24 Beam-based Correction and Optimization for Accelerators
The revolution time of a particle is determined by the path length of one
revolution and its speed, T =
C
v , hence
∆T
T
=
∆C
C
−
∆v
v
= (α c −
1
γ 2 )δ,
(1.84)
where γ is the Lorentz energy factor, δ =
∆p
p , and α c =
∆C
δC is the momentum
compaction factor. The ratio between
∆T
T and δ is the phase slippage factor,
η = α c −
1
γ 2 .
(1.85)
When the beam energy satisfies γ = γ T ≡
1
√
αc , the revolution time does
not depend on the momentum deviation to the first order. The corresponding
energy factor, γ T , is referred to as the transition gamma. There is no longitudinal focusing in this case as the phase coordinate is “frozen”. If η < 0, higher
energy particles complete a revolution in less time than the synchronous particle, and the opposite is true for η > 0. The η < 0 case is said to be below
transition, and η > 0 is above transition.
There is an oscillating electric field along the longitudinal direction across
the gap of an RF cavity. This electric field accelerates or decelerates the particles by adding or removing energy from them, respectively, as they traverse
the cavity gap. The net effect of the electric field across the gap can be characterized by an oscillating voltage
V (t) = V g sin(ω rf t + φ s ),
(1.86)
where ω rf is the angular resonant frequency of the cavity, φ s is the RF phase
when the synchronous particle crosses the gap, and V g is the RF gap voltage.
V g is related to the peak electric field, E, and the gap length g via
V g =
g/2
−g/2
E cos
ω rf z
βc
dz.
(1.87)
The resonant frequency of the RF cavity must be a multiple of the revolution
frequency, ω rf = hω 0 , with ω 0 = 2π/T 0 . T 0 is the revolution time for the
reference particle and h is an integer which is defined as the harmonic number.
In a synchrotron the beam energy changes with time as the beam picks
up energy from the RF cavity. The revolution frequency may vary with beam
energy for low or medium energy proton or heavy ion synchrotrons, which
requires the RF frequency to change accordingly. The RF voltage may also
ramp with time as the beam energy changes.
The longitudinal motion of an arbitrary particle can be described by its
energy and the RF phase at the time it traverses the RF cavity. The total
energy change of the particle in one revolution is determined by the energy
gain or loss in the RF cavity along with other energy losses, such as due to
The revolution time of a particle is determined by the path length of one
revolution and its speed, T =
C
v , hence
∆T
T
=
∆C
C
−
∆v
v
= (α c −
1
γ 2 )δ,
(1.84)
where γ is the Lorentz energy factor, δ =
∆p
p , and α c =
∆C
δC is the momentum
compaction factor. The ratio between
∆T
T and δ is the phase slippage factor,
η = α c −
1
γ 2 .
(1.85)
When the beam energy satisfies γ = γ T ≡
1
√
αc , the revolution time does
not depend on the momentum deviation to the first order. The corresponding
energy factor, γ T , is referred to as the transition gamma. There is no longitudinal focusing in this case as the phase coordinate is “frozen”. If η < 0, higher
energy particles complete a revolution in less time than the synchronous particle, and the opposite is true for η > 0. The η < 0 case is said to be below
transition, and η > 0 is above transition.
There is an oscillating electric field along the longitudinal direction across
the gap of an RF cavity. This electric field accelerates or decelerates the particles by adding or removing energy from them, respectively, as they traverse
the cavity gap. The net effect of the electric field across the gap can be characterized by an oscillating voltage
V (t) = V g sin(ω rf t + φ s ),
(1.86)
where ω rf is the angular resonant frequency of the cavity, φ s is the RF phase
when the synchronous particle crosses the gap, and V g is the RF gap voltage.
V g is related to the peak electric field, E, and the gap length g via
V g =
g/2
−g/2
E cos
ω rf z
βc
dz.
(1.87)
The resonant frequency of the RF cavity must be a multiple of the revolution
frequency, ω rf = hω 0 , with ω 0 = 2π/T 0 . T 0 is the revolution time for the
reference particle and h is an integer which is defined as the harmonic number.
In a synchrotron the beam energy changes with time as the beam picks
up energy from the RF cavity. The revolution frequency may vary with beam
energy for low or medium energy proton or heavy ion synchrotrons, which
requires the RF frequency to change accordingly. The RF voltage may also
ramp with time as the beam energy changes.
The longitudinal motion of an arbitrary particle can be described by its
energy and the RF phase at the time it traverses the RF cavity. The total
energy change of the particle in one revolution is determined by the energy
gain or loss in the RF cavity along with other energy losses, such as due to