62 Beam-based Correction and Optimization for Accelerators
The approximation of the continuous bending in the curved coordinate system with the propagation on a straight line introduces errors. Notably, an
on-energy particle at the phase space origin will be transported to non-zero
coordinates at the exit face, with
∆x 0 =
1
cos hL
− ρ, ∆p x,0 = sin hL, ∆z 0 = ρ tan hL − L.
(2.122)
These errors need to be subtracted from Eq. (2.121). There are other ways
to separate the dipole Hamiltonian for symplectic integration. Sometimes the
simple approach of replacing Eq. (2.121) with the solution of a drift space in
the straight coordinate system is used. The evaluation is faster, although it
comes with some loss of accuracy.
RF cavities:
An RF cavity can be modeled as a thin-lens element in which the momentum deviation coordinate is modified according to the z-coordinate of the
particle and the RF parameters, which gives
δ 2 = δ +
eV
β 2 E 0
sin(
2πhf 0 z
c
+ φ s ),
(2.123)
where V is the RF voltage, f 0 the revolution frequency, h the harmonic number, and φ s is the synchronous phase.
Radiation damping and quantum excitation:
Radiation damping and quantum excitation occur in electron storage rings
in which particles emit photons due to synchrotron radiation [106]. Since
higher energy particles lose more energy to photons, lower energy particles
lose less, and all particles on average gain the same amount of energy each
turn, the energies of all particles tend to converge to the same value. Similarly, as particles lose the transverse momenta to photon emissions and only
gain energy through work done in the longitudinal direction, the transverse
oscillations gradually decrease toward zero. Radiation damping can be implemented in the dipole symplectic integrator by making the particles lose the
correct amount of energy after each kick, and scale p x and p y to keep the x
and y
coordinates unchanged. With radiation damping, the beam motion is
no longer symplectic.
The emission of each photon gives the particle a kick in the momentum
coordinate. The kicks by the emission of photons put the particle on a random walk in the longitudinal phase space, resulting in increasing longitudinal
action variable. When photon emissions occur in a dispersive region, the particle will start to oscillate around the off-energy closed-orbit corresponding
to the new energy; hence the particle is excited in the transverse plane. The
excitation of beam motion by the impulsive kicks of photon emissions is called
quantum excitation or quantum diffusion. Quantum excitation causes particle
to deviate from the closed-orbit. It is balanced by radiation damping, leading
to an equilibrium beam distribution. To simulate quantum excitation, the energy loss of the particle at each kick is given by a random variable, which can
The approximation of the continuous bending in the curved coordinate system with the propagation on a straight line introduces errors. Notably, an
on-energy particle at the phase space origin will be transported to non-zero
coordinates at the exit face, with
∆x 0 =
1
cos hL
− ρ, ∆p x,0 = sin hL, ∆z 0 = ρ tan hL − L.
(2.122)
These errors need to be subtracted from Eq. (2.121). There are other ways
to separate the dipole Hamiltonian for symplectic integration. Sometimes the
simple approach of replacing Eq. (2.121) with the solution of a drift space in
the straight coordinate system is used. The evaluation is faster, although it
comes with some loss of accuracy.
RF cavities:
An RF cavity can be modeled as a thin-lens element in which the momentum deviation coordinate is modified according to the z-coordinate of the
particle and the RF parameters, which gives
δ 2 = δ +
eV
β 2 E 0
sin(
2πhf 0 z
c
+ φ s ),
(2.123)
where V is the RF voltage, f 0 the revolution frequency, h the harmonic number, and φ s is the synchronous phase.
Radiation damping and quantum excitation:
Radiation damping and quantum excitation occur in electron storage rings
in which particles emit photons due to synchrotron radiation [106]. Since
higher energy particles lose more energy to photons, lower energy particles
lose less, and all particles on average gain the same amount of energy each
turn, the energies of all particles tend to converge to the same value. Similarly, as particles lose the transverse momenta to photon emissions and only
gain energy through work done in the longitudinal direction, the transverse
oscillations gradually decrease toward zero. Radiation damping can be implemented in the dipole symplectic integrator by making the particles lose the
correct amount of energy after each kick, and scale p x and p y to keep the x
and y
coordinates unchanged. With radiation damping, the beam motion is
no longer symplectic.
The emission of each photon gives the particle a kick in the momentum
coordinate. The kicks by the emission of photons put the particle on a random walk in the longitudinal phase space, resulting in increasing longitudinal
action variable. When photon emissions occur in a dispersive region, the particle will start to oscillate around the off-energy closed-orbit corresponding
to the new energy; hence the particle is excited in the transverse plane. The
excitation of beam motion by the impulsive kicks of photon emissions is called
quantum excitation or quantum diffusion. Quantum excitation causes particle
to deviate from the closed-orbit. It is balanced by radiation damping, leading
to an equilibrium beam distribution. To simulate quantum excitation, the energy loss of the particle at each kick is given by a random variable, which can