!
FIGURE 2.20
Chromaticity of a focusing
quadrupole.
38 unifying physics of accelerators, lasers and plasma
We can also compute the so-called momentum compaction factor with
dC/C 1
D(s)
α c =
=
ds
(2.52)
dp/p
C
ρ(s)
which will be discussed further in Chapter 5, which deals
with longitudinal dynamics.
2.6.2 Betatron tunes and resonances
Taking into account the definition of the betatron phase in
Eq. 2.23, we can write the phase advance over one turn of a
circular machine as
ds
Δφ C = 2π Q =
(2.53)
β(s)
The quantity Q is called the betatron tune. The betatron tunes
are essential quantities used to analyze the stability of a circular accelerator. In particular, if Q is an integer number, resonance conditions occur, as a tiny disturbance at some place
along the orbit would repeat for many turns, accumulating
into a large disruption of the particle’s motion.
The general equations for resonance conditions of the betatron tunes can be written as follows:
m Q x + n Q z = k
(2.54)
where m, n and k are integer numbers and where |m| + |n|
is called the order of the resonance. Resonances of the lowest
orders are the most dangerous for the stability of the particle
motion and thus must be carefully avoided by proper machine optics design.
2.7 Aberrations and coupling
2.7.1 Chromaticity
Offsets of energy in the particles cause not only dispersion
but also result in different focusing strengths of the magnetic
elements (as illustrated in Fig. 2.20).
The quadrupole strength for off-energy particles in the
first order can be approximated as follows
e ∂B y
e
∂B y
k 0
k 1 =
=
=
≈ k 0 (1 − δ)
(2.55)
p ∂x
p 0 (1 + δ) ∂x
1 + δ
Taking into account the equations that describe beta function evolution, we can show that the betatron tunes shift in
correlation to changes in focusing strength:
1
ΔQ =
β(s)ΔK(s)ds
(2.56)
4π
FIGURE 2.20
Chromaticity of a focusing
quadrupole.
38 unifying physics of accelerators, lasers and plasma
We can also compute the so-called momentum compaction factor with
dC/C 1
D(s)
α c =
=
ds
(2.52)
dp/p
C
ρ(s)
which will be discussed further in Chapter 5, which deals
with longitudinal dynamics.
2.6.2 Betatron tunes and resonances
Taking into account the definition of the betatron phase in
Eq. 2.23, we can write the phase advance over one turn of a
circular machine as
ds
Δφ C = 2π Q =
(2.53)
β(s)
The quantity Q is called the betatron tune. The betatron tunes
are essential quantities used to analyze the stability of a circular accelerator. In particular, if Q is an integer number, resonance conditions occur, as a tiny disturbance at some place
along the orbit would repeat for many turns, accumulating
into a large disruption of the particle’s motion.
The general equations for resonance conditions of the betatron tunes can be written as follows:
m Q x + n Q z = k
(2.54)
where m, n and k are integer numbers and where |m| + |n|
is called the order of the resonance. Resonances of the lowest
orders are the most dangerous for the stability of the particle
motion and thus must be carefully avoided by proper machine optics design.
2.7 Aberrations and coupling
2.7.1 Chromaticity
Offsets of energy in the particles cause not only dispersion
but also result in different focusing strengths of the magnetic
elements (as illustrated in Fig. 2.20).
The quadrupole strength for off-energy particles in the
first order can be approximated as follows
e ∂B y
e
∂B y
k 0
k 1 =
=
=
≈ k 0 (1 − δ)
(2.55)
p ∂x
p 0 (1 + δ) ∂x
1 + δ
Taking into account the equations that describe beta function evolution, we can show that the betatron tunes shift in
correlation to changes in focusing strength:
1
ΔQ =
β(s)ΔK(s)ds
(2.56)
4π