Reciprocal lattice point symmetry

The true unit cell is not necessarily the smallest unit that will account for all the reciprocal lattice points it is also necessary that the cell chosen should conform to the crystal symmetry. The reflections of crystals with face-centred or body-centred lattices can be accounted for by unit cells which have only a fraction of the volume of the true unit cell, but the smallest unit cells for such crystals are rejected in favour of the smallest that conforms to the crystal symmetry. The... 

The experimental diffraction pattern (Fig. 5), has six-fold symmetry. Measurable intensity was found at all reciprocal lattice points of the reconstructed unit cell, even far away from bulk Bragg peaks, proving that there indeed is a genuine (4m x 4m)R 9° supercell formation with atomic relaxations. 

In other words, Np ss is the number of symmetrically independent points in the reciprocal lattice limited by a sphere with the diameter d N (= 1/t/v) as established by Eq. 5.3 after substituting the Bragg angle, 0, of the iV observed Bragg peak for Qhu. Additional restrictions are imposed on Nposs in high symmetry crystal systems when reciprocal lattice points are not related by symmetry but when they have identical reciprocal vector lengths due to specific unit cell shape (e.g. h05 and h34 in the cubic, or 05/ and 34/ in the... 

Since any family of planes hki has a plane normal that is both positive and negative, that is, in opposite directions in space, every family of planes also gives rise to a second reciprocal lattice point, —h — k — l. Thus reciprocal space will always contain a center of symmetry (see Figure 3.7) at its origin independent of the crystal from which it is derived. 

A theorem exists in Fourier mathematics that when transforms can be sampled at least one additional time between normally allowed reciprocal lattice points, phases for the allowed reflections can be determined. Use of this non-integral sampling approach is steeped in complex mathematical formulations and processes, but it has nonetheless been put to good use in a number of cases, particularly with viruses and other highly redundant complexes whose symmetry arrangement exceeds the space group symmetry. 

We suppose that jR is a symmetry operator that corresponds to some proper or improper rotation, and that r is a vector in the real lattice. The vector Rr is also a vector in the real lattice since K is a symmetry operator. There are as many points in reciprocal lattice space as in the direct lattice, and each direct lattice vector corresponds to a definite vector in the reciprocal lattice. It follows that Rr corresponds to a reciprocal lattice point if r is a reciprocal lattice vector. Thus the operators R, S,. . . , that form the rotational parts of a space group are also the rotational parts of the reciprocal lattice space group. It now follows that the direct and reciprocal lattices must belong to the same crystal class, although not necessarily to the same type of translational lattice (see Eqs. 10.28-10.31). 

Equations (17) and (18) further show a relationship between real and reciprocal space. The function F(hkl) is the Fourier transform of the unit cell contents, expressed in the reciprocal space coordinates h, k. and /. Because the. symmetry operation of translation holds for all three spatial directions in crystals, the Fourier transform of the entire crystal is zero, except at reciprocal lattice points. 

The cylindrically symmetric distribution of crystallites about the fiber axis also leads to a decrease in the quantity of data obtainable, although for an entirely different reason. The rotational character of the diffraction pattern causes all lattice plans whose reciprocal lattice points have the same and C coordinates to diffract into the same point in space, irrespective of the value of their angular reciprocal lattice coordinate (l). This leads to an overlapping of reflections which would otherwise be distinguishable were it not for the cylindrical symmetry of the polymeric fiber. Thus, in many instances two or more independent reflections produce only one observable diffraction maximum. 

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. 

Diffracted beam directions are determined by intersecting the reciprocal lattice points with the sphere of reflection. All the reciprocal lattice points lying in any one layer of the reciprocal lattice layer perpendicular to the axis of rotation intersect the sphere of the reflection in a circle. The height of the circle above the equatorial plane is proportional to the vertical reciprocal lattice spacing. By remounting the crystal successively around different axes, we can determine the complete distribution of reciprocal lattice points. Of course, one mounting is sufficient if the crystal is cubic, but two or more may be needed if the crystal has lower symmetry. 

The conditions described here also define the conditions for diffraction of electron waves at the Brillouin zone boimdaries. Likewise the Brillouin zones described in Chapter 2 are reciprocal-space objects with the symmetry of the reciprocal lattice rather than the real-space lattice. The reciprocal lattice points in Figure 2.5, for example, are located at points hbj, kb2, and lb3. The reciprocal lattice for a simple cubic system with basis vectors ai, a 2, and as has reciprocal lattice vectors parallel to the real space vectors. However, larger distances in real space correspond to shorter distances in reciprocal space. Thus, planes that are widely spaced in real space have closely spaced reciprocal lattice points and vice versa. One may determine by examination of Figure 4.2 that the (100) planes are V3 times farther apart than are the (111) planes. In general, the distance, d, between (hkl) lattice planes in a cubic system may be shown to be ... 

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