Ewald-sphere construction

What happens if the set of (hkl) lattice planes is not exactly at the Bragg orientation As shown on figure 2b, the position of the two spots is hardly affected but the intensity of the diffracted beam is strongly modified. This behavior can be explained by means of the Ewald sphere construction. 

Figure 3. The excitation and Ewald sphere construction of CBED.

Figure 4.5 The Ewald sphere construction in reciprocal space...

These examples show the great practical utility of the Ewald sphere construction. We did once hear Paul Ewald say, some sixty years after he laid the basis of X-ray scattering theory, that he wished people had named something else after him, as it was such a trivial idea ... 

Following Fraser et al. (4), we choose to represent the scattered intensity in terms of a cylindrically symmetric "specimen intensity transform" I (D), where D is a position vector in reciprocal space. Figure 10 shows the Ewald sphere construction, the wavelength of the radiation being represented by X. The angles p and X define the direction of the diffracted beam and are related to the reciprocal-space coordinates (R, Z) and the pattern coordinates (u,v) as follows ... 

Figure 10. Ewald sphere construction showing the relationship between the reciprocal space position vector D, the reciprocal space coordinates (R,Z), and the angles n and which define the direction of a scattered ray...

Figure 7.16. Illustration of Ewald sphere construction, and diffraction from reciprocal lattice points. This holds for both electron and X-ray diffraction methods. The vectors AO, AB, and OB are designated as an incident beam, a diffracted beam, and a diffraction vector, respectively.

The Ewald sphere construction is an elegant method for determining the diffracted wavevectors from a crystal, and will be used a great deal in this... 

Figure 3.10. The Ewald sphere construction in reciprocal space when there is a slight deviation from the exact Bragg angle.

Figure 3.16(a) shows the Ewald sphere construction for a systematic row of reflections when the first-order reflection g satisfies the exact Bragg condition. The corresponding reciprocal lattice point C] is thus on the Ewald sphere and s = 0. The reciprocal lattice point G2 (corresponding to the second-order reflection 2g) is outside the sphere, and so. S2g is negative. 

In the usual Ewald sphere construction shown in Figure 4.1, the center L of the sphere is determined such that the magnitude of the wavevectors to... 

Figure 4.1. The usual Ewald sphere construction in which LO is the incident wave-vector and LG the diffracted wavevector, both of magnitude l/. OG is the reciprocal lattice vector of the operating reflection and is of magnitude /d hkl).

If the construction is now performed taking into account the mean inner potential of the crystal, then the center Q of the sphere is determined such that the magnitude of the wavevectors to 0(000) and to G khl) is K. It can be seen from Eq. (4.15) that K>k, so the point Q is somewhere to the left of the Laue point L in Figure 4.1. The Ewald sphere construction with Q as center and radius K is shown in Figure 4.2. The diagram also includes the point L, but the distance between Q and L has been greatly... 

Figure 4.6. Detail of the Ewald sphere construction used in deriving Eq. (4.32).

Establishing kinematical BE and weak beam DF conditions. The procedures to be followed for establishing these diffracting conditions are best described in terms of the Ewald sphere construction (Sections 3.4 and 3.7). 

This construction places a reciprocal lattice point at one end of h. By definition, the other end of h lies on the surface of the sphere. Thus, Bragg s law is only satisfied, when another reciprocal lattice point coincides with the surface of the sphere. Diffraction is emanating from the sample in these directions. To detect the diffracted intensity, one simply moves the detector to the right position. Any vector between two reciprocal lattice points has the potential to produce a Bragg peak. The Ewald sphere construction additionally indicates which of these possible reflections satisfy experimental constraints and are therefore experimentally accessible. 

Although the diffraction patterns obtained from an electron microscope are easy to understand in terms of the reciprocal lattice, it is still necessary to allocate the appropriate hkl value to each spot in order to obtain crystallographic information. This is called indexing the pattern. A comparison of the real space situation, in an electron microscope, with the reciprocal space equivalent, the Ewald sphere construction, (Figure 6.5), shows that the relationship between the dhkt values of the spots on the recorded diffraction pattern is given by the simple relationship ... 

Fig. 11. The Ewald sphere construction, used to determine which Bragg reflections will be obtained for a given orientation of a crystal, (a) The geometry of diffraction, (b) The Ewald sphere and its relationship to the reciprocal lattice.

A very useful geometric construction that describes the diffraction condition is the Ewald sphere construction. A sphere is drawn of unit radius,... 

The number of reflections for which the intensity can be measured experimentally is limited since, as is seen from the Ewald sphere construction discussed in Section 1.5.3, the range of reciprocal space that can be explored with a radiation of wavelength k is limited by... 

The concept of the reciprocal lattice is very useful in discussing the diffraction of x-rays and neutrons from crystalline materials, especially in conjunction with the Ewald sphere construction discussed in Section 1.5.3. The regular arrangement of atoms and atomic groupings in a crystal can be described in terms of the crystal lattice, which is uniquely specified by giving the three unit cell vectors a, b9 and c. It turns out that the diffraction from a crystal is similarly associated with a lattice in reciprocal space. The reciprocal lattice is specified by means of the three unit cell vectors a, b, and c in the same way as the crystal lattice is based on a9 b, and c. In fact, the crystal lattice and the reciprocal lattice are related to each other by the Fourier transform relationship. 

Figure 6.4 Ewald Sphere Construction that demonstrates how the complete X-ray scattering pattern (all vertices of reciprocal lattice) can be visualised by adjusting i) the wavelength (X) of X-ray beam So incident upon a crystal mounted at position M ii) the orientation of the crystal relative to beam Sq. Each vertex of the reciprocal lattice corresponds with a different hW-reflection. A given hW-reflection may be visualised only when scattered beam s cuts the surface of the Ewald sphere at a position P coincident with a corresponding reciprocal lattice vertex. In principle, X and crystal orientation at M may be adjusted to visualise the vast majority of vertices of a reciprocal lattice and hence the vast majority if not all of the hW-reflections possible from a given mounted crystal (Laue Condition).

Rg. 4.13 The Ewald sphere construction for reflection from a set of hkl) planes. 

Figure 3 Ewald sphere construction for an electron incident normal to the surface on (A) a perfect 2D system and (B) a quasi-2D system. In (A), the reciprocal lattice vector Q2o is shown associated with the relevant scattered wave. The dashed scattered wave vectors propagate into the bulk and thus are not observable.

Fig. 2. Ewald sphere construction for the inelastic scattering events. For He atoms the energy loss is comparable to the impact energy of the particles, while for electrons it is negligible on the scale of the figure. The HREELS spectra are therefore effectively constant q, scans, while HATOF spectra run along the so-called scan curves and include losses with different q, values.

Figure 3.2.1.8 visualizes the diffraction scenario within the reciprocal lattice via the Ewald-sphere construction for two difierent electron energies and the LEED-typical... 

Figure 3.2.1.8 Ewald-sphere construction for two different electron energies.

As evident from the Ewald-sphere construction or from Eqs. 3.2.1.8, 3.2.1.15, the component of the final wave vector normal to the surface is also fixed, namely to... 

---