Distance, real-reciprocal space relationships

FIGURE 1.7 In (a) the object, again exposed to a parallel beam of light, is not a continuous object or an arbitrary set of points in space, but is a two-dimensional periodic array of points. That is, the relative x, y positions of the points are not arbitrary they bear the same fixed, repetitive relationship to all others. One need only define a starting point and two translation vectors along the horizontal and vertical directions to generate the entire array. We call such an array a lattice. The periodicity of the points in the lattice is its crucial property, and as a consequence of the periodicity, its transform, or diffraction pattern in (b) is also a periodic array of discrete points (i.e., a lattice). Notice, however, that the spacings between the spots, or intensities, in the diffraction pattern are different than in the object. We will see that there is a reciprocal relationship between distances in object space (which we also call real space), and in diffraction space (which we also call Fourier space, or sometimes, reciprocal space). 

What has been said here is true but obscures another fundamental property of the Fourier transform, one that complicates matters a bit but not hopelessly so. The Fourier transform fails to directly carry translational relationships from one space to another, in particular, from real space into reciprocal space. This means that the transform does not discriminate between asymmetric units based on the distances between them. The immediate relevance of this is that a set of asymmetric units related by a screw axis symmetry operator (which has translational components) in real space is transformed into diffraction space as though it simply contained a pure rotation axis. The translational components are lost. If our crystal has a 6i axis, we will see sixfold symmetry in the diffraction pattern. If we have 2i2j2i symmetry in real space, the diffraction pattern will exhibit 222 (or more properly, mmm) symmetry. 

The subperiodicity of the axial projection within the unit cell in real space of course has an effect in diffraction space. Recall the reciprocal relationship between the two. If a distance is halved in real space, it is doubled in reciprocal space if a periodicity is quartered in real space, it is multiplied fourfold in reciprocal space, and so on. Thus, along the corresponding reciprocal space axis c, or the 00/ line of diffraction intensities, reflection spacing must correspond to the increased periodicity, or smaller repeat distance in real space. In reciprocal space, reflections will appear less frequently, at greater intervals. For a 2i axis, the appearance of reflections will have double the normal periodic interval. For a 2i axis, reflections will occur along the 00/ line of reciprocal lattice points only for / = 2,4,6, 8.. 

In Table 7.1, where real and reciprocal cell dimensions, or other distances are related, an orthogonal system is assumed for the sake of simplicity. For nonorthogonal systems, the relationships are somewhat more complicated and contain trigonometric terms (as we saw in Chapter 3), since the unit cell angles must be taken into account. Rotational symmetry is preserved in going from real to reciprocal space, and translation operations create systematic absences of certain reflections in the diffraction pattern that makes them easily recognized. As already noted, because of Friedel s law a center of symmetry is always present in diffraction space even if it is absent in the crystal. This along with the absence of... 

The reciprocal lattice has all the properties, including symmetry, of the real (or direct) lattice, but a plane in direct space is only a point in reciprocal space. We can define unit cell axes in reciprocal space a, b, and c, with the angles between them designated as a, P, and y. The distance between reciprocal lattice points (100) and (200), for example, is equal to a. The subsequent discussion refers to a unit cell with axes which are mutually perpendicular. At right angles to the axis a, planes of reciprocal lattice points are formed of constant h. Thus, for the one including the origin, all lattice points have indexes — okl), the next one kl), and so on and, similarly, for the other directions. Table 4.3 contains the direct-reciprocal relationships for an orthorhombic unit cell. 

Figure 4.6 Shows the relationships among the incident, k , and scattered, k , wave vectors, the change in wave vector, Ak, the extra path lengths, x and x , of one part of the scattered wave front along the incident and outgoing waves, and the distance between two scattering centers (atoms) in real space, R. Also shown is the equivalent reciprocal space construction based on the Ewald sphere concept showing that diffraction occurs when Ak begins and ends on a reciprocal lattice point.

The inverse relationship between the crystal s real space lattice and its reciprocal lattice defines the distances between adjacent reflections along reciprocal lattice rows and columns in the diffraction pattern. Conversely, measurement of the reciprocal lattice spacings yields the unit cell parameters. Angles between the axes of the reciprocal lattice can similarly be used to determine unit cell axial angles. 

Figure 4.9 illustrates the construction of reciprocal lattice. The relationship between the reciprocal lattice vector and the interplanar distance in real-space is expressed as... 

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