40 Beam-based Correction and Optimization for Accelerators
with phase space coordinate X = (x, x
)
T , transfer matrix M, and d a 2element column vector such that δd represents the accumulated orbit shift
induced by the momentum deviation over the section. The vertical plane is
not considered here because there is no vertical dispersion in an ideal planar
accelerator. The same treatment can be extended to include the vertical plane.
Eq. (2.31) shows that a part of the phase space coordinate, X, is linearly
dependent on the momentum deviation. It is possible and desirable to separate
this part out from the usual betatron motion, with
X = X β + Dδ,
(2.32)
where the dispersion vector D=(D, D
)
T , and D is the dispersion function,
D
=
dD
ds . Inserting Eq. (2.32) into Eq. (2.31) and separating the terms dependent on δ, we obtain
X β = MX β,0 ,
(2.33)
D = MD 0 + d,
(2.34)
where X 0 = X β,0 + D 0 δ was used. The X β term represents the betatron
motion and is called the betatron coordinate.
Eq. (2.34) specifies how the dispersion vector is transported through an
accelerator section. At the exit face the dispersion vector consists of a term
that is transported from the initial dispersion vector by the transfer matrix
and a term that represents the contributions from the dipole fields in the
section itself.
In a one-pass system, the choice of the initial values of the dispersion
functions is somewhat arbitrary. Usually the initial dispersion is chosen to
match the expected beam distribution at the entrance point. In this case,
the dispersion vector obtained with Eq. (2.34) will be consistent with the
dispersion derived from the beam distribution at a downstream location. For a
transport line, the initial dispersion is usually given by the dispersion function
values at the extraction point of the upstream machine. For a linac, the initial
dispersion may be set to zero, assuming the initial transverse distribution
has no correlation with the momentum deviation. In general, given an initial
distribution, f (X, δ), the dispersion function can be found from
Dσ
2
δ =
δxf (X, δ)dXdδ, D
σ
2
δ =
δx
f (X, δ)dXdδ,
(2.35)
where σ
2
δ =
δ
2 f (X, δ)dXdδ and σ δ is defined as the rms momentum spread.
In a circular accelerator, there is a natural choice for the separation of the
betatron coordinates and the dispersion terms, which is to impose the periodic
condition on the dispersion function,
D(s + C) = D(s), D
(s + C) = D
(s),
(2.36)
with phase space coordinate X = (x, x
)
T , transfer matrix M, and d a 2element column vector such that δd represents the accumulated orbit shift
induced by the momentum deviation over the section. The vertical plane is
not considered here because there is no vertical dispersion in an ideal planar
accelerator. The same treatment can be extended to include the vertical plane.
Eq. (2.31) shows that a part of the phase space coordinate, X, is linearly
dependent on the momentum deviation. It is possible and desirable to separate
this part out from the usual betatron motion, with
X = X β + Dδ,
(2.32)
where the dispersion vector D=(D, D
)
T , and D is the dispersion function,
D
=
dD
ds . Inserting Eq. (2.32) into Eq. (2.31) and separating the terms dependent on δ, we obtain
X β = MX β,0 ,
(2.33)
D = MD 0 + d,
(2.34)
where X 0 = X β,0 + D 0 δ was used. The X β term represents the betatron
motion and is called the betatron coordinate.
Eq. (2.34) specifies how the dispersion vector is transported through an
accelerator section. At the exit face the dispersion vector consists of a term
that is transported from the initial dispersion vector by the transfer matrix
and a term that represents the contributions from the dipole fields in the
section itself.
In a one-pass system, the choice of the initial values of the dispersion
functions is somewhat arbitrary. Usually the initial dispersion is chosen to
match the expected beam distribution at the entrance point. In this case,
the dispersion vector obtained with Eq. (2.34) will be consistent with the
dispersion derived from the beam distribution at a downstream location. For a
transport line, the initial dispersion is usually given by the dispersion function
values at the extraction point of the upstream machine. For a linac, the initial
dispersion may be set to zero, assuming the initial transverse distribution
has no correlation with the momentum deviation. In general, given an initial
distribution, f (X, δ), the dispersion function can be found from
Dσ
2
δ =
δxf (X, δ)dXdδ, D
σ
2
δ =
δx
f (X, δ)dXdδ,
(2.35)
where σ
2
δ =
δ
2 f (X, δ)dXdδ and σ δ is defined as the rms momentum spread.
In a circular accelerator, there is a natural choice for the separation of the
betatron coordinates and the dispersion terms, which is to impose the periodic
condition on the dispersion function,
D(s + C) = D(s), D
(s + C) = D
(s),
(2.36)