42 Beam-based Correction and Optimization for Accelerators
D (m)
0
0.2
0.4
s (m)
0
50
100
150
200
H (m)
0
0.01
0.02
0.03
Figure 2.4 The dispersion function (top) and the dispersion invariant (bottom) in
the SPEAR3 storage ring.
orbit errors, the errors to the dispersion function are typically dominated by
the Fourier harmonics close to the betatron tune.
In a flat accelerator, by design all bending occurs on the horizontal plane.
Hence there is no vertical dispersion in the ideal scenario. However, vertical
orbit correctors and feed-down effects due to orbit offsets in quadrupole and
sextupole magnets give the beam small vertical orbit kicks, which generate
vertical dispersion. The horizontal dispersion is also coupled to the vertical
plane through skew quadrupoles located at dispersive regions or sextupoles
with both vertical orbit offsets and horizontal dispersion. The vertical dispersion generated due to these unintended sources is called the spurious vertical
dispersion. In electron storage rings, the spurious vertical dispersion increases
the vertical beam emittance.
The path length traveled by an off-momentum particle is closely related
to the horizontal dispersion function. The path length over an infinitesimal
distance along the reference orbit ds is given by
dl =
(1 + hx) 2 + x 2 + y 2 ds ≈ (1 + hx)ds,
(2.43)
where h =
1
ρ is the curvature of the reference orbit. For an off-momentum
particle with momentum deviation δ, from Eq. (2.32), we have x = x β +
Dδ. The path length difference between an off-momentum particle and the
reference particle is
∆C =
(dl − ds) ≈
hDδds,
(2.44)