The Ewald Construction

Draw the incident wave vector Sq. This points in the direction of the incident beam with length IjX. 

Draw a sphere centered on the tail of this vector with radius 1/2. The incident wave vector Sq defines the radius of the sphere. The scattered wave vector s, also of length 1/2, points in a direction from the sample to the detector. This vector is drawn also starting from the center of the sphere and also terminates at a point on the surface. The scattering vector h = s - So completes the triangle from the tip of s to the tip of Sq, both lying on the surface of the sphere. 

Draw the reciprocal lattice with the origin lying at the tip of Sq. 

Find all the places on the surface of the sphere, where reciprocal lattice points lie. 

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. 

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Figure A.4 The Ewald construction. Given an incident wave vector ki, a sphere of radius k, is drawn around the end point of kz. Diffraction peaks are observed only if the scattering vector q ends on this sphere.

The formalism of the reciprocal lattice and the Ewald construction can be applied to the diffraction at surfaces. As an example, we consider how the diffraction pattern of a LEED experiment (see Fig. 8.21) results from the surface structure. The most simple case is an experiment where the electron beam hits the crystal surface perpendicularly as shown in Fig. A.5. Since we do not have a Laue condition to fulfill in the direction normal to the surface, we get rods vertical to the surface instead of single points. All intersecting points between these rods and the Ewald sphere will lead to diffraction peaks. Therefore, we always observe diffraction... 

FIGURE 8 Comparison of the modification of the Ewald construction when going from ideally translational symmetric lattices to real lattices in which planes degenerate into lattice slabs. The quantity P is explained in Equation (2). 

Figure 6 (a) The Ewald construction illustrated, (b) Limiting sphere and sphere of reflection... 

In the Ewald construction (Figure 3.17), a circle with a radius proportional to 1/A and centered at C, called the Ewald circle, is drawn. In three dimensions it is referred to as the Ewald sphere or the sphere of reflection. The reciprocal lattice, drawn on the same scale as that of the Ewald sphere, is then placed with its origin centered at 0. The crystal, centered at C, can be physically oriented so that the required reciprocal lattice point can be made to intersect the surface of the Ewald sphere. 

We can see the diffraction pattern with our own eyes when we collect X-ray data because we obtain the image, the pattern of diffraction spots, on the face of our detector or film. We can t directly see the families of planes in the actual crystal, but we know, through the Ewald construction, how the diffraction pattern is related to the crystal orientation, and hence to the dispositions of the planes that pass through it. We also know from Ewald how to move the crystal about its center, once we know its orientation with respect to our laboratory coordinate system, in order to illuminate various parts of reciprocal space. In data collection we watch the diffraction pattern, not the crystal, and let the pattern of intensities guide us. 

From Equation (2), we deduce that diffraction is observed only when the indices h, k, l in d take integral values. These reciprocal space vectors form a lattice, the reciprocal lattice, and the mathematical relationship between the real and reciprocal lattices (and between other aspects of the diffraction pattern) is a FT, as we will explain below. The interpretation of the Ewald construction is that diffraction is observed when the scattering vector s-s0 is equal to a reciprocal space vector A bki with integral indices h, k, l. This occurs whenever such a... 

Figure 12 The Ewald construction drawn for the reflection (-2 2 0). The crystal is located at the origin O and the endpoint of the vector s lies at a lattice point of the reciprocal lattice (gray). The radius of the circle is A 1.

So how do we know the unit cell of the crystal and its orientation The first step in the collection of crystallographic data consists of taking one or two test images, from which the spot positions are determined. Each diffraction spot is then assigned indices h,k,l based on its position on the detector. This is called indexing and the unit cell parameters and crystal orientation are determined here. Once the diffraction pattern is indexed, we can use the Ewald construction to predict where spots should be observed. The prediction is important, since some of the spots may be so faint that detection would be impossible unless we knew where to expect them. 

The conditions for diffraction expressed by Eq. (7) may be represented graphically by the Ewald construction shown in Fig. A1-8. The vector SoM is drawn parallel to the incident beam and l/X in length. The terminal point O of this vector is taken as the origin of the reciprocal lattice, drawn to the same scale as the vector Sq/X. A sphere of radius 1 jX is drawn about C, the initial point of the incident-beam vector. Then the condition for diffraction from the hk ) planes is that the point hkl in the reciprocal lattice (point P in Fig. Al-8) touch the surface of the sphere, and the direction of the diffracted-beam vector S/A is found by joining C to P. When this condition is fulfilled, the vector OP equals both and (S — So)/A, thus satisfying Eq. (7). Since diffraction depends on a reciprocal-lattice point touching the surface of the sphere drawn about C, this sphere is known as the sphere of reflection. ... 

Fig. Al-8 The Ewald construction. Section through the sphere of reflection containing the incident and diffracted beam vectors.

As stated in Sec. 3-6, when monochromatic radiation is incident on a single crystal rotated about one of its axes, the reflected beams lie on the surface of imaginary cones coaxial with the rotation axis. The way in which this reflection occurs may be shown very nicely by the Ewald construction. Suppose a simple cubic crystal is rotated about the axis [001]. This is equivalent to rotation of the reciprocal lattice about the bs axis. Figure A1-9 shows a portion of the reciprocal lattice oriented in this manner, together with the adjacent sphere of reflection. 

Changing the orientation of the crystal reorients the reciprocal lattice bringing different reciprocal lattice points on to the surface of the Ewald sphere. An ideal powder contains individual crystallites in all possible orientations with equal probability. In the Ewald construction, every reciprocal lattice point is smeared out onto the surface of a sphere centered on the origin of reciprocal space. This is illustrated in Figure 1.9. The orientation of the vector is lost and the three-dimensional vector space is reduced to one dimension of the modulus of the vector A ti-... 

Figure 6.2 The Ewald construction (a) the reciprocal lattice (b) a vector of length /X, dawn parallel to the beam direction (c) a sphere passing through the 000 reflection, drawn using the vector in (b) as radius (d) the positions of the diffracted beams...

Figure 6.3 The geometry of the Ewald construction, showing it to be identical to Bragg s law...

Figure 6.5 Comparison of real space and reciprocal space formation of diffraction patterns (a) schematic formation of a diffraction pattern in an electron microscope (b) the Ewald construction of a diffraction pattern...

Before we can measure the intensity of a Bragg reflection, we need to determine where and from what direction to orient the X-ray detector. A geometrical description of diffraction, the Ewald sphere, allows us to calculate which Bragg reflections will be formed if we know the orientation of the crystal with respect to the incidentX-ray beam. In the Ewald construction (shown in two dimensions in Fig. 11), a sphere of radius 1/X is drawn with the crystal at its center and the reciprocal lattice on its surface. A Bragg reflection is produced when a reciprocal lattice point touches the surface of the Ewald sphere. As the orientation of the crystal is changed, so is the orientation of its reciprocal lattice. 

It is easy to show, using the Ewald construction, that the following combination of vectors, are true ... 

The fundamental equation (3.39), S = s — Sq = r, which describes the elastic diffraction by crystals, may be represented by two geometrical constructions. Bragg s law is a construction in physical space the Ewald construction is carried out in reciprocal space. These two representations are equivalent and facilitate the visualization of the diverse methods of diffraction. 

The Ewald construction obtains the direction of a diffracted wave by the intersection of two loci. The first of these loci is the Ewald sphere the relationships S = s —Sq, Ijsll = IISoil = 1/A (equation (3.21), Fig. 3.12) show that S is a secant joining two points on the surface of a sphere of radius 1/A. The second locus is defined by equation (3.39) the vector S coincides with the vector r of the reciprocal lattice. The construction goes through the following steps (Fig. 3.19) ... 

Fig. 3,19. The Ewald construction. The circle represents the intersection of the Ewald sphere and a plane of the reciprocal lattice passing through the origin and containing the lattice points hkl) with hU- - kV- - /W= 0 U, V, Wbeing coprime integers). The primitive translation of the direct lattice lUVW] is normal to this plane (Section 1.4.3). The reader can imagine other planes of this series which obey the relation hU- -kV - IW— n, n 0, above and below the plane of the figure...

Figure 3.23 illustrates the Laue method with the aid of the Ewald construction. All the reciprocal lattice points situated between the spheres of radius and... 

By making a small modification to the Ewald construction, we obtain a more informative representation of the Laue method. By multiplying all the dimensions of the construction by the wavelength A, we obtain a lattice -h fcb -h /c ) and a sphere of radius 1. Thus, for polychromatic radiation, we obtain a superposition of lattices of variable dimensions intersected by a single... 

The Ewald construction is based on the consideration of the scattering vector of Eqs. (5.6) or of (5.11) t5q>e, as being the vector which transforms the incident wave vector into the diffiacted one, on a sphere. Figure 5.12 of 1/2 radius such construction is possible according to the condition of reflection (D1 of preceding section) of the diffracted X-ray towards the incident one (wave vectors with equal fk modulus the present discussion follows Putz and Laciama (2005). 

A useful feature of the Ewald construction is the ability to see which planes will be able to meet the Laue condition. Consider the reciprocal lattice superimposed on the Ewald sphere shown in Figure 6.5a. In this case, the k is 2.4 times larger than the reciprocal lattice vector, which means the direct lattice spacing a is 2.4 times larger than the wavelength A. None of the reciprocal lattice points lie on the Ewald sphere (at least in the... 

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