2 Beam Dynamics
45
even later than it did on the previous turn. This seems to defeat the whole idea of
phase stability. Depending on how the synchrotron is designed and which particles
it accelerates, there can be a certain energy where our initial ideas of phase stability
break down. This is called the transition energy. Fortunately there is also a way of
ensuring stability above transition.
2.5.2 Transition Energy
The rigorous argument to resolve the question of velocity versus path length is to
examine how the revolution time (or its reciprocal, the revolution frequency) varies
as the particle is given extra acceleration. The revolution frequency is:
f =
βc
2πR
, (β = v/c) .
(2.63)
This revolution frequency, f, depends on two momentum dependent variables,
the relativistic β=v/c and R, the mean radius. The penultimate equation gives the
change in the radius. The momentum dependence of β is determined by:
p =
E 0 β
1 − β 2
.
(2.64)
The rate of “catching up” depends upon a “slip factor”, η, which is defined as
logarithmic differential of frequency as a function of momentum. The procedure of
partial derivatives tells us there must be two terms. Hence:
η rf =
Δf/f
Δp/p
=
p
β
dβ
dp
−
p
R
dR
dp
=
1
γ 2 −
D
R 0
.
(2.65)
The first term on the right-hand side describes the increase in speed with p and
the other (negative), how the path to be completed increases with p.
The second term is energy independent while the first term shrinks as acceleration
proceeds. At low energy this is largest and η is >0. But, since γ = E/E 0 , the first term
becomes smaller than the second at high energy so that η changes sign from positive
to negative. During the acceleration process there is a certain energy, the transition
energy, at which η is momentarily zero. At transition, the value of γ satisfies:
1
γ 2
tr
=
D
R
.
(2.66)
In proton synchrotron design this condition tends to be encountered mid–way
through the acceleration cycle and can only be avoided with some ingenuity in the
design of the lattice. This was a worry to the designers of the PS and AGS, the
45
even later than it did on the previous turn. This seems to defeat the whole idea of
phase stability. Depending on how the synchrotron is designed and which particles
it accelerates, there can be a certain energy where our initial ideas of phase stability
break down. This is called the transition energy. Fortunately there is also a way of
ensuring stability above transition.
2.5.2 Transition Energy
The rigorous argument to resolve the question of velocity versus path length is to
examine how the revolution time (or its reciprocal, the revolution frequency) varies
as the particle is given extra acceleration. The revolution frequency is:
f =
βc
2πR
, (β = v/c) .
(2.63)
This revolution frequency, f, depends on two momentum dependent variables,
the relativistic β=v/c and R, the mean radius. The penultimate equation gives the
change in the radius. The momentum dependence of β is determined by:
p =
E 0 β
1 − β 2
.
(2.64)
The rate of “catching up” depends upon a “slip factor”, η, which is defined as
logarithmic differential of frequency as a function of momentum. The procedure of
partial derivatives tells us there must be two terms. Hence:
η rf =
Δf/f
Δp/p
=
p
β
dβ
dp
−
p
R
dR
dp
=
1
γ 2 −
D
R 0
.
(2.65)
The first term on the right-hand side describes the increase in speed with p and
the other (negative), how the path to be completed increases with p.
The second term is energy independent while the first term shrinks as acceleration
proceeds. At low energy this is largest and η is >0. But, since γ = E/E 0 , the first term
becomes smaller than the second at high energy so that η changes sign from positive
to negative. During the acceleration process there is a certain energy, the transition
energy, at which η is momentarily zero. At transition, the value of γ satisfies:
1
γ 2
tr
=
D
R
.
(2.66)
In proton synchrotron design this condition tends to be encountered mid–way
through the acceleration cycle and can only be avoided with some ingenuity in the
design of the lattice. This was a worry to the designers of the PS and AGS, the