34
E. Wilson and B. J. Holzer
The kick, projected as an amplitude increment becomes:
δu = wβ x kk sin Q H θ cos Q V θ.
(2.43)
When we use:
sin A cos B =
1
2
sin (A − B) +
1
2
sin (A + B)
and ignore the second, high frequency term; we obtain the coupled equations for a
single passage:
δw
w = −
ε H
ε V
β x β z
2 kk sin (Q H − Q V ) θ,
δu
u =
ε V
ε H
β x β z
2 kk sin (Q H − Q V ) θ.
(2.44)
These are incremental equations which we must sum over the n turns as the
coupling enhances u at the expense of w.
Figure 2.14 shows diagrammatically the coupled motion. The vertical betatron
amplitude decrease from w + to w in one quarter period of the slow oscillation
which takes 1/4|Q H − Q v | turns. The mean value of the cosine is taken as 2/π. We
then arrive at the expressions for the maximum excursions in amplitude:
Δw
w =
ε H
ε V
√
β x β z
4π|Q H −Q v | kk,
Δu
u =
ε V
ε H
√
β x β z
4π|Q H −Q v | kk.
(2.45)
We now move from the phase plane into real space. Some machines were
designed to have a rectangular “vacuum chamber” which would accept particles
which simultaneously have large horizontal and vertical “emittances”. In this sense
emittance is defined for a single particle
ε H = πu
2 /β x , ε V = πw
2 /β y .
(2.46)
In the presence of coupling, the particle motion is a series of Lissajous figures
filling the rectangular cross-section but always touching it somewhere on each turn
(Fig. 2.15). It is inevitable therefore that if coupling increases either amplitude by
u/u or w/w, some fraction of particles will be lost.
A rigorous treatment of coupling is too lengthy to include here but the reader may
consult [9–11] for a complete description. This lengthier treatment leads to a model
in which the modes of betatron oscillations are no longer about the vertical and
horizontal planes but about two orthogonal principal planes inclined with respect to
the vertical and horizontal frame.
E. Wilson and B. J. Holzer
The kick, projected as an amplitude increment becomes:
δu = wβ x kk sin Q H θ cos Q V θ.
(2.43)
When we use:
sin A cos B =
1
2
sin (A − B) +
1
2
sin (A + B)
and ignore the second, high frequency term; we obtain the coupled equations for a
single passage:
δw
w = −
ε H
ε V
β x β z
2 kk sin (Q H − Q V ) θ,
δu
u =
ε V
ε H
β x β z
2 kk sin (Q H − Q V ) θ.
(2.44)
These are incremental equations which we must sum over the n turns as the
coupling enhances u at the expense of w.
Figure 2.14 shows diagrammatically the coupled motion. The vertical betatron
amplitude decrease from w + to w in one quarter period of the slow oscillation
which takes 1/4|Q H − Q v | turns. The mean value of the cosine is taken as 2/π. We
then arrive at the expressions for the maximum excursions in amplitude:
Δw
w =
ε H
ε V
√
β x β z
4π|Q H −Q v | kk,
Δu
u =
ε V
ε H
√
β x β z
4π|Q H −Q v | kk.
(2.45)
We now move from the phase plane into real space. Some machines were
designed to have a rectangular “vacuum chamber” which would accept particles
which simultaneously have large horizontal and vertical “emittances”. In this sense
emittance is defined for a single particle
ε H = πu
2 /β x , ε V = πw
2 /β y .
(2.46)
In the presence of coupling, the particle motion is a series of Lissajous figures
filling the rectangular cross-section but always touching it somewhere on each turn
(Fig. 2.15). It is inevitable therefore that if coupling increases either amplitude by
u/u or w/w, some fraction of particles will be lost.
A rigorous treatment of coupling is too lengthy to include here but the reader may
consult [9–11] for a complete description. This lengthier treatment leads to a model
in which the modes of betatron oscillations are no longer about the vertical and
horizontal planes but about two orthogonal principal planes inclined with respect to
the vertical and horizontal frame.