Ewald’s sphere

If expected narrow meridional WAXS reflections are not found, the reason may be the bending-away of Ewald s sphere (cf. Fig. 2.6, p. 28). In this case it may be convenient to tilt the sample with respect to the primary beam (cf. Fig. 2.7, p. 29). 

One of the basic features of high-energy ED is the short wavelength of electrons used 0.05 A (accelerating voltage lOOkV). Therefore Ewald s sphere practically degenerates into a plane, and the electron diffraction pattern (ED) is the planar cross-section of the reciprocal lattice (Fig.2.). 

The simple geometrical arrangement of the reciprocal lattice, Ewald s sphere, and three vectors (ko, ki, and d hki) in a straightforward and elegant fashion yields Braggs equation. From both Figure 2.27 and Figure 2.28, it is clear that vector ki is a sum of two vectors, ko and d hki ... 

The Ewald s sphere and the reciprocal lattice are essential tools in the visualization of the three-dimensional diffraction patterns from single crystals, as will be illustrated in the next few paragraphs. They are also invaluable in the understanding of the geometry of diffraction from polycrystalline (powder) specimens, which will be explained in the next section. 

Figure 2.28. The visualization of diffraction using the Ewald s sphere with radius MX and the two-dimensional reciprocal lattice with unit vectors a and b. The origin of the reciprocal lattice is located on the surface of the sphere at the end of ko. Diffraction can only be observed when a reciprocal lattice point, other than the origin, intersects with the surface of the Ewald s sphere [e.g. the point (13)], The incident and the diffracted beam wavevectors, k and ki, respectively, have common origin in the center of the Ewald s sphere. The two wavevectors are identical in length, which is the radius of the sphere. The unit cell of the reciprocal lattice is shown using double lines.

Figure 2.29. The illustration of a single crystal showing the orientations of the basis vectors corresponding to both the direct (a, b and c) and reciprocal (a, b and c ) lattices and the Ewald s sphere. The reciprocal lattice is infinite in all directions but only one octant (where h>0,k>0 and / > 0) is shown for clarity.

Figure 2.31. The origin of the powder diffraction eone as the result of the infinite number of the completely randomly oriented identical reciprocal lattice vectors, d hki, forming a circle with their ends placed on the surface of the Ewald s sphere, thus producing the powder diffraction cone and the corresponding Debye ring on the flat screen (film or area detector). The detector is perpendicular to both the direction of the incident beam and cone axis, and the radius of the Debye ring in this geometry is proportional to tan20.

Assuming that the diffracted intensity is distributed evenly around the base of each cone (see the postulations made above), there is usually no need to measure the intensity of the entire Debye ring. Hence, in a conventional powder diffraction experiment, the measurements are performed only along a narrow rectangle centered at the circumference of the equatorial plane of the Ewald s sphere, as shown in Figure 2.32 and indicated by the arc with an... 

The Lorentz factor takes into account two different geometrical effects and it has two components. The first is owing to finite size of reciprocal lattice points and finite thickness of the Ewald s sphere, and the second is due to variable radii of the Debye rings. Both components are functions of 0. 

In both cases (Eqs. 2.78 and 2.79) the preferred orientation factor Thu is proportional to the probability of the point of the reciprocal lattice, hkl, to be in the reflecting position (i.e. the probability of being located on the surface of the Ewald s sphere). In other words, this multiplier is proportional to the amount of crystallites with hkl planes parallel to the surface of the flat sample. 

The recipe for constructing Ewald s sphere is as follows (Figure 1.8) ... 

The distance by which the lattice node G is missed by Ewald s sphere is called the excitation error. It is a vector along the direction of the foil normal joining the reciprocal lattice node G to the intersection point with Ewald s sphere. It is positive when pointing in the sense of the incident beam and negative in the opposite case. 

FIGURE 5.1 The Ewald s sphere and the representation of the Bragg s equation in the approximation of the two (transmitted-diffracted) waves. 

FIGURE 5.12 The construction (left) of the Ewald s sphere of diffraction (right), after (Matter Diffraction, 2003 X-Rays, 2003 Putz Lacrama, 2005). 

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