6 Design and Principles of Synchrotrons and Circular Colliders
273
There is an optimum value of g for which Eq. (6.74) has a maximum. As to the
other parameters, N and Z are properties of the beam, W is a property of the cooling
system and M, ˜
M and U depend on the interplay of cooling system-, beam- and
storage ring characteristics. The term in the bracket can at best be 1 but is more like
1/10 to 1/100 in real systems, depending on how well the mixing and noise problems
are solved. The ideal cooling rate W/N can be interpreted as the maximum rate at
which information on single particles can be acquired. Note that the gain parameter
g (fractional sample error correction) should not be confounded with the electronic
gain of cooling system which is typically 120 db or 12 orders of magnitude in power.
Lattice parameters are especially important for the achievement of ‘good’ values
of M, ˜
M and U, maximising the bracket in Eq. (6.74). In addition to the struggle for
large bandwidth, the advance in stochastic cooling is intimately linked to progress in
dealing with the noise and mixing factors. In summary it can be said that present-day
systems are working with a bandwidth of around 1 GHz for an individual cooling
system with the possibility of extensions up to nearly 10 GHz by using several
cooling bands in the same ring. Limitations on W are discussed in [106].
Turning to the mixing dilemma discussed at length in [107], we note that
stochastic cooling only works if after each correction the samples (at least partly) rerandomise (desired mixing), and at the same time a particle on its way from pick-up
to kicker does not slip too much with respect to its own signal (undesired mixing).
The mixing rates 1/M and 1/ ˜
M are related to the fraction of the sample length
by which a particle with the typical momentum deviation slips with respect to the
nominal particle. Here M refers to the way from kicker to pick-up (‘K to P’), and ˜
M
to the way pick-up to kicker (‘P to K’). Both depend on the flight-time dispersion
which in turn is given by the local ‘off-momentum factors’,
η kp =
dT
T
/
dp
p
kp
,
(6.75)
and the similar quantity η pk respectively. For a regular lattice the beam paths ‘K to
P’ and ‘P to K’ consist of a number of identical cells and one has
η kp ≈ η pk ≈ η =
γ
−2
tr − γ
−2
,
(6.76)
i.e. the local η-factors are close to the off-momentum factor for the whole ring. In
this situation the ratio ˜
M/M is simply given by the corresponding path lengths (T pk
and T kp ). Then, e.g. in the case of the CERN AD (antiproton decelerator) where the
cooling loop cuts diagonally across the ring, one has ˜
M ≈ M instead of the desired
˜
M 1, M = 1. The usual compromise is to accept imperfect mixing, letting both
˜
M and M be in the range of 3–5, say. The price to pay is a slower cooling rate, for
273
There is an optimum value of g for which Eq. (6.74) has a maximum. As to the
other parameters, N and Z are properties of the beam, W is a property of the cooling
system and M, ˜
M and U depend on the interplay of cooling system-, beam- and
storage ring characteristics. The term in the bracket can at best be 1 but is more like
1/10 to 1/100 in real systems, depending on how well the mixing and noise problems
are solved. The ideal cooling rate W/N can be interpreted as the maximum rate at
which information on single particles can be acquired. Note that the gain parameter
g (fractional sample error correction) should not be confounded with the electronic
gain of cooling system which is typically 120 db or 12 orders of magnitude in power.
Lattice parameters are especially important for the achievement of ‘good’ values
of M, ˜
M and U, maximising the bracket in Eq. (6.74). In addition to the struggle for
large bandwidth, the advance in stochastic cooling is intimately linked to progress in
dealing with the noise and mixing factors. In summary it can be said that present-day
systems are working with a bandwidth of around 1 GHz for an individual cooling
system with the possibility of extensions up to nearly 10 GHz by using several
cooling bands in the same ring. Limitations on W are discussed in [106].
Turning to the mixing dilemma discussed at length in [107], we note that
stochastic cooling only works if after each correction the samples (at least partly) rerandomise (desired mixing), and at the same time a particle on its way from pick-up
to kicker does not slip too much with respect to its own signal (undesired mixing).
The mixing rates 1/M and 1/ ˜
M are related to the fraction of the sample length
by which a particle with the typical momentum deviation slips with respect to the
nominal particle. Here M refers to the way from kicker to pick-up (‘K to P’), and ˜
M
to the way pick-up to kicker (‘P to K’). Both depend on the flight-time dispersion
which in turn is given by the local ‘off-momentum factors’,
η kp =
dT
T
/
dp
p
kp
,
(6.75)
and the similar quantity η pk respectively. For a regular lattice the beam paths ‘K to
P’ and ‘P to K’ consist of a number of identical cells and one has
η kp ≈ η pk ≈ η =
γ
−2
tr − γ
−2
,
(6.76)
i.e. the local η-factors are close to the off-momentum factor for the whole ring. In
this situation the ratio ˜
M/M is simply given by the corresponding path lengths (T pk
and T kp ). Then, e.g. in the case of the CERN AD (antiproton decelerator) where the
cooling loop cuts diagonally across the ring, one has ˜
M ≈ M instead of the desired
˜
M 1, M = 1. The usual compromise is to accept imperfect mixing, letting both
˜
M and M be in the range of 3–5, say. The price to pay is a slower cooling rate, for