Lattice factor

Figure 8.44. Effect of paracrystalline distortions on a series of reflections in a scattering diagram after compensation of the decay according to POROD s law (lattice factor (1 /N) Z 2). The quadratic increase of integral breadths of the reflections is indicated by boxes of equal area and increasing integral breadth. L is the average long period...

Partial volume per unit mass associated with spedes si For a self-avoiding walk on a lattice, factor defining the attraction of adjacent points on the lattice [w = exp( — /U)] Hypervolume representing a three-body interaction In field theory, quantity defining the variation of the effective interaction with respect to the real interaction... 

Thus, the lattice factor Z(s), which, for an infinite crystal, is an infinite set of delta functions located at the reciprocal lattice points rkkl, is replaced for a finite crystal by... 

Z(s) (s), which is a similarly infinite set of slightly broadened peaks E(s) located at the same reciprocal lattice points. The lattice factor Z(s) E(s) 2 that articulates the intensity in (3.27) is a similar set of broadened peaks located at the reciprocal lattice points. The shape of the individual peaks for the intensity function is given by Z,2(s), which has a width, in terms of FWHM, about 1/V2 of that of E(s). 

The lattice factor z(dq/2n) expresses the fact that the square of the Fourier transform of z(x /d) is itself a lattice of period 2n/d (in q) in reciprocal space. A proportionality instead of an equality sign is used in (5.135) to acknowledge that the proportionality constant arising from the squaring of the delta functions in z q) is here not explicitly accounted for. With pu(x) given by (5.132), we obtain... 

Factor I gives rise to the 14 Bravais lattices Factor II generates the 32 point- groups Factor III creates the 232 space- groups... 

The other example of the metal-metal interactions influence on the intercalation capacity may be intercalates based on the Nb X chalcogenides having separated lattice channels in the structure. These compounds form LiNb S /l/,Li2Nb2Se, LiQ. Nb Te intercalates on the basis of Nb X that is evidence of the influence of both electronic and lattice factors on the compounds intercalation capacity. 

Lattice factor Function describing the effects of the crystal lattice structure on scattered x-ray... 

This equation can be simplified if we condense the result into three different factors, each contributing to the overall scattering intensity the form factor, structure factor, and the lattice factor. The form factor represents scattering from the electron density distribution in the material. 

Then, the lattice factor L modifies this scattering to include interference effects resulting from... 

For a regular lattice structure, the lattice factor modifies the amplitude in such a way that the scattered intensity is small except at certain values. This results in the well-defined intensity peaks characteristic of Bragg scattering. 

A paracrystal theory was first prcalculating scattering profiles of distorted crystals. The general equation of the lattice factor Z(q) is introduced for ID, 2D, and 3D cases, and the total scattering intensity I(q) is written as follows for spherical and monodisperse scatterers ... 

For the ID case, the lattice factor Z(q) is calculated numerically, and has been compared with experimental results. Cooper et al. calculated the Z(q) for 3D cases [43] applied to sc lattices. The general equation of Z(q) for 3D cases... 

Fig. 3-2. Paracrystalline lattice factor for a fee lattice. Curve f g = 0.05, curve 2 0.07, curve 3 0.09, curve 4 0.11, curve 5 0.13, curve 6 0.15. Taken from [45] with the permission of the American Physical Society...

Fig. 3-6. Comparison of the experimental interference function S(q) obtained by neutron scattering for latex dispersions by Cebida et al. [48] with theoretical lattice factors Zfq), The filled circles, triangles and open circles represent S(q) observed for a latex particle of R = 157 A at volume fractions 0.04,0.08, and 0.13, respectivdy. Curves I, 2. and 3 represent Z(q) for fee structures with [a = 830 A, g = 0.24], [a = 679 A, g = 0.21], [a = 539 A, g = 0.18], respet ely. Taken ftom [45] with the permission of the American Physical Society...

The first term of Eq.(1), a diffuse back-ground scattering is generally much smaller and less angularly dependent than the second term, and S (s ), the shape amplitude related to the size and shape of the assembly of the particles varies with scattering angle s much more rapidly than Z( ), the paracrystalline lattice factor, so that... 

The lattice factor can be considered to sample the structure factor at different points in reciprocal space and the observed diffraction pattern is essentially a two-dimensional projection of the square of this sampling. For samples which are polycrystalline in nature (Le, the grains are randomly oriented) the diffraction patterns are of the Debye-Scherrer type consisting of concentric rings. When the structural units are oriented along a particular axis the rings give way to broad spots, the distribution of which can reveal the orientation of the specimen. ... 

As any normal component Ak = gi is allowed, the full reciprocal lattice (as defined by the values Ak to produce a nonzero lattice factor) consists of structureless parallel rods normal to the surface whose lateral periodicity is given by gy = ghk as illustrated in Figure 3.2.1.6c. In a formal sense and as evident from Eq. (3.2.1.5), the reciprocal lattice is equivalent to the modulus square of the discrete Fourier transform of the two-dimensional lattice of the real-space unit-cell arrangement. 

The Structure Factor It is important to always have in mind that the reciprocal lattice defines only the directions in which diffraction spots can appear. Also, the lattice factor G = G(ghk) takes the same value independent of energy and spot hk. The intensity is determined solely by the structure factor I = F = [f(ko,kj) which, as defined in Eq. (3.2.1.4), describes the intensity contribution of each unit... 

F is dominated by multiple scattering, and when considering that properly, Eq. (3.2.1.4) is exact as already mentioned. In kinematic approximation, that means with each of the unit-cell atoms assumed to scatter only once, the structure factor can be expressed rather simply and in analogy to the lattice factor, namely as... 

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