synchrotron radiation 45
3.1.2 Cooling time
Knowledge of the SR-caused power losses immediately allow
us to estimate the SR cooling time of the beam in a storage
ring.
First of all, let’s — for convenience’s sake — rewrite the
exact formula for power lost per unit length
2 γ 4
dW 2 e
=
ds
3 R 2
in the way that does not depend on the systems of units:
dW 2 r e γ 4
=
mc
2
(3.7)
ds
3 R 2
The SR energy loss of a particle per turn is therefore:
4π r e γ 4
U 0 =
mc
2
(3.8)
3 R
The effect of particle cooling due to SR is based on the fact
that when an electron radiates a photon, its momentum decreases. Taking into account that while the beam of particles
can have a range of angles within the beam, the accelerating
RF cavity would restore only the longitudinal part of momentum, whereas the transverse degrees of freedom of the particles will be cooled down as illustrated in Fig. 3.3.
FIGURE 3.3
RF cavity restores only longitudinal momentum, thus other degrees of freedom are cooled due to synchrotron radiation.
Let’s estimate the cooling time τ of a particle with energy
E 0 cycling with revolution period T 0 in a circular machine as
τ = E 0 T 0 /U 0 , which yields
2π R γmc 2
τ ≈
(3.9)
c
U 0
After substitution, the inverse cooling time can be written as
2 c r e γ 3
τ
−1
≈
(3.10)
3 R 2