Beam dynamics topics 53
the perturbation Hamiltonian can be rewritten as
˜
H 1 = R
jklmn≥0
H jklmn (θ)z
j
1 ¯
z
k
1 z
l
2 ¯
z
m
2 δ
n ,
= R
H jklmn (θ)β
j+k
2
x
β
l+m
2
y
J
j+k
2
x
J
l+m
2
y
δ
n
e
i[(j−k)(ψx−νxθ)+(l−m)(ψy−νyθ)] e
i[(j−k)φx+(l−m)φy] ,
≡
h jklmn (θ)J
j+k
2
x
J
l+m
2
y
e
i[(j−k)φx+(l−m)φy] δ
n ,
(2.87)
where
h jklmn (θ) = RH jklmn (θ)β
j+k
2
x
β
l+m
2
y
e
i[(j−k)(ψx−νxθ)+(l−m)(ψy−νyθ)] . (2.88)
The Hamiltonian ˜
H 1 is said to be given in the resonance basis.
The functions h jklmn (θ) are periodic with respect to θ with the period of
2π and can be Fourier expanded,
h jklmn (θ) =
∞
p=−∞
h
(p)
jklmn e
ipθ ,
(2.89)
with the Fourier coefficients given by
h
(p)
jklmn =
1
2π
dθh jklmn (θ)e
−ipθ .
(2.90)
With the Fourier expansion, the perturbation Hamiltonian is now written as
˜
H 1 =
jklmn≥0
∞
p=−∞
h
(p)
jklmn J
j+k
2
x
J
l+m
2
y
e
i[(j−k)φx+(l−m)φy+pθ] δ
n .
(2.91)
The various terms in the perturbation Hamiltonian have different impact
on the beam motion. The terms with n > 0 affect off-momentum particles and
can be referred to as chromatic terms, while the n = 0 terms affect the onmomentum particle motion and are called geometric terms. The terms with
j = k and l = m are independent of the angle coordinates. Among these
terms, the ones with p = 0 cause the tunes to change, which can be seen from
the Hamilton’s equation
∆ν x =
∂ ˜
H 1
∂J x
,
∆ν y =
∂ ˜
H 1
∂J y
.
(2.92)
For example, the terms h
(0)
11000 and h
(0)
00110 correspond to betatron tune changes
due to quadrupole errors around the ring,
∆ν x = h
(0)
11000 ,
∆ν y = h
(0)
00110 .
(2.93)