Wakefield Calculations for 3D Collimators Page: 3 of 4
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kL Z (A) - Z6(0) , (8)
where A is beam offset from the axis.
For a monopole mode the steady-state field pattern does
not depend on pipe radius and both Eqs. 5 and 6 give
Ze =jln(-) (9)
2w3/2a- b) (10)
The dipole steady-state field pattern, however, depends on
the pipe radius and Eqs. 5 and 6 give different results. For
a long collimator (d -> o) we expect
kL 27 2
for a short collimator (d -> 0)
L 47T by
In order to test Eqs. 11, 12, and to obtain a feeling about
"long" and "short" we have calculated the loss and kick
factors for example collimators as functions of collimator
length d. For the first example a =r/2, bi 19 mm, and
b2 =1.9 mm; bunch lengths o- 0.5, 0.3, 0.1 mm. We
find that the numerically obtained loss factors are not sen-
sitive to collimator length d, and Eq. 9 approximates them
well. The calculated kick factors are shown in Fig. 2 (left
frame). The straight gray lines give the asymptotic (long,
short) approximations, Eqs. 11, 12. The numerical results
agree well with the analytical estimates. Also, the direct
sums of the kicks for "in"- and "out"-transitions (calculated
separately) are shown. We see that they agree with the kick
for a sufficiently long collimator, as expected.
(square) or 38 mm (rectangular), beam pipe half-height and
half-width bi 19 mm. The results are presented in Fig. 2
(the right frame) and Table 1. The gray lines in the figure
show the approximate kicks for the square collimator. We
see from the table that the rectangular collimator gives a
kick very similar to the round one. Of the three collima-
tors, the smallest kick is for the square one. We see that
Eqs. 11, 12, can be useful for all three types of collimators.
Table 1: Loss and kick factors as estimated by 2D electro-
static calculation. The bunch length o- 0.3 mm. "Short"
means using Eq. 6, "long" Eq. 5.
(12) From the above results we see that the kicks for long and
short step collimators are related by
Hence, in Eq. 3 we suggest one use A 1 for a long col-
limator (d -> o) and A for a short one (d -> 0). For
a collimator in the inductive regime, however, the result
should be independent of d.
The good agreement we have found between direct time-
domain calculation  and the approximations (5, 6), sug-
gests that the latter method can be used to approximate
short-bunch wakes for a large class of 3D collimators.
& a cm I
4.5 ;- - .
4.5 a - .n
3 .9 rih
A IV ;A mini
0.01 0.1 1 10 0.01 0.1
Figure 2: Kick factor vs. collimator length. A round
collimator (left), a square or rectangular collimator ( =
0.3 mm, right).
For 3D collimators, the loss factor for short and long col-
limators is, in general, no longer nearly the same. In this
case the energy impedance Ze can be found numerically by
solving the 2D problem (5') (see, e.g. ). We have per-
formed both 3D time-dependent and 2D electrostatic cal-
culations for square and rectangular collimators with aper-
ture half-height b2 1.9 mm, aperture width h 3.8 mm
In our simulations for 3D collimators we used a time-
domain numerical scheme  combined with an indirect
integration algorithm . We consider now two sets of col-
limators that were measured in experiment at SLAC. The
first set includes four collimators measured in 2001 .
The parameters are given in Table 2. All rectangular col-
limators have width h 2b1 38 mm and no flat region
(d = 0).
Table 2: Geometry of SLAC collimators of 2001.
Coll. # 1 2 3 4
Type rect. square rect. rect.
b2 [mm] 1.9 1.9 1.9 3.8
a [mrad] 168 335 335 298
In order to check the accuracy of the 3D discretization
we have calculated wakefields for round collimators with
bi 19 mm, b2 1.9 mm, a 335 mrad using the 2.5D
and 3D codes. The two results are nearly indistinguishable
(see Fig. 3 on the left). The wakes for square collimator
k [V/pC] kL [V/pC/mm]
Type short long short long
round 78 78 2.50 5.01
rect. 56 72 2.43 6.11
square 74 78 1.99 4.25
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Zagorodnov, I.; /DESY; Bane, K.L.F. & /SLAC. Wakefield Calculations for 3D Collimators, article, July 17, 2006; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc885694/m1/3/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.