Chemistry Reference
In-Depth Information
If the respective glancing angle
α
2
of the refractive beam becomes zero, the
refracted beam will emerge tangentially to the boundary surface. Consequently,
there is a minimum critical angle
α
1
=
α
crit
for which refraction is just possible.
According to Equation 1.52, this angle of incidence is determined by
cos
α
crit
n
2
(1.65)
For angles
α
1
even lower than
α
crit
, Equation 1.52 gives no real value for the
refraction angle
α
2
since the cosine cannot be
>
1. In this case, no beam enters
the second medium, but the boundary, like an ideal mirror, completely reflects
the incident beam back into the first medium, that is, vacuum or air. This
phenomenon is called“total
external
reflection.”In contrast to X-rays, visible
light can undergo“total
internal
reflection”when the light comes from a solid
medium below a critical angle. It does not enter the adjacent vacuum or air as
the second medium, but is totally reflected back into this first medium.
The
critical
angle
of
total
reflection
can
easily
be
calculated
from
Equation 1.65. Since
α
crit
is small, its cosine can be approximated by
α
crit
2
cos
α
crit
1
(1.66)
The combination with Equation 1.55 leads to the simple relation
p
2
δ
α
crit
(1.67)
Insertion of Equation 1.58 gives the approximation
r
Z
A
ρ
1
:
651
E
α
crit
(1.68)
where
E
has to be given in keV and
ρ
in g/cm
3
in order to get
α
crit
in degrees.
This formula may be converted into
p
C
m
5
10
α
crit
(1.69)
E
As already mentioned for Equation 1.58, this approximation is exactly valid for
photon energies above the decisive absorption edges of the material. Table 1.9
gives values for different media and photon energies frequently used for
excitation: 8.4 keV is the energy of W-L
α
photons, 17.44 keV is the energy
of Mo-K
α
photons, and 35 keV may represent the photon energy of the hump
appearing in a continuous spectrum of an X-ray tube. All values of
α
crit
lie
between 0.04
°
and 0.6
°
. For all other combinations of medium and photon
energy, the critical angle can simply be calculated after Equation 1.69 and by
use of Table 1.7.
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