Beam dynamics topics 49
satisfied and, correspondingly, Abs(||C||) r
2 ≈ 1. However, for the next generation storage ring light sources with the natural emittance at the diffraction
limit, it is desirable to operate near or on the linear difference resonance. In
this case, |ν x −ν y −p| ≤ |G − |, and hence the horizontal and vertical emittances
are nearly equal. The beam in this condition is called a round beam.
2.5 CHROMATIC EFFECT
As discussed in Section 2.3, the energy dependence of the bending angles by
dipole fields gives rise to dispersion, the orbit dependence on beam energy. As
the multipole field one order higher than the dipole, quadrupole fields determine the linear optics of the lattice. The focusing strength of a quadrupole
also depends on the beam energy. The dependence of the linear optics on the
beam energy is called the chromatic effect.
From Eq. (1.36) we see that the focusing gradient for an off-momentum
particle is K =
b1
1+δ , where b 1 =
1
Bρ
∂By
∂x is the focusing gradient for the reference particle, and δ is the momentum deviation of the off-momentum particle.
Therefore, the transfer matrix of the quadrupole for the off-momentum particle is different from that of the reference particle. In a circular accelerator, the
one-turn transfer matrix depends on the momentum deviation, which means
the betatron tunes, betatron phase advances, and Courant-Snyder parameters
all depend on the beam energy.
The focusing error of a quadrupole for an off-momentum particle is
∆K x = −(b 1 + 2h
2 )δ ≈ −K x δ, ∆K y = b 1 δ = −K y δ,
(2.75)
where K x = b 1 + h
2 , K y = −b 1 are the horizontal and vertical focusing functions, respectively, and h is the curvature of the reference orbit. The focusing
errors for off-momentum particles affect the linear optics of the off-momentum
particles in the same manner as the quadrupole errors we previously studied.
For example, the betatron tune of a ring would be changed by
∆ν ≈
1
4π
β(−Kδ)ds,
(2.76)
where −Kδ is the error of the focusing function for a particle with momentum deviation δ. The derivative of the momentum dependent tune shift with
respect to the momentum deviation is called the chromaticity,
C ≡
dν
dδ
.
(2.77)
The uncorrected chromaticity is called natural chromaticity,
C nat ≈ −
1
4π
Kβds.
(2.78)
The natural chromaticity is a negative quantity because a particle with higher
momentum receives less focusing by the quadrupoles.
satisfied and, correspondingly, Abs(||C||) r
2 ≈ 1. However, for the next generation storage ring light sources with the natural emittance at the diffraction
limit, it is desirable to operate near or on the linear difference resonance. In
this case, |ν x −ν y −p| ≤ |G − |, and hence the horizontal and vertical emittances
are nearly equal. The beam in this condition is called a round beam.
2.5 CHROMATIC EFFECT
As discussed in Section 2.3, the energy dependence of the bending angles by
dipole fields gives rise to dispersion, the orbit dependence on beam energy. As
the multipole field one order higher than the dipole, quadrupole fields determine the linear optics of the lattice. The focusing strength of a quadrupole
also depends on the beam energy. The dependence of the linear optics on the
beam energy is called the chromatic effect.
From Eq. (1.36) we see that the focusing gradient for an off-momentum
particle is K =
b1
1+δ , where b 1 =
1
Bρ
∂By
∂x is the focusing gradient for the reference particle, and δ is the momentum deviation of the off-momentum particle.
Therefore, the transfer matrix of the quadrupole for the off-momentum particle is different from that of the reference particle. In a circular accelerator, the
one-turn transfer matrix depends on the momentum deviation, which means
the betatron tunes, betatron phase advances, and Courant-Snyder parameters
all depend on the beam energy.
The focusing error of a quadrupole for an off-momentum particle is
∆K x = −(b 1 + 2h
2 )δ ≈ −K x δ, ∆K y = b 1 δ = −K y δ,
(2.75)
where K x = b 1 + h
2 , K y = −b 1 are the horizontal and vertical focusing functions, respectively, and h is the curvature of the reference orbit. The focusing
errors for off-momentum particles affect the linear optics of the off-momentum
particles in the same manner as the quadrupole errors we previously studied.
For example, the betatron tune of a ring would be changed by
∆ν ≈
1
4π
β(−Kδ)ds,
(2.76)
where −Kδ is the error of the focusing function for a particle with momentum deviation δ. The derivative of the momentum dependent tune shift with
respect to the momentum deviation is called the chromaticity,
C ≡
dν
dδ
.
(2.77)
The uncorrected chromaticity is called natural chromaticity,
C nat ≈ −
1
4π
Kβds.
(2.78)
The natural chromaticity is a negative quantity because a particle with higher
momentum receives less focusing by the quadrupoles.