Monoclinic unit cell, reciprocal

Figure 5 Relationship between reciprocal and direct lattices for a monoclinic unit cell shown in the ac (= a c ) plane...

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. 

FIGURE 3.21 A real, three-dimensional monoclinic unit cell (heavy lines) and its corresponding reciprocal unit call (thin lines). The unique b and b axes (unique because of the single twofold axis along b) are parallel for the monoclinic system, but the relationship of the a and a to the c and c axes depends on the angle between a and c. Note the inverse relationship between the real and reciprocal edge lengths. 

This might seem of scant use in protein crystallography, since we have no centric space groups. Crystals of biological macromolecules, as previously pointed out, cannot possess inversion symmetry. Sets of centric reflections frequently do occur in the diffraction patterns of macromolecular crystals, however, because certain projections of most unit cells contain a center of symmetry. The correlate of a centric projection, or centric plane in real space, is a plane of centric reflections in reciprocal space. A simple example is a monoclinic unit cell of space group P2. The two asymmetric units have the same hand, as they are related by pure rotation, and for every atom in one at xj, yj, Zj there is an equivalent atom in the other at —Xj, yj, —zj. If we project the contents of the unit cell on to a plane perpendicular to the y axis, namely the xz plane, by setting y = 0 for all atoms, however, then in that... 

The steps in the construction of a reciprocal lattice are given below and are illustrated for a section of a monoclinic unit cell. 

Suppose, however, that although the equatorial reflections fit a rectangular projected cell-base—that is, a rectangular zero-level reciprocal lattice net—the rest of the spots do not fall on row lines. This must mean that the remaining axis of the reciprocal lattice is (as in Fig. 90) not normal to the zero level in other words, the unit cell is monoclinic,... 

This idea has some useful consequences in terms of interpreting diffraction patterns. For example, consider the case of a twofold axis in real space, along b and perpendicular to the ac plane, as we would have in a monoclinic crystal of space group P2 or C2. The projection of all of the electron density in the unit cell having this dyad symmetry onto the ac plane, would of course also have twofold symmetry. Because this projection has dyad symmetry, the corresponding diffraction pattern, which is the MO plane of reflections in reciprocal space, would also have twofold symmetry, namely reflections F o = F-h-kO-... 

Figure 2.10 The construction of a reciprocal lattice (a) the a-c section of the unit cell in a monoclinic (mP) direct lattice (b) reciprocal lattice axes lie perpendicular to the end faces of the direct cell (c) reciprocal lattice points are spaced a = 1 /do o and c = 1 /V/0oi (d) the lattice plane is completed by extending the lattice (e) the reciprocal lattice is completed by adding layers above and below the first plane...

Figure C.l illustrates how the reciprocal vector c is related to the unit cell vectors a, b9 and c. The relationships a c =0 and b c = 0 show that the vector c is perpendicular to both a and b and hence to the base OACB of the unit cell. The relationship c c = 1 means that its length c is equal to the reciprocal of OP, the projection of c to c. In the special case in which the directions of c and c coincide, i.e., when c is perpendicular to both a and b9 c is equal to 1/ c. Figure C.2 illustrates the relationship between the crystal and reciprocal lattices, drawn for a monoclinic crystal where b and b are normal to the plane of the drawing. Note that...

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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