30 Beam-based Correction and Optimization for Accelerators
where ˆ
ϕ and ˆ
δ are the oscillation amplitudes in the ϕ, δ directions, respectively.
The phase space ellipses of the stable longitudinal motion are illustrated in
Figure 1.11. The ratio of the momentum and phase oscillation amplitudes is
ˆ
δ
ˆ
ϕ
=
ν s
h|η|
.
(1.114)
In a storage ring, the beam typically settles down to an equilibrium distribution. In phase space, the equilibrium distribution must conform to the
Hamiltonian contour in order not to exhibit any change with time while the
individual particles move along the contours. Therefore, the equilibrium distribution has the form of
ρ(ϕ, δ) ∼ ρ(H(ϕ, δ)) ∼ ρ( ˆ
ϕ),
(1.115)
where ˆ
ϕ is related to the Hamiltonian through H =
1
2
ω0ν
2
s
hη ˆ
ϕ
2 . In an electron
storage ring, the combination of radiation induced damping and diffusion creates a Gaussian distribution, which can be written as
ρ( ˆ
ϕ) =
1
2πσ 2
ˆ
ϕ
exp
−
1
2
ˆ
ϕ
2
σ 2
ˆ
ϕ
,
(1.116)
or expressed in the phase space coordinates φ and δ, rather than the phase
amplitude ˆ
φ,
ρ(ϕ, δ) =
1
2πσ ϕ σ δ
exp
−
1
2
ϕ
2
σ 2
ϕ
+
δ
2
σ 2
δ
,
(1.117)
where σ ϕ is the rms bunch length and σ δ is the rms momentum deviation.