54 Beam-based Correction and Optimization for Accelerators
The terms h
(0)
11001 and h
(0)
00111 correspond to tune changes for off-momentum
particles, namely, the chromaticities.
∆C x = h
(0)
11001 ,
C y = h
(0)
00111 .
(2.94)
The tune changes due to the above terms are independent of the action variables. For most terms, the tune shifts depend on the action variables, or equivalently, the oscillation amplitudes; such effects are called tune shifts with amplitude, amplitude-dependent detuning, or nonlinear detuning. The leading
terms in the perturbation Hamiltonian that give rise to nonlinear detuning
are
∆ ˜
H 1 = h
(0)
22000 J
2
x + h
(0)
11110 J x J y + h
(0)
00220 J
2
y .
(2.95)
The tune shifts with amplitude from these terms can be characterized by the
following coefficients.
∂ν x
∂J x
= 2h
(0)
22000 ,
∂ν x
∂J y
=
∂ν y
∂J x
= h
(0)
11110 ,
∂ν y
∂J y
= 2h
(0)
00220 .
(2.96)
There are also higher order terms that cause tune shifts dependence on the
action variables to higher orders.
The terms in Eq. (2.91) that have either j = k or l = m or both depend on
the angle coordinates φ x or φ y . Most of these terms have little impact over the
motion of the particles because they oscillate quickly with time and hence the
average effect is negligible. However, for some terms, the phase factor may be
slowly varying. The effect from these terms can build up and fundamentally
change the behavior of particle motion. These terms satisfy the resonance
condition
(j − k)
dφ x
dθ
+ (l − m)
dφ y
dθ
+ p = (j − k)ν x + (l − m)ν y + p,
= n 1 ν x + n 2 ν y + p ≈ 0,
(2.97)
where n 1 = j − k and n 2 = l − m. The order of the resonance is defined as
|n 1 | + |n 2 |. Beam motion around nonlinear resonances can become unstable,
as the resonances can drive the particles to large oscillation amplitudes and
cause beam loss. To alleviate the impact of the nonlinear resonances, it is
desirable to minimize the strengths of the resonance harmonics, h
(p)
jklmn .
Periodicity in a lattice can automatically set many systematic (i.e., inherent in the design) resonance terms to zero. If a ring consists of N repetitive,
identical cells, the resonance harmonics reduce to
h
(p)
jklm =
1
2π
cell
h jklm e
−jpθ dθ
N −1
q=0
e
−i2πq
p
N ,
=
1
2π
cell
h jklm e
−jpθ dθ
e
−iπp
N −1
N
sin pπ
sin(pπ/N )
,
(2.98)
The terms h
(0)
11001 and h
(0)
00111 correspond to tune changes for off-momentum
particles, namely, the chromaticities.
∆C x = h
(0)
11001 ,
C y = h
(0)
00111 .
(2.94)
The tune changes due to the above terms are independent of the action variables. For most terms, the tune shifts depend on the action variables, or equivalently, the oscillation amplitudes; such effects are called tune shifts with amplitude, amplitude-dependent detuning, or nonlinear detuning. The leading
terms in the perturbation Hamiltonian that give rise to nonlinear detuning
are
∆ ˜
H 1 = h
(0)
22000 J
2
x + h
(0)
11110 J x J y + h
(0)
00220 J
2
y .
(2.95)
The tune shifts with amplitude from these terms can be characterized by the
following coefficients.
∂ν x
∂J x
= 2h
(0)
22000 ,
∂ν x
∂J y
=
∂ν y
∂J x
= h
(0)
11110 ,
∂ν y
∂J y
= 2h
(0)
00220 .
(2.96)
There are also higher order terms that cause tune shifts dependence on the
action variables to higher orders.
The terms in Eq. (2.91) that have either j = k or l = m or both depend on
the angle coordinates φ x or φ y . Most of these terms have little impact over the
motion of the particles because they oscillate quickly with time and hence the
average effect is negligible. However, for some terms, the phase factor may be
slowly varying. The effect from these terms can build up and fundamentally
change the behavior of particle motion. These terms satisfy the resonance
condition
(j − k)
dφ x
dθ
+ (l − m)
dφ y
dθ
+ p = (j − k)ν x + (l − m)ν y + p,
= n 1 ν x + n 2 ν y + p ≈ 0,
(2.97)
where n 1 = j − k and n 2 = l − m. The order of the resonance is defined as
|n 1 | + |n 2 |. Beam motion around nonlinear resonances can become unstable,
as the resonances can drive the particles to large oscillation amplitudes and
cause beam loss. To alleviate the impact of the nonlinear resonances, it is
desirable to minimize the strengths of the resonance harmonics, h
(p)
jklmn .
Periodicity in a lattice can automatically set many systematic (i.e., inherent in the design) resonance terms to zero. If a ring consists of N repetitive,
identical cells, the resonance harmonics reduce to
h
(p)
jklm =
1
2π
cell
h jklm e
−jpθ dθ
N −1
q=0
e
−i2πq
p
N ,
=
1
2π
cell
h jklm e
−jpθ dθ
e
−iπp
N −1
N
sin pπ
sin(pπ/N )
,
(2.98)