Chemistry Reference
In-Depth Information
Different values for
E
crit
have already been listed in Table 1.6 provided that the
bending radius is chosen to be 3.34 m (at DELTA in Dortmund).
The integral of the Bessel function in Equation 1.32 can be approximated for
two special cases. For small energies (
E
E
crit
) of the emitted photons, the
increasing branch can be written as
≅
1
=
3
Δ
W
Δ
E
11
9
?
π
?
α
f
?
γ
?
E
E
crit
(1.34)
For high energies (
E
>>
E
crit
) of photons, the decreasing branch is
≅
1
=
2
Δ
W
Δ
E
2
3
?
π
?
α
f
?
γ
?
E
E
crit
E
E
crit
?
exp
(1.35)
Both asymptotes were calculated and also presented in Figure 1.19 as dimen-
sionless quantities (Watt/Watt). It can simply be shown that Equation 1.35 has a
maximum at 0.50
E
crit
and reaches a spectral flux (
Δ
W
/
Δ
E
) with a value of
about 0.9
α
f
γ
. The maximum of the actual curve in black, however, is located
at
E
max
≈
0.29
E
crit
[52,53] with a value of nearly 1.8
α
f
γ
according to
Equation 1.32.
As stated, the basic quantity of SR is the spectral flux, which can be given by
the number of photons with a certain energy emitted per unit time or second.
Usually a bandwidth of 0.1% of the respective average energy is chosen while
the beam current is normalized to 1 ampere and the electron energy is assumed
to be 1 GeV. However, for local investigations it is decisive that the radiation
source is a spot with a very small angular divergence. For that reason, the flux is
related to the respective solid angle and is called angular density of the flux or
spectral brightness [53]. When the photon flux is related to a small source area
(cross-section of the beam) in addition, it is called brilliance [56]. These
quantities are commonly used in the synchrotron literature though brilliance
is also called spectral brightness in English-speaking areas (USA, UK).
Brilliance is mostly related to number of photons per second, per mrad
2
,
per mm
2
, and per 0.1% spectral bandwidth but
not
to the SI units of 1 sr and of
1m
2
[67].
Figure 1.20 represents the brilliance derived from Figure 1.19 in a double
logarithmic plot with both asymptotic approximations. It represents the
average
brilliance integrated over the small vertical divergence. For low photon
energies, we find an increasing straight line with the slope 1/3;
1
for high photon
energies, we have a decreasing exponential curve. A comparison with the
radiation of a black body is described later.
1
In comparison to the average brilliance with a slope of 1/3 for the increasing branch, the in-plane
brilliance shows a slope of 2/3 and a maximum which is nearly doubled.
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