Diffraction Conditions

In the concepts developed above, we have used the kinematic approximation, which is valid for weak diffraction intensities arising from imperfect crystals. For perfect crystals (available thanks to the semiconductor industry), the diffraction intensities are large, and this approximation becomes inadequate. Thus, the dynamical theory must be used. In perfect crystals the incident X rays undergo multiple reflections from atomic planes and the dynamical theory accounts for the interference between these reflections. The attenuation in the crystal is no longer given by absorption (e.g., p) but is determined by the way in which the multiple reflections interfere. When the diffraction conditions are satisfied, the diffracted intensity ft-om perfect crystals is essentially the same as the incident intensity. The diffraction peak widths depend on 26 m and Fjjj and are extremely small (less than... 

What gives rise to streaks in a RHEED pattern from a real surface For integral-order beams, die explanation is atomic steps. Atomic steps will be present on nearly all crystalline surfaces. At the very least a step density sufficient to account for any misorientation of the sample from perfeedy flat must be included. Diffraction is sensitive to atomic steps. They will show up in the RHEED pattern as streaking or as splitdng of the diffracted beam at certain diffraction conditions that depend on the path difference of a wave scattered from atomic planes displaced by an atomic step height. If the path difference is an odd muldple of A./2, the waves scattered... 

Figure 3. Grating diffraction (a) geometry for the diffraction condition, (b) high-order diffraction for a grating with 7 mlings.

The mathematics necessary to understand the diffraction of X rays by a crystal will not be discussed in any detail here. Chapter 4 of reference 10 contains an excellent discussion. The arrangement of unit cells in a crystal in a periodic manner leads to the Laue diffraction conditions shown in equations 3.3 where vectors a, b, and c as well as lattice indices h, k, and l have been defined in Figure 3.5 and S is a vector quantity equal to the difference between the resultant vector s after diffraction and the incident X-ray beam wave vector So so that S = s - So-... 

If the specimen crystal is curved, there will be a range of positions where the diffraction conditions are satisfied even for a plane wave. The rocking curve is broadened. It is simple to reduce the effect of curvature by reducing the collimator aperture. For semiconductor crystals it is good practice never to mn rocking curves with a collimator size above 1 mm, and 0.5 mm is preferable. Curved specimens are common if a mismatched epilayer forms coherently on a substrate, then the substrate will bow to reduce the elastic strain. The effect is geometric and independent of the diffraction geometiy. Table 2.1 illustrates this effect. 

This very useful method also has the advantage that the equations do not contain anything about the material or diffraction conditions other than the Bragg angle and geometry. The independence from material parameters arises because the refractive index for X-rays is very close to unity. The equations are, of course, similar to those for optical interference from thin films, since the physics is the same, but in the optical case we do need to know the refractive index. 

Figure 4.8 The powder difTraction experiment, (a) Reciprocal space notation. The Ewald sphere is fixed, and the lattice is rotated about all angles about the origin. Only the rotations about [100] are shown in this two-dimensional section. Intersections with the Ewald sphere define the diffracting conditions, (b) The corresponding diffracted beams in real space...

Figure 4.11 The kinematic dispersion surface. The circles centred on the origin O and the relp H, with radius (1/ vacuum )(1+ 2) represent, in the plane shown, the allowable wavevectors in the crystal far from the diffracting condition. A section of the Ewald sphere is shown...

Show that in the LEED arrangement illustrated in Fig. 2.11, constructive interference between electrons scattered from different atoms gives the diffraction condition... 

Exercise 16.3-2 Show that if k is the wave vector of incident radiation (X-ray or neutron) or the wave vector of a particle or quasiparticle, then eq. (25) leads to the Bragg diffraction condition. 

The electron gas model adequately describes the conduction of electrons in metals however, it has a problem, that is, the electrons with energy near the Fermi level have wavelength values comparable to the lattice parameters of the crystal. Consequently, strong diffraction effects must be present (see below the diffraction condition (Equation 1.47). A more realistic description of the state of the electrons inside solids is necessary. This more accurate description is carried out with the help of the Bloch and Wilson band model [18],... 

Subsequently, introducing the diffraction condition in Equation 1.46, we will get [5]... 

---