LCLS Conceptual Design Report - Stanford Synchrotron Radiation ...

L C L S C O N C E P T U A L D E S I G N R E P O R T
2
13
⎛2a⎞ s
⎜
⎟
0 ≡⎜ ⎟
⎜⎝Zσ 0 ⎠
, (7.32)
the rms relative energy spread induced in a smooth cylindrical chamber of radius, a, and
conductivity, σ, is
σ δ
RW
2
e cNL Z0
≈ ( 0.
22)
. (7.33)
2 3 2
π aEσ
σ
z
This is for a gaussian bunch, which is long compared to s 0. To more accurately estimate the
RW energy spread generated after BC2, the beamline is broken into two discrete sections of
significantly small radius (Table 7.24). Remaining sections have much larger radii (40–400 mm)
and are ignored here. The short iris surfaces of the copper accelerating structures in L3 contribute
no significant component to the resistive wall wakefields [69].
Table 7.24 The two main beamline sections that transport the micro-bunch and contribute to a resistive
wall wake energy spread generated between BC2 and undulator entrance.
Beamline Section Material Conductivity
Ω –1 -m –1
Radius
mm
Length
Linac-3 non-accelerating chambers Stainless Steel 0.14×10 7 12.7 76 85
Linac-to-undulator beamline Aluminum 3.6×10 7 12.7 106 29
The 76-meters of 1-inch diameter stainless steel are distributed along Linac-3 as
quadrupole/BPM chambers, and other short non-accelerating sections including 22 meters beyond
sector-30 in the beam switchyard, before the aperture significantly increases. The aluminum
sections are new chambers for the DL2/ED2 beamline leading up to the undulator entrance,
which follow the large radius beam switchyard. The replacement of the existing 100 meters of
stainless chamber, which is presently used in the undulator hall, with aluminum removes an effect
which would otherwise increase the coherent energy spread by 0.05% rms, with an energy
gradient which is nearly linear across the bunch. The associated transverse resistive-wall
wakefields are insignificant for reasonable electron trajectories.
For these sections, the bunch (22 µm) is shorter than s 0, so the RW energy spread of the
short-bunch (σz/s 0 < 1) is calculated using the point-charge wake function [70],
2
⎛ ∞ 2 −x
ss0
⎞
0 ⎜ −ss0
⎟
⎜3s0π+ 8 ⎟
0
4qcZ 1 3s 2 dx x e
Ez() s =− e cos −
2 ∫ , (7.34)
6
π a x
⎝ ⎠
which is convoluted with the bunch distribution similar to that at bottom of Figure 7.7. This
estimate ignores the frequency dependence of the conductivity, an effect which is quite small.
The results for each of the two sections of Table 7.24 are shown in Figure 7.75.
7-108 ♦ A C C E L E R A T O R
m
s 0
µm