3
Synchrotron Radiation
3.1 SR on the back of
In this chapter we will consider one of the most important
an envelope
43 phenomena that governs the behavior of accelerators — syn3.2 SR effects on the
chrotron radiation (SR).
beam
48
SR can be both helpful, as it yields the creation of high
3.3 SR features
51 brightness radiation sources, and harmful, as it can deteriorate the beam by creating additional energy spread and beam
emittance growth.
Traditional derivations of SR equations are rather mathematically involved. However, in this chapter we will use simplified back-of-the-envelope style derivations, which nevertheless obtain all of the important characteristics of SR with
high accuracy.
3.1 SR on the back of an envelope
In our simple picture, the SR is the result of the charged particle leaving part of its fields behind when it is moving on a
curved trajectory. The part of the field that is left behind (or
radiated) cannot catch up with the motion of the particle, as
it cannot move faster than the speed of light.
Armed with this concept, let’s estimate the power loss due
to SR, the typical energy of the emitted photons, and other
parameters of synchrotron radiation as well as the most important effects that SR inflicts on the beam.
3.1.1 SR power loss
The straightforward concept of SR described above is represented in Fig. 3.1. In this instance, the particle moving with
velocity v (which is close to the speed of light) on a radius R
has its field lines pointing mostly transversely, and the part
of field moving further away (on the radius R + r) would be
left behind, as it cannot move faster than c.
The radius r can be evaluated as
c
R
r = R − 1 ≈
(3.1)
v
2γ 2
where we assumed that γ » 1 or β ≈ 1 and thus (1 − v/c) =
(1 − β)(1 + β)/(1 + β) ≈ (1 − β 2 )/2 = 1/(2γ 2 ).
43
DOI: 10.1201/b18696-3
Synchrotron Radiation
3.1 SR on the back of
In this chapter we will consider one of the most important
an envelope
43 phenomena that governs the behavior of accelerators — syn3.2 SR effects on the
chrotron radiation (SR).
beam
48
SR can be both helpful, as it yields the creation of high
3.3 SR features
51 brightness radiation sources, and harmful, as it can deteriorate the beam by creating additional energy spread and beam
emittance growth.
Traditional derivations of SR equations are rather mathematically involved. However, in this chapter we will use simplified back-of-the-envelope style derivations, which nevertheless obtain all of the important characteristics of SR with
high accuracy.
3.1 SR on the back of an envelope
In our simple picture, the SR is the result of the charged particle leaving part of its fields behind when it is moving on a
curved trajectory. The part of the field that is left behind (or
radiated) cannot catch up with the motion of the particle, as
it cannot move faster than the speed of light.
Armed with this concept, let’s estimate the power loss due
to SR, the typical energy of the emitted photons, and other
parameters of synchrotron radiation as well as the most important effects that SR inflicts on the beam.
3.1.1 SR power loss
The straightforward concept of SR described above is represented in Fig. 3.1. In this instance, the particle moving with
velocity v (which is close to the speed of light) on a radius R
has its field lines pointing mostly transversely, and the part
of field moving further away (on the radius R + r) would be
left behind, as it cannot move faster than c.
The radius r can be evaluated as
c
R
r = R − 1 ≈
(3.1)
v
2γ 2
where we assumed that γ » 1 or β ≈ 1 and thus (1 − v/c) =
(1 − β)(1 + β)/(1 + β) ≈ (1 − β 2 )/2 = 1/(2γ 2 ).
43
DOI: 10.1201/b18696-3