Ewalds Sphere

You may by now be convinced that the diffraction pattern and its underlying reciprocal net reveal the dispositions of families of planes, but it is probably not at all clear how this relationship can be used in the laboratory. Exactly how are the reciprocal lattice vectors 

1 Actually, he had no intention of helping us, because he invented his construction before the first X-ray diffraction data was ever collected. He must have invented it for other uses, or he must have been a man of almost supernatural vision. 

While the absolute distance between points in reciprocal space is a function of X and the reciprocal lattice parameters a, b, c, a, / , and y, the absolute distances between maxima observed on the film will be expanded in proportion to F, which acts as a constant magnification factor. 

The axes of the reciprocal lattice, remember, maintain a fixed orientation with respect to the real axes of the crystal by definition, regardless of the crystal s orientation. That is, if the crystal is rotated, the reciprocal lattice is rotated as well. If the crystal is continuously reoriented in a specific manner about its center by some constant motion, all of the points on a single reciprocal lattice plane, or region of reciprocal space, can be made to systematically pass through the sphere of reflection. If the film is maintained constantly parallel with a reciprocal lattice plane by mechanical linkage to the crystal, a magnified but otherwise undistorted replica of the reciprocal lattice plane will be recorded on the film. This principle, proposed by de Jong and Bouman (1938), was the basis for some of the more widely used 

An important property of Fourier transforms that we did not emphasize in the previous chapter is that spatial relationships in one space are maintained in the corresponding transform space. That is, specific relationships between the orientations in real space of the members of a set of objects are carried across into reciprocal space. This is particularly important in terms of crystallographic symmetry, and we will encounter it again when we consider the process known as molecular replacement (see Chapter 8). 

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Because of the much shorter wavelength of elecuon beams, the Ewald sphere becomes practically planar in elecU on diffraction, and diffraction spots are expected in this case which would only appear in X-ray diffraction if the specimen were rotated. 

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

The notion of a reciprocal lattice cirose from E vald who used a sphere to represent how the x-rays interact with any given lattice plane in three dimensioned space. He employed what is now called the Ewald Sphere to show how reciprocal space could be utilized to represent diffractions of x-rays by lattice planes. E vald originally rewrote the Bragg equation as ... 

The tangent plane approximation is valid the curvature of the Ewald sphere is negligible at small scattering angles. 

Figure 3.4 X-ray beam passing through the Ewald sphere and diffracted by planes in a single crystal produces reflection spots. (Adapted with permission from Figure 1.13 of Drenth, J. Principles of Protein X-ray Crystallography, 2nd ed., Springer-Verlag, New York, 1999. Copyright 1999 Springer-Verlag, New York.)...

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

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. 

Geometrically, electron diffraction patterns of crystals can be approximated as sections of the reciprocal lattice, since the Ewald sphere can be regarded as a plane (i.e. the radius of the Ewald sphere, 1/2, is much larger than the lengths of low-index reciprocal lattice vectors). 

Figure 2. Fonnation of ring and oblique texture patterns, a - several randomly rotated artificial crystallites and its Fourier transform (inset) b - reciprocal space with rings and zero tilt Ewald sphere c - 60° tilt of the Ewald sphere (reflection centers lie on the ellipse) d - the diffraction pattern as it is seen on the image plane.

If the tilt of the texture is in the y z -plane (due to cylindrical symmetry it does not matter in which direction we tilt the sample, so the Xc coordinate of the centre of the Ewald sphere can be assigned to zero) then the Ewald sphere equation can be written as ... 

Then the coordinates of the intersection of the Ewald sphere and the /-reflection sphere of radius i/are ... 

Oblique texture patterns have almost perfect 2mm symmetry and thus the whole set of diffraction spots is represented by the reflections in one quadrant. The arcs are exactly symmetrically placed relative to the major axis, being sections of the same spherical band in reciprocal space. The reflections on the lower half of the pattern are sections of reciprocal lattice rings, which are Friedel partners and thus equivalent to those giving the reflections of the upper half assuming a flat surface of the Ewald sphere. Actually, if the curvature of the Ewald sphere is taken into account, the upper and lower parts of a texture pattern will differ slightly. 

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