Beam dynamics topics 41
where C is the circumference. Applying the periodic condition to Eq. (2.34),
with M being the one-turn transfer matrix and d the derivative of the orbit
shift with respect to δ, the dispersion vector is found to be
D = (I − M)
−1 d.
(2.37)
The periodic dispersion function in circular accelerators is essentially the
derivative of the closed orbit for an off-momentum particle with respect to the
momentum deviation, i.e.,
D =
dX c (δ)
dδ
,
(2.38)
where the closed orbit satisfies the periodic condition
X c (δ) = MX c (δ) + δd.
(2.39)
The calculation of d can be done by sequentially applying Eq. (2.31) to all
elements of the ring. This calculation can be facilitated by using the extended
transfer matrix for the X e = (x, x
, δ)
T coordinates, for which Eq. (2.31) can
be rewritten as
X e = M e X e,0 , with M e =
M d
0 1
.
(2.40)
The extended transfer matrix of an accelerator section can be obtained by
concatenating the matrices of the individual elements,
M e = M e,n M e,n−1 · · · M e,2 M e,1 .
(2.41)
From Eq. (2.34), we see that the dispersion vector transports in the same
manner as phase space coordinates, except there are perturbations if the accelerator section contains bending fields. This is understandable because the
dispersion function is a part of the phase space coordinates. In fact, the dispersion function can be seen as orbit errors due to distributed dipole field
errors 1/ρ(s). Outside of dipole magnets, there are no orbit error sources and
hence the transportation of the dispersion follows D = MD 0 , which is identical to the transportation of betatron coordinates. Therefore, there exists an
invariant of motion similar to J in Eq. (1.63), which is defined by
H =
1
β
D
2 + (αD + βD
)
2
.
(2.42)
H is called the dispersion invariant, which is constant in regions without dipole
fields. Figure 2.4 shows the dispersion function and the dispersion invariant
in the SPEAR3 storage ring.
In a storage ring with a periodic lattice structure, the design dispersion
function is also periodic. However, if the bending fields or the linear optics have
errors, the dispersion function will deviate from the design. Like the closed
where C is the circumference. Applying the periodic condition to Eq. (2.34),
with M being the one-turn transfer matrix and d the derivative of the orbit
shift with respect to δ, the dispersion vector is found to be
D = (I − M)
−1 d.
(2.37)
The periodic dispersion function in circular accelerators is essentially the
derivative of the closed orbit for an off-momentum particle with respect to the
momentum deviation, i.e.,
D =
dX c (δ)
dδ
,
(2.38)
where the closed orbit satisfies the periodic condition
X c (δ) = MX c (δ) + δd.
(2.39)
The calculation of d can be done by sequentially applying Eq. (2.31) to all
elements of the ring. This calculation can be facilitated by using the extended
transfer matrix for the X e = (x, x
, δ)
T coordinates, for which Eq. (2.31) can
be rewritten as
X e = M e X e,0 , with M e =
M d
0 1
.
(2.40)
The extended transfer matrix of an accelerator section can be obtained by
concatenating the matrices of the individual elements,
M e = M e,n M e,n−1 · · · M e,2 M e,1 .
(2.41)
From Eq. (2.34), we see that the dispersion vector transports in the same
manner as phase space coordinates, except there are perturbations if the accelerator section contains bending fields. This is understandable because the
dispersion function is a part of the phase space coordinates. In fact, the dispersion function can be seen as orbit errors due to distributed dipole field
errors 1/ρ(s). Outside of dipole magnets, there are no orbit error sources and
hence the transportation of the dispersion follows D = MD 0 , which is identical to the transportation of betatron coordinates. Therefore, there exists an
invariant of motion similar to J in Eq. (1.63), which is defined by
H =
1
β
D
2 + (αD + βD
)
2
.
(2.42)
H is called the dispersion invariant, which is constant in regions without dipole
fields. Figure 2.4 shows the dispersion function and the dispersion invariant
in the SPEAR3 storage ring.
In a storage ring with a periodic lattice structure, the design dispersion
function is also periodic. However, if the bending fields or the linear optics have
errors, the dispersion function will deviate from the design. Like the closed