Evolving bunch and retardation in the impedance formalism Page: 1 of 3
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EVOLVING BUNCH AND RETARDATION IN THE IMPEDANCE FORMALISM *
R. Warnockt, SLAC, Stanford, CA 94309 and M. Venturini, LBNL, Berkeley, CA 94720
The usual expression for the longitudinal wake field in
terms of the impedance is exact only for the model in which
the source of the field is a rigid bunch. To account for a de-
forming bunch one has to invoke the complete impedance,
a function of both wave number and frequency. A compu-
tation of the corresponding wake field would be expensive,
since it would involve integrals over frequency and time
in addition to the usual sum over wave number. We treat
the problem of approximating this field in an example of
current interest, the case of coherent synchrotron radiation
(CSR) in the presence of shielding by the vacuum chamber.
Consider a rigid bunch moving on a circular trajectory
of radius R. A test particle feels a voltage
V(O, t)= woQ ) eng--wotZ(n)An , (1)
where 0 - wt is the azimuthal angle between the test par-
ticle and the reference particle, the latter having revolution
frequency w, o 3Oc/R. The impedance at azimuthal mode
number n (wave number n/R) is Z(n). The mode ampli-
tude An is the Fourier transform of the line density A(0) in
the bunch rest frame. With f A(0)dO 1 the total charge
is Q. For the case of a deforming bunch one's first inclina-
tion is merely to replace An in (1) by An(t). The resulting
formula (or its equivalent statement in terms of a wake po-
tential) has been used in dynamical calculations based on
the Vlasov-Fokker-Planck equation [1, 2, 3, 4], and in ear-
lier macroparticle simulations. In such work An (or A(0)) is
updated at each time step according to the values of exter-
nal and coherent forces at the previous step. Although the
calculations seem successful in many respects, the simple
replacement An - An (t) is a first approximation of un-
certain accuracy, especially in an unstable regime of rapid
Our object here is to derive this first approximation
and systematic corrections. We do so in an analytically
solvable model, the case of particles on circular trajecto-
ries between two infinite parallel plates, perfectly conduct-
ing. The plates represent the vacuum chamber, which sup-
presses CSR at wavelengths greater than a certain "shield-
ing cutoff'. This model has considerable utility in spite of
its simplified view of a real system; it led to interesting re-
* Work supported in part by Department of Energy contract DE-FG03-
sults on instabilities induced by CSR in the work of Refs.
The field equations are solved in cylindrical coordinates
(r, 0, y), with y-axis perpendicular to the plates and origin
midway between the plates of separation h. We allow an
arbitrary but fixed distribution of charge in the y-direction,
with density H(y) , f H(y)dy 1. The full charge den-
sity has the form p(O, t) QA(0-wt, t)H(y)(r -R)/R.
We make a Laplace transform of the Maxwell equations
and the charge/current density in time, assuming that the
charge and the fields are zero before t 0. We also
make Fourier transforms in 0 and y, the Fourier series in
y chosen to satisfy the boundary conditions of fields on
the plates. Then the transformed field equations can be
solved in terms of Bessel functions. The Fourier/Laplace
transform of the longitudinal electric field (averaged over
the vertical distribution H(y)) will be denoted as E(n, o).
Here o u + iv , v > 0 is a complex frequency,
while the Laplace transform variable conjugate to time is
s -io. By linearity of the field equations, -2wRE is
proportional to the corresponding transform of the current,
with a proportionality constant Z(n, o) called the complete
impedance: -2RE(n, o) Z(n, o)I(n, o).
The transform of the current is
Z~n, t) d(e'- o~An(t) ,
n(t) - 0 de-,noA(8 t)
" 2x .
Compare the case of a rigid bunch existing for all time, for
I(n, ,w) QwgSo6c-nwgo) .
The impedance has the form
Z(n, o) Zo;r) E A1.[ Jc n (%pR)H$4? (R)
k 2 Jini(1R)H ()( R)] .
Here HM1= Jn + iYn, where Jn and Y, are Bessel func-
tions of the first and second kinds, Zo 120f Q in m.k.s.
units, and a =rp/h ,y2 (K/c)2 - a. The sum
on p corresponds to modes in the Fourier expansion with
respect to y. The factor AP depends on the vertical dis-
tribution H(y), and is zero for even p if H is even. For
a Gaussian distribution with r.m.s. width oy < h, and
the y-average to define E taken over [-un , Qy], we have
AP = 2 sin(x)cx2/2/x x apuy.
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Warnock, Robert & Venturini, Marco. Evolving bunch and retardation in the impedance formalism, article, May 22, 2003; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc785099/m1/1/: accessed February 18, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.