3.3.5 Bend Magnet Angular Density of Spectral Flux
Many experiments seek to use all the available photons, in which case the quantity of
interest is the photon flux integrated over the vertical angle ψ. The final result is:
dℱ bm
dθ
¼
ffiffi ffi
3
p
2π
αγ
Δω
ω
I
e
ω
ω c
Z 1
ω=ω c
K 5=3 y
0
ð Þdy
0
ð3:20Þ
or, in practical units (Fig. 3.5):
dℱ bm
dθ
¼ 2:46 Â 10
13 E e GeV
½
ŠI A
½ ŠG 1 ω=ω c
ð
Þ
ð3:21Þ
where G 1 (ω/ω c ) (Fig. 3.5) is referred to as the synchrotron radiation universal
function:
G 1
ω
ω c
¼
ω
ω c
Z 1
ω=ω c
K 5=3 y
0
ð Þdy
0
ð3:22Þ
3.3.6 Angular Divergence
In our qualitative derivation, we saw that the apparent acceleration seen by an
observer in the laboratory frame depends critically on the observation angle. The
flux and spectral distribution will thus change with observation angle—higher
energy photons will have a narrower opening angle than those at lower energy.
The vertical angular divergence can be approximated by a Gaussian function, for
which the rms deviation σ evaluated at Ψ = 0 is given by:
σ ψ
ð Þ ¼
ffiffiffiffiffi
2π
3
r
1
γ
ω c
ω
À1
R 1
ω=ω c
K 5=3 y
0
ð Þdy
0
K
2
2=3
ω
2ω c
ð3:23Þ
The opening angle varies by about one order of magnitude for a 100-fold
variation in photon energy (Fig. 3.6).
46
3 Synchrotron Radiation Fundamentals