84
W. Herr and E. Forest
If the map for h eff corresponds to a one-turn-map, we can write for the tunes:
Q x (J x , J y , δ) =
1
2π
∂h eff
∂J x
(3.114)
Q y (J x , J y , δ) =
1
2π
∂h eff
∂J y
(3.115)
and the change of path length:
s = −
∂h eff
∂δ
= α c δ
(3.116)
In the non-linear case, particles with different J x , J y , δ have different tunes. Their
dependence on J x , J y is the amplitude detuning, the dependence on δ are the
chromaticities.
The effective Hamiltonian can always be written (here to 3rd order) in a form:
h eff = + μ x J x + μ y J y +
1
2
α c δ
2
(3.117)
+ c x1 J x δ + c y1 J y δ + c 3 δ
3
(3.118)
+ c xx J
2
x + c xy J x J y + c yy J
2
y + c x2 J x δ
2
+ c y2 J y δ
2
+ c 4 δ
4
(3.119)
and then tune depends on action J and momentum deviation δ:
Q x (J x , J y , δ) =
1
2π
∂h eff
∂J x
=
1
2π
⎛
⎜
⎝μx +
detuning
2c xx J x + c xy J y +
chromat icity
c x1 δ + c x2 δ
2
⎞
⎟
⎠
(3.120)
Q y (J x , J y , δ) =
1
2π
∂h eff
∂J y
=
1
2π
⎛
⎜
⎝μy +
detuning
2c yy J y + c xy J x +
chromat icity
c y1 δ + c y2 δ
2
⎞
⎟
⎠
(3.121)
The meaning of the different contributions are:
– μ x , μ y : linear phase advance or 2π· i.e. the tunes for rings
–
1
2 α c , c 3 , c 4 : linear and nonlinear “momentum compaction”
– c x1 , c y1 : first order chromaticities
– c x2 , c y2 : second order chromaticities