Three-dimensional reciprocal lattices

Electron crystallography of textured samples can benefit from the introduction of automatic or semi-automatic pattern indexing methods for the reconstruction of the three-dimensional reciprocal lattice from two-dimensional data and fitting procedures to model the observed diffraction pattern. Such automatic procedures had not been developed previously, but it is the purpose of this study to develop them now. All these features can contribute to extending the limits of traditional applications such as identification procedures, structure determination etc. 

The purpose of indexing texture patterns is the geometrical reconstruction of the three-dimensional reciprocal lattice from the two-dimensional distribution of H spacings. One advantage of texture patterns is the possibility to determine all unit cell parameters of a crystal unambiguously and index all the diffraction peaks from only a single texture... 

It should be noted that for a three-dimensional reciprocal lattice, a third vector (c ) is used that is perpendicular to both a and b axes of the real lattice. 

The powder diffraction experiment is the cornerstone of a truly basic materials characterization technique - diffraction analysis - and it has been used for many decades with exceptional success to provide accurate information about the structure of materials. Although powder data usually lack the three-dimensionality of a diffraction image, the fundamental nature of the method is easily appreciated from the fact that each powder diffraction pattern represents a one-dimensional snapshot of the three-dimensional reciprocal lattice of a crystal. The quality of the powder diffraction pattern is usually limited by the nature and the energy of the available radiation, by the resolution of the instrument, and by the physical and chemical conditions of the specimen. Since many materials can only be prepared in a polycrystalline form, the powder diffraction experiment becomes the only realistic option for a reliable determination of the crystal structure of such materials. 

The last few decades of the 20 century transformed the powder diffraction experiment from a technique familiar to a few into one of the most broadly practicable analytical diffraction experiments, particularly because of the availability of a much greater variety of sources of radiation -sealed and rotating anode x-ray tubes were supplemented by intense neutron and brilliant synchrotron radiation sources. Without a doubt, the accessibility of both neutron and synchrotron radiation sources started a revolution in powder diffraction, especially with respect to previously unimaginable kinds of information that can be extracted from a one-dimensional projection of the three-dimensional reciprocal lattice of a crystal. Yet powder diffraction fundamentals remain the same, no matter what is the brilliancy of the source of particles or x-ray photons employed to produce diffraction peaks, and how basic or how advanced is the method used to record the powder diffraction data. 

In powder diffraction, the very first step in solving the crystal structure, i.e. finding the true unit cell, may present considerable difficulties because the experimental data are a one-dimensional projection of the three-dimensional reciprocal lattice recorded as a function of a single independent... 

Consider, for example, crystals with face-centered cubic Bravais lattices. For the (001), (110) and (111) sections the plane lattices are square, rectangular and hexagonal, respectively. The basic translation vectors of the direct and reciprocal lattices for these three cases are given in Table 11.2 (ai and 02 are given in units a/2, Bi in units 27r/a, where a is the cubic lattice parameter). Note that for a cubic lattice the planes (100), (010) and (001) are equivalent. The equivalence takes place also for (110), (101), and (oil) planes. We see that the vectors Bi(i = 1,2) are now not the translation vectors of the three-dimensional reciprocal lattice. Therefore, the boundaries of BZ-2 do not coincide with those of BZ-3. 

Note that some nonequivalent points of BZ-3 become equivalent in BZ-2 (for example, the points X for the surface (001) and the points L for the (110) surface). Some points of BZ-2 have higher qrmmetry than in BZ-3 (for example, the vertices of BZ-2 for the (111) surface). These properties of BZ-2 arise because the unit-cell vectors in the two-dimensional reciprocal space B are not the lattice vectors of the three-dimensional reciprocal lattice. 

The purpose of pattern indexing is to reconstruct the three-dimensional reciprocal lattice of a crystalline solid from the radial distribution of lengths d =Hd) of the diffraction vectors. The basic equation used for indexing a powder diffraction pattern is obtained by squaring the reciprocal-lattice vectors =ha +kb +lc ), expressed in terms of the basis vectors of the reciprocal lattice a, b, c ) and hkl Miller indices,... 

Figure 3.2.1.6 Relation between real and reciprocal unit-cell vectors for (a) the general case and (b) the special case 5] 02- In panel (c) the three-dimensional reciprocal lattice is displayed.

As with the two-dimensional lattices, the three-dimensional (Bravais) lattices, the direct lattices, are said to occupy real space, and the reciprocal... 

When such features exist, they are penetrated by the electron beam so the material is represented by a three-dimensional point lattice and diffraction only occurs when the Ewald sphere intersects a point. This produces a transmission-type spot pattern. For smooth surfaces, the diffraction pattern appears as a set of streaks normal to the shadow edge on the fluorescent screen, due to the interaction of the Ewald sphere with the rods projecting orthogonally to the plane of the two-dimensional reciprocal lattice of the surface. The reciprocal lattice points are drawn out into rods because of the very small beam penetration into the crystal (2—5 atomic layers). We would emphasize, however, that despite contrary statements in the literature, the appearance of a streaked pattern is a necessary but not sufficient condition by which to define an atomically flat surface. Several other factors, such as the size of the crystal surface region over which the primary wave field is coherent and thermal diffuse scattering effects (electron—phonon interactions) can influence the intensity modulation along the streaks. 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

The reciprocal lattice of a mosaic crystal is a three-dimensional periodic system of points, each of which characterized by a vector Hhu = ha -l- kb -l-Ic, where a, b, c are axial vectors and h,k,l, are point indices. 

From a comparison of various spot electron diffraction patterns of a given crystal, a three-dimensional system of axis in the reeiproeal lattice may be established. The reeiproeal unit cell may be eompletely determined, if all the photographs indexed. For this it is sufficient to have two electron diffraction patterns and to know the angle between the seetions of the reeiproeal lattice represented by them, or to have three patterns which do not all have a particular row of points in common (Fig.5). Crystals of any compound usually grow with a particular face parallel to the surface of the specimen support. Various sections of the reciprocal lattice may, in this case, be obtained by the rotation method (Fig.5). 

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

The three-dimensional shape defined by the reciprocal lattice vectors is not always the same as the shape of the supercell in real space. For the fee primitive cell, we showed in Chapter 2 that... 

---