Total external reflection

For crystals which have flat faces which extend for a fraction of 1 ym, a new type of phenomenon may be observed. Electrons incident at the edge of the crystal parallel to the surface may be channelled along the surface. The potential field of the crystal extending into the vacuum deflects the electrons so that they tend to enter the surface but they are scattered out of the crystal by the surface atoms or by diffraction from the crystal lattice planes parallel to the surface. If the scattering angle is less than the critical angle for total external reflection, the scattered electrons can not surmount the external potential barrier and are deflected back into the crystal (figure 4 (a)). 

Figure 3.17. (a) Specular X-ray intensity of a DIP trim with 20.6 nm thickness. Bragg reflections up to the seventh order are seen, (b) Magnification of the specular scan from the region of total external reflection across the first specular DIP Bragg reflection. The inset displays a rocking scan across the DIP(OOl) reflection which exhibits FWHM as small as 0.0087°. Reprinted with permission from A. C. Diirr, F. Schreiber, M. Mtinch, N. Karl, B. Krause, V. Kruppa and H. Dosch. Applied Physics Letters, 81, 2276 (2002). Copyright 2002, American Institute of Physics. 

Calculations of the variations expected in the fluorescent-yield (FY) profiles as a function of the distribution model parameters are shown in Figure 7.19. When the species of interest resides predominantly at the solid surface, the FY profile shows a maximum at the critical angle for total external reflection. As the ratio of the surface-bound species to the total number of species in the solution volume adjacent to the surface decreases, the FY distribution broadens at the low angles. A similar effect is noted when a diffuse layer accumulation arises due to an interfacial electrostatic potential. 

Monochromatic radiation is again directed at the water surface at a small angle, a, less than the critical angle for total external reflection. The intensity of the reflected radiation is explored as a function of the... 

In the region of total external reflection, the incident and exit angle a, and ay are small enough to neglect the influence of the periodic crystal structure and to use a description based on one mean refractive index nj = 1 — Sj + ifl [22-24], Since the dispersion part... 

Systems with lower concentrations of heavy metal ions can be studied by means of gra2ing-incidence XAFS spectroscopy. In grazing-incidence geometry the X-ray beam penetrates only about 10 nm into the sample. This method can be applied to study sorption on specified crystallographic planes when single crystals are used rather than powders. Incident beam angles of a fraction of E assure total (external) reflection geometry. 

The reflectivity in the total external reflection region is that of the topmost layer. It should be noted that in this region, the electric field intensity is a continuous function of angle of incidence. At the surface of the multilayer, the electric field intensity is given by ... 

In general, the -field intensity in the total external reflection region can be determined from... 

In the case of thin films or monolayers, two different techniques can be employed these are the total external-reflection Bragg diffraction (TERBD) technique introduced by Eisenberger and Marra and the previously mentioned technique based on crystal truncation rods, introduced by Robinson, which can provide in-plane structural information or information on interfacial roughness, respectively. 

This technique (sometimes referred to as grazing incidence x-ray scattering) essentially involves conventional Bragg diffraction under conditions of total external reflection. 

As originally described by Eisenberger and Marra, the angle of incidence is kept below the critical angle for the material under study so that the x-ray beam undergoes total external reflection. As mentioned previously, this has two very important consequences. First of all, since only an evanescent wave penetrates the substrate, the sampling depth is very shallow and of the order of 10 to 20 A. 

The kinematic treatment is generally used to interpret surface diffraction data. However, one must remember that the total external reflection effect is dynamial in nature so that the kinematic treatment is strictly not applicable. However, as has been pointed out by Vineyard, a simple distorted wave approximation ean be used, and this can be quite adequately treated in a kinematical approach. This last point greatly simplifies data interpretation. 

Figure 7. Scattering geometry vertical scattering (reflectivity) and in-plane scattering (diffraction). The x-rays (or neutrons) with incident wave vector k strike the surface at a small angle of incidence (/), which is less than the critical angle for total external reflection, . In reflectivity measurements, the vertical incidence and exit angles are equal so the difference between k and the reflected wave vector k, is perpendicular to the scattering plane. In diffraction measurements, the difference k — k,i lies essentially in the plane.

---