2 Beam Dynamics
43
The chromaticity [12] arises because the focusing strength of a quadrupole has
(Bρ) in the denominator and is therefore inversely proportional to momentum:
k =
1
(Bρ)
dB z
dx
.
(2.58)
A small spread in momentum in the beam, ± causes a spread in focusing
strength:
Δk
k
= ∓
Δp
p
.
(2.59)
Integrated over all focusing (and defocusing) elements in the ring, we obtain a
change in the tune of the machine
ΔQ =
1
4π
β(s)δk(s)ds.
(2.60)
This enables us to calculate Q :
ΔQ =
1
4π
β(s)δk(s)ds =
−1
4π
β(s)k(s)ds
Δp
p
.
(2.61)
The quantity in square brackets is the chromaticity Q . To be clear, this is
called the natural chromaticity. For most alternating gradient machines, its value
is about −1.3Q. Of course there are two Q values relating to horizontal and vertical
oscillations and therefore two chromaticities. Chromaticity may be corrected with
sextupole magnets (see Chap. 3, and Sects. 6.1 and 8.1).
2.5 Longitudinal Motion
2.5.1 Stability of the Lagging Particle
Suppose two particles are well below the velocity of light. A particle A, that arrives
at the right moment to in the RF resonator and thus will obtain exactly the right
acceleration voltage. We call this particle “synchronous” (see Fig. 2.22). A second
particle, B, arrives late, and so receives an extra energy increment which will cause it
to speed up and overtake the synchronous particle, A. In so doing, its energy defect,
grows and, provided the amplitude is not too large, its trajectory will follow an
ellipse in phase space. This describes this motion up and down the r.f. wave (Fig.
2.22) and may remind some readers of the representation of a simple harmonic
oscillator, or pendulum. When plotted in a phase space diagram of velocity versus
longitudinal displacement we indeed obtain a shape that is elliptical. The trajectory
43
The chromaticity [12] arises because the focusing strength of a quadrupole has
(Bρ) in the denominator and is therefore inversely proportional to momentum:
k =
1
(Bρ)
dB z
dx
.
(2.58)
A small spread in momentum in the beam, ± causes a spread in focusing
strength:
Δk
k
= ∓
Δp
p
.
(2.59)
Integrated over all focusing (and defocusing) elements in the ring, we obtain a
change in the tune of the machine
ΔQ =
1
4π
β(s)δk(s)ds.
(2.60)
This enables us to calculate Q :
ΔQ =
1
4π
β(s)δk(s)ds =
−1
4π
β(s)k(s)ds
Δp
p
.
(2.61)
The quantity in square brackets is the chromaticity Q . To be clear, this is
called the natural chromaticity. For most alternating gradient machines, its value
is about −1.3Q. Of course there are two Q values relating to horizontal and vertical
oscillations and therefore two chromaticities. Chromaticity may be corrected with
sextupole magnets (see Chap. 3, and Sects. 6.1 and 8.1).
2.5 Longitudinal Motion
2.5.1 Stability of the Lagging Particle
Suppose two particles are well below the velocity of light. A particle A, that arrives
at the right moment to in the RF resonator and thus will obtain exactly the right
acceleration voltage. We call this particle “synchronous” (see Fig. 2.22). A second
particle, B, arrives late, and so receives an extra energy increment which will cause it
to speed up and overtake the synchronous particle, A. In so doing, its energy defect,
grows and, provided the amplitude is not too large, its trajectory will follow an
ellipse in phase space. This describes this motion up and down the r.f. wave (Fig.
2.22) and may remind some readers of the representation of a simple harmonic
oscillator, or pendulum. When plotted in a phase space diagram of velocity versus
longitudinal displacement we indeed obtain a shape that is elliptical. The trajectory