46 unifying physics of accelerators, lasers and plasma
3.1.3 Cooling time and partition
In the previous section we estimated the inverse cooling time
as τ −1 ≈ 2 c r e γ 3 /(3R 2 ). Traditionally, there is a factor of 2 in
the definition in the cooling time:
τ
−1
1 c r e γ 3
τ = 2E 0 T 0 /U 0 ⇒
=
(3.11)
3 R 2
We will use this latter definition in this section.
We can express the evolution of the beam emittance under
the influence of an SR damping as
ε(t) = ε 0 exp( − 2 t /τ )
(3.12)
Both transverse planes, as well as the longitudinal motion
in rings, are usually coupled. Thus we can expect that the
damping will be distributed between these degrees of freedom in some proportion depending on details of the optics.
Distribution of cooling between the degrees of freedom
is defined by the so-called partition numbers J x , J y and J E ,
which we mention here without derivations. The cooling time
of a degree of freedom is correspondingly
τ
τ i =
(3.13)
J i
The total radiated power due to SR is fixed and constant,
therefore
τ
−1 = const.
(3.14)
i
which corresponds to the partition theorem
J i = 4
(3.15)
for a typical accelerator
J x ≈ 1 , J y ≈ 1 , J E ≈ 2
(3.16)
and adjusting the optics of the machine changes the distribution of the partition numbers.
3.1.4 SR photon energy
For γ » 1 the emitted pho- In order to estimate the typical energy of the SR photons,
tons go into 1/γ cone.
we need to make an assumption that is based on relativistic
kinematics: the radiation of relativistic particles is emitted
into a cone with angular spread of 1/γ.
Let’s take this assumption into account when examining
the radiation emitted during motion along the curved trajectory shown in Fig. 3.4 and ask a question — during what time
interval Δt would the remote observer see the emitted fields?
3.1.3 Cooling time and partition
In the previous section we estimated the inverse cooling time
as τ −1 ≈ 2 c r e γ 3 /(3R 2 ). Traditionally, there is a factor of 2 in
the definition in the cooling time:
τ
−1
1 c r e γ 3
τ = 2E 0 T 0 /U 0 ⇒
=
(3.11)
3 R 2
We will use this latter definition in this section.
We can express the evolution of the beam emittance under
the influence of an SR damping as
ε(t) = ε 0 exp( − 2 t /τ )
(3.12)
Both transverse planes, as well as the longitudinal motion
in rings, are usually coupled. Thus we can expect that the
damping will be distributed between these degrees of freedom in some proportion depending on details of the optics.
Distribution of cooling between the degrees of freedom
is defined by the so-called partition numbers J x , J y and J E ,
which we mention here without derivations. The cooling time
of a degree of freedom is correspondingly
τ
τ i =
(3.13)
J i
The total radiated power due to SR is fixed and constant,
therefore
τ
−1 = const.
(3.14)
i
which corresponds to the partition theorem
J i = 4
(3.15)
for a typical accelerator
J x ≈ 1 , J y ≈ 1 , J E ≈ 2
(3.16)
and adjusting the optics of the machine changes the distribution of the partition numbers.
3.1.4 SR photon energy
For γ » 1 the emitted pho- In order to estimate the typical energy of the SR photons,
tons go into 1/γ cone.
we need to make an assumption that is based on relativistic
kinematics: the radiation of relativistic particles is emitted
into a cone with angular spread of 1/γ.
Let’s take this assumption into account when examining
the radiation emitted during motion along the curved trajectory shown in Fig. 3.4 and ask a question — during what time
interval Δt would the remote observer see the emitted fields?