a time independent solution of the Vlasov equa-tion describing
the beams transverse phase space, with theforces obtained
self-consistently from the Maxwell equa-tions using the beam charge
and current profiles. We beginwith a flat beam (a, >> by), as
these are the simplest, andmost likely to be found in a linear
collider. For the purposeof calculation the beams are assumed to be
uniform in 1:and L (at least locally), and have identical profiles
in y.
The vertical force on an ultra-relativistic particle is
thuss
YFy cc -24Ey) = -87re& Y(Y)dY0
where Y, normalized by s-, Ydy = 1, describes the verti-cal beam
profile, and Cb = N/~~T(T,~T, is the beam surfacecharge density. We
look for separable solutions to the timeindependent Vlasov
equation
afEz, ?L,,af=,at y Y YdPy where tjy = p,l-fm, i.e. solutions of
the fornl f =Y(YNP,). Th is f orm is in fact approached in true
ther-mal equilibrium. In order for this equilibrium to de-velop
through phase space mixing, more than one non-linear betatron
oscillation must occur during collision,D, N 2m,C~a~/a, > 1.
The solution to t,he momentum equation obtained in thismanner
is
x2 x2 2J%d = dz exp -21which is the Maxwell-Boltzmann form we
should expect.The separation constant X2 = ym/c$ is inversely
propor-tional to the temperature of the system. The solution to
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the corresponding coordinate equation isY(y) = ;sech(ory), (Y =
47~eC~X.
This profile is the one-dimensional analogue to the
Bennettprofile found in cylindrically symmetric
Maxwell-Vlasovequilibria. The separation constant X2 remains to be
cal-culatctl in this treatment. As a first attempt, one canuse the
fact that the distribution function at the origin inphase space is
stationary, by symmetry, that is af/& = 0at (Y:~J~,) = (0,O).
Thus f(O,O) is a constant of the mo-tion. Assuming an initial
bi-Gaussian distribution in phasespace, and equating its peak
density in phase space to thatof the 13rnnett-type profile found
upon equilibration, wehave
f(O.0) = Y(O)P(O) = &n
= (&b)Vp$
TVe thlrs have, solving directly for CY,
8yr,& 13YEI 12\Vith tllis relation we can compare the
luminosity thatcomes about by the transition to a pinch confined
Bennett-like state with that of the initial Gaussian beam. At
thispoint, w(x IlIake allowance for the fact that the beamsare not,
ulliforrn in z and redefine & -+ (c~)/ =It/v% 7rurm,.
Luminosity EnhancementThe lurrlinosity enhancement due to
pinch-confinement
can be calclllated by taking the luminosity integrals of thetwo
cases, assuming A, = a,/,/$ < 1, the depth of focuseffcct#s can
I)e ignored, and D, < 1,
Following t11is I)rescription, the luminosity enhancement
isYrecb[ 1132?r)/5 [ fk] 1362n Y
lliis results I-)(_.lie computer simulation of luminosity
en-hancelllc,nt, 1)~ C:lren and Yokoya[l] is reprinted in Figure1.
Our cxpr(~.+ion clearly has too strong a dependence onD, and A, to
model the simulation results correctly. Thisis bcrausc, CVCII
though we have invoked emittance growth(phase spact:
filn~nentation) as the mechanism behind col-lisionless allpr(Jtlch
to equilibrium, we have neglected tocalculat,t t,his
i>lllit.tancc growth.
The e:lllittallcc growth can be estimated by using amethocl
dcvelop~~tl by Anderson to examine space-chargedriven emittaucr:
growth.[4] We begin by assuming lami-11ar flow of bcz1111 articles
pinching down under the influ-ence of the opposing beam forces
which, ignoring for the
3;
2.5 i-
2:
al 5cl jI -r
1r
x _--x , I+- EJ-
:/+//
0.5 i --,-A&l Io L. .--... , ~_ _ . i 1 1 .L_.---I----L.
1 10 100D Y
Figure 1: Luminosity enhancement as a function of D, ant1A,,
found by computer simulation.
moment the time dependence of the collision, is assumedto be
undergoing an identical pinch. The force on thebeam particles can
then be written in terms of the enclosedbeam current, and thus the
initial transverse displacement,F = -8re2&,G([), where G(t) =
s, Y(y)dy is a constantof the motion under laminar flow conditions.
The equationof motion for the beam particles is thus y + I
