Synchrotron Motion
251
k s = b 2 /Bρ. For a thin slice of it, the effect is Δa = −k s dsx
2 . From eq. (11.1),
we obtain the change of the periodic solution of the second order dispersion
function at the location of the sextupole slice, which is
ΔD 2
ΔD
2
= −
k s dsD
2
2 sin (πν)
β cos (πν)
sin (πν) − α cos (πν)
,
where ν = θ/2π. Plugging it into eq. (10.6) and using eqs. (10.7) and (10.8),
we obtain
Δη
ph
2 (s) =
1
C
[(l|x)ΔD 2 + (l|a)ΔD
2 ]
= −
1
C
k s dsD
2
2 sin (πν)
{[(a|x)D + (1 − (x|x)) D
] β cos (πν)
− [(1 − (a|a)) D + (x|a)D
] (sin (πν) − α cos (πν))} .
After straightforward algebraic and trigonometric manipulations, we arrive at
a simple result, which is
Δη
ph
2 (s) =
1
C
k s D
3 ds,
and the total change of the second order phase slip factor is
Δη
ph
2 =
1
C
C
0
k s (s) D
3 ds.
Using the same method, we can easily obtain the result where both horizontal
and vertical dispersion are present but coupling is corrected. Here the kick of
the sextupole slice is
Δa
Δb
= −k s ds
x
2
− y
2
−2xy
,
and the change of the periodic solution of the second order dispersion function
at the location of the sextupole slice is
ΔD x2
ΔD
x2
= −
k s ds
D
2
x − D
2
y
2 sin (πν x )
β x cos (πν x )
sin (πν x ) − α x cos (πν x )
,
ΔD y2
ΔD
y2
=
k s dsD x D y
sin (πν y )
β y cos (πν y )
sin (πν y ) − α y cos (πν y )
.
Plugging it into the 4D version of eq. (10.6), using eqs. (10.7) and (10.8) and