Chemistry Reference
In-Depth Information
two amplitudes,
E
t
N
1
for the substrate and
E
0
for the vacuum, have to be
determined after connecting them by a matrix multiplication:
0
M
ν
;
ν
1
E
t
N
1
1
E
0
N
Π
(2.52)
0
ν
With the known amplitude
E
t
N
1
for the substrate, the remaining amplitudes for
all layers can then be calculated in accord with expression 2.50 in steps
backward. Beginning with the known amplitude
E
0
for the vacuum, the
remaining amplitudes can be calculated also in steps forward.
The primary intensity for a given outer glancing angle
α
0
in dependence on
the depth of the
v
th layer is then given by
2
≅
I
n
?
E
t
v
exp
ik
0
α
ν
z
E
r
v
exp
I
prim
;
ν
α
0
;
z
ik
0
α
v
z
i
φ
ν
(2.53)
for
z
values between
d
ν
/2 and
+
d
ν
/2. The phase shift
φ
ν
is taken into account
because of total reflection at the different interfaces.
By means of the primary intensity, the fluorescence intensity of an element
named
x
in the layer
v
can now be determined. The element may be present in
the
v
th layer with a mass fraction
c
x,v
. Its atoms may be excited to fluorescence
in the primary wave field,
I
prim,
v
, by a monochromatic radiation with photons of
energy
E
0
. At these conditions, the fluorescence intensity of the layer can be
derived from Equation 2.47 by replacing the fundamental parameters of the
first layer by those of the
v
th layer. This original fluorescence radiation with a
photon energy
E
<
E
0
will further be absorbed on its way to the detector, which
again should be mounted perpendicular to the stratified medium and close to it.
Absorption takes place in all layers
above
the relevant
ν
th layer, that is, in the
(
v
1. Consequently, the detector will
measure an intensity that results from Equation 2.47 by multiplication with an
absorption factor
1) layers with an index
j
=
1,...,
v
"
#
X
ν
1
A
ν
exp
μ
=
ρ
j
;
E
ρ
j
d
j
(2.54)
j
1
where (
μ
/
ρ
)
j,E
is the mass-attenuation coefficient of the
j
th layer for photons of
the respective energy
E
.
Again, three auxiliary quantities have been defined:
μ
=
ρ
ν
;
E
0
Q
t
;
ν
μ
=
ρ
(2.55a)
ν
;
E
α
ν
μ
=
ρ
ν
;
E
0
Q
r
;
ν
μ
=
ρ
(2.55b)
ν
;
E
α
ν
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