advanced beam manipulation, cooling, damping and stability 201
FIGURE 10.20
Flat beam collision in an IR of a typical linear collider.
For a beam with uniform density distribution and transverse sizes σ x,y , the maximum field at the boundary of the
beam can be estimated using the Gauss theorem Eds = 4πQ,
which thus gives for this maximum field:
eN
E ≈
(10.20)
(σ x σ z )
where we take into account that, for the beam with uniform
distribution in the longitudinal direction, the surface integral
ds turns into the contour integral de with charge Q taken
per unit of length.
Taking the contour integral in the above derivation along
the contour inside of the beam, we conclude that the field
inside of the beam grows linearly. The fields outside of the
beam can be estimated in the same manner. It is important
to note that, if the beam is very flat, the integral would be
almost independent of the offset from the beam in the y direction until the point when the vertical offset reaches values
comparable with σ x .
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FIGURE 10.21
Fields of the flat beam.
Therefore, for vertical offsets ranging between σ y and σ y , the
field will be almost constant and after that will start to decrease as 1/r, as illustrated in Fig. 10.21.
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