Chemistry Reference
In-Depth Information
For
transversal
coherence, the factor
R
/
D
is decisive. It can vary between
5
×
10
3
and 5
×
10
5
. For a medium value,
R
/
D
=
5
×
10
4
and
E
=
5 keV, this
length amounts to 6.2
μ
m but decreases inversely to
E
(1.5
μ
m for
E
=
20 keV).
For
longitudinal
coherence, the relative spectral resolution
E
/
Δ
E
is the key
factor. It may differ between 5
×
10
1
and 5
×
10
4
. For a medium value
E
/
Δ
E
=
5
×
10
3
and
E
=
5 keV, this quantity is limited to 0.6
μ
m, which is a
tenth of the mentioned transversal coherence length.
The structures to be analyzed—particulate contaminations, islands, or
thin layers—have to be within the triangle of Figure 4.17 with a base
d
determined by the detector's field of vision and by the height
h
determin-
ed by the width of the incoming beam
w
beam
; for small angles
h
is about
w
beam
/2.
Additionally, the structures should have a height
h
coh
, which corresponds to the
smaller value of
ξ
l
and
ξ
t
. In accord with Bragg's law, we find for the
longitudinal coherence
1
2
α
hc
0
2
E
E
Δ
E
h
coh
ξ
l
=
2 sin
α
(4.22)
Of course it is doubtful whether the coherence of the two interfering beams is
not destroyed or at least diminished by a specimen positioned in the relevant
triangle. This important question may be answered by GI-XRF experiments.
Remembering the vertical period of a standing wave above a substrate
defined by Equation 2.12a we find
a
vertical
E
Δ
E
h
coh
(4.23)
This relationship is illustrated in Figure 4.19 with varying angle
α
and para-
meters
E
and
E
/
Δ
E
. A small glancing angle, a high spectral resolution, and
a low excitation energy is beneficial for coherence. In the case of TXRF where
α
is chosen to be somewhat smaller than the critical angle (0.07
°
for a Si
substrate and hard X-rays of 20 keV) the height of coherence in Figure 4.19 is
about 65
μ
m for a medium spectral resolution of 5
×
10
3
. In the case of GI-XRF
with angles up to 1
°
, the layer thickness should be as small as 5
μ
m for hard
X-rays.
These conditions can be moderated significantly for soft X-rays and
smaller glancing angles as is shown in Figure 4.20. It is a supplement to
Figure 4.19 and shows the photon energy versus the glancing angle in a
double logarithmic plot. The straight lines represent the product in accord
with the formula
3
E
Δ
E
α
(4.24)
E
6
10
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