Beam dynamics topics 39
The betatron phase advance is closely related to the beta function via
dψ =
ds
β . The errors to the phase advance due to quadrupole errors can be
calculated from the beta beating, which gives
∆ψ = ∆
ds
β
= −
∆β
β 2 ds = −
∆β
β
ν 0 dφ,
=
J 0 ψ(s)
2ν 0
+
∞
n=1
1
n
ν
2
0 |J n |
ν 2
0 −
1
4 n 2 sin(nφ(s) + χ n ).
(2.30)
The n = 0 term gives the betatron tune shift, ∆ν =
J0
2 .
Linear optics errors have many detrimental effects and correction of the
optics errors is usually desired. By increasing the beta function at locations
of limiting apertures, beta beating may reduce the acceptance of the machine
and in turn cause a reduced tuning range, scrape off the beam on the vacuum
chamber, or reduce the injection efficiency. Deviations of beta functions from
the design may cause radiation protection issues as the beam loss distribution
in the machine is changed. In transport lines, errors of optics functions at the
extraction point cause a mismatch of the beam distribution with the optics of
the downstream accelerator and may reduce the injection efficiency. In a free
electron laser (FEL), optics mismatch in the undulator will reduce the FEL
power.
In a storage ring, optics errors may significantly impact the nonlinear beam
dynamics performance, resulting in a reduced dynamic aperture and local momentum apertures, which may in turn decrease the injection efficiency and the
Touschek lifetime, respectively. Typically the nonlinear beam dynamics of a
storage ring design makes use of a cancellation scheme of the nonlinear effects by the sextupoles. The cancellation scheme relies on the phase advances
between certain pairs of sextupoles being close to an odd multiple of π. As
optics errors distort the betatron phase advances, the cancellation scheme
does not work as expected. This may lead to increases of the nonlinear resonance strengths and changes to the tune footprint. The excitation of certain
nonlinear resonances may reduce the dynamic aperture or local momentum
apertures.
2.3 DISPERSION
Dipole fields, including the fields in dipole magnets and the feed-down dipole
components in quadrupoles and sextupoles, determine the beam orbit in an
accelerator. Because the bending angle of a dipole magnet for a beam is inversely proportional to the beam energy, the beam orbit depends on the beam
energy. The dependence of beam orbit on beam energy is called dispersion.
Eqs. (1.29) describes the particle motion through a dipole magnet, including the effect of energy errors. This form of equation can be extended to
describe the particle motion in a general accelerator section,
X = MX 0 + δd,
(2.31)
The betatron phase advance is closely related to the beta function via
dψ =
ds
β . The errors to the phase advance due to quadrupole errors can be
calculated from the beta beating, which gives
∆ψ = ∆
ds
β
= −
∆β
β 2 ds = −
∆β
β
ν 0 dφ,
=
J 0 ψ(s)
2ν 0
+
∞
n=1
1
n
ν
2
0 |J n |
ν 2
0 −
1
4 n 2 sin(nφ(s) + χ n ).
(2.30)
The n = 0 term gives the betatron tune shift, ∆ν =
J0
2 .
Linear optics errors have many detrimental effects and correction of the
optics errors is usually desired. By increasing the beta function at locations
of limiting apertures, beta beating may reduce the acceptance of the machine
and in turn cause a reduced tuning range, scrape off the beam on the vacuum
chamber, or reduce the injection efficiency. Deviations of beta functions from
the design may cause radiation protection issues as the beam loss distribution
in the machine is changed. In transport lines, errors of optics functions at the
extraction point cause a mismatch of the beam distribution with the optics of
the downstream accelerator and may reduce the injection efficiency. In a free
electron laser (FEL), optics mismatch in the undulator will reduce the FEL
power.
In a storage ring, optics errors may significantly impact the nonlinear beam
dynamics performance, resulting in a reduced dynamic aperture and local momentum apertures, which may in turn decrease the injection efficiency and the
Touschek lifetime, respectively. Typically the nonlinear beam dynamics of a
storage ring design makes use of a cancellation scheme of the nonlinear effects by the sextupoles. The cancellation scheme relies on the phase advances
between certain pairs of sextupoles being close to an odd multiple of π. As
optics errors distort the betatron phase advances, the cancellation scheme
does not work as expected. This may lead to increases of the nonlinear resonance strengths and changes to the tune footprint. The excitation of certain
nonlinear resonances may reduce the dynamic aperture or local momentum
apertures.
2.3 DISPERSION
Dipole fields, including the fields in dipole magnets and the feed-down dipole
components in quadrupoles and sextupoles, determine the beam orbit in an
accelerator. Because the bending angle of a dipole magnet for a beam is inversely proportional to the beam energy, the beam orbit depends on the beam
energy. The dependence of beam orbit on beam energy is called dispersion.
Eqs. (1.29) describes the particle motion through a dipole magnet, including the effect of energy errors. This form of equation can be extended to
describe the particle motion in a general accelerator section,
X = MX 0 + δd,
(2.31)