PRINCIPLES OF TOTAL REFLECTION XRF - Total-Reflection X-Ray Fluorescence Analysis and Related Methods - page 112

Figure2.23. Fluorescence intensity of a periodic multilayer plotted against the glancing angle α .
The multilayer of Section 2.1.2 was used for calculation, consisting of 15 bilayers of platinum and
cobalt ( d = 2.1 nm). A Cu-K α beam was assumed for excitation. The reflectivity of the multilayer is
represented by the dotted curve. Figure from Ref. [22], reproduced with permission. Copyright
1991 by Elsevier.
mostly between 0.15 and 0.3 nm, the Bragg angle is on the order of 10 ° for
wavelengths of about 0.1 nm. For that reason, Bragg reflection does not appear
under grazing but under steeper incidence.
2.4FORMALISMFORINTENSITYCALCULATIONS
The phenomena qualitatively described in Section 2.3 can rigorously be
calculated from theory. The primary beam can be described by a plane
incoming and reflected wave leading to a standing wave field. Its intensity
can be calculated from the optical theory of wave propagation in layers with flat
interfaces, especially by the Fresnel relations. The primaryintensity is a
function of the glancing angle α 0 at grazing incidence and is further dependent
on the depth z normal to the layer surface. Besides the primary intensity, the
intensity ratio of the reflected and the incoming wave can be calculated. This
ratio is defined as reflectivity and can directly be checked by reflectivity
measurements. All calculations can be based either on a recursive formalism
described by Parratt [23] or on an equivalent matrix formalism first proposed
by Abelès [24] and extensively described by Born and Wolf [2] and Król
etal . [25]. The latter is the more elegant method and is preferred here.