The apparent acceleration and electric field in the laboratory frame are increased
by γ
4 , which is more than 10
12 even for a modest 0.5 GeV storage ring! To estimate
the amount of power radiated, we remember (Eq. 3.2) that the power radiated per
unit area is proportional to the square of the observed acceleration (and hence γ
8 !).
However, for the average power over time and space, we lose factors of 1/γ
2 for the
reduced solid angle and 1/γ
2 for the reduced time interval. Overall, for a fixed bend
radius ρ, the average power of the emitted radiation is still enhanced by a very
respectable factor of γ
4 compared to our non-relativistic case.
P / γ
4
ð3:6Þ
We can also use qualitative arguments to estimate the angular extent over which
SR is observed and the average energy of the photons emitted. The angular opening
is on the order θ 1/γ, because it is only for small angles that the time-squeezing
factor κ is small. We thus get most of the radiation as the electron passes through an
arc of 1/γ, which takes an apparent time on the order of Δt ~ 2ρ/γ
3 c (Fig. 3.3).
From Fourier transform theory, the average frequency in a pulse of duration Δt is
ω typ ¼ 1/Δt. If we use γ ¼ 6000 for our generic 3 GeV storage ring and a bend radius
of 10 m, this yields a typical angular frequency of ω typ ¼ 3.2 Â 10
18 s
À1 , for a photon
energy of 13 keV. This turns out to be a surprisingly good estimate for the peak in the
synchrotron radiation spectrum. The key result is that the median photon energy
scales as E
3 and inversely with the bend radius.
E typ /
γ
3 c
ρ
ð3:7Þ
For starters, this is all you really need to know about source properties, but if you
want to go deeper, read on.
3.3 Bend Magnet Radiation: The Details
As mentioned before, detailed calculation of the radiation from a bend magnet is not
for the faint of heart. Here the key equations are presented for reference, without
rigorous step-by-step derivation. For those who want more, the details are available
in excellent articles [67,68], in classic works such as the paper by Schwinger [7] or
the text by Jackson [69], as well as in more recent books [70].
Our coordinate system describes the observation position by horizontal angle ϕ
and vertical angle ψ, as shown in Fig. 3.4. We also need to consider the polarization,
which can have horizontal ε ˆ σ and vertical ε ˆ π components (Fig. 3.4).
We now compare some exact results with those derived qualitatively.
42
3 Synchrotron Radiation Fundamentals
by γ
4 , which is more than 10
12 even for a modest 0.5 GeV storage ring! To estimate
the amount of power radiated, we remember (Eq. 3.2) that the power radiated per
unit area is proportional to the square of the observed acceleration (and hence γ
8 !).
However, for the average power over time and space, we lose factors of 1/γ
2 for the
reduced solid angle and 1/γ
2 for the reduced time interval. Overall, for a fixed bend
radius ρ, the average power of the emitted radiation is still enhanced by a very
respectable factor of γ
4 compared to our non-relativistic case.
P / γ
4
ð3:6Þ
We can also use qualitative arguments to estimate the angular extent over which
SR is observed and the average energy of the photons emitted. The angular opening
is on the order θ 1/γ, because it is only for small angles that the time-squeezing
factor κ is small. We thus get most of the radiation as the electron passes through an
arc of 1/γ, which takes an apparent time on the order of Δt ~ 2ρ/γ
3 c (Fig. 3.3).
From Fourier transform theory, the average frequency in a pulse of duration Δt is
ω typ ¼ 1/Δt. If we use γ ¼ 6000 for our generic 3 GeV storage ring and a bend radius
of 10 m, this yields a typical angular frequency of ω typ ¼ 3.2 Â 10
18 s
À1 , for a photon
energy of 13 keV. This turns out to be a surprisingly good estimate for the peak in the
synchrotron radiation spectrum. The key result is that the median photon energy
scales as E
3 and inversely with the bend radius.
E typ /
γ
3 c
ρ
ð3:7Þ
For starters, this is all you really need to know about source properties, but if you
want to go deeper, read on.
3.3 Bend Magnet Radiation: The Details
As mentioned before, detailed calculation of the radiation from a bend magnet is not
for the faint of heart. Here the key equations are presented for reference, without
rigorous step-by-step derivation. For those who want more, the details are available
in excellent articles [67,68], in classic works such as the paper by Schwinger [7] or
the text by Jackson [69], as well as in more recent books [70].
Our coordinate system describes the observation position by horizontal angle ϕ
and vertical angle ψ, as shown in Fig. 3.4. We also need to consider the polarization,
which can have horizontal ε ˆ σ and vertical ε ˆ π components (Fig. 3.4).
We now compare some exact results with those derived qualitatively.
42
3 Synchrotron Radiation Fundamentals