Weighted reciprocal lattice

Shmueli, U. and Wilson, A. J.C. (1996). Statistical properties of the weighted reciprocal lattice. In International Tables for Crystallography, Shmueli, U., ed., Vol. B, pp. 184-200. Kluwer Academic Pubhshers, Dordrecht. 

In the absence of anomalous scattering, Friedel s law holds. It states that X-rays are scattered with equal intensity from the opposite sides of a set of planes hkl. This is equivalent to the statement that the diffraction experiment adds a center of symmetry to the intensity-weighted reciprocal lattice, regardless of whether or not the crystal has an inversion center. The following equations apply ... 

Fig. 231. Coronene. (a) Form of the molecule. (6) hOl section of the weighted reciprocal lattice (areas of spots proportional to structure amplitudes), (c) Derivation of foreshortening of tilted molecule from peaks in reciprocal space. Alt A9, etc., are the positions of the benzene peaks derived from (b) perpendiculars to the A vectors are drawn at Blt B%, etc., at distances from the origin equal to reciprocals of A vectors these lines give the projected shape of the benzene ring.

As part of a study of biscyclic dibenzacridines, the crystal structure of 1,2-8,9-dibenzacridine (72) has been investigated by Mason (1957, 1960). The space group was shown to be Pna2v thus, with four molecules in the unit cell, no molecular symmetry is required. The structure was determined from an examination of the weighted reciprocal lattice and trial and error methods, refinement being by two-dimensional least-squares procedures. At the conclusion of the analysis,... 

As remarked previously, a crystal acts to decompose the continuous Fourier transform of the electron density in the unit cells into a discrete spectrum, the diffraction pattern, which we also call the weighted reciprocal lattice. Thus a crystal performs a Fourier analysis in producing its diffraction pattern. It remains to the X-ray crystallographer to provide the Fourier synthesis from this spectrum of waves and to recreate the electron density. 

The reciprocal lattice described in Chapter 2 consists of an array of points. Because a diffraction pattern is a direct representation of the reciprocal lattice, it is often useful to draw it as a weighted reciprocal lattice, in which the area allocated to each node is proportional to the structure factor FQikt) of each reflection, where... 

The weighted reciprocal lattice omits all reflections that are systematically absent, and so gives a clearer impression of the appearance of a diffraction pattern from a crystal (Figure 6.14). 

Figure 6.14 The hkO section of the weighted reciprocal lattice of rutile, Ti02. The lattice points are drawn as circles with radii proportional to the structure factor of the reflection...

Here t is a reciprocal lattice vector. The intensity of the scattering at a particular value of q is determined from the relative vibration amplitudes of the atoms taken in pairs over the unit cell and weighted by the scattering lengths of each atom. 

The remaining diffractions, which correspond to the cosets of the weighted reciprocal lattice with respect to the family sub lattice, are termed non-family reflections and are instead typical of each polytype they can be sharp or diffuse, depending on whether the polytype is ordered or not, i.e. on whether the distribution of subsequent p-operations is ordered or random. 

The square root of the intensities, partially reduced when necessary, gives an approximant of the structure factors. By dividing these by the Fourier transform of the layer, an un-weighted, un-scaled PID is obtained. The mean value of PID along several period of the same reciprocal lattice row is computed, and the result is brought on the same scale [see Appendix B, Eqn. (B.4)]. 

The intensity of radiation diffracted from a set of hkl planes depends on the relative phases of the waves from adjacent planes, which in turn depends on the symmetry of the structure and the types of atoms that are in the unit cell. When these waves are completely in step the diffracted beam is intense. When they are completely out of step, the diffracted intensity is zero (see also Section 14.7). The relative intensity of the diffracted radiation from a set of planes is portrayed by the weighted reciprocal lattice. 

In the original implementation of the PME algorithm [109,111], Lagrangian weight functions W2p(x) [116] were used for interpolation of the complex exponents, by using values at 2p points in the interval x < p. Unfortunately, these functions W2p(x) are only piecewise differentiable, so the approximate reciprocal lattice sum caimot be differentiated to arrive at the reciprocal part of the Coulomb forces. Thus, the forces were interpolated as well. 

Instead of the Lagrangian weight functions W2p(x), the Cardinal B-splines M (x) [117,118] were utilized in later versions of the PME method [110,112,113]. These weight functions are continuously differentiable and allow the forces to be obtained from analytical differentiation of the approximation of the reciprocal lattice sum. The Cardinal B-spline of the second order M2(x) gives the linear hat function M2(x)= 1 — x — 1 in the interval 02. The nth order B-spline satisfies the following properties ... 

Fig. 6.10. Weighted reciprocal lattice for (a) zero layer hexagonal or rhombohedral, (b) first layer rhombohedral, and (c) first layer hexagonal (By conrtesy of L.S. Dent Glasser)...

There exists an important relationship between the position of an atom or molecule in the lattice and the intensity of the diffracted X-radiation, and to incorporate the both in one diagram the concept of weighted reciprocal lattice is introduced here and its applications in structure analysis are discussed in appropriate chapter later. 

Fig. 7.10. (a) The weighted reciprocal lattice drawn say on a stretchable surface, (b) The axes are drawn from the origin away from each other with an intention to make them parallel to each other, (c) The axes are made parallel. The diffraction spots then resemble the Weissenberg photograph (replica) of Fig. 7.9, showing streamers and festoons (By courtesy of L.S. Dent Glasser)... 

Figure 9 shows the uniaxial orientation distribution functions qjiCp 0), determined from X-ray diffraction measurement, for a high-density polyethylene specimen stretched to an extension ratio of 1.4. Twelve different 7th reciprocal lattice vectors were observed. With qjiCp 0) for7 = 1 to 12, the coefficients Qj were calculated from equation (9) for any higher orders of I, and then the coefficients Wion were calculated from the simultaneous equations of equation (11) with respect to 7 by the weighted least-square method up to /= 18. 

Figure 10 compares the observed orientation distribution functions with those calculated for the respective reciprocal lattice vectors at the extension ratio of 1.4. That is, Q o were calculated in turn from WiQn, which were initially determined by the weighted least-squares method, by using equation (11), and further qjiCp 0) were calculated from the recalculated Q q by the use of equation (5). 

---