PRINCIPLES OF TOTAL REFLECTION XRF - Total-Reflection X-Ray Fluorescence Analysis and Related Methods - page 103

Equation 2.19 is not very transparent but it can be transformed into a more
comprehensible approximation as shown in Section 2.4.1:

I B α I n C 1 R α α (2.20)

where C is a quantity of the substrate mainly determined by 1/ β or 1 / [( μ / ρ ) ρ ],

respectively. This formula can easily be interpreted as follows [17,18].
If the glancing angle α is far above the critical angle α crit of total
reflection, the primary X-ray beam deeply penetrates into the substrate.
A thick layer is passed through with a thickness proportional to α or to the
sine of α if larger glancing angles are considered. The fluorescence signal
comes from this layer. Because the primary beam is nearly not reflected
( R ≈ 0), the signal is only dependent on α , that is, directly proportional to α .
In the region of total reflection, however, the primary beam is evanescent in
the substrate, as was demonstrated in Figure 2.11. A major part of the
primary beam is reflected ( R » 0) and only the remainder (1 R ) is decisive

for fluorescence. The quantity (1 R ) α in Equation 2.20 is called energy

transfer and defines that portion of the impinging energy that penetrates
into the substrate.
Equation 2.20 is demonstrated by Figure 2.17 for a Mo-K α beam striking a
thick and flat Si substrate. In general, the signal intensity linearly decreases as
glancing angles α decrease. At the critical angle of total reflection, however, a

Figure2.17. Signal intensity of a thick, flat, and smooth silicon substrate ( ——— ), calculated for an
impinging Mo-K α beam. In addition, the reflectivity R ( ) is shown, dependent on the glancing
angle α .Below α crit = 0.102 ° , total reflection occurs with a steplike increase in reflectivity and a steplike
decrease of the signal intensity. The oblique dashed line represents the intensity from a rough Si
substrate. Figure from Ref. [1], reproduced with permission. Copyright1996, John Wiley and Sons.