44 unifying physics of accelerators, lasers and plasma
)LHOGOHIW
EHKLQG
R R + r
)LHOGOLQHV
FIGURE 3.1
Synchrotron radiation — conceptual explanation.
The energy in the field that is left behind (radiated) can be
estimated (we use Gaussian units in this section) as a volume
integral of the field squared:
W ≈
2
E dV
(3.2)
The field E can be estimated as the field on the radius r from
the particle
e
E ≈ r 2
and the characteristic volume can be estimated (see Fig. 3.2)
as
≈
2
V r ds
(3.3)
where ds is an element of the path along the orbit, as we will
FIGURE 3.2
Illustrating the characteristic
volume used in Eq. 3.3.
eventually wish to find the power lost per unit of length.
The energy loss per unit length can thus be written as:
dW
2
≈
2 2
e
≈
2
E r
r
(3.4)
ds
r 2
and after substituting
R
r ≈ 2γ 2
we get an estimate of:
2 γ 4
dW e
≈
(3.5)
ds
R 2
which we can compare with the exact formula:
2 γ 4
dW 2 e
=
(3.6)
ds
3 R 2
and conclude that our rough estimations give a very reasonable result.