Reciprocal unit cells, orthorhombic

The unit cell parameters a and were 5.76 X and 13.20 respectively which agrees well with published data (14,15) based on fiber diagrams. Clearly the diffractogram is an hkO reciprocal lattice net as expected if the symmetry axis of the molecular helix is perpendicular to the crystal face. Since only even reflections are observed along the hOO and OkO directions the pgg base plane symmetry is confirmed in keeping with the proposed P2j2i2i space group for the orthorhombic unit cell (14,15). 

Table 1. Real and Reciprocal Space Relationship in an Orthorhombic Unit Cell...

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. 

Assuming that the equatorial reflections have been shown to fit a rectangular reciprocal lattice net, attention may be turned to the upper and lower layer lines. The values for all the spots are read off on Bernal s chart, and the reciprocal lattice rotation diagram is constructed from these values if the values for the upper and lower layer lines correspond with those of the equator—that is, row7 lines as well as layer lines are exhibited as in Fig. 80—then the unit cell must be orthorhombic. It should be noted that some spots may be missing from the equator, and it may be necessary to halve one or both of the reciprocal axes previously found to satisfy the equatorial reflections. The dimensions of the unit cell, and the indices of all the spots, follow immediately from the reciprocal lattice diagrams. 

If no external evidence is available, it is still possible to determine the unit cell dimensions of crystals of low symmetry from powder diffraction patterns, provided that sharp patterns with high resolution are avail able. Hesse (1948) and Lipson (1949) have used numerical methods successfully for orthorhombic crystals. (Sec also Henry, Lipson, and Wooster, 1951 Bunn 1955.) Ito (1950) has devised a method which in principle will lead to a possible unit cell for a crystal of any symmetry. It may not be the true unit cell appropriate to the crystal symmetry, but when a possible cell satisfying all the diffraction peaks on a powder pattern lias been obtained by Ito s method, the true unit cell can be obtained by a reduction process first devised by Delaunay (1933). Ito applies the reduction process to the reciprocal lattice (see p. 185), but International Tables (1952) recommend that the procedure should be applied to the direct space lattice. 

The reciprocal lattice, like the lattice of the crystal, may also be divided into unit cells with the reciprocal unit vectors a, b, andc as edges. Since reciprocal space of a crystal is zero everywhere except at lattice points, however, the interiors of the reciprocal unit cells will be vacant. The relation between orthorhombic and monoclinic unit cells, and the corresponding reciprocal unit cells derived from them are shown in Figures 3.20 and 3.21. The type of reciprocal unit cell will be the same as the real cell from which it arises, and the reciprocal unit cell, hence the reciprocal lattice, will manifest the symmetry and centering properties of the real crystal lattice. 

A modification of this model, introduced by Mardalen et al. [56] with reference to the intensity distribution of Figure 2.10, is a monoclinic cell (with a the unique axis). The reciprocal c-axis (c ) is in this case inclined relative to c, because the crystallographic angle a is only about 51 ° and the (>-axis takes the value 4,85 A keeping the interplanar distance equal to 3.78 A for 010. The volume of the unit cell, and thus also the mass density of the polymers remain the same as for the orthorhombic model. 

Figure 2.43. Comparison between a real monoclinic crystal lattice (a = b c) and the corresponding reciprocal lattice. Dashed lines indicate the unit cell of each lattice. The magnitudes of the reciprocal lattice vectors are not in scale for example, la l = 1/dioo, lc l = 1/dooi, IGioil = dioi, etc. Note that fra-orthogonal unit cells (cubic, tetragonal, orthorhombic), the reciprocal lattice vectras will be aligned parallel to the real lattice vectors. 2009 From Biomolecular Crystallography Principles, Practice, and Application to Structural Biology by Bernard Rupp. Reproduced by permission of Garland Science/ Taylor Francis Group LLC.

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