42
E. Wilson and B. J. Holzer
Fig. 2.20 The extra inward
force given to a low
momentum particle by the
dipoles is balanced by the
focusing the quadrupoles and
defines a dispersion function
Fig. 2.21 The beam cross
sections in real space for
beams of three different
momenta at a point where the
dispersion function is large
displacement of the closed orbit is:
x(s) = D(s)
Δp
p 0
.
(2.55)
In Fig. 2.21 we see how the effect of dispersion for off momentum orbits adds to
the betatron motion to widen the beam cross section. The betatron motion of each
of the three particles: < 0, = 0, and > 0, is within an ellipse in
physical (x, z) space. The ellipses for each momentum are separated by a distance
D(s) The semi-aperture required will be:
a V =
β V ε V , a H =
β H ε H + D(s)
Δp
p
.
(2.56)
2.4.2 Chromaticity
This effect is equivalent to the chromatic aberration in a lens. It is defined as a
quantity Q :
ΔQ = Q
Δp
p
.
(2.57)
E. Wilson and B. J. Holzer
Fig. 2.20 The extra inward
force given to a low
momentum particle by the
dipoles is balanced by the
focusing the quadrupoles and
defines a dispersion function
Fig. 2.21 The beam cross
sections in real space for
beams of three different
momenta at a point where the
dispersion function is large
displacement of the closed orbit is:
x(s) = D(s)
Δp
p 0
.
(2.55)
In Fig. 2.21 we see how the effect of dispersion for off momentum orbits adds to
the betatron motion to widen the beam cross section. The betatron motion of each
of the three particles: < 0, = 0, and > 0, is within an ellipse in
physical (x, z) space. The ellipses for each momentum are separated by a distance
D(s) The semi-aperture required will be:
a V =
β V ε V , a H =
β H ε H + D(s)
Δp
p
.
(2.56)
2.4.2 Chromaticity
This effect is equivalent to the chromatic aberration in a lens. It is defined as a
quantity Q :
ΔQ = Q
Δp
p
.
(2.57)