Symmetry in the diffraction pattern

Any symmetry in the packing of objects is related (in a reciprocal way) to symmetry in its diffraction pattern, and this symmetry in the diffraction pattern can be used to determine the crystal symmetry (see Tables 4.2 and 4.3). This is of great importance to the X-ray crystallo-grapher because this is the way the space group of a crystal is determined. 

TABLE 4.4. Some common space groups and the equivalent positions of objects in them. 

FIGURE 4.14. The symmetry operations involved the packing of molecules in two crystalline forms of citric acid (which has no asymmetric carbon atom), (a) Citric acid monohydrate with a primitive orthorhombic unit cell, space group F2i2i2i, with atoms at x,y,z j + z i + x, — y,—z —+ y, — z. All molecules, 

FIGURE 4.14. (b). Anhydrous citric acid with a primitive monoclinic unit cell space group P2i/a, with atoms at x,y,z —x, —y, —z — x, + y,—z — y,z. 

There is a twofold screw axis parallel to b. There is also a glide plane parallel to the plane of the paper at 6 = with translation o/2. Molecules A and C are (arbitrarily) conformationally of one handedness, and molecules B and D are conformationally of the opposite handedness. 

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Any symmetry in the intensities in the diffraction pattern other than that implied by Friedel s Law is called Laue symmetry (because it can be displayed on Laue X-ray diffraction photographs of an appropriately-aligned crystal, see Figure 4.16). Friedel s Law implies that there is a center of symmetry in the diffraction pattern. Therefore the Laue symmetry-displayed by the diffraction pattern is the point-group symmetry of the crystal with an additional center of symmetry (if this does not already exist). If a crystal is monoclinic then, the intensities I[hkl) and I hkl) are the same, although I hkl) does not equal I[hkl). Orthorhombic... 

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. 

Figure 8 X-ray diffraction images, (a) A precession photograph of muconate lactonizing enzyme. The fourfold symmetry in the diffraction pattern is clearly visible. This gives an undistorted view of the reciprocal lattice but are no longer used because they are not as efficient as rotation images, (b) A rotation image of hen s egg white lysozyme. This easily obtainable image gives a distorted projection of the reciprocal lattice, but this is no obstacle for modern programs.

S. Deviations from radial symmetry in the diffraction pattern u... 

Grain size in textured films. For films having a preferred growth direction—e.g., (111)—LEED can be used to determine the preferred direction and the grain size parallel to the surfrce. The preferred direction is obtained from the symmetry of the diffraction patterns, while the grain size is obtained from the shape in angle of diffracted beams. 

Equation (B. 11) implies that /(H ) = /(H), that is, the rotational symmetry of the space group, is repeated in the diffraction pattern. In addition, if the atomic scattering factors / are real, which is the case when resonance effects are negligible, a center of symmetry is added to the diffraction pattern, that is, /(H) = F(H) F (H) = /( —H) even in the absence of an inversion center, which is Friedel s law. 

The Laue symmetry of the diffraction pattern is now reduced to the symmetry of the non-centrosymmetric point group to which the crystal belongs (see Table 2 a listing of symmetry-equivalent reflections in the non-centrosymmetric point groups is available26). Under these circumstances, it is possible to determine different manifestations of non-centrosymmetry in a crystal, such as ... 

T(hxJrky.Jr z) for this atom is 180° when h- -k- -l is odd.) Therefore, reflections from planes having A+ +1 odd are not fotlnd in the diffraction pattern of a-iron. On the other hand, all planes such as 110, 200, 310, and 211 which have h+k+I even give strong reflections, because all the atoms lie on these planes. (In other words, the phase angle for the centre atom is 0° when h -rk- l is even.) It is important to note that this is true not only for body-centred cubic crystals but for a l other body-centred crystals, whatever their symmetry. 

The term diffraction symmetry, which is often called Laue symmetry because it can be most conspicuous in a type of X-ray photograph called a Laue photograph, is applied to those point groups that are recognizably different in the diffraction pattern. 

Notice the starlike six-fold symmetry of the diffraction pattern. Again, just accept this pattern as the diffraction signature of a hexagon of spheres. (Now you know enough to recognize two simple objects by their diffraction patterns.) Figure 2.10 depicts diffraction by these hexagonal objects in a lattice of the same dimensions as that in Fig. 2.8. 

Materials with the same framework type code (i.e. framework topology) may have very different diffraction patterns, so for some framework type codes several different reference materials have been included. Examples listed under FAU, GIS, MFI and NAT illustrate the extent of the differences observed in the diffraction patterns of materials with identical framework topologies but variations in composition and/or symmetry. 

Generally, lipids forming lamellar phase by themselves, form lamellar lipoplexes in most of these cases, lipids forming Hn phase by themselves tend to form Hn phase lipoplexes. Notable exceptions to this rule are the lipids forming cubic phase. Their lipoplexes do not retain the cubic symmetry and form either lamellar or inverted hexagonal phase [20, 24], The lamellar repeat period of the lipoplexes is typically 1.5 nm higher than that of the pure lipid phases, as a result of DNA intercalation between the lipid bilayers. In addition to the sharp lamellar reflections, a low-intensity diffuse peak is also present in the diffraction patterns (Fig. 23a) [81]. This peak has been ascribed to the in-plane positional correlation of the DNA strands arranged between the lipid lamellae [19, 63, 64, 82], Its position is dependent on the lipid-DNA ratio. The presence of DNA between the bilayers has been verified by the electron density profiles of the lipoplexes [16, 62-64] (Fig. 23b). 

The Klebsiella K8 polysaccharide has another modification, in which the molecule has a four-fold screw symmetry and packs tetragonally. Since the intensity distribution in the diffraction pattern of the tetragonal form is very similar to that in the orthorhombic form, we assumed (as a first approximation) that the molecule has a four-fold helical (4 or 4 ) symmetry, the same in both crystal forms. 

Space groups (or enantiomorphous pairs) that are uniquely determined from the symmetry of the diffraction pattern and systematic absences are shown in boldface type. 

As a consequence of Friedel s law, the diffraction pattern exhibits the symmetry of a centrosymmetric crystal class. For example, a crystal in class 2, on account of the 1 symmetry imposed on its diffraction pattern, will appear to be in class 2/m. The same result also holds for crystals in class m. Therefore, it is not possible to distinguish the classes 2, m, and 2/m from their diffraction patterns. The same effect occurs in other crystal systems, so that the 32 crystal classes are classified into only 11 distinct Laue groups according to the symmetry of the diffraction pattern, as shown in Table 9.4.1. 

The symmetry of the crystal is indicated in its diffraction pattern. Systematic absences in the diffraction pattern show that there are translational symmetry elements relating components in the unit cell. The translational component of the symmetry elements causes selective and predictable destructive interference to occur when the specific translation in the arrangement of atoms are simple fractions of the normal lattice... 

Laue symmetry Symmetry in the intensities of the diffraction pattern beyond that expected from Friedel s Law. The Laue symmetry of the diffraction pattern of a crystal is the point-group symmetry of the crystal plus, as Friedel noted, a center of symmetry. There are 11 Laue symmetry groups. 

Crystallites above about 50 nm (and smaller for electron diffraction) are large enough to yield broadened but characteristic diffraction patterns. By characteristic it is meant that the symmetry and geometric arrangement of the structure will be clearly reflected in the diffraction pattern. The normal way in which this is represented is via the crystallographic structure factor ... 

Radiation and particles, i.e. x-rays, neutrons and electrons, interact with a crystal in a way that the resulting diffraction pattern is always centrosymmetric, regardless of whether an inversion center is present in the crystal or not. This leads to another classification of crystallographic point groups, called Laue classes. The Laue class defines the symmetry of the diffraction pattern produced by a single crystal, and can be easily inferred from a point group by adding the center of inversion (see Table 1.10). 

Only even orders are taken into account in Eqs. 2.83 and 2.84 due to the presence of the inversion center in the diffraction pattern. The number of harmonic coefficients C and terms k(h) varies depending on lattice symmetry and desired harmonic order L. The low symmetry results in multiple terms (triclinic has 5 terms for L = 2) and therefore, low orders 2 or 4 are usually sufficient. High symmetry requires fewer terms (e.g. cubic has only 1 term for L = 4), so higher orders may be required to adequately describe preferred orientation. The spherical harmonics approach is realized in GSAS. ... 

The maximum size of the unit cell edges can be estimated from the d-spacing of the first Bragg peak (t/max) observed in the diffraction pattern. In the majority of low symmetry cases (triclinic through orthorhombic crystal systems), the maximum size of the unit cell edge should not exceed... 

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... 

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